^{3}

One of the pronounced characteristics of gravity, distinct from other interactions, is that there are no local observables which are independent of the choice of the spacetime coordinates. This property acquires crucial importance in the quantum domain in that the structure of the Hilbert space pertinent to different observers can be drastically different. Such intriguing phenomena as Hawking radiation and the Unruh effect are all rooted in this feature. As in these examples, the quantum effect due to such observer dependence is most conspicuous in the presence of an event horizon and there are still many questions to be clarified in such a situation. In this paper we perform a comprehensive and explicit study of the observer dependence of the quantum Hilbert space of a massless scalar field in the vicinity of the horizon of Schwarzschild black holes in four dimensions, both in the eternal (two-sided) case and in the physical (one-sided) case created by collapsing matter. Specifically, we compare and relate the Hilbert spaces of three types of observers, namely (i) the freely falling observer, (ii) the observer who stays at a fixed proper distance outside of the horizon, and (iii) the natural observer inside of the horizon analytically continued from outside. The concrete results we obtain have a number of important implications on black hole complementarity pertinent to the quantum equivalence principle and the related firewall phenomenon, on the number of degrees of freedom seen by each type of observer, and on the “thermal-type” spectrum of particles realized in a pure state.

A quantum black hole is a fascinating but as yet an abstruse object. Recent endeavors to identify it in a suitable class of conformal field theories (CFTs) in the AdS/CFT context [

Although the quantization of a black hole itself is still a formidable task, some analyses of quantum effects around a (semi-)classical black hole have been performed since a long time ago, and they have already uncovered various intriguing phenomena. Among them are the celebrated Hawking radiation [

More recently, a further unexpected quantum effect in the black hole environment was argued to occur, namely that a freely falling observer encounters excitations of high-energy quanta, termed a “firewall,” as he/she crosses the event horizon of a black hole [^{1}

At the bottom of these phenomena lies the strong dependence of the quantization on the frame of observers, which is one of the most characteristic features of quantum gravity. This is particularly crucial when the spacetime of interest contains event horizons as seen by some observers, and leads to the notion of black hole complementarity [

The main aim of the present work is to investigate this observer dependence in some physically important situations as explicitly as possible to gain some firm and direct understanding of the phenomena rooted in this feature. For this purpose, we shall study the quantization of a massless scalar field in the vicinity of the horizon of the Schwarzschild black hole in four dimensions as perfomed by three typical observers. They are (i) the freely falling observer crossing the horizon, (ii) the stationary observer hovering at a fixed proper distance outside the horizon (i.e. one under constant acceleration), and (iii) the natural analytically continued observer inside the horizon.

Such an investigation, we believe, will be important for at least two reasons. One is that we will deal directly with the states of the scalar fields as seen by different observers and will not rely on any indirect arguments alluded to above. This makes the interpretation of the outcome of our study quite transparent (up to certain approximations that we must make for computation). Another role of our investigation is that the concrete result we obtain should serve as the properties of quantum fields in the background of a black hole, which should be compared, in the semi-classical regime, to the results to be obtained by other means of investigation, notably and hopefully by the AdS/CFT duality.^{2}

We will perform our study both for the case of a two-sided eternal Schwarzschild black hole and for that of a one-sided physical black hole modeled by a simple Vaidya metric produced by collapsing matter or radiation at the speed of light^{3}

In making use of this flat space approximation to the near-horizon region of a black hole, an important care must be taken, however. Although the scalar field and its canonical conjugate momentum are locally well-approximated by those in the flat space for the region of our interest, and hence the canonical quantization can be performed without any problem, as we try to extract the physical modes that create and annihilate the quantum states, such local knowledge is not enough in general. This is because the notion of

One such problem, which, however, can be easily dealt with, stems from the simple fact that the approximation by the four-dimensional flat space includes that of the spherical surface of the horizon by a tangential plane around a point. Clearly, since the physical modes of the scalar field should better be classified by the angular momentum, not by the linear momentum, we shall use

The problem pointed out above of the extraction of the modes within the flat region is much more non-trivial in the near-horizon region of ^{4}

Although we cannot summarize here all the results on how the different observers see their quanta and how they are related, let us list two that are of obvious interest:

Under the assumption that the metric of the interior of a physical Schwarzschild black hole, in particular one large enough so that the curvature at the horizon is very small, can be described by a Vaidya-type solution, our results indicate that the equivalence principle still holds quantum mechanically near the horizon of the black hole, and the freely falling observer finds no surprise as he/she goes through the horizon.

