# Growth of a black hole on a self-gravitating radiation

## Abstract

We feed a black hole on a self-gravitating radiation and observe what happens during the process. Considering a spherical shell of radiation, we show that the contribution of self-gravity makes the thermodynamic interaction through the bottom of the shell be distinguished from thermodynamic interaction through its top. The growth of a black hole horizon appears to be a sudden jump rather than a sequential increase. We additionally show that much of the entropy will be absorbed into the black hole only at the last moment of the collapse.

How does a black hole grow and what happens when one injects substances into it? What is the minimal reaction of the gravity+matter system in this situation? Although gravity collapse has been studied extensively, gravity collapse is a nonlinear and highly dynamic process, so even in classical gravity theory, this question is far from an analytical answer. The worse, the gravitational collapse is also believed to be closely related with the unknown quantum gravitational phenomena. One of the main difficulties is the uncontrollable nonlinear nature of the gravitational collapse. To circumvent this difficulty in this work, we design a quasi-static injection process in a black hole, which consider the self-gravity of matters. By considering the quasi-static process, we convert the uncontrollable non-linear dynamics to a controllable one. Evidently, a quasi-static process will minimize the entropy incremental. In this sense, the process corresponds to the minimal reaction of the gravity.

A remaining task is to build the quasi-static self-gravitating system around a black hole. We introduce a self-gravitating radiation bounded by two (artificial) walls located at \(r= r_+\) and \(r_-\). We, then, put part of the sphere into the black hole in succession until all the energy enters into the black hole. In the mean while, we keep track of the records happening on the black hole and on the matters surrounding it. Such a self-gravitating system confined in a shell was studied recently in Ref. [1], which model is a generalization of the well-known system originally introduced by Sorkin et al. [2] in 1981 as a spherically symmetric solution which maximizes entropy. A self-gravitating isothermal sphere was dealt by Schmidt and Homann [3] where they called the geometry a ‘photon star’. The heat capacity and the stability of the solution were further analyzed in Refs. [4, 5, 6, 7]. The structures of spherically-symmetric self-gravitating system were further studied in diverse area [8, 9, 10, 11, 12, 13]. An analytic approximation was tried in Ref. [14] in analogy with the situation that a blackhole is in equilibrium with the radiations. Recently, a complete analysis and the classification of the solutions were done in Ref. [15]. A star solution may have a regular or a conically-singular center. In Ref. [16], the solution was argued to have an ‘approximate horizon’, which mimics an apparent horizon [15].

In other point of view, the model can be regarded as a self-gravitating generalization of the brick wall model, which was introduced by t’Hooft [17] to explain the area proportionality of the black hole entropy. The original brick wall model is not very successful because it requires an ultra-violet cutoff. Reasons for this failure are the ignorance of quantum gravity and disregard of the self-gravity of matter. For the former, there is no concrete answer but various speculations. For the latter, correct manipulation of general relativity for the matters would do the role, which is the purpose of the present work. First, we summarize the solution of self-gravitating radiations. Then, we write down some of new aspects of the system especially when the radiation interacts with the inside. By using the results, we study the growth of a black hole on a self-gravitating radiation.

*T*(

*r*) represents the local temperature of the radiations at

*r*and

*u*and

*v*,

*u*,

*v*) plane numerically. None of the solution curves pass the line \(v=0\) and \(u=1\). Therefore, physically relevant regions are restricted to \(v \ge 0\) and \(u \le 1\). A solution curve \(C_\nu \) is characterized by the orthogonal distance of the curve from the line \(u=1\),

*H*to denote the fact that the corresponding value is evaluated on an ‘approximate horizon’ defined in Refs. [1, 16]. A given solution curve is parameterized by a logarithmic radial coordinate

*u*,

*v*) plane can be equivalently coordinated by using the set \((\nu , \xi )\). Eventually, an isothermal sphere of radiation of radius \(r_+\) is determined by choosing a point \((u_+,v_+)\) on a curve \(C_\nu \).

*u*,

*v*) follows the equations of motionf, the variations at the outer wall \((\delta u_+, \delta v_+)\) must be related with those at the inner wall \((\delta u_-,\delta v_-)\). Even though one cannot obtain the solution curve analytically in general, the variations at the inner wall are related with those at the outer wall:

*u*and

*v*. To simplify the equation, we have modified the definition of

*B*from that in Ref. [1]. The inverse equation can be obtained once we perform \(+ \leftrightarrow -\).

*inward temperature*to distinguish it from the ordinary temperature \(\beta _+^{-1}\).

