CINXE.COM

<!DOCTYPE html> <html > <head> <meta charset="utf-8"> <meta rel="search" type="application/opensearchdescription+xml" href="/open_search.xml" title="Academia.edu"> <meta content="width=device-width, initial-scale=1" name="viewport"> <meta name="google-site-verification" content="bKJMBZA7E43xhDOopFZkssMMkBRjvYERV-NaN4R6mrs"> <meta name="csrf-param" content="authenticity_token" /> <meta name="csrf-token" content="bqf8glDIIMTRCG0rfn9mrVqOoas-j22I2IvVbaYId-oaGaGTcu5I8i9H5oS7G1ffUEA2nzvHwvkeOYx8rZWy0w" /> <meta name="citation_title" content="On the Natural Gauge Fields of Manifolds" /> <meta name="citation_publication_date" content="2000/01/01" /> <meta name="citation_journal_title" content="Modern Physics Letters A" /> <meta name="citation_author" content="Bijan Saha" /> <meta name="twitter:card" content="summary" /> <meta name="twitter:url" content="https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds" /> <meta name="twitter:title" content="On the Natural Gauge Fields of Manifolds" /> <meta name="twitter:description" content="The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the" /> <meta name="twitter:image" content="https://0.academia-photos.com/12165292/3489833/59681778/s200_bijan.saha.jpg" /> <meta property="fb:app_id" content="2369844204" /> <meta property="og:type" content="article" /> <meta property="og:url" content="https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds" /> <meta property="og:title" content="On the Natural Gauge Fields of Manifolds" /> <meta property="og:image" content="http://a.academia-assets.com/images/open-graph-icons/fb-paper.gif" /> <meta property="og:description" content="The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the" /> <meta property="article:author" content="https://jinr.academia.edu/BijanSaha" /> <meta name="description" content="The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the" /> <title>(PDF) On the Natural Gauge Fields of Manifolds</title> <link rel="canonical" href="https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds" /> <script async src="https://www.googletagmanager.com/gtag/js?id=G-5VKX33P2DS"></script> <script> window.dataLayer = window.dataLayer || []; function gtag(){dataLayer.push(arguments);} gtag('js', new Date()); gtag('config', 'G-5VKX33P2DS', { cookie_domain: 'academia.edu', send_page_view: false, }); gtag('event', 'page_view', { 'controller': "single_work", 'action': "show", 'controller_action': 'single_work#show', 'logged_in': 'false', 'edge': 'unknown', // Send nil if there is no A/B test bucket, in case some records get logged // with missing data - that way we can distinguish between the two cases. // ab_test_bucket should be of the form <ab_test_name>:<bucket> 'ab_test_bucket': null, }) </script> <script> var $controller_name = 'single_work'; var $action_name = "show"; var $rails_env = 'production'; var $app_rev = '2dbf19f283ec395370665dd1f7acd1f78b8fa59d'; var $domain = 'academia.edu'; var $app_host = "academia.edu"; var $asset_host = "academia-assets.com"; var $start_time = new Date().getTime(); var $recaptcha_key = "6LdxlRMTAAAAADnu_zyLhLg0YF9uACwz78shpjJB"; var $recaptcha_invisible_key = "6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj"; var $disableClientRecordHit = false; </script> <script> window.require = { config: function() { return function() {} } } </script> <script> window.Aedu = window.Aedu || {}; window.Aedu.hit_data = null; window.Aedu.serverRenderTime = new Date(1740164010000); window.Aedu.timeDifference = new Date().getTime() - 1740164010000; </script> <script type="application/ld+json">{"@context":"https://schema.org","@type":"ScholarlyArticle","abstract":"The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the displacement field have been derived. It has been shown that the energy–momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations, we have derived the general form of the displacement field in Minkowski space–time which is invariant under rotation and space and time inversion. With this ansatz we found spherically-symmetric solutions of the equations in question.","author":[{"@context":"https://schema.org","@type":"Person","name":"Bijan Saha","url":"https://jinr.academia.edu/BijanSaha"}],"contributor":[],"dateCreated":"2021-11-25","dateModified":"2021-11-26","datePublished":"2000-01-01","headline":"On the Natural Gauge Fields of Manifolds","image":"https://attachments.academia-assets.com/75171349/thumbnails/1.jpg","inLanguage":"en","keywords":[],"publication":"Modern Physics Letters A","publisher":{"@context":"https://schema.org","@type":"Organization","name":"World Scientific Pub Co Pte Lt"},"sourceOrganization":[{"@context":"https://schema.org","@type":"EducationalOrganization","name":"jinr"}],"thumbnailUrl":"https://attachments.academia-assets.com/75171349/thumbnails/1.jpg","url":"https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds"}</script><style type="text/css">@media(max-width: 567px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 32px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 30px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 30px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 24px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 24px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 18px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 32px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 20px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 32px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 40px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 24px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 26px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 48px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 52px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 48px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 58px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 80px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 64px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 568px)and (max-width: 1279px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 36px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 104px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}@media(min-width: 1280px){:root{--token-mode: Rebrand;--dropshadow: 0 2px 4px 0 #22223340;--primary-brand: #0645b1;--error-dark: #b60000;--success-dark: #05b01c;--inactive-fill: #ebebee;--hover: #0c3b8d;--pressed: #082f75;--button-primary-fill-inactive: #ebebee;--button-primary-fill: #0645b1;--button-primary-text: #ffffff;--button-primary-fill-hover: #0c3b8d;--button-primary-fill-press: #082f75;--button-primary-icon: #ffffff;--button-primary-fill-inverse: #ffffff;--button-primary-text-inverse: #082f75;--button-primary-icon-inverse: #0645b1;--button-primary-fill-inverse-hover: #cddaef;--button-primary-stroke-inverse-pressed: #0645b1;--button-secondary-stroke-inactive: #b1b1ba;--button-secondary-fill: #eef2f9;--button-secondary-text: #082f75;--button-secondary-fill-press: #cddaef;--button-secondary-fill-inactive: #ebebee;--button-secondary-stroke: #cddaef;--button-secondary-stroke-hover: #386ac1;--button-secondary-stroke-press: #0645b1;--button-secondary-text-inactive: #b1b1ba;--button-secondary-icon: #082f75;--button-secondary-fill-hover: #e6ecf7;--button-secondary-stroke-inverse: #ffffff;--button-secondary-fill-inverse: rgba(255, 255, 255, 0);--button-secondary-icon-inverse: #ffffff;--button-secondary-icon-hover: #082f75;--button-secondary-icon-press: #082f75;--button-secondary-text-inverse: #ffffff;--button-secondary-text-hover: #082f75;--button-secondary-text-press: #082f75;--button-secondary-fill-inverse-hover: #043059;--button-xs-stroke: #141413;--button-xs-stroke-hover: #0c3b8d;--button-xs-stroke-press: #082f75;--button-xs-stroke-inactive: #ebebee;--button-xs-text: #141413;--button-xs-text-hover: #0c3b8d;--button-xs-text-press: #082f75;--button-xs-text-inactive: #91919e;--button-xs-icon: #141413;--button-xs-icon-hover: #0c3b8d;--button-xs-icon-press: #082f75;--button-xs-icon-inactive: #91919e;--button-xs-fill: #ffffff;--button-xs-fill-hover: #f4f7fc;--button-xs-fill-press: #eef2f9;--buttons-button-text-inactive: #91919e;--buttons-button-focus: #0645b1;--buttons-button-icon-inactive: #91919e;--buttons-small-buttons-corner-radius: 16px;--buttons-small-buttons-l-r-padding: 20px;--buttons-small-buttons-height: 48px;--buttons-small-buttons-gap: 8px;--buttons-small-buttons-icon-only-width: 48px;--buttons-small-buttons-icon-size: 20px;--buttons-small-buttons-stroke-default: 1px;--buttons-small-buttons-stroke-thick: 2px;--buttons-large-buttons-l-r-padding: 32px;--buttons-large-buttons-height: 64px;--buttons-large-buttons-icon-only-width: 64px;--buttons-large-buttons-icon-size: 20px;--buttons-large-buttons-gap: 8px;--buttons-large-buttons-corner-radius: 16px;--buttons-large-buttons-stroke-default: 1px;--buttons-large-buttons-stroke-thick: 2px;--buttons-extra-small-buttons-l-r-padding: 8px;--buttons-extra-small-buttons-height: 32px;--buttons-extra-small-buttons-icon-size: 16px;--buttons-extra-small-buttons-gap: 4px;--buttons-extra-small-buttons-corner-radius: 8px;--buttons-stroke-default: 1px;--buttons-stroke-thick: 2px;--background-beige: #f9f7f4;--error-light: #fff2f2;--text-placeholder: #6d6d7d;--stroke-dark: #141413;--stroke-light: #dddde2;--stroke-medium: #535366;--accent-green: #ccffd4;--accent-turquoise: #ccf7ff;--accent-yellow: #f7ffcc;--accent-peach: #ffd4cc;--accent-violet: #f7ccff;--accent-purple: #f4f7fc;--text-primary: #141413;--secondary-brand: #141413;--text-hover: #0c3b8d;--text-white: #ffffff;--text-link: #0645b1;--text-press: #082f75;--success-light: #f0f8f1;--background-light-blue: #f4f7fc;--background-white: #ffffff;--premium-dark: #877440;--premium-light: #f9f6ed;--stroke-white: #ffffff;--inactive-content: #b1b1ba;--annotate-light: #a35dff;--annotate-dark: #824acc;--grid: #eef2f9;--inactive-stroke: #ebebee;--shadow: rgba(34, 34, 51, 0.25);--text-inactive: #6d6d7d;--text-error: #b60000;--stroke-error: #b60000;--background-error: #fff2f2;--background-black: #141413;--icon-default: #141413;--icon-blue: #0645b1;--background-grey: #dddde2;--icon-grey: #b1b1ba;--text-focus: #082f75;--brand-colors-neutral-black: #141413;--brand-colors-neutral-900: #535366;--brand-colors-neutral-800: #6d6d7d;--brand-colors-neutral-700: #91919e;--brand-colors-neutral-600: #b1b1ba;--brand-colors-neutral-500: #c8c8cf;--brand-colors-neutral-400: #dddde2;--brand-colors-neutral-300: #ebebee;--brand-colors-neutral-200: #f8f8fb;--brand-colors-neutral-100: #fafafa;--brand-colors-neutral-white: #ffffff;--brand-colors-blue-900: #043059;--brand-colors-blue-800: #082f75;--brand-colors-blue-700: #0c3b8d;--brand-colors-blue-600: #0645b1;--brand-colors-blue-500: #386ac1;--brand-colors-blue-400: #cddaef;--brand-colors-blue-300: #e6ecf7;--brand-colors-blue-200: #eef2f9;--brand-colors-blue-100: #f4f7fc;--brand-colors-gold-500: #877440;--brand-colors-gold-400: #e9e3d4;--brand-colors-gold-300: #f2efe8;--brand-colors-gold-200: #f9f6ed;--brand-colors-gold-100: #f9f7f4;--brand-colors-error-900: #920000;--brand-colors-error-500: #b60000;--brand-colors-success-900: #035c0f;--brand-colors-green: #ccffd4;--brand-colors-turquoise: #ccf7ff;--brand-colors-yellow: #f7ffcc;--brand-colors-peach: #ffd4cc;--brand-colors-violet: #f7ccff;--brand-colors-error-100: #fff2f2;--brand-colors-success-500: #05b01c;--brand-colors-success-100: #f0f8f1;--text-secondary: #535366;--icon-white: #ffffff;--background-beige-darker: #f2efe8;--icon-dark-grey: #535366;--type-font-family-sans-serif: DM Sans;--type-font-family-serif: Gupter;--type-font-family-mono: IBM Plex Mono;--type-weights-300: 300;--type-weights-400: 400;--type-weights-500: 500;--type-weights-700: 700;--type-sizes-12: 12px;--type-sizes-14: 14px;--type-sizes-16: 16px;--type-sizes-18: 18px;--type-sizes-20: 20px;--type-sizes-22: 22px;--type-sizes-24: 24px;--type-sizes-28: 28px;--type-sizes-30: 30px;--type-sizes-32: 32px;--type-sizes-40: 40px;--type-sizes-42: 42px;--type-sizes-48-2: 48px;--type-line-heights-16: 16px;--type-line-heights-20: 20px;--type-line-heights-23: 23px;--type-line-heights-24: 24px;--type-line-heights-25: 25px;--type-line-heights-26: 26px;--type-line-heights-29: 29px;--type-line-heights-30: 30px;--type-line-heights-32: 32px;--type-line-heights-34: 34px;--type-line-heights-35: 35px;--type-line-heights-36: 36px;--type-line-heights-38: 38px;--type-line-heights-40: 40px;--type-line-heights-46: 46px;--type-line-heights-48: 48px;--type-line-heights-52: 52px;--type-line-heights-58: 58px;--type-line-heights-68: 68px;--type-line-heights-74: 74px;--type-line-heights-82: 82px;--type-paragraph-spacings-0: 0px;--type-paragraph-spacings-4: 4px;--type-paragraph-spacings-8: 8px;--type-paragraph-spacings-16: 16px;--type-sans-serif-xl-font-weight: 400;--type-sans-serif-xl-size: 42px;--type-sans-serif-xl-line-height: 46px;--type-sans-serif-xl-paragraph-spacing: 16px;--type-sans-serif-lg-font-weight: 400;--type-sans-serif-lg-size: 32px;--type-sans-serif-lg-line-height: 38px;--type-sans-serif-lg-paragraph-spacing: 16px;--type-sans-serif-md-font-weight: 400;--type-sans-serif-md-line-height: 34px;--type-sans-serif-md-paragraph-spacing: 16px;--type-sans-serif-md-size: 28px;--type-sans-serif-xs-font-weight: 700;--type-sans-serif-xs-line-height: 25px;--type-sans-serif-xs-paragraph-spacing: 0px;--type-sans-serif-xs-size: 20px;--type-sans-serif-sm-font-weight: 400;--type-sans-serif-sm-line-height: 30px;--type-sans-serif-sm-paragraph-spacing: 16px;--type-sans-serif-sm-size: 24px;--type-body-xl-font-weight: 400;--type-body-xl-size: 24px;--type-body-xl-line-height: 36px;--type-body-xl-paragraph-spacing: 0px;--type-body-sm-font-weight: 400;--type-body-sm-size: 14px;--type-body-sm-line-height: 20px;--type-body-sm-paragraph-spacing: 8px;--type-body-xs-font-weight: 400;--type-body-xs-size: 12px;--type-body-xs-line-height: 16px;--type-body-xs-paragraph-spacing: 0px;--type-body-md-font-weight: 400;--type-body-md-size: 16px;--type-body-md-line-height: 20px;--type-body-md-paragraph-spacing: 4px;--type-body-lg-font-weight: 400;--type-body-lg-size: 20px;--type-body-lg-line-height: 26px;--type-body-lg-paragraph-spacing: 16px;--type-body-lg-medium-font-weight: 500;--type-body-lg-medium-size: 20px;--type-body-lg-medium-line-height: 32px;--type-body-lg-medium-paragraph-spacing: 16px;--type-body-md-medium-font-weight: 500;--type-body-md-medium-size: 16px;--type-body-md-medium-line-height: 20px;--type-body-md-medium-paragraph-spacing: 4px;--type-body-sm-bold-font-weight: 700;--type-body-sm-bold-size: 14px;--type-body-sm-bold-line-height: 20px;--type-body-sm-bold-paragraph-spacing: 8px;--type-body-sm-medium-font-weight: 500;--type-body-sm-medium-size: 14px;--type-body-sm-medium-line-height: 20px;--type-body-sm-medium-paragraph-spacing: 8px;--type-serif-md-font-weight: 400;--type-serif-md-size: 40px;--type-serif-md-paragraph-spacing: 0px;--type-serif-md-line-height: 48px;--type-serif-sm-font-weight: 400;--type-serif-sm-size: 28px;--type-serif-sm-paragraph-spacing: 0px;--type-serif-sm-line-height: 32px;--type-serif-lg-font-weight: 400;--type-serif-lg-size: 58px;--type-serif-lg-paragraph-spacing: 0px;--type-serif-lg-line-height: 68px;--type-serif-xs-font-weight: 400;--type-serif-xs-size: 18px;--type-serif-xs-line-height: 24px;--type-serif-xs-paragraph-spacing: 0px;--type-serif-xl-font-weight: 400;--type-serif-xl-size: 74px;--type-serif-xl-paragraph-spacing: 0px;--type-serif-xl-line-height: 82px;--type-mono-md-font-weight: 400;--type-mono-md-size: 22px;--type-mono-md-line-height: 24px;--type-mono-md-paragraph-spacing: 0px;--type-mono-lg-font-weight: 400;--type-mono-lg-size: 40px;--type-mono-lg-line-height: 40px;--type-mono-lg-paragraph-spacing: 0px;--type-mono-sm-font-weight: 400;--type-mono-sm-size: 14px;--type-mono-sm-line-height: 24px;--type-mono-sm-paragraph-spacing: 0px;--spacing-xs-4: 4px;--spacing-xs-8: 8px;--spacing-xs-16: 16px;--spacing-sm-24: 24px;--spacing-sm-32: 32px;--spacing-md-40: 40px;--spacing-md-48: 48px;--spacing-lg-64: 64px;--spacing-lg-80: 80px;--spacing-xlg-104: 104px;--spacing-xlg-152: 152px;--spacing-xs-12: 12px;--spacing-page-section: 152px;--spacing-card-list-spacing: 48px;--spacing-text-section-spacing: 80px;--spacing-md-xs-headings: 40px;--corner-radius-radius-lg: 16px;--corner-radius-radius-sm: 4px;--corner-radius-radius-md: 8px;--corner-radius-radius-round: 104px}}</style><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/single_work_page/loswp-fd2fcde21889491abfafcac2e33d795c8d15f5c18207be857e53e09b77f94215.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/body-170d1319f0e354621e81ca17054bb147da2856ec0702fe440a99af314a6338c5.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/button-bfbac2a470372e2f3a6661a65fa7ff0a0fbf7aa32534d9a831d683d2a6f9e01b.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/heading-95367dc03b794f6737f30123738a886cf53b7a65cdef98a922a98591d60063e3.css" /><link rel="stylesheet" media="all" href="//a.academia-assets.com/assets/design_system/text_button-d1941ab08e91e29ee143084c4749da4aaffa350a2ac6eec2306b1d7a352d911a.css" /><link crossorigin="" href="https://fonts.gstatic.com/" rel="preconnect" /><link href="https://fonts.googleapis.com/css2?family=DM+Sans:ital,opsz,wght@0,9..40,100..1000;1,9..40,100..1000&family=Gupter:wght@400;500;700&family=IBM+Plex+Mono:wght@300;400&family=Material+Symbols+Outlined:opsz,wght,FILL,GRAD@20,400,0,0&display=swap" rel="stylesheet" /> </head> <body> <div id='react-modal'></div> <div class="js-upgrade-ie-banner" style="display: none; text-align: center; padding: 8px 0; background-color: #ebe480;"><p style="color: #000; font-size: 12px; margin: 0 0 4px;">Academia.edu no longer supports Internet Explorer.</p><p style="color: #000; font-size: 12px; margin: 0;">To browse Academia.edu and the wider internet faster and more securely, please take a few seconds to <a href="https://www.academia.edu/upgrade-browser">upgrade your browser</a>.</p></div><script>// Show this banner for all versions of IE if (!!window.MSInputMethodContext || /(MSIE)/.test(navigator.userAgent)) { document.querySelector('.js-upgrade-ie-banner').style.display = 'block'; }</script> <div class="bootstrap login"><div class="modal fade login-modal" id="login-modal"><div class="login-modal-dialog modal-dialog"><div class="modal-content"><div class="modal-header"><button class="close close" data-dismiss="modal" type="button"><span aria-hidden="true">×</span><span class="sr-only">Close</span></button><h4 class="modal-title text-center"><strong>Log In</strong></h4></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><button class="btn btn-fb btn-lg btn-block btn-v-center-content" id="login-facebook-oauth-button"><svg style="float: left; width: 19px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="facebook-square" class="svg-inline--fa fa-facebook-square fa-w-14" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 448 512"><path fill="currentColor" d="M400 32H48A48 48 0 0 0 0 80v352a48 48 0 0 0 48 48h137.25V327.69h-63V256h63v-54.64c0-62.15 37-96.48 93.67-96.48 27.14 0 55.52 4.84 55.52 4.84v61h-31.27c-30.81 0-40.42 19.12-40.42 38.73V256h68.78l-11 71.69h-57.78V480H400a48 48 0 0 0 48-48V80a48 48 0 0 0-48-48z"></path></svg><small><strong>Log in</strong> with <strong>Facebook</strong></small></button><br /><button class="btn btn-google btn-lg btn-block btn-v-center-content" id="login-google-oauth-button"><svg style="float: left; width: 22px; line-height: 1em; margin-right: .3em;" aria-hidden="true" focusable="false" data-prefix="fab" data-icon="google-plus" class="svg-inline--fa fa-google-plus fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M256,8C119.1,8,8,119.1,8,256S119.1,504,256,504,504,392.9,504,256,392.9,8,256,8ZM185.3,380a124,124,0,0,1,0-248c31.3,0,60.1,11,83,32.3l-33.6,32.6c-13.2-12.9-31.3-19.1-49.4-19.1-42.9,0-77.2,35.5-77.2,78.1S142.3,334,185.3,334c32.6,0,64.9-19.1,70.1-53.3H185.3V238.1H302.2a109.2,109.2,0,0,1,1.9,20.7c0,70.8-47.5,121.2-118.8,121.2ZM415.5,273.8v35.5H380V273.8H344.5V238.3H380V202.8h35.5v35.5h35.2v35.5Z"></path></svg><small><strong>Log in</strong> with <strong>Google</strong></small></button><br /><style type="text/css">.sign-in-with-apple-button { width: 100%; height: 52px; border-radius: 3px; border: 1px solid black; cursor: pointer; } .sign-in-with-apple-button > div { margin: 0 auto; / This centers the Apple-rendered button horizontally }</style><script src="https://appleid.cdn-apple.com/appleauth/static/jsapi/appleid/1/en_US/appleid.auth.js" type="text/javascript"></script><div class="sign-in-with-apple-button" data-border="false" data-color="white" id="appleid-signin"><span   ="Sign Up with Apple" class="u-fs11"></span></div><script>AppleID.auth.init({ clientId: 'edu.academia.applesignon', scope: 'name email', redirectURI: 'https://www.academia.edu/sessions', state: "ad31f94e330e49a875fc2c5f18b7e9716a2b42cb2671d51d5aa80e0aeba6a59d", });</script><script>// Hacky way of checking if on fast loswp if (window.loswp == null) { (function() { const Google = window?.Aedu?.Auth?.OauthButton?.Login?.Google; const Facebook = window?.Aedu?.Auth?.OauthButton?.Login?.Facebook; if (Google) { new Google({ el: '#login-google-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } if (Facebook) { new Facebook({ el: '#login-facebook-oauth-button', rememberMeCheckboxId: 'remember_me', track: null }); } })(); }</script></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><div