^{1}

^{*}

^{2}

^{3}

The vacuum energy density of free scalar quantum field in a Rindler distributional space-time with distributional Levi-Cività connection is considered. It has been widely believed that, except in very extreme situations, the influence of acceleration on quantum fields should amount to just small, sub-dominant contributions. Here we argue that this belief is wrong by showing that in a Rindler distributional background space-time with distributional Levi-Cività connection the vacuum energy of free quantum fields is forced, by the very same background distributional space-time such a Rindler distributional background space-time, to become dominant over any classical energy density component. This semiclassical gravity effect finds its roots in the singular behavior of quantum fields on a Rindler distributional space-times with distributional Levi-Cività connection. In particular we obtain that the vacuum fluctuations have a singular behavior at a Rindler horizon . Therefore sufficiently strongly accelerated observer burns up near the Rindler horizon. Thus Polchinski’s account doesn’t violate the Einstein equivalence principle.

In March 2012, Joseph Polchinski claimed that the following three statements cannot all be true [

In this paper we argue that Polchinski was not wrong, but Unruh effect revision is needed.

Recall that the classical Cartan’s structural equations show in a compact way the relation between a connection and its curvature, and reveal their geometric interpretation in terms of moving frames. In order to study the mathematical properties of singularities, we need to study the geometry of manifolds endowed on the tangent bundle with a symmetric bilinear form it is allowed to become degenerate (singular).

Remark 1.1.1. But if the fundamental tensor is allowed to be degenerate (singular), there are some obstructions in constructing the geometric objects normally associated to the fundamental tensor. Also, local orthonormal frames and coframes no longer exist, as well as the metric connection and its curvature operator [

Remark 1.1.2. “Singular Semi-Riemannian Geometry”―the main brunch of contemporary semi-Riemannian geometry in which have been studied a smooth manifolds M furnished with a degenerate (singular) on a smooth submanifold M ′ ⊊ M metric tensor of arbitrary signature have been studied [

Remark 1.1.3. In order to solve problems of the gravitational singularity in classical general relativity the singular semi-Riemannian geometry based on Colombeau calculas and Colombeau generalized functions was much developed, see [

Remark 1.1.4. Let G ( M ′ ) be algebra of Colombeau generalized functions on M ′ ⊂ M , let ℝ ˜ be the ring of Colombeau generalized numbers [

Definition 1.1.1. Let G ( M ′ ) be algebra of Colombeau generalized functions on M ′ ⊂ M , and let ( g ε ( p ) ) ε be Colombeau generalized metric tensor on M such that ( g ε ( p ) ) ε is the Colombeau solution of the Einstein field Equation (1.3.19), (see Remark 1.3.7). We define now the Colombeau distributional scalar curvature R M ( p ) = [ ( R ε , M ( p ) ) ε ] (or distributional Ricci [

Then we say that: (i) gravitational field ( g ε ( p ) ) ε (or corresponding distributional spacetime) has a gravitational singularity on a smooth compact submanifold M c ⊂ M iff R M c ( p ) ∈ G ( M c ) \ C ∞ ( M c ) ; (ii) gravitational field ( g ε ( p ) ) ε has a gravitational singularity with compact support iff R M c ( p ) ∈ D ′ ( ℝ 3 ) .

Remark 1.1.5. It turns out that the distributional Schwarzschild spacetime has a gravitational singularity with compact support at origin { r = 0 } [

Definition 1.1.2. (i) Let G ( M ) be algebra of Colombeau generalized functions on M, and let ( g ε ( p ) ) ε be Colombeau generalized metric tensor on M such that ( g ε ( p ) ) ε is the Colombeau solution of the generalized Einstein field Equation (1.3.19). The generalized point value of ( g ε ( p ) ) ε at generalized point ( ( p ε ) ) ε is ( g ε ( p ε ) ) ε . (ii) We define now the generalized point value of the distributional scalar curvature R M ( p ) at generalized point p = [ ( ( p ε ) ) ε ] by formula R M ( p ) = [ ( R ε , M ( p ε ) ) ε ] .

As important example of Colombeau extension of the singular semi-Riemannian geometry mentioned above, we consider now Moller’s uniformly accelerated frame given by Moller’s line element [

d s 2 = − ( a + g x ) 2 d t 2 + d x 2 + d y 2 + d z 2 . (1.2.1)

Of couse Moller’s metric (1.2.1) degenerate at Moller horizon x h o r M o l = − ( a / g ) − 1 . Note that formally corresponding to the metric (1.2.1) classical Levi-Civitá connection is [

Γ 44 1 ( x ) = ( a + g x ) , Γ 14 4 ( x ) = Γ 41 4 ( x ) = 1 a + g x (1.2.2)

and therefore classical Levi-Civit’a connection (1.2.2) of couse is not available at Moller horizon x h o r M o l = − a ⋅ g − 1 . Recall that fundamental tensor corresponding to the metric (1.2.1) was obtained in Moller’s paper [

