# Is Relative Simultaneity Real?

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1. Apr 6, 2014

### andromeda

One problem in understanding Special Relativity is that it is intuitively hard to agree with relative simultaneity.

My major problem is that I cannot quite answer the question:

Is relative simultaneity a real effect or only a mathematical artifact of Lorentz transformation?

I am not the only one who has this kind of cognitive problems. This is quite common as well documented in peer reviewed journals [1].

The real or apparent contradiction arises in the following situation:

Lorentz transformation of two simultaneous distant events in a stationary system as defined by Einstein in [1] Part I § 1, without any doubt produce non simultaneous events in a moving system.

This is because the calculated times t1’ and t2` for events E1 and E2 at t1=t2(in the stationary) are not equal after transformation to the moving system. And that is the well known demonstration of relative simultaneity.

However if you have two particles approaching X axis in a manner that they are always on a parallel line to X, or simply a rigid rod descends in a motion parallel to X, both distant points cross X axis simultaneously by assumed arrangement in the stationary system.

If both points are aligned on X, they are automatically aligned on X’ which is the same line albeit moving.

How it is then possible that two points are both aligned with the axis and yet the times of alignment are different?

Where is the other point when one of them is on the axis at some time t'?

Is relative simultaneity real? Has it been experimentally proven?
Its not the question of time dilation which we know is real, but how does that relate to simultaneity? The equations alone do not provide these answers.

Any guidance will be highly appreciated.

[1] R., E. Scherr, P., S. Shaffer, S. Vokos "Student understanding of time in special relativity: Simultaneity and reference frames" Am. J. Phys. 69, S24 (2001);
[2] Albert Einstein, “On the Electrodynamics of Moving Bodies” (translation from original Annalen der Physik, 17(1905), pp. 891-921) published on the internet in http://www.fourmilab.ch/etexts/einstein/specrel/specrel.pdf

2. Apr 6, 2014

### Fredrik

Staff Emeritus
The distinction between "real" and "mathematical" isn't as clear as you seem to think. In fact, I'm not sure it's meaningful at all.

If time dilation and length contraction are real, then so is relativity of simultaneity. They are all things that show up when we compare how two different coordinate systems describe the same thing. So they all have the same status in the theory.

I don't understand the scenarios you describe. A line parallel to the x axis that crosses the x axis? If it's parallel to the x axis, it either is the x axis or never crosses it.

3. Apr 6, 2014

### andromeda

Thanks for your answer. Its a valid point do define real. Need to think about it a bit

I know the theory is internally consistent but the meaning of "real" (which I need to define better if possible) would lie outside the mathematical construct. Do not want to elaborate without thinking.

Sorry, confusing wording on my part. A line crosses the x axis is in the same sense as the bumper of my car crosses the rails on railway crossing. Moves parallel to them then aligns with them, and I called it "crosses the line". Should find a better word for this. I am not a native English speaker.

4. Apr 6, 2014

### ghwellsjr

This is totally wrong for the following reasons:

1) Simultaneity has to do with the Coordinate Times of two events in one Inertial Reference Frame (IRF), not one event as "viewed" from two different IRF's.

2) Simultaneity has nothing to do with the local time of observation, it has only to do with the Coordinate Time of events. If you're considering the time of observation of a distant event as the second event, then those two events can never be simultaneous.

Relative Simultaneity refers to the fact that two events that are simultaneous in one IRF may not be simultaneous in another IRF.

Last edited: Apr 6, 2014
5. Apr 6, 2014

### andromeda

Thank you all for the response.

Hope we can now agree on facts such that we do not need to repeat them again and use in subsequent reasoning.

1) Coordinate time is fully equivalent to clocks readings at arbitrary positions relative to origin of a
3D coordinate system, given the clocks are adequately synchronised.

2) It is clear now that if one event is detected and recorded in one place, it has the clock reading
at this place. The other one detected and recorded elsewhere, has the reading of a similar clock
at that clock's position.

3) The clocks have been synchronised in accordance with the procedure given by Einstein in his
1905 work assuming forward and backward speed of light being the same and constant in all
inertial frames.

4) Lorentz transformation is the consequence of the speed of light assumption and the
synchronisation procedure.

5) The simultaneity can be proven after the events by gathering event records from remote
locations. If the time of detected events in their respective places have the same numerical
values, then the events are called simultaneous. Otherwise they are successive.

I hope this is correct, and if not please indicate.

I hope to return to the thread in a day or two to rephrase my dilemma based on your comments and common understanding of terms and scientific facts described above.

6. Apr 6, 2014

### Staff: Mentor

I also like this article by Scherr, Shaffer, and Vokos:
http://www.aapt.org/doorway/TGRU/articles/Vokos-Simultaneity.pdf

In it they present some pedagogical approaches for teaching this concept. Unfortunately, very few students are lucky enough to be taught with this method, myself included and you also I assume.

7. Apr 6, 2014

### Staff: Mentor

X' is not parallel to the x' axis. I will assume that you know four-vector notation, and I will use units such that c=1.

