Download - Distributional SAdS BH spacetime-induced vacuum dominance 3ДОМRED1349769084-SDI Paper Template 2003
Distributional SAdS BH spacetime-induced vacuum dominance
Jaykov Foukzon 1*
[email protected] for Mathematical Sciences, Israel Institute of Technology, Haifa, Israel
Haifa, 3200003
Alexander Potapov 2
[email protected]’nikov Institute of Radioengineering and Electronics, Russian Academy of Sciences,
Moscow, RussiaMoscow, 125009
Elena Men’kova 3
[email protected] Research Institute for Optical and Physical Measurements, Moscow, Russia
Moscow, 119361.ABSTRACT
This paper dealing with extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2], [5]-[7] and [14]-[15]. The regularization of singularities presented in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically. In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. Using nonlinear distributional geometry and Colombeau generalized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates. However the Tolman formula for the total energy of a static and asymptotically flat spacetime, gives as it should be. The vacuum energy density of free scalar quantum field with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such distributional BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations has a singular
behavior on BHs horizon :
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Keywords: Colombeau nonlinear generalized functions, Distributional Riemannian Geometry, Distributional Schwarzschild Geometry, Schwarzschild singularity, Schwarzschild Horizon, smooth regularization, nonsmooth regularization, quantum fields on curved spacetime, vacuum fluctuations, vacuum dominance1. INTRODUCTION 1.1. The breakdown of canonical formalism of Riemann geometry for the
singular solutions of the Einstein field equationsEinstein field equations was originally derived by Einstein in 1915 in respect with canonical formalism of Riemann geometry, i.e. by using the classical sufficiently smooth metric tensor, smooth Riemann curvature tensor, smooth Ricci tensor, smooth scalar curvature, etc. However, singular solutions of the Einstein field equations with singular metric tensor and singular Riemann curvature tensor have soon been found. These singular solutions are formally accepted beyond rigorous canonical formalism of Riemannian geometry.
Remark 1.1. Note that if some components of the Riemann curvature tensor
become infinite at point one obtains the breakdown of canonical formalism of Riemann geometry in a sufficiently small neighborhood of the point i.e. in such
neighborhood Riemann curvature tensor will be changed by formula (1.7) see remark 1.2.
Remark 1.2. Let be infinitesimal closed contour and let be the corresponding surface
spanning by , see figure 1. We assume now that: (i) Christoffel symbol becomes
infinite at singular point by formulae
and (ii)
Let us derive now to similarly canonical calculation [3]-[4] the general formula for
the regularized change in a vector after parallel displacement around
infinitesimal closed contour This regularized change can clearly be written in the form
where
and where the integral is taken over the given
contour . Substituting in place of the canonical expression (see [4], Eq. (85.5)) we obtain
where
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Fig. 1. Infinitesimal closed contour and corresponding singular surface spanning by . Due to the degeneracy of the metric (1.12) at point , the Levi-
Civita’ connection is not available on
in canonical sense but only in an distributional sense
Fig. 2. Infinitesimal closed contour with a singularity at point on Horizon and corresponding singular surface spanning by . Due to the degeneracy of the metric (1.12) at , the Levi-Civita’ connection
is not available on horizon in
canonical sense but only in an distributional sense
Now applying Stokes' theorem (see [4], Eq. (6.19)) to the integral (1.3) and considering that the area enclosed by the contour has the infinitesimal value we get
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Substituting the values of the derivatives (1.4) into Eq. (1.5), we get finally:
where is a tensor of the fourth rank
Here is the classical Riemann curvature tensor. That is a tensor is clear from the
fact that in (1.6) the left side is a vector -- the difference between the values of vectors at one and the same point. Note that a similar results were obtained by many authors [5]-[17] by using Colombeau nonlinear generalized functions [1]-[2].
Definition 1.1. The tensor is called the generalized curvature tensor or the generalized Riemann tensor.
Definition 1.2. The generalized Ricci curvature tensor is defined as
Definition 1.3. The generalized Ricci scalar is defined as
Definition 1.4. The generalized Einstein tensor is defined as
Remark 1.3. (I) Note that the Schwarzschild spacetime is well defined only for . The boundary of the manifold in is the submanifold of
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diffeomorfic to a product This submanifold is called the event horizon, or simply the horizon [33], [34].
(II) The Schwarzschild metric (1.12) in canonical coordinates with ceases to be a smooth Lorentzian metric for because for such a value of the coefficient becomes zero while becomes infinite. For the metric (1.12) is again a smooth Lorentzian metric but is a space coordinate while is a time coordinate. Hence the metric cannot be said to be either spherically symmetric or static for [33].
(III) From consideration above obviously follows that on Schwarzschild spacetime the Levi-Civita connection
is not available in classical sense and that is well known many years from mathematical literature, see for example [22] section 6 and Remark 1.1 - Remark 1.2 above.
(IV) Note that [4]: (i) The determinant of the metric (1.12) is reqular on horizon, i.e. smooth and non-vanishing for
In addition:
(i) The curvature scalar is zero for
(ii) The none of higher-order scalars such as etc. blows up. For example
the quadratic scalar is regular on horizon, i.e. smooth and non-vanishing for
(V) Note that: (i) In physical literature (see for example [4], [33], [35],) it was wrongly assumed that a properties (i)-(iii) is enough to convince us that represent an non honest physical singularity but only coordinate singularity.
(VI) Such assumption based only on formal extensions
of the curvature scalar and higher-order scalars such as
on horizon and on origin by formulae
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however in the limit the Levi-Civitá connection becomes infinite [4]:
Thus obviously by consideration above (see Remark 1.1 - Remark 1.2) this extension given by Eq. (1.15) has no any sense in respect of the canonical Riemannian geometry.
(VII) From consideration above (see Remark 1.1 - Remark 1.2) obviously follows that the scalars such as has no any sense in respect of the canonical Levi-Civitá connection (1.11) and therefore cannot be said to be either honest physical singularity or only coordinate singularity in respect of the canonical Riemannian geometry.
Remark 1.4. Note that in physical literature the spacetime singularity usually is defined as location where the quantities that are used to measure the gravitational field become infinite in a way that does not depend on the coordinate system. These quantities are the classical scalar invariant curvatures of singular spacetime, which includes a measure of the density of matter.
Remark 1.5. In general relativity, many investigations have been derived with regard to singular exact vacuum solutions of the Einstein equation and the singularity structure of space-time. Such solutions have been formally derived under condition where
represent the energy-momentum densities of the gravity source. This for example is the case for the well-known Schwarzschild solution, which is given by, in the Schwarzschild coordinates
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where is the Schwarzschild radius with and being the Newton gravitational constant, mass of the source, and the light velocity in vacuum Minkowski space-time, respectively. The metric (1.12) describes the gravitational field produced by a point-like particle located at .
Remark 1.6. Note that when we say, on the basis of the canonical expression of the curvature square
formally obtained from the metric (1.12), that is a singularity of the Schwarzschild space-time, the source is considered to be point-like and this metric is regarded as meaningful everywhere in space-time.
Remark 1.7. From the metric (1.12), the calculation of the canonical Einstein tensor proceeds in a straightforward manner gives for
where Using Eq. (1.14) one formally obtains a boundary conditions
However as pointed out above the canonical expression of the Einstein tensor in a sufficiently small neighborhood of the point and must be replaced by the generalized Einstein tensor (1.10). By simple calculation easy to see that
and therefore the boundary conditions (1.15) is completely wrong. But other hand as pointed out by many authors [5]-[17] that the canonical representation of the Einstein tensor, valid only in a weak (distributional) sense, i.e. [12]:
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and therefore again we obtain Thus canonical definition of the Einstein tensor is breakdown in rigorous mathematical sense for the Schwarzschild solution at origin .
