Volume 156, number 1,2 PHYSICS LETTERS A 3 June 1991

On the propagation velocity of metric fluctuations

J.L. Rosales and J.L. Sfinchez-G6mez Departamento de Fisica Terrica, Universidad Autrhoma de Madrid, Madrid, Spain

Received 14 March 1991; accepted for publication 8 April 1991 Communicated by J.P. Vigier

Conformal metric fluctuations (introduced in a recent paper as a possible source of quantum fluctuations ) are studied regard- ing their momentum-energy characteristics. It is shown that such non-local fluctuations in fact conserve energy. A fluctuation velocity is defined which is found to be equal to the phase velocity in de Broglie's theory.

In a recent paper [ 1 ], we have presented a model for conformal ("classical") fluctuations of the space- time metric, arguing that such fluctuations could be closely related to the quantum behaviour at the deepest, most fundamental level; that is, to the quan- tum fluctuations undergone by an isolated (scalar) neutral particle. See refs. [2-5 ] for some recent rel- evant contributions to this subject.

Perhaps the most appealing feature of the quan- tum fluctuations is their non-locality, also a feature of the conformal fluctuations introduced in ref. [ 1 ]; the latter can be regarded as global, in the sense that it is the whole space-t ime metric that actually fluctuates.

In connection with this aspect, it seems worth- while to study in detail the momentum-energy prop- erties of such conformal fluctuations. Specifically we shall show that no actual energy is in fact involved; that is, the energy transferred by the vacuum to the particle is equal to the fluctuation in the potential energy of the gravitational vacuum field (this is ex- act at least for "slow" particles and to first order in the fluctuation parameter, as shown below). This fact should be, presumably, related to the well-known feature of the "min imal" (basic) quantum fluctua- tions that they do not carry any dynamical entities, and their non-locality cannot thereby be used to transmit a "superluminal" signal. We shall also show how a "propagation velocity" can be defined for the metric fluctuations, and that this velocity is equal to the phase velocity of the de Broglie waves; the latter

is greater than the velocity of light, but it must be recalled that no physical quantity whatever "moves" with the phase velocity (e.g. the particle moves with the group velocity).

According to ref. [ 1 ], the infinitesimal invariant interval for the "fluctuating" metric is

ds2=exp[ot(xU) ] d/-2-exp[7(xU) ] dtr 2 , (1)

where

a = ¢ e m?, 7 = a + 2 m i ' ,

being the (small) parameter which specifies the fluctuation, and m being - essentially - related to the cosmological constant in the corresponding de Sitter manifold.

Now the geodesic evolution of a particle in an "~- manifold" is given by

d2x p dx # dx ~ = 0 (2) ,

which for p = 0 reduces to

ds 2 + ½ (&+~)) -½~=0 (3)

(the dots denoting, as usual, time derivatives). For ~--,0, we obtain (/---,t)

d2 t (_.~)2 ds 2 + m - m = 0 . (4)

Then

0375-9601/91/$ 03.50 © 1991 - Elsevier Science Publishers B.V. (North-Holland) 17

Volume 156, number 1,2 PHYSICS LETTERS A 3 June 1991

dt - [1 + (Ae-mt) 2 ] 1/2, (5) ds

A being a constant. For e = 0, eq. ( 1 ) can be written as

d s Z = d l 2 - e 2mr do "2 , (6)

so that ds2=0 implies C=Coe -m', where Co ( = 1 in natural units) is the velocity of light in flat (Min- kowski) space. Then c is the velocity of light in the corresponding de Sitter manifold. Now, since mt << 1 (even for "cosmological" times), c,~ Co in any prac- tical, that is, observational, sense. Yet c is not math- ematically equal to Co, and this will be important later on.

Let us define u2= (dx /d t ) 2. Then

ds) dt] = l - ( u / c ) 2 (7)

and from (5)

(Me-m, )2= (U/C) 2 l_(u/c)2. (8)

We also define

dt ¢~-~"'~ dSS = [ 1 -- (H/C) 2 ] -1/2 (9)

and, analogously,

dF ~= ds" (10)

The energy transmitted to the particle by the fluc- tuation is (M is the rest mass)

Let us set

d/- dt d-s = ds + ~ " (12)

According to (3) we have 2 d2F

+ ( & + m ) l - ~ s ) - ( m + ½ & ) = 0 , (13, ds 2

that is,

d2 t d(~ [ ( ~ f ) 2 _~f] ds-- 5 +e --~- + ( & + m ) +2~q~

- ( m + ½ & ) = 0 .

Recalling (4), we are left with

[d__~ mt{dl'~ 2 dt ] E - - + m e ~-~) + 2 m ~ - ~ - ½ m e mt = 0 .

