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href="/nlab/show/physics">physics</a></strong>, <a class="existingWikiWord" href="/nlab/show/mathematical+physics">mathematical physics</a>, <a class="existingWikiWord" href="/nlab/show/philosophy+of+physics">philosophy of physics</a></p> <h2 id="surveys_textbooks_and_lecture_notes">Surveys, textbooks and lecture notes</h2> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">(higher) category theory and physics</a></em></p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics">geometry of physics</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/books+and+reviews+in+mathematical+physics">books and reviews</a>, <a class="existingWikiWord" href="/nlab/show/physics+resources">physics resources</a></p> </li> </ul> <hr /> <p><a class="existingWikiWord" href="/nlab/show/theory+%28physics%29">theory (physics)</a>, <a class="existingWikiWord" href="/nlab/show/model+%28physics%29">model (physics)</a></p> <p><a class="existingWikiWord" href="/nlab/show/experiment">experiment</a>, <a class="existingWikiWord" href="/nlab/show/measurement">measurement</a>, <a class="existingWikiWord" href="/nlab/show/computable+physics">computable physics</a></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/mechanics">mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass">mass</a>, <a class="existingWikiWord" href="/nlab/show/charge">charge</a>, <a class="existingWikiWord" href="/nlab/show/momentum">momentum</a>, <a class="existingWikiWord" href="/nlab/show/angular+momentum">angular momentum</a>, <a class="existingWikiWord" href="/nlab/show/moment+of+inertia">moment of inertia</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dynamics+on+Lie+groups">dynamics on Lie groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/rigid+body+dynamics">rigid body dynamics</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/field+%28physics%29">field (physics)</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lagrangian+mechanics">Lagrangian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/configuration+space">configuration space</a>, <a class="existingWikiWord" href="/nlab/show/state">state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a>, <a class="existingWikiWord" href="/nlab/show/Lagrangian">Lagrangian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/covariant+phase+space">covariant phase space</a>, <a class="existingWikiWord" href="/nlab/show/Euler-Lagrange+equations">Euler-Lagrange equations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+mechanics">Hamiltonian mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phase+space">phase space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+groupoid">symplectic groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multisymplectic+geometry">multisymplectic geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-symplectic+manifold">n-symplectic manifold</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+Lorentzian+manifold">smooth Lorentzian manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/special+relativity">special relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity">gravity</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a>, <a class="existingWikiWord" href="/nlab/show/dilaton+gravity">dilaton gravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/black+hole">black hole</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/classical+field+theory">Classical field theory</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+physics">classical physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/classical+mechanics">classical mechanics</a></li> <li><a class="existingWikiWord" href="/nlab/show/waves">waves</a> and <a class="existingWikiWord" href="/nlab/show/optics">optics</a></li> <li><a class="existingWikiWord" href="/nlab/show/thermodynamics">thermodynamics</a></li> </ul> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+mechanics">Quantum Mechanics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+mechanics+in+terms+of+dagger-compact+categories">in terms of ∞-compact categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+information">quantum information</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hamiltonian+operator">Hamiltonian operator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/density+matrix">density matrix</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kochen-Specker+theorem">Kochen-Specker theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bell%27s+theorem">Bell's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gleason%27s+theorem">Gleason's theorem</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantization">Quantization</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+quantization">deformation quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+integral">path integral quantization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semiclassical+approximation">semiclassical approximation</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/quantum+field+theory">Quantum Field Theory</a></strong></p> <ul> <li> <p>Axiomatizations</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/AQFT">algebraic QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Wightman+axioms">Wightman axioms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Haag-Kastler+axioms">Haag-Kastler axioms</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/operator+algebra">operator algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+net">local net</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+net">conformal net</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Reeh-Schlieder+theorem">Reeh-Schlieder theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Osterwalder-Schrader+theorem">Osterwalder-Schrader theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/PCT+theorem">PCT theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bisognano-Wichmann+theorem">Bisognano-Wichmann theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/modular+theory">modular theory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/boson">boson</a>, <a class="existingWikiWord" href="/nlab/show/fermion">fermion</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/FQFT">functorial QFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism">cobordism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category+of+cobordisms">(∞,n)-category of cobordisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cobordism+hypothesis">cobordism hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/extended+topological+quantum+field+theory">extended topological quantum field theory</a></p> </li> </ul> </li> </ul> </li> <li> <p>Tools</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/perturbative+quantum+field+theory">perturbative quantum field theory</a>, <a class="existingWikiWord" href="/nlab/show/vacuum">vacuum</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+quantum+field+theory">effective quantum field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/renormalization">renormalization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BV-BRST+formalism">BV-BRST formalism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-function+theory">geometric ∞-function theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/particle+physics">particle physics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/phenomenology">phenomenology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+%28in+particle+phyiscs%29">models</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/standard+model+of+particle+physics">standard model of particle physics</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fields+and+quanta+-+table">fields and quanta</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT">Grand Unified Theories</a>, <a class="existingWikiWord" href="/nlab/show/MSSM">MSSM</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/scattering+amplitude">scattering amplitude</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/on-shell+recursion">on-shell recursion</a>, <a class="existingWikiWord" href="/nlab/show/KLT+relations">KLT relations</a></li> </ul> </li> </ul> </li> <li> <p>Structural phenomena</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universality+class">universality class</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quantum+anomaly">quantum anomaly</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Green-Schwarz+mechanism">Green-Schwarz mechanism</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/instanton">instanton</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spontaneously+broken+symmetry">spontaneously broken symmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kaluza-Klein+mechanism">Kaluza-Klein mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integrable+systems">integrable systems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holonomic+quantum+fields">holonomic quantum fields</a></p> </li> </ul> </li> <li> <p>Types of quantum field thories</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/TQFT">TQFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2d+TQFT">2d TQFT</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Dijkgraaf-Witten+theory">Dijkgraaf-Witten theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/TCFT">TCFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-model">A-model</a>, <a class="existingWikiWord" href="/nlab/show/B-model">B-model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+mirror+symmetry">homological mirror symmetry</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/QFT+with+defects">QFT with defects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/conformal+field+theory">conformal field theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%281%2C1%29-dimensional+Euclidean+field+theories+and+K-theory">(1,1)-dimensional Euclidean field theories and K-theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-dimensional+Euclidean+field+theory">(2,1)-dimensional Euclidean field theory and elliptic cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/CFT">CFT</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/WZW+model">WZW model</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/6d+%282%2C0%29-supersymmetric+QFT">6d (2,0)-supersymmetric QFT</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+theory">gauge theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/field+strength">field strength</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a>, <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a>, <a class="existingWikiWord" href="/nlab/show/gauge+fixing">gauge fixing</a></p> </li> <li> <p>examples</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>, <a class="existingWikiWord" href="/nlab/show/QED">QED</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/electric+charge">electric charge</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/magnetic+charge">magnetic charge</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yang-Mills+field">Yang-Mills field</a>, <a class="existingWikiWord" href="/nlab/show/QCD">QCD</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Yang-Mills+theory">Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spinors+in+Yang-Mills+theory">spinors in Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+Yang-Mills+theory">topological Yang-Mills theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kalb-Ramond+field">Kalb-Ramond field</a></li> <li><a class="existingWikiWord" href="/nlab/show/supergravity+C-field">supergravity C-field</a></li> <li><a class="existingWikiWord" href="/nlab/show/RR+field">RR field</a></li> <li><a class="existingWikiWord" href="/nlab/show/first-order+formulation+of+gravity">first-order