# Inside the Schwarzschild-Tangherlini black holes

###### Abstract

The first-order semiclassical Einstein field equations are solved in the interior of the Schwarzschild-Tangherlini black holes. The source term is taken to be the stress-energy tensor of the quantized massive scalar field with arbitrary curvature coupling calculated within the framework of the Schwinger-DeWitt approximation. It is shown that for the minimal coupling the quantum effects tend to isotropize the interior of the black hole (which can be interpreted as an anisotropic collapsing universe) for and whereas for and the spacetime becomes more anisotropic. Similar behavior is observed for the conformal coupling with the reservation that for isotropization of the spacetime occurs during (approximately) the first of the lifetime of the interior universe. On the other hand, we find that regardless of the dimension, the quantum perturbations initially strengthen the grow of curvature and its later behavior depends on the dimension and the coupling. It is shown that the Karlhede’s scalar can still be used as a useful device for locating the horizon of the quantum-corrected black hole, as expected.

###### pacs:

04.62.+v, 04.70.Dy, 04.50.-h## I Introduction

Although described by the same line element, the classical interior of the Schwarzschild-Tangherlini Tangherlini (1963) black hole has entirely different properties than the region outside the event horizon and can be better understood as some sort of the anisotropic and nonstatic universe Novikov (1961); Brehme (1977); Doran et al. (2008). This interpretation (not mandatory) is helpful when one is forced to abandon usual, i.e., referring to the external world, interpretation of the coordinates, metric potentials and so on. In the -dimensional case much work have been done in this direction and we have a good understanding of the geometry and dynamics of the classical interior (See e.g., Refs. DiNunno and Matzner (2010); Frolov and Shoom (2007); Christodoulou and Rovelli (2015); Bengtsson and Jakobsson (2015) and the references therein). On the other hand, less is known about quantum processes inside black holes and their influence upon the background geometry.

In the recent paper Matyjasek and Sadurski (2015) we have studied influence of the quantized fields on the static spacetime of the Schwarzschild-Tangherlini black hole using the semi-classical Einstein field equations. Since the stress-energy tensor constructed in that paper functionally depends on the metric, one has a rare opportunity to analyze and compare the quantum corrections to the black hole characteristics (and the geometry itself) calculated for various spacetime dimensions. The purpose of this paper, which is a natural continuation of Refs. Matyjasek and Sadurski (2015); Hiscock et al. (1997), is to extend the study of the quantized fields to the interiors of the higher-dimensional black holes. It should be noted however, that now there are problems that do not appear in the external region. The first one is the problem of the central singularity and its closest vicinity. It is evident that the semicalssical Einstein field equations cannot be trusted there. The second difficulty is to some extend related to the previous one and may be stated as follows. The effective action of the quantized fields for ( is the coordinate of the event horizon) has been constructed for the positive-definite metric signature. Once the stress-energy tensor is calculated it can be transformed to the physical spacetime by analytic continuation. In the exterior region it is the familiar Wick rotation, which affects only the time coordinate. On the other hand, inside the event horizon the problem is more complicated.

The classical dimensional solution describing interior of the Schwarzschild-Tangherlini black hole with the event horizon located at is given by the line element

(1) |

where is a metric on a unit -dimensional sphere. Since only in case there is a simple linear relation between the mass and in the present paper we use (almost) exclusively the latter. The radius of the event horizon of the Schwarzschild-Tangherlini black hole characterized by the mass is given by

(2) |

where is -dimensional Newton constant and is the volume of the unit -dimensional sphere. If the -dimensional sphere is covered by a standard “angular” coordinates the metric can be written in the form

(3) |

Now, in order to construct the positive-definite metric let us replace by by and by The metric thus becomes:

(4) |

where

(5) |

Note that our transformation differs form that of Ref. Candelas and Jensen (1986), which results in the negative-definite metric.

