Notes on Black Hole Thermodynamics

General Relativity

Definitions & conventions

• $\mathbf{Convention.}$ A convenient system of units is adopted such that the speed of light $c=1$.

• $\mathbf{Convention.}$ Greek indices ($\mu,\ \nu,\ \lambda,\ \rho,\ \sigma$, etc.) are used to label $0,\ 1,\ 2,\ 3$; latin indices ($i,\ j,\ k$, etc.) are used to label $1,\ 2,\ 3$.

• $\mathbf{Convention.}$ Indices appearing as superscripts label contravariant components (e.g., $x^\mu=(t,\ x,\ y,\ z)$); indices appearing as subscripts label covariant components (e.g., $\partial_\mu=\bigg(\dfrac{\partial}{\partial t},\ \dfrac{\partial}{\partial x},\ \dfrac{\partial}{\partial y},\ \dfrac{\partial}{\partial z}\bigg)$).

• $\mathbf{Convention.}$ The elements of the flat-spacetime metric are

  $$\eta_{\mu\nu}=\begin{bmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{bmatrix}.$$

• $\mathbf{Convention.}$ The Einstein summation convention is used, e.g.,

  $$A_\mu B^\mu=\sum_{\mu=0}^3A_\mu B^\mu.$$

• $\mathbf{Definition.}$ Homeomorphism (同胚)

  Two topological spaces $(A,\,\mathcal N_A)$ and $(B,\,\mathcal N_B)$ are said to be homeomorphic if there is a homeomorphism between them, where the homeomorphism is a function $f\colon\,A\to B$ if

  • $f$ is a bijection (一一映射),
  • $f$ is continuous,
  • the inverse function $f^{-1}$ is continuous.

• $\mathbf{Definition.}$ Manifold (流形)

  A (topological) manifold $M$ is a second-countable Hausdorff space locally homeomorphic to Euclidean space, that is, each point $m\in M$ has an open neighborhood homeomorphic to an open neighborhood in Euclidean space.

• $\mathbf{Definition.}$ Metric (度规)

  A metric tensor is a symmetric and non-degenerate $(0,\ 2)$-tensor that maps a vector to its modulus through a quadratic form.

  $$g=g_{\mu\nu}\mathrm dx^\mu\wedge\mathrm dx^\nu$$

  such that

  $$\|A\|=g_{\mu\nu}A^\mu A^\nu$$

  and

  $$A\cdot B=g_{\mu\nu}A^\mu B^\nu.$$

  If the spacetime is flat, the metric degrades to $\eta_{\mu\nu}$.

  The contravariant components of the metric tensor are defined by the inverse of matrix.

  $$g_{\mu\lambda}g^{\lambda\nu}=\delta_\mu^\nu.$$

  The co- and contravariant indices are raised and lowered by the metric tensor, e.g.,

  $$A_\mu=g_{\mu\nu}A^\nu;\quad B^\mu=g^{\mu\nu}B_\nu.$$

  Note that the components $g_{\mu\nu}$ may vary under coordinate transformations, however the matrix's negative index of inertia ($n_-=1$) does not change and is a characteristic property of the metric tensor. Differential manifolds equipped with a metric field with exactly one negative element when diagonalized (like $\eta_{\mu\nu}$) are called pseudo-Riemannian manifold, which is used to describe the spacetime in Einstein's general relativity.

• $\mathbf{Definition.}$ Christoffel connection (Christoffel仿射联络)

  The problem arises when we try to translate a vector from one point to another on a non-flat manifold along one direction. Since the manifold itself has some certain curvature, the direction may change along the path. So we need to find the change of basis along one direction. We define the affine connection

  $$\nabla_\mu\hat e_\nu\equiv\nabla_{\hat e_\mu}\hat e_\nu=\Gamma_{\mu\nu}^\lambda\hat e_\lambda$$

  which means the change when translating $\hat e_\nu$ along $\hat e_\mu$. Here $\Gamma_{\mu\nu}^\lambda$ are called the connection coefficients, yet they do not construct a tensor. With this symbol we can write down a well-defined derivative of a vector field.

