Speaker
Description
Derivation of the Schwarzschild solution by substituting a spherically symmetric ansatz for the metric directly into the Einstein-Hilbert action instead of the Einstein field equations is known as the Weyl trick. Although it is a violation of the variational principle, surprisingly, this trick gives the correct result, which was later justified by the principle of symmetric criticality (PSC), first studied rigorously by Palais and later by Fels and Torre. PSC imposes two conditions on the infinitesimal symmetry group action (it should be stressed that these conditions are independent of the gravitational theory in question), and when satisfied, it allows one to symmetry-reduce any Lagrangian and guarantees that its field equations are fully equivalent to the reduced field equations.
We analyze all possible symmetry reductions of Lagrangians that yield fully equivalent field equations for any 4-dimensional metric theory of gravity. Specifically, we present a complete list of 44 infinitesimal group actions obeying PSC. We identify the corresponding invariant metrics and analyze their simplest form allowing for a successful symmetry reduction of Lagrangians. It turns out that PSC is satisfied not only by the infinitesimal symmetry group action of the Schwarzschild black hole, but also by, for example, the Taub-NUT solution, FLRW cosmologies, or NHEK geometry.
