New derivation of the variational principle for the dynamics of a gravitating spherical shell

The dynamics of a self-gravitating matter shell in general relativity is discussed in general. The case of a spherical shell composed of an arbitrary ideal fluid is then considered, and its Lagrangian function is derived from first principles. For this purpose, the standard Hilbert action is modified by an appropriate surface term at spatial infinity. The total Hamiltonian of the composed “$\mathrm{\text{shell}}+\mathrm{\text{gravity}}$” system is then calculated. Known results for the dust matter are recovered as particular cases. The above “surface renormalization” of the Hilbert action may be used for any spatially flat spacetime.

Figure 1

Given the shell position $\psi$ and velocity $\dot{\psi }$, there are two possible values of the angle $\mu$, satisfying constraint (3.14). Consequently, the Schwarzschild spacetime (see foliation $\{t=\mathrm{const}\}$ on the right-hand side) may be tailored with the Minkowski spacetime in two different ways (see possible two foliations $\{\tau =\mathrm{const}\}$ on the left-hand side).

Figure 2

Schematic view of the variation region $D_R$.