1.2. The distributional Schwarzschild geometry
General relativity as a physical theory is governed by particular physical equations; the focus of interest is the breakdown of physics which need not coincide with the breakdown of geometry. It has been suggested to describe singularity at the origin as internal point of the Schwarzschild spacetime, where the Einstein field equations are satisfied in a weak (distributional) sense [5]-[22].
1.2.1.The smooth regularization of the singularity at the origin
The two singular functions we will work with throughout this paper (namely the singular
components of the Schwarzschild metric) are and Since it
obviously gives the regular distribution . By convolution with a mollifier
adapted to the symmetry of the spacetime, (i.e. chosen radially symmetric) we embed it into the Colombeau algebra [22]:
Inserting (1.18) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:
Remark 1.8. Note that under regularization (1.18) for any the metric
obviously is a classical Riemannian object and there is no existed the breakdown of canonical formalism of Riemannian geometry for these metrics, even at origin It has been suggested by many authors to describe singularity at the origin as an internal point,
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where the Einstein field equations are satisfied in a distributional sense [5]-[22]. From the Colombeau metric (1.19) one obtains in a distributional sense [22]:
Hence, the distributional Ricci tensor and the distributional curvature scalar are of δ
-type, i.e.
Remark 1.9. Note that the formulae (1.20) should be contrasted with what is the expected result given by Eq. (1.17). However the equations (1.20) are obviously given in spherical coordinates and therefore strictly speaking this is not correct,
because the basis fields are not globally defined. Representing
distributions concentrated at the origin requires a basis regular at the origin. Transforming the formulae for into Cartesian coordinates associated with the spherical ones, i.e.,
, we obtain, e.g., for the Einstein tensor the expected result
given by Eq. (1.17), see [22].
1.2.2. The nonsmooth regularization of the singularity at the origin
The nonsmooth regularization of the Schwarzschild singularity at the origin is considered by N. R. Pantoja and H. Rago in paper [12]. Pantoja non smooth regularization of the Schwarzschild singularity is
Here is the Heaviside function and the limit is understood in a distributional sense. Equation (1.19) with as given in (1.21) can be considered as a regularized version of the Schwarzschild line element in curvature coordinates. From equation (1.21), the calculation of the distributional Einstein tensor proceeds in a straightforward manner. By simple calculation it gives [12]:
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and
which is exactly the result obtained in Ref. [9] using smoothed versions of the Heaviside function . Transforming now the formulae for into Cartesian coordinates
associated with the spherical ones, i.e., , we obtain for the generalized Einstein tensor the expected result given by Eq. (1.17)
see Remark 1.9.
1.2.3. The smooth regularization via Horizon
The smooth regularization via Horizon is considered by J. M. Heinzle and R. Steinbauer in
paper [22]. Note that A canonical regularization is the principal value
which can be embedded into [22]:
Inserting now (1.25) into (1.12) we obtain a generalized Colombeau object modeling the singular Schwarzschild spacetime [22]:
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Remark 1.10. Note that obviously Colombeau object, (1.26) is degenerate at because is zero at the horizon. However, this does not come as a surprise. Both and are positive outside of the black hole and negative in the interior. As a consequence any smooth regularization of (or ) must pass through zero somewhere and, additionally, this zero must converge to as the regularization parameter goes to zero.
Remark 1.11. Note that due to the degeneracy of Colombeau object (1.26), even the distributional Levi-Civitá connection obviously is not available by using the smooth regularization via horizon [22].
1.2.4.The nonsmooth regularization via Gorizon
In this paper we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. The question we are aiming at is the following: using distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to show that the horizon singularity of the Schwarzschild metric is not merely a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at the horizon in Schwarzschild coordinates.
The main focus of this work is a (nonlinear) superdistributional description of the Schwarzschild spacetime. Although the nature of the Schwarzschild singularity is much worse than the quasi-regular conical singularity, there are several distributional treatments in the literature [8]-[29], mainly motivated by the following considerations: the physical interpretation of the Schwarzschild metric is clear as long as we consider it merely as an exterior (vacuum) solution of an extended (sufficiently large) massive spherically symmetric body. Together with the interior solution it describes the entire spacetime. The concept of point particles -- well understood in the context of linear field theories -- suggests a mathematical idealization of the underlying physics: one would like to view the Schwarzschild solution as defined on the entire spacetime and regard it as generated by a point mass located at the origin and acting as the gravitational source.
This of course amounts to the question of whether one can reasonably ascribe distributional curvature quantities to the Schwarzschild singularity at the horizon.
The emphasis of the present work lies on mathematical rigor. We derive the physically expected result for the distributional energy momentum tensor of the Schwarzschild geometry, i.e., , in a conceptually satisfactory way. Additionally, we set up a unified language to comment on the respective merits of some of the approaches taken so far. In particular, we discuss questions of differentiable structure as well as smoothness and degeneracy problems of the regularized metrics, and present possible refinements and workarounds. These aims are accomplished using the framework of nonlinear supergeneralized functions (supergeneralized Colombeau algebras ). Examining
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the Schwarzschild metric (1.12) in a neighborhood of the horizon, we see that, whereas is smooth, is not even (note that the origin is now always excluded from
our considerations; the space we are working on is ). Thus, regularizing the Schwarzschild metric amounts to embedding into (as done in (3.2)). Obviously, (3.1) is degenerated at , because is zero at the horizon. However, this does not come as a surprise. Both and are positive outside of the black
hole and negative in the interior. As a consequence any (smooth) regularization (
) [above (below) horizon] of must pass through small enough vicinity
( )
of zeros set somewhere and, additionally, this vicinity
( ) must converge to as the regularization parameter goes to zero. Due to the degeneracy of (1.12), the Levi-Cività connection is not available. By appropriate nonsmooth regularization (see section 3) we obtain an Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon, i.e.,
Consider corresponding distributional connections
and
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Obviously coincides with the corresponding Levi-Cività connection
on as and
there. Clearly, connections in respect the regularized metric
i.e. . Proceeding in this manner, we obtain the nonstandard result
Investigating the weak limit of the angular components of the generalized Ricci tensor using the abbreviation
and let be the function ( ) where by (
) we denote the class of all functions with compact support such that
(i) supp (supp )
(ii) Then for any function we get:
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i.e., the Schwarzschild spacetime is weakly Ricci-nonflat (the origin was excluded from our considerations). Furthermore, the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime
with the determinant of the four dimensional metric and the coordinate volume element, gives
as it should be.
The paper is organized in the following way: in section II we discuss the conceptual as well as the mathematical prerequisites. In particular we comment on geometrical matters (differentiable structure, coordinate invariance) and recall the basic facts of nonlinear superdistributional geometry in the context of algebras of supergeneralized functions. Moreover, we derive sensible nonsmooth regularizations of the singular functions to be used throughout the paper. Section III is devoted to these approach to the problem. We present a new conceptually satisfactory method to derive the main result. In this final section III we investigate the horizon and describe its distributional curvature. Using nonlinear superdistributional geometry and supergeneralized functions it seems possible to show that the horizon singularity is not only a coordinate singularity without leaving Schwarzschild coordinates.
