Black hole thermodynamics

theory, to the spin connection one-formω. The second equation describes the curvature two form entering the Einstein-Hilbert action (3.1). From this construction we immediately see the non-renormalizability of the Einstein-Hilbert action by expanding about a flat reference spacetime g_µν “ η_µν `κh_µν. We redefined the quantum excitations to have a canonically normalized kinetic term

SEH“ ż

d⁴xpBhq²“

1`κh`κ²h²`. . .‰

, (3.5)

where in the IR (low-energy/long wavelength regime) there is a stable weak coupling fixed point κ^‹ “ 0 andκis an irrelevant scaling field. Thus, we can view classical gravity as an effective theory only valid up to energiesEκ „1 or even below which can be translated intoE „MP, whereMP„Op10²⁸eVqdenotes the Planck mass. Due to the irrelevant nature of the coupling in the IR, gravity is a non-renormalizable theory,i.e. adding additional irrelevant fields does not change the UV behavior, so we need an infinite number of counter terms to cure divergences in loop integrals. At high energiesE „MPthe low-energy effective theory must be replaced by a new theory containing UV degrees of freedom. A different resolution would be the existence of a UV fixed point in the theory.

Another interesting property of gravity is its unusual thermodynamic behavior,i.e. violating one of the three “axiomatic” properties that define thermodynamics, the so-called entropy postulates:

• Homogeneity,i.e. the entropy is a homogeneous function of order oneSpcXq “cSpXqfor cą0. This shows the extensivity of the entropy and insures that a thermodynamic system cannot be described by intrinsic variables alone (“the size of the system matters!”).

• Convexitydescribes a function with values larger or equal to a secant between two points or, if its derivative exists, a tangent that is larger or equal to the functional valuesf²pxq ě0.

Stability requires a convex entropy function implying the second law for conventional mat-ter.

• Superadditivityis the following propertySpx1`x2q ďSpx2q `Spx1qvalid for allx1,2 in the domain ofS. In particular, it implies the third law of thermodynamicsSp0q “0.

As can be shown, not all of the three postulates are independent. Two of the above properties imply the third one. Normal matter as microscopic constituents of statistical systems obeys all three postulates but gravitating matter such as stars or black holes² violate the convexity postulate. Note that this does not violate the second law of thermodynamics, but can rather be seen as a consequence of it. Uniformly distributed matter can be considered as the most ordered state regarding the gravitational interaction, whereas extreme density differences (e.g. randomly distributed massive objects in almost empty space) yield a very high disorder on long length scales where the gravitational interaction dominates. So introducing a small density fluctuation in a uniformly matter distribution, a gravitational system tends to amplify these deviations in contrast to conventional statistical systems. For example, the specific heat of huge gravitating objects is negative,i.e. adding more matter yields a decrease in temperature. Thus, massive black holes are colder than lighter black holes.

Taking the so-called Unruh effect into account, an accelerating observer sees a heat bath with

2A black hole can be viewed as a massive gravitational object that does not allow for light to escape its event horizon.

Strictly speaking, a black hole constitutes a causally disconnected spacetime region, where “information is trapped”

and observers inside the black hole cannot communicate or escape to the outside of a black hole. However, matters are more complicated as we will see once we look at the Unruh effect and the Hawking radiation. For a detailed discussion of the peculiarities of black holes see [81].

temperature

TUpxq “ apxq

2π , (3.6)

compared to a freely falling observer. This implies that the vacuum state of a quantum field theory depends on the state of the observer and cannot be compared since the canonical com-mutation relations are defined in the local coordinates of the free falling and the accelerated observer.³ If we apply the equivalence principle of acceleration and gravity, we would expect that a stationary observer close to the horizon of a black hole measures a black body tempera-ture given by the surface gravityκprHqat the horizon of the black hole

TH“ κprHq

2π . (3.7)

In fact this temperature has been independently discovered by Hawking, yet to date there is no conclusive even indirect experiment to measure this effect experimentally.⁴ Additionally to the thermal radiation, there is a related evaporation process, since the thermal heat bath consists of virtual particle/anti-particle pairs, created at the horizon. A consistent extension of the local heat bath allows for a leakage of some of the virtual particle pairs escaping to infinity and thus becoming real. Thus, a black hole emits radiation, loses mass/energy and increases its temper-ature in the process until it is completely evaporated (unless the mass influx from other sources leads to a positive net increase in the black hole energy).⁵ For a Schwarzschild black hole the Hawking temperature at the event horizonrH“2M G,κ“ p2rHq^´1is given byT “ p8πM Gq^´1 and we see that the temperature is inversely proportional to the black hole mass. This implies the following entropy relation, known as the Bekenstein-Hawking entropy

dM “TdS“ 1

8πM GdS ñ SBH“4πM²G“AprHq

4G , (3.8) suggesting that the information content of the black hole is encoded in its area andnotin its volume. The extensivity of the entropy in ordinary thermodynamic systems scales with the volume of the system and since the maximal entropy is a measure for the total degrees of freedom needed to describe the system, we conclude that the degrees of freedom to describe a quantum gravity system scale only with its area.