For a physical (one-sided) black hole, the vacuum^{5}^{6}

The plan of the rest of the paper is as follows: In

We begin by describing the quantization of a massless^{7}

Trajectories and equal-time slices of the Rindler observers in various wedges. The boundaries of the wedges W

The subject of quantization by Rindler observers has a long history [^{8}

Before getting to the quantization of a scalar field, we need to describe the relationship between the Minkowski coordinates and the Rindler coordinates in various wedges.

The

Since we will be mostly concerned with the first two coordinates, and the roles of the rest of the

As for the Rindler coordinates, we begin with the one in the right wedge W

Here, the symbol

The metric in terms of these variables is

Note that

This shows that

The coordinates

The metric takes exactly the same form as Eq. (

Next, consider the future and the past wedges, W

This means that in W

Just like

This interchange of the timelike and the spacelike natures also occurs in the past wedge W

In

One can easily check that these relations are compatible with the relations between the Minkowski variables and the Rindler wedge variables given above.

We now discuss the quantization of a massless scalar field

In this subsection, just for setting the notation, we summarize the simple case for the Minkowski coordinate. The action, the canonical momentum, and the equation of motion are given by

Here and throughout, we often denote ^{9}

Using the orthogonality of the exponential function, we can easily extract the mode operators and check that they satisfy the usual commutation relations:

Let us now begin the discussion of quantization in the Rindler coordinates in various wedges.

We first consider the Rindler wedges outside the horizon, namely W

The canonical momentum is given by

As it is a second-order differential equation, there are two independent solutions, which can be taken to be the exponential function times the modified Bessel functions, namely ^{10}

Let us make some remarks on this formula:

(1) For the Hermitian conjugate part, only the conjugation for the exponential part is needed since

(2) The normalization constant chosen here will lead to the canonical form of the commutation relations, as explained in

(3) The variable

Canonical quantization is performed by imposing the following equal-time commutation relation:

Using the orthogonality relation for the modified Bessel functions explained in

For some details of the calculations, see

The quantization in W

One can then check that the equal-time commutation relation

Next, consider the quantization in the Rindler wedges inside the horizon, i.e. in W

The action in the W

From the signs of various terms, it is clear that

The equation of motion takes the form

There are again two independent solutions, which can be taken as

To expand the scalar field in terms of these functions, care should be taken as to which function should be associated with the annihilation (resp. creation) modes. This is because, in contrast to the previous case,

To guess which Hankel function should be taken as describing the positive frequency part, it is physically natural to first look at the asymptotic behavior of

We see that

To check that such an association is actually the correct one, one must compute the “equal-time” (i.e. equal-

Using the orthogonality of the Hankel functions, we get the canonical form of the commutation relations for creation/annihilation operators:

See

We have seen that in W

From the action in Eq. (

Since

We now wish to express

Now let us consider the limit of large time,

This is independent of

We are ready to discuss the relation between the quantizations in W

First, since W

Actually, this expression for

Then we can immediately solve this relation for

Furthermore, the integration measures are related as

Using these definitions, the relation in Eq. (

Note that, as is well known, the annihilation operator ^{11}

The situation is different for the quantization in the W

As in the case of

Apart from a factor of

The fact that ^{12}

The important difference, however, is that the entities recognized as “particles” by the two observers are quite distinct and their wave functions have “dual” profiles.

We now make a useful observation that the Fourier transform exhibited above can be realized by a unitary transformation, in the sense to be described below.^{13}

Define the Fourier transform

The

Let us look for a special class of functions for which the functional forms of their Fourier transform are the same up to a proportionality constant. The simplest such function is obviously the following Gaussian, for which the proportionality constant is unity:

We know that such a function is the coordinate representation of the ground state of the one-dimensional harmonic oscillators

In what follows, we take the coordinate representations of

Now, as is well known, the

Inserting the unity

Thus, the functional form of the Fourier transform

Let us consider the number operator

Therefore, we can express the Fourier transform

Note that here the terminology “Fourier transform” refers to the transform of the

Exactly the same formulas hold for

Up to a constant, ^{14}

Now, in order to apply this formalism to the oscillators such as

Since so far we have realized the Fourier transform as a differential operation on the set of functions

Thus, expanding

In fact, one can easily verify that

So, the Fourier transform for the form of the operator is indeed reproduced.

