Introduction

“Relativity of simultaneity” and “time dilation”, are basic resultant effects of special relativity theory (SRT)^[1]. Both are determined by the character of the physical spacetime.

In the following we make a clear distinction between the time coordinate of an event in any inertial system and the time indicated by any clock (“proper time”), located at that system. They are not necessarily the same. Thus, we show that “absolute simultaneity” of events, based on clock times, is preserved in all inertial systems. On the other hand, “relativity of simultaneity”, based on time coordinates of events in spacetime cannot be denied. There is no contradiction here, and since we are doing physics with rulers and clocks, “absolute simultaneity” is what physically accounts. The same holds for “time dilation” effect - it is generally symmetric as a ratio between time coordinate of an event and its clock “proper time” but specifically-it is asymmetric between clock “proper times” measured at both systems. In this way, the “twin paradox”, for example, cannot even emerge, so there is no need to resolve it.

A brief review of SRT principles and effects

SRT deals with inertial systems. At the heart of SRT there are two basic principles:

a) The speed of light, c, is constant and it has the same magnitude in any inertial system.

b) All inertial systems are equivalent. There is no preferred “rest reference system”.

Thus, the laws of physics are covariant in all inertial systems. Lorentz transformation is profoundly essential to the SRT for it is compatible with the relativity principle: It maintains the covariance of physical laws.

Its expression in space time coordinates (or coordinates differences) is the following^[2]:

(2.1) Δx’ = (Δx-vΔt)γ,

(2.2) Δt’ = (Δt-vΔx/c²)γ,

(2.3) Δy’ = Δy,

(2.4) Δz’ = Δz,

 γ is the boost factor,

(2.5) γ = (1-v²/c²)^(-1/2) .

The inverse transformation is symmetric (for v ® -v):

(2.6) Δx = (Δx’+vΔt’)γ,

(2.7) Δt = (Δt’+vΔx’/c²)γ,

(2.8) Δy = Δy’,

(2.9) Δz = Δz’.

Three basic effects emerge out of these equations, which revolutionized our perception of space and time.

The first effect, "time dilation", is the relation between the proper time, Dt, and the laboratory time, Dt. It emerges simply from (2.1), (2.2) by putting Δt’ = Dt , Δx’=0 and is given by:

(2.10) Δτ = Δt/γ.

The second effect is “length contraction” which is the relation between the proper length L₀ and the laboratory length L and is expressed by:

(2.11) L = L₀ / γ.

The third effect, relativity of simultaneity, is expressed directly by equation (2.7).

Assuming for example that in the rest inertial system, (x’; t’), two events at a distance L₀ happen simultaneously so that:

(2.12) Δt’ ≡ Dt = 0.

We obtain in the laboratory inertial system (x; t):

(2.13) Δt = Δx v γ /c² = L₀ v γ /c².

The “Relativity of Simultaneity” issue-revisited

A railway car of length, L₀ = 2l =2cτ, (in the railway car “rest system”) is moving with velocity v relative to the laboratory system. A light pulse is sent at time, τ₀ = t₀ = 0, from a light source at the center of the railway car towards detectors at both rear and front ends.

Obviously, in the railway car reference system, the detection of the light pulses at both ends will occur simultaneously at time τ. It should be emphasized that all clocks in the railway car frame are synchronized and thus, any clock at any point in this frame presents the common time of all the clocks in the railway car reference system.

The observer in the laboratory system should agree about the simultaneity of detection events, since for him, the detection time, τ, in the rest frame, presented by any clock, is equivalent to detection time t in the laboratory frame, given by the “time dilation” effect:

(3.1) t = γτ.

And therefore:

(3.2) Δt = Δτ = 0.

If, according to the equations above, we have absolute simultaneity of events why does the standard interpretation of SRT argue for relativity of simultaneity?

Observing the pulse light test in the laboratory system, from another point of view (but based on the same Lorentz equations), reveals quite a different expected result. The front end of the railway car moves away from the light source while the rear end is approaching it, so the rear end detector will measure the light pulse first.

A simple calculation based on the Lorentz transformation leads to the following time difference in the laboratory system (as was shown in (2.12), (2.13) above):

(3.3) Δt = L₀ vγ/c².

It turns out that two contradicting conclusions, concerning simultaneity of events in the laboratory reference frame are derived from the same Lorentz equations. So in order to restore consistency, this contradiction should be resolved entirely within the framework of the Lorentz equations.

The solution is fortunately very simple. We must notice that we are talking about two different tests and that is the reason we are arriving at two different results. The test performed in the railway car system is not equivalent to the test performed in the laboratory system.

In the first test the detection of the light pulses at the rear and front ends will be simultaneous either in the railway car system as well as in the laboratory system (and in any other inertial reference frame). In the second test the detection events will not be simultaneous in any inertial system. The time difference we get in the laboratory system between events is according to (3.3) and respectively in the railway car system we obtain:

(3.4) Δτ = Δt/γ = L₀ v/c².

