TY - JOUR
T1 - Optimal metric regularity in General Relativity follows from the RT-equations by elliptic regularity theory in Lp-spaces
AU - Reintjes, Moritz
AU - Temple, Blake
PY - 2020
Y1 - 2020
N2 - 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 dΓ 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 δΓ, 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) ∈ Wm,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) ∈ Wm,p with Riem(Γ) ∈ Wm,p, m ≥ 1, n < p < ∞, given in x-coordinates, there always exists a coordinate transformation x → y such that Γ(y) ∈ Wm+1, p. This implies all discontinuities in m′th derivatives of δΓ 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 Lp-spaces applied to the RT-equations.
AB - 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 dΓ 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 δΓ, 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) ∈ Wm,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) ∈ Wm,p with Riem(Γ) ∈ Wm,p, m ≥ 1, n < p < ∞, given in x-coordinates, there always exists a coordinate transformation x → y such that Γ(y) ∈ Wm+1, p. This implies all discontinuities in m′th derivatives of δΓ 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 Lp-spaces applied to the RT-equations.
U2 - 10.4310/maa.2020.v27.n3.a1
DO - 10.4310/maa.2020.v27.n3.a1
M3 - RGC 21 - Publication in refereed journal
VL - 27
SP - 199
EP - 242
JO - Methods and Applications of Analysis
JF - Methods and Applications of Analysis
SN - 1073-2772
IS - 3
ER -