On spherically symmetric solutions of the Einstein-Euler equations. (English) Zbl 1351.35220

Makino, Tetu

Kyoto J. Math. 56, No. 2, 243-282 (2016).

The author of the present paper discusses the existence and specific properties of spherically symmetric solutions of the Einstein-Euler equations. The Einstein equations have the form: \[ R_{\mu\nu }-(1/2)g_{\mu\nu }R=8\pi c^{-4}GT_{\mu\nu }, \] where \(R_{\mu\nu }\) is the Ricci tensor of the matter, \(R\) is the scalar curvature, \(g^{\alpha\beta } R_{\alpha\beta}\) is associated with the metric \(ds^2 = g_{\mu\nu }dx^{\mu }dx^{\nu }\), \(T^{\mu\nu }\) is the energy-momentum tensor of the matter, \(G\) is the constant of gravitation, and \(c\) is the speed of light. These Einstein equations imply the Euler equations \(\nabla_{\nu }T^{\mu\nu }=0\). This is a mathematical model of gaseous star as an application of the General Relativity theory. It is considered also a barotropic equation of state. Further, spherically symmetric solutions to the considered equations are found. Equilibria of the spherically symmetric Einstein-Euler equations are given by the Tolman-Oppenheimer-Volkoff equations \[ dm/dr=4\pi r^2\rho , \;\;dP/dr = -(\rho +P/c^2)\frac{G(m+4\pi r^3P/c^2)} {r^2(1-2Gm/c^2r)}, \] where the variables depend on \(\rho \) and \(r\): \(F = F(\rho (r))\), \(H = H(r)\), \(\rho = \rho (r)\), \(P = P (\rho (r))\), \(V \equiv 0\), \( R \equiv r\). In the paper the exact expressions for the above stated variables are given in explicit form. Thus the time-periodic solutions of the considered equations are obtained. The solutions are defined around the equilibrium of the linearized equations.
The author discusses the existence of true solutions near the time-periodic approximations. The solutions defined on the physical boundary, i.e. satisfying the boundary conditions (at the free boundary with the vacuum) would be of interest here. To find the aforementioned solutions it is used an approach by the Nash-Moser theorem.
Finally, the author asserts that the present work can be considered “as a touchstone in order to estimate the universality of the method which was originally developed for the nonrelativistic Euler-Poisson equations”.

Reviewer: Dimitar A. Kolev (Sofia)

MSC:

35Q76	Einstein equations
35L05	Wave equation
35L52	Initial value problems for second-order hyperbolic systems
35L57	Initial-boundary value problems for higher-order hyperbolic systems
35L70	Second-order nonlinear hyperbolic equations
76N15	Gas dynamics (general theory)
83C05	Einstein’s equations (general structure, canonical formalism, Cauchy problems)
85A30	Hydrodynamic and hydromagnetic problems in astronomy and astrophysics

Keywords:

Einstein equations; spherically symmetric solutions; vacuum boundary; Nash-Moser theorem; Tolman-Oppenheimer-Volkoff equations

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