Definitions and Results

Abstract

Let S be a closed 2-dimensional surface embedded in a 3+1-dimensional spacetime (M, g). We assume that S has a compact filling by which we mean that there exists a Cauchy hypersurface ∑ containing S such that S is the boundary of a compact region of ∑.

Let γ be the induced metric on S,

$$\gamma (x,y) = g(x,y)$$

(3.1.1)

for all X,Y ∈ T S, the tangent space to S. We denote by dμ_γ the area element and by ∈ _ab its components relative to an orthonormal frame (e _a )a=1.2. We denote by |S| the area and by r(S) the radius of S,

$$r(S) = \sqrt {\frac{1} {{4\pi }}\left| S \right|.}$$

(3.1.2)