What is the difference between the two tests?

The observer in the laboratory and the observer on the train do not agree about the location of the “center”.

The first test is performed with the light source at the center of the railway car in the railway car reference frame (point A’ in Fig. 1 below). The second test is performed with the light source at the center of the railway car in the laboratory reference frame (point A in Fig. 1 below).

Fig.1: Railway car as viewed: 1) In laboratory frame (solid line-scaled by γ) 2) In car “rest” frame (dashed and solid lines)

Due to relativity effect induced by the Lorentz transformation, these two center points (as well as the whole railway car), are initially displaced (as demonstrated in Fig. 1 above) by a distance Δx given, in the laboratory frame, by:

(3.5) Δx = L₀ vγ/(2c) = l vγ/c.

In the first test, the light source is located at A’. As a result the light pulses will arrive also simultaneously, in the laboratory frame, to both rear and front detectors at time t given by:

(3.6) t= Δx/v = L₀ γ/(2c) = l γ/c = γτ.

This result is compatible with equation (3.1). So the detection events are simultaneous in the railway car and the laboratory frames (as well as in any other inertial frame).

On the other hand, in the second test, the light source is located at the center of the railway car in the laboratory frame (point A in Fig.1 above) which is displaced from point A’ by Δx’:

(3.7) Δx’ = L₀ v/(2c)=l v/c.

In this case the time difference, Δτ, in the railway car frame is given by:

(3.8) Δτ = L₀ v/c².

This result is compatible with time difference, Δt, in the laboratory system:

(3.9) Δt = L₀ v γ/c².

This result is again compatible with equation (3.3).

In this second test, the detection events will not be simultaneous at either of the inertial systems (the railway car or the laboratory or in any other frame). But the time dilation ratio still holds:

(3.10) Δt = γ Δτ.

We thus have a complementary interpretation to SRT, which restores “absolute simultaneity” of events, independent of the specific inertial reference system. Let us summarize what we have done here. We simply replaced the final time difference by an initial space difference (in this case of the whole railway car and adequately of its center point as shown in Fig 1. above). According to our relativity model, the relative motion between those frames, finally “closes” this initial distance, at any proper time, τ, (or, t = γ τ, in the laboratory frame respectively), thus restoring absolute simultaneity of events in all inertial frames.

There is an additional argument, which supports this complementary interpretation

Let us go back to the first test but now we will send the light also in the z direction (towards the upper wall of the railway car, which is at a distance of L₀ /2 = l from the center). The observer in the railway car will detect the light exactly at the center of the upper wall simultaneously with the detection at the rear and front ends. The laboratory observer should agree with this result and according to the new interpretation he actually does, preserving also the direction of the light pulse (in the z direction). The same applies also to the second test (in which the center is in the laboratory frame). Moreover, the same argument applies to light pulse sent in any direction, thus preserving the spherical front shape of a light wave in any inertial reference frame as required by SRT.

Till now we have analyzed the “relativity of simultaneity” issue. We have used the “time dilation” effect without saying anything about the “time dilation” issue. It is the subject of the next section.

The “Time Dilation” issue-revisited

The “time dilation” effect is considered by the standard interpretation of SRT to be symmetric. The main argument, which is supposed to support this symmetry, is that the “length contraction” effect is symmetric (to which we agree also in our complementary approach).

But the symmetry of the “length contraction” effect does not necessarily mean that “time dilation” effect is also symmetric.

To show this, let us consider again the railway car with length, L_0, (in the car rest frame), which has length, L, in the laboratory frame with:

(4.1) L=L₀ /γ.

At laboratory time, t=0, we send a light pulse from the rear to the front end, which is reflected back to the rear end.

The total time, Στ, Σt, in the railway car frame and in the laboratory frame respectively- required for the light pulse to reach the rear end will be:

(4.2) Στ = 2 L₀ /c,

(4.3) Σt=L/(c-v) + L/(c+v)=2Lc/(c²-v²) = (2L /c) γ².

By inserting the “length contraction” effect of (4.1) we obtain:

(4.4) Σt=(2L₀ /c) γ = γ Στ

So, in spite of the length contraction of the railway car in the laboratory frame, we obtain a longer time measured in the laboratory, which is expressed by the time dilation effect.

The inevitable conclusion is that “time dilation” effect” is inherently asymmetric.

This is a very important conclusion with four major consequences:

1. There is no “twin paradox” in special relativity theory.
2. We do not need the “relativity of simultaneity” effect to maintain the symmetry of the “time dilation” effect, which does not really exist. This, in turn, supports the “absolute simultaneity” argumentation of the previous section
3. Neither do we need any special mechanism to break the nonexistent symmetry of the time dilation effect in order to explain the nonexistent “twin paradox”^(2),(3).
4. Moreover, new theories like the conventionality of simultaneity (CS) thesis⁽⁴⁾ should not be physically appropriate any more.
   

