Generating perfect fluid spheres in general relativity

Ever since Karl Schwarzschild’s 1916 discovery of the spacetime geometry describing the interior of a particular idealized general relativistic star—a static spherically symmetric blob of fluid with position-independent density—the general relativity community has continued to devote considerable time and energy to understanding the general-relativistic static perfect fluid sphere. Over the last 90 years a tangle of specific perfect fluid spheres has been discovered, with most of these specific examples seemingly independent from each other. To bring some order to this collection, in this article we develop several new transformation theorems that map perfect fluid spheres into perfect fluid spheres. These transformation theorems sometimes lead to unexpected connections between previously known perfect fluid spheres, sometimes lead to new previously unknown perfect fluid spheres, and in general can be used to develop a systematic way of classifying the set of all perfect fluid spheres.

Figure 1

The solid lines show $B(r)$ for 10 reapplications of Theorem 1 onto the Minkowski metric. The dashed line corresponds to a single application with the specific choice ${\lambda }_{\mathrm{once}}=\sum _{}^{}{\lambda }_i$. It can be seen that 10 applications of Theorem 1 can be expressed by one application.

Figure 2

The solid lines show $\zeta (r)$ for 10 reapplications of Theorem 2 onto the Minkowski metric. The dashed line corresponds to a single application with the specific choice for ${\sigma }_{\mathrm{once}}$ and ${ϵ}_{\mathrm{once}}$ determined by the composition law of Theorem 2. It can be seen that 10 applications of Theorem 2 can be reexpressed by a single application.

Figure 3

Structure graph for Minkowski space as seed metric.

Figure 4

Numerical implementations of Theorem 3 and Theorem 4 can be used to investigate their behavior for $n$ applications, as the number of applications becomes large. For specific choices of parameters ${\lambda }_i$, ${\sigma }_i$ and ${ϵ}_i$ successive applications of Theorem 3 and Theorem 4 appear to be approaching a limit, in the sense that both metric components seem to converge toward fixed points.