class="hr-heading login-hr-heading"><span class="hr-heading-text">or</span></div></div></div></div><div class="modal-body"><div class="row"><div class="col-xs-10 col-xs-offset-1"><form class="js-login-form" action="https://www.academia.edu/sessions" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="9ZO-gTBcwJLii5MZbQgy0IOr8WyNUiIwRzdckoheZt6BLeOQEnqopBzEGLaobAOiiWVmWIgajUGBhQWDg8Oj5w" autocomplete="off" /><div class="form-group"><label class="control-label" for="login-modal-email-input" style="font-size: 14px;">Email</label><input class="form-control" id="login-modal-email-input" name="login" type="email" /></div><div class="form-group"><label class="control-label" for="login-modal-password-input" style="font-size: 14px;">Password</label><input class="form-control" id="login-modal-password-input" name="password" type="password" /></div><input type="hidden" name="post_login_redirect_url" id="post_login_redirect_url" value="https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds" autocomplete="off" /><div class="checkbox"><label><input type="checkbox" name="remember_me" id="remember_me" value="1" checked="checked" /><small style="font-size: 12px; margin-top: 2px; display: inline-block;">Remember me on this computer</small></label></div><br><input type="submit" name="commit" value="Log In" class="btn btn-primary btn-block btn-lg js-login-submit" data-disable-with="Log In" /></br></form><script>typeof window?.Aedu?.recaptchaManagedForm === 'function' && window.Aedu.recaptchaManagedForm( document.querySelector('.js-login-form'), document.querySelector('.js-login-submit') );</script><small style="font-size: 12px;"><br />or <a data-target="#login-modal-reset-password-container" data-toggle="collapse" href="javascript:void(0)">reset password</a></small><div class="collapse" id="login-modal-reset-password-container"><br /><div class="well margin-0x"><form class="js-password-reset-form" action="https://www.academia.edu/reset_password" accept-charset="UTF-8" method="post"><input type="hidden" name="authenticity_token" value="uXI4PxhLXD1Q36gRzwdH18Ew6rC2GfWrqEwPvgdOu-nNzGUuOm00C66QI74KY3aly_59hLNRWtpu_lavDNN-0A" autocomplete="off" /><p>Enter the email address you signed up with and we'll email you a reset link.</p><div class="form-group"><input class="form-control" name="email" type="email" /></div><input class="btn btn-primary btn-block g-recaptcha js-password-reset-submit" data-sitekey="6Lf3KHUUAAAAACggoMpmGJdQDtiyrjVlvGJ6BbAj" type="submit" value="Email me a link" /></form></div></div><script> require.config({ waitSeconds: 90 })(["https://a.academia-assets.com/assets/collapse-45805421cf446ca5adf7aaa1935b08a3a8d1d9a6cc5d91a62a2a3a00b20b3e6a.js"], function() { // from javascript_helper.rb $("#login-modal-reset-password-container").on("shown.bs.collapse", function() { $(this).find("input[type=email]").focus(); }); }); </script> </div></div></div><div class="modal-footer"><div class="text-center"><small style="font-size: 12px;">Need an account? <a rel="nofollow" href="https://www.academia.edu/signup">Click here to sign up</a></small></div></div></div></div></div></div><script>// If we are on subdomain or non-bootstrapped page, redirect to login page instead of showing modal (function(){ if (typeof $ === 'undefined') return; var host = window.location.hostname; if ((host === $domain || host === "www."+$domain) && (typeof $().modal === 'function')) { $("#nav_log_in").click(function(e) { // Don't follow the link and open the modal e.preventDefault(); $("#login-modal").on('shown.bs.modal', function() { $(this).find("#login-modal-email-input").focus() }).modal('show'); }); } })()</script> <div id="fb-root"></div><script>window.fbAsyncInit = function() { FB.init({ appId: "2369844204", version: "v8.0", status: true, cookie: true, xfbml: true }); // Additional initialization code. if (window.InitFacebook) { // facebook.ts already loaded, set it up. window.InitFacebook(); } else { // Set a flag for facebook.ts to find when it loads. window.academiaAuthReadyFacebook = true; } };</script> <div id="google-root"></div><script>window.loadGoogle = function() { if (window.InitGoogle) { // google.ts already loaded, set it up. window.InitGoogle("331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"); } else { // Set a flag for google.ts to use when it loads. window.GoogleClientID = "331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b"; } };</script> <div class="header--container" id="main-header-container"><div class="header--inner-container header--inner-container-ds2"><div class="header-ds2--left-wrapper"><div class="header-ds2--left-wrapper-inner"><a data-main-header-link-target="logo_home" href="https://www.academia.edu/"><img class="hide-on-desktop-redesign" style="height: 24px; width: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015-A.svg" width="24" height="24" /><img width="145.2" height="18" class="hide-on-mobile-redesign" style="height: 24px;" alt="Academia.edu" src="//a.academia-assets.com/images/academia-logo-redesign-2015.svg" /></a><div class="header--search-container header--search-container-ds2"><form class="js-SiteSearch-form select2-no-default-pills" action="https://www.academia.edu/search" accept-charset="UTF-8" method="get"><svg style="width: 14px; height: 14px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="search" class="header--search-icon svg-inline--fa fa-search fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M505 442.7L405.3 343c-4.5-4.5-10.6-7-17-7H372c27.6-35.3 44-79.7 44-128C416 93.1 322.9 0 208 0S0 93.1 0 208s93.1 208 208 208c48.3 0 92.7-16.4 128-44v16.3c0 6.4 2.5 12.5 7 17l99.7 99.7c9.4 9.4 24.6 9.4 33.9 0l28.3-28.3c9.4-9.4 9.4-24.6.1-34zM208 336c-70.7 0-128-57.2-128-128 0-70.7 57.2-128 128-128 70.7 0 128 57.2 128 128 0 70.7-57.2 128-128 128z"></path></svg><input class="header--search-input header--search-input-ds2 js-SiteSearch-form-input" data-main-header-click-target="search_input" name="q" placeholder="Search" type="text" /></form></div></div></div><nav class="header--nav-buttons header--nav-buttons-ds2 js-main-nav"><button class="ds2-5-button ds2-5-button--secondary js-header-login-url header-button-ds2 header-login-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"login-button--header"}" rel="nofollow">Log In</button><button class="ds2-5-button ds2-5-button--secondary header-button-ds2 hide-on-mobile-redesign react-login-modal-opener" data-signup-modal="{"location":"signup-button--header"}" rel="nofollow">Sign Up</button><button class="header--hamburger-button header--hamburger-button-ds2 hide-on-desktop-redesign js-header-hamburger-button"><div class="icon-bar"></div><div class="icon-bar" style="margin-top: 4px;"></div><div class="icon-bar" style="margin-top: 4px;"></div></button></nav></div><ul class="header--dropdown-container js-header-dropdown"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/login" rel="nofollow">Log In</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/signup" rel="nofollow">Sign Up</a></li><li class="header--dropdown-row js-header-dropdown-expand-button"><button class="header--dropdown-button">more<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-down" class="header--dropdown-button-icon svg-inline--fa fa-caret-down fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M31.3 192h257.3c17.8 0 26.7 21.5 14.1 34.1L174.1 354.8c-7.8 7.8-20.5 7.8-28.3 0L17.2 226.1C4.6 213.5 13.5 192 31.3 192z"></path></svg></button></li><li><ul class="header--expanded-dropdown-container"><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/about">About</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/press">Press</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/documents">Papers</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/terms">Terms</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/privacy">Privacy</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/copyright">Copyright</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://www.academia.edu/hiring"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="header--dropdown-row-icon svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg>We're Hiring!</a></li><li class="header--dropdown-row"><a class="header--dropdown-link" href="https://support.academia.edu/hc/en-us"><svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="header--dropdown-row-icon svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg>Help Center</a></li><li class="header--dropdown-row js-header-dropdown-collapse-button"><button class="header--dropdown-button">less<svg aria-hidden="true" focusable="false" data-prefix="fas" data-icon="caret-up" class="header--dropdown-button-icon svg-inline--fa fa-caret-up fa-w-10" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 320 512"><path fill="currentColor" d="M288.662 352H31.338c-17.818 0-26.741-21.543-14.142-34.142l128.662-128.662c7.81-7.81 20.474-7.81 28.284 0l128.662 128.662c12.6 12.599 3.676 34.142-14.142 34.142z"></path></svg></button></li></ul></li></ul></div> <script src="//a.academia-assets.com/assets/webpack_bundles/fast_loswp-bundle-51ada6d7e8fd4f6594cc8a39e5b209c6c5f9b45d97fd4f4728a73454252d9098.js" defer="defer"></script><script>window.loswp = {}; window.loswp.author = 12165292; window.loswp.bulkDownloadFilterCounts = {}; window.loswp.hasDownloadableAttachment = true; window.loswp.hasViewableAttachments = true; // TODO: just use routes for this window.loswp.loginUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F62389364%2FOn_the_Natural_Gauge_Fields_of_Manifolds%3Fauto%3Ddownload"; window.loswp.translateUrl = "https://www.academia.edu/login?post_login_redirect_url=https%3A%2F%2Fwww.academia.edu%2F62389364%2FOn_the_Natural_Gauge_Fields_of_Manifolds%3Fshow_translation%3Dtrue"; window.loswp.previewableAttachments = [{"id":75171349,"identifier":"Attachment_75171349","shouldShowBulkDownload":false}]; window.loswp.shouldDetectTimezone = true; window.loswp.shouldShowBulkDownload = true; window.loswp.showSignupCaptcha = false window.loswp.willEdgeCache = false; window.loswp.work = {"work":{"id":62389364,"created_at":"2021-11-25T12:55:59.183-08:00","from_world_paper_id":184770208,"updated_at":"2023-09-25T12:28:02.346-07:00","_data":{"abstract":"The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the displacement field have been derived. It has been shown that the energy–momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations, we have derived the general form of the displacement field in Minkowski space–time which is invariant under rotation and space and time inversion. With this ansatz we found spherically-symmetric solutions of the equations in question.","publisher":"World Scientific Pub Co Pte