G i k = R i k − 1 2 δ i k R = 0 , (1.2.3)

where R i k is the contracted Riemann-Christoffel tensor formally calculated by canonical way by using classical Levi-Civitá connection (1.2.2) and R = R i i . Using Dingle’s formula [

G 2 2 ( x ) = G 3 3 ( x ) = − 1 2 Δ ( x ) { Δ ″ ( x ) − [ Δ ′ ( x ) ] 2 2 Δ ( x ) } , Δ ( x ) = ( a + g x ) 2 , (1.2.4)

where Δ ′ ( x ) = ∂ Δ ( x ) / ∂ x and all other components of G i k vanishes identically. Note that

Δ ′ ( x ) = 2 g ( a + g x ) , Δ ″ ( x ) = 2 g 2 . (1.2.5)

Thus for any x ≠ − a ⋅ g − 1 we get a classical result

G 2 2 ( x ) = G 3 3 ( x ) = − 1 2 Δ ( x ) { 2 g 2 − 4 g 2 ( a + g x ) 2 2 Δ ( x ) } ≡ 0. (1.2.6)

Let { x n } n ∈ ℕ be a sequence such that lim n → ∞ x n = − a ⋅ g − 1 , x n ≠ − a ⋅ g − 1 , n ∈ ℕ . Then for any n ∈ ℕ we get

ℑ ( x n ) = G 2 2 ( x n ) = G 3 3 ( x n ) = − 1 2 Δ ( x n ) { 2 g 2 − 4 g 2 ( a + g x n ) 2 2 Δ ( x n ) } ≡ 0 , (1.2.7)

and therefore lim n → ∞ ℑ ( x n ) ≡ 0 . However

lim n → ∞ Γ 14 4 ( x n ) = l i m n → ∞ Γ 41 4 ( x n ) = l i m n → ∞ 1 a + g x n = ∞ , (1.2.8)

i.e. classical Levi-Civit’a connection given by (1.2.2) unavailable at Moller horizon.

Remark 1.2.1. In order to avoid difficultness mentioned above, we consider now the regularized Moller’s metric

d ε s 2 = − Δ ε ( x ) d t 2 + d x 2 + d y 2 + d z 2 , Δ ε ( x ) = [ ( a + g x ) 2 + ε 2 ] , ε ∈ ( 0 , 1 ] . (1.2.9)

Using now Dingle’s formula [

ℑ ( x ; ε ) = G 2 2 ( x ; ε ) = G 3 3 ( x ; ε ) = − 1 2 Δ ε ( x ) { Δ ″ ε ( x ) − [ Δ ′ ε ( x ) ] 2 2 Δ ε ( x ) } , Δ ε ( x ) = [ ( a + g x ) 2 + ε 2 ] . (1.2.10)

Note that

Δ ′ ε = 2 g ( 1 + g x ) , Δ ″ ε = 2 g 2 (1.2.11)

and therefore

ℑ ( x ; ε ) = − 1 2 Δ ε ( x ) { 2 g 2 − 2 g 2 ( a + g x ) 2 Δ ε ( x ) } = − 1 2 Δ ε ( x ) { 2 g 2 − 2 g 2 [ ( a + g x ) 2 + ε 2 ] − 2 g 2 ε 2 Δ ε ( x ) } = − g 2 ε 2 Δ ε 2 ( x ) . (1.2.12)

Remark 1.2.2. (i) Note that ( ℑ ( x ; ε ) ) ε , ε ∈ ( 0,1 ] is Colombeau generalized function such that

c l [ ( ℑ ( x ; ε ) ) ε ] ∈ G ( ℝ ) and c l [ ( ℑ ( − ( a / g ) − 1 ; ε ) ) ε ] = c l [ ( ε − 2 ) ε ] ∈ ℝ ˜ .

Remark 1.2.3. Note that: (i) at any point x ˜ ∈ ℝ ˜ such that x ∈ ℝ and x ≠ − a ⋅ g − 1 one obtains ( ℑ ( x ˜ ; ε ) ) ε ≈ ℝ ˜ 0 ˜ (see Definition 1.5.0 (i)) and therefore the Ricci tensor as well as the Ricci scalar are infinite small beyond Moller horizon x h o r M = − a ⋅ g − 1 ˜ . Thus at any point x ˜ such that x ∈ ℝ and x ≠ − a ⋅ g − 1 we obtain the disered result in a good agriment with formall canonical calculation (see for example [

Remark 1.2.4. (I) Thus Colombeau generalized fundamental tensor ( g i k ( ε ) ) ε corresponding to Colombeau metric

( d ε s 2 ) = − ( Δ ε ( x ) d t 2 ) ε + d x 2 + d y 2 + d z 2 , ( Δ ε ( x ) ) ε = ( [ ( a + g x ) 2 + ε 2 ] ) ε , ε ∈ ( 0,1 ] (1.2.13)

that is non vacuum Colombeau solution (see [

For Rindler metric

Definition 1.2.1. Distributional Moller’s geometry that is Colombeau extension of the classical Moller’s spacetime given by Colombeau generalized fundamental tensor (1.2.13).

