### On lifespan of solutions to the Einstein equations

#### Abstract

We investigate the issue of existence of maximal solutions to the vacuum

Einstein solutions for asymptotically flat spacetime. Solutions are

established globally in time outside a domain of influence of a suitable large

compact set, where singularities can appear. Our approach shows existence of

metric coefficients which obey the following behavior:

$g_{\alpha\beta}=\eta_{\alpha\beta}+O(r^{-\delta})$ for a small fixed

$\delta > 0$ at infinity (where $\eta_{\alpha\beta}$ is the Minkowski metric).

The system is studied in the harmonic (wavelike) gauge.

Einstein solutions for asymptotically flat spacetime. Solutions are

established globally in time outside a domain of influence of a suitable large

compact set, where singularities can appear. Our approach shows existence of

metric coefficients which obey the following behavior:

$g_{\alpha\beta}=\eta_{\alpha\beta}+O(r^{-\delta})$ for a small fixed

$\delta > 0$ at infinity (where $\eta_{\alpha\beta}$ is the Minkowski metric).

The system is studied in the harmonic (wavelike) gauge.

#### Keywords

Einstein equations; hyperbolic system; existence of maximal solutions; initial value problem; domains of dependence

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