A conjectured upper bound on the Choptuik critical exponents

Near-critical type II gravitational collapse is characterized by the formation of arbitrarily small black holes whose horizon radii are described by the simple scaling law r BH ∝ (p − p ∗ ) γ , where γ is the matter-dependent Choptuik critical exponent and (cid:3)p ≡ p − p ∗ is the phase space distance from the exact self-similar critical evolution. We point out that all matter models studied thus far in the physics literature are characterized by the upper bound γ ≤ 1. We conjecture that this is a generic feature of non-linear gravitational collapse scenarios.


Introduction
Almost three decades ago, Choptuik [1] studied numerically the non-linear gravitational collapse of a real self-gravitating massless scalar field and revealed an intriguing critical phenomenon in general relativity at the threshold of black-hole formation. Characterizing the strength of the initial field configuration by a parameter p [2], Choptuik [1] has demonstrated numerically that there is a critical field parameter p = p * for which the collapse is characterized by a self-similar behavior which leads to the formation of a central naked singularity. Table 1 Critical phenomena in gravitational collapse. Remarkably, one finds that all self-gravitating matter models reported in the physics literature over the past three decades [1, are characterized by the simple relation γ ≤ 1.
In particular, it has been demonstrated explicitly in [1, that the exact value of the critical exponent γ in the black-hole scaling relation (1) depends on a number of physical factors, such as: (1) the type of collapsed matter, (2) the symmetry of the initial matter configuration, and (3) the spacetime dimension.

The conjectured upper bound on the Choptuik critical exponents
Despite the fact that the critical scaling exponent γ in (1) is model-dependent [1,, in the present compact paper we would like to raise the following physically intriguing question: Is there some underlying physical principle that bounds the possible numerical values of the Choptuik critical exponents?
In order to provide evidence that the answer to the above stated physically interesting question may be 'yes', we summarize in a compact form in Table 1 the main body of work [1, that has been mounting in the physics literature on the Choptuik critical phenomena for various selfgravitating matter models. What we find most intriguing is the fact that, for all matter models studied in the physics literature over the past three decades [1,, the critical exponents are all bounded from above by the simple relation The observation (2) [see the numerical data presented in Table 1] may indicate that there is some yet unknown universal mechanism which, regardless of the physical characteristics of the various collapse models (the assumed matter content, symmetry, and the spacetime dimension), bounds the possible values of the Choptuik critical exponents in dynamical gravitational collapse scenarios. We hope that the present compact paper would encourage physicists and mathematicians to further explore the (in)validity of the conjectured universal relation (2) for the Choptuik critical exponents.

Declaration of competing interest
The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper.