Varying Constants, Black Holes, and Quantum Gravity

Tentative observations and theoretical considerations have recently led to renewed interest in models of fundamental physics in which certain “constants” vary in time. Davies, Davis and Lineweaver [1] have argued that the laws of black hole thermodynamics disfavor models in which the fundamental electric charge e changes. I show that similar considerations severely constrain “varying speed of light” models, unless we are prepared to abandon cherished assumptions about quantum gravity. ∗email: [email protected] The idea that the fundamental “constants” of our Universe may vary in time dates back at least to Dirac’s Large Number Hypothesis [2]. Until recently, physically interesting variations seemed to be excluded by observation. Over the past several years, however, Webb et al. [3,4] have reported evidence that the fine structure constant α may have been slightly smaller in the early Universe. While this claim is still far from being established, the possibility, along with work on cosmological implications of “varying constants,” has inspired renewed interest in models in which either the elementary charge e or the speed of light c is dynamical [5–11]. In a Brief Communication to Nature [1], Davies, Davis and Lineweaver contend that black hole thermodynamics favors models with a varying speed of light. Their basic argument is simple. The Bekenstein-Hawking entropy of a charged black hole is S/k = πG ~c [ M + √ M2 −Q2/G ]2 . (1) Suppose α is indeed increasing in time, as Webb et al. suggest. If this variation comes from an increase in e, the resulting increase in Q will cause the entropy of such a black hole to decrease, evidently violating the generalized second law of thermodynamics. If, on the other hand, the change comes from a decrease in c, the entropy will increase with time, as it should. This is an intriguing argument. But as I shall show below, a slight rephrasing leads to much more radical implications for our current understanding of quantum gravity and black hole thermodynamics, implications perhaps unpalatable enough to argue against “varying speed of light” models. Consider a Reissner-Nordstrom black hole with charge Q = re and mass M = √ sMP , where MP = (~G/c) 1/2 is the Planck mass. Quantization of charge requires that r be an integer. In simple models of black hole thermodynamics (see [12] for a review), s is an integer, or a fixed constant multiple of an integer, as well. More elaborate approaches to quantum gravity lead to more complicated black hole spectra: for instance, neutral black holes in loop quantum gravity have