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</div> <div class="control"> <button class="button is-small is-link">Go</button> </div> </div> </form> </div> </div> <ol class="breathe-horizontal" start="1"> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.21093">arXiv:2412.21093</a> <span> [<a href="https://arxiv.org/pdf/2412.21093">pdf</a>, <a href="https://arxiv.org/ps/2412.21093">ps</a>, <a href="https://arxiv.org/format/2412.21093">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> A Lorentz invariant formulation of artificial viscosity for the relativistic Euler equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Chaddha%2C+A">Adhiraj Chaddha</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.21093v1-abstract-short" style="display: inline;"> The vanishing (artificial) viscosity method has played a fundamental role in the theory of classical shock waves, both by providing a mollified limit that identifies the correct physical (Lax admissible) shock waves, and as a guiding principle in the design of numerical difference schemes for simulating shock waves. However, for relativistic fluid flow, the underlying dissipation mechanism based o… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.21093v1-abstract-full').style.display = 'inline'; document.getElementById('2412.21093v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.21093v1-abstract-full" style="display: none;"> The vanishing (artificial) viscosity method has played a fundamental role in the theory of classical shock waves, both by providing a mollified limit that identifies the correct physical (Lax admissible) shock waves, and as a guiding principle in the design of numerical difference schemes for simulating shock waves. However, for relativistic fluid flow, the underlying dissipation mechanism based on the Euclidean Laplace operator violates Lorentz invariance (and hence the speed of light bound) -- the fundamental principle of Special Relativity. In this paper we introduce a simple dissipation mechanism for the relativistic Euler equations which is Lorentz invariant and consistent with the laws of Special Relativity. To establish basic consistency of the model for the study of shock waves, we prove existence and decay of Fourier Laplace mode solutions (implying dissipation), and we prove that 1-D shock profiles (viscous travelling wave approximations) exist if and only if the approximated shock waves are Lax admissible. Our analysis of shock profiles reveals an interesting simplification over classical artificial viscosity, leading to a one dimensional fixed point problem, due to the speed of light bound of Relativity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.21093v1-abstract-full').style.display = 'none'; document.getElementById('2412.21093v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 30 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">18 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 76L05 (Primary); 35L67; 35L70; 83A05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2412.08928">arXiv:2412.08928</a> <span> [<a href="https://arxiv.org/pdf/2412.08928">pdf</a>, <a href="https://arxiv.org/ps/2412.08928">ps</a>, <a href="https://arxiv.org/format/2412.08928">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> The essential regularity of singular connections in Geometry </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2412.08928v1-abstract-short" style="display: inline;"> We introduce a natural, mathematically consistent definition of the essential (highest possible) regularity of an affine connection -- a geometric property independent of atlas -- together with a checkable necessary and sufficient condition for determining whether connections are at their essential regularity. This condition is based on the relative regularity between the connection and its Rieman… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08928v1-abstract-full').style.display = 'inline'; document.getElementById('2412.08928v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2412.08928v1-abstract-full" style="display: none;"> We introduce a natural, mathematically consistent definition of the essential (highest possible) regularity of an affine connection -- a geometric property independent of atlas -- together with a checkable necessary and sufficient condition for determining whether connections are at their essential regularity. This condition is based on the relative regularity between the connection and its Riemann curvature. Based on this, we prove that authors' theory of the RT-equations provides a computable procedure for constructing coordinate transformations which simultaneously lift connection and curvature components to essential regularity, and lift the atlas of coordinate charts to the regularity required to preserve the essential regularity of the connection. This provides a definitive theory for determining whether or not singularities in a geometry are essential or removable by coordinate transformation, together with an explicit procedure for lifting removable singularities to their essential regularity, both locally and globally. The theory applies to general affine connections with components in $L^p$, $p>n$, which naturally includes shock wave singularities in General Relativity as well as continuous metrics with infinite gradients, (both obstacles to numerical simulation), but not yet singularities at the lower regularity $p\leq n$ associated with discontinuous metric components, for example, the event horizon of a black hole. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2412.08928v1-abstract-full').style.display = 'none'; document.getElementById('2412.08928v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 11 December, 2024; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2024. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58K30; 83C75 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2304.04444">arXiv:2304.04444</a> <span> [<a href="https://arxiv.org/pdf/2304.04444">pdf</a>, <a href="https://arxiv.org/ps/2304.04444">ps</a>, <a href="https://arxiv.org/format/2304.04444">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> Strong Cosmic Censorship with Bounded Curvature </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2304.04444v1-abstract-short" style="display: inline;"> In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextentability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in $L^p$. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arg… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.04444v1-abstract-full').style.display = 'inline'; document.getElementById('2304.04444v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2304.04444v1-abstract-full" style="display: none;"> In this paper we propose a weaker version of Penrose's much heeded Strong Cosmic Censorship (SCC) conjecture, asserting inextentability of maximal Cauchy developments by manifolds with Lipschitz continuous Lorentzian metrics and Riemann curvature bounded in $L^p$. Lipschitz continuity is the threshold regularity for causal structures, and curvature bounds rule out infinite tidal accelerations, arguing for physical significance of this weaker SCC conjecture. The main result of this paper, under the assumption that no extensions exist with higher connection regularity $W^{1,p}_\text{loc}$, proves in the affirmative this SCC conjecture with bounded curvature for $p$ sufficiently large, ($p>4$ with uniform bounds, $p>2$ without uniform bounds). <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2304.04444v1-abstract-full').style.display = 'none'; document.getElementById('2304.04444v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 10 April, 2023; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> April 2023. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2202.09535">arXiv:2202.09535</a> <span> [<a href="https://arxiv.org/pdf/2202.09535">pdf</a>, <a href="https://arxiv.org/ps/2202.09535">ps</a>, <a href="https://arxiv.org/format/2202.09535">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1098/rspa.2022.0444">10.1098/rspa.2022.0444 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Optimal Regularity and Uhlenbeck Compactness for General Relativity and Yang-Mills Theory </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2202.09535v3-abstract-short" style="display: inline;"> We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of General Relativity. Compactness is the essential tool of mathematical analysis for establishing validity of approximation schemes. Our proofs are based on the the… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.09535v3-abstract-full').style.display = 'inline'; document.getElementById('2202.09535v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2202.09535v3-abstract-full" style="display: none;"> We announce the extension of optimal regularity and Uhlenbeck compactness to the general setting of connections on vector bundles with non-compact gauge groups over non-Riemannian manifolds, including the Lorentzian metric connections of General Relativity. Compactness is the essential tool of mathematical analysis for establishing validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with $L^p$ curvature. Solutions of the RT-equations furnish coordinate and gauge transformations which give a non-optimal connection a gain of one derivative over its Riemann curvature, (i.e., to optimal regularity). The RT-equations are elliptic regardless of metric signature, and regularize singularities in solutions of the hyperbolic Einstein equations. As an application, singularities at GR shock waves are removable, implying geodesic curves, locally inertial coordinates and the Newtonian limit all exist. By the extra derivative we extend Uhlenbeck compactness from Uhlenbeck's setting of vector bundles with compact gauge groups over Riemannian manifolds, to the case of compact and non-compact gauge groups over non-Riemannian manifolds. Our version of Uhlenbeck compactness can also be viewed as a "geometric" improvement of the Div-Curl Lemma, improving weak continuity of wedge products to strong convergence. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2202.09535v3-abstract-full').style.display = 'none'; document.getElementById('2202.09535v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 16 February, 2023; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 19 February, 2022; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> February 2022. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Version 3: Section 7 added. Version 2: Minor rewordings in abstract and introduction</span> </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/2105.10765">arXiv:2105.10765</a> <span> [<a href="https://arxiv.org/pdf/2105.10765">pdf</a>, <a href="https://arxiv.org/ps/2105.10765">ps</a>, <a href="https://arxiv.org/format/2105.10765">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> </div> </div> <p class="title is-5 mathjax"> On the Optimal Regularity Implied by the Assumptions of Geometry II: Connections on Vector Bundles </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="2105.10765v4-abstract-short" style="display: inline;"> We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fib… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.10765v4-abstract-full').style.display = 'inline'; document.getElementById('2105.10765v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="2105.10765v4-abstract-full" style="display: none;"> We extend authors' prior results on optimal regularity and Uhlenbeck compactness for affine connections to general connections on vector bundles. This is accomplished by deriving a vector bundle version of the RT-equations, and establishing a new existence theory for these equations. These new RT-equations, non-invariant elliptic equations, provide the gauge transformations which transform the fibre component of a non-optimal connection to optimal regularity, i.e., the connection is one derivative more regular than its curvature in $L^p$. The existence theory handles curvature regularity all the way down to, but not including, $L^1$. Taken together with the affine case, our results extend optimal regularity of Kazden-DeTurck and the compactness theorem of Uhlenbeck, applicable to Riemannian geometry and compact gauge groups, to general connections on vector bundles over non-Riemannian manifolds, allowing for both compact and non-compact gauge groups. In particular, this extends optimal regularity and Uhlenbeck compactness to Yang-Mills connections on vector bundles over Lorentzian manifolds as base space, the setting of General Relativity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('2105.10765v4-abstract-full').style.display = 'none'; document.getElementById('2105.10765v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 January, 2024; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 22 May, 2021; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2021. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Version 4: Improved local and (new) global results; curvature regularity down to L1. Version 3: More details of proof in Section 5. Inclusion of Theorems 2.6 and 2.7. Version 2: New title; improvements to presentation; slightly weaker regularity assumption for the optimal regularity result, and slightly stronger assumption for Uhlenbeck compactness; otherwise results unchanged</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 58A05 (Primary); 83C99 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Theoretical and Mathematical Physics, Vol. 28, Issue 5 (2024), pp. 1425-1486 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1812.06795">arXiv:1812.06795</a> <span> [<a href="https://arxiv.org/pdf/1812.06795">pdf</a>, <a href="https://arxiv.org/ps/1812.06795">ps</a>, <a href="https://arxiv.org/format/1812.06795">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> </div> <p class="title is-5 mathjax"> How to smooth a crinkled map of spacetime: Uhlenbeck compactness for $L^\infty$ connections and optimal regularity for general relativistic shock waves by the Reintjes-Temple-equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1812.06795v5-abstract-short" style="display: inline;"> We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $螕$ to optimal regularity, one derivative smoother than the Riemann curvature tensor ${\rm Riem}(螕)$. As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we est… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.06795v5-abstract-full').style.display = 'inline'; document.getElementById('1812.06795v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1812.06795v5-abstract-full" style="display: none;"> We present authors' new theory of the RT-equations, nonlinear elliptic partial differential equations which determine the coordinate transformations which smooth connections $螕$ to optimal regularity, one derivative smoother than the Riemann curvature tensor ${\rm Riem}(螕)$. As one application we extend Uhlenbeck compactness from Riemannian to Lorentzian geometry; and as another application we establish that regularity singularities at GR shock waves can always be removed by coordinate transformation. This is based on establishing a general multi-dimensional existence theory for the RT-equations, by application of elliptic regularity theory in $L^p$ spaces. The theory and results announced in this paper apply to arbitrary $L^\infty$ connections on the tangent bundle $T\mathcal{M}$ of arbitrary manifolds $\mathcal{M}$, including Lorentzian manifolds of General Relativity. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1812.06795v5-abstract-full').style.display = 'none'; document.getElementById('1812.06795v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 14 December, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This paper summarises the theory developed in arXiv:1805.01004, arXiv:1808.06455 and arXiv:1912.12997. V5: Improved wording and minor corrections in Section 7. V4: Cor 3.1 added. V3: Thms 2.6 - 2.8, addressing Uhlenbeck compactness and $L^\infty$ connections, added. Improved discussion section and introduction. V2: Improved presentation; Curvature bound added to Thm 2.4 and 2.5</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C99 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. R. Soc. A 476: 20200177, 2020 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1808.06455">arXiv:1808.06455</a> <span> [<a href="https://arxiv.org/pdf/1808.06455">pdf</a>, <a href="https://arxiv.org/ps/1808.06455">ps</a>, <a href="https://arxiv.org/format/1808.06455">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> </div> <p class="title is-5 mathjax"> Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in $L^p$-spaces </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1808.06455v4-abstract-short" style="display: inline;"> Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $螕$ and curvature $Riem(螕)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the conditio… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.06455v4-abstract-full').style.display = 'inline'; document.getElementById('1808.06455v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1808.06455v4-abstract-full" style="display: none;"> Shock wave solutions of the Einstein equations have been constructed in coordinate systems in which the gravitational metric is only Lipschitz continuous, but the connection $螕$ and curvature $Riem(螕)$ are both in $L^{\infty}$. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. Here we address the mathematical problem as to whether the condition that $Riem(螕)$ has the same regularity as $螕$, is sufficient for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of $未螕$, thereby raising the regularity of the connection and the metric by one order--a subtle problem. We have now discovered, in a framework much more general than GR shock waves, that the regularization of non-optimal connections is determined by a nonlinear system of elliptic equations with matrix valued differential forms as unknowns, the Regularity Transformation equations, or RT-equations. In this paper we establish the first existence theory for the nonlinear RT-equations in the general case when $螕, {\rm Riem}(螕)\in W^{m,p}$, $m\geq1$, $n<p< \infty$, where $螕$ is any affine connection on an $n$-dimensional manifold. From this we conclude that for any such connection $螕(x) \in W^{m,p}$ with ${\rm Riem}(螕) \in W^{m,p}$, $m\geq1$, $n<p< \infty$, given in $x$-coordinates, there always exists a coordinate transformation $x\to y$ such that $螕(y) \in W^{m+1,p}$. That is, $螕$ exhibits optimal regularity in $y$-coordinates. The problem of optimal regularity for the hyperbolic Einstein equations is thus resolved by elliptic regularity theory in $L^p$-spaces applied to the RT-equations. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1808.06455v4-abstract-full').style.display = 'none'; document.getElementById('1808.06455v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 31 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 20 August, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">46 pages. Version 4: Simplified use of elliptic PDE theory in proofs; minor improvements of wording; results unchanged. Version 3: Improved wording, results unchanged. Version 2: Minor changes to title and wording in introduction; results unchanged</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1805.01004">arXiv:1805.01004</a> <span> [<a href="https://arxiv.org/pdf/1805.01004">pdf</a>, <a href="https://arxiv.org/ps/1805.01004">ps</a>, <a href="https://arxiv.org/format/1805.01004">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> The Regularity Transformation Equations: An elliptic mechanism for smoothing gravitational metrics in General Relativity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1805.01004v5-abstract-short" style="display: inline;"> A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature t… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.01004v5-abstract-full').style.display = 'inline'; document.getElementById('1805.01004v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1805.01004v5-abstract-full" style="display: none;"> A central question in General Relativity (GR) is how to determine whether singularities are geometrical properties of spacetime, or simply anomalies of a coordinate system used to parameterize the spacetime. In