Lt","publication_date":"2000,,","publication_name":"Modern Physics Letters A"},"document_type":"paper","pre_hit_view_count_baseline":null,"quality":"high","language":"en","title":"On the Natural Gauge Fields of Manifolds","broadcastable":true,"draft":null,"has_indexable_attachment":true,"indexable":true}}["work"]; window.loswp.workCoauthors = [12165292]; window.loswp.locale = "en"; window.loswp.countryCode = "SG"; window.loswp.cwvAbTestBucket = ""; window.loswp.designVariant = "ds_vanilla"; window.loswp.fullPageMobileSutdModalVariant = "control"; window.loswp.useOptimizedScribd4genScript = false; window.loginModal = {}; window.loginModal.appleClientId = 'edu.academia.applesignon'; window.userInChina = "false";</script><script defer="" src="https://accounts.google.com/gsi/client"></script><div class="ds-loswp-container"><div class="ds-work-card--grid-container"><div class="ds-work-card--container js-loswp-work-card"><div class="ds-work-card--cover"><div class="ds-work-cover--wrapper"><div class="ds-work-cover--container"><button class="ds-work-cover--clickable js-swp-download-button" data-signup-modal="{"location":"swp-splash-paper-cover","attachmentId":75171349,"attachmentType":"pdf"}"><img alt="First page of “On the Natural Gauge Fields of Manifolds”" class="ds-work-cover--cover-thumbnail" src="https://0.academia-photos.com/attachment_thumbnails/75171349/mini_magick20211125-23122-lm1j63.png?1637873999" /><img alt="PDF Icon" class="ds-work-cover--file-icon" src="//a.academia-assets.com/images/single_work_splash/adobe_icon.svg" /><div class="ds-work-cover--hover-container"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span><p>Download Free PDF</p></div><div class="ds-work-cover--ribbon-container">Download Free PDF</div><div class="ds-work-cover--ribbon-triangle"></div></button></div></div></div><div class="ds-work-card--work-information"><h1 class="ds-work-card--work-title">On the Natural Gauge Fields of Manifolds</h1><div class="ds-work-card--work-authors ds-work-card--detail"><a class="ds-work-card--author js-wsj-grid-card-author ds2-5-body-md ds2-5-body-link" data-author-id="12165292" href="https://jinr.academia.edu/BijanSaha"><img alt="Profile image of Bijan Saha" class="ds-work-card--author-avatar" src="https://0.academia-photos.com/12165292/3489833/59681778/s65_bijan.saha.jpg" />Bijan Saha</a></div><div class="ds-work-card--detail"><p class="ds-work-card--detail ds2-5-body-sm">2000, Modern Physics Letters A</p><div class="ds-work-card--work-metadata"><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">visibility</span><p class="ds2-5-body-sm" id="work-metadata-view-count">…</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">description</span><p class="ds2-5-body-sm">11 pages</p></div><div class="ds-work-card--work-metadata__stat"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">link</span><p class="ds2-5-body-sm">1 file</p></div></div><script>(async () => { const workId = 62389364; const worksViewsPath = "/v0/works/views?subdomain_param=api&work_ids%5B%5D=62389364"; const getWorkViews = async (workId) => { const response = await fetch(worksViewsPath); if (!response.ok) { throw new Error('Failed to load work views'); } const data = await response.json(); return data.views[workId]; }; // Get the view count for the work - we send this immediately rather than waiting for // the DOM to load, so it can be available as soon as possible (but without holding up // the backend or other resource requests, because it's a bit expensive and not critical). const viewCount = await getWorkViews(workId); const updateViewCount = (viewCount) => { try { const viewCountNumber = parseInt(viewCount, 10); if (viewCountNumber === 0) { // Remove the whole views element if there are zero views. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); return; } const commaizedViewCount = viewCountNumber.toLocaleString(); const viewCountBody = document.getElementById('work-metadata-view-count'); if (!viewCountBody) { throw new Error('Failed to find work views element'); } viewCountBody.textContent = `${commaizedViewCount} views`; } catch (error) { // Remove the whole views element if there was some issue parsing. document.getElementById('work-metadata-view-count')?.parentNode?.remove(); throw new Error(`Failed to parse view count: ${viewCount}`, error); } }; // If the DOM is still loading, wait for it to be ready before updating the view count. if (document.readyState === "loading") { document.addEventListener('DOMContentLoaded', () => { updateViewCount(viewCount); }); // Otherwise, just update it immediately. } else { updateViewCount(viewCount); } })();</script></div><p class="ds-work-card--work-abstract ds-work-card--detail ds2-5-body-md">The gauge symmetry inherent in the concept of manifold has been discussed. Within the scope of this symmetry the linear connection or displacement field can be considered as a natural gauge field on the manifold. The gauge-invariant equations for the displacement field have been derived. It has been shown that the energy–momentum tensor of this field conserves and hence the displacement field can be treated as one that transports energy and gravitates. To show the existence of the solutions of the field equations, we have derived the general form of the displacement field in Minkowski space–time which is invariant under rotation and space and time inversion. With this ansatz we found spherically-symmetric solutions of the equations in question.</p><div class="ds-work-card--button-container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--work-card","attachmentId":75171349,"attachmentType":"pdf","workUrl":"https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds"}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--work-card","attachmentId":75171349,"attachmentType":"pdf","workUrl":"https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds"}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div><div class="ds-signup-banner-trigger-container"><div class="ds-signup-banner-trigger ds-signup-banner-trigger-control"></div></div><div class="ds-signup-banner ds-signup-banner-control"><div id="ds-signup-banner-close-button"><button class="ds2-5-button ds2-5-button--secondary ds2-5-button--inverse"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">close</span></button></div><div class="ds-signup-banner-ctas"><img src="//a.academia-assets.com/images/academia-logo-capital-white.svg" /><h4 class="ds2-5-heading-serif-sm">Sign up for access to the world's latest research</h4><button class="ds2-5-button ds2-5-button--inverse ds2-5-button--full-width js-swp-download-button" data-signup-modal="{"location":"signup-banner"}">Sign up for free<span class="material-symbols-outlined" style="font-size: 20px" translate="no">arrow_forward</span></button></div><div class="ds-signup-banner-divider"></div><div class="ds-signup-banner-reasons"><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Get notified about relevant papers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Save papers to use in your research</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Join the discussion with peers</span></div><div class="ds-signup-banner-reasons-item"><span class="material-symbols-outlined" style="font-size: 24px" translate="no">check</span><span>Track your impact</span></div></div></div><script>(() => { // Set up signup banner show/hide behavior: // 1. If the signup banner trigger (a 242px-high* invisible div underneath the 'See Full PDF' / 'Download PDF' buttons) // is already fully scrolled above the viewport, show the banner by default // 2. If the signup banner trigger is fully visible, show the banner // 3. If the signup banner trigger has even a few pixels scrolled below the viewport, hide the banner // // * 242px is the empirically determined height of the signup banner. It's better to be a bit taller than // necessary than too short, so it's fine that the mobile (small breakpoint) banner is shorter. // First check session storage for the signup banner's visibility state const signupBannerHidden = sessionStorage.getItem('ds-signup-banner-hidden'); if (signupBannerHidden === 'true') { return; } const signupBanner = document.querySelector('.ds-signup-banner'); const signupBannerTrigger = document.querySelector('.ds-signup-banner-trigger'); if (!signupBannerTrigger) { window.Sentry.captureMessage("Signup banner trigger not found"); return; } let footerShown = false; window.addEventListener('load', () => { const rect = signupBannerTrigger.getBoundingClientRect(); // If page loaded up already scrolled below the trigger (via scroll restoration), show the banner by default if (rect.bottom < 0) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } }); // Wait for trigger to fully enter viewport before showing banner (ensures PDF CTAs are never covered by banner) const observer = new IntersectionObserver((entries) => { entries.forEach(entry => { if (entry.isIntersecting && !footerShown) { footerShown = true; signupBanner.classList.add('ds-signup-banner-visible'); } else if (!entry.isIntersecting && footerShown) { if (signupBannerTrigger.getBoundingClientRect().bottom > 0) { footerShown = false; signupBanner.classList.remove('ds-signup-banner-visible'); } } }); }); observer.observe(signupBannerTrigger); // Set up signup banner close button event handler: const signupBannerCloseButton = document.querySelector('#ds-signup-banner-close-button'); signupBannerCloseButton.addEventListener('click', () => { signupBanner.classList.remove('ds-signup-banner-visible'); observer.unobserve(signupBannerTrigger); // Store the signup banner's visibility state in session storage sessionStorage.setItem('ds-signup-banner-hidden', 'true'); }); })();</script></div></div></div><div data-auto_select="false" data-client_id="331998490334-rsn3chp12mbkiqhl6e7lu2q0mlbu0f1b" data-doc_id="75171349" data-landing_url="https://www.academia.edu/62389364/On_the_Natural_Gauge_Fields_of_Manifolds" data-login_uri="https://www.academia.edu/registrations/google_one_tap" data-moment_callback="onGoogleOneTapEvent" id="g_id_onload"></div><div class="ds-top-related-works--grid-container"><div class="ds-related-content--container ds-top-related-works--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="0" data-entity-id="79309023" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/79309023/A_Gauge_theoretical_Treatment_of_the_Gravitational_Field_Classical">A Gauge-theoretical Treatment of the Gravitational Field: Classical</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="25024933" href="https://perimeterinstitute.academia.edu/HenriqueGomes">Henrique Gomes</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2008</p><p class="ds-related-work--abstract ds2-5-body-sm">In the geometrodynamical setting of general relativity one is concerned mainly with Riemannian metrics over a manifold M . We show that for the space M := Riem(M), we have a natural principal fiber bundle (PFB) structure Diff(M) →֒ M π → M/Diff(M), first hinted at in [1]. This construction makes the gravitational field amenable to exactly the same gauge-theoretic treatment given in [2], where it is used to separate rotational and vibrational degrees of freedom of n-particle systems, both classically and quantum mechanically. Furthermore, we show how the gauge connection in this PFB setting can be seen as a realization of Mach’s Principle of Relative Motion, in accordance with Barbour’s et al work on timeless gravitational theories [3] using best-matching. We show Barbour’s reconstruction of GR is obtained by requiring the connection to be the one induced by the deWitt metric in M. As a simple application of the gauge theory, we put the ADM lagrangian in a Kaluza-Klein context, in wh...