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arXiv:1111.0805v1

[hep-ph]

3Nov2011

Relativity accommodates superluminal mean velocities

B. AllesINFN Sezione di Pisa, Pisa, Italy

Contrarily to a widespread belief, measures of velocity can yield a value larger than c, the lightspeed in vacuum, without contradicting Einsteins relativity. Nevertheless, the effect turns out tobe too small to explain the recently claimed superluminal velocity by the OPERA collaboration.

04.20.-q; 04.20.Cv; 04.90.+e; 14.60.Lm

The recent observation of superluminal neutrinos bythe OPERA team [1] (see also the MINOS collabora-tion [2]) has prompted an intense discussion. Particlephysics arguments [3] or astrophysics data [4,5] seem toindicate that neutrinos could hardly attain such veloc-ities. Moreover, and generally speaking, it is acceptedthat within the present physics knowledge, such veloci-ties cannot be contemplated if the validity of the basiclaws of relativity is to be maintained. In particular thecausal structure of spacetime would be badly violated if

superluminal particles existed.However it should not be forgotten that such a causal

structure holds good in special relativity, namely, in anideal world where gravity is absent. When instead grav-ity is taken into account, a special relativistic spacetimearises only locally, in a close neighbourhood of everyspacetime event P. Strictly speaking, with infinite math-ematial precision, this is true exactly at P (although,considering the inevitable inaccuracy of measurement de-vices, the region of validity of such an approximationturns out to be more or less extended, depending on themagnitude of the spacetime curvature at P). Thus, itshould not come as a surprise that measurements carried

out in a nonlocal way may apparently violate the maintenets of special relativity.

We are going to show that, although the instantaneousvelocity of a particle cannot be larger than c, relativityconsents that the mean velocity of the same particle maywell exceed c. The interesting and appealing aspect of theabove statement is that what the OPERA collaborationhas measured is precisely the neutrino mean velocity.

To illustrate the above effect it will be assumed thatthe spacetime is satisfactorily described by the metricds2 = g00dt

2 gijdxidxj where (x1, x2, x3) (x, y, z)are general spatial coordinates and t is the coordinatetime. Throughout the paper mass and time units will

be chosen such that c and the Newton constant G are 1.Also, the Einstein summation convention for pairs of re-peated indices will be used with latin indices indicat-ing spatial components and greek indices referring indis-tinctly to both spatial and time components. It will bealso supposed that all metric components are time in-dependent, g/t = 0, and that the timespace onesvanish, g0i = 0. This last condition is not essential (weinclude it to avoid lengthy mathematical expressions) andcan be omitted without changing the conclusions of the

paper [6].Consider a massless (or nearly massless) particle

travelling along a spatial trajectory parametrized bya variable and described by the three functions(x(), y(), z()). It travels from the spatial point P1(where the parameter takes the value 1) to the spatialpoint P2 (value 2). After having measured the physicaldistance between the two points, for instance by lyingrods in succession (the metric is time independent), anobserver at P2 registers the time used by the parti-

cle to complete the trip. This time reading requires aprevious synchronization among the clocks at P1 and P2.Then, the observer at P2 defines the mean velocity of theparticle as

v

. (1)

Let us see what general relativity predicts for this quan-tity.

The physical distance is given by

=

21

d

gij xixj , (2)

where xi

dxi

/d. Viewed as a quadratic form, gij ispositive definite. On the other hand, one of the severalequations that govern the motion of the particle stemsfrom the nullification of its proper time, ds2 = 0 =g00dt2 gijdxidxj , whence

t =

21

d

gijxixj

g00. (3)

Expression (3) gives the coordinate time interval em-ployed by the particle during the trip from P1 to P2.To recover the time reading on the observers clockat P2 one has to multiply =

g00(2)t. Therefore

the mean velocity recorded by the observer is

v

c=

=

1g00(2)

2

1d

gijxixj21

d

gijxixj/g00. (4)

The dependence of g in (2)(4) on comes throughtheir dependence on the spatial coordinates.

The relevant fact that we wish to emphasize is that themathematical framework of general relativity does notcontain any mechanism whatsoever able to oblige (4) tostay equal or less than 1. Rather, as remarked above, (4)

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tends to 1 only when the velocity measurement becomeslocal, to wit, when points P1 and P2 tend to coincide. In-deed, since

gij xixj is positive, the mean value theorem

can be applied and it states that a value , comprised be-

tween 1 and 2, exists such that21

d

gijxixj/g00 =

(1/

g00())21

d

gijxixj . Inserting this expression

in the denominator of (4) we obtain

v

c=

g00()g00(2)

, (5)

which tends to unity as P1 P2 because in this limit 2 also.

