Continuation criterion for solutions to the Einstein equations. (English) Zbl 1533.83006
Summary: We prove a continuation condition in the context of 3 + 1 dimensional vacuum Einstein gravity in Constant Mean extrinsic Curvature (CMC) gauge. More precisely, we obtain quantitative criteria under which the physical spacetime can be extended in the future indefinitely as a solution to the Cauchy problem of the Einstein equations given regular initial data. In particular, we show that a gauge-invariant \(H^2\) Sobolev norm of the spacetime Riemann curvature remains bounded in the future time direction provided the so-called deformation tensor of the unit timelike vector field normal to the chosen CMC hypersurfaces verifies a spacetime \(L^\infty\) bound. To this end, we implement a novel technique to obtain this refined estimate by using Friedlander’s parametrix for tensor wave equations on curved spacetime and Moncrief’s subsequent improvement. We conclude by providing a physical explanation of our result as well as its relation to the issues of determinism and weak cosmic censorship.
©2024 American Institute of Physics
©2024 American Institute of Physics
MSC:
| 83C05 | Einstein’s equations (general structure, canonical formalism, Cauchy problems) |
| 35Q75 | PDEs in connection with relativity and gravitational theory |
| 83C75 | Space-time singularities, cosmic censorship, etc. |
| 53Z05 | Applications of differential geometry to physics |
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