particular, it is an open problem whether there always exist coordinate transformations which smooth a gravitational metric to optimal regularity, two full derivatives above the curvature tensor, or whether regularity singularities exist. We resolve this open problem above a threshold level of smoothness by proving in this paper that the existence of such coordinate transformations is equivalent to solving a system of nonlinear elliptic equations in the unknown Jacobian and transformed connection, both viewed as matrix valued differential forms. We name these the Regularity Transformation equations, or RT-equations. In a companion paper we prove existence of solutions to the RT-equations for connections $螕\in W^{m,p},$ curvature ${\rm Riem}(螕) \in W^{m,p}$, assuming $m\geq1$, $p>n$. Taken together, these results imply that there always exist coordinate transformations which smooth arbitrary connections to optimal regularity, (one derivative more regular than the curvature), and there are no regularity singularities, above the threshold $m\geq1$, $p>n$. Authors are currently working on extending these methods to the case of GR shock waves, when gravitational metrics are only Lipschitz continuous, ($m=0$, $p=\infty$), and optimal regularity is required to recover basic properties of spacetime. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1805.01004v5-abstract-full').style.display = 'none'; document.getElementById('1805.01004v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 November, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 2 May, 2018; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2018. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">Version 5: Subsections are promoted to sections. Version 3 & 4: Improved wording, results unchanged. Version 2: We improved our main result, Theorem 1.1, by replacing the problematic boundary data (1.5) with linear data (eqn (1.5) in V.2), and we replaced eqns (1.1) - (1.2) of V.1 with the Poisson equation (3.53) of V.1. We modified the rest of the paper to adapt to our improved main result</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1708.09643">arXiv:1708.09643</a> <span> [<a href="https://arxiv.org/pdf/1708.09643">pdf</a>, <a href="https://arxiv.org/ps/1708.09643">ps</a>, <a href="https://arxiv.org/format/1708.09643">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4310/ATMP.2018.v22.n8.a3">10.4310/ATMP.2018.v22.n8.a3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Fermionic Signature Operator and Space-Time Symmetries </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Finster%2C+F">Felix Finster</a>, <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1708.09643v2-abstract-short" style="display: inline;"> We show that and specify how space-time symmetries give rise to corresponding symmetries of the fermionic signature operator and generalized fermionic projector states. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1708.09643v2-abstract-full" style="display: none;"> We show that and specify how space-time symmetries give rise to corresponding symmetries of the fermionic signature operator and generalized fermionic projector states. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1708.09643v2-abstract-full').style.display = 'none'; document.getElementById('1708.09643v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 June, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 31 August, 2017; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> August 2017. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages, LaTeX, 1 figure, minor improvements (published version)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv.Theor.Math.Phys. 22 (2018) 1907-1937 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1610.02390">arXiv:1610.02390</a> <span> [<a href="https://arxiv.org/pdf/1610.02390">pdf</a>, <a href="https://arxiv.org/ps/1610.02390">ps</a>, <a href="https://arxiv.org/format/1610.02390">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Differential Geometry">math.DG</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1007/s00205-019-01456-8">10.1007/s00205-019-01456-8 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Shock Wave Interactions and the Riemann-flat Condition: The Geometry behind Metric Smoothing and the Existence of Locally Inertial Frames in General Relativity </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1610.02390v5-abstract-short" style="display: inline;"> We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regu… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1610.02390v5-abstract-full').style.display = 'inline'; document.getElementById('1610.02390v5-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1610.02390v5-abstract-full" style="display: none;"> We prove that the essential smoothness of the gravitational metric at shock waves in GR, a PDE regularity issue for weak solutions of the Einstein equations, is determined by a geometrical condition which we introduce and name the {\it Riemann-flat condition}. The Riemann-flat condition determines whether or not the essential smoothness of the gravitational metric is two full derivatives more regular than the Riemann curvature tensor. This provides a geometric framework for the open problem as to whether {\it regularity singularities} (points where the curvature is in $L^\infty$ but the essential smoothness of the gravitational metric is only Lipschitz continuous) can be created by shock wave interaction in GR, or whether metrics Lipschitz at shocks can always be smoothed one level to $C^{1,1}$ by coordinate transformation. As a corollary of the ideas we give a proof that locally inertial frames always exist in a natural sense for shock wave metrics in spherically symmetric spacetimes, independent of whether the metric itself can be smoothed to $C^{1,1}$ locally. This latter result yields an explicit procedure (analogous to Riemann Normal Coordinates in smooth spacetimes) for constructing locally inertial coordinates for Lipschitz metrics, and is a new regularity result for GR solutions constructed by the Glimm scheme. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1610.02390v5-abstract-full').style.display = 'none'; document.getElementById('1610.02390v5-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 28 October, 2019; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 7 October, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> October 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">V5: Improved presentation, in particular, to Section 6. Results unchanged. V4: We shortened the presentation, added Def 3.1 and removed last section of previous version. V3: We extended results from connections of bounded variation to connections bounded in $L^\infty$, otherwise main results remain identical. V2: Result of Theorem 1.5 was extended from 2-D to spherically symmetric space-times</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Arch. Rat. Mech. Anal. 235 (2020), 1873-1904 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1601.08106">arXiv:1601.08106</a> <span> [<a href="https://arxiv.org/pdf/1601.08106">pdf</a>, <a href="https://arxiv.org/ps/1601.08106">ps</a>, <a href="https://arxiv.org/format/1601.08106">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> A note on incompressibility of relativistic fluids and the instantaneity of their pressures </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1601.08106v2-abstract-short" style="display: inline;"> We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfi… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.08106v2-abstract-full').style.display = 'inline'; document.getElementById('1601.08106v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1601.08106v2-abstract-full" style="display: none;"> We introduce a natural notion of incompressibility for fluids governed by the relativistic Euler equations on a fixed background spacetime, and show that the resulting equations reduce to the incompressible Euler equations in the classical limit as $c\rightarrow \infty$. As our main result, we prove that the fluid pressure of solutions of these incompressible "relativistic" Euler equations satisfies an elliptic equation on each of the hypersurfaces orthogonal to the fluid four-velocity, which indicates infinite speed of propagation. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1601.08106v2-abstract-full').style.display = 'none'; document.getElementById('1601.08106v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 14 June, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 29 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2016. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">7 pages. Version 2: Improved wording and presentation</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C99 (Primary); 76B99 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1511.08183">arXiv:1511.08183</a> <span> [<a href="https://arxiv.org/pdf/1511.08183">pdf</a>, <a href="https://arxiv.org/ps/1511.08183">ps</a>, <a href="https://arxiv.org/format/1511.08183">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> A Proposal of a Damping Term for the Relativistic Euler Equations </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1511.08183v1-abstract-short" style="display: inline;"> We introduce a damping term for the special relativistic Euler equations in $3$-D and show that the equations reduce to the non-relativistic damped Euler equations in the Newtonian limit. We then write the equations as a symmetric hyperbolic system for which local-in-time existence of smooth solutions can be shown. </span> <span class="abstract-full has-text-grey-dark mathjax" id="1511.08183v1-abstract-full" style="display: none;"> We introduce a damping term for the special relativistic Euler equations in $3$-D and show that the equations reduce to the non-relativistic damped Euler equations in the Newtonian limit. We then write the equations as a symmetric hyperbolic system for which local-in-time existence of smooth solutions can be shown. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1511.08183v1-abstract-full').style.display = 'none'; document.getElementById('1511.08183v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 November, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> November 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">11 pages</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83A05 (primary); 76N10 (secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.04081">arXiv:1506.04081</a> <span> [<a href="https://arxiv.org/pdf/1506.04081">pdf</a>, <a href="https://arxiv.org/ps/1506.04081">ps</a>, <a href="https://arxiv.org/format/1506.04081">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1098/rspa.2014.0834">10.1098/rspa.2014.0834 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> No Regularity Singularities Exist at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.04081v1-abstract-short" style="display: inline;"> We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spa… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.04081v1-abstract-full').style.display = 'inline'; document.getElementById('1506.04081v1-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.04081v1-abstract-full" style="display: none;"> We give a constructive proof that coordinate transformations exist which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of points of shock wave collision in General Relativity. The proof applies to collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. Our result here implies that spacetime is locally inertial and corrects an error in our earlier RSPA-publication, which led us to the false conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Thus, our result implies that regularity singularities, (a type of mild singularity introduced in our RSPA-paper), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes. Our result generalizes Israel's celebrated 1966 paper to the case of such shock wave interactions but our proof strategy differs fundamentally from that used by Israel and is an extension of the strategy outlined in our original RSPA-publication. Whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations remains open. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.04081v1-abstract-full').style.display = 'none'; document.getElementById('1506.04081v1-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">21 pages. In this paper we summarise the results and proofs in arXiv:1409.5060. The result here corrects the wrong conclusion in arXiv:1105.0798 and arXiv:1112.1803</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. R. Soc. A 471 : 20140834, 2015 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1506.04074">arXiv:1506.04074</a> <span> [<a href="https://arxiv.org/pdf/1506.04074">pdf</a>, <a href="https://arxiv.org/ps/1506.04074">ps</a>, <a href="https://arxiv.org/format/1506.04074">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> "Regularity Singularities" and the Scattering of Gravity Waves in Approximate Locally Inertial Frames </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1506.04074v2-abstract-short" style="display: inline;"> It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but n… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.04074v2-abstract-full').style.display = 'inline'; document.getElementById('1506.04074v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1506.04074v2-abstract-full" style="display: none;"> It is an open question whether solutions of the Einstein-Euler equations are smooth enough to admit locally inertial coordinates at