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The observations of refs 1, 2 suggest that the fine-structure constant, α, may have been slightly smaller in the early Universe. As α = e2/ℏc depends on e, c and Planck's constant, ℏ, it is natural to ask which of these values varies. Davies et al. offer an ingenious argument. A black hole with mass M and charge Q = ne has an entropy7 S = πG/ℏc[M + (M2 − n2e2/G)1/2]2. Evidently, an increase in e will reduce the entropy, violating the generalized second law of thermodynamics, whereas a decrease in ℏ or c will increase the entropy. This argument involves assumptions that may not be valid for all models6,8,9, but it offers an interesting starting point.

As Davies et al. note, however, such an argument should consider not just the black hole, but also its surroundings. An isolated black hole is never in thermal equilibrium: it decays by Hawking radiation and, if it is charged, by spontaneous emission of charged particles10. These processes reduce S, but do not violate the second law of thermodynamics because there is a compensating increase in the entropy of the environment.

To investigate the thermodynamics of varying 'constants', one should study a black hole that is in equilibrium with its environment. This can be done by considering a black hole in a 'box' of radius rB, with fixed boundary temperature T and charge Q (the canonical ensemble) or electrostatic potential φ (the grand canonical ensemble). Note that rB can be altered only by doing work on the system.

In the canonical ensemble, the entropy is given by S = πrB2x2, where x is determined by the seventh-order equation11 x5(x − q2)(x − 1) + b2(x2 − q2)2 = 0, where q = √√(GQ/rBc2) and b = ℏc/4πrBkT. Figure 1 shows a plot of S/rB2 against q2 and b. It is apparent — and may be confirmed numerically — that the entropy increases with increasing α. For the grand canonical ensemble, exact analytical results lead to the same conclusion. Black-hole thermodynamics thus militates against models in which the fundamental charge, e, decreases, but places no restriction on increasing e.

Figure 1: Entropy, SB, of a black hole and its surroundings as a function of charge, q2, and inverse temperature, b.
figure 1

For any fixed temperature, an increase in the fine-structure constant increases q2 and therefore SB.

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To compare this result with that of Davies et al., note first that the Hawking temperature of a charged black hole decreases with increasing e. A black hole will thus cool below the ambient temperature of the heat bath and will absorb heat, thereby increasing its mass. According to the first law of thermodynamics, the net change in entropy is dS = 1/T(dE − φ dQ), and it may be verified that the increase in the energy, E, dominates.

Of course, such thermodynamic arguments only describe relationships among equilibria, and not the transitions between equilibria. A more detailed analysis, however, requires an explicit, dynamic model. In particular, any theory with a variable fine-structure constant necessarily contains a new scalar field, α itself, the entropy of which cannot be neglected during dynamic processes in which α is changing. Jacobson (personal communication) has suggested that a suitable dynamic version of the second law of thermodynamics12 will ensure that entropy increases during such a process. Black-hole thermodynamics is therefore insufficient to constrain theories in which α increases.