Existence and properties of spherically symmetric static fluid bodies with a given equation of state

Abstract

It is shown that for a given equation of state and a given value of the central pressure there exists a unique global solution of the Einstein equations representing a spherically symmetric static fluid body. For the proof a new theorem on singular ordinary differential equations is established which is of interest in its own right. For a given equation of state and central pressure, the fluid will either fill the entire space or be finite in extent with a vacuum exterior. Criteria are given which allow one to decide for certain equations of state which of these two cases occurs. This generalizes well known results in Newtonian theory and is proved by showing that the relativistic model inherits the property of having a finite radius from a Newtonian model. Parameter-dependent families of relativistic solutions are constructed which have a Newtonian limit in a rigorous sense. The relationship between relativistic and Newtonian equations of state is examined by looking at the example of a degenerate Fermi gas.

Time-Independent Gravitational Fields

“…However, the system is singular at r = and the first step is to demonstrate that for each value of the central density there is a unique solution such that the spacetime is regular at the center. This is shown in [65] or in [50]. This solution defines a neighborhood of a regular center and can be extended as long as (1 − 2G c 2 r 2 w) remains positive.…”

“…Equation (5.25) can easily be integrated: It is also useful to introduce the following quantity which is related to the 'mean density up to r' w(r) = r −3 m(r) . In [65] the following theorem is proved.…”

“…In [65] it is shown that the radius of the star is finite if p0 o dp/ρ(p) 2 is finite. Conversely, o dp/(ρ(p)c −2 p < ∞ implies that the matter distribution is infinitely extended.…”

On spherically symmetric solutions of the Einstein–Euler equations

“…See [14]. Even if 1 < γ ≤ 6/5, it is possible that there are short equilibria, since Proposition 3 guarantees existence of tails of short equilibria in any case and we can arbitrarily modify the equation of state in the higher density region.…”

On the stability of (M theory) stars against collapse: Role of anisotropic pressures

“…2 This box is needed since the radius of the star will be infinite otherwise [33]. As explained in [29], enclosing the star within a box prevents the evaporation of its constituents and makes its total mass finite.…”