The worldsheet of X is given by $X=(t,a,ut,0)$ where -1<u<1 is the velocity of X in the unprimed frame, t is coordinate time in the unprimed frame and a is a parameter which picks out a given point along the line at any given t.

Transforming to the primed frame we get $X'=(\gamma(t-av),\gamma(a-tv),ut,0)$. Now, taking $t'=\gamma(t-av)$ and $a'=\gamma(a-tv)$, solving for t and a, and substituting back and simplifying we get:

$X'=(t',a',u't'+a'u'v,0)$ where $u'=\gamma u$

Note that the distance to the x axis is no longer $ut$, but $u't'+a'u'v$ where the second term means that it is not parallel to the x' axis.

I agree with Fredrik that the question about "real" requires a good rigorous definition of "real". If you come up with such a definition, whether it is a generally accepted definition which could be discussed here or a personal one which would have to be analyzed elsewhere, I would expect that you would find that simultaneity itself is not real.

In other words, the Lorentz transform shows that the universe "cares" about causality. Things that are causally related in one frame share the same causal relationship in any other frame. But the universe simply does not "care" about simultaneity. As soon as you realize that you can stop being distracted by things that are physically unimportant (simultaneity) and focus on what is physically important (causality).

The relativity of simultaneity is a consequence of the Lorentz transforms, which are well-established, scientifically:
http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html

The experimental evidence is overwhelming.

Last edited: Apr 6, 2014
8. Apr 6, 2014

### andromeda

This thread is getting interesting.
Before I come back with something worth publishing, I have to say that the only thing one needs to reconcile with the common sense in order to accept Einstein theories, is just one single concept of relative simultaneity.
All the rest is easy (relative to how much math one can handle).

9. Apr 7, 2014

### georgir

I think a useful step in coming to terms with relativity is not trying to categorize events as "simultaneous" or "not simultaneous", as that is obviously dependent on reference frame, but instead "space-like separated" (i.e. simultaneous in some reference frame) and "time-like separated" (i.e. co-located in some reference frame), and the edge-case of "light-like separated" which are always absolute, frame-independent relations between any two events.

10. Apr 7, 2014

### andromeda

To clarify the issue: in my original post I (rightly or wrongly) referred to X as an axis, that is a straight line, and now in your argument you use X as a 4 dimensional vector representation of a traveling line (in old fashion 3D terminology, when you drop first coordinate and t is a variable scalar parameter).

I understand that your argument can be verbalised in a notation tolerant manner like this:

A line descending towards "x axis" in the stationary system such that it remains parallel to "x axis" at all times (t), it remains not parallel to "x' axis" at all times (t') in the moving system.

Naturally, X axis is always parallell to X' axis (using my original notation).

Perhaps for further discussion I will keep your capital X as you use it and refer to x,y,x in lower case as axes, but then it will be difficult to refer to x,y,z as coordinates based on those axes although it can be avoided using indexed notation.

11. Apr 7, 2014

### jkl71

12. Apr 7, 2014

### Staff: Mentor

Feel free to use whatever notation you prefer. I may have misinterpreted your intended usage of x vs X, and if you wish to alter your notation for clarity then that is certainly fine. I will do my best to use your notation or I will try to clearly define any notation that I employ.

The notation doesn't change the geometry. A line which is parallel to the "boost" axis and moving in one frame will not be parallel to the "boost" axis in another frame.

13. Apr 8, 2014

### andromeda

Thank you DaleSpam for supporting the parallel lines argument.

The reason of this reply is that I do not fully agree with your conclusion although your mathematical presentation is correct. Its all about the interpretation not just only maths. Furthermore my main question whether relative simultaneity is real or apparent hinges on understanding of parallel line translation in the lateral direction to x axis

This post will be a bit long but I cannot do it any simpler at this stage.

Notation Remark:
The coordinates are expressed in positional association form and the notation $X=(t,x1,y1,z1)$ represents a point in the space time or in 3d mentality a point in euclidean 3d sub-space which is associated with "coordinate 1" when its value is equal to t.

Let the traveling straight line parallel to x axis be represented by:
$X=(t,a,ut,0)$ ​
where:
t is a scalar parameter representing a value of "time coordinate 1" in the unprimed frame​
X[1]=ct where c is the speed of light set to 1 for convenience.​
a is a scalar parameter which picks out all values of spatial "coordinate 2" comonly referred as x coordinate.​
-1<u<1 is the velocity component of X in the direction of spatial "coordinate 3" commonly referred as y coordinate in the unprimed frame, ​
spatial "coordinate 4" commonly referred as z coordinate is permanently set to 0

Lorentz Transformation matrix $L$ can be written as:

| γ,-vγ,0,0|
|-vγ, γ,0,0|
| 0, 0,1,0|
| 0, 0,0,1|​

The transformation from unprimed to primed is represented by matrix multiplication as follows
$X'=LX$​
where $X$ should in fact be a column vector but to conserve space we show it in text horizontally dropping the transposition notation.

Transforming to the primed frame we get $X'=(\gamma(t-av),\gamma(a-vt),ut,0)$.

