Thermodynamical Aspects of Gravity: New insights
The fact that one can associate thermodynamic properties with horizons brings together principles of quantum theory, gravitation and thermodynamics and possibly offers a window to the nature of quantum geometry. This review discusses certain aspects of this topic concentrating on new insights gained from some recent work. After a brief introduction of the overall perspective, Sections 2 and 3 provide the pedagogical background on the geometrical features of bifurcation horizons, path integral derivation of horizon temperature, black hole evaporation, structure of Lanczos-Lovelock models, the concept of Noether charge and its relation to horizon entropy. Section 4 discusses several conceptual issues introduced by the existence of temperature and entropy of the horizons. In Section 5 we take up the connection between horizon thermodynamics and gravitational dynamics and describe several peculiar features which have no simple interpretation in the conventional approach. The next two sections describe the recent progress achieved in an alternative perspective of gravity. In Section 6 we provide a thermodynamic interpretation of the field equations of gravity in any diffeomorphism invariant theory and in Section 7 we obtain the field equations of gravity from an entropy maximization principle. The last section provides a summary.
1 Introduction and perspective
Soon after Einstein developed the gravitational field equations, Schwarzschild found the simplest exact solution to these equations, describing a spherically symmetric spacetime. When expressed in a natural coordinate system which makes the symmetries of the solution obvious, it leads to the line interval:
where with , in units with . It was immediately noticed that this metric exhibits a curious pathology. One of the metric coefficients, vanished on a surface , of finite area , given by , while another metric coefficient diverged on the same surface. After some initial confusion, it was realized that the singular behaviour of the metric is due to bad choice of coordinates and that the spacetime geometry is well behaved at . However, the surface acts as a horizon blocking the propagation of information from the region to the region . This leads to several new features in the theory, many of which, even after decades of investigation, defies a complete understanding. The most important amongst them is the relationship between physics involving horizons and thermodynamics.
This connection, between black hole dynamics involving the horizon and the laws of thermodynamics, became apparent as a result of the research in early 1970s. Hawking proved  that in any classical process involving the black holes, the sum of the areas of a black hole horizons cannot decrease which, with hindsight, is reminiscent of the behaviour of entropy in classical thermodynamics. This connection was exploited by Bekenstein in his response to a conundrum raised by John Wheeler. Wheeler pointed out that an external observer can drop material with non-zero entropy into the inaccessible region beyond the horizon thereby reducing the entropy accessible to outside observers.111John Wheeler posed this as a question to Bekenstein: What happens if you mix cold and hot tea and pour it down a horizon, erasing all traces of “crime” in increasing the entropy of the world? This is based on what Wheeler told me in 1985, from his recollection of events; it is also mentioned in his book, see page 221 of Ref. . I have heard somewhat different versions from other sources. Faced with this difficulty, Bekenstein came up with the idea that the black hole horizon should be attributed an entropy which is proportional to its area [3, 4, 5]. It was also realized around this time that one can formulate four laws of black hole dynamics in a manner analogous to the laws of thermodynamics . In particular, if a physical process (say, dropping of small amount of matter into the Schwarzschild black hole) changes the mass of a black hole by and the area of the event horizon by , then it can be proved that
where is called the surface gravity of the horizon. This suggests an analogy with the thermodynamic law with and . However, classical considerations alone cannot determine the proportionality constants. (We will see later that quatum mechanical considerations suggest and , which is indicated in the second equality in Eq. (2).)
In spite of this, Bekenstein’s idea did not find favour with the community immediately; the fact that laws of black hole dynamics have an uncanny similarity with the laws of thermodynamics was initially considered to be only a curiosity. (For a taste of history, see e.g .) The key objection at that time was the following. If black holes possess entropy as well as energy (which they do), then they must have a non-zero temperature and must radiate — which seemed to contradict the view that nothing can escape the black hole horizon. Investigations by Hawking, however, led to the discovery that a non-zero temperature should be attributed to the black hole horizon . He found that black holes, formed by the collapse of matter, will radiate particles with a thermal spectrum at late times, as detected by a stationary observer at large distances. This result, obtained from the study of quantum field theory in the black hole spacetime, showed that one can consistently attribute to the black hole horizon an entropy and temperature.
An immediate question that arises is whether this entropy is the same as the “usual entropy”. If so, one should be able to show that, for any processes involving matter and black holes, we must have which goes under the name generalized second law (GSL). One simple example in which the area (and thus the entropy) of the black hole decreases is in the emission of Hawking radiation itself; but the GSL holds since the thermal radiation produced in the process has entropy. It is generally believed that GSL always holds though a completely general proof is difficult to obtain. Several thought experiments, when analyzed properly, uphold this law (see, for example, Ref. ) and a proof is possible under different sets of assumptions . All these suggest that the area of the black hole corresponds to an entropy which is same as the “usual entropy”.
