The Hookean Law of Black Holes and Fragmentation: Insights from Maximum Force Conjecture and Ruppeiner Geometry

Abstract

We show that the notion of the maximum force conjecture F 1/4 in general relativity, when applied to asymptotically flat singly spinning Myer-Perry black holes in any dimension, reveals the underlying thermodynamic instability in a number of ways. In particular, the "Hookean force law" F1 = kx, suitably defined, is bounded by the conjectured limit, but in d 6 it is further bounded by a dimensional dependent value less than 1/4, which remarkably corresponds to the Emparan-Myers fragmentation (splitting of a black hole into two becomes thermodynamically preferable). Furthermore, we define another "force" as the square of the angular momentum to entropy ratio (F2 = J 2 /S 2 ). In dimensions d 6, the positive Ruppeiner scalar curvature region in the thermodynamic phase space is marked by the upper boundary F2 = 1 12 d−3 d−5 and the lower boundary F2 = 1 4 d−3 d−5, the latter corresponds to a black hole that suffers from Gregory-Laflamme instability. Surprisingly, the upper and lower boundaries correspond to F = 1/4 when d = 6 and d → ∞, respectively. We discuss how the maximum force may be relevant to the underlying black hole microstructure and its relationship to cosmic censorship.

Tests For Maximum Force and Maximum Power

“…When the topic was explored in more detail, the factor 1/4, which is the force limit in natural units, was deduced by Gibbons [7] and others [8], and studied further [9][10][11][12][13][14][15][16].…”

“…The first law (11) describes how a changing horizon area δA induces a changing horizon energy δE for a given surface gravity a. In other words, the first law describes the dynamics of every horizon.…”

“…To achieve this formulation, one introduces the general surface element dΣ and the local boost Killing vector field k that generates the horizon (with a suitable norm). These two quantities allow rewriting the left hand side of the first law (11) as…”

“…In particular, the Raychaudhuri equation does not contain any physics of space-time or of gravitation.) Combining the generalizations of both sides of the first law (11) yields the equation…”

“…Each step in the previous derivation of the field equations can be reversed: one can return from the field equations (15) to the first law (11) and then, using maximum speed, to the maximum force used in equation (8).…”