On static shells and the Buchdahl inequality for the spherically symmetric Einstein-Vlasov system

TL;DR

This paper constructs static shell solutions for the spherically symmetric Einstein-Vlasov system, demonstrating that the ratio 2M/R_1 can approach 8/9, aligning with numerical bounds and extending previous theoretical results.

Contribution

It proves the existence of static shell solutions with ratios approaching the theoretical maximum for Vlasov matter, confirming the sharpness of the Buchdahl inequality in this context.

Findings

Static shells of Vlasov matter can have 2M/R_1 arbitrarily close to 8/9.

Constructed a sequence of solutions with R_1/R_0 approaching 1.

Confirmed the upper bound of 2M/R_1 for Vlasov matter matches the theoretical limit.

Abstract

In a previous work \cite{An1} matter models such that the energy density $\rho\geq 0,$ and the radial- and tangential pressures $p\geq 0$ and $q,$ satisfy $p+q\leq\Omega\rho, \Omega\geq 1,$ were considered in the context of Buchdahl's inequality. It was proved that static shell solutions of the spherically symmetric Einstein equations obey a Buchdahl type inequality whenever the support of the shell, $[R_0,R_1], R_0>0,$ satisfies $R_1/R_0<1/4.$ Moreover, given a sequence of solutions such that $R_1/R_0\to 1,$ then the limit supremum of $2M/R_1$ was shown to be bounded by $((2\Omega+1)^2-1)/(2\Omega+1)^2.$ In this paper we show that the hypothesis that $R_1/R_0\to 1,$ can be realized for Vlasov matter, by constructing a sequence of static shells of the spherically symmetric Einstein-Vlasov system with this property. We also prove that for this sequence not only the limit supremum of $2M/R_1$ is bounded, but that the limit is $((2\Omega+1)^2-1)/(2\Omega+1)^2=8/9,$ since $\Omega=1$ for Vlasov matter. Thus, static shells of Vlasov matter can have $2M/R_1$ arbitrary close to $8/9,$ which is interesting in view of \cite{AR2}, where numerical evidence is presented that 8/9 is an upper bound of $2M/R_1$ of any static solution of the spherically symmetric Einstein-Vlasov system.