firstname.lastname@example.orgInstitute of Physics, Maria Curie-Skłodowska University
pl. Marii Curie-Skłodowskiej 1, 20-301 Lublin, Poland
Institute of Physics, Maria Curie-Skłodowska University
pl. Marii Curie-Skłodowskiej 1, 20-301 Lublin, Poland
July 12, 2019
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 D=4 and 5, whereas for D=6 and 7 the spacetime becomes more anisotropic.
Similar behavior is observed for the conformal coupling with the reservation
that for D=5 isotropization of the spacetime occurs during (approximately) the first 1/3
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.
04.62.+v, 04.70.Dy, 04.50.-h
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 D=4-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 r+≥0 (r+ 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 D−dimensional solution describing interior of the Schwarzschild-Tangherlini
black hole with the event
horizon located at T=r+ is given by
the line element
where dΩ2D−2 is a metric on a unit (D−2)-dimensional sphere. Since only in D=4
case there is a simple
linear relation between the mass and r+ 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 M is given by
where G(D) is D-dimensional Newton constant and ωD−2 is the volume of the unit
If the (D−2)-dimensional sphere is covered by a standard “angular” coordinates
the metric T2dΩ2D−2 can be written in the form
Now, in order to construct the positive-definite metric let us replace T by iT,θ1 by iθ1 and r+
by ir+. The metric thus becomes:
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, λC, associated
with the mass of the field, m, is much smaller
than the characteristic radius of the curvature of the spacetime L, 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 W(1). 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
W(1) 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:
and ⌊x⌋ denote the floor function, i.e., it gives
the largest integer less than or equal to x. Here [ak] is the
coincidence limit of the k-th Hadamard-DeWitt coefficient constructed
from the Riemann tensor, its covariant derivatives up to (2k−2)-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
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 m−2 term in D=4 and to m−1 in
D=5 whereas for the analogous results in D=6 and D=7 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 λC/L≪1 (with the physical constants reinserted)
where s=(D−1)(D−2)2(D−3) and T is given in seconds111This
is a generalization of the condition T≫(M/m2)1/3 employed
in the D=4-dimensional back reaction calculations reported in Ref. Hiscock et al. (1997).
taking D=4,r+ equal to the Schwarzschild radius of the Sun and
m=10−30 kg one has T≫10−16 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 r+/c 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 D-dimensional Schwarzschild-Tangehrlini black holes.
In Sec. III we construct and formally solve the D-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 4≤D≤7.
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
and, in the neighborhood of a point (θ(0)1,θ(0)2,...,θ(0)D−2),
the locally Euclidean coordinates
Near the singularity the Schwarzschild-Tangherlini metric can be approximated by the Kasner metric
It can easily be checked that both Kasner conditions are satisfied. Indeed,
On the other hand,
near the event horizon the Schwarzschild-Tangherlini metric asymptotically approaches
Once again it is the Kasner metric with p1=1 and vanishing remaining Kasner exponents.
Finally observe, that the line element (15) can be formally obtained from the Rindler solution
by using the complex coordinate transformation.
Now, let us consider two points at the same coordinate instant separated by ΔX.
While the coordinate distance remains constant the physical distance between two points on the X− coordinate line is given by
and grows as the coordinate time decreases.
On the other hand, the proper distance between two points separated by dΩD−2 is given by
and it decreases as the coordinate time goes from r+ 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
as it has been done in Eq. (9), the proper time monotonically grows with the coordinate time T.
Conversely, taking the negative root, the proper time increases as the coordinate time goes from r+ to 0. Since the functional
relations between τ and T are not very illuminating we present them graphically (Fig I), demanding
for both types of the universe τ=π/2 as T=r+. 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 τ/r+ decreases with the dimension.
Table 1: The lifetime of the interior universe.
Iii The back reaction
The classical Schwarzschild-Tangherlini line element is a solution of the
D-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
where Tab 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
where f1(T) and h1(T) are unknown functions, and ε 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 D=4 case) with the small parameter
of Ref. Hiscock et al. (1997). The parameter ε should be set to 1 at the end of the calculations.
The functions f0(T) and h0(T) are given by −gTT and gXX of the line
element (4), respectively.
The resulting semi-classical Einstein field equations for the line element (22-24)
are given by
It can easily be shown that the integration constant C1 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 C1 can be absorbed
in the second equation. Let us analyze this problem more closely. First, consider the function f(T), which can be written as
and observe that introducing the renormalized radius of the event horizon,
¯r+, defined by the equation
the integration constant can be relegated in the first-order calculations.
The same transformation can be used to renormalize r+ in the second metric potential
where O(ε) terms containing integrals of
the stress-energy tensor have not been displayed explicitly.
To determine the second integration constant, C2, additional
piece of information is needed. Fortunately,
considered characteristics of the quantum-corrected interior of
the Schwarzschild-Tangherlini black holes are independent of C2.
Since C1 and r+ have no independent physical meaning, in what follows,
for notational simplicity, we shall replace ¯r+ with r+ and treat
r+ 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 r+/T, and hence the calculations of the functions f1 and h1
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 pab defined as
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)
and its trace is D=Drr.
