Russian Physics Journal, Vol. 47, No. 4, 2004

LARGE-SCALE TWO-DIMENSIONAL QUANTUM FLUCTUATIONS

OF FIVE-DIMENSIONAL AND FOUR-DIMENSIONAL

SPACE-TIME METRICS

M. S. Shapovalova UDC 530.12:531.51

Large-scale two-dimensional quantum fluctuations of five-dimensional space-time metric are constructed andthe effect of the fluctuations on the nested four-dimensional worlds is studied. In doing so, the fluctuationsaffect not all four-dimensional worlds but only a part of them. The energy-momentum tensor of four-dimensional space-time has a physical form both in the absence and in the presence of fluctuations; it meansthat the fluctuations can be realized by real matter. A spatial region occupied by the fluctuations constructedin this work can be infinitely large and the fluctuations can occur during a long period of time. Therefore,we refer to these fluctuations as large-scale fluctuations.

INTRODUCTION

Quantum fluctuations of metric were first studied by Wheeler [1, 2], who showed that these fluctuationsare significant at the Planckian distances. In [3–5], Modanese pointed to the possibility of large-scale quantumfluctuations of metric tensor, that is, such fluctuations which can exist at arbitrary large distances both in timeand space coordinates.

In this work, we consider the methods of constructing quantum two-dimensional (2D) fluctuations of five-dimensional (5D) space-time metric and the effect of these fluctuations on four-dimensional (4D) space-time.

We assume an infinite multitude of physical 4D worlds nested in 5D space-time. The 5D space-time V 5 hastwo time and three space coordinates. A set of 4D worlds V 4 (having one time and three space coordinates) formsa foliation of codimension 1 in the 5D space-time V 5 [6–10].

The coordinates in V 5 are represented by (x0, x1, x2, x3, x4), the coordinates in V 4 – by (y0, y1, y2, y3). Thecoordinates x0 and x1 are the time coordinates in V 5.

The foliation in the space-time V 5 is specified in the plane (x0, x1) by the formula

x0 = α exp(x1) + 1, (1)

where α (0 < α < ∞) is the real parameter determining the 4D worlds V 4α1

, V 4α2

, etc. Thus, 4D worlds differ by theparameter α, each 4D world V 4

αihaving its own value of the parameter αi. The foliation in the plane (x0, x1) is

shown in Fig. 1.The coordinates of the 5D space-time V 5 are related with the coordinates of the 4D space-time V 4 by the

following relations:

x0 = 1 + α exp(y0),x1 = y0,

x2 = y1, x3 = y2, x4 = y3.

It is shown in Fig. 1 that in the space-time V 5, 4D worlds with different values of the parameter α come close inthe past (with decrease in the coordinate y0) and move apart in the future (with increase in the coordinate y0).

Omsk State University. Translated from Izvestiya Vysshykh Uchebnykh Zavedenii, Fizika, No. 4, pp. 20–24,April, 2004. Original article submitted November 23, 2002.

1064-8887/04/4704-0359 c© 2004 Springer Science+Business Media, Inc. 359

Fig. 1. Foliation in 5D space-time.

1. THE 5D SPACE-TIME METRIC IN THE ABSENCE OF FLUCTUATIONS

Let us consider the 5D space-time V 5 of the following form:

V 5 = (x0, x1, x2, x3, x4) ∈ R5 : x0 > 1.

The 4D space-time V 4α becomes

V 4 = (y0, y1, y2, y3) ∈ R4.Let us specify the metric in the space-time V 5

dI2 = GIKdxIdxK = dx02+ x02

dx12 − F 2(r)dr2 − dΩ2. (2)

The signature of the metric is (+ + − − −). The function F (r) in Eq. (2) is an arbitrary function chosen so thatthe volume of 3D space equal to

V =

∞∫

0

π∫

0

2π∫

0

F (r) sin θdrdθdφ = 4π

∞∫

0

F (r)dr,

is finite. For example, as the function F (r), we can take

F (r) = r0 exp(−r), (3)

where r0 is a positive constant comparable with a spatial dimension of the Universe by the order of magnitude.In this case, the volume of 3D space is finite and is equal to V = 4πr0. The 3D space is noncompact and has theEuclidean topology R3.

