The Schwarzschild Mass in General Relativity

TL;DR

This paper discusses defining the Schwarzschild mass in general relativity using a coordinate-invariant average density, challenging traditional definitions based on observer-dependent central or surface densities.

Contribution

It introduces a new, coordinate-invariant way to define the central and surface densities in static, spherical mass configurations in general relativity.

Findings

Average density depends only on total mass and radius

Central and surface densities are independent of each other

Traditional EOSs may not satisfy the new density definition

Abstract

The central (surface) energy-density, $E_0 (E_R)$, which appears in the expression of total static and spherical mass, $M$ (corresponding to the total radius $R$) is defined as the density measured only by one observer located at the centre (surface) in the Momentarily Co-moving Reference Frame (MCRF). Since the mass, $M$, depends only on the central (surface) density for most of the equations of state (EOSs) and/or exact analytic solutions of Einstein's field equations available in the literature, the central (surface) density measured in the preferred frame (that is, in the MCRF) appears to be not in agreement with the coordinate invariant form of the field equations that result for the source mass, $M$. In order to overcome the use of any preferred coordinate system (the MCRF) defined for the central (surface) density in the literature, we argue for the first time that the said density may be defined in the coordinate invariant form, that is, in the form of the average density ($3M/4\pi R^3)$ of the the configuration which turns out to be independent of the radial coordinate $`r'$ and depends only on the central (surface) density of the configuration. In this connection, we further argue that the central (surface) density of the structure should be {\em independent} of the density measured on the other boundary (surface/central) because there exists no a priori relation between the radial coordinate $`r'$ and the proper distance from the centre of the sphere to its surface \cite{Ref1}. In the light of this reasoning, the various EOSs and analytic solutions of Einstein's field equations in which the central and the surface density are {\em interdependent} can not fulfill the definition of central (surface) density measured only by one observer located in the MCRF at the centre (surface) of the configuration.