formulation of gravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/general+covariance">general covariance</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/D%27Auria-Fre+formulation+of+supergravity">D'Auria-Fre formulation of supergravity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravity+as+a+BF-theory">gravity as a BF-theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/particle">particle</a>, <a class="existingWikiWord" href="/nlab/show/relativistic+particle">relativistic particle</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particle">fundamental particle</a>, <a class="existingWikiWord" href="/nlab/show/spinning+particle">spinning particle</a>, <a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string">string</a>, <a class="existingWikiWord" href="/nlab/show/spinning+string">spinning string</a>, <a class="existingWikiWord" href="/nlab/show/superstring">superstring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/membrane">membrane</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/AKSZ+theory">AKSZ theory</a></p> </li> </ul> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+theory">String Theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/string+theory+results+applied+elsewhere">string theory results applied elsewhere</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/number+theory+and+physics">number theory and physics</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Riemann+hypothesis+and+physics">Riemann hypothesis and physics</a></li> </ul> </li> </ul> <div> <p> <a href="/nlab/edit/physicscontents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="mass">Mass</h1> <div class='maruku_toc'> <ul> <li><a href='#definitions'>Definitions</a></li> <li><a href='#formulas'>Formulas</a></li> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#References'>References</a></li> </ul> </div> <h2 id="definitions">Definitions</h2> <p>The <strong>mass</strong> of a <a class="existingWikiWord" href="/nlab/show/physics">physical</a> system is its intrinsic <a class="existingWikiWord" href="/nlab/show/energy">energy</a>.</p> <p>It is a somewhat arbitrary choice, made necessary by the theory of <a class="existingWikiWord" href="/nlab/show/relativity">relativity</a>, to decide whether the old term ‘mass’ should be used for intrinsic energy or total energy; one may clarify that intrinsic energy is <strong>rest mass</strong> while total energy is <strong>relativistic mass</strong>.</p> <p>Traditionally, mass and energy are measured in different units related by the <a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>=</mo><mi>m</mi><msup><mi>c</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E = m c^2</annotation></semantics></math>. Thus one may call a system's mass its <strong>rest energy</strong> when it is measured in units of energy; conversely, one may call a systems's total energy its <strong>relativistic mass</strong> when it is measured in units of mass.</p> <p>The <strong><a class="existingWikiWord" href="/nlab/show/kinetic+energy">kinetic energy</a></strong> of a system is the difference between its total energy and its mass (or rest energy). Thus the kinetic energy is the energy due to the motion of the system (or better, due to the relative motion of the system and the observer).</p> <p>When a system is thought of as the combination of several <a class="existingWikiWord" href="/nlab/show/subsystem">subsystems</a>, then its mass also may be broken into several pieces:</p> <ul> <li>for each subsystem, its mass,</li> <li>for each subsystem, its kinetic energy due to relative motion of the subsystem,</li> <li>for each pair of subsystems, the <a class="existingWikiWord" href="/nlab/show/potential+energy">potential energy</a> of their interaction.</li> </ul> <p>The latter entry may be positive (in which case the system is liable to separate into its component parts, at least at sufficiently high <a class="existingWikiWord" href="/nlab/show/temperature">temperature</a>, giving off energy) or negative (in which case the system is <strong>bound</strong> and can be expected to remain together until additional energy is added).</p> <h2 id="formulas">Formulas</h2> <p>Mass <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> may be given in terms of energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/linear+momentum">linear momentum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>p</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{p}</annotation></semantics></math> as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><msqrt><mrow><msup><mi>E</mi> <mn>2</mn></msup><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow></msqrt><mo>.</mo></mrow><annotation encoding="application/x-tex"> m = \sqrt {E^2 - {|\mathbf{p}|}^2} .</annotation></semantics></math></div> <p>As <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo>=</mo><mi>E</mi><mstyle mathvariant="bold"><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{p} = E \mathbf{v}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math> the linear <a class="existingWikiWord" href="/nlab/show/velocity">velocity</a>, we also have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mi>E</mi><msqrt><mrow><mn>1</mn><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow></msqrt><mo>.</mo></mrow><annotation encoding="application/x-tex"> m = E \sqrt {1 - {|\mathbf{v}|}^2} .</annotation></semantics></math></div> <p>(For nonrelativistic units, change <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">E/c^2</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>p</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{p}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\mathbf{p}/c</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{v}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">/</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">\mathbf{v}/c</annotation></semantics></math>.)