Implanting soft hairs on Kaluza-Klein black holes

In this work we investigate current research on quasilocal energy and conservation laws in general relativity. We explore the derivations and motivations for the Brown and York quasilocal energy and the Epp invariant quasilocal energy. We obtain expressions for the quasilocal energy of the radially inhomogeneous Lemaître-Tolman geometry via both the Brown and York and the Epp definitions. We then make a perturbative compar- ison between the energy predicted by Newtonian cosmology and the quasilocal energy of a Friedmann-Lemaître-Robertson-Walker universe transformed into locally inertial Fermi normal coordinates. It is found that by transforming to Fermi normal coordinates the magnitude of the difference in energy between these cosmological models is reduced. Recent developments on the utility of a rigid quasilocal frame (RQF) in quasilocal conservation laws are investigated. We apply the RQF construction to a Lemaître-Tolman universe and prove the existence of such a frame at the center of spherical symmetry. We obtain an explicit function in terms of the components of the metric that allow a congruence of observers to remain expansion- and shear-free for all time