Equivalence principle and generalized accelerating black holes from binary systems

The Einstein equivalence principle in general relativity allows us to interpret accelerating black holes as a black hole immersed into the gravitational field of a larger companion black hole. Indeed it is demonstrated that C-metrics can be obtained as a limit of a binary system where one of the black holes grows indefinitely large, becoming a Rindler horizon. When the bigger black hole, before the limiting process, is of Schwarzschild type we recover usual accelerating black holes belonging to the Plebanski-Demianski class, thus type D. Whether the greater black hole carries some extra features, such as electric charges or rotations, we get generalized accelerating black holes which belong to a more general class, the type I. In that case the background has a richer structure, reminiscent of the physical features of the inflated companion, with respect to the standard Rindler spacetime. This insight allow us to build a general type D metric, describing an accelerating Kerr-NUT black black hole. It has well defined limits to all the type-D black holes of general relativity, including the elusive (type-D) accelerating Taub-NUT spacetime. Extension to the presence of the cosmological constant is also provided.

Figure 1

(a) Rod diagram for a collinear binary black hole system, associated to the Bach-Weyl metric. The $w_4\to +\infty$ limit precisely gives the rod representation of the uncharged C-metric, or a single accelerating black hole (b) Black segments on the time-like line represent event horizons while the semi-infinite blue line the Rindler horizon.

Figure 2

Embedding in the three dimensional Euclidean flat space of the event horizons of a Bach Weyl metric, which describes a couple of Schwarzschild black holes. In the pictures configuration the gauge freedom on $C_f$ is fixed to remove the conical singularity between the black holes, while $w_i=i$. Nevertheless conical singularities, interpreted as semi-infinite strings, are present on the axis of symmetry, in the external region to balance an equilibrium configuration.

Figure 3

The $w_4\to \infty$ limit brings a binary black hole system (illustrated in Fig. 2) into an accelerating black hole metric (1.4), whose embedding is pictured here. For larger values of $w_4$ the right element of the binary grows bigger and its event horizon becomes an accelerating horizon. If the right black hole of the binary carries no charges, then after the limit, the metric turns of special algebraic type, D, and the accelerating horizon is a usual Rindler one. When the right black hole of the binary carries some charges, the resulting accelerating metric remains of general type I, as the binary, and the accelerating background is endowed with that charge. The conical singularities on the symmetry axis eventually can be removed introducing external electromagnetic or gravitational fields, as done in [8] or [15].

Figure 4

Embedding in ${\mathbb{E}}^3$ of the event horizons of the metric (1.8)–(1.11). In the presence of an external gravitational field the binary black hole can be regularized and freed from conical singularities. In the picture $w_i=i,C_f=144,b_1=5/4\log (4/3),b_2=-\log (4/3)/4$.

Figure 5

The limit for $w_4\to \infty$ can be carried on also for the regularized binary metric (1.8)–(1.11), pictured in Fig. 4. The embedding of the horizons of the resulting metric, described by (1.12)–(1.15), are pictured here, $b_2=-1/15$.