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A BRIEF OVERVIEW ON THE CONSEQUENCES OF CONSIDERING A LOCALLY COVARIANT AETHER

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Abstract

The cause of incompatibility between the Lorentz transformation and the classical definition of rigid bodies in translational motion is the synchronisation condition of clocks of inertial frames. By changing this condition it is possible: a) To break out this incompatibility within the framework of SR preserving the local covariance of the Lorentz transformation at each point. b) To prove the existence of an aether compatible with such covariance. This can be done without considering any coupled field. Christian Corda showed in 2019 that this effect of clock synchronization is a necessary condition to explain the Mössbauer rotor experiment(Honorable Mention at the Gravity Research Foundation 2018). A brief overview of the problem is set out here as a summary and complement to [4].
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
A BRIEF OVERVIEW ON THE CONSEQUENCES OF
CONSIDERING A LOCALLY COVARIANT AETHER
v.5
Summary
The cause of incompatibility between the Lorentz transformation and the
classical definition of rigid bodies in translation is the synchronization
condition of clocks of inertial frames. By changing this condition it is
possible:
a) To break out this incompatibility within the framework of SR,
preserving the local covariance of the Lorentz transformation at each
point of the flat space-time.
b) To prove the existence of an aether compatible with such covariance.
This can be done without considering any coupled field.
Christian Corda showed in 2019 that this effect of clock synchronization is
a necessary condition to explain the Mössbauer rotor experiment
(Honorable Mention at the Gravity Research Foundation 2018).
A brief overview of the problem is set out here as a summary and
complement to [4].
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
1. Relativity of vertical motion
Applying the Lorentz transformation to the vertical motion of a particle can
lead to the absurd situation that its height be different depending on how
the horizontal axes are labeled. Figure 1 and figure 2 show the same particle
A moving upwards in the same inertial frame S with constant speed U. In
Fig. 1 the axes of S are labeled with the space-like coordinates (X,Y) and
the position of A is defined by the equations of motion X = L and Y = UT
where L is a constant value.
Fig 1: position of particle A in S (X,Y,T): (L, UT).
Fig 2: position of particle A in S (X1,Y,T): (L + M, UT).
If we assume (fig.1) that S(X,Y,T) moves with constant velocity (v, 0) with
respect to another reference frame s(x,y,t) whose origin coincides with the
origin of S at the instant t = T = 0, inserting the Lorentz transformation (Y
= y, T = (t-vx/c2), x = vt + L/) into Y = UT we get that the height “y” of
particle A at t = 0 in s is [*]:
1) y = - UvL / c2
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
In figure 2, we have considered two constant values L and M and we have
labeled the axes of frame S with another coordinate system (X1,Y) where
X1 = X+M. The equations of motion of particle A are then X = L+M and Y =
UT. Labeling the horizontal axis of s so that its origin coincides with that of
S(X1,Y,T) at t = T = 0 and inserting the Lorentz transformation (Y = y, T =
(t-vx1/c2), x1 = vt + (L+M)/) again into Y = UT, we get that the height
“y” of particle A at t = 0 in s is different [**]:
2) y = - Uv (L+M) / c2
Which of the two heights 1) or 2) is the correct one? How should we label
the horizontal axes? Can the relativity principle hold if the height “y” of a
particle depends on how to label the horizontal axes?
Taking into account this situation, an observer moving along a
perpendicular direction to the trajectory of a particle can find some
difficulties in defining its position. Is this a well-known effect? Must we
consider it to be an indetermination?
It is to be noted that the Minkowsky diagram of the particle A (fig. 1) can
only be depicted if we take L = 0. For any other value of L we get two
different diagrams. Inserting the Lorentz transformation (Y = y, T = (t-
vx/c2), x = vt + L/ , u = U/) into Y = UT we get y = ut - UvL/c2 and depicting
these two equations in a Minkowsky diagram, the following figure is
obtained:
Fig 2b: position of particle A in a tT yY diagram.
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
We observe that we can only define a single world line if the particle is
placed at the origin of S (L = 0). Regardless of the relationship between t
and T, taking L to be different from 0, the two blue lines will be always apart
by an amount UvL/c2 along the y ,Y direction. Which one is the world line of
particle A?
2. Violation of Born´s rigidity
The classical definition of rigid motion of two particles says that translation
is only possible if the distance between them remains constant. However,
it turns out that applying the Lorentz transformation we find that the
distance can only be constant in a unique single inertial frame of reference.
Let's see it with a simple example. Consider two particles A and B in
translation with uniformly accelerated vertical motion in frame S(X,Y,T)
such that the segment AB with length L remains always parallel to the
horizontal axis X (XA = 0, XB = L). If we assume that the acceleration vector
(0, E) is constant and we take the height of both particles to be defined by
the expressions YA = YB = 0.5 ET2, we have that the vertical distance between
A and B in S is always (fig.3):
3) YB - YA = 0
Fig 3: height of particles AB in S (X, Y, T): YA = YB = 0.5 ET2. L = constant.
