## ADMMass

Frank Löﬄer

2012-05-07

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 [?] as a surface integral over a sphere with inﬁnite radius:

$${M}_{\text{ADM}}=\frac{1}{16\pi}{\oint}_{\infty}\sqrt{\gamma}{\gamma}^{ij}{\gamma}^{kl}\left({\gamma}_{ik,j}-{\gamma}_{ij,k}\right)\mathrm{\text{d}}\phantom{\rule{0.3em}{0ex}}{S}_{l}$$ | (1) |

. This is, assuming $\alpha =1$
at inﬁnity, equivalent to

$${M}_{\text{ADM}}=\frac{1}{16\pi}\int {\left(\alpha \sqrt{\gamma}{\gamma}^{ij}{\gamma}^{kl}\left({\gamma}_{ik,j}-{\gamma}_{ij,k}\right)\right)}_{,l}\mathrm{\text{d}}{\phantom{\rule{0.3em}{0ex}}}^{\phantom{\rule{0.3em}{0ex}}3}\phantom{\rule{0.3em}{0ex}}\phantom{\rule{0.3em}{0ex}}x$$ | (2) |

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

$${M}_{\text{ADM}}=\frac{1}{16\pi}{\oint}_{\infty}\alpha \sqrt{\gamma}{\gamma}^{ij}{\gamma}^{kl}\left({\gamma}_{ik,j}-{\gamma}_{ij,k}\right)\mathrm{\text{d}}\phantom{\rule{0.3em}{0ex}}{S}_{l}.$$ | (3) |

This diﬀers from equation (1) by the factor $\alpha $
inside the integral. For evaluations of those equations at inﬁnity,
$\alpha =1$ 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 [?]. 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