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J. Bell’s Relativistic Paradox and Equivalence Principle

Bogomolov G.D.

P.L. Kapitsa Institute of Physical Problems, Russian Academy of Sciences, Moscow

Morozov V.B.

P.N. Lebedev Physical Institute,RussianAcademyofSciences,Moscow

The Einsteinian equivalence principle allows – by natural way – resolving the J. Bell’s

relativistic paradox. The metrics were found of uniformly accelerated relativistic systems. An

experimental verification of the relations obtained is proposed.

1. The special relativity theory allows describing motion of a uniformly accelerated

single body. The world line of a body located in origin of coordinates of Minkowsky plane [1] is

described by the hyperbole

cosh

1,

sinh

,

here the parameter is proper time.

However, an attempt of motion describing of several accelerated bodies brings to a

paradox. For example, if two bodies start moving at the same time with the same acceleration

(controlled by accelerometers) and in the same direction, our intuition suggests us that the

additional body would move in line with the similar law

cosh

1

,

sinh

(1)

and a distance between the bodies would not change from point of view of a “motionless”

observer. On the other side, according to the Lorentz formula, in a convected reference system, to

which these bodies belong, a distance between them

1

grows infinitely with velocity growth. We see that the effect is opposite to the Lorentzian

compression. This paradox was named Bell’s relativistic paradox [2, 3]. The strange result caused

an insipid polemics [4] and served repeatedly as a pretext for relativistic theory revision attempts.

The experience suggests that usually in the relativistic physics, an intuition is a tricky

helper. But in this particular case, is it possible to put trust in intuition or not?

2. Among other things, the equivalence principle states as follows: a reference system

of accelerated bodies Σ1 moving with the same acceleration – according to the Einsteinian

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equivalence principle – is equivalent to a system of motionless bodies Σ2 located in a uniform

gravitational field1.

Lemma. In an eigensystem of uniformly accelerated points, distances do not change.

Indeed, the equivalence principle establishes identity of laws of a system of equally accelerated

points and a uniform gravitational field but a distance between motionless points in a gravitational

field does not depend on time2.

Conclusion. We must not only refuse from intuitive suggestions but also apply methods of

the general relativity theory instead of the special one for investigation of non-inertial reference

systems.

3. In the non-relativistic physics, a description of a uniform field of any nature is not

difficult at all. Meantime, a description of a uniform field of accelerations or of an equivalent-to-

it gravitational field is beyond scope of possibilities of the special relativity theory. In particular,

readings of standard clocks in an accelerated system depend not only on their velocity but also on

their spatial location [5, 6]. Formally, this law is put in the form of the asymptotical equation for

the proper time in the points of the accelerated system and time in a convected system

1

⁄

, (2)

where is acceleration. This ratio explains why the motion equation (1) is inappropriate for the

accelerated systems.

The methods of the general relativity theory allow describing a motion in the fullest form.

For a transfer from the motion equation of the special relativity theory to the motion equation of

the general relativity theory, it is needed – in the equation – to replace the ordinary differentiation

with the covariant one:

⟶

.

In order to obtain a motion equation in electromagnetic fields, it is necessary to replace

with [6]. Now as a component in the motion equation, there is a metric of a reference

system. More to it, for a motion description in curvilinear coordinates, we need nothing except for

1“Asitisknown,physicallawsrelatedtoΣ1donotdifferfromlawsrelatedtoΣ2;thisisconnectedwiththefactthat

inagravitationalfield,allbodiesareacceleratedequally.Thatiswhyasforthemodernstateofourknowledge,

there’snoevidenceforconsideringthatthereferencesystemsΣ1andΣ2differfromeachotherinanyrelation;so

lateron,weshallpresupposeawholephysicalequivalencyofagravitationalfieldandthecorrespondingtoit

accelerationofareferencesystem.Thispresuppositionextrapolatestherelativityprincipletothecaseofa

uniformlyacceleratedstraight‐linemotionofareferencesystem.Theheuristicvalueofthispresuppositionconsists

inthefactthatitallowsreplacingauniformfieldofgravitywithauniformlyacceleratedreferencesystem,which‐

toacertainextent‐lendsitselftotheoreticalconsideration”,Einstein[5].

2TheproofbelongstoG.D.Bogomolov.

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knowing the metric. And on contrary. Always it is possible to find a metric satisfying this equation

at given accelerations or forces acting to the bodies system.