  $$\begin{align} \nabla_\mu(A^\nu\hat e_\nu)&=(\partial_\mu A^\nu)\hat e_\nu+A^\nu\nabla_\mu\hat e_\nu\\ &=(\partial_\mu A^\nu+\Gamma_{\mu\lambda}^\nu A^\lambda)\hat e_\nu \end{align}$$

  or simply denoted as

  $$\nabla_\mu A^\nu=\partial_\mu A^\nu +\Gamma_{\mu\lambda}^\nu A^\lambda.$$

  This well-defined derivative is the so-called covariant derivative of a vector (tensor) field. Incidentally, we can verify that $\nabla_\mu A^\nu$ is a $(1,\ 1)$-tensor.

  The fundamental theorem of Riemannian geometry states that, given the metric tensor field, there is a unique connection that satisfies

  • it's torsion-free: $\Gamma_{\mu\nu}^\lambda=\Gamma_{\nu\mu}^\lambda$;
  • it's compatible with the metric: $\nabla_\lambda g_{\mu\nu}=0$.

  This unique connection is called the Christoffel connection, given by

  $$\Gamma_{\mu\nu}^\lambda=\frac12 g^{\lambda\sigma}(\partial_\mu g_{\nu\sigma}+\partial_\nu g_{\sigma\mu}-\partial_\sigma g_{\mu\nu}).$$

• $\mathbf{Definition.}$ Intrinsic derivative (内禀导数)

  If a vector is defined along a paramatrized curve (e.g., 4-momentum along the worldline of a particle)

  $$V(\lambda)=V^\mu(\lambda)\hat e_\mu,$$

  where $\lambda$ is the parameter of the curve. We can only define the derivative along the curve

  $$\nabla_{\frac{\mathrm d}{\mathrm d\lambda}}(V^\mu\hat e_\mu)=\bigg(\frac{\mathrm dV^\mu}{\mathrm d\lambda}+\Gamma_{\nu\sigma}^\mu\frac{\mathrm d x^\nu}{\mathrm d\lambda} V^\sigma\bigg)\hat e_\mu$$

  or simply denoted as

  $$\frac{\mathrm DV^\mu}{\mathrm d\lambda}=\frac{\mathrm dV^\mu}{\mathrm d\lambda}+\Gamma_{\nu\sigma}^\mu\frac{\mathrm d x^\nu}{\mathrm d\lambda} V^\sigma,$$

  which is called the intrinsic derivative of a vector along the curve.

• $\mathbf{Definition.}$ Geodesic (测地线)

  Recall how we define a line in the Euclidean geometry: a curve whose tangent vector holds invariant along the curve. We can easily generalize this definition to the Riemann geometry: a curve whose tangent vector has a constant zero intrinsic derivative along the curve. Such a curve is called a geodesic and can be roughly understood as the "shortest" path connecting two points on the manifold. According to general relativity, a particle moves along a geodesic if no external force is exerted.

  A geodesic is commonly described by a parametrized curve $x^\mu(\lambda)$ where $\lambda$ is the parameter. By definition, it satisfies

  $$\frac{\mathrm D}{\mathrm d\lambda}\frac{\mathrm d x^\mu}{\mathrm d\lambda}=0$$

  or explicitly,

  $$\frac{\mathrm d^2 x^\mu}{\mathrm d\lambda^2}+\Gamma_{\nu\sigma}^\mu\frac{\mathrm dx^\nu}{\mathrm d\lambda}\frac{\mathrm dx^\sigma}{\mathrm d\lambda}=0.$$

  This is the famous geodesic equation in the Riemann geometry.

• $\mathbf{Definition.}$ Riemann curvature tensor (Riemann曲率张量)

  Now that the affine connection offers a description on how to translate a vector, we are interested in how it changes a vector by translating it along a closed loop. We know it relates to the curvature enclosed inside the loop; anyway recall what the Gauss-Bonnet theorem states.