(14)

(15)

But from (5), (8), (9) and taking into account that

dt ds= - - (16) ¢

(recalling that ¢= [ 1 + (Ae-mt) 2 ] 1/2, we easily obtain

dqb 2~¢ A ~2_ ½ d-~ + 1_¢2 (¢2_1)1/2¢2_1 ~ O ~ (17)

To solve this equation, let us set

/ / 2=¢2-1 . (18)

Then, from (9)

//2= (U/C)2[I ..}_ (U/C)2..]_...] ,

so that (17) can be written as (there is a sign am- biguity; the plus sign has been chosen for definiteness)

d ~ 2 ~ A(f l2+½) dfl + -fl- +/ /2(1 +//)1/2 = 0 , (19)

which up to order ~2 (i.e., in fact u2/c 2) has the solution

(~-- -- ~ (1 "~ 1//2) _{_C///2. (20)

Now, from (8)

A =emt(u /c ) [ 1 + ½ ( U / C ) 2 - ] - . . . ] .

In order to get rid of divergences, we set the constant C= 0 in (20), so that we obtain

( I ) = - - ½ e m t [ l " F l ( u / c ) 2 ] + O [ ( u / c ) 4 ] . (21)

The particle fluctuation energy, taking into account (11) and (12), is

5El= - ½eMc~emt[ 1 + ½ (u / c ) 2 ] . (22)

On the other hand, the energy fluctuation of the vac-

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Volume 156, number 1,2 PHYSICS LETTERS A 3 June 1991

uum gravitational field can be obtained upon re- calling that

goo = 1 + 2 V J c 2 = 1 + ¢emt+o(¢ 2) .

Then, for a "static" ( u = 0 ) particle

8Eg=MVg = ' i . fa2. .mt (23)

so that (22) gives forS--O (u--O)

8 ( E l + E g ) = 0 . (24)

This equation states the character of energy-conser- vation that the fluctuation indeed has. (We have to remark again that our treatment is mathematically restricted to "non-relativistic" particles. It is easy to see that the energy conservation holds also at order U2/C 2 by just performing a simple Lorentz transfor- mation, and taking the appropriate limit. )

Let us now return to (22), which is appropriate for slow (u << c) particles. We can define, in a nat- ural way, a "propagation velocity" for the fluctua- tion as follows (E=Ef ) ,

5E(u) du 5E(u) v(u) = - (25)

d [SE(u) ] d [ S E ( u ) ] / d u "

Then from (22) we obtain

c~ u 2 c~e -2m ' - + ~ u

v( u ) = e-~Gm, u + ~U -- U (26)

that is,

v (u )=c2 /u+½u. (27)

Notice that (27) coincides with the non-relativistic limit of the phase velocity for a free particle, whose "kinetic" (group) velocity is u, in de Broglie's theory:

E MC2+ ½Mu 2 v ( u ) - -

p Mu

=c2/u+½u. (28)

Let us summarize. First, we have shown that the conformal metric fluctuations are non-dissipative, in the sense that the fluctuation energy of the particle equals the corresponding fluctuation energy of the vacuum field; that is, there is an energetic particle- field equilibrium. This is important, since, as is well known, the quantum fluctuations are non-dissipative. Second, if we "observed" just the particle (ignoring the gravitational vacuum), we would see that, in spite of being isolated, it fluctuates. Then we could define, in a sensible way, a "fluctuation velocity" which would indicate how the fluctuations "propagate" in momentum-energy space. This velocity is unphysi- cal, in the sense that it does not describe any "phys- ical" motion, rather it should represent a kind of "information velocity"; we thus think that its being equal to the phase velocity appearing in de Broglie's theory (for very recent work on extensions of this theory see ref. [ 6 ] and references therein) could help us to find a truly real - objective - source of the "primordial" quantum fluctuations, related to the conformal fluctuations of the space-time metric.

This work has been particularly supported by C.I.C.Y.T. (Spain), under contract no. PB 88-0173. We are grateful to R.F. Alvarez-Estrada, S. Bergia, and J.P. Vigier for some useful discussions.

References

[ 1] J.L. Rosales and J.L. S~lnchez-G6mez, Phys. Lett. A 142 (1989) 207.

[2] C. Frederick, Phys. Rev. D 13 (1976) 3183. [ 3 ] J.P. Vigier, Astron. Nachr. 303 (1982) 55. [ 4 ] K. Namsrai, Non-local quantum field theory and stochastic

quantum mechanics (Reidel, Dordrecht, 1986 ). [ 5 ] S. Bergia, F. Cannata and A. Passini, Phys. Lett. A 137 (1989)

21. [6] J.P. Vigier, Phys. Lett. A 135 (1989) 99; Found. Phys. 21

(1991) 125.

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