The Bekenstein-Hawking entropy of black holes in Einstein’s theory of gravity is equal to a quarter of the horizon area in units of Newton’s constant. Wald has proposed that in general theories of gravity the entropy of stationary black holes with bifurcate Killing horizons is a Noether charge which is in general different from the Bekenstein-Hawking entropy. We show that the Noether charge entropy is equal to a quarter of the horizon area in units of the effective gravitational coupling on the horizon defined by the coefficient of the kinetic term of a specific metric perturbation polarization on the horizon. We present several explicit examples of static spherically symmetric black holes.

Supernova searches have been been suggested as a method for determining precisely the current value and time variation of the equation of state, $w$, of the dark energy component responsible for the accelerated expansion of the Universe. We show that the method is fundamentally limited by the fact that luminosity distance depends on $w$ through a multiple integral relation that smears out information about $w$ and its time variation. The effect degrades the resolution of $w$ that can be obtained from current data.

We explore various pitfalls and challenges in determining the equation of state $\mathrm{}\left(w\right)\mathrm{}$ of the dark energy component that dominates the universe and causes the current accelerated expansion. We demonstrated in an earlier paper the existence of a degeneracy that makes it impossible to resolve well the value of w or its time derivative with supernovae data. Here we consider standard practices, such as assuming that w is constant or greater than $-1,$ and show that they also can lead to gross errors in estimating the true equation of state. We further consider combining measurements of the cosmic microwave background anisotropy and the Alcock-Paczynski test with supernovae data and find that the improvement in resolving the time derivative of w is marginal, although the combination can constrain its present value perhaps to 20% uncertainty.