1.2.5. Distributional Eddington-Finkelstein spacetime
In physical literature for many years the belief that Schwarzschild spacetime is extendible exists, in the sense that it can be immersed in a larger
spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with In physical literature [4], [33], [34], [35] one considers the formal change of
coordinates obtained by replacing the canonical Schwarzschild time by "retarded time" above horizon given when by
From (1.31) it follows for
The Schwarzschild metric (1.12) above horizon (see section 3) in this coordinate obviously takes the form
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When we replace (1.33) below horizon by
From (1.36) it follows for
The Schwarzschild metric (1.12) below horizon (see section 3) in this coordinate obviously takes the form
Remark 1.12. (i) Note that the metric (1.33) is defined on the manifold and obviously it is regular Lorentzian metric: its coefficients are smooth.
(ii) The term ensures its non-degeneracy for
(iii) Due to the nondegeneracy of the metric (1.32) the Levi-Civita connection
obviously now available and therefore nonsingular on horizon in contrast Schwarzschild metric (1.12) one obtains [36]:
(iv) In physical literature [4], [33], [34], [35] by using properties (i)-(iii) this spacetime wrongly convicted as a rigorous mathematical extension of the Schwarzschild spacetime.
Remark 1.13. Let us consider now the coordinates: (i) and
(ii)
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Obviously both transformations given by Eq. (1.33) and Eq. (1.36) are singular because the both Jacobian of these transformations is singular at
and
Remark 1.14. Note first (i) such singular transformations not allowed in conventional Lorentzian geometry and second
(ii) both Eddington-Finkelstein metrics given by Eq. (1.35) and by Eq. (1.38) again not well defined in any rigorous mathematical sence at
Remark 1.15. From consideration above follows that Schwarzschild spacetime is not extendible, in the sense that it can be immersed in a larger
spacetime whose manifold is not covered by the canonical Schwarzschild coordinate with Thus Eddington-Finkelstein spacetime cannot be considered as an extension of the
Schwarzschild spacetime in any rigorous mathematical sense in respect conventional Lorentzian geometry. Such "extension" is the extension by abnormal definition and nothing more.
Remark 1.16. From consideration above it follows that it is necessary a regularization of the Eq. (1.34) and Eq. (1.37) on horizon. However, obviously only nonsmooth regularization via horizon is possible. Under nonsmooth regularization (see section 3) Eq. (1.34) and Eq. (1.37) takes the form
and
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correspondingly. Therefore Eq. (1.41) - Eq. (1.42) take the form
and
From Eq. (1.43) - Eq. (1.44) one obtains generalized Eddington-Finkelstein transformations such as
and
Therefore, Eq. (1.45) - Eq. (1.46) take the form
and
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At point one obtains
and
where Thus generalized Eddington-Finkelstein transformations (1.47) - (1.48) are well defined in sense of Colombeau generalized functions. Therefore Colombeau generalized object modeling the classical Eddington-Finkelstein metric (1.35) – (1.36) above and below gorizon takes the form
It is easily to verify by using formula A.2 (see appendix) that the distributional curvature scalar is singular at as in the case of the distributional Schwarzschild spacetime given by Eq. (1.28). However this is not surprising because the classical Eddington-Finkelstein spacetime and generalized Eddington-Finkelstein specetime given by Eq. (1.53) are essentially different geometrical objects.
2. GENERALIZED COLOMBEAU CALCULUS
2.1 Notation and basic notions from standard Colombeau theory
We use [1], [2], [7] as standard references for the foundations and various applications of standard Colombeau theory. We briefly recall the basic Colombeau construction. Throughout the paper will denote an open subset of . Standard Colombeau generalized functions on are defined as equivalence classes of nets of smooth functions
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(regularizations) subjected to asymptotic norm conditions with respect to for their derivatives on compact sets.
The basic idea of classical Colombeau's theory of nonlinear generalized functions [1], [2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter . Let with for all
where is a separable, smooth orientable Hausdorff manifold of dimension .
Definition 2.1. The classical Colombeau's algebra of generalized functions on is defined as the quotient:
of the space of sequences of moderate growth modulo the space of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where denoting the space of smooth vector fields on ):
Remark 2.1. In the definition the Landau symbol appears, having the following
meaning:
Definition 2.2. Elements of G are denoted by:
Remark 2.2. With componentwise operations ( ) is a fine sheaf of differential algebras with respect to the Lie derivative defined by .
The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., iff for all charts ,
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where on the open set in the respective estimates Lie derivatives are replaced by partial derivatives.
The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., iff for all charts ,
where on the open set in the respective estimates Lie derivatives are replaced by partial derivatives.
Remark 2.3. Smooth functions are embedded into simply by the constant embedding , i.e., , hence is a faithful subalgebra of
.
Point Values of a Generalized Functions on . Generalized Numbers
Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field or . Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these numbers.
Definition 2.3. Inserting into yields a well defined element of the ring of constants (also called generalized numbers) (corresponding to resp. ), defined as the set of moderate nets of numbers ( with for some )
modulo negligible nets ( for each ); componentwise insertion of points of
into elements of yields well-defined generalized numbers, i.e., elements of the ring of constants:
(with = or = for = or = ), where
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Generalized functions on are characterized by their generalized point values, i.e., by their values on points in the space of equivalence classes of compactly supported nets
with respect to the relation for all ,
where denotes the distance on induced by any Riemannian metric.
Definition 2.4. For and the point value of at the point is
defined as the class of in
Definition 2.5. We say that an element is strictly nonzero if there exists a representative and a such that for sufficiently small. If is strictly
nonzero, then it is also invertible with the inverse The converse is true as well.
Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.
Definition 2.6. Let be an open subset of . On a set
we introduce an equivalence relation:
and denote by the set of generalized points. The set of points with compact support is
Definition 2.7. A generalized function is called associated to zero, on in L. Schwartz sense if one (hence any) representative converges to zero
weakly, i.e.
We shall often write:
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The -module of generalized sections in vector bundles-especially the space of generalized tensor fields is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields
where denotes the space of smooth tensor fields and the tensor product is taken
over the module . Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [1]-[2], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as -multilinear maps taking generalized vector and
covector fields to generalized functions, i.e., as -modules we have
In particular a generalized metric is defined to be a symmetric, generalized -tensor
field (with its index independent of and) whose determinant is
invertible in . The latter condition is equivalent to the following notion called strictly
nonzero on compact sets: for any representative of we have
for all small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: is a generalized metric iff (on every relatively
compact open subset of ) there exists a representative of such that for
fixed (small enough) (resp. ) is a classical pseudo-Riemannian
metric and is invertible in the algebra of generalized functions. A generalized
metric induces a -linear isomorphism from to and the inverse metric
is a well defined element of (i.e., independent of the
representative ). Also the generalized Levi-Civita connection as well as the generalized Riemann-, Ricci- and Einstein tensor of a generalized metric are defined simply by the usual coordinate formulae on the level of representatives.
2.2 Generalized Colombeau Calculus
We briefly recall the basic generalized Colombeau construction. Colombeau supergeneralized functions on where are defined as equivalence
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classes of nets of smooth functions where (regularizations) subjected to asymptotic norm conditions with respect to for their derivatives on compact sets.
The basic idea of generalized Colombeau's theory of nonlinear supergeneralized functions [1], [2] is regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter . Let with such that:
(i) and
(ii) for all where is a separable, smooth orientable Hausdorff manifold of dimension .
Definition 2.8. The supergeneralized Colombeau's algebra of supergeneralized functions on where is defined as the quotient:
of the space of sequences of moderate growth modulo the space of negligible sequences. More precisely the notions of moderateness resp. negligibility are defined by the following asymptotic estimates (where is denoting the space of smooth vector fields on ):
where is denoting the weak Lie derivative in L. Schwartz sense. In the definition the
Landau symbol appears, having the following meaning:
Definition 2.9. Elements of are denoted by:
Remark 2.4. With componentwise operations ( ) is a fine sheaf of differential algebras with respect to the Lie derivative defined by .
