Abstract
We develop the large D limit of general relativity for spherically symmetric scalar fields in both asymptotically flat and asymptotically anti-de Sitter spaces. The leading order equations in the 1/D expansion can be solved analytically, providing a large D description of oscillating soliton stars. When the amplitude reaches a critical threshold, certain divergences occur which we interpret as signal of horizon formation. We estimate the size of the resulting black hole and obtain a scaling exponent. We speculate on some connections to Choptuik critical collapse.
Highlights
In this paper, we develop the large D limit of general relativity in situations where horizons may or may not exist
We have developed the large D limit for spherically symmetric scalar fields in both flat space and in anti-de Sitter space (AdS), arriving at an effective set of large D equations
This matching at higher orders requires terms that are polynomial in time τ which may possibly cause the breakdown of the perturbation series when τ ∼ O(D), which can be analogously compared to the t ∼ 1/ 2 timescale where perturbation theory in AdS breaks down
Summary
To gain intuition on the nature of the large D limit we are taking, we start by considering a probe real scalar field in flat spacetime, under spherical symmetry:. The equation (2.1) suggests that in order to have√nontrivial equations in the large D limit we need to perform a rescaling of time to τ = D − 2t. Note that the resulting equation is parabolic, with the roles of space and time swapped. This will be the case for all equations we obtain in the large D limit, below. We can compare the result to the exact solutions of (2.1) with m = 0 which are given by (a sum of) the Bessel functions φ. One can show analytically using asymptotic properties of Bessel functions that (2.3) and (2.4) (suitably normalised) agree in the large D limit
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