The space-time V 5 is topologically homeomorphic (R × S1) × R3 and the space-time V 4 is topologicallyhomeomorphic R× R3.

The scalar curvature 5R of the space-time V 5 is 5R = −2.

The Einstein action 5S for the space-time V 5 has the following form:

5S =c3

16πk

∫5R(−G)1/2d5x =

c3πr0

2k

(x02

2 − x02

1

). (4)

2. THE 4D SPACE-TIME METRIC IN THE ABSENCE OF FLUCTUATIONS

The induced metric gik of the space-time V 4 in the coordinates (y0, r, θ, φ) takes the form

ds2α = gα

ikdyidyk

360

=(α2 exp(2y0) + (α exp(y0) + 1)2

)dy02 − F 2(r)dr2 − dy22 − dΩ2. (5)

The scalar curvature 4R of the space-time V 4 is 4R = 5R = −2.Nonzero components of the energy-momentum tensor Tik of the space-time V 4 are

T00 =1κ

(α2 exp(2y0) + (α exp(y0) + 1)2

), T11 = − 1

κF 2(r) = − 1

κr20 exp(−2r).

The energy-momentum tensor Tik of the space-time V 4 is physical, that is, it can correspond to real physicalmatter, provided

T00 > 0, (6)

and there is such a time-like 4-vector wi that the following energy condition is fulfilled:

Tikwidwk > 0. (7)

Condition (6) for the tensor Tik is evidently satisfied. Condition (7) is fulfilled if , for example, the time-likevector wi = (1, 0, 0, 0) is taken as the vector wi.

3. FLUCTUATIONS OF METRIC IN THE 5D SPACE-TIME

Let us consider two-dimensional fluctuations of the metric in the space-time V 5 for which the metric tensorGIK becomes

dI2 = GIKdxIdxK

= dx02+

(x02

+ H(x0, x1))

dx12 − F 2(r)dr2 − dθ2 − sin2 θdφ2. (8)

The fluctuations are specified by the function H(x0, x1) depending on two time coordinates x0 and x1. Let thefunction H(x0, x1) be such that it can be represented in the form

H(x0, x1) = h2(x0)q2(x1) + 2x0h(x0)q(x1).

Then metric (8) can be written as

dI2 = dx02+

(x0 + h(x0)q(x1)

)2dx12 − F 2(r)dr2 − dθ2 − sin2 θdφ2. (9)

The fluctuations occur in the region U of the space-time V 5

U = a0 < x0 < b0, a1 < x1 < b1, 0 < r < ∞, 0 6 θ < π, 0 6 φ < 2π,

where a0, b0, b1, and a1 are the boundary points of the fluctuation region and 1 < a0 < b0, 0 < (b1 − a1) 6 2π.The region U encompasses all 3D space, therefore, the fluctuations occur in the complete space. The boundaryconditions on the functions h(x0) and q(x1) are of the form

h(a0) = h(b0) = 0, (10)

q(a1) = q(b1) = 0. (11)

Outside the region U , one assumes h(x0) ≡ 0 and q(x1) ≡ 0. In the region U , the functions h(x0) and q(x1) cantake any finite values.

Metric (9) has singularities at the points for which h(x00)q(x

10) = −x0, since at these points G11 = 0. In

particular, it is at these points where the scalar curvature 5R can turn to infinity.The volume of the 3D space V 3 is not changed under fluctuations.

