Black hole interior mass formula
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Abstract
We argue by explicit computations that, although the area product, horizon radii product, entropy product, and irreducible mass product of the event horizon and Cauchy horizon are universal, the surface gravity product, the surface temperature product and the Komar energy product of the said horizons do not seem to be universal for Kerr–Newman black hole spacetimes. We show the black hole mass formula on the Cauchy horizon following the seminal work by Smarr [Phys Rev Lett 30:71 (1973), Phys Rev D 7:289 (1973)] for the outer horizon. We also prescribe the four laws of black hole mechanics for the inner horizon. A new definition of the extremal limit of a black hole is discussed.
Keywords
Black Hole Event Horizon Entropy Product Black Hole Mass Outer Horizon1 Introduction
An intriguing feature of stationary axially symmetric black holes is that the product of the horizon areas are often independent of the mass of the black hole. Rather such products depend on the charge and angular momentum of the black hole. They may also be formulated in terms of the proper radii of the Cauchy horizon and event horizon.
It is also known that every regular axisymmetric and stationary spacetime of an Einstein–Maxwell system with surrounding matter has a regular Cauchy horizon (\(\mathcal{H}^\)) inside the event horizon (\(\mathcal{H}^+\)) if and only if the angular momentum \(J\) and charge \(Q\) do not both vanish. In contrast, the Cauchy horizon becomes singular and approaches a curvature singularity in the limit \(J\rightarrow 0\), \(Q \rightarrow 0\) [3, 4, 5].
The fact is that the Cauchy horizon is an “infinite blueshift” region and classically unstable due to the linear perturbation. Thus when an observer crosses the Cauchy horizon \(r=r_{}\), he/she observes all of the events which occur at “RegionI” and also the electromagnetic and gravitational field oscillations at infinite frequency are seen which actually occur at finite frequency in “RegionI” [14].

We prove in Sect. 2, like the area and entropy product, that the surface gravity product, surface temperature or black hole temperature product and the Komar energy product of both inner horizon and outer horizon do not show any global properties due to the mass dependence. Such products are not universal in nature.
 We explicitly show in Sect. 3 that the black hole mass or ADM mass can be expressed in terms of the area of the Cauchy horizon \((\mathcal{H}^{})\):and we prove that the mass can be expressed as a sum of the surface energy, the rotational energy, and the electromagnetic energy of the Cauchy horizon \((\mathcal{H}^{})\):$$\begin{aligned} \mathcal{M}^2&= \frac{\mathcal{A}_{}}{16\pi }+\frac{4\pi J^2}{\mathcal{A}_{}} +\frac{Q^2}{2}+\frac{\pi Q^4}{\mathcal{A}_{}} , \end{aligned}$$(4)$$\begin{aligned} \mathcal{M}&= \mathcal{E}_{s}+ \mathcal{E}_{r} + \mathcal{E}_{em}. \end{aligned}$$(5)
 Also we find in Sect. 4 that the Christodoulou–Ruffini [15] mass formula may be expressed in terms of the area of the Cauchy horizon \((\mathcal{H}^{})\):$$\begin{aligned} \mathcal{M}^2&= \left( \mathcal{M}_{\mathrm{irr} }+\frac{Q^2}{4 \mathcal{M}_{\mathrm{irr} } }\right) ^2+\frac{J^2}{4 (\mathcal{M}_{\mathrm{irr} })^2}. \end{aligned}$$(6)

We further investigate the laws of black hole mechanics for the inner horizon in Sect. 5.
 We also point out that the product of Christodoulou’s irreducible mass of the inner horizon (Cauchy horizon) and the outer horizon (event horizon) are independent of the mass, i.e.,$$\begin{aligned} \mathcal{M}_{\mathrm{irr}+} \mathcal{M}_{\mathrm{irr}}&= \sqrt{\frac{\mathcal{A}_{+} \mathcal{A}_{}}{16\pi }} = \sqrt{\frac{J^2+\frac{Q^4}{4}}{4}}. \end{aligned}$$(7)

We also shortly derive the identity \(K_{\chi ^{\mu }}=2\mathcal{S}_{} T_{}\) on the Cauchy horizon in Sect. 6.

The entropy of the Cauchy horizon may be expressed in the form \(\mathcal{S}_{}=\frac{E_{}}{2T_{}}\) as described in Sect. 7.
2 Charged rotating black hole
3 Smarr formula for Cauchy horizon (\(\mathcal{H}^{}\))
4 Christodoulou’s irreducible mass for Cauchy horizon
A reversible process is characterized by an unchanged irreducible mass, whereas an irreversible process is characterized by an increase in irreducible mass of a black hole. It should be noted that there exists no process which will decrease the \(\mathcal{M}_{\mathrm{irr}}\) for a Cauchy horizon.
5 The four laws of black hole mechanics on the event horizon \((\mathcal{H}^{+})\) and Cauchy horizon \((\mathcal{H}^{})\)

