G en eral Relativ ity an d G ravi tation , Vol. 29, No. 10, 1997

Time Multidimension in Grav ity

V. S. Barashenkov ,1 ,2 A. B. Pestov 1 and M. Z. Yur’iev 3

Rece ived Apr il 7, 199 7

T he in¯ uence of possible addit ional (hidden) component s of time on a

body ’ s mot ion in the ® eld of a gravit at ional wave is considered . Contrary

to the one-time theory, oscillat ions of the body ª height º and ª widthº

sizes in a plane perpendicular to the direct ion of the wave propagat ion

occur independent ly from one another. T his pecu liarity can be used for

the exp erim ental check of emission of gravit at ional waves with dist inct

t ime tra ject ories in cosm ic cat acly sms. An interest ing analogy between

elect rom agnet ic and grav itat ion quant ities is discussed in the contex t of

t ime multidimension.

KEY WORDS : 3-dimensional t ime ; gravit at ional waves

1. INTRODUCTION

The hypothesis of t ime mult idimension is a direct generalizat ion of Ein-

steinian special relat ivity assuming greater symmetry of the t ime and space

coordinat es when time is considered as a three-dimensional subspace and

every event is characterized by a six-dimensional vector4

x = (x1 , x2 , x3 , ct1 , ct2 , ct3 ). (1)

Dirac, Fock and Podolsky introduced a proper time for each material

body [1], and Tomonaga was the ® rst who introduced that time for ev-

ery spat ial point x [2]. Though these generalizat ions themselves did not

1 J oint Inst itute for Nuclear Research, Dubna Moscow Region, 141980, Russia2 E-mail: barashenkov@lcta30.j inr.dubna.su3

Firm YURIDOR, Moscow, Russia4 In what follows the Lat in and Greek indices take the values k = 1, ..., 3, m = 1, ..., 6) .

All mat rices will be denoted by capit al let t ers.

1 3 4 5

0001-7701/ 97/ 1000-1345$12.50/ 0 1997 P lenum Publishing Corporation

1 3 4 6 B a r a sh e n kov , P e s t ov a n d Yu r ’ i e v

discover any new physical eŒects, they improved the theory and allowed

one to formulate the condit ion of compatibility for equat ions of mot ion ex-

cluding superluminal velocit ies and to develop a consistent renormalizat ion

procedure. It is interesting to follow the path of space-t ime symmetrizat ion

further and to consider time as a three-dimensional vector [3± 6].5

Analysis has shown that this approach is at variance with all the

present ly known experimental facts [6]. In part icular, the di� cult ies with

negat ive energies mentioned by Dorking and Demers [8,9] can be avoided

by means of the principle of time irreversibility [10,11]. Nevertheless, we

do not observe macroscopic bodies moving along time trajectories dist inct

from ours because the energy necessary for creation of such objects is

huge [3]. Bodies with diverse t-trajectories would appear in cosmic cata-

clysms where enormous amounts of energy are produced. What is more, in

very strong gravitational ® elds the concept of energy itself loses it s sense

and the energy conservat ion law becomes inexact . All that must in¯ u-

ence the properties of emitted gravit ational waves which acquire the ª time

component º .6

How does this aŒect the behavior of gravitation detectors which are

now under const ruct ion in many countries? Discussion of this quest ion

is the main goal of the present considerat ion. It should be noted that

both aspects of the problem, the discovery of the time mult i-dimension or,

on the contrary, the proof of inconsist ency for that hypothesis, are very

important.

Section 2 is devoted to calculat ions of the metric tensor determining

a plane gravitational wave in six-dimensional space-t ime. We show that

in contrast to the customary one-t ime gravity the transversal tensor com-

ponent s are mutually independent. In Section 3 where tidal forces are

considered it is shown that due to this fact the ª height º and ª widthº of

a gravitat ion detector also have to oscillat e independently. The observa-

tion of such a detector behavior may be considered as an indicat ion of

the time mult idimension. An interesting analogy with electrodynamics is

drawn and a possibility of longit udinal component s of gravitational waves

is considered. In the last sect ion we discuss our results.

5 Several authors have used the mult i-time hypothesis in connect ion with at tempts to

bypass the di� cult ies in nonlocal theories [7].6 One can expect the appearan ce of ob ject s with a ª turned t imeº on the level of

m icroscopic space-t ime intervals where energy necessary for their creat ion is ab out

their rest -mass. T his asp ect demands special invest igat ion, however, one can predict

that if the dev iat ion of t ime tra jectories is signi ® cant , then the t ime of an interact ion

of such part icles with their surrounding is very short [12,13].

T im e M u lt id im e n s ion in G rav i ty 1 3 4 7

2. PLANE GRAVITATIONAL WAVES

According to the standard procedure (see, e. g., Ref. 14) we represent

the metric tensor as a sum

gm u = gmu + hmu , (2)

where

g =± I O

O I, (3)

I is the three-dimensional unit matrix and hmu is a small addit ion.

