Energy decomposition within Einstein-Born-Infeld black holes

We analyze the consequences of the recently found generalization of the Christodoulou-Ruffini black hole mass decomposition for Einstein-Born-Infeld black holes [characterized by the parameters $(Q,M,b)$, where $M=M(M_{\text{irr}},Q,b)$, $b$ scale field, $Q$ charge, $M_{\text{irr}}$ “irreducible mass,” physically meaning the energy of a black hole when its charge is null] and their interactions. We show in this context that their description is largely simplified and can basically be split into two families depending upon the parameter $b|Q|$. If $b|Q|\le 1/2$, then black holes could have even zero irreducible masses and they always exhibit single nondegenerated horizons. If $b|Q|>1/2$, then an associated black hole must have a minimum irreducible mass (related to its minimum energy) and has two horizons up to a transitional irreducible mass. For larger irreducible masses, single horizon structures raise again. By assuming that black holes emit thermal uncharged scalar particles, we further show in light of the black hole mass decomposition that one satisfying $b|Q|>1/2$ takes an infinite amount of time to reach the zero temperature, settling down exactly at its minimum energy. Finally, we argue that depending on the fundamental parameter $b$, the radiation (electromagnetic and gravitational) coming from Einstein-Born-Infeld black holes could differ significantly from Einstein-Maxwell ones. Hence, it could be used to assess such a parameter.

Figure 1

Mass formula (thick plus dotted curves), Eq. (16), when the parameter $b|Q|$ satisfies Eq. (17), chosen here as 2. The dashed curve represents $b{M}_0$, as given by Eq. (14). The dot-dashed curve is the asymptote to $M$, $M_{\text{irr}}$. Besides, $bM$ exhibits a minimum at the critical point $M_{\text{irr}}^{c}b\approx 0.97$ (where the horizons become degenerated) and for $M_{\text{irr}}^{c}b\le M_{\text{irr}}b<M_{\text{irr}}^{t}b\approx 3.18$, we have the range of irreducible masses that generalize Reissner-Nordström black holes. For $M_{\text{irr}}\ge M_{\text{irr}}^t$, there is a sole horizon (not degenerated), whose radial coordinate inside of it is always spacelike. The irreducible masses associated with the outer horizon are $M_{\text{irr}}\ge M_{\text{irr}}^c$. The dotted curve is related to the inner horizon solutions (for given configurations) and is not relevant to the analyses concerning the black hole mass decomposition.

Figure 2

Mass decomposition when the parameter $b|Q|$ does not satisfy Eq. (17) and is chosen to be 0.4. The curves have the same meaning as the ones in Fig. 1. From the solid curve we see that $M$ is a monotonic function and always larger than $M_0$. This means that such a case characterizes a scenario where there is always a sole event horizon and there is no classical analogue to it.

Figure 3

Mass formula for selected values of the parameter $b|Q|$ (numbers on the curves) that encompasses all physically distinct classes of black holes for the Born-Infeld Lagrangian. The dotted curve represents the mass formula for the Maxwell Lagrangian. The dot-dashed curve demarcates the transition from two horizon solutions (as given by the thick curve) to a single one (as given by the dashed curve), where its associated inner horizon is null. The branches related to the inner horizons were removed.

Figure 4

Einstein-Born-Infeld black hole temperature as a function of the irreducible mass for selected values of the parameter $b|Q|$. The temperature goes to infinity as the irreducible mass tends to zero whenever $2b|Q|\le 1$ (thick curve). Whenever $2b|Q|>1$ (dashed curve), it decreases with the decrease of the irreducible mass (keeping the charge constant), always being null for a finite value of the latter. The temperature experiences a transitional behavior for $2b|Q|=1$ (dot-dashed curve), being null just when the irreducible mass of the system is so [see Eq. (20)], albeit it cannot be seen directly from this plot. Finally, $b|Q|\to \infty$ (dotted curve) corresponds to the Reissner-Nordström case.

Figure 5

Total radiation (gravitational plus electromagnetic) $W_{\text{rad}}/Q$ released in the process of coalescence of two identical Einstein-Born-Infeld black holes with $\alpha =0.95$ under the assumption it leads to another one of the same type with the same parameters as their classical counterparts. The thick curve represents such a case. The dashed curve stands for the radiation encountered in the Einstein-Maxwell theory, $W_{\text{rad}(\text{clas})}/Q$. The associated radiation tends to its classical counterpart when $bQ$ goes to infinity. The energy released in the case of nonlinear black hole interaction is always larger than the one coming from its classical counterpart, for a given charge $Q$.