Having Euclidean version of the geometry of the black hole interior on can construct the one-loop approximation to the effective action of the quantized massive fields in a large mass limit. Indeed, for a sufficiently massive fields, i.e., when the Compton length, associated with the mass of the field, is much smaller than the characteristic radius of the curvature of the spacetime the contribution of the vacuum polarization to the effective action dominates and the contribution of real particles is negligible. One can therefore make use of the Schwinger-DeWitt asymptotic expansion that approximates the effective action This approach has been successfully applied in a number of interesting cases and the background spacetimes range from black holes Frolov and Zel’nikov (1982, 1983); Frolov and Zelnikov (1984); Matyjasek (2000, 2001, 2006); Thompson and Lemos (2009); Belokogne and Folacci (2014) to cosmology Matyjasek and Sadurski (2013); Matyjasek et al. (2014) and from wormholes Taylor et al. (1997) to topologically nontrivial spacetimes Kofman and Sahni (1983); Sahni and Kofman (1986); Kofman et al. (1983); Matyjasek and Tryniecki (2009). For the purposes of the present paper, the most relevant are the results presented in Refs. Hiscock et al. (1997); Matyjasek et al. (2013); Matyjasek and Sadurski (2015)

In Ref. Matyjasek and Sadurski (2015) it has been shown that the approximate one-loop effective action of the quantized massive scalar field in a large mass limit can be constructed from the (asymptotic) Schwinger-DeWitt representation of the Green function and in the lowest order it can be written in the following form:

(6) |

where and denote the floor function, i.e., it gives the largest integer less than or equal to Here is the coincidence limit of the -th Hadamard-DeWitt coefficient constructed from the Riemann tensor, its covariant derivatives up to -order and contractions. For the technical details concerning construction of the Hadamard-DeWitt coefficients the reader is referred to Refs. Sakai (1971); Gilkey (1975); Amsterdamski et al. (1989); Avramidi (2000). The (regularized) stress-energy tensor can be calculated from the standard definition

(7) |

There is one-to-one correspondence between the order of the WKB approximation and the order of the Schwinger-DeWitt expansion. For example, the sixth-order WKB approximation is equivalent to term in and to in whereas for the analogous results in and the eight-order WKB approximation is required.

On general grounds one expects that the lowest-order (nonvanishing) term of the Schwinger-DeWitt expansion is the most important. The condition (with the physical constants reinserted) leads to

(8) |

where and is given in seconds^{1}^{1}1This
is a generalization of the condition employed
in the -dimensional back reaction calculations reported in Ref. Hiscock et al. (1997).
For example,
taking equal to the Schwarzschild radius of the Sun and
kg one has which is many orders of magnitude
smaller than the coordinate time of the event horizon. It follows than
that in our calculations we can go fairly close to the central singularity.
Note that the coordinate time goes form to 0. In the rest of the
paper we use the geometric units and the adopted conventions are those
of Misner, Thorne and Wheeler Misner et al. (1973).

The paper is organized as follows. In Sec. II we study some aspects of the classical interior of the -dimensional Schwarzschild-Tangehrlini black holes. In Sec. III we construct and formally solve the -dimensional semiclassical Einstein field equation and analyze the problem of the finite renormalization. In Sec. III.1 we show how to construct the appropriate measure of anisotropy and investigate the two useful scalars: the Kretschmann scalar and the Karlhede scalar. Finally, taking the stress-energy tensor of the quantized massive scalar field, in Sec. III.2 we study the semiclassical equations and analyze the influence of quantum perturbations on the black hole interior for

## Ii Interior of classical Schwarzschild-Tangherlini black hole

To gain a better understanding of the classical interior of the Schwarzschild-Tangherlini black hole let us introduce the proper time

(9) |

and, in the neighborhood of a point the locally Euclidean coordinates

(10) |

Near the singularity the Schwarzschild-Tangherlini metric can be approximated by the Kasner metric

(11) |

where

(12) |

It can easily be checked that both Kasner conditions are satisfied. Indeed,

(13) |

and

(14) |

On the other hand, near the event horizon the Schwarzschild-Tangherlini metric asymptotically approaches

(15) |

where

(16) |

Once again it is the Kasner metric with and vanishing remaining Kasner exponents. Finally observe, that the line element (15) can be formally obtained from the Rindler solution

(17) |

by using the complex coordinate transformation.

