Lightlike Singularities in Compactified Supergravity

We discuss the (causal) structure of a recently found black hole solution of eompactffied d = 11 supergravity. It is shown that the singularity is in fact lightlike and coincides with the horizon. Consequences are that the Hawking temperature is undetermined and that there is no other universe connected to the singularity. In a previous paper in collaboration with P. van Nieuwenhuizen [ 1 ], we have constructed a number of black hole solutions ofd = 11 supergravity compacti-fled to four dimensions over a seven-sphere. These solutions tend asymptotically to the well known Freund-Rubin [2] or Englert [3] solutions but near the core of the black hole they differ substantially from the conventional Schwarzschild or Reissner-NordstriSm solutions. In this letter we discuss in detail the core structure of the solution which tends asymptotically to the Freund-Rubin solution (case II in ref. [1 ]) and reveal some novel features, notably the existence of a lightlike singularity. As a starting point the following ansatz for the metric was considered:-ds 2 =-B(r) dt 2 + A (r) dr 2 + r2(d[22) 2 + R (r)2(d~7) 2 , (d~n) 2 = de 2 + sin2 cn(d~n_l) 2. (1) It describes a static metric with R 1 × SO 3 X SO8 symmetry. For the three-index photon field AMN P of d = 1 1 supergravity we choose a Freund-Rubin an-satz [2]: Fmnpq = ib (r) emnpq , (2) where b is allowed to depend on r and m, n, p and q are four-dimensional fiat indices, the other components of Fare put equal to zero. For large r we want the extra seven-dimensional space to be compact with radius R~, which implies that b andR have to go to constants related by: b 2 = ~ R 7. 2. (3) The only nontrivial Maxwell equation is integrable and yields the relation bR 7 = constant. So that after rescal-ing the variables, such that R** = 1, one has that (the sign of b is irrelevant): b 2 = 3-~12 R-14. (4) This allows us to eliminate b from the Einstein equations. The Einstein equations provide us with three independent equations for A, B and R. For reasons which will become clear shortly we choose the following particular combinations: A'/A + B'/B = (7rR"/R)/(1 + 7rR'/2R), (5) A'/A-B'/B = (2/r)(1-A)+ 14R'/R-24rA/R 14, (6) A = r[r(ln R)']' (7) • [-r(ln R)'(1 + 12r2R-14) + 6r2(R-2-R-14)]-1. …