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Gauge-theoretical Treatment of the Gravitational Field: Classical","attachmentId":86067491,"attachmentType":"pdf","work_url":"https://www.academia.edu/79309023/A_Gauge_theoretical_Treatment_of_the_Gravitational_Field_Classical","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/79309023/A_Gauge_theoretical_Treatment_of_the_Gravitational_Field_Classical"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="1" data-entity-id="33195778" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/33195778/Differential_Geometry_and_Relativity_Theories_vol_1">Differential Geometry and Relativity Theories vol. 1</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33058671" href="https://ucriverside.academia.edu/DavidCarfi">David Carfi</a><span>, </span><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="6703645" href="https://hiu.academia.edu/DavidCarf%C3%AC">David Carfì</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><p class="ds-related-work--abstract ds2-5-body-sm">In this book, we focus on some aspects of smooth manifolds, which appear of fundamental importance for the developments of differential geometry and its applications to Theoretical Physics, Special and General Relativity, Economics and Finance. In particular we touch basic topics, for instance definition of tangent vectors, change of coordinate system in the definition of tangent vectors, action of tangent vectors on coordinate systems, structure of tangent spaces, geometric interpretation of tangent vectors, canonical tangent vectors determined by local charts, tangent frames determined by local charts, change of local frames, tangent vectors and contravariant vectors, covariant vectors, the gradient of a real function, invariant scalars, tangent applications, local Jacobian matrices, basic properties of the tangent map, chain rule, diffeomorphisms and derivatives, transformation of tangent bases under derivatives, paths on a manifold, vector derivative of a path with respect to a re-parametrization, tangent derivative versus calculus derivative, vector derivative of a path in local coordinates, existence of a path with a given initial tangent vector and other topics.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Differential Geometry and Relativity Theories vol. 1","attachmentId":53276944,"attachmentType":"pdf","work_url":"https://www.academia.edu/33195778/Differential_Geometry_and_Relativity_Theories_vol_1","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/33195778/Differential_Geometry_and_Relativity_Theories_vol_1"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="2" data-entity-id="127587983" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/127587983/The_problem_of_motion_in_gauge_theories_of_gravity">The problem of motion in gauge theories of gravity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="333927722" href="https://independent.academia.edu/SerhiiSamokhvalov">Serhii Samokhvalov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">General Relativity and Gravitation, 2021</p><p class="ds-related-work--abstract ds2-5-body-sm">In this article we consider the problem to what extent the motion of gauge-charged matter that generates the gravitational field can be arbitrary, as well as what equations are superimposed on the gauge field due to conditions of compatibility of gravitational field equations. Considered problem is analyzed from the point of view symmetry of the theory with respect to the generalized gauge deformed groups without specification of Lagrangians. In particular it is shown, that the motion of uncharged particles along geodesics of Riemannian space is inherent in an extremely wide range of theories of gravity and is a consequence of the gauge translational invariance of these theories under the condition of fulfilling equations of gravitational field. In the cause of gauge-charged particles, the Lorentz force, generalized for gauge-charged matter, appears in equations of motion as a consequence of the gauge symmetry of the theory under the condition of fulfilling equations of gravitational and gauge fields. In addition, we found relationships of equations for some fields that follow from the assumption about fulfilling of equations for other fields, for example, relationships of equations of the gravitational field and the gauge field of internal symmetry which follow from the assumption about fulfilling of equations of matter fields. In particular, we obtained the identity that generalizes in the case of arbitrary gauge field (and in the presence of gauge-charged matter) the identity found by Hilbert for the electromagnetic field. At the end of the article there is an Appendix, which briefly describes the main provisions and facts from the theory of generalized gauge deformed groups and presents the main ideas of a single group-theoretical interpretation of gauge fields of both external (space-time) and internal symmetry, which is an alternative to their geometric interpretation.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"The problem of motion in gauge theories of gravity","attachmentId":121299926,"attachmentType":"pdf","work_url":"https://www.academia.edu/127587983/The_problem_of_motion_in_gauge_theories_of_gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/127587983/The_problem_of_motion_in_gauge_theories_of_gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="3" data-entity-id="101171572" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory">Aspects Of Gauge Field Theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="267962040" href="https://independent.academia.edu/TarunPalla1">Tarun Palla</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Deccan Education Society's Willingdon College,Sangli, 2022</p><p class="ds-related-work--abstract ds2-5-body-sm">To describe successfully the properties and dynamics of the particle, we need field theory. The dynamics of gauge bosons are described by Gauge Field Theory, a type of quantum field theory. In this project, we work in four-dimensional Minkowski space. Gauge field theory is characterized by the presence of a vector field, we defined the Lagrangian for the gauge field (photon field) and showed that it is Lorentz invariant and gauge invariant. Any physical theory must be Lorentz invariant. We derived the equation of motion for the gauge field and derived all Maxwell's equations in terms of the electric and magnetic fields. We check the invariance of the action under global and local symmetry transformations using Noether's theorem. We derived the stress-energy tensor that gives us energy density, and the momentum of the field and determined the conserved charges associated with the gauge field. We then quantize the gauge field by treating each field as a harmonic oscillator. We derived the expression for vector field in terms of annihilation and creation operators. The gauge invariance implies that in four dimensions the photon has only two physical degrees of freedom</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Aspects Of Gauge Field Theory","attachmentId":101785541,"attachmentType":"pdf","work_url":"https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/101171572/Aspects_Of_Gauge_Field_Theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="4" data-entity-id="30738546" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/30738546/On_the_Motion_of_Matter_in_Spacetime">On the Motion of Matter in Spacetime</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39395900" href="https://independent.academia.edu/TuckerRobin">Robin Tucker</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the Motion of Matter in Spacetime","attachmentId":51184109,"attachmentType":"pdf","work_url":"https://www.academia.edu/30738546/On_the_Motion_of_Matter_in_Spacetime","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/30738546/On_the_Motion_of_Matter_in_Spacetime"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="5" data-entity-id="78935614" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/78935614/Gravitational_field_on_the_Lorentz_tangent_bundle_generalized_paths_and_field_equations">Gravitational field on the Lorentz tangent bundle: generalized paths and field equations</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="217823952" href="https://independent.academia.edu/AlkiviadisTriantafyllopoulos">Alkiviadis Triantafyllopoulos</a></div><p class="ds-related-work--metadata ds2-5-body-xs">The European Physical Journal Plus, 2020</p><p class="ds-related-work--abstract ds2-5-body-sm">We investigate the dynamics of gravitational field and particles in a generalized framework of a Lorentz tangent bundle. By variating an appropriate action for each case, we obtain generalized forms of paths and generalized field equations for a Sasaki type metric. We show that Stokes theorem is modified with respect to general relativity due to local anisotropy and the presence of a nonlinear connection which induces an adapted basis in our space.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Gravitational field on the Lorentz tangent bundle: generalized paths and field equations","attachmentId":85819498,"attachmentType":"pdf","work_url":"https://www.academia.edu/78935614/Gravitational_field_on_the_Lorentz_tangent_bundle_generalized_paths_and_field_equations","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/78935614/Gravitational_field_on_the_Lorentz_tangent_bundle_generalized_paths_and_field_equations"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="6" data-entity-id="71922959" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/71922959/On_General_Solutions_for_Field_Equations_in_Einstein_and_Higher_Dimension_Gravity">On General Solutions for Field Equations in Einstein and Higher Dimension Gravity</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="7910585" href="https://independent.academia.edu/VacaruSergiu">Sergiu Vacaru</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><p class="ds-related-work--abstract ds2-5-body-sm">We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (sele...