Applied to the oscillators

In using the operators

Finally, let us relate the modes in the W

Again, since

However, recall that the “full” Rindler spacetime has the left wedge W

Note that the right-hand side contains

Intuitively, this is a reflection of the fact that the region W

As an application of the formulas in Eqs. (

The solution is
^{15}

Let us make a few remarks on the relation between the field expressed in the Minkowski frame and in the combined W

In the context of the discussions of the entanglement and the entropy thereof, instead of the expression in Eq. (

By using the relations in Eqs. (^{16}

Evidently, the Poincaré symmetry of the flat Minkowski space is a fundamental symmetry governing, above all, the structure of correlation functions. Although the quantum generators of the Poincaré algebra are well known in the ordinary Minkowski frame, their forms are non-trivial in terms of the modes of observers in the W

As described in the next section, the vicinity of the horizon of the four-dimensional Schwarzschild black hole we are interested in has the structure of the

As usual, the Poincaré generators can be constructed in terms of the energy–momentum tensor, which for a scalar field takes the form

Here,

The subscripts

Recall that the relation between the Minkowski modes

Note that this is diagonal in

Next, let us rewrite the Hamiltonian operator in terms of the Unruh modes. Using the rapidity representation, with

Since the integral over

Then, expanding

The remaining Gaussian integral produces a

In an entirely similar manner, the

These operators are understood to be used within a matrix element such that the object is infinitely differentiable with respect to

In

Having derived the expressions of the generators in terms of W

As for the generator of the angular momentum, it cleanly separates into a W

Two remarks are in order:

First,

Second, the relative minus sign between

These remarks are expressed by the following simple relations:

Thus, acting on the field, the boost generator indeed induces the Rindler time evolution in each wedge as shown below:

In contrast, the generators

Note that the last term in both

As emphasized in the introduction, the main aim of this paper is to study the structure of the Hilbert spaces of the scalar field near the horizon of the Schwarzschild black hole quantized in the frames of different observers. This is made possible largely because such near-horizon geometry has the structure of the flat Minkowski space, to be recalled shortly. This allows us to make use of the knowledge of the quantization in various frames which has been reviewed, with some additional new information, in the previous section. As we shall discuss, however, we must take due care that our computations should be performed in such a way that the approximation used is legitimate.

Now, in studying the quantization around the horizon of a black hole, it will be important to distinguish two cases, namely the case of the eternal (i.e. two-sided) black hole and the more physical one where the (one-sided) black hole is produced by a collapse of matter (or radiation). There are essential differerences between the two.

In this section, we analyze the simpler case of the eternal Schwarzschild black hole.

Let us first recall how the flat geometry emerges in the vicinity of the event horizon of a Schwarzschild black hole.

We denote the metric for the four-dimensional Schwarzschild black hole of mass

The notations are standard, except that we set the Newton constant

First, we consider the region W

Near the horizon at

Now, if we keep up to the second term of this expansion, the Schwarzschild metric becomes

Further, focusing on the small two-dimensonal region perpendicular to the radial direction around

Then in this region the metric further simplifies and becomes identical to the Rindler metric for W^{17}

To see the region of validity of this approximation, let us find out the condition under which we can neglect the third term of the expansion in Eq. (

For the other regions W

The approximation of the vicinity of the horizon as a four-dimensional flat Minkowski space is certainly a great advantage, as long as we are interested only in the quantities determined by the local properties of the fields. However, as we have repeatedly emphasized, in a quantum treatment the concept of states created by the mode operators is a global one, and that is precisely what we are interested in. It turns out that the inadequacy of the flat approximation is particularly troublesome for the two-dimensional transverse space, since the orthogonality relation needed to extract the modes from the fields requires integration over the entire range of

The obvious cure for this part of the problem is to replace the expansion in terms of the plane waves by the spherical harmonics

Explicitly, we can write the general expansions of a massless scalar and its conjugate in the vicinity of the horizon in the form

The equal-time canonical commutation relation takes the form

The orthogonality for

Using the orthogonality relation, we can extract the modes as

From the canonical commutation relation in Eq. (

In summary, the expansion in flat space described in

As the behavior of the scalar field on the transverse spherical surface near the horizon is treated exactly as above, we need only be concerned with the dependence on the remaining two dimensions,

Among the many interesting questions that stem from the observer dependence of the quantization around a black hole, perhaps the most provocative one is whether the freely falling observer, hereafter abbreviated as FFO, sees a different Hilbert space structure for the quantized scalar field before and after he/she passes through the horizon. In other words, whether the equivalence principle for the field is affected by the quantum effects or not.