These ideas can be extended to black hole solutions in more general theories than just Einstein’s gravity. The temperature can be determined by techniques like, for example, analytic continuation to imaginary time (see Sec. 3.2) which depends only on the metric and not on the field equations which led to the metric. But the concept of entropy needs to generalized in these models and will no longer be one quarter of the area of horizon. This could be done by using the first law of black hole dynamics itself, say, in the form . Since the temperature is known, this equation can be integrated to determine . This was done by Wald and it turns out that the entropy can be related to a conserved charge called Noether charge which arises from the diffeomorphism invariance of the theory . Thus, the notions of entropy and temperature can be attributed to black hole solutions in a wide class of theories.
This raises the question: What are the degrees of freedom responsible for the black hole entropy? There have been several attempts in the literature to answer this question both with and without inputs from quantum gravity models. A statistical mechanics derivation of entropy was originally attempted in ; the entropy has been interpreted as the logarithm of: (a) the number of ways in which black hole might have been formed [4, 13]; (b) the number of internal black hole states consistent with a single black hole exterior  and (c) the number of horizon quantum states [2, 15, 16]. There are also other ideas which are more formal and geometrical [17, 18, 19, 20], or based on thermo-field theory [21, 22, 23] just to name two possibilities. In addition, considerable amount of work has been done in calculating black hole entropy based on different candidate models for quantum gravity. (We will briefly summarize some aspects of these in Sec. 4.2; more extensive discussion as well as references to original literature can be found in the reviews [24, 25]).
Clearly, there is no agreement in literature as regards the degrees of freedom which contribute to black hole entropy and the attempts mentioned above sample the divergent views. In fact, once the answer is known, it seems fairly easy to come up with very imaginative derivations of the result.
A crucial new dimension is added to this problem when we study horizons which are not associated with black holes. Soon after Hawking’s discovery of a temperature associated with the black hole horizon, it was realized that this result was not confined to black holes alone. The study of quantum field theory in any spacetime with a horizon showed that all horizons possess temperatures [26, 27, 28]. In particular, an observer who is accelerating through the vacuum state in flat spacetime perceives a horizon and will attribute to it  a temperature proportional to her acceleration . (For a review, see Refs.[30, 31, 32, 33, 34, 35, 36].) The situation regarding entropy — especially whether one should attribute entropy to all horizons — remains unclear. In fact, many of the attempts to interpret black hole entropy mentioned earlier cannot be generalized to interpret the entropy of other horizons. So a unified understanding of horizon thermodynamics encompassing both entropy and temperature still remains elusive — which will be one of the key issues we will discuss in detail in this review.
This is also closely related to the question of whether gravitational dynamics — in particular, the field equations — have any relation to the horizon thermodynamics. Given a spacetime metric with a horizon (which may or may not be a solution to Einstein’s equations) one can study quantum field theory in that spacetime and discover that the horizon behaves like a black body with a given temperature. At no stage in such an analysis do we need to invoke the gravitational field equations. So it is reasonable to doubt whether the dynamics of gravity has anything to do with horizon thermodynamics and one may — at first — think that there is no connection between the two.
Several recent investigations have shown, however, that there is indeed a deeper connection between gravitational dynamics and horizon thermodynamics (for a recent review, see Ref. ). For example, studies have shown that:
Gravitational field equations in a wide variety of theories, when evaluated on a horizon, reduce to a thermodynamic identity . This result, first pointed out in Ref., has now been demonstrated in several cases like the stationary axisymmetric horizons and evolving spherically symmetric horizons in Einstein gravity, static spherically symmetric horizons and dynamical apparent horizons in Lovelock gravity, three dimensional BTZ black hole horizons, FRW cosmological models in various gravity theories and even in the case Horava-Lifshitz gravity (see Sec. 5.1 for detailed references). If horizon thermodynamics has no deep connection with gravitational dynamics, it is not possible to understand why the field equations should encode information about horizon thermodynamics.
Gravitational action functionals in a wide class of theories have a a surface term and a bulk term. In the conventional approach, we ignore the surface term completely (or cancel it with a counter-term) and obtain the field equation from the bulk term in the action. Therefore, any solution to the field equation obtained by this procedure is logically independent of the nature of the surface term. But when the surface term (which was ignored) is evaluated at the horizon that arises in any given solution, it gives the entropy of the horizon! Again, this result extends far beyond Einstein’s theory to situations in which the entropy is not proportional to horizon area. This is possible only because there is a specific holographic relationship [39, 40, 41] between the surface term and the bulk term which, however, is an unexplained feature in the conventional approach to gravitational dynamics. Since the surface term has the thermodynamic interpretation as the entropy of horizons, and is related holographically to the bulk term, we are again led to suspect an indirect connection between spacetime dynamics and horizon thermodynamics.
Based on these features — which have no explanation in the conventional approach — one can argue that there is a conceptual reason to revise our perspective towards spacetime (Sec. 6 and 7) and relate horizon thermodynamics with gravitational dynamics. This approach should work for a wide class of theories far more general than just Einstein gravity. This will be the new insight which we will focus on in this review.
To set the stage for this future discussion, let us briefly describe this approach and summarize the conclusions. We begin by examining more closely the implications of the existence of temperature for horizons.
In the study of normal macroscopic systems — like, for example, a solid or a gas — one can deduce the existence of microstructure just from the fact that the object can be heated. The supply of energy in the form of heat needs to be stored in some form in the material which is not possible unless the material has microscopic degrees of freedom. This was the insight of Boltzmann which led him to suggest that heat is essentially a form of motion of the microscopic constituents of matter. That is, the existence of temperature is sufficient for us to infer the existence of microstructure without any direct experimental evidence.
The thermodynamics of the horizon shows that we can actually heat up a spacetime, just as one can heat up a solid or a gas. An unorthodox way of doing this would be to take some amount of matter and arrange it to collapse and form a black hole. The Hawking radiation emitted by the black hole can be used to heat up, say, a pan of water just as though the pan was kept inside a microwave oven. In fact the same result can be achieved by just accelerating through the inertial vacuum carrying the pan of water which will eventually be heated to a temperature proportional to the acceleration. These processes show that the temperatures of the horizons are as “real” as any other temperature. Since they arise in a class of hot spacetimes, it follows à la Boltzmann that the spacetimes should possess microstructure.
In the case of a solid or gas, we know the nature of this microstructure from atomic and molecular physics. Hence, in principle, we can work out the thermodynamics of these systems from the underlying statistical mechanics. This is not possible in the case of spacetime because we have no clue about its microstructure. However, one of the remarkable features of thermodynamics — in contrast to statistical mechanics — is that the thermodynamic description is fairly insensitive to the details of the microstructure and can be developed as a fairly broad frame work. For example, a thermodynamic identity like has a universal validity and the information about a given system is only encoded in the form of the entropy functional . In the case of normal materials, this entropy arises because of our coarse graining over microscopic degrees of freedom which are not tracked in the dynamical evolution. In the case of spacetime, the existence of horizons for a particular class of observers makes it mandatory that these observers integrate out degrees of freedom hidden by the horizon.
To make this notion clearer, let us start from the principle of equivalence which allows us to construct local inertial frames (LIF), around any event in an arbitrary curved spacetime. Given the LIF, we can next construct a local Rindler frame (LRF) by boosting along one of the directions with an acceleration . The observers at rest in the LRF will perceive a patch of null surface in LIF as a horizon with temperature . These local Rindler observers and the freely falling inertial observers will attribute different thermodynamical properties to matter in the spacetime. For example, they will attribute different temperatures and entropies to the vacuum state as well as excited states of matter fields. When some matter with energy moves close to the horizon — say, within a few Planck lengths because, formally, it takes infinite Rindler time for matter to actually cross — the local Rindler observer will consider it to have transfered an entropy to the horizon degrees of freedom. We will show (in Sec. 6) that, when the metric satisfies the field equations of any diffeomorphism invariant theory, this transfer of entropy can be given  a geometrical interpretation as the change in the entropy of the horizon.
This result allows us to associate an entropy functional with the null surfaces which the local Rindler observers perceive as horizons. We can now demand that the sum of the horizon entropy and the entropy of matter that flows across the horizons (both as perceived by the local Rindler observers), should be an extremum for all observers in the spacetime. This leads  to a constraint on the geometry of spacetime which can be stated, in , as
for all null vectors in the spacetime. The general solution to this equation is given by where has to be a constant because of the conditions . Hence the thermodynamic principle leads uniquely to Einstein’s equation with a cosmological constant in 4-dimensions. Notice, however, that Eq. (3) has a new symmetry and is invariant [44, 45] under the transformation which the standard Einstein’s theory does not posses. (This has important implications for the cosmological constant problem  which we will discuss in Sec. 7.5.) In , the same entropy maximization leads to a more general class of theories called Lanczos-Lovelock models (see Sec. 3.6).
We can now remedy another conceptual shortcoming of the conventional approach. An unsatisfactory feature of all theories of gravity is that the field equations do not have any direct physical interpretation. The lack of an elegant principle which can lead to the dynamics of gravity (“how matter tells spacetime to curve”) is quite striking when we compare this situation with the kinematics of gravity (“how spacetime makes the matter move”). The latter can be determined through the principle of equivalence by demanding that all freely falling observers, at all events in spacetime, must find that the equations of motion for matter reduce to their special relativistic form.
In the alternative perspective, Eq. (3) arises from our demand that the thermodynamic extremum principle should hold for all local Rindler observers. This is identical to the manner in which freely falling observers are used to determine how gravitational field influences matter. Demanding the validity of special relativistic laws for the matter variables, as determined by all the freely falling observers, allows us to determine the influence of gravity on matter. Similarly, demanding the maximization of entropy of horizons (plus matter), as measured by all local Rindler observers, leads to the dynamical equations of gravity.
In this review, we will examine several aspects of these features and will try to provide, in the latter part of the review, a synthesis of ideas which offers an interesting new perspective on the nature of gravity that makes the connection between horizon thermodynamics and gravitational dynamics obvious and natural. In fact, we will show that the thermodynamic underpinning goes far beyond Einstein’s theory and encompasses a wide class of gravitational theories.
The review is organized as follows. In Section 2 we will review several features of horizons which arise in different contexts in gravitational theories. In particular, we will describe some generic features of the spacetimes with horizons which will be important in the later discussions. In Section 3 we shall provide a simple derivation of the temperature of a (generic) horizon using path integral methods. Section 3.1 introduces the basic concepts related to path integrals and Section 3.2 applies them to a spacetime with horizon to obtain the temperature. Some alternate ways of obtaining the temperature of horizons are discussed in Section 3.3. The origin of Hawking radiation from matter that collapses to form a black hole is discussed in Section 3.4. The next two subsections describe the generalization of these ideas to theories other than Einstein’s general relativity. The relationship between horizon entropy and the Noether charge is introduced in Section 3.5 and the structure of Lanczos-Lovelock models is summarized in Section 3.6. Section 4 discusses several conceptual issues raised by the existence of temperature and entropy of the horizons, concentrating on the nature of degrees of freedom which contribute to the entropy and the observer dependence of the concept of entropy. This discussion is continued in the next section where we take up the connection between horizon thermodynamics and gravitational dynamics and describe several peculiar features which have no simple interpretation in the conventional approach. In Section 6 we provide a thermodynamic interpretation of the field equations of gravity in any diffeomorphism invariant theory. This forms the basis for the alternative perspective of gravity described in Section 7 in which we obtain the field equations of gravity from an entropy maximization principle. The last section provides a summary.
We will use the signature and units with . The Greek superscripts and subscripts will run over the spatial coordinates while the Latin letters will cover time coordinate as well as spatial coordinates.
2 Gravity and its horizons
2.1 The Rindler horizon in flat spacetime
The simplest context in which a horizon arises for a class of observers occurs in the flat spacetime itself. Consider the standard flat spacetime metric with Cartesian coordinates in the plane given by
where is the line element in the transverse space. The lines divide the plane into four quadrants (see Fig. 1) marked the right () and left () wedges as well as the past () and future () of the origin. We now introduce two new coordinates () in place of in all the four quadrants through the transformations:
for with the positive sign in and negative sign in and
for with the positive sign in and negative sign in . Clearly, is used in and . With these transformations, the metric in all the four quadrants can be expressed in the form
Figure 1 shows the geometrical features of the coordinate systems from which we see that: (a) The coordinate is timelike and is spacelike in Eq. (7) only in and where with their roles reversed in and with . (b) A given value of corresponds to a pair of points in and for and to a pair of points in and for . (c) The surface acts as a horizon for observers in . In particular, observers who are stationary in the new coordinates with constant, constant will follow a trajectory in the plane. These are trajectories of observers moving with constant proper acceleration in the inertial frame who perceive a horizon at . Such observers are usually called Rindler observers and the metric in Eq. (7) is called the Rindler metric. (The label in Fig. 1 corresponds to .)
2.2 The Rindler frame as the near-horizon limit
The Rindler frame will play a crucial role in our future discussions for two reasons. First, one can introduce Rindler observers even in curved spacetime in any local region. To do this, we first transform to the locally inertial frame (with coordinates ) around that event and then introduce the local Rindler frame with coordinates by the transformations in Eq. (5) and Eq. (6). Such a local notion is approximate but will prove to be valuable in our future discussions because it can be introduced around any event in any curved spacetime. Second, the Rindler (like) transformations work for a wide variety of spherically symmetric solutions to gravitational field equations. This general class of spacetimes can be expressed by a metric of the form
where the function has a simple zero at some point with a non-zero first derivative . A Taylor series expansion of near gives with . It is, therefore, obvious that near the horizon, located at , all these metrics can be approximated by the Rindler metric in Eq. (7). Hence Rindler metric is useful in the study of spacetime near the horizon in several exact solutions. (We are assuming that ; there are certain solutions — called extremal horizons — in which this condition is violated and ; we will not discuss them in this review.)
In the above analysis we started from a flat spacetime expressed in standard inertial coordinates and then introduced the transformation to Rindler coordinates. This transformation, in turn, brought in a pathological behaviour for the metric at . Alternatively, if we were given the metric in Rindler coordinates in the form of Eq. (7) we could have used the transformation in Eq. (5) and Eq. (6) to remove the pathological behaviour of the metric. In such a process we would have also discovered that a given value of () actually corresponds to a pair of events in the full spacetime thereby ‘doubling up’ the manifold. This process is called analytic extension.
In the case of metrics given by Eq. (8) the pathology at is similar to the pathology of the Rindler metric at . Just as one can eliminate the latter by analytic extension, one can also eliminate the singularity at in the metric in Eq. (8) by suitable coordinate transformation. Consider for example, the transformations from to by the equations
This leads to a metric of the form
where needs to be expressed in terms of using the coordinate transformations. The horizon now gets mapped to ; but it can be shown that the factor remains finite at the horizon.
The similarity between the coordinate transformations in Eq. (9) and Eq. (5) is obvious. (As in the case of Eq. (5), one can introduce another set of transformations to cover the remaining half of the manifold by interchanging and factors.) The curves of constant in the original spherically symmetric metric in Eq. (8) become hyperbolas in the plane, just as in the case of transformation from Rindler to inertial coordinates.
2.3 Horizons in static spacetimes
In the Rindler frame (as well as near the horizon in a curved spacetime), one can introduce another coordinate system which often turns out to be useful. This is done by transforming from () to where . Then the Rindler metric in Eq. (7) reduces to the form
and the coordinate transformation transformation corresponding to Eq. (5) becomes
The form of the metric in Eq. (11) also arises in a wide class of static (and stationary, though we will not discuss this case) spacetimes with the following properties: (i) The metric is static in the given coordinate system, ; (ii) vanishes on some 2-surface defined by the equation , (iii) is finite and non zero on and (iv) all other metric components and curvature remain finite and regular on . The line element will now be:
The comoving observers in this frame have trajectories constant, four-velocity and four acceleration which has the purely spatial components . The unit normal to the constant surface is given by . A simple computation now shows that the normal component of the acceleration , ‘redshifted’ by a factor , has the value
where the last equation defines the function . From our assumptions, it follows that on the horizon , this quantity has a finite limit ; the is called the surface gravity of the horizon.
These static spacetimes, however, have a more natural coordinate system defined in terms of the level surfaces of . That is, we transform from the original space coordinates in Eq.(13) to the set by treating as one of the spatial coordinates — which is always possible locally. The denotes the two transverse coordinates on the constant surface. The line element in the new coordinates will be:
where etc. are the components of the acceleration in the new coordinates. The original 7 degrees of freedom in are now reduced to 6 degrees of freedom in , because of our choice for . In Eq.(15) the spacetime is described in terms of the magnitude of acceleration , the transverse components and the metric on the two surface and maintains the independence. The is now merely a coordinate and the spacetime geometry is described in terms of all of which are, in general, functions of . In spherically symmetric spacetimes with horizon, for example, we will have if we choose . Important features of dynamics are usually encoded in the function . Near the surface, , the surface gravity, and the metric reduces to
where the second equality is applicable close to . This is the same metric as in Eq.(11) if we set . Therefore a wide class of metrics with horizon can be mapped to the Rindler form near the horizon.
The form of the metric in Eq. (11) is particularly useful to study the analytic continuation to imaginary values of time coordinate. If we denote , then, in the right wedge , the transformations become
which are just the coordinate transformation from the Cartesian coordinates to the polar coordinates in a two dimensional plane. To avoid a conical singularity at the origin, it is necessary that is periodic with period , which — in turn — requires to be periodic with period . We will see later that such a periodicity in the imaginary time signals the existence of non-zero temperature.
2.4 Exponential redshift and thermal power spectrum
Another generic feature of the horizons we have defined is that they act as surfaces of infinite redshift. To see this, consider the redshift of a photon emitted at , where is close to the horizon surface , and is observed at . The frequencies at emission and detection are related by . The radial trajectory of the out-going photon is given by which integrates to
where we have approximated the integral by the dominant contribution near . This gives , leading to the exponentially redshifted frequency
as detected by an observer at a fixed as a function of .
Such an exponential redshift is also closely associated with the emergence of a temperature in the presence of a horizon. To see this, let us consider how an observer in Rindler frame (or, more generally, in the spherically symmetric frame with the metric given by Eq. (8)) will view a monochromatic plane wave moving along the axis in the inertial frame (or, more generally, in the analytically extended coordinates). Such a scalar wave can be represented by with . Any other observer who is inertial with respect to the constant observer will see this as a monochromatic wave, though with a different (Doppler-shifted) frequency. But an accelerated observer, at constant using her proper time will see the same mode as varying as
where we have used Eq. (9) and defined . This is clearly not monochromatic and has a frequency which is being exponentially redshifted in time. The power spectrum of this wave is given by where is the Fourier transform of with respect to :
with and . Because of the exponential redshift, this power spectrum will not vanish for leading to . Evaluating this Fourier transform (by changing to the variable and analytically continuing to Im ) one gets:
This leads to the the remarkable result that the power, per logarithmic band in frequency, at negative frequencies is a Planckian at temperature :
Though in Eq. (22) depends on , the power spectrum is independent of ; monochromatic plane waves of any frequency (as measured by the freely falling observers with constant) will appear to have Planckian power spectrum in terms of the (negative) frequency , defined with respect to the proper time of the accelerated observer located at constant. The scaling of the temperature is precisely what is expected in general relativity for temperature. (Similar results also arise in the case of real wave with ; see ref. [47, 48].)