</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On General Solutions for Field Equations in Einstein and Higher Dimension Gravity","attachmentId":81068184,"attachmentType":"pdf","work_url":"https://www.academia.edu/71922959/On_General_Solutions_for_Field_Equations_in_Einstein_and_Higher_Dimension_Gravity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/71922959/On_General_Solutions_for_Field_Equations_in_Einstein_and_Higher_Dimension_Gravity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="7" data-entity-id="125846009" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/125846009/Fibre_bundles_connections_general_relativity_and_Einstein_Cartan_theory">Fibre bundles, connections, general relativity, and Einstein-Cartan theory</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="329124018" href="https://independent.academia.edu/SocolovskyM">Miguel Socolovsky</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Fibre bundles, connections, general relativity, and Einstein-Cartan theory","attachmentId":119816256,"attachmentType":"pdf","work_url":"https://www.academia.edu/125846009/Fibre_bundles_connections_general_relativity_and_Einstein_Cartan_theory","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/125846009/Fibre_bundles_connections_general_relativity_and_Einstein_Cartan_theory"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="8" data-entity-id="113564428" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/113564428/Covariant_Canonical_Gauge_Theory_of_Classical_Gravitation_for_Scalar_Vector_and_Spin_1_2_Particle_Fields">Covariant Canonical Gauge Theory of Classical Gravitation for Scalar, Vector, and Spin-1/2 Particle Fields</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="174117521" href="https://fias.academia.edu/DavidVasak">David Vasak</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2023</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Covariant Canonical Gauge Theory of Classical Gravitation for Scalar, Vector, and Spin-1/2 Particle Fields","attachmentId":110491899,"attachmentType":"pdf","work_url":"https://www.academia.edu/113564428/Covariant_Canonical_Gauge_Theory_of_Classical_Gravitation_for_Scalar_Vector_and_Spin_1_2_Particle_Fields","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/113564428/Covariant_Canonical_Gauge_Theory_of_Classical_Gravitation_for_Scalar_Vector_and_Spin_1_2_Particle_Fields"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-wsj-grid-card" data-collection-position="9" data-entity-id="66340140" data-sort-order="default"><a class="ds-related-work--title js-wsj-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/66340140/J_ul_2_00_1_On_the_Motion_of_Matter_in_Spacetime">J ul 2 00 1 On the Motion of Matter in Spacetime</a><div class="ds-related-work--metadata"><a class="js-wsj-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="39395900" href="https://independent.academia.edu/TuckerRobin">Robin Tucker</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2003</p><p class="ds-related-work--abstract ds2-5-body-sm">We argue that the geodesic hypothesis based on auto-parallels of the Levi-Civita connection may need refinement in theories of gravity with additional scalar fields. This argument is illustrated with a re-formulation of the Brans-Dicke theory in terms of a spacetime connection with torsion determined dynamically in terms of the gradient of the Brans-Dicke scalar field. The perihelion shift in the orbit of Mercury is calculated on the alternative hypothesis that its worldline is an auto-parallel of such a connection. If scalar fields couple significantly to matter and spinless test particles move on such worldlines, current time keeping methods based on the conventional geodesic hypothesis may need refinement.</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"J ul 2 00 1 On the Motion of Matter in Spacetime","attachmentId":77571724,"attachmentType":"pdf","work_url":"https://www.academia.edu/66340140/J_ul_2_00_1_On_the_Motion_of_Matter_in_Spacetime","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-wsj-grid-card-view-pdf" href="https://www.academia.edu/66340140/J_ul_2_00_1_On_the_Motion_of_Matter_in_Spacetime"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div><div class="ds-sticky-ctas--wrapper js-loswp-sticky-ctas hidden"><div class="ds-sticky-ctas--grid-container"><div class="ds-sticky-ctas--container"><button class="ds2-5-button js-swp-download-button" data-signup-modal="{"location":"continue-reading-button--sticky-ctas","attachmentId":75171349,"attachmentType":"pdf","workUrl":null}">See full PDF</button><button class="ds2-5-button ds2-5-button--secondary js-swp-download-button" data-signup-modal="{"location":"download-pdf-button--sticky-ctas","attachmentId":75171349,"attachmentType":"pdf","workUrl":null}"><span class="material-symbols-outlined" style="font-size: 20px" translate="no">download</span>Download PDF</button></div></div></div><div class="ds-below-fold--grid-container"><div class="ds-work--container js-loswp-embedded-document"><div class="attachment_preview" data-attachment="Attachment_75171349" style="display: none"><div class="js-scribd-document-container"><div class="scribd--document-loading js-scribd-document-loader" style="display: block;"><img alt="Loading..." src="//a.academia-assets.com/images/loaders/paper-load.gif" /><p>Loading Preview</p></div></div><div style="text-align: center;"><div class="scribd--no-preview-alert js-preview-unavailable"><p>Sorry, preview is currently unavailable. You can download the paper by clicking the button above.</p></div></div></div></div><div class="ds-sidebar--container js-work-sidebar"><div class="ds-related-content--container"><h2 class="ds-related-content--heading">Related papers</h2><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="0" data-entity-id="54294393" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54294393/Reconsiderations_on_the_formulation_of_general_relativity_based_on_Riemannian_structures">Reconsiderations on the formulation of general relativity based on Riemannian structures</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="169882" href="https://buap-mx.academia.edu/ManuelSolano">Manuel Solano</a></div><p class="ds-related-work--metadata ds2-5-body-xs">General Relativity and Gravitation, 2011</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Reconsiderations on the formulation of general relativity based on Riemannian structures","attachmentId":70727888,"attachmentType":"pdf","work_url":"https://www.academia.edu/54294393/Reconsiderations_on_the_formulation_of_general_relativity_based_on_Riemannian_structures","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/54294393/Reconsiderations_on_the_formulation_of_general_relativity_based_on_Riemannian_structures"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="1" data-entity-id="98861440" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/98861440/Riemannian_Geometry_Definitions_Pictures_and_Results">Riemannian Geometry: Definitions, Pictures, and Results</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="262444114" href="https://independent.academia.edu/AdamMarsh30">Adam Marsh</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2014</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Riemannian Geometry: Definitions, Pictures, and Results","attachmentId":100098826,"attachmentType":"pdf","work_url":"https://www.academia.edu/98861440/Riemannian_Geometry_Definitions_Pictures_and_Results","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/98861440/Riemannian_Geometry_Definitions_Pictures_and_Results"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="2" data-entity-id="40559154" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/40559154/Equations_of_motion_in_the_theory_of_relativistic_vector_fields">Equations of motion in the theory of relativistic vector fields</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5962652" href="https://independent.academia.edu/SergeyFedosin">Sergey G Fedosin</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Letters of Chemistry, Physics and Astronomy, Vol. 83, pp. 12-30, 2019</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Equations of motion in the theory of relativistic vector fields","attachmentId":61171557,"attachmentType":"pdf","work_url":"https://www.academia.edu/40559154/Equations_of_motion_in_the_theory_of_relativistic_vector_fields","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/40559154/Equations_of_motion_in_the_theory_of_relativistic_vector_fields"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="3" data-entity-id="69097099" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/69097099/A_pr_2_01_7_Equations_of_Motion_as_Covariant_Gauss_Law_The">A pr 2 01 7 Equations of Motion as Covariant Gauss Law : The</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="65263234" href="https://independent.academia.edu/MArshadMomen">M. Arshad Momen</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2017</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A pr 2 01 7 Equations of Motion as Covariant Gauss Law : The","attachmentId":79326057,"attachmentType":"pdf","work_url":"https://www.academia.edu/69097099/A_pr_2_01_7_Equations_of_Motion_as_Covariant_Gauss_Law_The","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/69097099/A_pr_2_01_7_Equations_of_Motion_as_Covariant_Gauss_Law_The"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="4" data-entity-id="33656090" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/33656090/New_Supplementary_Conditions_for_a_Non_Linear_Field_Theory_General_Relativity">New Supplementary Conditions for a Non-Linear Field Theory: General Relativity</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="33695437" href="https://independent.academia.edu/GiampieroEsposito">Giampiero Esposito</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1999</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"New Supplementary Conditions for a Non-Linear Field Theory: General