*inward temperature*? First, for a given mass transfer \(\delta M_{\mathrm{rad}}\) through the inner wall, the entropy change of the system may not be the same as the expectation of the outside observer because of Eq. (13) and its ordinary analogue,

*inward heat capacity*is defined by

*inward heat capacity*is related with \({\mathcal {V}}\) by \(\beta _- C_{V-}^{-1} = (T_- {\mathcal {V}}_-)^{-1} +(r_- \chi _-^2)^{-1}.\) Therefore, we find

^{1}

*S*satisfying \(du/dv = 1\), where \(v_S \approx \varepsilon ^{2/3}/2 \). Note that \(r_- \rightarrow r_H\) when \(v_- \rightarrow 1/2\). The temperature of the system is given by \(\beta _+^{-1} = (8\pi \sigma )^{-1/4} \sqrt{\varepsilon /r_H}\). The mass of the radiation is

*inward heat capacity*is positive definite and is independent of the physical quantities at the outer wall. On the other hand, the heat capacity for the interactions through the outer wall in Eq. (18) becomes

*inward temperature*and the ordinary temperature:

*inward temperature*is higher than the ordinary temperature. Now, by using \(S_\pm =r_\pm \beta _\pm /3 \times ( 2v_\pm /3+ u_\pm ) \) and Eqs. (19), (21), and (23), the entropy of the radiation becomes

*u*,

*v*, and \(\beta \) represent the (intermediate) values at the inner wall. Instead, \(u_-\), \(v_-\), and \(\beta _-\) represent the initial values before the collapse. A drop of \(\delta M\) reduces the mass of the radiation to \(M_{\mathrm{rad}} - \delta M\) and increases the mass inside the inner wall to \(M+ \delta M\). From the definition of the

*inward heat capacity*(15) and Eq. (22), the variation of

*inward temperature*satisfies

*u*tends to increase but

*v*tends to recede from \(v= 1/2\) during the collapse.

*u*arrives at one, which implies the formation of a black hole. Setting \(v(\alpha ) = v_-\), the temperature at the time of \(\alpha \) fraction fallen can be found from Eq. (25):

In summary, we have studied the (classical) growth of a black hole by absorbing a spherical shell of self-gravitating radiations. We have shown two important facts. First, the *inward temperature* describing thermodynamic interactions of the shell with the interior is higher than that with the outside, where the details of the interior are not important. The *inward temperature* is different from the Tolman temperature and reflects the effect of self-gravity of the radiation and the curvature effect. For a given energy transfer from the radiation, the information transmission through the inner wall is suppressed by the ratio between the ordinary to *inward temperature*. Second, during the gravitational collapse, the temperature of the radiation monotonically decreases to zero. At the time the temperature vanishes the horizon forms. Because of this, much information will be absorbed into the black hole only at the last moment of the collapse.

From the experience of the brick wall model, one usually expect that the density of particles at the bottom diverges when it hits the surface of a black hole. Such case corresponds to the \(v_-\rightarrow \infty \) limit of the present case. For other cases, the density at the bottom takes a finite value even at the end of the collapse. A biggest difference from the brick wall model is that the present model includes the self-gravity of the radiations rather than considering only the background gravity of a black hole. Notice that the density varies greatly at the thin region around \(r_-\sim 2M_-\). As given in Ref. [15], the solution curve follows Eq. (19) for \(\varepsilon ^{2/3}< v_-< \varepsilon ^{-2/3}\). In this range, the radius is changed only by \(O(\varepsilon ^{2/3})\). There is no curvature singularity too because the curvature components at the bottom of the box must be in order of \(O(v_-/r_-^2)\).

*O*(1) number for all \(v_-\). The term in the square bracket is, by using \(S_{\mathrm{bh}} = 4\pi M_{\mathrm{bh}}^2\), nothing but the ratio of the Hawking temperature relative to the radiation’s

*inward temperature*: \(T_H/\beta _-^{-1}\). This implies that when the

*inward temperature*of the radiation is close to the Hawking’s, the quantum effect should be taken into account. An important difference from the existing expectation is that the

*inward temperature*is a main concern rather than the ordinary temperature. The ordinary temperature can be lower than the Hawking’s because of Eq. (23).

A problem caused by considering the quantum effect is that the Hawking radiation can interfere with the final absorption process, where the temperature goes to zero. However, as noticed in Ref. [16], the self-gravitating radiation cannot be continuously connected to a static black hole unless the Einstein field equation breaks down on a macroscopic near-horizon shell. Let us assume that the quantum nature is described by a field \(\phi \). In the zero temperature limit, the kinetic energy term will goes to zero which implies \({\dot{\phi }} \rightarrow 0\). Then, the field may take the form of a static non-radiating field similar to that of the Coulomb field. In this case, the radial pressure becomes negative of the density and a natural stable static configuration of matters with a black hole exists as shown in Ref. [19].

## Footnotes

## Notes

### Acknowledgements

This work was supported by Korea National University of Transportation 2017.

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