The Ricci tensor

Rmu =1

26hm u +

1

2

¶ 2

¶ xm ¶ xsF

su +

¶ 2

¶ xu ¶ xsF

sm , (4)

where h = gm uhmu , the operator 6 º ¶ 2 / ¶ xm ¶ xm and

Fsm = h

sm ± 1

2 h d sm . (5)

The expression in bracket s can be turned into zero if we impose the con-

dit ion

¶ Fsu / ¶ xs = 0 (6)

by means of the gauge transformation

hm u ® hmu + ¶ m ju + ¶ u jm (7)

when

Fmu ® Fmu ± ¶ m ju ± ¶ ujm + gmu ¶ s js . (8)

To prove that Fm u , besides eq. (6), is also constrained by the condit ions

F º gmu

Fmu = 0, Fm uUu

= 0, (9)

where Um are component s of a constant unit vector (Um Um = 1) de® ning

our gauge complet ely, we pass to the momentum space:

Fmu = ei ks x

s

Fmu(k)d6k. (10)

Then relat ions (6), (9) and wave equat ion

6hm u = 6Fmu = 0 (11)

1 3 4 8 B a r a sh e n kov , P e s t ov a n d Yu r ’ i e v

are reduced to the system of algebraic equat ions

gmu

Fmu (k) = 0, (12)

ks

Fsu (k) = 0, (13)

Us

Fsu (k) = 0, (14)

(ks ks

)Fmu (k) = 0 . (15)

Turning the axes one can reduce the momentum vector to the form

k = (1, 0, 0, 1, 0, 0) . Hence eqs. (12) ± (15) can be rewritten as follows:

Fm 1 = Fm 4 , (16)

F41 = F42 = F43 = F45 = F46 = 0, (17)

F11 + F22 + F33 ± F44 ± F55 ± F66 = 0. (18)

Vector Um is taken here in the form Um = (0, 0, 0, 1, 0, 0) .We satisfy these equat ions with the help of the gauge transformat ion

(8) which has now the form

Fmu (k) ® Fm u(k) ± ikm ju (k) ± iku jm (k) + igmu ks js

(k) (19)

or, for the special form of the component s km chosen above,

F42 ® F42 ± ij2 , F43 ® F43 ± ij3 , (20)

F45 ® F45 ± ij5 , F46 ® F46 ± ij6 . (21)

With j2 , j3 , j5 , j6 given, one can turn F42 , F43 , F45 , F46 into zero.

Now we must satisfy only the equat ions F41 = 0 and (18) .

Using again the transformat ion (19) we get

F / 4 ® F / 4 + ij1 ± ij4 , F41 ® F41 ± ij1 ± ij4 . (22)

If we assume F = F41 = 0, then we obtain a consistent system of equat ions

for the calculat ions of j1 and j4 .

So, we convinced ourselves that the tensor

h =

0 0 0 0 0 0

0 h 22 h 23 0 h 25 h 26

0 h 23 h 33 0 h 35 h 36

0 0 0 0 0 0

0 h 25 h 35 0 h 55 h 56

0 h 26 h 36 0 h 56 h 66

, (23)

where h 22 + h 33 ± h 55 ± h66 = 0, satis® es all necessary condit ions.

In comparison with the customary one-t ime theory where every plane

gravitational wave in the linear approxim ation is completely de® ned by

only two quant ities h 23 and h 22 = ± h 33 , the mult i-time wave depends on

nine independent quant it ies.

T im e M u lt id im e n s ion in G rav i ty 1 3 4 9

3. TIDAL FORCES

Such forces acting on part icles of a body moving along a geodesic

curve, i. e. ª free-fallingº together with an observer in the gravit ational

® eld considered, are described by the so-called deviat ion equat ion [15],

d2L

m / dt2

+ Rm

4s 4Ls

= 0 . (24)

Here L is the vector linking two point s with the same proper time on

closely related geodesic curves,7 and R is the curvature tensor.

The displacement vector can be represent as a sum of two terms,

Lm

= F m+ D F m

(25)

where F m is a constant component and D F m is a small addit ion changing

during the mot ion. In this case eq. (24) assumes the form

d2D F m / dt

2+ R

m4s4 F s

= 0 . (26)

(We omitted the term Rm4s4 D F s since we con® ne ourselves only to the

® rst approximat ion.)

Using the (transverse, traceless) gauge considered above one can write

Rm

4s 4 = gmm

Rm 4s 4 = ± 12 gmm ¶ 2 hms / ¶ t 2 (27)

(the summation over m is absent here). So,

d2D F m / dt

2= ( F k / 2)g

mm ¶ 2hm s / ¶ t

2 F s . (28)

Let us suppose that the body considered is a parallepiped with a cross

section F 2 £ F 3 . Inserting F m = d m k F k , D F m = D F k d m k and k = 2, 3, into

eq. (27) we get the equat ion determining the change of the body ª widthº

and ª height º :

d2 D F k / dt

2= ± ( F k / 2)d

2h kk / dt

2 . (29)

Hence

D lk / lk = ± 12 h k k , k = 2, 3. (30)

We see that there is a quant itative diŒerence between one- and mult i-

time cases: in the latter there is no correlat ion of the detector ª widthº and

7 T he proper ( scalar) t ime t = (t21 + t2

2 + t23 ) ± 1 / 2 is counted along the body t ime

tra ject ory [6].