In this subsection we will perform some preparatory computations in the frame of an FFO who crosses the horizon along various directions in the Penrose diagram, i.e. with various velocities.

First, let us briefly describe how the geodesic of a massive classical particle (which represents an FFO) near the horizon maps to the motion in the flat coordinate system obtained by the non-linear transformation of the previous subsection. Although the final answer should be a straight line in the flat coordinate system, as the geodesic should map to a geodesic, it is instructive to see the physical meaning of this mapping.

Consider first the motion in W

Here,

This differential equation for ^{18}

Now let us rewrite this motion of Eq. (

As expected, this describes a family of timelike straight line trajectories with velocity

Then the relation to the canonical Minkowski variables

This is nothing but the Lorentz boost by the velocity

One can perform a similar analysis of the geodesic in the W

With this preparation, let us now discuss the quantization and the mode expansion of the free scalar field by an FFO in the vicinity of the horizon where the flat space approximation for the dependence on

Approximately flat regions, shown in gray, near the horizon of the two-sided Schwarzschild black hole and the corresponding region in the Rindler coordinates of the flat spacetime.

In this region the general solution for a scalar field as seen by an FFO is

This expression is perfectly valid

The observation that allows us to overcome this obstacle is that regions of infinite range do exist around the horizon along the lightcones in the

As the argument for W

In the approximately flat region near the horizon, the scalar field

Contrary to the case of the Minkowski frame discussed above, the extraction of the modes in W

This means that in the flat region around the horizon, the number of modes is the same between a W

It is easy to see that this is indeed compatible with the Fourier transform relation in Eq. (

We now come to the more difficult situation of the quantization from the viewpoints of a W

What we can check easily is that, if we assume the canonical form of the commutation relations for the modes as in the flat space, then by using the completeness relation,

Actually, we can argue that the relation between

(1) In the flat region, using completeness, we can re-expand the field

(2) Another argument goes as follows: For simplicity, consider the case where we try to use the orthogonality integral along the spacelike straight line at

These arguments indicate that, as far as the flat region near the horizon is concerned, the relations between the modes for the FFO and the observers in various Rindler frames should be the same as those already exhibited in

Black holes of more physical interest are the ones formed by a collapse of matter, as actually occurs in nature. They are “one-sided” and have rather different spacetime structures compared with the two-sided eternal black holes discussed in the previous section.

In this section, we investigate how the observers in various frames quantize a massless scalar field in the simplest model of a Schwarzschild black hole of such a type, namely the so-called Vaidya spacetime [^{19}

Let us begin by recalling the basics of such a Vaidya spacetime. After a black hole is formed by spherical collapse, by Birkohoff’s theorem the metric outside the horizon is always that of the Schwarzschild black hole. On the other hand, for the simplest situation above, the metric inside is isomorphic to part of the flat Minkowski space. Thus the Penrose diagram of the entire spacetime is obtained by gluing these two types of geometries along the lightlike line representing the falling shell, as shown in

Penrose diagram of the simplest Vaidya spacetime. It consists of two parts: one is the flat region inside the matter shell (

The Vaidya metric is a solution of the Einstein equation

The delta function at

The metric above consists of two parts, one of which corresponds to the region inside the shock wave,

Notice that the two time coordinates,

In the subsequent subsections we mainly focus on the special limit of the Vaidya spacetime, for which the collapse of the matter has taken place a very long time ago so that the flat space region in

In order to be able to study the quantization of a scalar field in an explicit manner, in what follows we shall (i) make a reasonable assumption about the effect of the matter shock wave on the field, and (ii) implement it in a well-defined way by making a regularization which replaces the lightlike trajectory by a slightly timelike one.

As for (i), since we focus on the Schwarzschild region outside of the locus of the shock wave, the effect of the shock wave on the scalar field ^{20}^{21}^{22}

Next, let us elaborate on point (ii). If we take the boundary to be strictly lightlike, i.e. along

Another advantage of such a regularization is the following. As will become evident, the effect of the boundary condition on the quantization can easily be taken into account in the frame of an FFO moving in the direction of the shock wave. When this direction is slightly timelike, we can change it by a Lorentz transformation into the case for a general FFO moving with any velocity. On the other hand, even if we could manage to treat the case of the strictly lightlike shock wave and an FFO moving along such a direction, we cannot relate such an observer by a Lorentz boost to a general FFO moving with a finite velocity.