We saw earlier (see Eq. (18)) that waves propagating from a region near the horizon will undergo exponential redshift. An observer detecting this exponentially redshifted radiation at late times , originating from a region close to will attribute to this radiation a Planckian power spectrum given by Eq. (23). This result lies at the foundation of associating a temperature with a horizon.
The Planck spectrum in Eq. (23) is in terms of the frequency and has the (correct) dimension of time; no appears in the result. If we now switch the variable to energy and write , then one can identify a temperature which scales with . This “quantum mechanical” origin of temperature is superficial because it arises merely because of a change of units from to . An astronomer measuring frequency rather than photon energy will see the spectrum in Eq. (23) as Planckian without any quantum mechanical input. The real role of quantum theory is not in the conversion of frequency to energy but in providing the complex wave in the inertial frame. It represents the vacuum fluctuations of the quantum field.
2.5 Field theory near the horizon: Dimensional reduction
The fact that on the horizon leads to several interesting conclusions regarding the behaviour of any classical (or quantum) field near the horizon. Consider, for example, an interacting scalar field in a background spacetime described by the metric in Eq.(15), with the action:
where denotes the contribution from the derivatives in the transverse directions including cross terms of the type . Near , with , the action reduces to the form
where we have changed variable to defined in Eq. (9) (which behaves as near ) and ignored terms that vanish as . Remarkably enough this action represents a two dimensional free field theory in the coordinates which has the enhanced symmetry of invariance under the conformal transformations (see e.g., Section 3 of ,). The solutions to the field equations near are plane waves in the coordinates:
These modes are the same as where is the solution to the classical Hamilton-Jacobi equation; this equality arises because the divergence of factor near the horizon makes the WKB approximation almost exact near the horizon. The mathematics involved in this phenomenon is fundamentally the same as the one which leads to the “no-hair-theorems” (see, eg., ) for the black hole. These solutions possess several other symmetry properties which are worth mentioning:
To begin with, the metric and the solution near are invariant under the rescaling , in the sense that this transformation merely adds a phase to . This scale invariance can also be demonstrated by studying the spatial part of the wave equation  near , where the equation reduces to a Schrodinger equation for the zero energy eigenstate in the potential . This Schrodinger equation has the natural scale invariance with respect to which is reflected in our problem.
Second, the relevant metric in the plane is also invariant, up to a conformal factor, to the metric obtained by :
Since the two dimensional field theory is conformally invariant, if is a solution, then is also a solution. This is clearly true for the solution in Eq. (26). Since is a coordinate in our description, this connects up the infrared behaviour of the field theory with the ultraviolet behaviour.
More directly, we note that the symmetries of the theory enhance significantly near the hypersurface. Conformal invariance, similar to the one found above, occurs in the gravitational sector as well. Defining by , we see that near the horizon, where The space part of the metric in Eq.(15) becomes, near the horizon which is conformal to the metric of the anti-De Sitter (AdS) space. The horizon becomes the surface of the AdS space. These results hold in any dimension. There is a strong indication that most of the results related to horizons will arise from the enhanced symmetry of the theory near the surface (see e.g. [53, 54, 55, 56] and references cited therein). One can construct the metric in the bulk by a Taylor series expansion, from the form of the metric near the horizon, along the lines of Exercise 1 (page 290) of  to demonstrate the enhanced symmetry. These results arise because, algebraically, makes certain terms in the diffeomorphisms vanish and increases the symmetry. This fact will prove to be useful in Sec. 5.2.
2.6 Three specific examples of horizons
For the sake of reference, we briefly describe three specific solutions to Einstein’s equations with horizons having a metric of the form in Eq. (8), viz. the Rindler, Schwarzschild and de Sitter spacetimes. In each of these cases, the metric can be expressed in the form of Eq. (8) with different forms of given in the Table 1. All these cases have only one horizon at some surface and the surface gravity is well defined. (We have relaxed the condition that the horizon occurs at ; hence is defined as evaluated at the location of the horizon, .) The coordinate transformations relevant for analytic extension of these three spacetimes are also given in Table 1. The coordinates are well behaved near the horizon while the original coordinate system is singular at the horizon. Figure 1 describes all the three cases of horizons which we are interested in, with suitable definition for the coordinates.