1 3 5 0 B a r a sh e n kov , P e s t ov a n d Yu r ’ i e v

ª height º oscillat ions in a plane perpendicular to the direct ion of gravita-

tional wave propagat ion . In the customary one-t ime theory the amplitudes

of these oscillat ions are equal. That can be used for the experimental check

of possible hidden mult i-dimensional ity of time in our world.

A few words about longit udinal waves are necessary. In the mult i-time

world plane electromagnet ic waves possess longit udinal components [10].

Gravitational waves also have such component s. Indeed, in the linear ap-

proximation (2) there is a remarkable analogy between the electromagnet ic

tensor Fmu and the curvature tensor Rab mu . The latter can be split into

three groups of components,

ei k j = R i ,j + 3 ,k , j + 3 , Hi j = 14 e i k l e j m n Rk lm ,n + 3 , (31)

Gi j =14 e i k l e j m n Rk + 3 , l+ 3 ,m + 3 ,n + 3 , (32)

which are similar to the electric, magnet ic, and ª time-magnet icº ® elds

E i k = F i ,k + 3 , Hi =12 e i j k F j k , (33)

G i = ± 12 e i j k F j + 3 ,k + 3 (34)

(see Refs. 10,16) . The ® elds e, H, G satisfy relat ions analogous to the

generalized mult i-time Maxwell equat ions8 and, therefore, as in the case

of the electromagnet ic ® eld, the gravitational ® eld should have longit udinal

component s.

4. CONCLUSION

The mult i-time gravitational waves diŒer in many aspects from one-

time ones. In part icular, interesting eŒects are stipulat ed by the longit u-

dinal wave component . In fact, most of the diŒerences cannot be observed

at the level of recent experimental accuracy. For the time being the ques-

tion concerns an unpretentious registrat ion of gravitational pulses from

8 In part icular, the identit ies

Fm u = - F u m , ¶ m Fu s + ¶ u F s m + ¶ s F m u = 0

have the grav itat ional similarity

R m u s l = - R u m s l = - R m u l s = R s l m u , R m u s l + R u s m l + R s m u l = 0,

¶ t R m u s l + ¶ m R u t s l + ¶ u R t m s l = 0 .

T im e M u lt id im e n s ion in G rav i ty 1 3 5 1

any enormous cosmic events. Therefore, one may hope to ® x only the

possible diŒerence of the detector amplit ude oscillat ions in perpendicular

direct ions. However, the observat ion of only this eŒect would be an in-

triguing indicat ion of the existence of hidden time propert ies and would be

a powerful booster of further experimental invest igat ions. The absence of

the eŒect is also an important result , making more precise our ideas about

the most puzzling property of our world, time.

ACKNOWLEDGEMENT

We are grateful to B.F.Kostenko for useful discussions.

REFERENCES

1. Dirac, P. A. M., Fock, V . A., Podolsky B . (1932) . Zs. Sow jetun ion 2 , 168.

2. Tom onaga, S. (1946) . Prog. Theor . Phys. 1 , 27.

3. Cole, E. A. B . (1980) . J. Phys. A : Math. G en . 1 3 , 109.

4. Cole, E. A. B ., and Buchanan, S. A. (1982) . J. Phys . A: Math. G en . 1 5 , L255.

5. Cole, E. A. B . (1985) . In. Proc. Sir Ar thur Eddin gton Cen ten ary Sum pos ium , Y.

Choquet-Bruhat and T . M. Rarade, eds. (World Scient i ® c, Singap ore) , vol. 2, p. 178.

6. Barashenkov, V . S., and Yur’ iev , M. Z. (1996) . J INR preprint E2-96-246.

7. Recam i, E. (1986) . Riv. Nuovo Cim . 9 , 1.

8. Dorlind, J . (1970) . Am er. J. Phys. 3 8 , 539.

9. Dem ers, P. (1975) . Can ad. J . Phys . 5 3 , 687.

10. Barashenkov, V . S., and Yur’ iev , M. Z. (1997) . Nuovo Cim . B to appear.

11. Barashenkov, V . S. (1996) . J INR preprint E2-96-10 .

12. Cole, E. A. B ., and Starr, I. M. (1985) . Lett. Nuovo Cim . 4 3 , 38.

13. Barashenkov, V . S., and Yur’ iev , M. Z. (1996) . J INR preprint E2-96-109.

14. Landau, L. D., and Lifshitz, E. M. (1971) . T he Classical Theory of F ields (Addison-

Wesley, Reading, Mass., and Pergam on, Oxford).

15. Weber, J . (1961) . G en era l Relativ ity an d G ravitation al W ave s (W iley-Interscience,

New York) .

16. Cole, E. A. B . (1980) . Nuovo Cim . A 6 0 , 1.