## ADMMass

2012-05-07

Abstract

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 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:

• Surface Integral (over rectangular domain)
• ADMMass_distance_from_grid_boundary: speciﬁes the distance between the physical domain boundary and the integration domain. If this is set, this fully speciﬁes the domain boundary.
• ADMMass_surface_distance: speciﬁes a distance of the integration boundary from a given point, speciﬁed using ADMMass_x_pos.
• Otherwise, ADMMass_[xyz]_[min|max] specify a rectangular integration domain.
• Volume Integral (over sphere)
• If ADMMass_use_all_volume_as_volume_radius is set, the whole volume is used for integration.
• If ADMMass_use_surface_distance_as_volume_radius is set and ADMMass_volume_radius is not (negative), ADMMass_surface_distance is used to specify the integration radius.
• Otherwise, ADMMass_volume_radius speciﬁes this radius.
• ADMMass_[xyz]_pos specify the position of the integration sphere.
• Use ADMMass_Excise_Horizons to exclude domains where thorn OutsideMask didn’t specify domain as outside. This can be used to, e.g. excise black hole apparent horizons.

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.