TY - JOUR
T1 - Optimal regularity and Uhlenbeck compactness for general relativity and Yang-Mills theory
AU - Reintjes, Moritz
AU - Temple, Blake
PY - 2023/3/29
Y1 - 2023/3/29
N2 - 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 (GR). Compactness is the essential tool of mathematical analysis for establishing the validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with Lp 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.
AB - 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 (GR). Compactness is the essential tool of mathematical analysis for establishing the validity of approximation schemes. Our proofs are based on the theory of the RT-equations for connections with Lp 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.
KW - shock waves
KW - Yang-Mills theories
KW - general relativity
KW - apparent singularities
KW - compensated compactness
KW - SHOCK-WAVE SOLUTIONS
KW - GRAVITATIONAL COLLAPSE
KW - EINSTEIN EQUATIONS
KW - EXISTENCE
KW - METRICS
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UR - https://www.scopus.com/record/pubmetrics.uri?eid=2-s2.0-85149685701&origin=recordpage
U2 - 10.1098/rspa.2022.0444
DO - 10.1098/rspa.2022.0444
M3 - RGC 21 - Publication in refereed journal
VL - 479
JO - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
JF - Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences
SN - 1364-5021
IS - 2271
M1 - 20220444
ER -