In this section we explicitly perform the quantization of a scalar field with the boundary condition imposed along a slightly timelike line from the point of view of FFOs traversing the horizon with various velocities.

In what follows we will concentrate on the flat two-dimensional portion in

We shall take the boundary line to be the one expressed by (see

Slightly timelike boundary line (shown in red).

We should remark that this corresponds to the case where the shell of matter collapses along the line for which the so-called tortoise light-cone coordinate

The third set of coordinates to be introduced is

It will also be convenient to relate the frame

We begin with the quantization in the

The quantized scalar field that vanishes for

This clearly shows that the mode with negative ^{23}

Let us define for convenience the following combinations of the mode operators:

Clearly, the operators with plus and minus superscripts commute with each other. Then, from the discussion above, the Hilbert space

This structure will be important in the discussion of the Unruh-like effect near the horizon of a physical Schwarzschild black hole in

We now consider the quantization by an FFO in a general frame ^{24}

Now let us compute the

(1) Since

(2) The second point to keep in mind is that from the Lorentz transformation we easily find that

With these facts in mind, the equal-

Note that for the the exponents involving

Since the rest of the calculations are somewhat tedious but more or less straightforward, we shall describe some intermediate steps in

The terms which contain

On the other hand, for the terms containing the difference

In contrast, when the difference is of order

Together, this is nothing but the behavior of the

In the other limit, where

Then, we are left with the sum over

Combining, we find that the commutator is proportional to the desired product of

Two remarks are in order:

(1) Although the correct

(2) Nevertheless, the fact that the quantization for a general FFO with the boundary condition

We now consider the quantization in the W

Since

Now we impose the vanishing condition for

Using the relation

We now impose the boundary condition

Notice that this argument is valid even when we take the value of

Thus, the conclusion is that the boundary condition places the relations in Eq. (

Finally, let us consider the quantization by an observer in the W

If we take the same special slightly timelike boundary line which goes through the origin of the coordinate frame

Although this is a valid argument, it certainly depends crucially on the special choice of the boundary line. Therefore we should also consider the case where the boundary line is slightly shifted to the positive

A slightly timelike boundary line (shown in red), which is shifted infinitesimally in the positive

As discussed in ^{25}

Since Eq. (

Note that this oscillates wildly as

Therefore, the conclusion should be the same as in the case of the W

Having analyzed and compared the quantizations of a scalar field by different natural observers in a concrete manner, we now consider the implications of our results.

One of the clear results is that the degrees of freedom of the modes that the observer sees are in general different, both for the case of the two-sided eternal black hole and the more physical one-sided one. Explicitly, the FFO and the W

The fact that the sizes of the quantum Hilbert spaces are halved for FFO and W

Whether the equivalence principle holds quantum mechanically is quite a different question. It asks whether the FFO, upon crossing the “horizon,” which does not exist for him/her

Needless to say, this conclusion is valid under the assumption that the metric of the interior of the Schwarzschild black hole is essentially given by the Vaidya-type metric. If the interior of the black hole is such that it cannot be specified just by the information of the metric, the conclusion may differ. However, as far as classical Schwarzschild black holes produced by the collapse of matter are concerned, our assumption is conservative and should be reasonable.

Thus, for a large enough black hole that itself can be treated classically, with a small value of curvature at the horizon, our explicit computations for the quantum effects of the massless scalar field as seen by the three types of observers should be reliable; in particular, the freely falling observer does not encounter the so-called firewall phenomenon.

The Unruh effect [

Evidently, this coincides with the thermal distribution of bosons at temperature

Although in this example the background is taken to be the flat spacetime to begin with, one might expect a similar phenomenon to be seen by a stationary observer just outside the horizon of a physical Schwarzschild black hole, since the spacetime there is well-approximated by the right Rindler wedge of a flat Minkowski space.