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The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., iff for all charts ,
where on the open set in the respective estimates Lie derivatives are replaced by partial derivatives.
The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e., iff for all charts ,
where on the open set in the respective estimates Lie derivatives are replaced by partial derivatives.
Remark 2.5. Smooth functions are embedded into simply by the constant embedding , i.e., , hence is a faithful subalgebra of
.
Point Values of a Supergeneralized Functions on . Supergeneralized Numbers
Within the classical distribution theory, distributions cannot be characterized by their point values in any way similar to classical functions. On the other hand, there is a very natural and direct way of obtaining the point values of the elements of Colombeau's algebra: points are simply inserted into representatives. The objects so obtained are sequences of numbers, and as such are not the elements in the field or . Instead, they are the representatives of Colombeau's generalized numbers. We give the exact definition of these numbers.
Definition 2.10. Inserting into yields a well defined element of the ring of constants (also called generalized numbers) (corresponding to resp. ), defined as the set of moderate nets of numbers ( with for
some ) modulo negligible nets ( for each ); componentwise insertion of
points of into elements of yields well-defined generalized numbers, i.e., elements of the ring of constants:
(with = or = for = or = ), where
Supergeneralized functions on are characterized by their generalized point values, i.e., by their values on points in the space of equivalence classes of compactly supported
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nets with respect to the relation for all
, where denotes the distance on induced by any Riemannian metric.
Definition 2.11. For and the point value of at the point
is defined as the class of in
Definition 2.12. We say that an element is strictly nonzero if there exists a representative and a such that for sufficiently small. If is strictly
nonzero, then it is also invertible with the inverse The converse is true as well.
Treating the elements of Colombeau algebras as a generalization of classical functions, the question arises whether the definition of point values can be extended in such a way that each element is characterized by its values. Such an extension is indeed possible.
Definition 2.13. Let be an open subset of . On a set
we introduce an equivalence relation:
and denote by the set of supergeneralized points. The set of points with compact support is
Definition 2.14. A supergeneralized function is called associated to zero, on in L. Schwartz's sense if one (hence any) representative converges
to zero weakly, i.e.
We shall often write:
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Definition 2.15. The -module of supergeneralized sections in vector bundles - especially the space of generalized tensor fields - is defined along the same lines using analogous asymptotic estimates with respect to the norm induced by any Riemannian metric on the respective fibers. However, it is more convenient to use the following algebraic description of generalized tensor fields
where denotes the space of smooth tensor fields and the tensor product is taken
over the module . Hence generalized tensor fields are just given by classical ones with generalized coefficient functions. Many concepts of classical tensor analysis carry over to the generalized setting [], in particular Lie derivatives with respect to both classical and generalized vector fields, Lie brackets, exterior algebra, etc. Moreover, generalized tensor fields may also be viewed as -multilinear maps taking generalized vector and
covector fields to generalized functions, i.e., as -modules we have
In particular a supergeneralized metric is defined to be a symmetric, supergeneralized -tensor field (with its index independent of and) whose
determinant is invertible in . The latter condition is equivalent to the
following notion called strictly nonzero on compact sets: for any representative
of we have for all small enough. This notion captures the intuitive idea of a generalized metric to be a sequence of classical metrics approaching a singular limit in the following sense: is a generalized metric iff (on every relatively compact open subset of ) there exists a representative
of such that for fixed (small enough) (resp. ) is a
classical pseudo-Riemannian metric and is invertible in the algebra of generalized
functions. A generalized metric induces a -linear isomorphism from
to and the inverse metric is a well defined element of
(i.e., independent of the representative ). Also the supergeneralized Levi-Civita connection as well as the supergeneralized Riemann, Ricci and Einstein tensor of a supergeneralized metric are defined simply by the usual coordinate formulae on the level of representatives.
2.3 Distributional general relativity
We briefly summarize the basics of distributional general relativity, as a preliminary to latter discussion. In the classical theory of gravitation one is led to consider the Einstein field equations which are, in general, quasilinear partial differential equations involving second order derivatives for the metric tensor. Hence, continuity of the first fundamental form is
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714715716717718719720721722723724725726727728729730731732733734735736737738739
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expected and at most, discontinuities in the second fundamental form, the coordinate independent statements appropriate to consider 3-surfaces of discontinuity in the spacetime manifolfd of General Relativity.
In standard general relativity, the space-time is assumed to be a four-dimensional differentiable manifold endowed with the Lorentzian metric
.
At each point of space-time , the metric can be diagonalized as with , by choosing the coordinate system
appropriately.
In superdistributional general relativity the space-time is assumed to be a four-dimensional differentiable manifold where endowed with the Lorentzian supergeneralized metric
At each point , the metric can be diagonalized as
by choosing the generalized coordinate system appropriately.
The classical smooth curvature tensor is given by
with being the smooth Christoffel symbol. The supergeneralized nonsmooth curvature
tensor is given by
with being the supergeneralized Christoffel symbol. The fundamental classical
action integral is
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8081
where is the Lagrangian density of a gravitational source and is the gravitational Lagrangian density given by
Here is the Einstein gravitational constant and is defined by
with . There exists the relation
with
Thus the supergeneralized fundamental action integral is
where is the supergeneralized Lagrangian density of a gravitational source and
is the supergeneralized gravitational Lagrangian density given by
Here is the Einstein gravitational constant and is defined by
with . There exists the relation
with
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8384
Also, we have defined the classical scalar curvature by
with the smooth Ricci tensor
From the action the classical Einstein equation
follows, where is defined by
with
being the energy-momentum density of the classical gravity source. Thus we have defined the supergeneralized scalar curvature by
with the
supergeneralized Ricci tensor
From the action
the generalized Einstein equation
follows, where
is defined by
with
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being the
supergeneralized energy-momentum density of the supergeneralized gravity source. The classical energy-momentum pseudo-tensor density of the gravitational field is defined by
with
. The supergeneralized energy-momentum pseudo-tensor density of the gravitational field is defined by
with
.
3. DISTRIBUTIONAL SCHWARZSCHILD GEOMETRY FROM NONSMOOTH REGULARIZATION VIA HORIZON
3.1 Calculation of the stress-tensor by using nonsmooth regularization via Horizon
In this section we leave the neighborhood of the singularity at the origin and turn to the singularity at the horizon. The question we are aiming at is the following: using distributional geometry (thus without leaving Schwarzschild coordinates), is it possible to show that the horizon singularity of the Schwarzschild metric is not merely only a coordinate singularity. In order to investigate this issue we calculate the distributional curvature at horizon in
Schwarzschild coordinates. In the usual Schwarzschild coordinates the Schwarzschild metric (1.12) takes the form above horizon and below horizon
correspondingly
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842843844845
846
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Remark 3.1. Following the above discussion we consider the metric coefficients and as an element of and embed it into
by replacements above horizon and below horizon correspondingly
Remark 3.2. Note that, accordingly, we have fixed the differentiable structure of the manifold: the usual Schwarzschild coordinates and the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (3.1) are extended on through the horizon. Therefore we have above horizon and below horizon correspondingly
Inserting (3.2) into (3.1) we obtain a generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e.