Let us consider an instance of the effect just ex-posed by studying the radial motion of a particle in aSchwarzschild metric (it characterizes the vacuum out-side a spherically symmetric mass distribution with totalmass m) in standard coordinates, ds2 = g00dt

2grrdr2r2d2 where d is the solid angle and g00 = 1/grr =1 2m/r. The massless or almost massless particle willbe made travel radially from r1 to r2 (we do not specify

yet whether r1 < r2 or r2 < r1). Because g00 increaseswith r, following (5), we deduce v < c if r2 > r1 andv > c if r2 < r1. The detachment from the light speedcan be better appreciated by plotting v/c against r1 andr2. Specifically, for the Schwarzschild metric we have(calling i ri/(2m), i = 1, 2)

2m=

22 2

21 1

+1

2log

(

1 1/2 + 1)(

1 1/1 1)(

1 1/2 1)(

1 1/1 + 1)

,

2m =

1 1/2 2 1 + log2

1

1 1 . (6)

The absolute values in (6) make these expressions validfor both cases, r2 > r1 and r2 < r1. This is an importantremark because the fact that the time intervals for theparticle to go from r1 to r2 or for coming from r2 to r1are the same allows to reliably synchronize clocks withthe exchange of light signals.

In Figs. 1 and 2 the mean velocity v/c is plotted against1/1 for various values of2. Note (i) that we have delib-erately excluded the region 1/1 > 1 because otherwisethe particle would enter the horizon of the correspondingblack hole and (ii) that all plots touch the line v/c = 1

when 1 coincides with 2, confirming the validity of spe-cial relativity at short distances.The case r1 > r2 (the particle approaching the spheri-

cal distribution of mass) is shown in Fig 1. As viewed bythe observer in r2, v turns out to be always larger than c.This effect is particularly pronounced when the observeris close to the Schwarzschild radius rs 2m (lower valuesof 2) and for large separations r1 r2. For 2 > 100 theresulting v/c is so close to 1 that the related plots cannotbe seen within the scale of the vertical axis.

In Fig. 2 the case r1 < r2 (the particle departing fromthe spherical massive object) is treated. Again v tendsto c whenever the radial coordinate r1 of the position ofthe particles departure is close to the radial coordinater2 of the position of arrival, but now all mean velocitiescome smaller than c. Note the tendency of v to becomenull as r1 approaches rs for every r2.

A terrestrial experimental setup prepared to detect

a value for v different from c would necessarily be con-strained to use r1, r2 > r, the Earth radius. With sucha condition and in (6) can be approximated withexcellent accuracy to linear order in m/ri (m is theEarth mass) giving

v

c 1 + m

r m

r1 rlog

r1r

> 1 , (7)

where the observer has been put on the Earths surface,r2 = r, and we have taken r1 > r. The percentage ofexcess of velocity is a meagre (v c)/c 1010.

0 0.2 0.4 0.6 0.8 1

1/1

1

2

3

4

5

v/c

2

=1.05

2

=2.0

2

=3.0

2

=4.0

2

=5.0

2

=10.0

2

=100.0

2

=1000.0

2

r2.

0 0.2 0.4 0.6 0.8 1

1/1

0.6

0.7

0.8

0.9

1

v/c

2

=1.05

2

=2.0

2

=3.0

2=4.0

2

=5.0

2

=10.0

2

=100.0

2

=1000.0

2

>1

FIG. 2. Mean velocity v/c of the particle travelling fromr1 = 2m1 to r2 = 2m2 in a Schwarzschild metric for severalvalues of 2 as a function of 1/1 and always with r2 > r1.

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To exemplify the use of the above findings, we ap-ply them in two cases: firstly to the already mentionedOPERA experiment and secondly to the determinationof the distance EarthMoon by laser ranging.

In the OPERA experiment a beam of muon neutri-nos was produced at CERN SPS and sent to the GranSasso underground laboratories (LNGS) in Italy wherethey were revealed [1]. After an accurate synchroniza-

tion between clocks of both laboratories [8], the baselinelength was divided by the time of flight of neutrinos toobtain their average velocity. The result was larger thanc by an amount [(v c)/c]experiment = 2.480.28(stat)0.30(syst) 105. However, the abovedescribed mech-anism is unable to explain this excess. This can beeasily seen by adopting the simplifying hypothesis thatneutrinos travel over the sphere of the Earths surfaceand taking advantage of the fact that the angular sec-tor of the Schwarzschild metric in standard coordinatesis flat. Then equals the Euclidean result r ( isthe angular separation between CERN and LNGS [10])

while t is r/1 2m/r. Hence v/c is always 1.Eliminating the previous simplifying hypothesis (neutri-nos actually traversed the Earths crust following theimaginary chord that joins CERN and LGNS) adds anegligible contribution (to verify this assertion an inte-rior metric was derived at first order in m assuming aplanet Earth with uniform mass density). The inclusionof the spin of Earth (for example by using the LenseThirring metric) produces an even smaller contribution(while m/r 1010, J/r2 1016, J being theEarths angular momentum). Only the rotation of theobservers laboratory at Gran Sasso induces a reveal-able effect but it affects the synchronization of clocks andtherefore it has nothing to do with the topic described inthis paper [9].

These considerations seem to imply that the obser-vation of a superluminal velocity for the neutrinos ofOPERA must be likely ascribable to other reasons: ei-ther purely experimental oversights [11,12] or really newphysics peeping out (see [1327] for a partial list of the-oretical suggestions and criticisms).