As another important example of Colombeau extension of the singular semi-Riemannian geometry we consider now classical singular Schwarzschild spacetime furnished with a degenerate and singular Schwarzschild metric

Remark 1.3.1. Note that formally corresponding to the metric (1.3.1) classical Levi-Civitá connection given by canonical Christoffel symbols are [

i.e. classical Levi-Civita connection given by Equation (1.3.2) unavailable at Schwarzschild horizon.

Remark 1.3.2. Nevertheless in classical handbooks [

By Equation (1.3.2) it is mistakenly pointed out that the Schwarzschild metric has only a coordinate singularity at

Remark 1.3.3. Note that canonical formal calculation gives

Assume that

Equation (1.3.4) one obtains directly

Remark 1.3.4. Notice that: if

A. Einstein emphasized that uncertainty of the form 0/0 mentioned above that is a fundamental mathematical problem, see [

However Equation (1.3.7) doesn’t holds at

Remark 1.3.5. Thus from Equation (1.3.4) for

and we get nothing at Schwarzschild horizon. Therefore semi-Riemannian geometry break down at Schwarzschild horizon [

Remark 1.3.6. Recall that canonical derivation of the canonical singular Schwarzschild metric in classical handbooks is always based on assumption that:

Assumption 1.3.1. Classical semi-Riemannian geometry holds on the whole semi-Riemannian manifold, see for example [

Let

where

(i) all

The equations

and

and

From Equation (1.3.11)-Equation (1.3.12) one obtains

Therefore AB = constant. Since at

and by integration Equation (1.3.15) one obtains

From Equation (1.3.16) and consideration above (see Remark 1.3.4) Assumption 1.3.1 wrong, otherwise one obtains the contradiction.

Remark 1.3.7. In order to avoid this difficulty:

(i) we have introduced instead a classical Einstein field equations

[where the sign of the energy-momentum tensor is defined by (

apropriate Colombeau generalization of the Equation (1.3.17)-Equation (1.3.18) such that

where the sign of the distributional energy-momentum tensor is defined by

see [

(ii) we have introduced instead of Assumption 1.3.1 the following assumption.

Assumption 1.3.2. Distributional semi-Riemannian geometry holds on the whole distributional semi-Riemannian manifold.

Definition 1.3.1. Let

(i)

Let

and let

(i) all

and

and

Weak distributional limit in

Remark 1.3.8. It turns out that the distributional Schwarzschild metric (1.3.22) has a gravitational singularity with compact support at origin

Let us consider now nonclassical spacetime furnished with a degenerate at horizon

Nonsingular metric (1.3.27) is obtained by the coordinate transformation:

i.e. classical Levi-Civitá connection given by (1.3.28) of course unavaluble at horizon

Remark 1.3.9. In order to avoid difficulty with the degeneracy of the metric (1.3.27) mentioned above, we consider now the corresponding distributional Colombeau metric which reads

where

Definition 1.3.2. Distributional Schwarzschild geometry in isotropic coordinates which is Colombeau extension of the classical spacetime (1.3.27), given by Colombeau generalized fundamental tensor (1.3.30).

Colombeau generalized metric (1.3.30) nondegenerate at horizon in Colombeau sence and distributional Levi-Civitá connection now available on the whole distributional Schwarzschild spacetime in isotropic coordinates. Notice that generalized metric (1.3.30) has the form given by Equation (A.1) (see Appendix A) with

From Equation (A.2) (see Apendix A2) and Equation (1.3.31) in the limit

Compare the equation

Remark 1.3.10. Notice that in contrast with result of naive formal calculation mentioned above (see Equation (1.3.29)) we get:

Let us perform the following coordinate transformation

to the classical singular Schwarzschild metric

we get

In Equation (1.4.2), m is the central mass,

which is distributional Rindler’s spacetime if we neglect the angular contribution. The condition

By using simple coordinate transformations it could be shown that (1.4.5) again becomes the distributional Rindler metric when we take

Remark 1.4.1. At this stage of consideration, it is already clear that near horizon Schwarzschild black hole geometry has a gravitational singularity at horizon. Notice that in classical literature (see, for example, [

Designation 1.5.1. We denote by

Designation 1.5.2. In the sequel we denote by:

Definition 1.5.1. The elements of

is the algebra of Colombeau generalized functions on

Remark 1.5.0. Note that: (i) there exists natural embedding

Definition 1.5.2. (i) Let

or

(ii) We say that

if

(iii) We say that

Definition 1.5.3. Let

correspondingly. We introduce equivalence relation given by

and denote by

Note that if the

Definition 1.5.4. We denote by

Definition 1.5.5. Let

Definition 1.5.6. A real vector bundle consists of:

1) Topological spaces X (base space) and E (total space)

2) A continuous surjection

3) For every x in X, the structure of a finite-dimensional vector space over Colombeau ring

The open neighborhood U together with the homeomorphism

The Cartesian product

The basic idea of Colombeau’s theory of generalized functions is a regularization by sequences (nets) of smooth functions and the use of asymptotic estimates in terms of a regularization parameter

of the space

Elements of

With componentwise operations

The spaces of moderate resp. negligible sequences and hence the algebra itself may be characterized locally, i.e.,

Let

The Colombeau tangent vector at the point x may then be defined as

Let

1)

2)

3)

Let M be a differentiable manifold and let

1)

Note that the derivation will by definition have the Leibniz property

2)

Colombeau vector field

Suppose now that M is a

which is modeled on the product rule of calculus.

