All conformally flat Einstein–Gauss–Bonnet static metrics

Abstract

It is known that the standard Schwarzschild interior metric is conformally flat and generates a constant density sphere in any spacetime dimension in Einstein and Einstein–Gauss–Bonnet (EGB) gravity. This motivates the questions: in EGB does the conformal flatness criterion yield the Schwarzschild metric? Does the assumption of constant density generate the Schwarzschild interior spacetime? The answer to both questions turn out in the negative in general. In the case of the constant density sphere, a generalised Schwarzschild metric emerges. When we invoke the conformal flatness condition the Schwarschild interior solution is obtained as one solution and another metric which does not yield a constant density hypersphere in EGB theory is found. For the latter solution one of the gravitational metrics is obtained explicitly while the other is determined up to quadratures in 5 and 6 dimensions. The physical properties of these new solutions are studied with the use of numerical methods and a parameter space is located for which both models display pleasing physical behaviour.