Quantum mechanically, this poses the problem of naïvely violating some of the fundamental laws of nature, i.e. the conservation of information. Unitary evolution does not allow a pure state to evolve into a mixed quantum state characterizing a thermal ensemble. So an observer outside of the black hole should in principle retrieve information (after the initial entropy is halved due to evaporation) that once entered the black hole. The problem intensifies once we take the quantum no-cloning theorem into account which asserts that a quantum state cannot be duplicated without violating the linearity of quantum mechanics and hence the Heisenberg

3So far, it is not clear if the Unruh effect is real and measurable, but there are experiments devised for possible direct detection, see [82].

4However, there exists an analogous table-top experiment, in the sense of emergent gravity, using phonons in perfect fluids where the fluid is flowing faster than the local speed of sound trapping the phonons. The change from super-sonic speed of the fluid to subsuper-sonic speed creates a event horizon where the frequency of the phonons approaches zero. A possible realization using Bose-Einstein condensates might be expandable to detect a phononic Hawking radiation [83]

5Interestingly, a free falling observer crossing the event horizon would not detect any thermal heat bath. Every space-time point on its infalling trajectory close to the horizon would consist of the usually vacuum state. Furthermore, the event horizon is not special since the free falling observer “sees” only a weakly curved locally flat Minkowski spacetime.

heat bath of vacuum fluctuations

at the horizon

Hot conducting membrane

Partitioned cells of areaℓ²_P

Figure 3.1. Due to the Unruh effect, a stationary observer close to the black hole, will ob-serve a heat bath with temperatureTU{H. Hawking radiation allows for emission of virtual particles becoming real particles away from the black hole horizon, thus rendering the black hole a thermal ideal black body (left panel). Additionally in-falling matter is observed by a distant observer as frozen in time, “heated up” and spread over the black hole horizon because of the IR/UV connection ∆x∆t „ℓ²_P. This stretched horizon can be described by a hydrodynamic theory incorporating dissipation effects such as viscosity and resistivity (middle panel). Statistically, we can view the black hole horizon as a “holographic” projection (see also Infobox Holographic Principleon page74) of the degrees of freedom residing “inside” the black hole, which are partitioned into Plank areaℓ²_Pcells (right panel).

uncertainty principle [84]. We could imagine an external observer B, sufficiently close to the horizon, collecting information by observing an infalling observer A and eventually entering the black hole to retrieve again the identical/duplicated information from observer A. This is equivalent to duplicate the information at the horizon, where one copy of the information enters the black hole and the other copy is sent out, thus violating the no-cloning theorem. A possible resolutions to this paradox, which violates neither the no-cloning theorem nor the conservation of information, is called black hole complementarity stating that the external observer “sees” the black hole as a complex system with information stored in its degrees of freedom, counted by the area scaling of the entropy, where the infalling observer carries its information freely through the horizon. The information paradox is resolved by taking into account that sending the information requires at least one quantum, such as a photon, with a definite frequency/energy. In order for A to send the information before B hits the singularity, the energy required by the observer is larger than the black hole mass. So we conclude that in principle it is not possible for the external observer to receive the same information outside of the horizon and inside the black hole. In this sense, the black hole complementarity avoids a violation of quantum mechanics and the equivalence principle. An alternative approach to resolve the puzzle, the black hole firewalls [85], sparked a still ongoing hot debate c.f. [86]. It departs from the idea that the horizon is a simple coordinate singularity and propose that once the black hole reaches the

“age” where information can be retrieved (after evaporating half of its initial entropy), the event horizon turns into an impenetrable firewall burning the infalling observer. In the following, we will assume that there exist two different viewpoints of black holes which are complementary in the above sense of avoiding any violation of the fundamental laws observed and verified so far.

• For a distant observer (in asymptotically flat spacetime at infinity) the black hole can be described by a nearly perfect fluid with dissipative properties, such as viscosity and elec-trical resistivity in addition to its temperature/entropy, covering the black hole horizon.