However, the analysis cannot be the same for the following reasons. First, there is no W

Thus, the question of interest is the distribution of the W^{26}

Unfortunately, in general this computation cannot be performed accurately due to our lack of knowledge of the fields outside the approximately flat region. The required calculation is of the form

There is, however, a class of modes for which the computation can be performed sufficiently accurately using the function for the flat space region. These are the ones with large angular momentum

Now we look at the region

The solution is well known and is given, with a certain normalization, by

This shows that for

To perform the computation of Eq. (

Therefore, after the removal of the angular part, what we need to compute is

To perform the differentiation with respect to

These expressions can be further simplified by introducing the parametrization

Then, we can write

Now we consider the integral over

Further, it is convenient to use the rapidity-based oscillators

Now, using these formulas, it is straightforward to compute the right-hand side of the formula in Eq. (

One can check that they satisfy the correct commutation relation,

Finally, with the expressions in Eq. (

Several remarks are in order:

(1) We recognize that the first factor is of the same form as the familiar “thermal” distribution. We emphasize, however, that in this case it is not genuinely thermal since W

(2) The last integral represents the coherent sum over an infinite number of rapidities which contribute to the W

(3) As the last remark, note that the dependence on

In any case, we have found that, even in the case of the one-sided black hole, the Unruh-like effect does exist.

It is instructive to compare this with the case of the usual Unruh effect. From Eq. (

In this work we have made a detailed study of the issue of observer dependence for the quantization of fields in a curved spacetime, which is one of the crucial problems that one must deal with whenever one discusses quantum gravity. Understanding this issue is particularly important in cases where an event horizon exists for some of the observers. Explicitly, we have focused on the quantization of a scalar field in the most basic such configuration, namely the spacetime in the vicinity of the horizon of a four-dimensional Schwarzschild black hole, including the interior as well as the exterior. Detailed and comprehensive analyses were performed for the three typical observers, clarifying how the modes they observe are related. We studied both the two-sided eternal case and the more physical one-sided case produced by the falling shell, or a shock wave. For the latter, the effect of the collapsing matter upon the scalar field outside of the shell is represented by an effective boundary condition along the shock wave.

One important conclusion obtained from such explicit calculations is that as long as the interior of a large black hole can be described more or less by a metric like that of Vaidya, the free-falling ovserver sees no change in the Hilbert space structure of the quantized field as he/she crosses the horizon. In other words, the equivalence principle holds quantum mechanically as well, at least in the above sense.

Another result worth emphasizing is that in the one-sided case despite the fact that there are no counterparts of the left Rindler modes in the vacuum of the freely falling observer, and hence no tracing procedure over them is relevant, there still exists an Unruh-like effect. Namely, in such a vacuum the number density of the W

In addition to these results, comprehensive and explicit knowledge of the properties and the relations of the Hilbert spaces for the different observers have been obtained, and we believe this will be of use in better understanding the quantum properties of gravitational physics.

Evidently, the problem of observer dependence that we studied in the semi-classical regime in this work is of universal importance in any attempt to understand quantum gravity. In particular, it would be extremely interesting to see how this problem appears and should be treated in the construction of the “bulk” from the “boundary” in the AdS/CFT correspondence, which is anticipated to give important hints for formulating quantum gravity and understanding quantum black holes. Although there have been some attempts to address this question, it is not well understood how the change of frame (i.e. the choice of “time”) for the quantization, both in the bulk and the boundary, is expressed and controlled in the AdS/CFT context. The best place to look into would be the AdS

In this work we have concentrated on the relations between the modes seen by different observers and have not touched upon the correlation functions between the fields. Some two-point correlation functions in the Rindler wedges of the Minkowski space have been studied [

Y.K. acknowledges T. Eguchi for a useful discussion. The research of K.G. is supported in part by a JSPS Research Fellowship for Young Scientists, while that of Y.K. is supported in part by Grant-in-Aid for Scientific Research (B) No. 25287049, both from the Japan Ministry of Education, Culture, Sports, Science and Technology.

Open Access funding: SCOAP

For various basic computations performed in the main text using the expansions in terms of the eigenmodes, the orthogonality and the completeness of the modified Bessel functions of imaginary order are essential. In this appendix, we give some useful comments on such relations previously obtained in the literature and provide additional information.