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9293
The
generalized Ricci tensor above horizon may now be calculated componentwise using the classical formulae
From (3.2) we
obtain
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h r r 2m
r r 2m2 2 1/2
r 2m2 2 1/2
r2,
rh 1 h
r r 2mr r 2m2 2 1/2
r 2m2 2 1/2
r2 1
r 2m2 2r
r 2mr 2m2 2 1/2
r 2m2 2 1/2
r 1 r 2m2 2
r
r 2mr 2m2 2 1/2
1.
hr r 2m
r r 2m2 2 1/2
r 2m2 2 1/2
r2
1r r 2m2 2 1/2
r 2m2
r r 2m2 2 3/2 r 2mr2 r 2m2 2 1/2
r 2mr2 r 2m2 2 1/2
2 r 2m2 2 1/2
r3.
r2h 2rh
r2 1r r 2m2 2 1/2
r 2m2
r r 2m2 2 3/2 r 2mr2 r 2m2 2 1/2
r 2mr2 r 2m2 2 1/2
2 r 2m2 2 1/2
r3
2r r 2mr r 2m2 2 1/2
r 2m2 2 1/2
r2
rr 2m2 2 1/2
rr 2m2
r 2m2 2 3/2 r 2m
r 2m2 2 1/2
r 2mr 2m2 2 1/2
2 r 2m2 2 1/2
r
2r 2m
r 2m2 2 1/22 r 2m2 2 1/2
r
rr 2m2 2 1/2
rr 2m2
r 2m2 2 3/2.
3.5
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Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation
and
let be the function where by we denote the class of all
functions with compact support such that:
(i) supp
(ii) (ii)
Then for any function we get:
By
replacement from (3.6) we obtain
By replacement from (3.7) we obtain the expression
From Eq. (3.8) we obtain
where we have expressed the function as
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with . Equations (3.9) - (3.10) give
Thus in
where from Eq. (3.11) we obtain
For we get:
By replacement from (3.13) we obtain
By replacement from (3.14) we obtain
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From Eq. (3.15)
we obtain
where we have expressed the function as
with . Equation (3.17) gives
where use is made of the relation
Thus in we obtain
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The supergeneralized Ricci tensor below horizon may now be calculated
componentwise using the classical formulae
From Eq. (3.21) we obtain
Investigating the weak limit of the angular components of the Ricci tensor (using the abbreviation
where is a function with compact support such that
we get:
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By replacement from Eq. (3.23) we obtain
By replacement from (3.23) we obtain
which is calculated to give
where we have expressed the function as
with . Equation (3.27) gives
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Thus in where from Eq. (3.28) we obtain
For we get:
By replacement from (3.30) we obtain
By replacement from (3.31) we obtain
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which is calculated to give
where we have expressed the function as
with . Equation (3.34) gives
where use is made of the relation
Thus in we obtain
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Using Eqs. (3.12), (3.20), (3.29), (3.37) we obtain
Thus the Tolman formula [3], [4] for the total energy of a static and asymptotically flat spacetime with the determinant of the four dimensional metric and the coordinate volume element, gives
We rewrite now the Schwarzschild metric (3.3) in the form
Using Eq. (A.5) from Eq. (3.40) one obtains for
and
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3.2. Examples of distributional geometries. Calculation of the distributional quadratic scalars by using nonsmooth regularization via Horizon
Let us consider again the Schwarzschild metric (3.1)
We revrite now the Schwarzschild metric (3.43) above Horizon ( ) in the form
Following the above discussion we consider the singular metric coefficient as an element of and embed it into by replacement
Thus above Horizon ( ) the corresponding distributional metric takes the form
We rewrite now the Schwarzschild metric (3.43) below Horizon ( ) in the form
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Following the above discussion we consider the singular metric coefficient as an element of and embed it into by replacement
Thus below Horizon ( ) the corresponding distributional metric takes the form
From Eq. (3.46) one obtains
From Eq. (3.46) using Eq. (A.5) one obtains
From Eq. (3.51) for one obtains
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Remark 3.3. Note that curvature scalar again is nonzero but nonsingular.
Let us introduce now the general metric which has the form [11]:
where
Remark 3.4. Note that the coordinates and are time and space coordinates, respectively, only if
In the Cartesian coordinate system with
x 1 rcos sin ,x 2 rsin sin ,x 3 rcos , 3.56
the metric (3.53)-(3.55) takes the form
with given by
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From Eq. (3.54) one obtains
Regularizing the function above gorizon (under condition ) such as
with from Eq. (3.59)-Eq. (3.60) one obtains
Regularizing the function below gorizon (under condition ) such as
with
from Eq. (3.59), Eq. (3.62) one obtains
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Remark 3.5. Finally the metric (3.57) becomes the Colombeau object of the form
with given by
Using
now Eq. A2 one obtains that the Colombeau curvature scalars in terms of
Colombeau generalized functions is expressed as
Remark 3.6. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers
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(ii) for Colombeau scalars well defined and becomes to infinite small
Colombeau generalized numbers
Using now Eq. A2 one obtains that the Colombeau scalars in terms of
Colombeau generalized functions is expressed as
Remark 3.7. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers,
(ii) for Colombeau scalars well defined and becomes to infinite small Colombeau generalized numbers.
Using now Eq. A2 one obtains that the Colombeau scalars in terms of
Colombeau generalized functions is expressed as
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Remark 3.8. Note that (i) on horizon Colombeau scalars well defined and becomes to infinite large Colombeau generalized numbers,
(ii) for Colombeau scalars finite
and tends to zero in the limit
Remark 3.9. Note that under generalized transformations such as
and
the metric given by Eq. (3.61) - Eq. (3.64) becomes to Colombeau metric of the form
4. QUANTUM SCALAR FIELD IN CURVED DISTRIBUTIONAL SPACETIME
4.1 Canonical quantization in curved distributional spacetime
Much of formalism can be explained with Colombeau generalized scalar field. The basic concepts and methods extend straightforwardly to distributional tensor and distributional spinor fields. To being with let's take a spacetime of arbitrary dimension with a metric
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of signature The action for the Colombeau generalized scalar field
is
The corresponding equation of motion is
Here
With explicit, the mass should be replaced by Separating out a time coordinate we can write the action as
The canonical momentum at a time is given by
where labels a point on a surface of constant x , the x argument of is suppressed,
is the unit normal to the surface, and is the determinant of the induced spatial
metric . To quantize, the Colombeau generalized field and its conjugate
momentum are now promoted to hermitian operators and required to satisfy the canonical commutation relation,
Here for any scalar function without the use of a metric volume element. We form now a conserved bracket from two complex Colombeau solutions to the scalar wave equation (4.2) by
where
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1143
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This bracket is called the generalized Klein-Gordon inner product, and the
generalized Klein Gordon norm of The generalized current density is
divergenceless, i.e. when the Colombeau generalized functions
and satisfy the KG equation (4.2), hence the value of the integral in (4.7) is
independent of the spacelike surface Σ over which it is evaluated, provided the functions vanish at spatial infinity. The generalized KG inner product satisfies the relations
We define now the annihilation operator associated with a complex Colombeau solution by the bracket of with the generalized field operator
It follows from the hermiticity of that the hermitian conjugate of is given by
From Eq. (4.5) and CCR (4.6) one obtains
Note that from Eq. (4.11) follows
Note that if is a positive norm solution with unit norm hf with, then and
satisfy the commutation relation Suppose now that is a
normalized quantum state satisfying then for each the state
is a normalized eigenstate of the number operator
with eigenvalue The span of all these states defines a Fock
space of the distributional - wavepacket -particle excitations above the state If we want to construct the full Hilbert space of the field theory in curved distributional spacetime, how can we proceed? We should find a decomposition of the space of complex Colombeau solutions to the wave equation (4.2) into a direct sum of a positive norm subspace and its complex conjugate such that all brackets between solutions from the two subspaces vanish. That is, we must find a direct sum decomposition:
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such that
and
The condition (4.15) implies that each in can be scaled to define its own harmonic oscillator sub-albegra. The second condition implies, according to (4.13), that the annihilators and creators for and in the subspace commute amongst themselves:
Given such a decompostion a total Hilbert space for the field theory can be defined as the space of finite norm sums of possibly infinitely many states of the form
where is a state such that for all in The state as in
classical case, is called a Fock vacuum and Hilbert space, is called a Fock space. The representation of the field operator on this Fock space is hermitian and satisfies the canonical commutation relations in sense of Colombeau generalized function.