  Imagine there is an infinitesimal parallelogram spanned by two small vectors $A^\mu$ and $B^\mu$. Another vector $V^\mu$ initially at the origin travels back to the origin around the parallelogram, and it changes to $\widetilde V^\mu$. It must take the form

  $$\widetilde V^\mu=R^\mu_{\nu\sigma\rho}V^\nu A^\sigma B^\rho$$

  since an infinitesimal transform should be linear in dimensions.

  Here we define the Riemann curvature tensor $R^\mu_{\nu\sigma\rho}$, which can be expressed in terms of the Christoffel symbols.

  $$R _ { \nu \rho \sigma } ^ { \mu } = \partial _ { \rho } \Gamma _ { \sigma \nu } ^ { \mu } - \partial _ { \sigma } \Gamma _ { \rho \nu } ^ { \mu } + \Gamma _ { \rho \lambda } ^ { \mu } \Gamma _ { \sigma \nu } ^ { \lambda } - \Gamma _ { \sigma \lambda } ^ { \mu } \Gamma _ { \rho \nu } ^ { \lambda }.$$

  This $(1,\ 3)$-tensor describes the local curvature of the manifold.

  There are some (anti-)symmetric pairs of indices:

  • $R _ { \mu \nu \rho \sigma } = - R _ { \mu \nu \sigma \rho }$
  • $R _ { \mu \nu \rho \sigma } = - R _ { \nu \mu \rho \sigma }$
  • $R _ { \mu \nu \rho \sigma } = R _ { \rho \sigma \mu \nu }$

• $\mathbf{Definition.}$ Ricci tensor & Ricci scalar (Ricci张量和Ricci曲率标量)

  The Ricci tensor is the only non-trivial contraction derived from the Riemann tensor.

  $$R_{\mu\nu}=R^{\lambda}_{\mu\lambda\nu}.$$

  It is a symmetric tensor, i.e., $R_{\mu\nu}=R_{\nu\mu}$. The Ricci scalar is the trace of the Ricci tensor.

  $$R=g^{\mu\nu}R_{\mu\nu}.$$

• $\mathbf{Definition.}$ Killing vector (Killing矢量)

  Every symmetry of the spacetime manifold is related to a Killing vector (field). A Killing vector field is a vector field such that

  $$\nabla_\mu\xi_\nu+\nabla_\nu\xi_\mu=0.$$

  A simple way to specify a Killing vector is to note that, if a particular coordinate does not explicitly appear in $g_{\mu\nu}$, the direction in this coordinate is a Killing vector.

  If $V$ is a Killing vector field, its inner product with the 4-momentum of a particle is conserved along its geodesic.

Einstein's equations & Schwarzschild metric

• $\mathbf{Definition.}$ Time-like & space-like & null vectors (类时、类空、零矢量)

  With signature $(-,\ +,\ +,\, +)$, a vector $A^\mu$ is said to be time-like if

  $$g_{\mu\nu}A^\mu A^\nu<0,$$

  or space-like if

  $$g_{\mu\nu}A^\mu A^\nu>0,$$

  or null if

  $$g_{\mu\nu}A^\mu A^\nu=0.$$

  The fact that a material particle cannot move at a speed that exceeds the speed of light $c=1$ requires that the tangent vector of the worldline of any material particle be time-like. Actually, the 4-velocity is by definition always "normalized"

  $$\frac{\mathrm d x^\mu}{\mathrm d\tau}\frac{\mathrm dx_\mu}{\mathrm d\tau}\equiv-1.$$

  As for photons, they travel along null-geodesics, i.e.,

  $$\frac{\mathrm d x^\mu}{\mathrm d\lambda}\frac{\mathrm dx_\mu}{\mathrm d\lambda}\equiv0$$

  where $\lambda$ is an arbitrary affine parameter that parametrizes the geodesic.

• $\mathbf{Theorem.}$ Given the energy-momentum tensor field $T_{\mu\nu}$ of the spacetime manifold, the metric field $g_{\mu\nu}$ is determined by solving the Einstein's equations:

  $$R_{\mu\nu}-\frac12g_{\mu\nu}R=8\pi GT_{\mu\nu}.$$

  From the definitions of the Ricci tensor and Ricci scalar, we know the Einstein's equations are a set of highly-nonlinear PDEs.