4. The other well-known paradox is the clocks paradox (or twins’ one) [7]. It has a

simple solution in a case of not too high accelerations [1]. However for a rigorous solution of this

problem, it is necessary to have a description of the accelerated system. Historically, the first

accelerated system allowing solving the clocks paradox was the Moeller (Rindler) reference

system [8]. The metric of such system is

1

, (3)

This is one of possible versions of description of the accelerated motion in the clocks paradox.

However the Moeller’s system can be considered as a uniform one only in the limit →1 as a

system acceleration (3) depends on coordinates. For a precise description of a uniformly

accelerated system, it is possible to make use of the motion equation in curvilinear coordinates [6]

Γ

0.

An ordinary acceleration at zero velocity – i.e. an acceleration measured by an accelerometer –

looks fairly simply [9]:

Γ

.

If the acceleration is directed along and it is constant, then:

2

.

As far as must be asymptotically equal to (2), the solution of this differential equation is chosen

in the form:

.

Then the metric of the uniformly accelerated motion takes the form

. (4)

Here, a free choice stays in a selection of functions and . For instance, in the paper [10],

the following motion law was found on the Minkowski plane

′

2cosh

1, ′

2sinh

(5)

with the following metric in the proper reference system

. (6)

The points motion law at a uniform acceleration (5) differs from the “evident” one (1), see Fig. 1.

Actually, a distance between two points change according to the law

4

1/,

i.e. a distance between uniformly accelerated points is reduced according to the Lorentz law.

Both the metric (6) and the metric (3) belong to the pseudo-Euclidean space. But the metric (6) is

not a spatially isotropic one. The space is isotropic [11] even in a stationary gravitational field.

Then as it is applied to a gravitational field, the metric (4) should be written in the form

.

The metric represents a reference system of a uniform gravitational field [12], which can be

considered as a succession of convected reference systems differing with their scaling factor (Fig.

2). This metric belongs to the class of the metrics (see [13]) found based on relativity principle or

(if to put it more precisely) on the condition of local light velocity constancy. It is possible that the

unusual properties of this metric would help understanding some peculiarities of the general

relativity theory [14], [15].

References

1. Pauli W. Theory of relativity (Pergamon Press, Oxford, 1958) [Pauli W Theory of

relativity. (Moscow, Publishing House Nauka, 1991)].

2. Bell J.S. Speakable and unspeakable in quantum mechanics. Cambridge University

Press, p.67. (1993).

3. Gherstein C.C., Logunov A.A. J.С. Bell’s problem (Preprint of Institute of Physics of

High Energies, 1996).

4. Bell's spaceship paradox. From Wikipedia. (Nov 2017) URL:

https://en.wikipedia.org/wiki/Bell%27s_spaceship_paradox

5. Einstein A Über das Relativitätsprinzip und die aus demselben gezogenen

Folgerungen. Jahrb. d. Radioaktivitat u. Elektronik, 4, 411—462 (1907).

6. Landau L D, Lifshitz E M The Classical Theory of Fields (Pergamon Press, Oxford,

(1975).

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1918, 6, 697-702. [Einstein A, Dialogue concerning objections against the relativity

theory. Collection of scientific papers. Vol.1 (Moscow, Publishing House Nauka,

1965) p. 616].

8. Moeller C On homogeneous gravitational fields in the general theory of relativity and

the clock paradox. (Published in. Ejnar Munksgaard 1943).

9. Sazhin M V General relativity theory for astronomers. URL:

http://www.astronet.ru/db/msg/1170927.

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10. Lass, H. Accelerating Frames of Reference and the Clock Paradox, American Journal

of Physics, Vol. 31, pp. 274-276, 1963.

11. Blokhintsev D.I. Space and time in micro-world. Moscow, Publishing House Nauka,

1982

12. Morozov V B. A note on the equivalence principle applicability to the general theory

of relativity arXiv:1404.3083 [physics.gen-ph].

13. Einstein A, Fokker A D. Die Nordströmsche Gravitationstheorie vom Standpunkt des

absoluten Differentialkalküls. Ann. Phys., 44, 321. (1914),

14. Morozov V B. Initial principles of the general theory of relativity. Gravitational field

equation. New solutions. ResearchGate. (2018).

15. Morozov V B. Cosmology of Uniform Empty Space. ResearchGate. (2018).