361

It is the curvature of the space-time V 5 that changes. Now the scalar curvature is

5R = 5R + ∆5R = −2− 2q(x1)

x0 + h(x0)q(x1)d2h(x0)

dx02 . (12)

The action 5S for the space-time V 5 under fluctuations is

5S =c3r0

2k

b0∫

a0

b1∫

a1

(1 +

q(x1)(x0 + h(x0)q(x1))

d2h(x0)dx02

) (x0 + h(x0)q(x1)

)dx0dx1.

The fluctuations introduce no distorting interference to the contribution of the 5D space-time V 5 being anextremal of the Euler equations (i.e., in this case, the Einstein equations) to the Feynmann integral with respect tothe trajectories ∫

exp(

i5S

~

)D[5G] (13)

provided, that the action 5S of space-time under fluctuations differs from the action 5S of space-time in the absenceof fluctuations by no more than an order of magnitude ~. That is

|5S − 5S| < ~. (14)

Obviously, this inequality is satisfied, providing5S = 5S. (15)

To fulfil this condition, one must properly choose the functions h(x0) and q(x1).It follows from condition (15) that

b0∫

a0

b1∫

a1

(1 +

q(x1)(x0 + h(x0)q(x1))

d2h(x0)dx02

) (x0 + h(x0)q(x1)

)dx0dx1 =

b0∫

a0

b1∫

a1

x0dx0dx1.

Rearranging the integral from the right side of the equation to the left side and cancelling, we get

b0∫

a0

b1∫

a1

q(x1)(

h(x0) +d2h(x0)

dx02

)dx0dx1 = 0.

Therefore, condition (15) is fulfilled atb1∫

a1

q(x1)dx1 = 0 (16)

orb0∫

a0

(h(x0) +

d2h(x0)dx02

)dx0 = 0. (17)

Equality (16) is valid if, for example, the function q(x1) is taken as

q(x1) = sin(c1x1),

where c1 is a constant and c1 6= 0. In this case, the boundary points a1 and b1 of the fluctuations are

a1 =πm

c1, b1 =

πn

c1,

362

where m and n are real numbers. Then

b1∫

a1

q(x1)dx1 = − (cos(πn)− cos(πm)) = 0

and, hence, 5S = 5S. As h(x0), one can take any function satisfying the boundary conditions (10) alone.Equality (17) is obviously fulfilled if the function h(x0) satisfies the following differential equation:

h(x0) +d2h(x0)

dx02 = 0.

The solution to this equation ish(x0) = c1 cosx0 + c2 sin x0,

where c1 and c2 are constants found from the boundary conditions (10),

−c2

c1= tan a0 = tan b0.

Hence, the boundary points a0 and b0 are not arbitrary. These are the points where tan a0 = tan b0, that is

b0 − a0 = πn,

where n is a real number. One can take any function satisfying only the boundary conditions (11) as q(x1).

4. THE EFFECT OF FLUCTUATIONS OF FIVE-DIMENSIONAL METRICON THE FOUR-DIMENSIONAL SPACE-TIME METRIC

Under fluctuations of the 5D metric, the induced metric of the space-time V 4 becomes

ds2α = gα

ikdyidyk

=(α2 exp(2y0) + (α exp(y0) + 1)2 + H(y0)

)dy02 − F 2(r)dr2 − dΩ2. (18)

The fluctuations of the 5D metric GIK are also fluctuations for the 4D metric gik. In the 4D case, thefluctuations depend on one time coordinate y0 alone (that is, they are one-dimensional despite the fact that thefluctuations of the 5D metric are two-dimensional) and are described by the function H(y0).

If b1− a1 6= 2π, that is, the region of fluctuations U is not a ring in the plane (x0, x1), not all 4D worlds V 4α

fall into the region of fluctuations V 4α . The condition that the space-time V 4

α with a certain specified value of theparameter α would not get into the region of fluctuations, has the form

b0

exp(a1) + 1< α <

a0

exp(b1) + 1.

