energy and relativistic clock rates in five dimensions

G en eral Relativ ity an d G ravi tation , Vol. 30, No. 11, 1998

Energy and Relativ istic Clock Rates in Five

Dimensions

Fabio Cardone,1 Mauro Francav iglia2 and Roberto Mignani3 ,4

Rece ived May 11, 199 8

In the framework of a Kaluza± Klein-like schem e, based on a ® ve-dim en-

sional Riem annian space in which energy plays the role of the ® fth di-

m ension, we discuss a class of solut ions of the ® ve-dimensional Einstein

equat ions in vacuum, which allows us to recover the energy -dependent

phenomenological m et ric for grav itat ion, recent ly derived from the ana-

lysis of some exp erimental data concern ing the slowing down of clock

rat es in the gravit at ional ® eld of Earth.

KEY WORDS : Broken Lorent z invariance ; deform ed Minkowski

space

The geometrical structure of the physical world Ð both at a large and a

small scale Ð has been debated for a long time; after Einstein, the gener-

ally accepted view is that physical phenomena occur in a four-dimensional

spacet ime, endowed with a global Riemannian structure, which is assumed

to be locally ¯ at (i.e. having a Minkowskian geometry).

1 Universit Áa della Tuscia, Ist ituto di Gen io Rurale, V ia S. Camillo De Lellis, I-01100

V iterbo, Italy and C.N.R. - GNFM2 Dipart imento di Matem at ica, Universit Áa di Torino, Via C. Alberto, 10, I-10123 Torino,

Italy and C.N.R. - GNFM3 Dipart imento di Fisica ª E. Am aldiº , Universit Áa di Roma ª Rom a Treº , V ia della Vasca

Navale, 84, I-00146 Roma, Italy. E-mail: mignani@amaldi. ® s.uniroma3.it4 I.N.F.N. - Sezione di Rom a 1, c/ o Dipart imento di Fisica, Universit Áa di Roma ª La

Sapienzaº , P.le A.Moro 2, I-00185 Rom a, Italy

1 6 1 9

0001-7701/ 98/ 1100-1619$15.00/ 0 1998 P lenum Publishing Corporation

1 6 2 0 C a r d o n e , Fr an c av ig lia an d M ig n a n i

It is however well known that many attempts at generalizing this four-

dimensional picture have been made, with the main scope of int roducing

uni® ed schemes for the fundamental interactions, e.g. by assuming the

existence of further dimensions [1± 8]. The most celebrated theory of this

type is due to Kaluza [1] and Klein [2], who assumed a ® ve-dimensional

metric in order to unify gravitation and electromagnet ism in a single ge-

ometrical structure. The Kaluza± Klein formalism lat er was extended to

higher dimensions, in order to produce uni® ed theories of all four fun-

damental interact ions, i.e. including weak and strong forces [5± 8]. We

also recall that a (constant) non-Minkowskian metric was introduced for

weak interactions on a phenomenologica l basis, in order to accommodate

possible violat ions of the Lorentz invariance at distances greater than the

Planck length [9].

More recently, some experimental data concerning physical phenom-

ena ruled by diŒerent fundamental interactions seemed to provide evidence

for a local departure from the Minkowski metric [10± 15]: among them the

lifet ime of the (weakly decaying) K 0s meson [16], the Bose± Einstein cor-

relat ion in (strong) pion product ion5 and the superluminal propagat ion

of electromagnet ic waves in waveguides [18]. These phenomena seemingly

show a (local) breakdown of Lorentz invariance, together with a plausi-

ble inadequacy of the Minkowski metric; on the other hand, they can be

int erpreted in terms of a deformed Minkowski spacet ime, with metric co-

e� cient s depending on the energy of the process considered [10± 15].