If we define addition and scalar multiplication on the set of derivations at x by

where

Given any vector space

Let M be a smooth manifold and let x be a point in M. Let

Suppose now that M is a

We can then define the differential map

(i)

Let

The

Remark 1.5.1. Smooth sections of

Since

Moreover we have the following algebraic characterization of the space of generalized sections

where

Here

A generalized

(i)

(ii)

We call a separable, smooth Hausdorff manifold M furnished with a generalized pseudo-Riemannian metric

A generalized metric

Note that condition (ii) above is precisely equivalent to invertibility of

The inverse metric

Moreover if

From now on we denote the inverse metric by

Notice that

Let

Generalized connection

(D1)

(D2)

(D3)

Let ^{i}. The generalized Christoffel symbols for this chart are given by the

Theorem. [

(D4)

hold for all

(II) On every chart

The generalized Christoffel symbols are given by

or by using representative

We define now the action of a classical (smooth) connection D on generalized vector fields

(III) Let

(i) If

(ii) If

(iii) Let

Let

(i) The generalized Riemannian curvature tensor

(ii) The generalized Ricci curvature tensor is defined by

(iii) The generalized Ricci scalar is defined by

(iv) Finally we define the generalized Einstein tensor by

Examining now the Schwarzschild metric (1.3.1) (note that the origin is now excluded from our considerations, the space we are working on is

Here

Obviously, (1.6.1) is degenerate at

Obviously the generalized pseudo-connection

where

Remark 1.6.1. In paper [

Remark 1.6.2. Due to the degeneracy of any smooth regularization of the metric (1.3.1) no canonical Levi-Civitá connection could be defined. In order to avoid such difficultnes in our papers [

The basic idea of the theory of super generalized functions is regularization by sequences (nets) of appropriate classes of non smooth and discontinuous functions or classical distributions and the use of asymptotic estimates in terms of a regularization parameter

of the space

Here

We denote by

Let

of the space

The

(

dimension of the fibers) satisfying

Remark 1.6.3. Smooth sections of

Since

Moreover we have the following algebraic characterization of the space of super generalized sections

where

Here

A super generalized

(i)

(ii)

We call a separable, smooth Hausdorff manifold M furnished with a super generalized pseudo-Riemannian metric

A super generalized metric

Note that condition (ii) above is precisely equivalent to invertibility of

Let

Super generalized connection

(D1)

(D2)

(D3)

Let

Theorem. (I) Let

(D4)

hold for all

(II) On every chart

The super generalized Christoffel symbols are given by

or by using representative

We define now the action of a classical (smooth) connection D on super generalized vector fields

(III) Let

(i) If

(ii) If

(iii) Let

Let

(i) The super generalized Riemannian curvature tensor

(i) The super generalized Ricci curvature tensor is defined by

(iii) The super generalized Ricci scalar is defined by

4) Finally we define the super generalized Einstein tensor by

Singular space-times present one of the major challenges in general relativity. Originally it was believed that their singular nature is due to the high degree of symmetry of the well-known examples ranging from the Schwarzschild geometry to the Friedmann-Robertson-Walker cosmological models. However, Penrose and Hawking [

in a conceptually satisfactory way.

Remark 2.1.1. The result (2.1.1) can be easily obtained by using apropriate nonsmooth regularization of the Schwarzschild singularity at the origin

The nonsmooth regularization of the Schwarzschild singularity at the origin

Here

and the limit

with

and

In papers [

Remark 2.1.2. The nonsmooth regularization of the Schwarzschild singularity above the horizon

Here

The truncated distributional Schwarzschild geometry.

There exist two different types of distributional Schwarzschild blackhole geometry corresponding to classical Schwarzschild solution. That is: (i) full distributional Schwarzschild blackhole geometry, given by Colombeau generalized object, for example by Equation (1.3.30), see

Remark 2.1.3. In a nutshell, there is a widespread but mistaken belief that there exist true gravitational singularities, for example at origin

Schwarzschild spacetime, and non principal and non gravitational, i.e. purely coordinate singularities, for example at horizon

originally defined by singular and degenerate Schwarzschild metric [

by using apropriate singular coordinate change [

Remark 2.1.4. Note that: (i) metric (2.1.11) is singular and degenerate at Schwarzschild horizon

(ii) however in physical literature (see for example [

Remark 2.1.5. (see [

Remark 2.1.6. Notice that consideration above meant the following definition of the gravitational singularity.

Definition 2.1.1. There is no gravitational singularity at

Remark 2.1.7. Notice that at singular point

however in the limit

and therefore the Definition 2.1.1 is not sound and even does not any sense under canonical semi-Riemannian geometry.

