# Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in *L*^{p}-spaces

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Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review

## Author(s)

## Detail(s)

Original language | English |
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Pages (from-to) | 199–242 |

Number of pages | 43 |

Journal / Publication | Methods and Applications of Analysis |

Volume | 27 |

Issue number | 3 |

Publication status | Published - 2020 |

Externally published | Yes |

## Link(s)

DOI | ## DOI |
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Permanent Link | https://scholars.cityu.edu.hk/en/publications/publication(18cdea37-170c-4d1c-bd53-bf22ae93550c).html |

## Abstract

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

**Γ**and curvature*Riem*(**Γ**) are both in*L*^{∞}, the curvature being one derivative smoother than the curvature of a general Lipschitz metric. At this low level of regularity, the physical meaning of such gravitational metrics remains problematic. In fact, the Einstein equations naturally admit coordinates in which**Γ**has the same regularity as*Riem*(**Γ**) because the curvature transforms as a tensor, but the connection does not. Here we address the mathematical problem as to whether the condition that*Riem*(**Γ**) has the same regularity as**Γ**, or equivalently the exterior derivatives*d***Γ**have the same regularity as**Γ**, is sufficient to allow for the existence of a coordinate transformation which perfectly cancels out the jumps in the leading order derivatives of*δ***Γ**, 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**Γ**, Riem(**Γ**) ∈*,***W**^{m,p}*m*≥ 1,*n*<*p*< ∞, where**Γ**is any affine connection on an*n*-dimensional manifold. From this we conclude that for any such connection**Γ**(*x*) ∈*with***W**^{m,p}**Riem**(**Γ**) ∈*,***W**^{m,p}*m*≥ 1,*n*<*p*< ∞, given in*x*-coordinates, there always exists a coordinate transformation*x*→*y*such that**Γ**(*y*) ∈**W**^{m+1},*p*. This implies*all*discontinuities in*m*′th derivatives of*δ***Γ***cancel out*, the transformation*x*→*y*raises the connection regularity by one order, and**Γ**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*-spaces applied to the RT-equations.^{p}## Citation Format(s)

**Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in**/ Reintjes, Moritz; Temple, Blake.

*L*-spaces.^{p}In: Methods and Applications of Analysis, Vol. 27, No. 3, 2020, p. 199–242.

Research output: Journal Publications and Reviews (RGC: 21, 22, 62) › 21_Publication in refereed journal › peer-review