What is interesting for us here is the fact that an analogous energy-

dependent metric seems to occur when analyzing some classical experi-

mental data due to Alley [19,20], concerning the slowing down of clocks

embedded into the gravitational ® eld of Earth. More recent ly, a work of

van Flandern [21] about the accelerat ion of binary systems (the same con-

cerned in the analysis of Taylor and Hulse, Ref. 22) seems to show again a

breakdown of Lorentz invariance, which in our opinion can be int erpreted

in terms of the energy-dependent gravitational metric ensuing from our ® t

of Alley’ s data [15,23].

All the above facts suggested to us a (four-dimensional) generaliza-

tion of the (local) space-t ime structure based on an energy-dependent ª de-

formationº of the usual Minkowski geometry, whereby the corresponding

deformed metrics ensuing from the ® t to the experimental data seem to

provide an eŒective dynamical description of the relevan t in teractions (at

the energy scale and in the energy range considered ).

5 For experimental as well as theoret ical reviews on the Bose± Einstein eŒect in mult i-

boson product ion, see e.g. [17].

E n e r g y a n d R e la t iv i s t i c C lo ck R a t e s in F iv e D im e n s io n s 1 6 2 1

On the basis of this four-dimensional analysis we have therefore drawn

the following conclusions: (i) the energy of the process considered (which is

to be understood as the energy measured by the detectors via their electro-

magnet ic interaction in the usual Minkowski space) plays the role of a true

dyn amical variable ; ( ii) energy represents a characteristic param eter of the

phenomenon under consideration (and therefore, for any given process, it

cannot be changed at will) . In other words, when describing a process, the

deformed geometry of the interaction region of spacetime is ª frozenº by

those values of the metric coe� cien ts corresponding to the energy value of

the process . From a geometrical viewpoint , we can restate this by saying

that we are actually working on ª slicesº (sections) of a ® ve-dimension al

space, in which the ® fth dimension is just represen ted by the energy.

In recent invest igat ions of ours (see the more extended paper, Ref. 24)

we have taken this viewpoint , namely that the four-dimensional energy-

dependent spacet ime is just a manifestation (or a ª shadowº , to use the

famous word of Minkowski) of a larger space in which energy plays the

role of a ® fth dimension. A direct announcement of the results concerning

the four fundamental interact ions is contained in a short letter of ours [25].

We shall here brie¯ y give the main results concerning the applicat ion of

our formalism to the clock experiments.

The four-dimensional ª deformedº metric scheme introduced in [10±

15] is based on the assumpt ion that spacet ime, in a preferred frame which

is ® xed by the scale of energy E , is endowed with a metric of the form

ds2= b2

0 (E )c2 dt2± b2

1 (E )dx2± b2

2 (E )dy2± b2

3 (E )dz2(1)

with xm = (x0 , x1 , x2 , x3 ) = (ct, x, y, z), c being the usual speed of light in

vacuum.

The use of an energy-dependent spacet ime metric is not new since it

can be traced back to Einstein himself. In order to account for the modi® ed

rate of a clock in the presence of a gravitational ® eld, Einstein was in fact

the ® rst to generalize the special-relat ivist ic interval by introducing a ª time

curvatureº as follows [19]:

ds2= 1 +

2w

c2c2 dt2

± dx2± dy2

± dz2, (2)

where w is the Newtonian gravitational potential.

The metric (1) is supposed to hold locally, i.e. in the spacet ime region

where the process occurs. It is supposed moreover to play a dynamical role,

and to provide a geometric descript ion of the int eract ion, in the sense that


each interaction produces its own metric, through diŒerent specializations

of the parameters bm (E ). The spacet ime described by (1) is ¯ at and the

geometrical descript ion of the fundamental interactions based on it dif-

fers from the general relat ivist ic one. Although for each interaction the

corresponding metric reduces to the Minkowskian one for a suitable value

of the energy E0 (which, as we said, is characteristic of the interaction

considered) , the energy of the process is ® xed and cannot be changed at

will. Thus, in spite of the fact that formally it would be possible to recover

Minkowski space by a suitable change of coordinat es (e.g. by a rescaling) ,

this would amount , in our new framework, to be just a mathematical oper-

ation devoid of any physical meaning. In the ® ve-dimensional vision which

we work in, in fact, the physics of the interaction lies in the curvature of

a ® ve-dimensional metric depending on energy, while the four-dimensional

sections at E = const. turn out to be ª mathematically ¯ atº (spaces of

four-dimensional zero curvature), since the metric coe� cients depend only

on E .