Now, let us consider two points at the same coordinate instant separated by While the coordinate distance remains constant the physical distance between two points on the coordinate line is given by

(18) | |||||

and grows as the coordinate time decreases. On the other hand, the proper distance between two points separated by is given by

(19) |

and it decreases as the coordinate time goes from to 0. This behavior is independent of the dimension

Let us return to the proper time: It should be noted that taking a positive sign of the root of the equation

(20) |

as it has been done in Eq. (9), the proper time monotonically grows with the coordinate time Conversely, taking the negative root, the proper time increases as the coordinate time goes from to 0. Since the functional relations between and are not very illuminating we present them graphically (Fig I), demanding for both types of the universe as This can always be done by suitable choice of the integration constant. The universe inside the event horizon (in both time scales) has a finite lifetime. From the results collected in Table I one sees that decreases with the dimension.

Dimension | Proper time | Coordinate time |
---|---|---|

## Iii The back reaction

The classical Schwarzschild-Tangherlini line element is a solution of the -dimensional vacuum Einstein field equations. In this section we shall analyze the corrections to the characteristics of the classical black hole interior caused by the quantum fields. The semi-classical field equations have the form

(21) |

where is the properly regularized stress-energy tensor of the quantized field(s) and all remaining symbols have their usual meaning. We have chosen, for simplicity, to work with the minimal generalization of the Einstein equations. Other curvature invariants can be added to the action functional, but the resulting equations can be treated in precisely the same way as the “minimal” theory.

### iii.1 General considerations

We shall analyze how far one can go with the semiclassical Einstein field equations without defining explicitly the stress-energy tensor of the quantized fields. The only requirement placed on the stress-energy tensor is its regularity on the event horizon and the absence of the net fluxes. Unfortunately, except for metrically simple manifolds with a high degree of symmetry, the equations (21) cannot be solved exactly. However, assuming the expected quantum corrections to be small, one can try to solve the equations perturbatively and concentrate on the first-order calculations (with the zeroth-order being the classical solution). To achieve this let us consider the general line element

(22) |

with

(23) |

and

(24) |

where and are unknown functions, and is a small dimensionless parameter, which helps to keep track of the order of the terms in complicated expansions. It must not be confused (in case) with the small parameter of Ref. Hiscock et al. (1997). The parameter should be set to 1 at the end of the calculations. The functions and are given by and of the line element (4), respectively.

The resulting semi-classical Einstein field equations for the line element (22-24) are given by

(25) |

and

(26) |

The first equation can formally be integrated to give

(27) |

It can easily be shown that the integration constant has no independent meaning and can be absorbed into the definition of the renormalized (dressed) radius of the event horizon. Moreover, by the very same procedure, the constant can be absorbed in the second equation. Let us analyze this problem more closely. First, consider the function which can be written as

(28) |

and observe that introducing the renormalized radius of the event horizon, defined by the equation

(29) |

the integration constant can be relegated in the first-order calculations. The same transformation can be used to renormalize in the second metric potential

(30) |

where terms containing integrals of the stress-energy tensor have not been displayed explicitly. To determine the second integration constant, additional piece of information is needed. Fortunately, considered characteristics of the quantum-corrected interior of the Schwarzschild-Tangherlini black holes are independent of Since and have no independent physical meaning, in what follows, for notational simplicity, we shall replace with and treat as the renormalized (dressed) radius of the event horizon.

On general grounds one expects that the components of the stress-energy tensor constructed within the framework of the Schwinger-DeWitt approximation are simple polynomials in and hence the calculations of the functions and reduce to two elementary quadratures. Now, in order to better understand the influence of the quantized fields on the black hole interior, we shall study the trace of the rate of the deformations tensor and the ratio of the Hubble parameters. Similarly, to study the influence of the quantized fields on the curvature we calculate the Kretschmann scalar. Additionally we will check if the Karlhede’s scalar is still a useful device for detecting the event horizon.