If S moves with constant velocity (v, 0) with respect to another reference
s(x,y,t) whose origin coincides with the origin of S at t = T = 0, applying the
Lorentz transformation we get that the vertical distance between A and B
in s is:
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
4A) yB - yA = 0.5 E (L2v2/c4- 2Lv t /c2 )
which shows us that, at each instant of time "t" the distance yB - yA is
different despite being always constant in S and, therefore, in s the two
particles cannot be in translational motion despite being in translational
motion in S. This symmetry violation is one of the reasons why the classical
concept of rigidity cannot be used for translational motion of systems of
particles and extended rigid bodies.
Moreover, if we make use of hyperbolic motion in the frame S(X,Y,T) we
may write (1)(2):
4B) J2 = Y2 - c2 T2
where “J” is the high in the corresponding commoving inertial frames of a
particle rising with constant proper acceleration. Using Born´s relativistic
definition of rigidity, “J” must be invariant under Lorentz transformations.
It is straightforward to see that this is the case just for boosts along the Y
direction. For a boost along X we may insert Y = y, T = (t - vx/c2) into 4B)
to get:
4C) J2 = y2 - c2 (t - vx/c2)2
which is clearly not invariant (only x2 - ct2 is invariant for inertial comoving
frames moving along the horizontal direction). Thus, we get a violation of
Born´s rigidity which suggest that a re-synchronization of clocks of inertial
frames might be a necessary condition to considerer rigid translations in the
framework of SR.
3. Relative synchronism
In the two previous sections, we have applied the Lorentz transformation
under the implicit hypothesis that the clocks of inertial frames are
synchronized such that they all show the same time regardless of their
position (Einstein synchronization). However, when we apply the Lorentz
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
transformation assuming that each clock of an inertial frame s(x,y,t) has an
offset of vx/c2 we get that [4]:
1- The indeterminacy of sections 1 and the symmetry violation of section
2 disappear.
2- We need a preferred reference frame to apply the Lorentz
transformation.
Specifically, if we consider that an inertial frame s(x,y,t) moves with
constant speed (-v, 0) with respect to another inertial frame S(X,Y,T) such
that the origins coincide at the instant t = T = 0 and we consider that the
clocks of s have an offset vx/c2, we may write:
5) t = t0 + vx/c2
where “t0" is the time-like coordinate of the clock located at x = 0 and "t" is
the time-like coordinate of another clock located at an arbitrary position x.
Inserting 5) into the time-like component of the Lorentz transformation we
have that:
a) The height of particle A of section 1 in s is always y = 0 at time t0= 0
regardless of how we label the horizontal axes [***].
b) In s, the vertical distance yB-yA of the two particles AB of section 2 is
always constant with the same value yB-yA=0. Thus, we conclude that AB
remain in translation also in s. (inserting t0 = t - vxA/c2 = t - vxB/c2 into yB -yA
= 0.5 E (2(t-vxB/c2)2 - 2(t-vxA/c2)2 yields yB-yA = 0).
c) We need to consider a preferred reference frame due to the fact that the
relative speed “v” between frames appears in the synchronization
condition 5), and it only has physical meaning if we consider that the speed
of S is v = 0. Such a frame is the preferred one and it performs the role of
an aether such that the twins paradox vanishes and we get that the older
twin is always the one which remains in the preferred frame.
It is worth noting that the proposed synchronization condition does not
cause any violation of the local invariance of the Lorentz transformation in
the neighborhood (dx,0,0) of each point (x,0,0) of the horizontal trajectory
of a particle since in such neighborhood we have to read the increment dt
with a unique single clock placed at (x,0,0). Therefore we have that the
displacement four-vector (dt,dx,0,0) and the rest of derived four-vectors
(velocity, acceleration, force, ...) will have a clearly covariant behavior at
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
each point of the trajectory and the postulates (relativity principle and
constancy of the speed of light) and all the results of SR will hold locally.
On the other hand, if we consider a long horizontal displacements of a high-
speed particle we cannot subtract the time of two distant clocks (t) in
order to calculate an average velocity x/t since all clocks of the inertial
frame have a different offset.
4. A local Lorentz transformation
We may consider the time of a clock H placed at an arbitrary coordinate x
to be t and the time of a clock P placed at an arbitrary coordinate xP to be
tP. Let the offset (t tP) between the two clocks be:
6) (t tP) = v (x - xP)/c2
where (x-xP) is a constant value. If we insert 6) into the time-like component
T = (t - vx/c2) of the Lorentz transformation for H, we get:
7) T = (tP - vxP/c2)
On the other hand, if we assume that the origins coincide (x = X = 0) at time
tP = 0 we may write the space-like component of the Lorentz transformation
as:
8) X = (x - vtP)
Assuming that both clocks are placed at the same point x = xP, inserting x =
xP , X = XP , T = TP into 7)8) yields:
9) XP = (xP - vtP)
10) TP = (tP - vxP/c2)
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
which is the local Lorentz transformation for an event happening at point
P. On the other hand, if the distance between x and xP is different from 0
and the clock P is placed at the origin of coordinates, we may insert xP = 0
into 7)8) to get:
11) X = (x - vtP)
12) T = tP
which is a change of coordinates that:
- It is compatible with GPS simultaneity.