This so-called “stretched horizon” [87] yields a hot conducting membrane with properties explicitly computed by the membrane paradigm [88,89]. The reason for this membrane picture to work lies in the gravitational redshift of the infalling matter. For the distant

ob-server the infalling matter never reaches the black hole horizon in finite time, but is frozen infinitesimally close to the horizon, forming a membrane stretched over the horizon. The properties of this membrane can be described by an effective hydrodynamic description using a non-gravitational quantum field theory defined on the lower dimensional horizon with ddimensions, say. Adding more matter to the black hole is seen as disturbing the stretched horizon which gives rise to wave-like excitations of the fluid. In particular, due to the quite unintuitive IR/UV connection, which states that with increasing energies we are probinglarger(!) length scales (for a detailed discussion see [81]),⁶the information stored in the infalling matter will be uniformly spread over the stretched horizon, as shown in Figure3.1.

• On the other hand, an infalling observer does not observe any of the properties of the stretched horizon. On the contrary, such an observer will only approach an effectively zero-temperature gravitating object with increasing tidal forces which can be described by the fulld`1gravitational theory.

Furthermore, these two pictures can be related to each other by the holographic principle, which states that the full information content encoded in the entropy and its distribution over the (in-ternal) degrees of freedom in the theory, characterized by the partition function, are equivalent if we reduce the dimensionality of one of the two descriptions. As the name suggests we draw from the analogue of optical holograms storing three dimensional information in an effectively two dimensional object.

Holographic Principle

The holographic principle⁷asserts that the maximal information of a physical system is en-coded in its boundary area measured in units of¯h. For black holes/gravity this can be under-stood in terms of a collapsing shell of matter which is teleologically⁸ equivalent to a black hole of the same mass and entropySBH “^A{^4G. Therefore, any thermodynamic system sat-isfying the second law of thermodynamics with given energy and entropy is bounded by the mass of the black hole of area A and entropy SBH. The microscopic degrees of freedom of such a thermodynamic system are distributed over the boundary of the system with a partition of one degree of freedom per Planck areaℓ²_P. Two physical descriptions are equiva-lent if the same information is encoded,i.e. the maximal entropy, the number of degrees of freedom and the partition functions must be equal. Therefore, the partition function of the thermodynamic system, described by (Euclidean) quantum field theory, must be defined in one dimension lower than the gravitating mass distribution forming the black hole.

The emergence of the holographic principle draws heavily from semi-classical considerations.

Thus, to understand these peculiarities, a true theory of quantum gravity is needed⁹. There have

6Resolving distances of the order of the Planck mass requires energies comparable to black hole masses, which in turn creates a singularity shielded by the event horizon. This yields the relations∆x„Eand the corresponding spacetime uncertainty∆x∆t„ℓ²_Pand so we are not able to probe distances smaller than the Planck lengthℓ_P.

7A nice review about the holographic principle and its realization in terms of the AdS/CFT correspondence can be found in [90] and an explicit AdS/CFT calculation involving the stretched horizon is conducted in [14], determining the viscosity of black membranes.

8The Schwarzschild radiusr_s“2M Gcan be viewed as “real” for any mass distribution that eventually will collapse in a spherical region with a smaller radiusrărs.

9Due to the Weinberg-Witten theorem [91] a perturbative QFT cannot describe classical gravity. Thus, the hydrodynamic QFT of the hot membrane must describe a non-local or strongly correlated system. On the other hand the near horizon area of a black hole cannot be completely captured by classical gravity.

been many attempts to formulate a quantum theory of gravity, such as loop quantum gravity [46,92–94], string theory [95–99], non-commutative geometry [100], string-net condensates [101–104], just to name a few. To my knowledge, only string theory has been successful to incorporate the holographic principle and allows for explicit realizations, for example

• Matrix Theory: M-Theory¹⁰is conjectured to be dual to D0-branes (particles) in ten di-mensions, providing a second quantization scheme of M-Theory in flat space in a certain set of limits (the infinite momentum frame^N{^RÑ 8[105]). Thus, the eleven dimensional membranes (extended objects of M-theory) emerge from point like fundamental degrees of freedom in ten dimensions described in terms of supersymmetric (NˆNmatrix) quantum mechanics withN Ñ 8. The additional dimension is removed by compactification on a circle with radiusR “gsℓs related to the string couplinggs and the string lengthℓs. The eleven dimensional M-theory is recovered by the strong coupling limitRÑ 8. For more details the reader is urged to consult the comprehensive review [106]. A major difference to the AdS/CFT correspondence lies in the fact that matrix theory deals with an infinite vol-ume Minkowski space, whereas the AdS spacetime possess a boundary with a well defined Cauchy problem,i.e. the boundary conditions of the “AdS box” introduce sources for the interacting bulk fields. An overview of the connection between matrix theory and AdS/CFT can be found in [107].

• AdS/CFT Correspondence: The AdS5/CFT4 correspondence connects holographically a conformalN “4supersymmetricSUpN_cqYang-Mills theory in four dimensions to type IIB string theory on AdS5ˆS⁵. This holographic duality will be explained in the subsequent sections and extended in a way to geometrize the renormalization group method.