Angular Momentum Fluctuations

TL;DR

This paper investigates angular momentum fluctuations of near-static Schwarzschild black holes in thermal equilibrium, revealing that fluctuations originate from both the black hole and surrounding radiation, with implications for quantum area quantization.

Contribution

It introduces a semi-classical analysis of black hole angular momentum fluctuations considering small but non-zero angular momentum, linking fluctuations to horizon area and radiation effects.

Findings

Radiation contribution depends on the cutoff distance from the horizon.

Black hole fluctuations are proportional to the horizon area.

Results suggest a new quantization rule for the event horizon area.

Abstract

In this paper, we consider angular momentum fluctuations of a Schwartzschild black hole in thermal equilibrium with radiation which, for the sake of simplicity is here modeled by a scalar field. Important, we do not set the black hole angular momentum $J$ identically to zero at the outset; we allow it to have a small value (in the sense that $J/M<<1$) and then study the conditions for thermodynamical equilibrium; only then take the $J\rightarrow 0$ limit. We calculate the black hole's angular momentum fluctuations which turn out to have two independent contributions: one that comes from the black hole itself, with no respect to the radiation, and another one that arises from the radiation. The result is astonishingly simple: the radiation contribution depends exclusively on the cut-off proper distance from the horizon (or equivalently, the width of the brick wall), while the black hole contribution is proportional to its event horizon area. Accordingly, there are no strictly static black holes in nature, they randomly rotate in all possible directions. Since a black hole is nothing but geometry, we are dealing with geometry fluctuations -- our results are of quantum-gravitational nature (albeit at a semi-classical level). Interestingly enough, if we apply to the black hole fluctuations component the (quantum) rules of angular momentum we obtain an event horizon area quantization rule, albeit with a different spectrum from an equally spaced area spectrum which is widely accepted in the literature.

Equations

$$g_{tt}$$

$$g_{\phi \phi }$$

$$g_{t\phi }$$

$$g_{\theta \theta }$$

$$g_{rr}$$

$$\Sigma$$

$$\Delta$$

$$r_{+}=M+\sqrt{M^2-a^2}$$

$$A=8\pi M{r}_{+}.$$

$$\mathcal{E}$$

$$\mathcal{J}$$

$$\beta F=\sum _m\int \sqrt{-g}drd\theta d\phi \int n(ϵ,m,r,\theta )\ln \left(1-e^{-\beta (ϵ-\hslash m\omega )}\right)dϵ.$$

$$J-\left(\frac{\partial F}{\partial \omega }\right)=0.$$

$$\mathcal{E}=M+\omega J+\left(\frac{\partial (\beta F)}{\partial \beta }\right).$$

$$\mathcal{S}=\frac{2\pi M{r}_{+}}{\hslash }+{\beta }^2\left(\frac{\partial F}{\partial \beta }\right).$$

$$F=-\sum _m\int \int \sqrt{-g}drd\theta d\phi \frac{\gamma (ϵ,m,r,\theta )}{e^{\beta (ϵ-m\hslash \omega )}-1}dϵ,$$

$${\tilde{g}}_{tt}$$

$${\tilde{g}}_{t\phi }$$

$$\frac{1}{\sqrt{-\tilde{g}}}{\partial }_a\left({\tilde{g}}^{ab}\sqrt{-\tilde{g}}{\partial }_b\Phi \right)-\xi R\Phi =0,$$

$${\widetilde{ϵ}}^2\frac{{\tilde{g}}_{\tilde{\phi }\tilde{\phi }}}{D}+2\frac{{\tilde{g}}_{\tilde{\phi }t}\widetilde{ϵ}}{D}m-m^2\frac{(-{\tilde{g}}_{tt})}{D}-\left[\frac{1}{g_{rr}}{p}_r^2+\frac{1}{g_{\theta \theta }}{p}_{\theta }^2\right]=0,$$

$$\gamma (\widetilde{ϵ},m)=\frac{1}{(2\pi \hslash {)}^3}\int d\tilde{\phi }d{p}_{\theta }d\theta d{p}_{r}dr.$$

