A new dynamical approach to black hole thermodynamics

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In the last decade or so, there has been major mathematical advances in how dynamical systems theory connects with statistical mechanics. Most notably, there is the 1997 seminal paper by Hans Henrik Rugh that presents a new dynamical approach to thermodynamics in the microcanonical ensemble. In this approach, the temperature of a Hamiltonian dynamical system is computed as a time average of a particular function evaluated on the energy surface. The function itself is obtained from derivatives of the Hamiltonian. It has long been assumed that Hamiltonian dynamical systems exhibit some sort of ergodicity, where time-averages are viewed as being equivalent to space-averages over the microcanonical ensemble. However, until recently, an explicit mathematical formula for the temperature that reflected this was missing. The new formula not only provides an algorithm by which to compute the temperature but furnishes a long-sought connection between dynamical systems theory and the statistical mechanics of Hamiltonian systems. Moreover, the mathematical formalism has now been extended to other conserved quantities besides the energy (e.g. angular momentum). One of my objectives is to apply these new ideas and computational algorithms stemming from statistical mechanics to black hole (BH) thermodynamics. Black holes are ideally suited for this formalism as they are Hamiltonian dynamical systems described by three conserved quantities: mass (energy) M, charge Q and angular momentum J. Each of these conserved quantities can be expressed as a surface integral and has an associated thermodynamic variable that can be calculated as a time-average. In particular, the temperature of a BH would be computed as a time-average of a function evaluated from the Hamiltonian only. As with recent computations of the free energy, the temperature could be evaluated numerically in a gravitational collapse scenario. This would be a novel contribution to BH thermodynamics.