In those 4D worlds V 4 which get into the region of fluctuations, the latter occur over the complete 3D space.In each particular 4D space-time determined by the parameter α, the fluctuations can be excluded by a cer-

taintransformation of the coordinates y0 → y0′ = f(y0), which is found from the condition

∫ (α2 exp(2y0) + (α exp(y0) + 1)2 + H(y0)

)1/2dy0

=∫ (

α2 exp(2y0) + (α exp(y0) + 1)2))1/2

dy0′.

The 4D space-time metric (18) in the presence of fluctuations in the coordinates (y0′, y1, y2, y3) takes the same formas the initial 4D metric (5).

363

Physically, these fluctuations of 4D metric correspond to deceleration or acceleration of time in the space-time V 4.

The scalar curvature of 4D space-time in the presence of fluctuations is unaffected, 4R = −2.Nonzero components of the energy-momentum tensor under fluctuations are

T00 =1κ

(α2 exp(2y0) + (α exp(y0) + 1)2 + H(y0)

),

T11 = − 1κ

F 2(r) = − 1κ

r20 exp(−2r).

Conditions (6) and (7), under which the energy-momentum tensor of the 4D space-time V 4 retains its physicalmeaning, will be satisfied as before.

CONCLUDING REMARKS

In the paper, large-scale two-dimensional quantum fluctuations of the 5D space-time metric are constructedand the effect of the fluctuations on 4D space-time is studied.

Despite the fact that the fluctuations of the 5D metric are two-dimensional, the induced fluctuations of the4D space-time metric depend on one coordinate alone, that is, they are one-dimensional. The effect of fluctuationsof the 5D space-time metric on the 4D space-time metric is exhibited as a spontaneous change of the scale alongthe time axis, that is, as either deceleration or acceleration of time (and, hence, they can be eliminated by thetransformation of coordinates). This change occurs in 4D space-time for a certain period of time in the complete3D space.

In 4D worlds, induced fluctuations of the metric take place for a certain period of time in the complete 3Dspace simultaneously. These fluctuations affect not all 4D worlds, but only a certain part of them.

Under fluctuations, the curvature of 5D space-time changes, while the curvature of 4D space-time is unaf-fected in this case.

The energy-momentum tensor of 4D space-time has a physical meaning both in the absence and in thepresence of fluctuations. This implies that these fluctuations can be realized by real particles (physical matter) andassume neither negative mass, for example, nor any other nonphysical sources of the gravitational field.

The fluctuations constructed in this work are quantum, despite the fact that they can occur for an infinitelylarge period of time, affect the complete 3D space, and can have an infinitely large amplitude. These fluctuationsare manifested not only at the Planckian but also at the macroscopic scale. Thus, the space-time geometry canchange not only at the Planckian distances but also in macroscopic regions.

The fluctuations constructed cannot be observed in 4D space-time since the changes in the scale along thetime axis take place simultaneously in the complete 3D space.

REFERENCES

1. J. A. Wheeler, Einstein’s Foresight [Russian translation], Mir, Moscow (1970).2. J. A. Wheeler, Gravitation, Neutrino, and the Universe [Russian translation], Nauka, Moscow (1962).3. G. Modanese, Large “Dipolar” Vacuum Fluctuations in Quantum Gravity. Paper gr-qc/0005009, (2000).4. G. Modanese, Phys. Lett. B, 460, 276 (1999).5. G. Modanese, Phys. Lett. D, 460, (2000).6. M. S. Shapovalova, Abstract of paper at Sci. Students’ Conf., OmSU, Omsk, OmSU, 2000.7. I. Tamura, Topology of Foliatons [Russian translation], Mir, Moscow (1979).8. M. S. Shapovalova, Mat. Strukt. Model., No. 7, 104 (2001).9. A. K. Gutz and M. S. Shapovalova, Large Fluctuations of Time and Change of Space-Time Signature. Los

Alamos E-print Paper: gr-qc/0001076 (2000).10. M. S. Shapovalova, Gravitation & Cosmology, 7, No. 3, 193 (2001).

364