Note on counterterms in asymptotically flat spacetimes

Abstract

We consider in more detail the covariant counterterm proposed by Mann and Marolf in asymptotically flat spacetimes. With an eye to specific practical computations using this counterterm, we present explicit expressions in general d dimensions that can be used in the so-called 'cylindrical cut-off' to compute the action and the associated conserved quantities for an asymptotically flat spacetime. As applications, we show how to compute the action and the conserved quantities for the NUT-charged spacetime and for the Kerr black hole in four dimensions.

Quasilocal formalism and thermodynamics of asymptotically flat black objects

“…In other words, can an asymptotic observer distinguish between a black hole or a black ring with the same conserved charges? The answer is obviously yes: analogous to an electric dipole whose moments can be read off from a multipole expansion at infinity, the subleading terms in the boundary stress tensor should encode the information necessary to distinguish between black objects with different horizon topologies in the bulk 18 .…”

“…although a similar formalism should be valid for any spacetime dimension (see [18] for a similar analysis in four dimensions).…”

Explicit form of the Mann-Marolf surface term in (3+1) dimensions

“…Mathematically this is also the most difficult case to deal with, as d = 4 does not exhibit the many technical simplifications that occur for d > 4 [2,3,4,5,6]. However it is only in this physically most relevant situation of d = 4 that we will succeed in inverting R(K) to obtainK(R).…”

“…The Mann-Marolf counterterm has already been extensively compared with other counterterms appearing in the literature [2,3,4,5], so at this stage the only point of consistency checking is to verify that our explicit formulae make sense and yield the expected results. i) Simply from the way it is defined, it is clear that if the boundary is isometrically embeddable in Minkowski spacetime, then the Mann-Marolf procedure automatically reproduces the Gibbons-Hawking prescription [1].…”

“…Unfortunately there are many interesting situations (e.g., the Kerr spacetime) where any boundary placed at "finite distance" has a complicated intrinsic geometry that cannot be isometrically embedded into flat Minkowski spacetime, and in these situations another reference prescription is called for. A particularly promising recent suggestion is the Mann-Marolf boundary term introduced in [2], and further discussed in [3,4,5,6,7].…”