Abstract
Our proof relies on a new formulation of Einstein’s equations: we use a constant-mean-curvature foliation, and the unknowns are the scalar field, the lapse, the components of the spatial connection and second fundamental form relative to a Fermi–Walker transported spatial orthonormal frame, and the components of the orthonormal frame vectors with respect to a transported spatial coordinate system. In this formulation, the PDE evolution system for the structure coefficients of the orthonormal frame approximately diagonalizes in a way that sharply reveals the significance of the Kasner exponent sub-criticality condition for the dynamic stability of the flow: the condition leads to the time-integrability of many terms in the equations, at least at the low derivative levels. At the high derivative levels, the solutions that we study can be much more singular with respect to t, and to handle this difficulty, we use t-weighted high order energies, and we control non-linear error terms by exploiting monotonicity induced by the t-weights and interpolating between the singularity-strength of the solution’s low order and high order derivatives. Finally, we note that our formulation of Einstein’s equations highlights the quantities that might generate instabilities outside of the sub-critical regime.
| Original language | American English |
|---|---|
| Pages (from-to) | 827-916 |
| Number of pages | 90 |
| Journal | Journal of the American Mathematical Society |
| Volume | 36 |
| Issue number | 3 |
| DOIs | |
| State | Published - 2023 |
ASJC Scopus subject areas
- General Mathematics
- Applied Mathematics
Keywords
- Big Bang
- Fermi–Walker transport
- Hawking’s theorem
- Kasner solutions
- constant mean curvature
- curvature singularity
- geodesically incomplete
- maximal globally hyperbolic development
- singularity theorem
- stable blowup
- transported spatial coordinates