Choptuik scaling in six dimensions

Critical behavior in gravitational collapse, as first found by Choptuik @1#, occurs at and near the threshold of black hole formation @2#. For a one-parameter family of initial data slightly above the threshold, the mass of the black hole, M BH , scales like (p2p*) . Here p is the parameter, p* is its critical value and g is a constant that depends on the type of matter, but not on the family of data. For initial data slightly below the threshold, the maximum curvature scales like (p*2p) where g is the same constant as in the black hole mass scaling law @3#. @The definition of the scaling exponent g must be slightly generalized in n spacetime dimensions, where M BH has dimension (length) ; see Sec. II.# The critical solution (p5p*) has either continuous self similarity or discrete self similarity, depending on the type of matter. While critical gravitational collapse has been studied in many types of matter, the work has, in general, been done in 4 spacetime dimensions. ~The exception is work on analogs of gravitational collapse in 3 spacetime dimensions @4#.! One might therefore wonder whether critical behavior occurs in gravitational collapse in n spacetime dimensions for n.4, and if so, how the properties of the critical behavior depend on n. In this work, we perform numerical simulations of the collapse of a spherically symmetric scalar field in 6 spacetime dimensions. We find that the critical solution has discrete self-similarity. We find the scaling exponent g and the self-similarity period D. Section II reviews the Schwarzschild solution in n dimensions and shows how the definition of the g exponent should be generalized to the n-dimensional case. Section III gives the equations for the evolution of the scalar field and the metric in a form suitable for our numerical simulations. The numerical method is presented in Sec. IV and our results in Sec. V.