Remark 2.1.8. Notice that:

(i) in order to fix the problem with singularity and degeneracy of the Schwarzschild metric (2.1.11) at Schwarzschild horizon

(ii) the change (2.1.14) of Schwarzschild coordinates is singular at Schwarzschild horizon

(iii) under the singular change (2.1.14) Schwarzschild metric (2.1.11) becomes to well known regular and nondegenerate Eddington-Finkelstein metric [

(iv) in physical literature many years exist abnormal belief that by formal singular change (2.1.15) the singular and degenerate Schwarzschild spacetime

with regular and non degenerate metric tensor

(v) from statement (iii) it was mistakenly assumed that there is no gravitational singularity at BH horizon.

We remind now canonical definitions.

Definition 2.1.2. Let

Remark 2.1.9. Notice that such isometric embedding is a mathematical definition only and does not mean the equivalence

Definition 2.1.3. [

Remark 2.1.10. Notice that such extension is a mathematical definition only and therefore it is not always apropriate as extension of the Lorentzian manifolds

Remark 2.1.11. In order to obtain example for the statement mentioned and Remark 2.1.8 and Remark 2.1.9 we are going to prove below that the geometry of Schwarzschild spacetime

We remind now canonical definitions.

Definition 2.1.4. Let

Definition 2.1.5. (I) Let

(II) Let

Remark 2.1.12. (I) Note that the geometry of Schwarzschild spacetime

above Schwarzschild horizon

above Eddington-Finkelstein horizon

(II) Note that Schwarzschild spacetime

Thus the geometry of spacetime

Remark 2.1.13. Note that from Remark 2.1.11 it follows that Eddington-Finkelstein spacetime does not hold in rigorous mathematical sense as extension of the Schwarzschild spacetime

Remark 2.1.14. It is clear that nonregularity condition (2.1.23) arises not only from singularity of the function

Remark 2.1.15. We remind now that the relations (see [

give the connection between the metric of real space

and the metric of the four-dimensional space-time

For Eddington-Finkelstein metric (2.1.15) metric of the corresponding real space is

Remark 2.1.16. Notice that the Eddington-Finkelstein metric (2.1.15) is regular at the horizon and therefore the infalling observer encounters nothing unusual at the horizon. However from Equation (2.1.17) it follows that the infalling observer encounters singularity on horizon. But this is a contradiction.

Remark 2.1.17. Note that in order to deal with singular Schwarzschild metric (2.1.11) using mathematically and logically soundness approach, one applies contemporary distributional geometry based on Colombeau generalized functions [

(i)

(ii) for any

Distributional Schwarzschild spacetime.

Remark 2.1.18. Note that in the case of Schwarzschild spacetime the conditions (i) and (ii) mentioned above (see Remark 2.1.13) are satisfied only by using non smooth regularization of the singular and degenerate Schwarzschild metric

By apriporiate nonsmooth regularization one obtain Colombeau generalized object modeling the singular Schwarzschild metric above and below horizon [

Remark 2.1.19. Let us rewrite now the metric (2.1.24) (above horizon) in the form

and define a new generalized Colombeau coordinates

Remark 2.1.20. Notice that:

(i) Colombeau generalized coordinates (2.1.26) are the Colombeau extension of the canonical Eddington-Finkelstein coordinates (2.1.14) by Colombeau generalized function.

(ii) In contrast with canonical Eddington-Finkelstein coordinates (2.1.14) (see Remark 2.1.7), Colombeau generalized coordinates (2.1.26) holds at

Schwarzschild horizon

Rewriting now the metric (2.1.25) in terms of the Colombeau generalized coordinates

We rewrite now Colombeau metric (2.1.27) in the equivalent form

Colombeau metric (2.1.28) define the distributional Eddington-Finkelstein space-time

above the Eddington-Finkelstein horizon

Remark 2.1.21. Notice that

Of course at horizon

Remark 2.1.22. Note that:

(i) under coordinate change (2.1.26) the distributional curvature scalars of the distributional Schwarzschild space-time given by metric (2.1.24), does not changes because these scalars depend only on variable

(ii) in contrast with classical Eddington-Finkelstein space-time

distributional Eddington-Finkelstein space-time has a gravitational singularity at horizon.

Remark 2.1.23. Note that for the case of the distributional space-time the relations (2.1.24) obviously takes the form

where (2.1.30) give the connection between the Colombeau metric of the distributional real space

and the Colombeau metric of the four-dimensional distributional space-time

For distributional Eddington-Finkelstein metric (2.1.29) above horizon of the corresponding Colombeau metric of the distributional real space is

Remark 2.1.24. Notice since the distributional Eddington-Finkelstein space-time (2.1.29) has a gravitational singularity (see Definition 1.1.1) at horizon, there is no contradiction mentioned above for the case of the regular classical Eddington-Finkelstein metric (2.1.15) and the corresponding singular metric (2.1.17), see Remark 2.1.15.