The interior of the Schwarzschild-Tangherlini black holes is nonstatic and anisotropic. Following Novikov’s paper Novikov (1961) this can be analyzed using the rate of deformations tensor. Let us introduce the tensor defined as

(31) |

where Let the indices from the second half of the Latin alphabet denote spatial coordinates. The deformation rate tensor, which has only spatial components, is given by Frolov and Novikov (1998)

(32) |

and its trace is Now, let us consider a volume element where and construct the quantity

(33) |

with a natural interpretation as the speed of the relative change of the volume element of the space. For the quantum-corrected Schwarzschild-Tangherlini black hole the trace of the rate of deformation tensor is given by

(34) |

where

(35) |

and

(36) |

Te trace is independent of the integration constant It should be noted that in the closest vicinity of the event horizon the correction to the trace is practically independent of the function On the other hand, the behavior of the correction i.e., the question if the rate of deformations grows or decreases depends on the sign of the

The conclusions that can be drawn from the analysis of the classical part of the tensor and its
trace are qualitatively similar to those of the Novikov’s paper.^{2}^{2}2It should be noted that English
translation of the Novikov’s paper is not always correct. Regardless of the dimension is
always negative in the vicinity of the event horizon, whereas it is always positive for
is the smallest root of

Although the information yielded by the rate of deformation tensor is accurate, it is simultaneously hard to visualize and we need something somewhat simpler. The useful measure of the anisotropy is the ratio of the Hubble parameters

(37) |

where is any of the angular coordinates Making use of (23) and (24), one obtains

(38) |

where the first term in the right hand side of the above equation is the unperturbed part of Consequently, the second term, which we denote by , depending on the sign can make the black hole interior more isotropic or anisotropic. Further analysis of the role played by must be postponed until we solve the semicalssical Einstein field equations.

It could be easily shown that the simple differential curvature invariant which is very useful in detecting the location of (sometimes called Karlhede’s invariant Karlhede et al. (1982))

(39) |

vanishes on the event horizon of the Schwarzschild-Tangherlini black hole and is positive inside and negative outside. Because of their properties such invariants have become popular recently, see e.g., Page and Shoom (2015); Abdelqader and Lake (2015); Moffat and Toth (2014). Now, making use of the functions and one has

(40) | |||||

To obtain the classical term it suffices to set to zero the functions and Because of the presence of the factor the invariant of the quantum-corrected black hole always vanishes at the event horizon provided the functions and are regular in its vicinity. In view of the foregoing discussion it is expected that in the case at hand the condition is satisfied.

Finally, let us consider the simplest curvature invariant, namely the Kretschmann scalar, defined as the “square” of the Riemann tensor

(41) |

which, for the quantum-corrected interior of the -dimensional Schwarzschild-Tangherlini black hole has the following form:

(42) | |||||

The first term in the right-hand-side of the above equation gives the classical Kretschmann scalar and the remaining ones are the quantum corrections, which we denote by Although the semiclassical Einstein field equation are certainly incorrect as and should be replaced by the (unknown as yet) quantum gravity, it is of some interest to study the tendency exhibited by in this very limit. This, however, requires explicit knowledge of the functions and which is the subject of the next subsection.

### iii.2 The back reaction of the quantized massive fields

Now, let us return to the semiclassical Einstein field equations and solve (25) and (26) with the stress-energy tensor of the quantized massive scalar field. The relevant components of are listed in Appendix A. The angular components can easily be calculated form the covariant conservation equation For any considered dimension, the components of the tensor are simple polynomials (in ), the difference factorizes as

(43) |

and the function is regular for Indeed, after some algebra, one has

(44) |

and

(45) |

where for

(46) |

(47) |

and where the coefficients and the integration constants are listed in Appendix B. This result is not new: The stress-energy tensor has been obtained in the early 80’s by Frolov and Zel’nikov Frolov and Zel’nikov (1982, 1983) and subsequently used in Ref. Hiscock et al. (1997). To the best of our knowledge the results for higher dimensional geometries are new. Now, making use of the stress-energy tensor constructed in the Schwarzschild-Tangherlini spacetime, one has

(48) |

(49) |

Both tensors have been calculated from the effective action constructed from the Similarly, making use the effective action constructed form the coefficient one obtains

(50) |

(51) |

and

(52) |

(53) |

for and respectively. It should be noted that for the functions loose their polynomial character, but they are still regular except for

