Two-dimensional Black Holes and Planar General Relativity

The Einstein-Hilbert action with a cosmological term is used to derive a new action in 1+1 spacetime dimensions. It is shown that the two-dimensional theory is equivalent to planar symmetry in General Relativity. The two-dimensional theory admits black holes and free dilatons, and has a structure similar to two-dimensional string theories. Since by construction these solutions also solve Einstein’s equations, such a theory can bring two-dimensional results into the four-dimensional real world. In particular the two-dimensional black hole is also a black hole in General Relativity. General Relativity is thought to be the correct theory from the largest conceivable scales up to the Planck length, 10−33cm. A place to test these tinyest radii can be found in the late stages of black hole evaporation. This is a non-trivial issue and one is still probing regions where a semi-classical approximation is valid. The two-dimensional (2D) black holes found in the context of string theories [1,2] are being used to compute and analyze the back-reaction of the radiation on the geometry [3,4,5]. Such 2D theories are, in principle, toy models and it is important to construct a link with the four-dimensional (4D) world. With this purpose, we consider the following action, S = 1 2π ∫ dx √−ge−2φ ( R + 2 (∇φ) + 4λ ) , (1)