Recall that the classical Kruskal-Szekeres coordinates are defined, from the classical Schwarzschild coordinates

It follows that the Schwarzschild radius r, in terms of Kruskal-Szekeres coordinates, is implicitly given by

for both interior and exterior regions, i.e.

The location of the event horizon (

Remark 2.1.25. Note that the metric (2.1.37) ofcourse is perfectly well defined and non-singular at the event horizon. The curvature singularity is located at

Remark 2.1.26. In contrast with Eddington-Finkelstein coordinates the classical Kruskal-Szekeres coordinates holds at Schwarzschild horizon, but however the differentials

Remark 2.1.27. In order to avoid these difficulties one can apply instead of the Kruskal-Szekeres coordinates (2.1.35)-(2.1.36) the following distributional Kruskal-Szekeres coordinates to Colombeau generalized metric (2.1.8)

Therefore for both interior and exterior regions we get

Remark 2.1.28. Note that in contrast with (2.1.37) at horizon

In these new distributional coordinates the Colombeau metric (2.1.8) of the distributional Schwarzschild black hole manifold above horizon is given by formula

Here

Remark 2.1.29. Note that in contrast with (2.1.36) Colombeau generalized metric (2.1.39) is non degenerate at horizon

Remark 2.2.1. Note that due to the degeneracy of the metric (2.1.11) at Schwarzschild horizon, the classical Levi-Civit’a connection on whole Schwarzschild spacetime is not available [

Remark 2.2.2. In order to avoid difficulties with classical Levi-Civit’a connection mentioned above in Remark 2.2.1, in papers [

Obviously distributional connections

Remark 2.2.3. As expected, the distributional Ricci tensor as well as the distributional Ricci scalar vanish identically on

where

For

We remind now that 2D Rindler spacetime is a patch of Minkowski spacetime, see

Remark 2.2.4. Due to the degeneracy of the metric (2.2.5) at Rindler gorizon

and all other components being zero.

Remark 2.2.5. We emphazize that Rindler space-time _{>}, see Remark 2.1.12. Let

Remark 2.2.6. We emphazize that in physical literature the Rindler metric (2.2.5) mistakenly were considered as is just a part of the Minkowski space-time

from the space

Remark 2.2.7. Note that in order to avoid this difficultnes mentioned above (see Remark 2.2.4-2.2.5), the origin in classical consideration the Rindler horizon is always excluded from the space

following Moller [

where the accents indicate differentiation with respect variable R, and all other components of

Remark 2.2.8. By calculations mentioned above, from Mo̸ ller’s times until nowdays, Rindler metrical tensor was mistakenly considered in physical literature as an vacuum solution of the Einstein’s field equations,e.g.,solution for empty space,see Møller [

Remark 2.2.9. Note that Levi-Cività connection on the whole space ℝ^{3}^{.1} is available only in Colombeau sense under smooth regularization

Then for Einstein distributional tensor [

we get

Thus,

where

We remind that originally Einstein’s gravity was formulated by using classical pseudo Riemannian geometry with classical Levi-Civit’a connection. In classical pseudo Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold. More specifically, it is the torsion-free metric connection, i.e., the torsion-free connection on the tangent bundle (an affine connection) preserving a given (pseudo-Riemannian) Riemannian metric. The fundamental theorem of classical Riemannian geometry states that there is a unique connection which satisfies these properties.

Remark 2.3.1. Note that classical Einstein “Equivalence Principle” asserts the equivalence between inertial and gravitational forces of acceleration. The classical Einstein equivalence principle is the heart and soul of gravitational theory, for it is possible to argue convincingly that if EEP is valid, then gravitation must be a “curved spacetime” phenomenon, in other words, gravity must be governed by a “metric theory of gravity”, whose postulates are:

1) Spacetime is endowed with a symmetric Lorentzian metric.

2) The trajectories of freely falling test bodies are geodesics of that metric.

3) In local freely falling reference frames, the non-gravitational laws of physics are those written in the language of special relativity.

In order to obtain appropriate generalization of EEP based on distributional Colombeau geometry [

1) Spacetime in general case is endowed with a symmetric distributional Lorentzian metric.

2) The trajectories of freely falling test bodies are geodesics of that distributional metric.

3) In local freely falling distributional reference frames, the non-gravitational laws of physics are those written in the language of special relativity.

In a recent work [

Much of formalism can be explained with Colombeau generalized scalar field [

Here

Here

With

The canonical momentum at a time

where

Here

where

Using equation of motion Equation (3.1.2) one obtains corresponding Colombeau generalization of the canonical Green functions equations. In particular for the Colombeau distributional propagator

one obtains directly

We obtan now an adiabatic expansion of

where

and its Colombeau-Fourier transform

where

where

and we are using the symbol

where

with all geometric quantities on the right-hand side of Equation (3.1.17) evaluated at

In Equation (3.16), then the

The function

Using Equation (3.1.12), Equation (3.1.18) gives a representation of

where

In the normal coordinates about

with

Remark 3.1.1. Note that the Expansions (3.1.19) and (3.1.22) are, however, only asymptotic approximations in the limit of large adiabatic parameter T.