• $\mathbf{Theorem}.$ There is only one spherically-symmetric solution to the Einstein's equations in vacuum ($T_{\mu\nu}=0$), and it is analytically given by the famous Schwarzschild metric:

  $$\mathrm ds^2=-\bigg(1-\frac{2GM}{r}\bigg)\mathrm dt^2+\bigg(1-\frac{2GM}{r}\bigg)^{-1}\mathrm dr^2+r^2\mathrm d\Omega^2.$$

  Equivalently, the metric can be written in matrix

  $$g_{\mu\nu}=\begin{bmatrix} -\bigg(1-\dfrac{2GM}{r}\bigg)&0&0&0\\ 0&\bigg(1-\dfrac{2GM}{r}\bigg)^{-1}&0&0\\ 0&0&r^2&0\\ 0&0&0&r^2\sin^2\theta \end{bmatrix}.$$

  Note that ostensibly there are two singularities: $r=0,\ r=2GM$. However, it can be proved $r=2GM$ is not a real geometrical singularity; instead, it appears due to the particular choice of coordinates. On the other hand, $r=0$ is a real singularity and cannot be removed by any coordinate transformation.

  The fact that $g_{\mu\nu}$ is independent of $t$ implies that $\partial_t$ is a Killing vector. The fact that the spacetime is spherically-symmetric implies there are $3$ more Killing vectors, one of which is obvious, $\partial_\phi$.

  The conservation corresponding to $\partial_t$ is the energy (per unit mass)

  $$E=-\frac{\mathrm dx^\mu}{\mathrm d\lambda}\partial_\mu\cdot\partial_t=\bigg(1-\dfrac{2GM}{r}\bigg)\frac{\mathrm dt}{\mathrm d\lambda}.$$

  The conservation corresponding to $\partial_\phi$, if the particle is confined to $\theta=\pi/2$, is the magnitude of the orbital angular momentum (per unit mass)

  $$L=\frac{\mathrm dx^\mu}{\mathrm d\lambda}\partial_\mu\cdot\partial_\phi=r^2\sin^2\theta\frac{\mathrm d\phi}{\mathrm d\lambda}=r^2\frac{\mathrm d\phi}{\mathrm d\lambda}.$$

  Although $r=2GM$ is not an intrinsic singularity, we do find that this point is a little peculiar. Note that traversing this point makes $g_{tt}$ and $g_{rr}$ shift their signs. Thus inside the sphere determined by $r=2GM$, known as the event horizon (事件视界) in the Schwarzschild spacetime, $\partial_t$ is no longer a time-like vector, though still a Killing vector. In other words, the metric becomes "time"-dependent inside the horizon.

  This critical and remarkable transition brings about radical changes in the properties of the spacetime. From outside the horizon to inside, $r$ becomes the major role that determines the causality, instead of $t$. The light-cone then directs in $r$ and the decreasing of $r$ corresponds to the absolute future. (That's why the event horizon is also known as the point of no return.)

Black Hole Thermodynamics

Schwarzschild black hole

• $\mathbf{Keypoint.}$ What is a black hole? What is a Schwarzschild black hole?

  Definition quoted from Wikipedia:

  A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it.

  A black hole is a literally black massive spacetime region, since there is a boundary (usually a codimension-$\mathtt{1D}$ hypersurface) where nothing inside the boundary can exert causal effects on anything outside the boundary. Consequently, even light rays cannot escape beyond the boundary. Thus, classically, no information can be carried outward and the spacetime region looks absolutely BLACK from outside.

  Such a boundary is called the event horizon (not to be confused with the Killing horizon, etc.) of the black hole.

  A black hole is usually highly condensed since the radius of the event horizon is usually small.

  A Schwarzschild black hole is a rotationless and chargeless black hole, the exterior of which can be described by Schwarzschild spacetime. There is exactly one event horizon

  $$r=2GM.$$

• $\mathbf{Keypoint.}$ Event horizon.

