...sciel

Supported in part by the Polish Research Council grant KBN 2 P03B 073 15. E-mail: chrusciel@univ-tours.fr

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... Nagy

Supported by a grant from Région Centre. E-mail: nagy@gargan.math.univ-tours.fr

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... satisfy

The summation convention is used throughout. We use Greek indices for coordinate components and lower-case Latin indices for the tetrad ones; upper-case Latin indices run from $2$ to $n$ and are associated either to coordinates or to frames on $M$, in a way which should be obvious from the context. Finally Greek bracketed indices $(\alpha)$ etc. refer to objects defined outside of space-time, such as the Killing algebra ${\mycal K}_{{\mycal S}^{\perp}}$, or an exterior embedding space, and are also summed over.

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...:

The integral over $\partial {\mycal S}$ should be understood by a limiting process, as the limit as $R$ tends to infinity of integrals over the sets $t=0$, $r=R$. $d S_{\alpha\beta}$ is defined as $\frac{\partial}{\partial x^\alpha}\rfloor \frac{\partial}{\partial x^\beta}\rfloor \,{ d}x^0 \wedge\cdots \wedge\,{ d}x^{n}$, with $\rfloor$ denoting contraction; $g$ stands for the space-time metric unless explicitly indicated otherwise. Square brackets denote antisymmetrization with an appropriate numerical factor ($1/2$ for two indices), and $\mathring{\nabla}$ denotes covariant differentiation with respect to the background metric $b$.

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... scalar

Related results under less restrictive conditions can be found in [13].

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... invariant

If the scalar curvature of $h$ vanishes, then Einstein equations require the constant $k$ in the metric to vanish. In that case there arises an ambiguity in the definition of mass related to the possibility of rescaling $t$ and $r$ without changing the form of the metric, which rescales the mass. (This freedom does not occur when $k$ is non-zero.) This ambiguity can be removed by arbitrarily choosing some normalization for the $h$-volume of $M$, e.g. $4\pi$ in dimension $n+1$=4.

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