Frank Löffler



Thorn ADMMass can compute the ADM mass from quantities in ADMBase.

1 Introduction

The ADM mass evaluated as either surface integrals at infinity or volume integrals over entire hypersurfaces give a measure of the total energy in the spacetime.

The ADM mass can be defined [1] as a surface integral over a sphere with infinite radius:

MADM = 1 16π γγijγkl(γ ik,j γij,k)dSl (1)

. This is, assuming α = 1 at infinity, equivalent to

MADM = 1 16παγγijγkl(γ ik,j γij,k),ld3x (2)

. In practice, the following equation can also be used within the thorn:

MADM = 1 16π αγγijγkl(γ ik,j γij,k)dSl. (3)

This differs from equation (1) by the factor α inside the integral. For evaluations of those equations at infinity, α = 1 is assumed, and they are equal. For evaluations at a finite distance, however, this is usually not the case and the approximation of the ADM mass is gauge-dependent [1]. Depending on circumstances, either (1), (3 or  (2) might give better results.

2 Using This Thorn

Multiple measurements can be done for both volume and surface integral, but the limit for both is 100 (change param.ccl if you need more). You need to specify the number of integrations with ADMMass_number (and ADMMass will perform both integrations that many times).

Also note that this thorn uses the ADMMacros for derivatives. Thus, converegence of results is limited to the order of these derivatives (ADMMacros::spatial_order).

ADMMass provides several possibilities to specify the (finite) integration domain, both for surface and volume integral, which we list in the following:

You should output ADMMass::ADMMass_Masses for the result of the integrations, which will include the results for the volume integral, the usual surface integral and the sorface integral including the lapse.