  An event horizon is mathematically defined as a null hypersurface (the normal vector to which is null) where null geodesics cannot reach the time-like future infinity. Typically, this condition is equivalent to finding the root of

  $$g^{rr}=0\quad\text{or}\quad g_{rr}\to\infty.$$

  Note that in the Schwarzschild coordinates, $r=2GM$ not only results in $g_{rr}\to\infty$, but also makes $g_{tt}=0$. It can be proved that the observed redshift a photon experiences traveling from point $A$ to point $B$ is determined by the square root of the ratio of $g_{tt}$ between $A$ and $B$, i.e.,

  $$\frac{\nu_B}{\nu_A}=\root\of{\frac{g_{tt}(A)}{g_{tt}(B)}},$$

  as long as $\partial_t$ is a Killing vector. On the event horizon in the Schwarzschild spacetime, $g_{tt}=0$ indicates that any photon emitted here and observed by a static observer (outside the horizon) is infinitely redshifted, hence it helps to understand why the black hole looks so black. Be careful, however, this is not always the case; the infinite-redshift surface (aka. the Killing horizon of $t$) usually does not coincides with the event horizon, while in the Schwarzschild spacetime there is a coincidence. (Somehow it is because that the Schwarzschild black hole is the most simple black hole.)

  Through some simple calculation (consider a radial inward geodesic) we shall see a picture of fun:

  If a poor astronaut is to fall into a Schw. BH, and if he is small enough in size so that he is not torn into parts by the tide force all the way to his end, his colleagues outside the BH will see him approaching the event horizon slower and slower, his visage dimmer and dimmer, his watch (if he has one) elapsing more and more damped, and it shall take him infinitely long time to traverse the horizon, before which he will have become so dark that no one else can see.

  However, it will be a totally different story for the astronaut himself. He falls into the black hole accelerating, and penetrates the horizon with no unusual feeling, and keeps falling, and finally hits the $r=0$ singularity at a certain (finite) proper time (固有时).

• $\mathbf{Keypoint.}$ Kruskal coordinates. Wormholes.

  The Schwarzschild coordinates are asymptotically Minkowski coordinates when $r\to\infty$. However, as we have discussed above, they are not well-behaved, especially in that $t$ becomes space-like inside the horizon. As a result, the light-cone experiences a discontinuity at the horizon. This flaw is obstructive to our understanding of the whole picture.

  Take the coordinate transformation

  $$\left\{\begin{align} u&=\bigg(\frac{r}{2GM}-1\bigg)^{1/2}\exp(r/4GM)\cosh(t/4GM)\\ v&=\bigg(\frac{r}{2GM}-1\bigg)^{1/2}\exp(r/4GM)\sinh(t/4GM) \end{align}\right.\quad\text{for }r>2GM,$$

  and

  $$\left\{\begin{align} u&=\bigg(1-\frac{r}{2GM}\bigg)^{1/2}\exp(r/4GM)\sinh(t/4GM)\\ v&=\bigg(1-\frac{r}{2GM}\bigg)^{1/2}\exp(r/4GM)\cosh(t/4GM) \end{align}\right.\quad\text{for }r<2GM.$$

  Under such transformation, the Schwarzschild metric takes a well-behaved form

  $$\mathrm ds^2=\frac{32G^3M^3}{r}\exp(-r/2GM)(-\mathrm d v^2+\mathrm du^2)+r^2\mathrm d\Omega^2$$

  where $r$ is defined as

  $$u^2-v^2=\bigg(\frac{r}{2GM}-1\bigg)\exp(r/2GM).$$

  We can see that the sub-manifold with constant radius $r$ is represented by a hyperbola in the $(u,v)$-plane (every point in this plan represents $\mathtt{2D}$-sphere). Another astonishing fact is that null geodesics are always $45^\circ$ to the axes. Hence the light-cone looks everywhere the same even across the horizon. The horizon, naturally, corresponds to $u^2-v^2=0$. As for constant $t$, we notice the relation that $\dfrac{v}{u}=\tanh(t/4GM)$, so it is represented by a line passing through the origin.