4.2 Defining distributional outgoing modesFor illustration we consider the non-rotating, uncharged -dimensional SAdS BH with a distributional line element
where
where is the
metric of the ( )-sphere, and the AdS curvature radius squared is related to the cosmological constant by The parameter is proportional to the
mass of the spacetime: where
The distributional Schwarzschild geometry corresponds to The corresponding equation of motion (4.2) for massless case are
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1206
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The time-independence and the spherical symmetry of the metric imply the canonical decomposition
where denotes the -dimensional scalar spherical harmonics, satisfying
where is the Laplace-Beltrami operator. Substituting the decomposition (4.22) into Eq. (6) one gets a radial wave equation
We define now a tortoise distributional coordinate by the relation
By using a tortoise distributional coordinate the Eq. (4.24) can be written in the form of a Schrödinger equation with the potential
Note that the tortoise distributional coordinate becomes to infinite Colombeau
constant at the horizon, i.e. as but its behavior at infinity is
strongly dependent on the cosmological constant: for asymptotically-flat
spacetimes, and finite Colombeau constant for the SAdS geometry.
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4.2.1. Boundary conditions at the horizon of the distributional SAdS BH geometry
For most spacetimes of interest the potential as i.e.
and in this limit solutions to the wave equation (4.26) behave as
Note that classically nothing should leave the horizon and thus classically only ingoing modes (corresponding to a plus sign) should be present, i.e.
Note that for non-extremal spacetimes, the tortoise coordinate tends to
where Therefore near the horizon, outgoing modes behave as
where Now Eq. (4.30) shows that outgoing modes is Colombeau
generalized function of class
5. ENERGY-MOMENTUM TENSOR CALCULATION BY USING COLOMBEAU DISTRIBUTIONAL MODELS
We shall assume now any distributional spacetime which is conformally static in both the asymptotic past and future. We will be considered distributional spacetime which is conformally flat in the asymptotic past, i.e.
where , are smooth functions and
, are the components of an arbitrary distributional spatial
metric. Note that we use the same labels and for coordinates in the asymptotic past and future only for simplicity; they are obviously defined on non-intersecting
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regions of the spacetime.) In each of these asymptotic regions the distributional field
can be written as where satisfies
where is the flat Laplace operator, is the Laplace operator associated with
the spatial metric , and the effective potential is given by
with , the scalar curvature associated with the spatial
distributional metric .
We assume now this condition: (i) the massless ( ) field with arbitrary coupling in spacetimes which are asymptotically flat in the past and asymptotically static in the future, i.e. and , as those describing the formation of a static BH from matter initially scattered throughout space, and
(ii) the massless, conformally coupled field ( and ). With this assumptions for
the potential, two different sets of positive-norm distributional modes, and ,
can be naturally defined by the requirement that they are the solutions of Eq. (5.2) which satisfy the asymptotic conditions:
and
where , , , and are Colombeau solutions of
satisfying the normalization
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on a Cauchy surface in the asymptotic future. Note that each can be chosen to be real without loss of generality. There are reasonable situations where the distributional modes , given in Eq. (5.5), together with distributional modes fail to form a complete set of distributional normal modes. This happens whenever the operator
in Eq. (5.2) happens to possess normalizable, i.e. satisfying Eq.
(5.7) eigenfunctions with negative eigenvalues, . In this case, additional
positive-norm modes with the asymptotic behavior
and their complex conjugates are necessary in order to expand an arbitrary Colombeau solution of Eq. (5.1). As a direct consequence, at least some of the in-modes
(typically those with low ) eventually undergo an exponential growth. This
asymptotic divergence is reflected on the unbounded increase of the vacuum fluctuations,
where is the eigenfunction of Eq. (5.6) associated with the lowest negative eigenvalue allowed, is some positive constant, and is a dimensionless constant (typically of order unity) whose exact value depends globally on the spacetime structure
(since it crucially depends on the projection of each on the mode whose
; also depends on the initial state, here assumed to be the vacuum ). As one would expect, these wild quantum fluctuations give an important contribution to the vacuum energy stored in the field. In fact, the expectation value of its distributional energy-momentum tensor, in the asymptotic future is found to be dominated by this exponential growth:
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where is the derivative operator compatible with the distributional metric (so that
), is the associated distributional Ricci tensor so that
and we have omitted the subscript in and for simplicity. The Eqs. (5.10-5.12), together with Eq. (5.9), imply that on time scales determined by , the vacuum fluctuations of the field should overcome any other classical source of energy, therefore taking control over the evolution of the background geometry through the semiclassical Einstein equations (in which is included as a source term for the distributional Einstein tensor). We are then confronted with a startling situation where the quantum fluctuations of a field, whose energy is usually negligible in comparison with classical energy components, are forced by the distributional background spacetime to play a dominant role. We are still left with the task of showing that there exist indeed well-behaved distributional background spacetimes in which the operator
possesses negative eigenvalues condition on which depends Eq (5.9). Experience from quantum mechanics tells us that this typically occurs when gets sufficiently negative over a sufficiently large region. It is easy to see from Eq. (5.3) that, except for very special geometries (as the flat one), one can generally find appropriate values of which make as negative as would be necessary in order to guarantee the existence of negative eigenvalues. For distributional BH spacetime using Eq. (5.9) - Eq. (5.12) one obtains
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1321132213231324132513261327132813291330133113321333133413351336133713381339
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Remark 5.1. Note that in spite of the unbounded growth at in Eq. (5.13) - Eq. (5.16),
is covariantly conserved: . In the static case
, for instance for distributional BH geometry, this implies that the total vacuum energy is kept constant, although it continuously flows from spatial regions where its density is negative to spatial regions where it is positive.
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Remark 5.2. Note that the singular behavior at appearing in Eq. (5.13) - Eq. (5.16) leads only to asymptotic divergences, i.e. all the quantities remain finite everywhere except horizon.
6. DISTRIBUTIONAL SADS BH SPACETIME-INDUCED VACUUM DOMINANCE
6.1. Adiabatic expansion of Green functions
Using equation of motion Eq. (5.2) one can obtain corresponding distributional generalization of the canonical Green functions equations. In particular for the distributional propagator
one obtains directly
Special interest attaches to the short distance behaviour of the Green functions, such as in the limit with a fixed We obtain now an adiabatic
expansion of Introducing Riemann normal coordinates for the point
with origin at the point we have expanding
where is the Minkowski metric tensor, and the coefficients are all evaluated at Defining now
and its Colombeau-Fourier transform by
where one can work in a sort of localized momentum space. Expanding
(6.2) in normal coordinates and converting to -space, can readily be solved by iteration to any adiabatic order. The result to adiabatic order four (i.e., four derivatives of the metric) is
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where
and
we are using the symbol to indicate that this is an asymptotic expansion. One ensures that Eq. (6.5) represents a time-ordered product by performing the integral along the appropriate contour in figure 3. This is equivalent to replacing by Similarly, the adiabatic expansions of other Green functions can be obtained by using the other contours in figure 3. Substituting Eq. (6.6) into Eq. (6.5) gives
where
and, to adiabatic order 4,
with all geometric quantities on the right-hand side of Eq. (6.10) evaluated at
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Fig. 3. The contour in the complex plane to be used in the evaluation of the
integral giving . The cross indicates the pole at .