The horizons with the above features arise in Einstein’s theory as well as in more general theories of gravity. While the detailed properties of the spacetimes in which these horizons occur are widely different, there are some key features shared by all the horizons we are interested in, which is worth summarizing:
In all these cases, there exists a Killing vector field which is timelike in part of the manifold with the components in the Schwarzschild-type static coordinates. The norm of this field vanishes on the horizon that acts as a bifurcation surface . Hence, the points of are fixed points of the Killing field. Further the surface gravity of the horizon can be defined using the ‘acceleration’ of the Killing vector by:
When defined in this manner, the value of depends on the normalization chosen for . (If we rescale , the surface gravity also scales as ). Very often, however, we will be interested in the combination , which is invariant under this scaling.
In these spacetimes, there exists a spacelike hypersurface which includes and is divided by into two pieces and , the intersection of which is in fact . In the case of black hole manifold, for example, is the surface, and are parts of it in the right and left wedges and corresponds to the surface. The topology of and depends on the details of the spacetime but is assumed to have a non-zero surface gravity. Given this structure it is possible to generalize most of the results we will be discussing in the later sections.
Finally, to conclude this section, we shall summarize a series of geometrical facts related to the Rindler frame and the Rindler horizons which will turn out to be useful in our future discussions. Though we will present the results in the context of a 2-dimensional Rindler spacetime, most of the ideas have a very natural generalization to other bifurcation horizons. We begin with the metric for the Rindler spacetime expressed in different sets of coordinates:
The coordinate transformations relating these have been discussed earlier. In particular, note that the null coordinates in the two frames are related by . We will now introduce several closely related vectors and their properties.
(i) Let be a future directed null vector with components proportional to in the inertial frame. The corresponding affinely parameterized null curve can be taken to be with being the affine parameter and subscript (or ) indicates the components in the inertial (or Rindler) frame.
(ii) We also have the natural Killing vector corresponding to translations in the Rindler time coordinate. This vector has the components, and . This shows that the bifurcation horizon is at the location where . The “acceleration” of this Killing vector is given by and hence, on the horizon, consistent with Eq. (28). It is also easy to see that, on the horizon, with .
(iii) Another natural vector which arises in the Rindler frame is the four-velocity of observers, moving along the orbits of the Killing vector . On the horizon , this four-velocity has the limiting behaviour .
(iv) Lastly, we introduce the unit normal to constant surface, which also has the limiting behaviour when we approach the horizon. It therefore follows that and all tend to vectors proportional to on the horizon. These facts will prove to be useful in our later discussions.
3 Thermodynamics of horizon: A first look
We shall now provide a general argument which associates a non-zero temperature with a bifurcation horizon. This argument, originally due to T.D. Lee, (; also see ref. ) is quite powerful and elegant and applies to all the horizons which we will be interested in. It uses techniques from path integral approach to quantum field theory, which we shall first review briefly.
3.1 Review of Path Integral approach
It is known in quantum mechanics that the net probability amplitude for the particle to go from the event to the event is obtained by adding up the amplitudes for all the paths connecting the events:
where (path) is the action evaluated for a given path connecting the end points and . The addition of the amplitudes allows for the quantum mechanical interference between the paths. The quantity contains the full dynamical information about the quantum mechanical system. Given and the initial amplitude for the particle to be found at , we can compute the wave function at any later time by the usual rules for combining the amplitudes:
The above expressions continue to hold even when we deal with several degrees of freedom which may still be collectively denoted as ; it is understood that the integral in Eq. (31) has to be performed over all the degrees of freedom.
To obtain the corresponding results in field theory, one needs to go from a discrete set of degrees of freedom (labeled by ) to a continuum of variables denoting the coordinates in a spacelike hypersurface. In this case the dynamical variable at time is the field configuration . (For every value of we have one degree of freedom.) The integral in Eq. (31) now becomes a functional integral over the initial field configuration and Eq. (31) becomes
We shall, however, not bother to indicate this difference between field theory and point quantum mechanics and will continue to work with latter since the generalizations will be quite obvious by context.