Now, let us consider a volume element
where p=det(prs), and construct the quantity
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 D is given by
Te trace D is independent of the integration constant C2. It should be noted that in the closest vicinity
of the event horizon the correction to the trace D is practically independent of the function h′1.
On the other hand, the behavior of the correction D1, i.e., the question if the rate of deformations
grows or decreases depends on the sign of the f1.
The conclusions that can be drawn from the analysis of the classical part of the tensor Drs and its
trace are qualitatively similar to those of the Novikov’s paper.222It should be noted that English
translation of the Novikov’s paper is not always correct. Regardless of the dimension D0 is
always negative in the vicinity of the event horizon, whereas it is always positive for T<Tmin≈0.75r+.Tmin is the smallest root of D0(T)=0.
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
where θ is any of the angular coordinates
Making use of (23) and (24), one obtains
where the first term in the right hand side of the above equation is the unperturbed part of
α. Consequently, the second term, which we denote by δα, depending on the sign can make the black hole interior
more isotropic or anisotropic. Further analysis of the role played by δα
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 r+ (sometimes called Karlhede’s invariant Karlhede et al. (1982))
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 f and h one has
To obtain the classical term it suffices to set to zero the functions f1(T) and h1(T).
Because of the presence of the factor h0(T), the invariant I of the quantum-corrected
black hole always vanishes at the event horizon provided the functions f1(T) and h1(T)
are regular in its vicinity. In view of the foregoing discussion it is expected that
in the case at hand the condition I(r+)=0 is satisfied.
Finally, let us consider the simplest curvature invariant, namely the Kretschmann scalar,
defined as the “square” of the Riemann tensor
which, for the quantum-corrected interior of the D-dimensional Schwarzschild-Tangherlini black hole
has the following form:
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 δK. Although the semiclassical Einstein
field equation are certainly incorrect as T→0, and should be replaced by the (unknown as yet)
quantum gravity, it is of some interest to study the tendency exhibited by δK in this very limit.
This, however, requires explicit knowledge of the functions f1(T) and h1(T), 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 Tba are listed in Appendix A.
The angular components can easily be calculated form the covariant conservation
For any considered dimension, the components of the tensor are simple
polynomials (in x=r+/T), the difference TTT−TXX
and the function F(T) is regular for T>0.
Indeed, after some algebra, one has
where for D=4
where the coefficients α(D)i and the integration constants K(D) 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 D=5 Schwarzschild-Tangherlini
spacetime, one has
Both tensors have been calculated from the effective action constructed from the [a3].
Similarly, making use the effective action constructed form the coefficient [a4], one obtains
for D=6 and D=7, respectively.
It should be noted that for D=7 the functions loose their polynomial character, but they are still regular except for T=0.
Having constructed the functions f1(T) and h1(T) one can analyze the quantum corrections to the
Kretschmann scalar and anisotropy of the black hole interior. First, let us consider α.
Inspection of the unperturbed part of α shows that it is always negative. The sign of α
is positive if the internal geometry expands or contracts in all spatial directions. Of course,
for isotropic evolution one has α=1. The negative sign of α 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 α0<0 the anisotropy is weaken for δα>0 and strengthen in the regions
where δα<0. This, however, depends on the coupling constant ξ and the coordinate time T in a quite complicated way.
The results of the numerical calculations has been plotted in Figs 2 and 3. Specifically, for D=4 and D=5 the
δα is negative above the (δα=0)-curves and positive below. On the other hand, for D=6 and D=7
the perturbation is negative below the curves and positive above.
more isotropic for x>0.365
Table 2: The influence of δα on the black hole interior. Depending on the coupling
ξ and the dimension the quantum corrections can make the spacetime more isotropic or more anisotropic.
Now, we shall analyze in some details the behavior of the corrections to the Kretschmann scalar and the measure
of the anisotropy α for the physical values of the coupling parameter, i.e., for the minimal coupling
ξ=0 and the conformal coupling ξc=(D−2)/(4D−4).
(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 I=0 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 D=5 the perturbation δα>0 for ξ=0 whereas for ξc=3/16 isotropization occurs only for T>0.365r+.
In turn, for D=6 and D=7, the perturbation δα 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.
δK>0 for x>0.986
δK always positive
δK always positive
δK always positive
δK always positive
δK>0 for x>0.543
δK always positive
δK>0 for x>0.599
Table 3: The sign of the quantum corrections to the Kretschmann scalar.
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 T 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,
δK 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 δα=0
and δK=0 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
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 (T,T) and (X,X) components of the stress-energy tensor for 4≤D≤7.
The angular components are not displayed as they can easily be calculated from the covariant conservation equation.
Appendix B Coefficients of the functions f1(T) and h1(T)
Here we list the coefficients of the functions f1(T) and h1(T).
(See Eqs. (46 -53)). The integration constants C(D)2
are left unspecified and should be determined from the physically
motivated boundary conditions. All quantities considered in the main text,
such as α,D,K and I are independent of the integration
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