As far as the phenomenology is concerned, it is important to recall that

a local breakdown of Lorentz invariance may be envisaged for all the four

fundamental interactions (electromagnet ic, weak, strong and gravitational)

whereby one gets evidence for a departure of the spacetime metric from the

Minkowskian one (in the energy range examined) . The explicit functional

form of the metric (1) for the ® rst three interactions can be found in [10±

15].

The analysis of Alley’ s data [19,20] referring to the experimental re-

sults on the relat ive rates of clocks at diŒerent height s suggested to us

[23] that also the gravitational interaction (at least in a neighborhood of

Earth) can be described in terms of an energy-dependent metric. In this

case one ® nds

b20 (E ) = 1 +

EE0

2

, (3)

while no informat ion can be derived from the experimental data about

the space parameters [23]. The constant E0 is the threshold energy at

which the metric becomes Minkowskian, and its value obtained in the

gravitational case from the ® t to the experimental data is

E0 ’ 20 meV. (4)

Intriguingly enough, this is approximately of the same order of magni-

tude as the thermal energy corresponding to the 2.7±K cosmic background

radiat ion in the universe [25].


Examining the phenomenological metrics derived for all the four in-

teractions, we see that energy plays in fact a dual role. On one side, it is

to be considered as a dynamical variable, while, on the other hand, a ® xed

value of E determines the spacet ime structure of the interaction region at

that given energy . In this respect, therefore, the energy of the process has

to be considered as a geometrical quan tity int imately relat ed to the very

geometrical structure of the physical world. The simplest way of taking

this into account is to assume that energy does in fact represent an extra

dimension [24].

We assume then a 5-dimensional spacet ime with an energy-dependent

metric

ds2( 5) º a(E )c2dt2

± b(E )dx2± c(E )dy2

± d(E )dz2+ f (E ),2

0dE 2, (5)

where all coe� cients are funct ions of at most the energy E alone, and ,0

is a constant with suitable dimensions.

This metric is assumed to satisfy the vacuum Einstein equat ions with

a possible ª cosmological constantº L:

RA B ± 12 g( 5 )

A B R = Lg( 5)

A B (A , B = 1, ..., 5). (6)

As shown in [24], in the case of spat ial isotropy (b(E ) = c(E ) = d(E )),

these equat ions can be reduced to the following form:

3( ± 2b9 9 f + b9 f 9 ) = 4Lbf 2,

f [a2(b9 )2

± 2aa9 bb9 ± 4a2 bb9 9 ± 2aa9 9 b2+ b2

(a9 )2]

+ abf 9 (2ab9 + a9 b) = 4La2 b2 f 2,

3b9 (ab) 9 = ± 4Lab2 f ,

(7)

where a prime denotes derivat ion with respect to E and we put c = ,0 = 1.

Among the solut ions corresponding to L = 0, we ® nd6

f (E ) = const.,

a(E ) = 1 +EE0

2

,

b(E ) = c(E ) = d(E ) = const.,

(8)

6 Since L is relat ed to the vacuum energy in General Relat ivity and experimental

ev idence shows that L ¼ 3 ¢ 10 ± 5 2 m ± 2 , we assume L = 0 since we are not interested

here in quantum eŒect s.


where E0 is a constant having the dimensions of energy. This solut ion is

in fact the unique solut ion corresponding to a constant coe� cient f (E ) in

a larger family in which a(E ) is arbit rary and (f , a) satisfy the ordinary

diŒerential equat ion

± 2aa9 9 f + (a9 )2 f + aa9 f 9 = 0 . (9)

Moreover, the metric (8) coincides with the phenomenological gravit ational

metric (3) in the hypothesis of spat ial isotropy.