The Zeroth Law: The surface gravity, \(\kappa _{\pm }\) of a stationary black hole is constant over both the event horizon (\(\mathcal{H }^{+}\)) and the Cauchy horizon (\(\mathcal{H }^{}\)), respectively.
 The First Law: Any perturbation of a stationary black holes, a change of mass (change of energy), is related to a change of mass, angular momentum, and electric charge byIt can be seen that \(\frac{{\kappa }_{\pm }}{8\pi }\) is analogous to the temperature of \(\mathcal{H}^{\pm }\) in the same way as \(\mathcal{A}_{\pm }\) is analogous to entropy. It should be noted that \(\frac{{\kappa }_{\pm }}{8\pi }\) and \(\mathcal{A}_{\pm }\) are distinct from the temperature and entropy of the black hole. The above expression \(\frac{{\kappa }_{\pm }}{8\pi }\) can be derived from Eq. (35) in the following way. The effective surface tension can be rewritten as$$\begin{aligned} \mathrm{d}\mathcal{M}&= \frac{{\kappa }_{\pm }}{8\pi } \mathrm{d}\mathcal{A}_{\pm } + \Omega _{\pm } \mathrm{d}J +\Phi _{\pm }\mathrm{d}Q. \end{aligned}$$(75)and$$\begin{aligned} \mathcal{T}_{\pm }&= \frac{{\kappa }_{\pm }}{8\pi } = \frac{\partial \mathcal{M}}{\partial \mathcal{A}_{\pm }} \end{aligned}$$(76)$$\begin{aligned} \Omega _{\pm }&= \frac{4\pi J}{\mathcal{M}\mathcal{A}_{\pm }}=\frac{\partial \mathcal{M}}{\partial J} \end{aligned}$$(77)$$\begin{aligned} \Phi _{\pm }&= \frac{1}{\mathcal{M}} \left( \frac{Q}{2}+\frac{2\pi Q^3}{\mathcal{A}_{\pm }} \right) = \frac{\partial \mathcal{M}}{\partial Q}. \end{aligned}$$(78)
 The Second Law: The area \(\mathcal{A}_{\pm }\) of both event horizon \((\mathcal{H}^{+})\) and Cauchy horizon \((\mathcal{H}^{})\) never decreases, i.e.or$$\begin{aligned} \mathrm{d}\mathcal{A}_{\pm }&= \frac{4 \mathcal{A}_{\pm }}{r_{\pm }r_{\mp }} \left( \mathrm{d}\mathcal{M}\varvec{\Omega }_{\pm }.\mathrm{d}\varvec{J} \Phi _{\pm } \mathrm{d}Q \right) \ge 0 \end{aligned}$$(79)The change in irreducible mass of both event horizon \((\mathcal{H}^{+})\) and Cauchy horizon \((\mathcal{H}^{})\) can never be negative. It follows immediately from the above equation that$$\begin{aligned} \mathrm{d}\mathcal{M}_{\mathrm{irr}\pm }&= \frac{2 \mathcal{M}_{\mathrm{irr}\pm }}{r_{\pm }\!\!r_{\mp }} \left( \mathrm{d}\mathcal{M}\!\!\varvec{\Omega }_{\pm }.\mathrm{d}\varvec{J} \!\!\Phi _{\pm } \mathrm{d}Q \right) \!\ge \! 0. \end{aligned}$$(80)$$\begin{aligned} \mathrm{d}\mathcal{M} > \varvec{\Omega }_{\pm }.\mathrm{d}\varvec{J}+\Phi _{\pm } \mathrm{d}Q . \end{aligned}$$(81)

The Third Law: It is impossible by any mechanism, no matter how idealized, to reduce \(\kappa _{\pm }\), the surface gravity of both the event horizon \((\mathcal{H}^{+})\) and the Cauchy horizon \((\mathcal{H}^{})\), to zero by a finite sequence of operations.
6 \(\hbox {Komar conserved quantity}=2\times \hbox {entropy on }\mathcal{H}^{\pm }\times \text{ temperature } \text{ on }\,\,\mathcal{H}^{\pm } \) or \(K_{\chi ^{\mu }\pm }=2\mathcal{S}_{\pm } T_{\pm }\)
It is well known [27] that on the \(\mathcal{H}^{+}\) the Komar conserved quantity \((K_{\chi ^{\mu }+})\) corresponding to a null Killing vector \({\chi ^{\mu }}_+\) is equal to twice the product of the entropy (\(\mathcal{S}_{+}\)) on \(\mathcal{H}^{+}\) and the temperature (\(T_{+}\)) on \(\mathcal{H}^{+}\). Here we shall derive a similar expression to the one that holds on \(\mathcal{H}^{}\). Thus we have to prove the identity \(K_{\chi ^{\mu }}=2 \mathcal{S}_{} T_{}\) on the \(\mathcal{H}^{}\).
7 Generalized Smarr formula for mass on the Cauchy horizon
8 Degenerate black hole or extremal black hole
It may be noted that the Komar energy goes to zero at the extremal limit, i.e., \(E_{+}=E_{}=0\). This may imply that this is another way of seeing the discontinuity between the extremal spacetime and nonextremal spacetime.
9 Discussions
In this work, we have derived the Smarr formula on the Cauchy horizon \((\mathcal{H}^{})\). We have proposed the four laws of black hole mechanics for the inner horizon \((\mathcal{H}^{})\). We have found, in contrast to some earlier work [10, 30], particularly for the first law of the inner horizon \((\mathcal{H}^{})\), complete consistency between our results and these results.
We have also demonstrated that the area product, horizon radii product, entropy product, and irreducible mass product of the event horizon and Cauchy horizon are universal, although the surface gravity product, surface temperature product, and the Komar energy product are not universal for a Kerr–Newman black hole.
We have also defined the Christodoulou and Ruffini mass on the Cauchy horizon. We have further showed that the identity \(K_{\chi ^{\mu }}=2 \mathcal{S}_{} T_{}\) is valid on the inner horizon (\(\mathcal{H}^{}\)) and also the Komar energy in a compact form \(E_{}=2 \mathcal{S}_{} T_{}\). This also relates the generalized Smarr formula \( E_{}= \mathcal{M}2J\Omega _{}QV_{}\) on \(\mathcal{H}^{}\).
Another interesting point we have found is that the Komar energy goes to zero at the extremal limit, which may display a discontinuity between extremal spacetime and nonextremal spacetime.
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