Relativity","attachmentId":53664645,"attachmentType":"pdf","work_url":"https://www.academia.edu/33656090/New_Supplementary_Conditions_for_a_Non_Linear_Field_Theory_General_Relativity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/33656090/New_Supplementary_Conditions_for_a_Non_Linear_Field_Theory_General_Relativity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="5" data-entity-id="78173836" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/78173836/Tensor_Tensor_Theory_of_Gravitation">Tensor-Tensor Theory of Gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="24599623" href="https://independent.academia.edu/MerabGogberashvili">Merab Gogberashvili</a></div><p class="ds-related-work--metadata ds2-5-body-xs">1996</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Tensor-Tensor Theory of Gravitation","attachmentId":85313783,"attachmentType":"pdf","work_url":"https://www.academia.edu/78173836/Tensor_Tensor_Theory_of_Gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/78173836/Tensor_Tensor_Theory_of_Gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="6" data-entity-id="19082966" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/19082966/Gravitation_and_Electromagnetism_as_Geometrical_Objects_of_a_Riemann_Cartan_Spacetime_Structure">Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5172094" href="https://unicamp.academia.edu/WaldyrRodrigues">Waldyr A Rodrigues</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Advances in Applied Clifford Algebras, 2012</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Gravitation and Electromagnetism as Geometrical Objects of a Riemann-Cartan Spacetime Structure","attachmentId":40421243,"attachmentType":"pdf","work_url":"https://www.academia.edu/19082966/Gravitation_and_Electromagnetism_as_Geometrical_Objects_of_a_Riemann_Cartan_Spacetime_Structure","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/19082966/Gravitation_and_Electromagnetism_as_Geometrical_Objects_of_a_Riemann_Cartan_Spacetime_Structure"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="7" data-entity-id="22822026" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/22822026/12_Paper_3_SCI_EXPANDED_Z_Kasap_Weyl_Euler_Lagrange_Equations_of_Motion_on_Flat_Manifold_Advances_in_Mathematical_Physics_ISSN_1687_9120_http_dx_doi_org_10_1155_2015_808016_2015_1_11">12. Paper: 3. SCI-EXPANDED, Z. Kasap, Weyl-Euler-Lagrange Equations of Motion on Flat Manifold, Advances in Mathematical Physics, (ISSN:1687-9120),, http://dx.doi.org/10.1155/2015/808016, (2015), 1-11.</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="20237978" href="https://pamukkale.academia.edu/zekikasap">Zeki Kasap</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"12. Paper: 3. SCI-EXPANDED, Z. Kasap, Weyl-Euler-Lagrange Equations of Motion on Flat Manifold, Advances in Mathematical Physics, (ISSN:1687-9120),, http://dx.doi.org/10.1155/2015/808016, (2015), 1-11.","attachmentId":43368151,"attachmentType":"pdf","work_url":"https://www.academia.edu/22822026/12_Paper_3_SCI_EXPANDED_Z_Kasap_Weyl_Euler_Lagrange_Equations_of_Motion_on_Flat_Manifold_Advances_in_Mathematical_Physics_ISSN_1687_9120_http_dx_doi_org_10_1155_2015_808016_2015_1_11","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/22822026/12_Paper_3_SCI_EXPANDED_Z_Kasap_Weyl_Euler_Lagrange_Equations_of_Motion_on_Flat_Manifold_Advances_in_Mathematical_Physics_ISSN_1687_9120_http_dx_doi_org_10_1155_2015_808016_2015_1_11"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="8" data-entity-id="123662751" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/123662751/Geometry_Symmetries_and_Classical_Physics_A_Mosaic">Geometry, Symmetries, and Classical Physics: A Mosaic</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="292502683" href="https://uni-heidelberg.academia.edu/ManousosMarkoutsakis">Manousos Markoutsakis</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Geometry, Symmetries, and Classical Physics: A Mosaic, 2021</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Geometry, Symmetries, and Classical Physics: A Mosaic","attachmentId":118065360,"attachmentType":"pdf","work_url":"https://www.academia.edu/123662751/Geometry_Symmetries_and_Classical_Physics_A_Mosaic","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/123662751/Geometry_Symmetries_and_Classical_Physics_A_Mosaic"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="9" data-entity-id="33361755" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/33361755/Geodesic_invariant_equations_of_gravitation_Geodesic_invariant_equations_of_gravitation">Geodesic-invariant equations of gravitation Geodesic-invariant equations of gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3050126" href="https://independent.academia.edu/LeonidVerozub">Leonid Verozub</a><span>, </span><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5669558" href="https://independent.academia.edu/LVerozub">Leonid Verozub</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Geodesic-invariant equations of gravitation Geodesic-invariant equations of gravitation","attachmentId":53418954,"attachmentType":"pdf","work_url":"https://www.academia.edu/33361755/Geodesic_invariant_equations_of_gravitation_Geodesic_invariant_equations_of_gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/33361755/Geodesic_invariant_equations_of_gravitation_Geodesic_invariant_equations_of_gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="10" data-entity-id="30956588" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/30956588/On_the_Poincar_e_Gauge_Theory_of_Gravitation">On the Poincar´e Gauge Theory of Gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="32536896" href="https://unina.academia.edu/SalvatoreCapozziello">Salvatore Capozziello</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2009</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On the Poincar´e Gauge Theory of Gravitation","attachmentId":51387863,"attachmentType":"pdf","work_url":"https://www.academia.edu/30956588/On_the_Poincar_e_Gauge_Theory_of_Gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/30956588/On_the_Poincar_e_Gauge_Theory_of_Gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="11" data-entity-id="93964405" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/93964405/Mathematics_of_Differential_Geometry_and_Relativity">Mathematics of Differential Geometry and Relativity</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="7424508" href="https://amu-in.academia.edu/zafarahsan">zafar ahsan</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2016</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Mathematics of Differential Geometry and Relativity","attachmentId":96555595,"attachmentType":"pdf","work_url":"https://www.academia.edu/93964405/Mathematics_of_Differential_Geometry_and_Relativity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/93964405/Mathematics_of_Differential_Geometry_and_Relativity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="12" data-entity-id="54830828" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/54830828/Conserved_quantities_from_the_equations_of_motion_with_applications_to_natural_and_gauge_natural_theories_of_gravitation">Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="4693682" href="https://unito.academia.edu/MarcoFerraris">Marco Ferraris</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Classical and Quantum Gravity, 2003</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Conserved quantities from the equations of motion: with applications to natural and gauge natural theories of gravitation","attachmentId":71003277,"attachmentType":"pdf","work_url":"https://www.academia.edu/54830828/Conserved_quantities_from_the_equations_of_motion_with_applications_to_natural_and_gauge_natural_theories_of_gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/54830828/Conserved_quantities_from_the_equations_of_motion_with_applications_to_natural_and_gauge_natural_theories_of_gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="13" data-entity-id="49864476" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/49864476/Geodesic_invariant_equations_of_gravitation">Geodesic-invariant equations of gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3050126" href="https://independent.academia.edu/LeonidVerozub">Leonid Verozub</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Annalen der Physik, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Geodesic-invariant equations of gravitation","attachmentId":68064652,"attachmentType":"pdf","work_url":"https://www.academia.edu/49864476/Geodesic_invariant_equations_of_gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/49864476/Geodesic_invariant_equations_of_gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="14" data-entity-id="19083010" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/19083010/A_MAXWELL_LIKE_FORMULATION_OF_GRAVITATIONAL_THEORY_IN_MINKOWSKI_SPACE_TIME">A MAXWELL-LIKE FORMULATION OF GRAVITATIONAL THEORY IN MINKOWSKI SPACE–TIME</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="5172094" href="https://unicamp.academia.edu/WaldyrRodrigues">Waldyr A Rodrigues</a></div><p class="ds-related-work--metadata ds2-5-body-xs">International Journal of Modern Physics D, 2007</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A MAXWELL-LIKE FORMULATION OF GRAVITATIONAL THEORY IN MINKOWSKI SPACE–TIME","attachmentId":40421264,"attachmentType":"pdf","work_url":"https://www.academia.edu/19083010/A_MAXWELL_LIKE_FORMULATION_OF_GRAVITATIONAL_THEORY_IN_MINKOWSKI_SPACE_TIME","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/19083010/A_MAXWELL_LIKE_FORMULATION_OF_GRAVITATIONAL_THEORY_IN_MINKOWSKI_SPACE_TIME"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="15" data-entity-id="1382435" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/1382435/Some_subtleties_of_Riemannian_geometry">Some subtleties of Riemannian geometry</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="544278" href="https://ipmnet.academia.edu/YuriRylov">Yuri Rylov</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Arxiv preprint arXiv:0812.2833, 2008</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Some subtleties of Riemannian geometry","attachmentId":31155840,"attachmentType":"pdf","work_url":"https://www.academia.edu/1382435/Some_subtleties_of_Riemannian_geometry","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/1382435/Some_subtleties_of_Riemannian_geometry"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="16" data-entity-id="19645293" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/19645293/A_Gauge_Theory_of_Gravity_in_Curved_Phase_Spaces">A Gauge Theory of Gravity in Curved Phase-Spaces</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="3337625" href="https://independent.academia.edu/CarlosCastroPerelman">Carlos Castro Perelman</a></div><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"A