Spherical Systems in Gravity

We have completed a prescription, originally begun by Synge, to solve these equations in case the energy density of the system and one pressure are known. In such a case, we can provide the most general solution to the problem. We have also solved for the junction conditions.

The junction conditions tell us how the solution must behave at certain boundaries. For example, between various layers in an idealized star or at the surface of a star. Of course, the system doesn't have to be a star. Any spherically symmetric gravitating system is fair game. There are two conditions that are physically reasonable at the junction. One is known as Synge's junction condition which dictates continuity of various energy fluxes and pressures at the junction. The other is the Israel-Sen-Lanczos-Darmois (ISLD) condition, which tells us that certain derivatives of the metric or combinations thereof must be continuous. There are cases when both these conditions can hold and there are cases where only one can hold. We've managed to derive mathematical conditions for all cases. By the way, locating the boundary is no trivial task.

The boundary may be located by considering a function M(r,t) which is constant on the boundary. This will define a surface and it is the junction. We have mathematically proved, in a rigorous way, that such a function must exist.