We will next obtain a relation between the ground state wave function of the system and the path integral kernel which will prove to be useful. In the conventional approach to quantum mechanics, using the Heisenberg picture, we will describe the system in terms of the position and momentum operators and . Let be the eigenstate of the operator with eigenvalue . The kernel — which represents the probability amplitude for a particle to propagate from to — can be expressed, in a more conventional notation, as the matrix element:
where is the time-independent Hamiltonian describing the system. This relation allows one to represent the kernel in terms of the energy eigenstates of the system. We have
where is the n-th energy eigenfunction of the system under consideration. Equation (34) allows one to express the kernel in terms of the eigenfunctions of the Hamiltonian. For any Hamiltonian which is bounded from below it is convenient to add a constant to the Hamiltonian so that the ground state — corresponding to the term in the above expression — has zero energy. We shall assume that this is done. Next, we will analytically continue the expression in Eq. (34) to imaginary values of by writing . The Euclidean kernel obtained from Eq. (34) has the form
Suppose we now set , in the above expression and take the limit . In the large time limit, the exponential will suppress all the terms in the sum except the one with which is the ground state for which the wave function is real. We, therefore, obtain the result
Hence the ground state wave function can be obtained by analytically continuing the kernel into imaginary time and taking a suitable limit. The proportionality constant in Eq. (36) is irrelevant since it can always be obtained by normalizing the wave function . Hence we have
where, in the arguments of , the first one refers to Euclidean time and the second one refers to the dynamical variable. The last equality is obtained by noting that in Eq. (36) we can take the limit either by or by . This result holds for any closed system with bounded Hamiltonian. Expressing the kernel as a path integral we can write this result in the form
This formula is also valid in field theory if is replaced by the field configuration .
The analytic continuation to imaginary values of time also has close mathematical connections with the description of systems in thermal bath. To see this, consider the mean value of some observable of a quantum mechanical system. If the system is in an energy eigenstate described by the wave function , then the expectation value of can be obtained by integrating over . If the system is in a thermal bath at temperature , described by a canonical ensemble, then the mean value has to be computed by averaging over all the energy eigenstates as well with a weightage . In this case, the mean value can be expressed as
where is the partition function and we have defined a density matrix by
in terms of which we can rewrite Eq. (39) as
where the trace operation involves setting and integrating over . This standard result shows how contains information about both thermal and quantum mechanical averaging. Comparing Eq. (40) with Eq. (34) we find that the density matrix can be immediately obtained from the Euclidean kernel by:
with the Euclidean time acting as inverse temperature.
3.2 Horizon temperature from a path integral
We shall now consider the quantum field theory in a spacetime with a horizon which can be described in two different coordinate systems. The first one () is a global coordinate system which covers the entire spacetime manifold (which could be the inertial Cartesian coordinate system in flat spacetime or the Kruskal-like coordinate system in the case of spherically symmetric metrics with horizon). The second one () covers the four different quadrants of the spacetime and is related to the first set by a set of transformations similar to Eq. (9). We shall now show that the global vacuum state defined on the surface appears as a thermal state to observers confined to the right wedge with a temperature .
On analytic continuation to imaginary time, the two sets of coordinates behave as shown in Fig. 2. The key new feature is that becomes an angular coordinate having a periodicity . While the evolution in (effected by the inertial Hamiltonian ) will take the field configuration from to , the same time evolution gets mapped in terms of into evolving the “angular” coordinate from to and is effected by the Rindler Hamiltonian . (This should be clear from Fig. 2.) It is obvious that the entire upper half-plane is covered in two completely different ways in terms of the evolution in compared to evolution in . In coordinates, we vary in the range for each and vary in the range . In coordinates, we vary in the range for each and vary in the range . This fact allows us to prove that
as we shall see below.
To provide a simple proof of Eq. (43), let us consider the ground state wave functional in the extended spacetime expressed as a path integral. From Eq. (38) we know that the ground state wave functional can be represented as a Euclidean path integral of the form
where is the Euclidean time coordinate and we have denoted the field configuration on the hypersurface by . But we know that this field configuration can also be specified uniquely by specifying with and with . Hence we can write the above result in terms of and as
From Fig. 2 it is obvious that this path integral could also be evaluated in the polar coordinates by varying the angle from 0 to . When the field configuration corresponds to and when the field configuration corresponds to . Therefore Eq. (45) can also be expressed as
But in the Heisenberg picture, ‘rotating’ from to is a time evolution governed by the Rindler Hamiltonian . So the path integral Eq. (46) can be represented as a matrix element of the Rindler Hamiltonian giving us the result:
proving Eq. (43).
If we denote the proportionality constant in Eq. (43) by , then the normalization condition
fixes the proportionality constant , allowing us to write Eq. (43) in the form:
From this result, we can compute the density matrix for observations confined to the Rindler wedge by tracing out the field configuration on the left wedge. We get:
Thus, tracing over the field configuration in the region behind the horizon leads to a thermal density matrix for the observables in .
The main ingredients which have gone into this result are the following. (i) The singular behaviour of the coordinate system near divides the hypersurface into two separate regions. (ii) In terms of real coordinates, it is not possible to distinguish between the points and but the complex transformation maps the point to the point . That is, a rotation in the complex plane (Re , Im ) encodes the information contained in the full plane.
In fact, one can obtain the expression for the density matrix directly from path integrals along the following lines. We begin with the standard relation in Eq. (37) which gives
where, in the arguments of , the first one refers to Euclidean time and the second one refers to the dynamical variable. The density matrix used by the observer in the right wedge can be expressed as the integral