J. Bell’s Relativistic Paradox and Equivalence Principle

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DOI: 10.13140/RG.2.2.35232.69123/1
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Abstract
The Einsteinian equivalence principle allows – by natural way – resolving the J. Bell’s relativistic paradox. The metrics were found of uniformly accelerated relativistic systems.
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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

1Asitisknown,physicallawsrelatedtoΣ1donotdifferfromlawsrelatedtoΣ2;thisisconnectedwiththefactthat
inagravitationalfield,allbodiesareacceleratedequally.Thatiswhyasforthemodernstateofourknowledge,
there’snoevidenceforconsideringthatthereferencesystemsΣ1andΣ2differfromeachotherinanyrelation;so
lateron,weshallpresupposeawholephysicalequivalencyofagravitationalfieldandthecorrespondingtoit
accelerationofareferencesystem.Thispresuppositionextrapolatestherelativityprincipletothecaseofa
uniformlyacceleratedstraightlinemotionofareferencesystem.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).
7. Einstein A. Eine Dialog йЪег Einwdnde gegen die Relativitdtstheorie. Naturwiss.,
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).
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