Symmetry-Asymmetry issues-resolution

We have shown above that “time dilation” effect is inherently asymmetric. We still have to resolve the issue of how such asymmetric effect could emerge out of the Lorentz symmetric transformation. The resolution lies in the very essence of spacetime itself and it should be explained by the distinction between time coordinate of an event in any inertial reference frame and the clock time of that event in the same frame. Generally they do not coincide. Let us explain it in more detail.

The “time dilation” effect, t= γ τ for any two inertial systems is the ratio between ”time coordinate”, t at one system and the “proper time”, τ at the other.

The “proper time” is the time measured by a clock, attached to the observer in the relevant system. This “time dilation” ratio is always symmetric.

There is one and a single reference frame in which the “time coordinate” of any event and the frame “clock time” of that event coincide.

It happens in the “laboratory system”, in which the time span between any two events is a maximum. In this “laboratory system” the “time dilation” ratio is obviously asymmetric.

Thus, we have to separate between two types of “time dilation”:

1. “Time coordinate”/ “proper time” relation, which is always symmetric between any two inertial systems.
2. “laboratory clock time”/”proper time” relation between laboratory proper time and any inertial frame “proper time” which is inherently, asymmetric.

The only physical “time dilation” effect physically measured is between clocks in the “laboratory system” and in any other inertial system.

This effect is mathematically and physically asymmetric.

To clarify this distinction between time coordinate and clock time, let us take, for example, two inertial observers moving in the laboratory frame with velocity, v at opposite directions (Fig. 2 below)

Certainly, each observer will record on his clock an equal “proper time”, τ related to “laboratory time”, t by the “time dilation” ratio, t= γ τ

The clock “proper time” relation of both observers doesn’t fit the “time dilation” ratio with the appropriate relative velocity.

Yet, the “proper time”, τ of each observer will be related to the “coordinate time”, t’ of the other observer by, t’= γ τ.

The above resolution has a direct impact upon the “relativity of simultaneity” issue. We can now say that “relativity of simultaneity” of events exists when speaking only from the “time coordinates” viewpoint of those events, thus maintaining the symmetry of the “time coordinate/proper time dilation” effect. But for our, physically measured, “laboratory clock time/proper time dilation” effect, which is asymmetric -“absolute simultaneity” must hold.

Fig. 2-Two particles moving in laboratory frame with the same velocity in opposite directions

Summary

The complementary interpretation that we have established, so far, restores clock absolute simultaneity of events in all inertial systems. This is physically very important since, any “body” (especially one which is alive), is defined by the “simultaneity” of all its “point events”. Actually, one single clock, attached to the “body”, is sufficient to represent the “time of the body” (“body”, in a sense, should be considered as a “generalized event”). Thus, the very existence of a “body” should be independent of any reference frame. Absolute simultaneity takes care for this state of affairs.

The new complementary approach also solves the “twin paradox” by proving that the clock “time dilation” effect is inherently asymmetric so that, a priori, there is no paradox. Moreover, we do not need, as it appears in the standard interpretation, a mechanism outside SRT, which breaks a nonexistent symmetry.

An inevitable question arises as to the particular feature of the “laboratory system”, which causes this inherent asymmetry. The answer is twofold. First, in our specific “laboratory system” the time coordinate of any event coincides with laboratory clock time. Secondly, clock time difference, Δt, between events is a maximum, so that in any other inertial system the clock time difference, Δτ, is less than Δt. The issue of the “laboratory frame” we use here, worth a more detailed discussion, which is beyond the scope of the present paper. Anyway, and this is very important to emphasize, we all should agree about the existence, of such a reference frame with maximum time span between events. We also do not exclude “relativity of simultaneity” and symmetric “time dilation” effect from the viewpoint of time coordinates in spacetime.

Thus, to sum up, while providing an analysis of the clock “time dilation” asymmetry, the main contribution of our new approach to special relativity theory, is the restoration of clock time “absolute simultaneity”, which seems to represent more accurately our existent physical reality.

It is to the honor of Albert Einstein, if after one hundred years, we can still shed a new light on his great “Special Relativity Theory”.

References

[1] Albert Einstein, “On the electrodynamics of moving bodies”. (1905).

[2] Richard P. Feynman, “The Feynman lectures on physics”.16-1 to16-10, Addison-Wesley Publishing Company, California, 1963 (reprint 1977).

[3] Bernard F. Schutz, “A first course in general relativity”. 1-30, Cambridge University Press, Cambridge, 1985 (reprint 2002).

[4] Mamone Capria “On the conventionality of simultaneity in special relativity”.775-818. Foundation of Physics, Vol 31, No. 5, 2001