Negative Komar mass of single objects in regular, asymptotically flat spacetimes

We study two types of axially symmetric, stationary and asymptotically flat spacetimes using highly accurate numerical methods. One type contains a black hole surrounded by a perfect fluid ring and the other a rigidly rotating disc of dust surrounded by such a ring. Both types of spacetime are regular everywhere (outside of the horizon in the case of the black hole) and fulfil the requirements of the positive energy theorem. However, it is shown that both the black hole and the disc can have a negative Komar mass. Furthermore, there exists a continuous transition from discs to black holes even when their Komar masses are negative. PACS numbers: 04.70.Bw, 04.40.−b, 04.25.Dm A great many definitions for mass in general relativity have been proposed. Many of them consider only the spacetime as a whole, whereas others assign a mass locally to a portion of the spacetime. Even the latter are not always applicable to a single object in a spacetime containing multiple ones. In [1], Bardeen considers an axially symmetric, stationary spacetime containing a black hole surrounded by a perfect fluid matter distribution and assigns to each of the two objects a mass based on the Komar integral [2]. In a similar way, he assigns an angular momentum to each of the objects. Various other local mass definitions include the Hawking mass [3], the Bartnik mass [4], the Christodoulou mass [5], and a related mass used for isolated horizons, which can be generalized to dynamical ones [6, 7]. For a review article see [8]. A mass definition applicable both to matter and black holes, and which can be used for single components of a many-body problem in stationary spacetimes, is that based on the Komar integral. We will refer to it here as the Komar mass even when applied to single objects. 0264-9381/06/240081+07$30.00 © 2006 IOP Publishing Ltd Printed in the UK L81 L82 Letter to the Editor In this letter, we study axially symmetric and stationary spacetimes containing either a black hole or a rigidly rotating disc of dust, each surrounded by a perfect fluid ring. We find that there exists a continuous transition from the disc to the black hole. If one is interested in being able to talk about the mass of the individual objects in such stationary spacetimes and since the transition from the disc to the black hole exists, then one is led to look for a definition that is applicable in either scenario. We thus choose to examine the behaviour of the Komar mass and find that it can become negative both for the black hole3 and for the disc although the total mass of the spacetime is of course positive as guaranteed by the positive energy theorem. We find moreover that the continuous transition mentioned above exists even when the Komar mass of each of the two central objects is negative. The Poisson equation in Newtonian gravity ∇2U = 4πε (1) relates the potential U to the mass density ε.4 The mass contained in any volume V of space can be defined by integrating equation (1) over that volume and applying the divergence theorem