If one uses the canonical integral representation
in Eq. (6.8), then the integration may be interchanged with the integration, and performed explicitly to yield (dropping the )
The function which is one-half of the square of the proper distance between
and while the function has the following asymptotic adiabatic expansion
Using Eq. (6.4), equation (6.12) gives a representation of
where is the distributional Van Vleck determinant
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In the normal coordinates about that we are currently using, reduces to
The full asymptotic expansion of to all adiabatic orders
are
with the other being given by canonical recursion relations which enable their adiabatic expansions to be obtained. The expansions (6.13) and (6.16) are, however, only asymptotic approximations in the limit of large adiabatic parameter
If (6.16) is substituted into (6.14) the integral can be performed to give the adiabatic expansion of the Feynman propagator in coordinate space:
in which, strictly, a small imaginary part i should be subtracted from . Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (6.2), the expansion (6.17) does not determine the particular vacuum state in (6.1). In particular, the " " in the expansion of only ensures that (6.17) represents the expectation value, in some set of states, of a time-ordered product of fields. Under some circumstances the use of " " in the exact representation (6.14) may give additional information concerning the global nature of the states.
6.2. Effective action for the quantum matter fields in curved distributional spacetime
As in classical case one can obtain Colombeau generalized quantity called the effective action for the quantum matter fields in curved distributional spcetime, which, when functionally differentiated, yields
To discover the structure of , let us return to first principles, recalling the Colombeau path-integral quantization procedure such as developed in []. Our notation will imply a treatment for the scalar field, but the formal manipulations are identical for fields of higher spins. Note that the generating functional
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was interpreted physically as the vacuum persistence amplitude The
presence of the external distributional current density can cause the initial vacuum
state to be unstable, i.e. it can bring about the production of particles. In flat
space, in the limit no particles are produced, and one have the normalization condition
However, when distributional spacetime is curved, we have seen that, in general,
even in the absence of source currents J. Hence (6.19) will no longer apply. Path-integral quantization still works in curved distributional spacetime; one simply treats in
(6.19) as the curved distributional spacetime matter action, and as a current density
(a scalar density in the case of scalar fields). One can thus set = 0 in (6.19) and examine
the variation of
Note that
From (6.22) and (6.23) one obtains directly
Noting that the matter action appears exponentiated in (6.19), one obtains directly
and
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Following canonical calculation one obtains
where the proportionality constant is metric-independent and can be ignored. Thus we obtain
In (6.28) is to be interpreted as an Colombeau generalized operator which acts on an
linear space of generalized vectors normalized by
in such a way that
Remark 6.1.Note that the trace of an Colombeau generalized operator which acts on a linear space is defined by
Writing now the Colombeau generalized operator as
by Eq. (6.14) one obtains
Now, assuming to have a small negative imaginary part, we obtain
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where is the exponential integral function.
Remark 6.2. Note that for
where is the Euler's constant. Substituting now (6.35) into (6.34) and letting we obtain
which is correct up to the addition of a metric-independent infinite large Colombeau constant that can be ignored in what follows. Thus, in the generalized De Witt-Schwinger
representation (6.33) or (6.14) we have
where the integral with respect to brings down the extra power of that appears in
Eq. (6.36). Returning now to the expression (6.28) for using Eq. (6.37) and Eq.(6.31) we get
Interchanging the order of integration and taking the limit one obtains
Colombeau quantity is called as the one-loop effective action. In the case of fermion effective actions, there would be a remaining trace over spinorial indices. From Eq. (6.39) we may define an effective Lagrangian density by
whence one get
6.3. Stress-tensor renormalization
Note that diverges at the lower end of the integral because the damping
factor in the exponent vanishes in the limit . (Convergence at the upper end is guaranteed by the that is implicitly added to in the De Witt-Schwinger
representation of In four dimensions, the potentially divergent terms in the DeWitt-
Schwinger expansion of are
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where the coefficients , and are given by Eq. (6.9) - Eq. (6.10). The remaining
terms in this asymptotic expansion, involving and higher, are finite in the limit
Let us determine now the precise form of the geometrical terms, to compare them with the conventional gravitational Lagrangian that appears in (2.38). This is a delicate matter because (6.48) is, of course, infinite. What we require is to display the divergent terms in the form geometrical object . This can be done in a variety of ways. For
example, in dimensions, the asymptotic (adiabatic) expansion of is
of which the first terms are divergent as If is treated as a variable which can be analytically continued throughout the complex plane, then we may take the limit
From Eq. (6.44) it follows we shall wish to retain the units of as (length) , even when It is therefore necessary to introduce an arbitrary mass scale and to rewrite Eq. (6.44) as
If the first three terms of Eq. (6.45) diverge because of poles in the - functions:
Denoting these first three terms by we have
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The functions and are given by taking the coincidence limits of (6.9)-(6.10)
Finally one obtains
Special interest attaches to field theories in distributional spacetime in which the classical action is invariant under distributional conformal transformations, i.e.
From the definitions one has
From Eq. (6.51) one obtains
and it is clear that if the classical action is invariant under the conformal transformations (6.50), then the classical stress-tensor is traceless. Because conformal transformations are essentially a rescaling of lengths at each spacetime point the presence of a mass and
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hence a fixed length scale in the theory will always break the conformal invariance. Therefore we are led to the massless limit of the regularization and renormalization procedures used in the previous section. Although all the higher order terms in the DeWitt-Schwinger expansion of the effective Lagrangian (6.45) are infrared divergent at
as we can still use this expansion to yield the ultraviolet divergent terms arising from and in the four-dimensional case. We may put immediately in the
and terms in the expansion, because they are of positive power for These terms therefore vanish. The only nonvanishing potentially ultraviolet divergent term is therefore
which must be handled carefully. Substituting for with from (6.48), and rearranging terms, we may write the divergent term in the effective action arising from (6.53) as follows
where
and
Finally one obtains
Note that from Eq. (3.42) for follows that
Thus for the case of the distributional Schwarzchild spesetime given by the distributional metric (3.40) using Eq. (6.57) and Eq. (6.58) for one obtains
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This result is in a good agreement with Eq. (5.14) - Eq. (5.16).
7. COMCLUSIONS AND REMARKS
This paper dealing with an extension of the Einstein field equations using apparatus of contemporary generalization of the classical Lorentzian geometry named in literature Colombeau distributional geometry, see for example [1]-[2], [5]-[7] and [14]-[15]. The regularizations of singularities present in some solutions of the Einstein equations is an important part of this approach. Any singularities present in some solutions of the Einstein equations recognized only in the sense of Colombeau generalized functions [1]-[2] and not classically.
In this paper essentially new class Colombeau solutions to Einstein field equations is obtained. We have shown that a successful approach for dealing with curvature tensor valued distribution is to first impose admissible the nondegeneracy conditions on the metric tensor, and then take its derivatives in the sense of classical distributions in space.
The distributional meaning is then equivalent to the junction condition formalism. Afterwards, through appropriate limiting procedures, it is then possible to obtain well behaved distributional tensors with support on submanifolds of as we have shown for the energy-momentum tensors associated with the Schwarzschild spacetimes. The above procedure provides us with what is expected on physical grounds. However, it should be mentioned that the use of new supergeneralized functions (supergeneralized Colombeau algebras ) in order to obtain superdistributional curvatures, may renders a more rigorous setting for discussing situations like the ones considered in this paper.