points of shock wave interaction, or whether "regularity singularities" can exist at such points. The term {\it regularity singularity} was proposed by the authors as a point in spacetime where the gravitational metric tensor is Lipschitz continuous ($C^{0,1}$), but no smoother, in any coordinate system of the $C^{1,1}$ atlas. An existence theory for shock wave solutions in $C^{0,1}$ admitting arbitrary interactions has been proven for the Einstein-Euler equations in spherically symmetric spacetimes, but $C^{1,1}$ is the requisite smoothness required for space-time to be locally flat. Thus the open problem of regularity singularities is the problem as to whether locally inertial coordinate systems exist at shock waves within the larger $C^{1,1}$ atlas. To clarify this open problem, we identify new "Coriolis type" effects in the geometry of $C^{0,1}$ shock wave metrics and prove they are essential in the sense that they can never be made to vanish within the atlas of {\it smooth} coordinate transformations, the atlas usually assumed in classical differential geometry. Thus the problem of existence of regularity singularities is equivalent to the question as to whether or not these Coriolis type effects are essentially non-removable and `real', or merely coordinate effects that can be removed, (in analogy to classical Coriolis forces), by going to the less regular atlas of $C^{1,1}$ transformations. If essentially non-removable, it would argue strongly for a `real' new physical effect for General Relativity, providing a physical context to the open problem of regularity singularities. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1506.04074v2-abstract-full').style.display = 'none'; document.getElementById('1506.04074v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 22 June, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 12 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> June 2015. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages. Version 2: Corrections of some typographical errors and improvements of wording. Results are unchanged</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Methods and Applications of Analysis, Vol. 23, No. 3 (2016), pp. 233-258 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1409.5060">arXiv:1409.5060</a> <span> [<a href="https://arxiv.org/pdf/1409.5060">pdf</a>, <a href="https://arxiv.org/ps/1409.5060">ps</a>, <a href="https://arxiv.org/format/1409.5060">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> Spacetime is Locally Inertial at Points of General Relativistic Shock Wave Interaction between Shocks from Different Characteristic Families </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1409.5060v3-abstract-short" style="display: inline;"> We prove that spacetime is locally inertial at points of shock wave collision in General Relativity. The result applies for collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. We give a constructive proof that there exist coordinate transformations which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.5060v3-abstract-full').style.display = 'inline'; document.getElementById('1409.5060v3-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1409.5060v3-abstract-full" style="display: none;"> We prove that spacetime is locally inertial at points of shock wave collision in General Relativity. The result applies for collisions between shock waves coming from different characteristic families, in spherically symmetric spacetimes. We give a constructive proof that there exist coordinate transformations which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighborhood of such points of shock wave interaction, and a $C^{1,1}$ metric regularity suffices for locally inertial frames to exist. This result corrects an error in our earlier RSPA-publication, which led us to the wrong conclusion that such coordinate transformations, which smooth the metric to $C^{1,1}$, cannot exist. Our result here proves that regularity singularities, (a type of mild singularity introduced in our RSPA-publication), do not exist at points of interacting shock waves from different families in spherically symmetric spacetimes, and this generalizes Israel's famous 1966 result to the case of such shock wave interactions. The strategy of proof here is an extension of the strategy outlined in our RSPA-paper, but differs fundamentally from the method used by Israel. The question whether regularity singularities exist in more complicated shock wave solutions of the Einstein Euler equations still remains open. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1409.5060v3-abstract-full').style.display = 'none'; document.getElementById('1409.5060v3-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 7 February, 2017; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 17 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> September 2014. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">79 pages. The result here corrects the wrong conclusion in arXiv:1105.0798 and arXiv:1112.1803. This paper contains the proofs of the results announced in arXiv:1506.04081. V2: Minor improvements of wording; correction of a minor error in Lemma 8.3. Main results are unchanged. V3: Improvements of wording</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75 (Primary); 76L05 (Secondary) </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1312.7209">arXiv:1312.7209</a> <span> [<a href="https://arxiv.org/pdf/1312.7209">pdf</a>, <a href="https://arxiv.org/ps/1312.7209">ps</a>, <a href="https://arxiv.org/format/1312.7209">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4310/ATMP.2016.v20.n5.a2">10.4310/ATMP.2016.v20.n5.a2 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds II -- Space-Times of Infinite Lifetime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Finster%2C+F">Felix Finster</a>, <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1312.7209v4-abstract-short" style="display: inline;"> The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter gene… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7209v4-abstract-full').style.display = 'inline'; document.getElementById('1312.7209v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1312.7209v4-abstract-full" style="display: none;"> The previous functional analytic construction of the fermionic projector on globally hyperbolic Lorentzian manifolds is extended to space-times of infinite lifetime. The construction is based on an analysis of families of solutions of the Dirac equation with a varying mass parameter. It makes use of the so-called mass oscillation property which implies that integrating over the mass parameter generates decay of the Dirac wave functions at infinity. We obtain a canonical decomposition of the solution space of the massive Dirac equation into two subspaces, independent of observers or the choice of coordinates. The constructions are illustrated in the examples of ultrastatic space-times and de Sitter space-time. