A Penrose-Type Inequality for Static Spacetimes

Abstract We establish a lower bound on the total mass of the time slices of $$(n+1)$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>n</mml:mi> <mml:mo>+</mml:mo> <mml:mn>1</mml:mn> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> -dimensional asymptotically flat standard static spacetimes under the timelike convergence condition (TCC). The inequality can be viewed equivalently as a Minkowski-type inequality in these spaces, i.e. as a lower bound on the total mean curvature of the boundary, and thus extends inequalities from [4, 20, 23], and [11]. Equality is achieved only by slices of Schwarzschild space and is related to the characterization of quasi-spherical static vacuum metrics from [11]. As a notable special case, we obtain the Riemannian Penrose inequality in all dimensions for static spaces under the TCC.