If (3.1.22) is substituted into (3.1.20) the integral can be performed to give the adiabatic expansion of the Feynman propagator in coordinate space:

which, strictly, a small imaginary part

Remark 3.1.2. Since we have not imposed global boundary conditions on the distributional Green function Colombeau solution of (3.1.10), the expansion (3.1.23) does not determine the particular vacuum state in (3.1.9). In particular, the “

As in classical case one can obtain Colombeau generalized quantity

Note that the generating functional

was interpreted physically as the vacuum persistence amplitude

Following canonical calculation one obtains [

where the proportionality constant is metric-independent and can be ignored. Thus we obtain

In (3.2.4)

in such a way that

Remark 3.2.1. Note that the trace

Writing now the Colombeau generalized operator

by Equation (3.1.20) we obtain

Proceeding in standard manner we get [

Interchanging now the order of integration and taking the limit

Colombeau generalized quantity

whence one get

Note that

where the coefficients

Let us determine now the precise form of the geometrical

of which the first

From Equation (3.3.3) it follows we shall wish to retain the units of ^{−}^{4}, even when

If

Denoting these first three terms by

The functions

Finally one obtains [

Remark 3.3.1. All the higher order

which must be handled carefully. Substituting for

where

Finally we obtain [

Therefore for the case of the distributional Schwarzchild spesetime using Equation (2.2.4) and Equation (3.3.12)

Finally from Equation (3.3.13) for

Remark 3.3.2. Thus QFT in distributional curved spacetime predict that the infalling observer burns up at the BH horizon.

Remark 3.3.3. In order to avoid singularity at horizon

defined as an evolving constant (i.e. a Dirac observable), must correspond to a selfadjoint operator at the quantum level. Classically,

At the singularity, i.e.

Therefore, this argument strongly suggests that the classical singularity will be resolved at the quantum level since

Remark 3.3.4. Let

grows infinitely near the horizon [

Therefore Stefan-Boltzmann law under formal calculation by using classical Schwarzschild geometry is evidently violated. Let us remind that the acceleration

The acceleration points along the radius and is directed toward the center; as

From Equation (3.3.20) and Equation (3.3.14) as

Therefore Stefan-Boltzmann law under rigorous calculation by using distributional Schwarzschild geometry evidently is not violated.

We remind now that a black holes have an approximate Rindler region near the Schwarzschild horizon. For the the distributional Schwarzschild solution (2.1.8) by coordinate transformation

we obtain

The

Therefore, sufficiently strongly accelerated observer burns up near the Rindler horizon. Thus, Polchinski’s account is not a violation of the Einstein equivalence principle.

Remark 3.4.1. Note that by using Equation (A.8) and Equation (A.9) (see Appendix A) one obtains Equation (3.4.3) directly from distributionel Möller metric (1.2.13) and distributionel Rindler metric (2.2.10).

The Unruh effect is the prediction that an accelerating observer will observe blackbody radiation where an inertial observer would observe none. The Unruh effect was first described by Stephen Fulling in 1973, Paul Davies in 1975 and W. G. Unruh in 1976 [

where g is the local acceleration, ^{20} m/sec^{2} corresponds approximately to a temperature of 1 K. Notice that for a proper acceleration of 2.47 × 10^{20} m/sec^{2} the event horizon very close to observer by distance^{−1} = 3. 6387 × 10^{−4} m. It is currently not clear whether the Unruh effect has actually been observed, since the claimed observations are disputed. There is also some doubt about whether the Unruh effect implies the existence of Unruh radiation. Although Unruh’s prediction that an accelerating detector would see a thermal bath is not controversial, the interpretation of the transitions in the detector in the nonaccelerating frame is. It is widely, although not universally, believed that each transition in the detector is accompanied by the emission of a particle, and that this particle will propagate to infinity and be seen as Unruh radiation. The existence of Unruh radiation is not universally accepted. Some claim that it has already been observed [

Thus observer with a proper acceleration of 2.47 × 10^{20} m/sec^{2} burns up near the Mӧller horizon.

On a Riemannian or a semi-Riemannian manifold, the metric determines invariants like the Levi-Civita connection and the Riemann curvature. If the metric becomes degenerate (as in singular semi-Riemannian geometry), these constructions no longer work, because they are based on the inverse of the metric, and on the related operations like the contraction between covariant indices. In order to avoid these difficulties distributional geometry by using Colombeau generalized functions [

Such generalization of classical GR based on appropriate generalization of the Einstein equivalence principle (GEEP) is mentioned above in subsection 2.3. Using Rindler distributional geometry Unruh effect revisited. We pointed out that GEEP avoid the contradiction which was mentioned by Z. Merali in paper [

We thank the Editor and the referee for their comments.

