Abstract:We analyze the problem of defining the black hole entropy when Chern-Simons terms are present in the action. Extending previous works, we define a general procedure, valid in any odd dimensions both for purely gravitational CS terms and for mixed gaugegravitational ones. The final formula is very similar to Wald's original formula valid for covariant actions, with a significant modification. Notwithstanding an apparent violation of covariance we argue that the entropy formula is indeed covariant.

The role of the observers is frequently obscured in the literature, either by writing equations in a coordinate system implicitly pertaining to some specific observer or by entangling the invariance and the observer dependence of physical quantities. Using examples in relativistic kinematics and classical electrodynamics, we clarify the confusion underlying these misconceptions.

We present a direct, geometric derivation of the generalized Smarr formula for the stationary axially symmetric black holes with nonlinear electromagnetic fields. The additional term is proven to be proportional to the integral of the trace of the electromagnetic energy-momentum tensor and can be written as a product of two conjugate variables. From the novel relation we can deduce all previously proposed forms of the generalized Smarr formula, which were derived only for the spherically symmetric black holes, and provide the lowest order quantum correction to the classical relation from the Euler-Heisenberg Lagrangian. We use adjective "geometric" just to emphasize that in these derivations auxiliary physical assumptions are avoided by the approach based on the quantities of geometric origin.

We analyse near-horizon solutions and compare the results for the black hole entropy of five-dimensional spherically symmetric extremal black holes when the N = 2 SUGRA actions are supplied with two different types of higher-order corrections: (1) supersymmetric completion of gravitational Chern-Simons term, and (2) Gauss-Bonnet term. We show that for large BPS black holes lowest order α ′ corrections to the entropy are the same, but for non-BPS are generally different. We pay special attention to the class of prepotentials connected with K3 × T 2 and T 6 compactifications. For supersymmetric correction we find beside BPS also a set of non-BPS solutions. In the particular case of T 6 compactification (equivalent to the heterotic string on T 4 × S 1 ) we find the (almost) complete set of solutions (with exception of some non-BPS small black holes), and show that entropy of small black holes is different from statistical entropy obtained by counting of microstates of heterotic string theory. We also find complete set of solutions for K3 × T 2 and T 6 case when correction is given by Gauss-Bonnet term. Contrary to four-dimensional case, obtained entropy is different from the one with supersymmetric correction. We show that in Gauss-Bonnet case entropy of small "BPS" black holes agrees with microscopic entropy in the known cases.