A note on Smarr relation and coupling constants

Abstract

The Smarr relation plays an important role in black hole thermodynamics. It is often claimed that the Smarr relation can be written down simply by observing the scaling behavior of the various thermodynamical quantities. We point out that this is not necessarily so in the presence of dimensionful coupling constants, and discuss the issues involving the identification of thermodynamical variables. I. SMARR RELATION AND THE FIRST LAWThe fact that black holes behave like a thermodynamical system has dramatically changed our understanding of black holes ever since its conception in 1973 [1]. For an asymptotically flat Kerr-Newman black hole, the first law of black hole mechanics takes the formwhere M denotes the ADM mass of the black hole, S its Bekenstein-Hawking entropy, T its Hawking temperature, Q its electrical charge and J its angular momentum. The first law thus relates the various differential quantities. In some applications, one would like to work directly with the black hole parameters instead of their differentials. Fortunately, there is the Smarr relation [2]:where Φ denotes the electrical potential, while Ω denotes the angular velocity of the black hole. Smarr relations such as this have been widely studied in the literature, beyond the Kerr-Newman family. A good rule of thumb for writing down the Smarr relation for a given black hole is to look at the scaling (i.e. the dimensions) of the various thermodynamical quantities. See, e.g., Sec.2 of [3]. For example, in 4-dimensions, and in the units = k B = c = 1, we have M, Q ∝ L, and J, S ∝ L 2 , where L is a length scale. Due to Euler's theorem of quasi-homogeneous function (see below), we can simply write down M = M (S, Q, J) asFrom the first law (and the chain rule), one could identify the various partial derivatives and arrive at the Smarr relation, Eq.(2). Similarly, in the extended black hole

Black holes in double-Logarithmic nonlinear electrodynamics

“…Due to this property, the magnetic black hole solution is analyzed in more detail. The first law of thermodynamics, Hawking radiation, the extension of Smarr's formula and thermal stability are worked out for the magnetic black hole solution [59][60][61][62][63][64][65].…”

Black Holes in Double-Logarithmic Nonlinear Electrodynamics

“…Due to this property, the magnetic black hole solution is analyzed in more detail. The first law of thermodynamics, Hawking radiation, the extension of Smarr's formula and thermal stability are worked out for the magnetic black hole solution [59][60][61][62][63][64].…”

Energy dependent topological Black holes in Massive gravity coupled to Double-Logarithmic Electrodynamics

“…Furthermore, it is also shown that first law is satisifed and the extension of Smarr's relation in the present scenerio of massive gravity's rainbow coupled to double-Logarithmic electrodynamics is also constructed. [143][144][145][146][147][148].…”