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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.