# The black hole that went away

###### Abstract

A purported black hole solution in -dimensions is shown to be nothing more than flat space viewed from an accelerated frame.

###### pacs:

04.20.-q, 04.50.+h^{†}

^{†}preprint: UM-P-96/74

[ ]

In a recent paper, Kawai [1] describes a remarkable vacuum solution to Einstein’s equations in -dimensions. The result is remarkable because it goes against all conventional wisdom. The claim is that the solution describes a vacuum black hole with a working Newtonian limit. In contrast, the usual BTZ black hole [2] requires a negative cosmological constant, and general relativity should not have a Newtonian limit [3].

The Kawai “black hole” has a metric given by [1, 4]

(1) |

The coordinates are taken from the conventional range , , and . The metric appears to describe a singularity at , surrounded by an event horizon at .

A quick check confirms that (1) is indeed a solution to the vacuum Einstein equations . But this is where things start to unravel. Recall that in -dimensions the full Riemann curvature tensor can always be expressed in terms of the Ricci scalar via the relation [5]

(2) |

Consequently, we know that everywhere (there is no conical singularity at ). As a result, (1) cannot support a Newtonian limit.

There is another well known spacetime that is everywhere flat, has an event horizon, and a coordinate singularity at the origin – the Rindler wedge. A simple coordinate transform reveals the Kawai solution to be nothing more than a Rindler wedge. Transforming to the cartesian coordinates , and rescaling , the metric becomes

(3) |

Both and have the usual range , while is topologically compact with period . A further set of standard coordinate transformations reduces the Kawai solution to Minkowski space. The black hole has gone away.

## References

- [1] T. Kawai, Prog. Theor. Phys. 94 1169, (1995).
- [2] M. Banados, C. Teitelboim and J. Zanelli, Phys. Rev. Lett. 69 1849, (1992).
- [3] J. R. Gott III and M. Alpert, Gen. Rel. Grav. 16, 243 (1984); S. Giddings, J. Abbot and K. Kuchar̂, Gen. Rel. Grav. 16, 751 (1984), N. J. Cornish and N. E. Frankel, Phys. Rev. D43, 2555 (1991).
- [4] T. Kawai, Phys. Rev. D48, 5668 (1993).
- [5] S. Deser, R. Jackiw and G ’t Hooft, Ann. Phys. (NY), 152, 220 (1984).