Gauge Theory of Gravity in Curved Phase-Spaces","attachmentId":40756001,"attachmentType":"pdf","work_url":"https://www.academia.edu/19645293/A_Gauge_Theory_of_Gravity_in_Curved_Phase_Spaces","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/19645293/A_Gauge_Theory_of_Gravity_in_Curved_Phase_Spaces"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="17" data-entity-id="52964413" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/52964413/Conservation_laws_on_Riemann_Cartan_Lorentzian_and_teleparallel_spacetimes_II_Appendices_Clifford_bundles_Maxwell_theory_and_teleparallel_spacetimes">Conservation laws on Riemann-Cartan, Lorentzian and teleparallel spacetimes. II: Appendices: Clifford bundles, Maxwell theory, and teleparallel spacetimes</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="37078879" href="https://independent.academia.edu/QSouza">Q. Souza</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2007</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Conservation laws on Riemann-Cartan, Lorentzian and teleparallel spacetimes. II: Appendices: Clifford bundles, Maxwell theory, and teleparallel spacetimes","attachmentId":69972250,"attachmentType":"pdf","work_url":"https://www.academia.edu/52964413/Conservation_laws_on_Riemann_Cartan_Lorentzian_and_teleparallel_spacetimes_II_Appendices_Clifford_bundles_Maxwell_theory_and_teleparallel_spacetimes","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/52964413/Conservation_laws_on_Riemann_Cartan_Lorentzian_and_teleparallel_spacetimes_II_Appendices_Clifford_bundles_Maxwell_theory_and_teleparallel_spacetimes"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="18" data-entity-id="28520930" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/28520930/On_a_Generalized_Theory_of_Relativity">On a Generalized Theory of Relativity</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="9135532" href="https://nust-ac-zw.academia.edu/GoldenGadziraiNyambuya">Golden Gadzirai Nyambuya</a></div><p class="ds-related-work--metadata ds2-5-body-xs">2010</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"On a Generalized Theory of Relativity","attachmentId":48873509,"attachmentType":"pdf","work_url":"https://www.academia.edu/28520930/On_a_Generalized_Theory_of_Relativity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/28520930/On_a_Generalized_Theory_of_Relativity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="19" data-entity-id="43183354" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/43183354/Leibnizian_Galilean_and_Newtonian_structures_of_space_time">Leibnizian, Galilean and Newtonian structures of space–time</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="23423358" href="https://uni.academia.edu/MiguelAngelPonceSanchez">Miguel Angel Ponce Sanchez</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Journal of Mathematical Physics, 2003</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Leibnizian, Galilean and Newtonian structures of space–time","attachmentId":63449542,"attachmentType":"pdf","work_url":"https://www.academia.edu/43183354/Leibnizian_Galilean_and_Newtonian_structures_of_space_time","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/43183354/Leibnizian_Galilean_and_Newtonian_structures_of_space_time"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="20" data-entity-id="18885365" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/18885365/Towards_a_Covariant_Theory_of_Gravitation">Towards a Covariant Theory of Gravitation</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38999594" href="https://ccny1-cuny.academia.edu/AlexanderPoltorak">Alexander Poltorak</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Abstracts of Contributed Papers for the 9th International Conference on General Relativity and Gravitation, Jena, GDR, 1980</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Towards a Covariant Theory of Gravitation","attachmentId":40310009,"attachmentType":"pdf","work_url":"https://www.academia.edu/18885365/Towards_a_Covariant_Theory_of_Gravitation","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/18885365/Towards_a_Covariant_Theory_of_Gravitation"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="21" data-entity-id="101601301" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/101601301/Metric_Gauge_Fields_in_Deformed_Special_Relativity">Metric Gauge Fields in Deformed Special Relativity</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="35590555" href="https://independent.academia.edu/APetrucci1">A. Petrucci</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv (Cornell University), 2013</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Metric Gauge Fields in Deformed Special Relativity","attachmentId":102097705,"attachmentType":"pdf","work_url":"https://www.academia.edu/101601301/Metric_Gauge_Fields_in_Deformed_Special_Relativity","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/101601301/Metric_Gauge_Fields_in_Deformed_Special_Relativity"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="22" data-entity-id="86417463" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/86417463/Some_Mathematical_Issues_Pertaining_to_Translational_Gauge_Theories">Some Mathematical Issues Pertaining to Translational Gauge Theories</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="64582908" href="https://independent.academia.edu/DavidDelphenich">David Delphenich</a></div><p class="ds-related-work--metadata ds2-5-body-xs">Eprint Arxiv Gr Qc 0401066, 2004</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"Some Mathematical Issues Pertaining to Translational Gauge Theories","attachmentId":90876140,"attachmentType":"pdf","work_url":"https://www.academia.edu/86417463/Some_Mathematical_Issues_Pertaining_to_Translational_Gauge_Theories","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/86417463/Some_Mathematical_Issues_Pertaining_to_Translational_Gauge_Theories"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div><div class="ds-related-work--container js-related-work-sidebar-card" data-collection-position="23" data-entity-id="93325720" data-sort-order="default"><a class="ds-related-work--title js-related-work-grid-card-title ds2-5-body-md ds2-5-body-link" href="https://www.academia.edu/93325720/From_PDE_Systems_and_Metrics_to_Generalized_Field_Theories">From PDE Systems and Metrics to Generalized Field Theories</a><div class="ds-related-work--metadata"><a class="js-related-work-grid-card-author ds2-5-body-sm ds2-5-body-link" data-author-id="38243373" href="https://independent.academia.edu/MirceaNeagu">Mircea Neagu</a></div><p class="ds-related-work--metadata ds2-5-body-xs">arXiv: Differential Geometry, 2001</p><div class="ds-related-work--ctas"><button class="ds2-5-text-link ds2-5-text-link--inline js-swp-download-button" data-signup-modal="{"location":"wsj-grid-card-download-pdf-modal","work_title":"From PDE Systems and Metrics to Generalized Field Theories","attachmentId":96094043,"attachmentType":"pdf","work_url":"https://www.academia.edu/93325720/From_PDE_Systems_and_Metrics_to_Generalized_Field_Theories","alternativeTracking":true}"><span class="material-symbols-outlined" style="font-size: 18px" translate="no">download</span><span class="ds2-5-text-link__content">Download free PDF</span></button><a class="ds2-5-text-link ds2-5-text-link--inline js-related-work-grid-card-view-pdf" href="https://www.academia.edu/93325720/From_PDE_Systems_and_Metrics_to_Generalized_Field_Theories"><span class="ds2-5-text-link__content">View PDF</span><span class="material-symbols-outlined" style="font-size: 18px" translate="no">chevron_right</span></a></div></div></div></div></div></div><div class="footer--content"><ul class="footer--main-links hide-on-mobile"><li><a href="https://www.academia.edu/about">About</a></li><li><a href="https://www.academia.edu/press">Press</a></li><li><a href="https://www.academia.edu/documents">Papers</a></li><li><a href="https://www.academia.edu/topics">Topics</a></li><li><a href="https://www.academia.edu/hiring"><svg style="width: 13px; height: 13px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="briefcase" class="svg-inline--fa fa-briefcase fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M320 336c0 8.84-7.16 16-16 16h-96c-8.84 0-16-7.16-16-16v-48H0v144c0 25.6 22.4 48 48 48h416c25.6 0 48-22.4 48-48V288H320v48zm144-208h-80V80c0-25.6-22.4-48-48-48H176c-25.6 0-48 22.4-48 48v48H48c-25.6 0-48 22.4-48 48v80h512v-80c0-25.6-22.4-48-48-48zm-144 0H192V96h128v32z"></path></svg> <strong>We're Hiring!</strong></a></li><li><a href="https://support.academia.edu/hc/en-us"><svg style="width: 12px; height: 12px; position: relative; bottom: -1px;" aria-hidden="true" focusable="false" data-prefix="fas" data-icon="question-circle" class="svg-inline--fa fa-question-circle fa-w-16" role="img" xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512"><path fill="currentColor" d="M504 256c0 136.997-111.043 248-248 248S8 392.997 8 256C8 119.083 119.043 8 256 8s248 111.083 248 248zM262.655 90c-54.497 0-89.255 22.957-116.549 63.758-3.536 5.286-2.353 12.415 2.715 16.258l34.699 26.31c5.205 3.947 12.621 3.008 16.665-2.122 17.864-22.658 30.113-35.797 57.303-35.797 20.429 0 45.698 13.148 45.698 32.958 0 14.976-12.363 22.667-32.534 33.976C247.128 238.528 216 254.941 216 296v4c0 6.627 5.373 12 12 12h56c6.627 0 12-5.373 12-12v-1.333c0-28.462 83.186-29.647 83.186-106.667 0-58.002-60.165-102-116.531-102zM256 338c-25.365 0-46 20.635-46 46 0 25.364 20.635 46 46 46s46-20.636 46-46c0-25.365-20.635-46-46-46z"></path></svg> <strong>Help Center</strong></a></li></ul><ul class="footer--research-interests"><li>Find new research papers in:</li><li><a href="https://www.academia.edu/Documents/in/Physics">Physics</a></li><li><a href="https://www.academia.edu/Documents/in/Chemistry">Chemistry</a></li><li><a href="https://www.academia.edu/Documents/in/Biology">Biology</a></li><li><a href="https://www.academia.edu/Documents/in/Health_Sciences">Health Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Ecology">Ecology</a></li><li><a href="https://www.academia.edu/Documents/in/Earth_Sciences">Earth Sciences</a></li><li><a href="https://www.academia.edu/Documents/in/Cognitive_Science">Cognitive Science</a></li><li><a href="https://www.academia.edu/Documents/in/Mathematics">Mathematics</a></li><li><a href="https://www.academia.edu/Documents/in/Computer_Science">Computer Science</a></li></ul><ul class="footer--legal-links hide-on-mobile"><li><a href="https://www.academia.edu/terms">Terms</a></li><li><a href="https://www.academia.edu/privacy">Privacy</a></li><li><a href="https://www.academia.edu/copyright">Copyright</a></li><li>Academia ©2025</li></ul></div> </body> </html>