  This coordinate system is known as the Kruskal-Szekeres coordinates. Behold, the whole spacetime geometry of the Schwarzschild solution is shown below.

  

  Region $\mathbf I$ corresponds to our living spacetime, outside the black hole, asymptotically flat. Region $\mathbf{II}$ corresponds to the interior of a Schw. BH, terminating at a future singularity. Region $\mathbf{IV}$ corresponds to the interior of a Schwarzschild white hole, generating from a past singularity. At last, region $\mathbf{III}$ corresponds to another asymptotically flat spacetime region, which nobody can enter.

  Using this picture, we can further give a elementary discussion on the fabulous term: the wormhole.

Kerr black hole

• $\mathbf{Keypoint.}$ The structure of Kerr black hole.

  When a chargeless black hole is rotating... Well, despite all the possible difficulties, we are pretty sure of one thing: the spherical symmetry $\text{SO(3)}$ is broken to cylindrical symmetry $\text{U(1)}$. However, by properly choosing the rotation axis, the two familiar Killing vectors, $\partial_t$ and $\partial_\phi$, can be preserved.

  In this situation, we do have an analytical solution, the Kerr metric, named after Roy Kerr, proposed in 1963. But frankly speaking, I am reluctant to transcribe that metric here, since it generally looks like

  $$\mathrm ds^2=-\mathrm dt^2+\frac{\rho^2}{\Delta}\mathrm dr^2+\rho^2\mathrm d\theta^2+(r^2+a^2)\sin^2\theta\,\mathrm d\phi^2+\frac{2GMr}{\rho^2}(a\sin^2\theta\,\mathrm d\phi-\mathrm dt)^2$$

  where $\Delta=r^2-2GMr+a^2$, $\rho^2=r^2+a^2\cos^2\theta$, and $a=J/M$. Here $J$ is the total angular momentum of the black hole and $M$ is the mass of the black hole.

  You don't really... I mean, you don't need to make a copy of this formula in your mind. All you need to know is a rough picture of the Kerr BH:

  • The intrinsic singularity is somehow like a ring with radius $a$.
  • There are two event horizons residing respectively at $r_{\pm}$, where $r_\pm$ are two roots of $\Delta=0$.
  • There is an infinite-redshift surface ($\partial_t$ Killing horizon) outside the outer event horizon $r_+$. Inside the infinite-redshift surface $\partial_t$ becomes a space-like Killing vector.
  • The region between the outmost infinite-redshift surface and the outer event horizon is called the ergosphere. Particles in the ergosphere cannot rest at a fixed $(r,\theta,\phi)$ point anyhow; however, they still have chances to escape from the gravity of the black hole.

  Here is a structural sketch of a Kerr BH. (View from $\theta=\pi/2$.)

  

• $\mathbf{Keypoint.}$ Penrose process.

  The reason why we introduce the Kerr spacetime is to explain the so-called Penrose process, through which we can literally extract energy from a black hole!

  Consider dumping an unstable particle $A$ from the infinity (the flat region). The particle falls in the ergosphere, and suddenly decays into two separate parts $B$ and $C$. Then particle $C$ manages to escape from the ergosphere and flies to the static observer at the infinity.

  It is possible for particle $B$ either to keep falling to the abyss or to escape just like $C$ does. Since the spacetime is locally homeomorphic to the Minkowski spacetime, we have the conservation of 4-momentum for the decay process,

  $$\boldsymbol P_\mathrm{decay}(A)=\boldsymbol P_\mathrm{decay}(B)+\boldsymbol P_\mathrm{decay}(C).$$

  Take the inner product with $\partial_t$,

  $$g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(A)=g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(B)+g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(C).$$

  Since $\partial_t$ is a Killing vector throughout the spacetime, we have

  $$g_{tt,\ \infty}P_\infty^t(A)=g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(B)+g_{tt,\ \infty}P_\infty^t(C)$$

  where $\infty$ indicates quantity valued at the infinity. Since the energy of a particle $E=-\langle\partial_t,\ \boldsymbol P\rangle$, we have

  $$E_\infty(A)+g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(B)=E_\infty(C).$$