</p> <p>For a particle travelling at the <a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a>, we therefore have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m = 0</annotation></semantics></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbf{v} = 0</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">m = E</annotation></semantics></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow><mo>≪</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">0 \lt {|\mathbf{v}|} \ll c</annotation></semantics></math>, we still have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≈</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> m \approx E</annotation></semantics></math>. However, in nonrelativistic physics, it is much more useful to use <a class="existingWikiWord" href="/nlab/show/kinetic+energy">kinetic energy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><mi>E</mi><mo>−</mo><mi>m</mi></mrow><annotation encoding="application/x-tex">K = E - m</annotation></semantics></math> instead of the total energy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math>; then we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><mi>K</mi><mfrac><mrow><mn>1</mn><mo>−</mo><msup><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup><mo>+</mo><msqrt><mrow><mn>1</mn><mo>−</mo><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><msup><mo stretchy="false">|</mo> <mn>2</mn></msup></mrow></mrow></msqrt></mrow><mrow><msup><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> m = K \frac { 1 - {|\mathbf{v}|}^2 + \sqrt {1 - {|\mathbf{v}|^2}} } { {|\mathbf{v}|}^2 } </annotation></semantics></math></div> <p>exactly and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≈</mo><mn>2</mn><mfrac><mi>K</mi><mrow><msup><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow> <mn>2</mn></msup></mrow></mfrac></mrow><annotation encoding="application/x-tex"> m \approx 2 \frac K { {|\mathbf{v}|}^2 } </annotation></semantics></math></div> <p>when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo><</mo><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow><mo>≪</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">0 \lt {|\mathbf{v}|} \ll c</annotation></semantics></math>. Notice that this last expression makes sense already in nonrelativistic units.</p> <p>For a system made up of subsystems (indexed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>,</mo><mi>…</mi></mrow><annotation encoding="application/x-tex">i,j,\ldots</annotation></semantics></math>), we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>m</mi> <mi>i</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>i</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>,</mo></mrow><annotation encoding="application/x-tex"> m = \sum_i m_i + \sum_i \tilde{K}_i + \sum_{i,j} U_{i,j} ,</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math> is the mass of subsystem <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{K}_i</annotation></semantics></math> is the kinetic energy of subsystem <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> relative to the whole system, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i,j}</annotation></semantics></math> is the potential energy of the <a class="existingWikiWord" href="/nlab/show/forces">forces</a> between system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math>. The sum</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>m</mi> <mi>i</mi></msub><mo>+</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>i</mi></msub><mo>−</mo><mi>m</mi></mrow><annotation encoding="application/x-tex"> - \sum_{i,j} U_{i,j} = \sum_i m_i + \sum_i \tilde{K}_i - m </annotation></semantics></math></div> <p>is the <strong><a class="existingWikiWord" href="/nlab/show/binding+energy">binding energy</a></strong> of the system; when it is positive, it gives the minimum energy that must be added to the system to break it into components that do not interact. In many cases, the relative kinetic energies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>K</mi><mo stretchy="false">˜</mo></mover> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\tilde{K}_i</annotation></semantics></math> are negligible, so we can calculate binding energy by subtracting masses as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></munder><msub><mi>U</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo>≈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>m</mi> <mi>i</mi></msub><mo>−</mo><mi>m</mi><mo>.</mo></mrow><annotation encoding="application/x-tex"> - \sum_{i,j} U_{i,j} \approx \sum_i m_i - m .</annotation></semantics></math></div> <p>In nonrelativistic physics, where kinetic energy is always negligible, the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub></mrow><annotation encoding="application/x-tex">U_{i,j}</annotation></semantics></math> are <em>also</em> negligible compared to the masses, so it is not possible at all to calculate the sign of the binding energy using masses. Instead, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo>≈</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><msub><mi>m</mi> <mi>i</mi></msub><mo>,</mo></mrow><annotation encoding="application/x-tex"> m \approx \sum_i m_i ,</annotation></semantics></math></div> <p>the statement of nonrelativistic <strong>conservation of mass</strong>.