$$\gamma (\widetilde{ϵ},m)=\frac{1}{(2{\pi }^2\hslash {)}^3}\int d\tilde{\phi }d{p}_{\theta }d\theta dr\sqrt{g_{rr}}{\left[\frac{{\tilde{g}}_{\tilde{\phi }\tilde{\phi }}{\widetilde{ϵ}}^2+2{\tilde{g}}_{t\tilde{\phi }}m\widetilde{ϵ}-m^2(-{\tilde{g}}_{tt})}{D}-\frac{p_{\theta }^2}{g_{\theta \theta }}\right]}^{1/2}.$$

$$\gamma (\widetilde{ϵ},m)=\frac{1}{16{\pi }^2{\hslash }^3}\int d\theta dr\frac{\sqrt{g_{rr}{g}_{\theta \theta }}}{D}\left({\tilde{g}}_{\tilde{\phi }\tilde{\phi }}{\widetilde{ϵ}}^2+2{\tilde{g}}_{t\tilde{\phi }}m\widetilde{ϵ}-m^2(-{\tilde{g}}_{tt})\right).$$

$$\gamma (\widetilde{ϵ})=\frac{1}{12{\pi }^2{\hslash }^3}\int \sqrt{-g}d\tilde{\phi }d\theta dr\frac{1}{(-{\tilde{g}}_{tt}{)}^2}{\widetilde{ϵ}}^3,$$

$$\gamma (\widetilde{ϵ},r,\theta )=\frac{1}{12{\pi }^2{\hslash }^3(-{\tilde{g}}_{tt}{)}^2}{\widetilde{ϵ}}^3,$$

$$F=-\frac{{\pi }^2}{120{\hslash }^3{\beta }^4}V,$$

$$V={\int }_{r_{+}+h}^{R}dr\int d\theta \frac{\sqrt{-g}}{{\tilde{g}}_{tt}^2},$$

$${\tilde{g}}_{tt}=g_{tt}+({\omega }^2-2\omega \Omega )g_{\phi \phi }.$$

$$\mathcal{E}=M+\omega J+\frac{{\pi }^2}{40{\hslash }^3{\beta }^4}V$$

$$\frac{30{\hslash }^3{\beta }^4}{{\pi }^2}J+{\int }_{r_{+}+h}^{R}dr\int d\theta \frac{\sqrt{-g}}{(-{\tilde{g}}_{tt}{)}^3}{g}_{t\phi }+\omega {\int }_{r_{+}+h}^{R}dr\int d\theta \frac{\sqrt{-g}}{(-{\tilde{g}}_{tt}{)}^3}{g}_{\phi \phi }=0.$$

$$\mathcal{S}=\frac{4\pi M{r}_{+}}{\hslash }+\frac{{\pi }^2}{30{\hslash }^3{\beta }^3}V.$$

$$\beta =\frac{4\pi M}{\hslash }\left(\frac{M}{\sqrt{M^2-J^2/M^2}}+1\right).$$

$$g_{t\phi }\approx -\frac{2J{\sin }^2\theta }{r},$$

$$\frac{30{\hslash }^3{\beta }_0^4}{{\pi }^2}J-2J\int d\theta {\sin }^3\theta {\int }_{2M+\delta }^R\frac{r^{4}dr}{(r-2M{)}^3}+\omega \int d\theta {\sin }^3\theta {\int }_{2M+\delta }^R\frac{r^{7}dr}{(r-2M{)}^3}=0.$$

Full text

PACS numbers : 04.70Dy, 05.40.-a, 05.70-a,52.25 Kn

I A static black hole in equilibrium with radiation

A rotating black hole of mass $M$ and angular momentum $J$ is described by the Kerr geometry. In Boyer-Lindquist coordinates the metric coefficients take the form:

[TABLE]

where

[TABLE]

with $a=J/M$. The black hole event horizon is located at

[TABLE]

and the corresponding area is

[TABLE]