- Solves the Sagnac effect in the framework of SR without the need of using
either GR or the Langevin metric.
- It is compatible with the classical definition of extended rigid bodies in
translation.
- It allows to solve the problems of section 1 and section 2 of this document.
- It can only be applied to space time event coordinates. For tensor fields
we must clearly apply the Lorentz transformation as it can be easily
concluded from 9)10).
- It can only be applied to compare events happening at x and xP and cannot
be applied either to define an average velocity or to define a velocity
addition formula.
Thus, we may claim that, considering the synchronization condition 6):
a) We get Lorentz invariance at each point of flat space-time (eqs. 9,10)
when we use a unique single clock.
b) The Lorentz invariance is broken out when we use two clocks to measure
time intervals for long distances (eqs. 11,12).
c) We need to define the velocity v of the synchronization condition (eq. 6)
with respect to another frame. This frame has v = 0 and it plays the role of
an absolute preferred frame.
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
a)b)c) suggest that the Thomas precession is a local quantum effect that
cannot manifest for long displacements.
It is important to emphasize that (t tP) is not a function of x since (x - xP) is
a constant value corresponding to the distance between the two clocks. As
a consequence:
- Eq. 6) is just a re-parametrization of the t coordinate of each inertial
frame such that we can easily depict 11) 12) in a Minkowsky diagram
where the time-like axis of the non-preferred inertial frame is
relabeled in accordance to the offset 6). For each couple of values of
x and xP we can repeat the same process.
- The Minkowsky metric of flat space-time is preserved for any inertial
frame.
5. Conclusions
Assuming that all clocks of an inertial frame s(x,y,t) have an offset of vx/c2
we find that:
i) We have local covariance of the Lorentz transformation at each point
of the trajectory of a particle.
ii) This method of synchronization is a necessary condition for:
A) Being able to apply the Lorentz transformation to the
classical definition of translational motion of any system of
particles.
B) Making the vertical movement to be independent of the
choice of the coordinate system.
C) Explaining some empirical facts of Mechanics of Particles.
D) Explaining the Mossbauer Effect (5).
In order to apply this condition, we need to define a stationary preferred
frame (aether) which is the only one with v = 0 and in which all clocks show
the same time (T = T0 + vX/c2, v = 0 => T = T0). This method of synchronism
is global and avoids the need to launch rays of light at every single inertial
frame.
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J.L. Junquera –“Some consequences of considering a locally covariant aether”
References
[1] C.H. Misner, Kip. S. Thorne, J.A. Wheeler. “Gravitation”, W.H. Freeman
and company, 1973.
[2] Wolfgang Rindler, Relativity, Special, General, and Cosmological,
Oxford University Press, 2006.
[3] Patricia and John Schwarz, Special Relativity, Oxford University Press,
2005.
[4] Jose Luis Junquera. "Some kinematic considerations on the need for a
preferred frame". Researchgate. November2021. DOI:10.13140/
RG.2.2.26314.29126. License CC BY-NC 4.0.
[5] Christian Corda. Mössbauer rotor experiment as new proof of general
relativity: Rigorous computation of the additional effect of clock
synchronization. International Journal of Modern Physics D. Vol. 28, No.
10, 2019.
[*] Inserting the Lorentz transformation (Y = y, T = (t - vx/c2), x = vt + L/)
into Y = UT we have at t = 0: y = - UvL/c2.
[**] Inserting the Lorentz transformation (Y = y, T = (t-vx/c2), x = vt +
(L+M)/) into Y = UT we have at t = 0: y = - Uv (L+M)/c2.
[***] Inserting the Lorentz transformation (Y = y, T = (t-vx/c2)) and eq.5) t0
= t-vx/c2 into Y = UT we have at y = Ut0. At t0 = 0 we get y = 0. This result is
independent of the coordinate x.

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Article
We received an Honorable Mention at the Gravity Research Foundation 2018 Awards for Essays on Gravitation by showing that a correct general relativistic interpretation of the Mössbauer rotor experiment represents a new, strong and independent, proof of Einstein's general theory relativity (GTR). Here we correct a mistake which was present in our previous computations on this important issue by deriving a rigorous computation of the additional eect of clock synchronization. Finally, we show that some recent criticisms on our general relativistic approach to the Mössbauer rotor experiment are incorrect, by ultimately conrming our important result.
  • Wolfgang Rindler
Wolfgang Rindler, "Relativity, Special, General, and Cosmological", Oxford University Press, 2006.
Some kinematic considerations on the need for a preferred frame
  • Jose Luis
Jose Luis Junquera. "Some kinematic considerations on the need for a preferred frame". Researchgate. November2021. DOI:10.13140/ RG.2.2.26314.29126. License CC BY-NC 4.0.