  If $B$ never comes out, since $\partial_t$ is a space-like vector, there is no restriction on the sign of $P_\mathrm{decay}^t(B)$. When $P_\mathrm{decay}^t(B)>0$, $g_{tt,\ \mathrm{decay}}P_\mathrm{decay}^t(B)>0$, so we arrive at

  $$E_\infty(C)>E_\infty(A).$$

  This might be counterintuitive, however nothing forbids its existence. Therefore, we say a certain amount of energy is extracted from the Kerr BH.

  It can be proved that the extraction of energy results in the extraction of the angular momentum, and the change in $M$ and $J$ follows an inequality:

  $$\delta M>\Omega_H\delta J,$$

  where $\delta M,\delta J<0$, and $\Omega_H=\dfrac{a}{2GMr_+}$.

Black hole thermodynamics

• $\mathbf{Keypoint.}$ The non-decreasing area of the event horizon.

  The most inspiring fact we are going to point out is that

  (Hawking's area theorem.) The area of the event horizon of a black hole is non-decreasing with respect to time.

  Note that so far no quantum mechanical effect is considered.

  This law is straightforward for the Schw. BH. The area of horizon of a Schw. BH. is

  $$A_\text{Schw.}=\oint_{r=2GM}\root\of{|\det g|}\,\mathrm d\theta\wedge\mathrm d\phi=4\pi r^2=16\pi G^2M^2$$

  which is proportional to $M^2$. Since matter can only enter the BH, the mass is non-decreasing. Thus the area of horizon $A$ is non-decreasing.

  For the Kerr B.H., however, it is not that trivial, provided that the Penrose process is possible. The area of horizon of a Kerr BH. is

  $$A_\text{Kerr}=\oint_{r=r_+}\root\of{|\det g|}\,\mathrm d\theta\wedge\mathrm d\phi=2GMr_+\int_0^\pi\sin\theta\,\mathrm d\theta\int_0^{2\pi}\mathrm d\phi=8\pi GMr_+.$$

  Recall $r_+=GM+\root\of{G^2M^2-a^2}$. Let $GM=\mu$,

  $$A_\text{Kerr}=8\pi\mu(\mu+\root\of{\mu^2-a^2}).$$

  If a Penrose process takes place, the change of the area is

  $$\delta A_\text{Kerr}=\frac{16\pi G\mu r_+}{\root\of{\mu^2-a^2}}(\delta M-\Omega_H\delta J)>0.$$

  Hence we have shown that the law holds for the Kerr BH.

  Furthermore, this law holds for all classical situations.

• $\mathbf{Keypoint.}$ Introducing the entropy and the "thermodynamical" differential form.

  Let's rewrite $\delta A_\text{Kerr}$ in a clearer form.

  $$\delta A_\text{Kerr}=\frac{8\pi G}{\kappa}(\delta M-\Omega_H\delta J),$$

  or

  $$\delta M=\frac{\kappa}{8\pi G}\delta A+\Omega_H\delta J.$$

  Here, $\kappa=\dfrac{\root\of{\mu^2-a^2}}{2\mu r_+}$ is called the surface gravity.

  This looks familiar... Hummmm... The fundamental thermodynamic relation! Let's compare it with what we obtained just now.

  $$\mathrm d U=T\mathrm dS-p\mathrm d V.$$

  If we propose the counterparts:

  $$M\sim U,\quad \frac{\kappa}{8\pi G}\sim T,\quad A\sim S,\quad \Omega_H\delta J\sim-p\mathrm dV,$$

  we can establish the thermodynamics for a black hole, since we already have the "second law": Hawking's area theorem!

• $\mathbf{Keypoint.}$ Laws of black hole thermodynamics.

Salutations to Stephen Hawking

Hawking radiation

• $\mathbf{Keypoint.}$ Schwinger pair production.

Black hole informatics - Thorne–Hawking bet

Micro black hole