In this geometry, particles are dragged with angular velocity $\Omega=-g_{\varphi t}/g_{\varphi\varphi}$. Now consider such a black hole in thermal equilibrium with a bath of scalar particles within a confining vessel of radius $R>>2M$, observed by an an observer rotating with a constant angular velocity $\omega$ . The need of a rotating system will become clear shortly. The total energy and angular momentum of the system are

[TABLE]

where $n(\epsilon,m,r,\theta)$ represents the density of states per unit volume with a fixed azimuthal quantum number $m$ in the rest frame. The free energy of the radiation is:

[TABLE]

We assume that the total angular momentum ${\cal J}$ vanishes, the black hole’s angular momentum fluctuations result from the absorption and emission of quanta from the radiation. The angular momentum conservation condition (eq.(11)) can be expressed in terms of the free energy as:

[TABLE]

Similary, the energy conservation condition (eq. (10) )reads landaumechanics:

[TABLE]

The total entropy is given by

[TABLE]

The first term in the last equation corresponds to the Bekenstein-Hawking entropy. Integrating eq.(12) by parts

[TABLE]

where $\gamma(\epsilon,m,r,\theta)$ is the total number of states (per unit volume) for a given energy and azimuthal number $m$, $n=d\gamma/d\epsilon$. In the rotating frame, the azimuthal angle is $\tilde{\varphi}=\varphi-\omega t$ and the metric elements reads

[TABLE]

all other metric coefficients remaining the same as the in the non-rotating frame. The energy measured in this frame is $\tilde{\epsilon}=\epsilon-m\hbar\omega$.

In what follows, for the sake of completeness we follow the discussion given by Chang-Young et all Chang. A massless scalar field satisfies the wave equation :

[TABLE]

where $\xi$ represents the coupling constant. Neglecting back-reaction of the geometry and a semi-classical approximation $\Phi=\Phi_{0}e^{i(-\tilde{\epsilon}t+m\tilde{\varphi}+S(r,\theta))/\hbar}$. Then, it follows that

[TABLE]

with $p_{r}=\frac{\partial S}{\partial r}$, $p_{\theta}=\frac{\partial S}{\partial\theta}$, $D=-g_{rr}\tilde{g}_{tt}+\tilde{g}_{t\tilde{\varphi}}^{2}$ and we inverted also the metric In the semi-classical approximation the number of states for a fixed value of $m$ in the rotating frame is the volume in phase space thooft-minho :

[TABLE]

Performing the immediate integrations over $p_{r}$ and $\tilde{\varphi}$, and inserting the value of $p_{r}(r,\theta)$ obtained from eq. (20)

[TABLE]

Then, integrating over the classically allowed region it follows that

[TABLE]

The total number of states for all $m$ is given by summing $\sum_{m}\gamma(\tilde{\epsilon},m)$ . Approximating the sum over $m$ by an by integral over the region where the integrand is positive yields

[TABLE]

where $-g=g_{rr}g_{\theta\theta}D$ is the determinant of the Kerr metric.

Inspecting this expression, it is easy to identify the density of states in the rotating frame.

[TABLE]

Inserting this density of states in eq.(16) and performing the Bose-Einstein-like integration , the free energy boils down to a simple result

[TABLE]

where we defined

[TABLE]

with

[TABLE]

Inspecting eqs. (26),(27) we notice that the relevant (inverse) temperature in the free function is is $\beta\sqrt{-g_{tt}}$ which is nothing but Tolman’s inverse temperature, the local temperature measured by the rotating observer tolman-mattvisser. Following t’Hooft, we introduced a cut-off $h$ at the horizon that can either represents a brick wall $\Phi(r_{+}+h)=0$ or our ignorance of how to properly renormalize the divergences at the horizon. The total energy reads

[TABLE]

where $\omega$ is to be regarded as a chemical potential that implements angular momentum conservation (eq.(13)):

[TABLE]

Equivalently $\omega$ can be regarded as the angular velocity of a rotating observer must have such that the total angular momentum vanishes in his frame.