</p> <p>Many of the formulas above rely on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>v</mi></mstyle><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>c</mi></mrow><annotation encoding="application/x-tex">{|\mathbf{v}|} \leq c</annotation></semantics></math>, or equivalently <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">|</mo></mrow><mo>≤</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">{|\mathbf{p}|} \leq E</annotation></semantics></math>. For <a class="existingWikiWord" href="/nlab/show/tachyon">tachyons</a>, where this is violated, the mass becomes an <a class="existingWikiWord" href="/nlab/show/imaginary+number">imaginary number</a>. For this reason, it may make more sense to work with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>m</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">m^2</annotation></semantics></math> (which is always real) than with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math> itself. We also implicitly assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E \gt 0</annotation></semantics></math>; for <span class="newWikiWord">exotic particles<a href="/nlab/new/exotic+particle">?</a></span> in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">E \lt 0</annotation></semantics></math>, it is convenient also to take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo><</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">m \lt 0</annotation></semantics></math>. (In this case, of course, it is <em>not</em> sufficient to work only with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>m</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">m^2</annotation></semantics></math>.)</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/mass+term">mass term</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mass+gap">mass gap</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Higgs+mechanism">Higgs mechanism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ADM+mass">ADM mass</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/massive+Yang-Mills+theory">massive Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/space">space</a>, <a class="existingWikiWord" href="/nlab/show/time">time</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a></p> </li> </ul> <div> <p><strong>fundamental <a class="existingWikiWord" href="/nlab/show/scales">scales</a></strong> (fundamental/<a class="existingWikiWord" href="/nlab/show/natural+unit">natural</a> <a class="existingWikiWord" href="/nlab/show/physical+units">physical units</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/speed+of+light">speed of light</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck%27s+constant">Planck's constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">\hbar</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/gravitational+constant">gravitational constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mi>N</mi></msub><mo>=</mo><msup><mi>κ</mi> <mn>2</mn></msup><mo stretchy="false">/</mo><mn>8</mn><mi>π</mi></mrow><annotation encoding="application/x-tex">G_N = \kappa^2/8\pi</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+scale">Planck scale</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+length">Planck length</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>p</mi></msub><mo>=</mo><msqrt><mrow><mi>ℏ</mi><mi>G</mi><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>3</mn></msup></mrow></msqrt></mrow><annotation encoding="application/x-tex">\ell_p = \sqrt{ \hbar G / c^3 } </annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Planck+mass">Planck mass</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>p</mi></msub><mo>=</mo><msqrt><mrow><mi>ℏ</mi><mi>c</mi><mo stretchy="false">/</mo><mi>G</mi></mrow></msqrt></mrow><annotation encoding="application/x-tex">m_p = \sqrt{\hbar c / G}</annotation></semantics></math></p> </li> </ul> </li> <li> <p>depending on a given <a class="existingWikiWord" href="/nlab/show/mass">mass</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Compton+wavelength">Compton wavelength</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>m</mi></msub><mo>=</mo><mi>ℏ</mi><mo stretchy="false">/</mo><mi>m</mi><mi>c</mi></mrow><annotation encoding="application/x-tex">\lambda_m = \hbar / m c</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Schwarzschild+radius">Schwarzschild radius</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>m</mi><mi>G</mi><mo stretchy="false">/</mo><msup><mi>c</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">2 m G / c^2</annotation></semantics></math></p> </li> <li> <p>depending also on a given <a class="existingWikiWord" href="/nlab/show/charge">charge</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Schwinger+limit">Schwinger limit</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mi>crit</mi></msub><mo>=</mo><msup><mi>m</mi> <mn>2</mn></msup><msup><mi>c</mi> <mn>3</mn></msup><mo stretchy="false">/</mo><mi>e</mi><mi>ℏ</mi></mrow><annotation encoding="application/x-tex">E_{crit} = m^2 c^3 / e \hbar</annotation></semantics></math></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GUT+scale">GUT scale</a></p> </li> <li> <p><a href="string+theory#Scales">string scale</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/string+tension">string tension</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo><mn>1</mn><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><msup><mi>α</mi> <mstyle scriptlevel="0"><mo>′</mo></mstyle></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">T = 1/(2\pi \alpha^\prime)</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+length+scale">string length scale</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub><mo>=</mo><msqrt><mrow><mi>α</mi><mo>′</mo></mrow></msqrt></mrow><annotation encoding="application/x-tex">\ell_s = \sqrt{\alpha'}</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+coupling+constant">string coupling constant</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>s</mi></msub><mo>=</mo><msup><mi>e</mi> <mi>λ</mi></msup></mrow><annotation encoding="application/x-tex">g_s = e^\lambda</annotation></semantics></math></p> </li> </ul> </li> </ul> </div> <h2 id="References">References</h2> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Mass">Mass</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 19, 2023 at 00:46:59. 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