A Maximum Mass-to-Size Ratio in Scalar-Tensor Theories of Gravity

TL;DR

This paper extends the Buchdahl inequality to scalar-tensor theories of gravity, revealing that the maximum mass-to-size ratio can surpass general relativity limits and approach the black hole threshold, depending on matter conditions.

Contribution

It derives a theory-independent inequality in scalar-tensor gravity and shows the maximum mass-to-size ratio can exceed known limits under certain conditions.

Findings

Mass-to-size ratio can surpass Buchdahl's limit in scalar-tensor theories.

The ratio can approach the black hole limit under specific matter conditions.

Imposing  ho-3p4 5 on matter restricts the ratio below Buchdahl's limit.

Abstract

We derive a modified Buchdahl inequality for scalar-tensor theories of gravity. In general relativity, Buchdahl has shown that the maximum value of the mass-to-size ratio, $2M/R$, is 8/9 for static and spherically symmetric stars under some physically reasonable assumptions. We formally apply Buchdahl's method to scalar-tensor theories and obtain theory-independent inequalities. After discussing the mass definition in scalar-tensor theories, these inequalities are related to a theory-dependent maximum mass-to-size ratio. We show that its value can exceed not only Buchdahl's limit, 8/9, but also unity, which we call {\it the black hole limit}, in contrast to general relativity. Next, we numerically examine the validity of the assumptions made in deriving the inequalities and the applicability of our analytic results. We find that the assumptions are mostly satisfied and that the mass-to-size ratio exceeds both Buchdahl's limit and the black hole limit. However, we also find that this ratio never exceeds Buchdahl's limit when we impose the further condition, $\rho-3p\ge0$, on the density, $\rho$, and pressure, $p$, of the matter.