At last, the total entropy reads

[TABLE]

The first derivatives $(\frac{\partial{\cal S}}{\partial M})_{J}=0$ together with the energy constraint (eq.(14)) gives the radiation temperature

[TABLE]

In order to study thermodynamical fluctuations of a Schwatzschild we need to expand the entropy up to the second order in $J$ (or $a$). At the lowest order in $J$:

[TABLE]

so the second term in eq.(30) is at least linear in $J$, and so must be also $\omega$. Solving this equation at the first order in $J$, we replace the Kerr metric coefficients by the Schwartzschild metric with the exception of $g_{t\varphi}$, which takes the above value. Then :

[TABLE]

were $\beta_{0}=8\pi M/\hbar$ . The density of states is highly peaked near the horizon Chang, so most of the contribution to the integral comes from the lower integration limit. After some algebra

[TABLE]

Notice that

[TABLE]

Expanding eq.(31) to the second order in $a$ is a bit more sweaty. Let $f(r,\theta)$ represent the integrand in eq.(27),then

[TABLE]

where $f_{0}(r,\theta),f_{2}(r,\theta)$ represent the zeroth and second order expansion in terms of $a^{2}$. Since

[TABLE]

and at the relevant order

[TABLE]

then

[TABLE]

At last,

[TABLE]

where

[TABLE]

The density of states is very very large near the horizon Chang; minho and we take only the contribution from the lower limit of the integral where the divergence occurs. Putting all the pieces together (eqs.(31,36,3742) the total entropy boils down to

[TABLE]

where ${\cal S}_{0}$ is the total entropy for $J=0$. Clearly the equilibrium condition $(\frac{\partial S}{\partial J})_{M}=0$ is satisfied identically. Angular momentum fluctuations are given by landaulifshitz :

[TABLE]

Keeping only the most divergent term and expressing the coordinate distance in terms of the proper distance $\delta=\Delta^{2}/8M$, we can write

[TABLE]

where these terms represent black hole and radiation fluctuations:

[TABLE]

This is an amazingly simple result. Part of the fluctuations has origin in the black hole’s quantum atmosphere, which is the (quantum) field within a small shell of proper-width $\Delta$ around the horizon. The black hole fluctuations are a bit mysterious. Mathematically it originates from the Bekenstein-Hawking entropy - thus being a true property of the black hole. Let us consider only this angular momentum fluctuation. Assume $\overline{m}=0$ , then, from the basic properties of angular momentum

[TABLE]

were we approximated the sum by an integral. Equating $\Delta J_{BH}^{2}=\hbar^{2}\Delta m^{2}$ , if follows that the event horizon is quantized:

[TABLE]

This result is at odds with the linear spacing area spectrum vindicated by most authors bhspectroscopy-dreyer. Anyway it is very surprising that the black hole area quantization results from the quantum rules of angular momentum.

II Concluding Remarks

Angular momentum fluctuations emerge from a thin shell of Planckian width and from the black hole itself. .The former is Universal, does not depend on the black hole mass but only upon the width of this quantum atmosphere; the latter depends on the event horizon area. Being so different, they must have different physical origins. That is to say, a Schwatzschild black hole rotates randomly in all possible directions. The physical meaning of the fluctuations remains elusive. Surprisingly, the rules of angular momentum implies that the event horizon is quantized and a for large mass, the mass spectrum depends (nearly) linearly on the quantum number $l$ which also relates to the angular momentum fluctuations. Furthermore assuming that the cut-off parameter is Planckian, we can write $\Delta^{4}=\mu\hbar^{2}/15\pi^{2}$ for $\mu$ a numerical value of order one. Accordingly the (averaged) Cauchy horizon $\overline{r_{-}}=\overline{J^{2}}/2M^{3}\sim\mu\hbar^{2}/(8\pi(M^{2}+\mu\hbar)M)$ never vanishes. Since a black hole is nothing but geometry, our results should be regarded as being of (semi-classical ) quantum gravity nature.