The vacuum energy density of free scalar quantum field with a distributional background spacetime also is considered. It has been widely believed that, except in very extreme situations, the influence of gravity on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is false by showing that there exist well-behaved spacetime evolutions where the vacuum energy density of free quantum fields is forced, by the very same background distributional spacetime such BHs, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on curved spacetimes. In particular we obtain that the vacuum fluctuations has a singular behavior on BHs horizon :
We argue that this vacuum dominance may bear important astrophysical implications.
ACKNOWLEDGMENTSTo reviewers provided important clarifications.
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REFERENCES
1. Colombeau JF. New Generalized Functions and Multiplication of Distributions. North Holland, Amsterdam: 1984.
2. Colombeau JF. Elementary Introduction to New Generalized Functions. North Holland, Amsterdam: 1985.
3. Tolman RC. Relativity, Thermodynamics and Cosmology. Clarendon Press, Oxford: 1962.
4. Landau LD, Lifschitz EM. The Classical Theory of Fields. Pergamon Press, Oxford: 1975.
5. Vickers JA, Wilson JP. A nonlinear theory of tensor distributions. gr-qc/9807068.6. Vickers JA, Wilson JP. Invariance of the distributional curvature of the cone under
smooth diffeomorphisms. Class. Quantum Grav. 1999; 16: 579-588.7. Vickers JA. Nonlinear generalized functions in general relativity. In: Grosser M,
Hörmann G, Kunzinger M, Oberguggenberger M., editors. Nonlinear Theory of Generalized Functions, Chapman & Hall/CRC Research Notes in Mathematics. 401, 275-290, eds. Chapman & Hall CRC, Boca Raton; 1999.
8. Geroch R, Traschen J. Strings and other distributional sources in general relativity. Phys. Rev. D 1987; 36: 1017-1031.
9. Balasin H, Nachbagauer H. On the distributional nature of the energy-momentum tensor of a black hole or what curves the Schwarzschild geometry? Class. Quant. Grav. 1993; 10: 2271-2278.
10. Balasin H, Nachbagauer H. Distributional energy-momentum tensor of the Kerr-Newman space-time family. Class. Quant. Grav. 1994; 11: 1453-1461.
11. Kawai T, Sakane E. Distributional energy-momentum densities of Schwarzschild space-time. Prog. Theor. Phys. 1997; 98: 69-86.
12. Pantoja N, Rago H. Energy-momentum tensor valued distributions for the Schwarzschild and Reissner-Nordstrøm geometries. Preprint gr-qc/9710072, 1997.
13. Pantoja N, Rago H. Distributional sources in General Relativity: Two point-like examples revisited. Preprint gr-qc/0009053, 2000.
14. Kunzinger M., Steinbauer R. Nonlinear distributional geometry. Acta Appl.Math. 2001.
15. Kunzinger M., Steinbauer R. Generalized pseudo-Riemannian Geometry. Preprint mathFA/0107057, 2001.
16. Grosser M, Farkas E, Kunzinger M., Steinbauer R. On the foundations of nonlinear generalized functions I, II. Mem. Am. Math. Soc. 2001; 153: 729.
17. Grosser M, Kunzinger M., Steinbauer R, Vickers J. A global theory of nonlinear generalized functions. Adv. Math. 2001; to appear.
18. Schwartz L. Sur l'impossibilité de la multiplication des distributions. C. R. Acad. Sci. Paris 1954; 239: 847—848.
19. Gelfand IM, Schilov GE. Generalized functions. Vol. I: Properties and operations. Academic Press, New York, London: 1964.
20. Parker P. Distributional geometry. J. Math. Phys. 1979; 20: 1423--1426.21. Debney G, Kerr R, Schild A. Solutions of the Einstein and Einstein-Maxwell
equations. J. Math. Phys. 1969; 10: 1842-1854. 22. Heinzle JM, Steinbauer R. Remarks on the distributional Schwarzschild geometry.
Preprint gr-qc/0112047, 2001.23. Abramowitz M, Stegun IA, editors. Handbook of Mathematical Functions with
Formulas, Graphs, and Mathematical Tables. 9th printing. Dover, New York: 1972.24. Bracewell R. Heaviside's Unit Step Function. In: The Fourier Transform and Its
Applications. 3rd ed. McGraw-Hill, New York: 1999, pp. 57-61.25. Israel W. Nuovo Cimento 1966; 44 1; Nuovo Cimento 1967; 48 463.
* Tel.: +xx xx 265xxxxx; fax: +xx aa 462xxxxx.E-mail address: [email protected].
205
1656
1657165816591660166116621663166416651666166716681669167016711672167316741675167616771678167916801681168216831684168516861687168816891690169116921693169416951696169716981699170017011702170317041705170617071708
206207
26. Parker PE. J. Math. Phys. 1979; 20: 1423. 27. Raju CK. J. Phys. A: Math. Gen. 1982; 15: 1785.28. Geroch RP, Traschen J. Phys. Rev. D 1987; 38: 1017.29. Kawai T, Sakane E. Distributional Energy-Momentum Densities of Schwarzschild
Space-Time. Preprint grqc/9707029, 1998.30. Jacobson T. Introduction to Quantum Fields in Curved Spacetime and the Hawking
Effect. Available: http://arxiv.org/abs/0905.2975v2.31. Berti E, Cardoso V, Starinets AO. Quasinormal modes of black holes and black
branes. Available: http://arxiv.org/abs/gr-qc/0308048v3. 32. Lima WCC, Vanzella DAT. Gravity-induced vacuum dominance. Phys. Rev. Lett.
2010; 104: 161102. Available: http://arxiv.org/abs/1003.3421v1. 33. Choquet-Bruhat Y. General Relativity and the Einstein Equations. OUP Oxford: Dec
4, 2008 - Mathematics - 816 pages.34. Biswas T. Physical Interpretation of Coordinates for the Schwarzschild Metric.
Available: http://arxiv.org/abs/0809.1452v1.35. Hajicek P. An Introduction to the Relativistic Theory of Gravitation. Springer Science
& Business Media: Aug 26, 2008 - Mathematics - 280 pages.36. Müller T, Grave F. Catalogue of Spacetimes, Institut für Visualisierung und
Interaktive Systeme. Available: http://xxx.lanl.gov/abs/0904.4184v2, http://www.vis.uni-stuttgart.de/~muelleta/CoS/.
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APPENDIXExpressions for the Colombeau quantities and
in terms of and .
Let us introduce now Colombeau generalized metric which has the form
Th
e Colombeau scalars and in terms of
Colombeau generalized functions are expressed as
R A
2r 2A
A 3B
B
2r2AC D
2
AB A
A 2 B
B
12B
B
2 2A
B
AB 1
2A
A B
B
,
R R A2
212A
A 14
A
A 12
AB
AB 1r
A
A
2
2 A2
21r12
A
A 2B
B 1r2AC D
2
AB 12
AB
AB
12B
B 14
B
B
2
A2
212A
A 14
A
A 12
AB
AB B
B 12
B
B
2
12B
B 1r
A
A
2B
B
2
,
R R
A2
2A
A 12
A
A
2 2A
2
21rA
A 12
AB
AB
2
4A2
2
1rB
B 1r2AC D
2
AB 14
B
B
2 2
2 A2
21rA
A 2B
B
12
AB
AB B
B
12B
B
2 12
B
B
2
.
A. 2
Here
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211
176217631764
1765
1766
1767
17681769
17701771
1772
212213
Assume that
From Eq. (A.2) - Eq. (A.4) one obtains
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214
1773
1774
1775
1776
1777
1778
1779
1780
215216
2171781
218