Foukzon, J., Potapov, A. and E. Men’kova (2018) Was Polchinski Wrong? Colombeau Distributional Rindler Space-Time with Distributional Levi-Cività Connection Induced Vacuum Dominance. Unruh Effect Revisited. Journal of High Energy Physics, Gravitation and Cosmology, 4, 361-440. https://doi.org/10.4236/jhepgc.2018.42023

Let us introduce now Colombeau generalized metric which has the form

The Colombeau scalars

Remark A1.1. Note that the Colombeau scalars

The distributional Mӧller’s metric is

In order to aply Equation (A1.2) directly we chose now

We choose now in the Equation (A1.2):

Note that

From Equations (A1.5)-(A1.7) by Equation (A1.2) we get

From Equation (A1.8) in the limit

Remark A1.2. Note that: (1) Equation (1.2.14) in a nice agriment with Equation (A1.9), see Remark 1.2.2-Remark 1.2.4. (2) For

see Definition 1.5.2. (i). (3) At horizon

see Definition 1.5.2. (ii).

Remark A1.3. Let

From Equations (A1.5)-(A1.7) by formulae (A1.2) we get

From Equation (A1.13) in the limit

Remark A1.4. At horizon

see Definition 1.5.2. (ii).

Remark A1.5. Let

From Equation (A1.4)-Equation (A1.6) by formulae (A1.2) we get

In the limit

Remark A1.6. At horizon

see Definition 1.5.2. (ii).

Remark A1.7. Let

Remark A1.8. We assume now there exist a fundamental generalized length

such that

where parameter

By using (A1.21) we get the estimate

Let us consider now distributional Colombeau metric given by Equation (1.3.30) with

where

We choose now

We assume now that

From Equation (A2.3) by formulae (A1.2) we get

From Equation (A2.4) in the limit

Remark A2.1. Note that: (1) Equation (A2.5) in a nice agriment with Equation (A1.9). For

see Definition 1.5.2. (i). (3) At horizon

see Definition 1.5.2. (ii).

Remark A2.2. Let

From Equation (A2.3) by formulae (A1.2) we get

From Equation (A2.9) in the limit

Remark A2.3. Note that: (1) For

see Definition 1.5.2. (i). (2) At horizon

see Definition 1.5.2. (ii).

Remark A2.4. Let

From Equation (A2.3) by formulae (A1.2) we get

From Equation (A2.14) in the limit

where

Remark A2.5. Note that: (1) For

see Definition 1.5.2. (i). (2) At horizon

see Definition 1.5.2. (ii).

Remark A2.6. Let

Remark A2.7. We assume now there exist a fundamental generalized length

such that

where parameter η is a classical thickness of BH horizon.

By using (A2.19) we get the estimate

We calculate now the distributional curvature at Schwarzschild horizon. In the usual Schwarzschild coordinates

Metric takes the form above horizon

Remark B.1. Following the above discussion we consider the metric coefficients

Note that, accordingly, we have fixed the differentiable structure of the manifold: the Cartesian coordinates associated with the spherical Schwarzschild coordinates in (B.1) are extended through the origin. We have above

Inserting (B.4) into (B.2) we obtain a generalized object modeling the singular Schwarzschild metric above (below) gorizon, i.e.,

The generalized Ricci tensor above horizon

From (B.4) by differentiation we obtain

angular components of the Ricci tensor (using the abbreviation

and let

(i)

Then for any function

By replacement

By replacement

From Equation (B.11) we get

where we have expressed the function

with

Equations (B.12)-(3.13) give

Since

For

where use is made of the relation

Finally we obtain

The Colombeau generalized Ricci tensor below horizon

From (B.4) we obtain

Investigating the weak limit of the angular components of the Ricci tensor

(using the abbreviation

function

(i) (ii)

Then for any function

By replacement

By replacement

which is calculated to give

where we have expressed the function

with

Since

For

By replacement

By replacement

which is calculated to give

where we have expressed the function

with

where use is made of the relation

Thus

We calculate now the distributional Colombeau scalars

and rewrite Equation (A.1) in the following equivalent form

where

From Equation (A.2) and Equation (C.3) we obtain

Finally we obtain the following expression for the distributional Colombeau scalar

Remark C.1. Note that from Equation (C.5) follows that:

Definition 1.5.2. (i).

We assume now that

Remark C.2. Note that from Equation (C.6) at horizon

see Definition 1.5.2. (ii).

Remark C.3. Note that from Equation (C.5) follows that:

Remark C.4. Let

From Equation (A.2) and Equation (C.3) we obtain

Remark C.5. Note that from Equation (C.10) follows that:

see Definition 1.5.2. (i).

We assume now that

Remark C.6. Note that from Equation (C.10) at horizon

see Definition 1.5.2. (ii).

Remark C.7. Let

From Equation (A.2) and Equation (C.3) we obtain

Remark C.8. Note that from Equation (C.15) follows that:

see Definition 1.5.2. (i).

We assume now that

Remark C.9. Let

Remark C.10. Note that from Equation (C.15) at horizon r = 2m follows that:

see Definition 1.5.2. (ii).

Remark C.11. We assume now there exist a fundamental generalized length

such that

where parameter η is a classical thickness of BH horizon.

By using (C.20) we get the estimate