Other interesting classes of solut ions to eqs. (6) for L = 0 can be

obtained by the ansatz that all the coe� cients in (5) are pure powers of

E , i.e.

a(E ) = (E / E0 )q,

b(E ) = (E / E0 )m

,

c(E ) = (E / E0 )n

,

d(E ) = (E / E0 )p,

(10)

while for the dimensional parameter f (E ) we assume simply

f (E ) = E r. (11)

In this case, as shown in [24,25], the Einstein equat ions reduce to the

algebraic system

(2 + r) (p + m + n) ± m2± n2

± p2± mn ± mp ± np = 4LE r+ 2

,

(2 + r)(p + q + n) ± n2± p2

± q2± np ± nq ± pq = 4LE r+ 2

,

(2 + r)(p + q + m) ± m2± p2

± q2± mp ± mq ± pq = 4LE r+ 2

,

(2 + r)(q + m + n) ± m2± n2

± q2± mn ± mq ± nq = 4LE r+ 2

,

mn + mp + mq + np + nq + pq = ± 4LE r+ 2,

(12)

which admit (at least) twelve classes of solut ions. They include as special

cases all the phenomenological metrics discussed in [10± 15]. In part icular,

the metric (3) can be obtained as the only metric which lives in the inter-

section of three of the relevant classes; namely, it is obtained by setting

q = 2, m = n = p = r = 0, (13)

which obviously reduces to (3) [and to (8)] by a rescaling and a translat ion

of the energy parameter E0 .


Let us mention that the metric (8) suggest s introducing a modi® ed

proper time function t (t, E ) by setting

t = 1 +EE0

t. (14)

With this posit ion, the gravit at ional metric (8) takes the form

ds2( 5 ) º dt

2± b(E ) [dx2

+ dy2+ dz2

]

+ f (E ) +t 2

(E + E0 )2dE 2

± 2t

(E + E0 )dt dE , (15)

which shows a ® ve-dimensional ª Gaussian behaviorº (with lapse funct ion

equal to one).

Moreover, we would like to mention the following. As is well known,

Einstein suggest ed a Gedankenexperiment to criticize the Heisenberg un-

certainty principle. 7 The experiment involves a device composed of a clock

and a light source embedded in a gravitational potential. Bohr8 replied to

Einstein’ s criticism by showing that the validity of the uncertainty princi-

ple is preserved in Einstein’ s experiment, provided one uses the appropriat e

time spread D t which can be derived from g00 as given by the metric (2).

We remark that the same argument of Bohr can be applied here to our

gravitational metric (3), or (8), by expanding b0 (E ) º a(E ) as

a(E ) = 1 + 2EE0

+EE0

2

¼ 1 +EE 90

, (16)

with (E / E0 ) ¿ 1 and E 90 = E0 / 2.9 In the pure general relat ivist ic picture,

this would amount to expanding (1 + (2w / c2 ))2 as 1 + (2 u / c2 ) by putting

u = 2w and neglect ing terms of the order c - 4 .

From a purely theoretical side, we stress that the basic assumptions

of our ® ve-dimensional formalism, i.e. the energy as a ® fth dimension and

the consequent deformation of the four-dimensional spacet ime seen by the

observers, diversify our scheme from a standard Kaluza± Klein one, and

give rise to an entirely new class of generalizat ions of Relat ivity, which we

believe to deserve further invest igat ion.

7 T he exp erim ent was proposed by Einstein at the Sixth Solvay Conference in 1930.

See e.g. [26].8 For Bohr’ s reply to Einstein ’ s crit icism see [27].9 This last position corresponds of course to the possibility we st ill have of ® x ing the

energy scale.


ACKNOWLEDGEMENTS

Thanks are due to M. Ferraris, for his precious help in the use of

the symbolic manipulat ion programs, whereby calculat ions have been per-

formed, and to M. Gaspero for useful discussions.

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energy and relativistic clock rates in five dimensions

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