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1312.7209v4-abstract-full').style.display = 'none'; document.getElementById('1312.7209v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 12 October, 2020; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 27 December, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, LaTeX, statement of Proposition 3.4 modified, footnote on page 8 added</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv.Theor.Math.Phys. 20 (2016) 1007-1048 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1301.5420">arXiv:1301.5420</a> <span> [<a href="https://arxiv.org/pdf/1301.5420">pdf</a>, <a href="https://arxiv.org/ps/1301.5420">ps</a>, <a href="https://arxiv.org/format/1301.5420">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Functional Analysis">math.FA</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.4310/ATMP.2015.v19.n4.a3">10.4310/ATMP.2015.v19.n4.a3 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> A Non-Perturbative Construction of the Fermionic Projector on Globally Hyperbolic Manifolds I - Space-Times of Finite Lifetime </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Finster%2C+F">Felix Finster</a>, <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1301.5420v4-abstract-short" style="display: inline;"> We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dira… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.5420v4-abstract-full').style.display = 'inline'; document.getElementById('1301.5420v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1301.5420v4-abstract-full" style="display: none;"> We give a functional analytic construction of the fermionic projector on a globally hyperbolic Lorentzian manifold of finite lifetime. The integral kernel of the fermionic projector is represented by a two-point distribution on the manifold. By introducing an ultraviolet regularization, we get to the framework of causal fermion systems. The connection to the "negative-energy solutions" of the Dirac equation and to the WKB approximation is explained and quantified by a detailed analysis of closed Friedmann-Robertson-Walker universes. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1301.5420v4-abstract-full').style.display = 'none'; document.getElementById('1301.5420v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 25 January, 2016; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 23 January, 2013; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2013. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">29 pages, LaTeX, 1 figure, minor improvements (published version)</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Adv.Theor.Math.Phys. 19 (2015) 761-803 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1112.1803">arXiv:1112.1803</a> <span>  </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> </div> <p class="title is-5 mathjax"> The 'Regularity Singularity' at Points of General Relativistic Shock Wave Interaction </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1112.1803v6-abstract-short" style="display: inline;"> A proof that a new kind of non-removable {\it "regularity singularity"} forms when two shock waves collide within the theory of General Relativity, was first announced in ProcRoySoc A \cite{ReintjesTemple}. In the present paper we give complete proofs of the claims in \cite{ReintjesTemple} and extend the results on the regularity of the Einstein curvature tensor to the full Riemann curvature tenso… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.1803v6-abstract-full').style.display = 'inline'; document.getElementById('1112.1803v6-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1112.1803v6-abstract-full" style="display: none;"> A proof that a new kind of non-removable {\it "regularity singularity"} forms when two shock waves collide within the theory of General Relativity, was first announced in ProcRoySoc A \cite{ReintjesTemple}. In the present paper we give complete proofs of the claims in \cite{ReintjesTemple} and extend the results on the regularity of the Einstein curvature tensor to the full Riemann curvature tensor. The main result is that, in a neighborhood of a point where two shock waves collide in a spherically symmetric spacetime, the gravitational metric tensor cannot be lifted from C0,1 to C1 within the class of C1,1 coordinate transformations. This contrasts Israel's celebrated theorem \cite{Israel}, which states that around each point on a {\it single} shock surface there exist a coordinate system in which the metric is C1,1 regular. Moreover, at points of shock wave interaction, delta function sources exist in the second derivatives of the gravitational metric tensor in all coordinate systems of the C1,1-atlas, but due to cancellation, the Einstein and Riemann curvature tensor remain sup-norm bounded. We conclude that points of shock wave interaction are a new kind of spacetime singularity, (which we name "regularity singularity"), singularities that can form from the evolution of smooth initial data for perfect fluids and that lie in physical spacetime, but at such points {\it locally inertial} coordinates fail to exist. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1112.1803v6-abstract-full').style.display = 'none'; document.getElementById('1112.1803v6-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 17 September, 2014; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 8 December, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> December 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This article has been withdrawn since the main result is wrong due to an computational error. See arXiv:1409.5060 for a correction of this error and a proof of the opposite statement. This article has been withdrawn, since the main result and conclusion of this paper are wrong due to an arithmetic error. This error is corrected in arXiv:1409.5060, where we in fact prove an result opposite to the incorrect one in this paper. Namely, we give a constructive proof that there exist coordinate transformations which raise the regularity of the gravitational metric tensor from $C^{0,1}$ to $C^{1,1}$ in a neighbourhood of points of shock wave interaction, between shocks from different families. The result in arXiv:1409.5060 thus shows that regularity singularities do not exist at such points of shock interaction, contradicting the wrong result here</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Advances in Theoretical and Mathematical Physics, Volume 17, Number 4, 2014 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/1105.0798">arXiv:1105.0798</a> <span>  </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="Analysis of PDEs">math.AP</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1098/rspa.2011.0360">10.1098/rspa.2011.0360 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> Points of General Relativisitic Shock Wave Interaction are "Regularity Singularities" where Spacetime is Not Locally Flat </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a>, <a href="/search/gr-qc?searchtype=author&query=Temple%2C+B">Blake Temple</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="1105.0798v2-abstract-short" style="display: inline;"> We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ within the class of $C^{1,1}$ coordinate transformations in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.0798v2-abstract-full').style.display = 'inline'; document.getElementById('1105.0798v2-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="1105.0798v2-abstract-full" style="display: none;"> We show that the regularity of the gravitational metric tensor in spherically symmetric spacetimes cannot be lifted from $C^{0,1}$ to $C^{1,1}$ within the class of $C^{1,1}$ coordinate transformations in a neighborhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel's Theorem which states that such coordinate transformations always exist in a neighborhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of singularity for perfect fluids evolving in spacetime, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the spacetime is not locally Minkowskian under any coordinate transformation. In particular, at such singularities, delta function sources in the second derivatives of the gravitational metric tensor exist in all coordinate systems of the $C^{1,1}$ atlas, but due to cancelation, the curvature tensor remains uniformly bounded. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('1105.0798v2-abstract-full').style.display = 'none'; document.getElementById('1105.0798v2-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 15 June, 2015; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 4 May, 2011; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> May 2011. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">This article has been withdrawn since the main result is wrong due to an computational error. See arXiv:1506.04081 and arXiv:1409.5060 for a correction of this error and a proof of the opposite statement</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">MSC Class:</span> 83C75; 83C40; 35L65 </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Proc. R. Soc. A, 2012, vol. 468 no. 2146, 2962-2980 </p> </li> <li class="arxiv-result"> <div class="is-marginless"> <p class="list-title is-inline-block"><a href="https://arxiv.org/abs/0901.0602">arXiv:0901.0602</a> <span> [<a href="https://arxiv.org/pdf/0901.0602">pdf</a>, <a href="https://arxiv.org/ps/0901.0602">ps</a>, <a href="https://arxiv.org/format/0901.0602">other</a>] </span> </p> <div class="tags is-inline-block"> <span class="tag is-small is-link tooltip is-tooltip-top" data-tooltip="Mathematical Physics">math-ph</span> <span class="tag is-small is-grey tooltip is-tooltip-top" data-tooltip="General Relativity and Quantum Cosmology">gr-qc</span> </div> <div class="is-inline-block" style="margin-left: 0.5rem"> <div class="tags has-addons"> <span class="tag is-dark is-size-7">doi</span> <span class="tag is-light is-size-7"><a class="" href="https://doi.org/10.1088/0264-9381/26/10/105021">10.1088/0264-9381/26/10/105021 <i class="fa fa-external-link" aria-hidden="true"></i></a></span> </div> </div> </div> <p class="title is-5 mathjax"> The Dirac Equation and the Normalization of its Solutions in a Closed Friedmann-Robertson-Walker Universe </p> <p class="authors"> <span class="search-hit">Authors:</span> <a href="/search/gr-qc?searchtype=author&query=Finster%2C+F">Felix Finster</a>, <a href="/search/gr-qc?searchtype=author&query=Reintjes%2C+M">Moritz Reintjes</a> </p> <p class="abstract mathjax"> <span class="has-text-black-bis has-text-weight-semibold">Abstract</span>: <span class="abstract-short has-text-grey-dark mathjax" id="0901.0602v4-abstract-short" style="display: inline;"> We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Frie… <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0901.0602v4-abstract-full').style.display = 'inline'; document.getElementById('0901.0602v4-abstract-short').style.display = 'none';">▽ More</a> </span> <span class="abstract-full has-text-grey-dark mathjax" id="0901.0602v4-abstract-full" style="display: none;"> We set up the Dirac equation in a Friedmann-Robertson-Walker geometry and separate the spatial and time variables. In the case of a closed universe, the spatial dependence is solved explicitly, giving rise to a discrete set of solutions. We compute the probability integral and analyze a space-time normalization integral. This analysis allows us to introduce the fermionic projector in a closed Friedmann-Robertson-Walker geometry and to specify its global normalization as well as its local form. <a class="is-size-7" style="white-space: nowrap;" onclick="document.getElementById('0901.0602v4-abstract-full').style.display = 'none'; document.getElementById('0901.0602v4-abstract-short').style.display = 'inline';">△ Less</a> </span> </p> <p class="is-size-7"><span class="has-text-black-bis has-text-weight-semibold">Submitted</span> 6 February, 2013; <span class="has-text-black-bis has-text-weight-semibold">v1</span> submitted 6 January, 2009; <span class="has-text-black-bis has-text-weight-semibold">originally announced</span> January 2009. </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Comments:</span> <span class="has-text-grey-dark mathjax">22 pages, LaTeX, sign error in equation (3.7) corrected</span> </p> <p class="comments is-size-7"> <span class="has-text-black-bis has-text-weight-semibold">Journal ref:</span> Class.Quant.Grav.26:105021,2009 </p> </li> </ol> <div class="is-hidden-tablet">  <span class="help" style="display: inline-block;"><a href="https://github.com/arXiv/arxiv-search/releases">Search v0.5.6 released 2020-02-24</a>  </span> </div> </div> </main> <footer> <div class="columns is-desktop" role="navigation" aria-label="Secondary">  <div class="column"> <div class="columns"> <div class="column"> <ul class="nav-spaced"> <li><a href="https://info.arxiv.org/about">About</a></li> <li><a href="https://info.arxiv.org/help">Help</a></li> </ul> </div> <div class="column"> <ul class="nav-spaced"> <li> <svg xmlns="http://www.w3.org/2000/svg" viewBox="0 0 512 512" class="icon filter-black" role="presentation"><title>contact arXiv</title><desc>Click here to contact arXiv</desc><path d="M502.3 190.8c3.9-3.1 9.7-.2 9.7 4.7V400c0 26.5-21.5 48-48 48H48c-26.5 0-48-21.5-48-48V195.6c0-5 5.7-7.8 9.7-4.7 22.4 17.4 52.1 39.5 154.1 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