The orthogonality relations are needed in extracting each mode from the expansion of the scalar field appropriate for various coordinate frames. Such a relation for

The corresponding relations for the Hankel functions

The completeness relation for the function

Since

This relation is equivalent to the inverse of the so-called Kontorovich–Lebedev (KL) transform [

Then,

If we take

In fact, without resorting to the KL formula, there is a rather elementary derivation of Eq. (

For

By making a rescaling

Comparing with Eq. (

The corresponding completeness relations for the Hankel functions are given by

In this appendix we provide some details of the computations concerning the extraction of the modes and their relations described in

We are interested in a

Hereafter, we will set

In this case, we can identify

The solutions of the Klein–Gordon equation in this coordinate frame are

Let us compute the Klein–Gordon inner product of such functions explicitly. We get

Recall that the scalar field in the right Rindler wedge can be expanded as

The modes

In the future Rindler wedge, the metric is given by

In this case,

Then the solutions of the Klein–Gordon equation which damp at large

The Klein–Gordon inner product of these functions is given by

To get to the last line, we used the identity

By similar manipulations, it is easy to get the following inner products:

Recall that the scalar field in the future Rindler wedge can be expanded as

Taking the inner product with

We shall first derive several useful integrals involving

By extending the region of ^{27}

We now act

We now make the substitution

This formula is obtained simply from

The free scalar field in the right Rindler wedge can be expanded as in Eq. (

Let us now use the convenient parametrization

Now, by using Formula I given in Eq. (

This is the formula quoted in Eq. (

As in W

Using the Klein–Gordon inner product, we can extract

We now use the following integral representations [

Note that formula (i) can be obtained by the analytic continuation

For the part of Eq. (

In the last step, we used the identity in Eq. (

Together with the similar result for

This is the relation quoted in Eq. (

In this appendix we give a sketch of the proof that the scalar field in the Minkowski space

As in Eq. (

Now, substitute the expression of

We must study the conditions under which this integral exists. First, for

Then, we can make use of the formula in Eq. (10.9.16) of Ref. [

To express

We then get

Now, rather than displaying the complete expression for

Let us note that

Consider now the region W

First consider the region of W

Using this relation, it is easy to see that

Next, consider the region in W

Combining, we have shown that

In this appendix we shall demonstrate that the generators

First, consider the commutator

In an entirely similar manner, with

Finally, the fact that

Recall that the unitary transformation

As an application of this operation, let us show that it transforms

First, expand the unitary transformation as the sum of multiple commutators in the usual way:

The single commutator can be computed as

Based on this result, the double commutator is calculated as

Since this is in the original form of

Making the replacements

Here we supply some details of the quantization in different Lorentz frames with a slightly timelike boundary condition as discussed in

What we shall describe are the computations of the two terms in Eqs. (

First, the sum over

Then, the result for

Consider first the four terms in the third and the fourth lines, which contain

To analyze this expression we must distinguish two regions:

(1) If

(2) Thus, a non-vanishing result can possibly be obtained if and only if

Combining, this shows that the sum of terms containing the

Thus, we can write

^{1}It is practically impossible to list all such papers on this subject. We refer the reader to those citing the basic papers, Refs. [

^{2}As far as the vicinity of the horizon is concerned, the Schwarzschild black hole and the AdS black hole have the same structure.

^{3}Actually, we shall make an infinitesimal regularization to make the trajectory of the matter slightly timelike in order to avoid a certain singularity.

^{4}Evidently this corresponds to the case of slightly massive falling matter, which is physically reasonable.

^{5}The vacuum referred to here will be explained in

^{6}For related work, though in a different setting, see Ref. [

^{7}The massive case can be treated in an entirely similar manner.

^{8}For a review article closely related to this section, see Ref. [

^{9}

^{10}Let us make a remark on the boundary condition at

^{11}To avoid any confusion, let us stress that what we mean by a “Minkowski observer in

^{12}The vacuum

^{13}For related references, see Refs. [

^{14}There are different conventions for the normalization of the Hermite polynomials. Our definition is the most standard one.

^{15}The normalization constant

^{16}In the basic literature such as Refs. [

^{17}The approximation of taking

^{18}Actually, there is another solution with the sign in front of the

^{19}For a review, see, for example, Ref. [

^{20}Actually, there is another possibility: that the scalar field does not interact with the shock wave. In such a case, one needs to smoothly connect the solutions in the different spacetimes inside and outside the matter shell, and in

^{21}For a massless scalar, by using the invariance of the action under a constant shift, we can do so without loss of generality.

^{22} The boundary condition we introduce here should not be confused with the one considered in the so-called moving mirror model in the two-dimensional gravity theory discussed in the literature (see, for example,

^{23}Since

^{24}To get the explicit form of

^{25}Due to the use of the spherical harmonics instead of the plane wave in the transverse directions, the normalization factor

^{26}The reason for focusing on the FFO in the

^{27}This is purely as a mathematical equality. The physical energy