Having constructed the functions and one can analyze the quantum corrections to the Kretschmann scalar and anisotropy of the black hole interior. First, let us consider Inspection of the unperturbed part of shows that it is always negative. The sign of is positive if the internal geometry expands or contracts in all spatial directions. Of course, for isotropic evolution one has The negative sign of means that it is contracting in one dimension and expanding in the other. Depending on the sign, the quantum perturbation can strengthen or weaken the anisotropy. Since the anisotropy is weaken for and strengthen in the regions where This, however, depends on the coupling constant and the coordinate time in a quite complicated way. The results of the numerical calculations has been plotted in Figs 2 and 3. Specifically, for and the is negative above the -curves and positive below. On the other hand, for and the perturbation is negative below the curves and positive above.

Dimension | ||
---|---|---|

more isotropic | more isotropic | |

more isotropic | more isotropic for | |

more anisotropic | more anisotropic | |

more anisotropic | more anisotropic |

Now, we shall analyze in some details the behavior of the corrections to the Kretschmann scalar and the measure of the anisotropy for the physical values of the coupling parameter, i.e., for the minimal coupling and the conformal coupling (There is no need to perform such analysis for Karlhede’s scalar as its main role is to serve as a useful device for detecting location of the event horizon. Inspection of (40) and (46-53) shows that at the quantum-corrected event horizon, as expected.)

As have been observed earlier in Ref. Hiscock et al. (1997), the quantum corrections for the minimal and conformal coupling always tend to isotropize the interior of the Schwarzschild black hole. On the other hand however, in higher dimension the pattern is more complicated. Indeed, for the perturbation for whereas for isotropization occurs only for In turn, for and the perturbation is always negative for the minimal and the conformal coupling, i.e., the quantum effects make the black hole interior more anisotropic. These results are tabulated in Table 2.

Dimension | ||
---|---|---|

for | always positive | |

always positive | always positive | |

always positive | for | |

always positive | for |

Let us analyze how the growth of the curvature are affected by the quantum processes. And since the classical part of the Kretschmann scalar is positive, one concludes that the growth of curvature (as decreases) is weakened if the perturbation is negative. Inspection of the Figs. 4 and 5 as well as the Table 3 shows that initially, regardless of the dimension, is always positive for the both types of couplings. This is very important message as it refers to the region where the quality of the approximation is expected to be high.

A few words of comment are in order here. First, it should be emphasized once more that although we have plotted functions and for all allowable values of the coordinate time the approximation certainly does not work in the region close to the central singularity. Therefore our results show the tendency in behavior of the quantum corrections as the central singularity is approached (which can be misleading) rather than their actual run. Of course the answer to the question of how close the singularity can be approached depends on many factors, such as dimension, the ‘radius’ of the event horizon and the type of the quantized field. Each case should be analyzed separately. The second observation is less obvious and is, roughly speaking, related to the question of how long the first-order approximation dominates the higher-orders terms inside the event horizon. Once again this problem goes to the very core of the Schwinger-DeWitt asymptotic expansion. And once again there is no better answer than to recall its principal assumptions. Finally, let us observe that although the quantum corrections caused by a solitary field is expected to be small in the domain of applicability of the approximation, they can be made arbitrarily large for a large number of fields.

## Appendix A The stress-energy tensor of the quantized massive scalar field in the interior of the higher-dimensonal Schwarzschild-Tangherlini black hole

In this appendix we list the and components of the stress-energy tensor for The angular components are not displayed as they can easily be calculated from the covariant conservation equation.

(54) |

(55) |

(56) |

(57) |

(58) |

(59) |

(60) |

(61) |

## Appendix B Coefficients of the functions and

Here we list the coefficients of the functions and (See Eqs. (46 -53)). The integration constants are left unspecified and should be determined from the physically motivated boundary conditions. All quantities considered in the main text, such as and are independent of the integration constant

(62) |

(63) |

(64) |

(65) |

(66) |

(67) |

(68) |

(69) |

(70) |

(71) |

(72) |

(73) |

(74) |

(75) |

(76) |

(77) |

(78) |

(79) |

(80) |

(81) |

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