Consider now the lunar laser ranging [28,29] that al-lows to determine the EarthMoon distance. It consistsin sending a laser pulse from Earth to the lunar sur-face where it is reflected (several manned missions in thepast left corner reflectors on the Moons surface) and re-ceived back on Earth. Multiplying the time employed

by the light in its round trip by the speed of light yieldsthe EarthMoon distance. However, assuming that themetric is wellapproximated by the Schwarzschild one atlinear order in m/r, the mean speed of the light rayturns out to be larger than c and so, the distance shouldcome out less than what it really is. To remedy this in-convenient and obtain at least an order of magnitude ofthe necessary correction, we resort again to expressions(6).

Establishing that the radial coordinate of the labora-

tory is the Earths radius r, it remains to find the radialcoordinate on the Moon, r1. This is achieved by invert-ing the formula for in (6). After some algebra we getthe equation

exp

r m+ 1

= e , (8)

where r1/r, r/(2m) and is half ofthe time (measured by terrestrial clocks) spent duringthe round trip. Eq.(8) can be solved for in termsof the Lambert Wfunction [30]. However, on accountof the fact that 1, we can approximately set 1 + /(r m). Inserting it into the first of(6) leads to

EarthMoon distance + 2mr

, (9)

the correction being 2m /r 53 cm.In conclusion we have seen that mean velocities in gen-

eral do not conform to wellknown special relativity prin-

ciples. In particular, albeit amazing, the average velocityat which a particle has travelled for a long time can bedifferent from the instantaneous velocities that the par-ticle attained at every point along the trajectory, evenwhen the latter was the same at all points. Indeed, spe-cial relativity adequately describes physics only locallywhile the average over large distances of a velocity mustnecessarily be introduced as a nonlocal quantity. Uti-lizing these considerations for understanding the resultsfrom the OPERA collaboration, we conclude that thereal anomaly is not that (v c)/c be positive but that itis a rather large number.

We have analyzed the case in which the observer stays

at rest at one end of the particles trajectory. But ofcourse other experimental dispositions are possible: theobserver staying in the middle of the particles trajectoryor even at a point outside it; the observer not at rest, etc.Also many possible definitions of mean velocity, otherthan the one used by OPERA (1), can be conceived. Inall cases bizarre results should be carefully interpreted.

Acknowledgements

It is a pleasure to thank Mihail Mintchev and Gian-carlo Cella for stimulating discussions.

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181803, (2011).[4] L. GonzalezMestres, arXiv:1109.6630.[5] D. Fargion, arXiv:1109.5368.

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[6] Was the metric to contain nonzero timespace termsg0i, the spatial distance in (2) would be evaluated withthe positive definite quadratic form g0ig0j/g00 + gij [7]and the synchronization of coordinate times at distantpoints would require the inclusion of a lag defined by

21

d g0ixi/g00. Apart from these complications, the

effect presented in the text would be, mutatis mutandis,qualitatively the same.

[7] C. Mller, The theory of relativity, Clarendon Press,Oxford (1972).[8] PhysikalischTechnische Bundesanstalt (PTB), Rela-

tive calibration of the GPS calibration time link betweenCERN and LNGS, Report calibration CERNLNGS,(2011).

[9] An event occurring at coordinate time tCERN at CERNis simultaneous with another event occurring at GranSasso at coordinate time tLNGS if tLNGS tCERN (r cos )

2 2.3 ns ( is the angular speedof the Earths rotation, is the latitude of the CERNor Gran Sasso assumed equal and is the differ-ence in longitude of both laboratories). 2.3 0.9 ns isprecisely the systematic error identified in [8].

[10] is obtainable from the geographical coordinates ofGeneva (latitude=46o 12, longitude=6o 9) and GranSasso (latitude=42o 28, longitude=13o 33).

[11] H. Bergeron, arXiv:1110.5275.[12] S. Dado, A. Dar, arXiv:1110.6408.[13] G. AmelinoCamelia et al., arXiv:1109.5172.[14] F. Tamburini, M. Laveder, arXiv:1109.5445.[15] F. R. Klinkhamer, arXiv:1109.5671.[16] G. J. Giudice, S. Sibiryakov, A. Strumia, arXiv:

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arXiv:1109.5917.[20] M. Li, T. Wang, arXiv:1109.5924.[21] J. Alexandre, J. Ellis, N. E. Mavromatos, arXiv:

1109.6296.[22] A. Nicolaidis, arXiv:1109.6354.[23] F. R. Klinkhamer, G. E. Volokik, arXiv:1109.6624.[24] E. Ciuffoli, J. Evslin, J. Liu, X. Zhang, arXiv:1109.6641.[25] M, M. Anber, J. F. Donoghue, arXiv:1110.0132.[26] M. Pavsic, arXiv:1110.4754.[27] J. Bramante, arXiv:1110.4871.[28] J. G. Williams et al., Phys. Rev. D53, 6730, (1996).[29] J. D. Anderson et al., Class. Quant. Grav., 18, 2447,

(2001).[30] R. M. Corless et al., Adv. Comput. Math., 5, 329, (1996)

and references therein.

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