Up to this point I assume everyone would agree.

Before going further, the above equations can be interpreted as follows:

$X$ represents an instant of a travelling line in the unprimed system such that when $X[1]=t$ the instant of this line in 3d space is $(a,ut,0)$, where ut is a fixed instance of coordinate y at clock time t while a (as assumed before) represents "all x'es". Coordinate z remains 0.

Note that at t=0 which is the synchronisation time appropriate for the above form of $L$ Lorentz transformation matrix, the $X$ at t=0 is identical with the x axis.

Similar but not identical reasoning can be used about $X'$

$X'$ represents an instant of a travelling line in the primed system such that when $X'[1]=\gamma(t-av)$ the instant of this line in 3d space is $(\gamma(a-vt),ut,0)$ where ut is a fixed instance of coordinate y at clock time t. Coordinate z remains 0. Coordinate x' is obviously different than x because of the motion of the system.
The difference is that the "time coordinate 1" is different for every value of a that represents arbitrary position in x direction, which is not the case in the unprimed system. This means clocks in the primed system are synchronised that way.

At time t=0 $X$ is exactly aligned with x axis so it is aligned with x' axis because x ad x' axis are on the same straight line.
The fact that the "time coordinate 1" varies in the primed system is because of choice Einstein's synchronisation method.

My conclusion is:
the traveling line is parallel to x axis and to x' axis at each instant of its existence including t=0 and (t'=0 at x'=0) when it fully coincides with both.

We have to live with the fact that clocks in the moving system are phase shifted and running at different rate than in the uprimed system and equal time at distant locations does not mean temporal coincidence. Unfortunately all dictionaries agree that simultaneous means equal times so the usage of "temporal coincidence" is justified although it is used as a synonym of simultaneity.

That situation would be similar to a degree, if we used time of the primed system as an output of a precise sundial synchronised at 0 GMT and travelling along the equator east-west and reasoning about kinematics perceived while in motion. At each instant of the travelling line motion form the perspective of the moving system every point of that line would have different sundial time.

Continuing with the originator's argument:

Now, taking $t'=\gamma(t-av)$ and $a'=\gamma(a-tv)$, solving for t and a, and substituting back and simplifying we get:

$X'=(t',a',u't'+a'u'v,0)$ where $u'=\gamma u$

Note that the distance to the x axis is no longer $ut$, but $u't'+a'u'v$ where the second term means that it is not parallel to the x' axis.

This is all correct mathematically when you wish to express $X'$ in a consistent set of parameters. The grouping by t' parameter for phase shifted clocks will show the line as not parallel because for the same t' in extreme case one point would be say "now" the other where the line was yesterday and this is definitely not parallel line.

There is a little known publication in an official academic peer reviewed journal dating back to 1972 which describes time phase shift extensively but I am afraid I would be banned for referring to something not adhering to majority views.

14. Apr 8, 2014

### Staff: Mentor

Then your disagreement is illogical.

If the math is correct (as you concede) and if the premises are correct (the Lorentz transforms are presumed correct on this forum) then the conclusion is correct. The only way to dispute the conclusion is by showing the math incorrect or by disputing the premises.

I will go through your reply in detail, but it is rather lengthy so it may take some time, and the outcome is guaranteed by the rules of logic.

15. Apr 8, 2014

### Staff: Mentor

Actually, it took less time than I anticipated.
Presumably then $X'=(t',x1',y1',z1')$ would be the notation for a point in the primed frame.

Then, by your own notation in the primed frame $t'=\gamma(t-av)$, and all of the rest of my previous post follows.

This simply incorrect. An "instant" by definition is a single value of time in the given system. $X'[1]=\gamma(t-av)$ is not a single value in the primed system, it is potentially all time in that system depending on the range of t and a. Now, it is possible for you to fix a relationship between t and a such that the expression above evaluates to a single value. However, if you do this you will either wind up with a single point (which cannot be parallel to anything) or you will wind up with a line which is not parallel to the x axis.

This is the logic error. You called something an "instant" in the primed frame which does not meet the definition of "instant" in the primed frame. All of the rest of your logic from this point on fails. I encourage you to work through the math to see that if you fix a and t such that you do get a single instant then you do indeed not obtain a parallel line.

Last edited: Apr 8, 2014
16. Apr 8, 2014

### andromeda

I am happy that the discussion is progressing and I accept the challenge. It may take me a while in order to write something free from errors and ambiguities or at least significantly better.

I see the travelling line problem we are discussing can be resolved to mutual satisfaction this way or another in finite time. Either outcome will satisfy my internal curiosity department. This is possible because of the simplicity of the stated problem, only linear algebra and all framework of Special Relativity unchallenged.
The reason two intelligent people disagree while acting in good faith is they either do not understand each other or at least one fails to recognise his own mistakes.
All this can be improved gradually.

17. Apr 8, 2014

### Staff: Mentor

Sounds good. I look forward to the next installment whenever you get around to it.

18. Apr 9, 2014

### darkhorror

How about look at it simply, take 4 clocks, 2 in a stationary FOR, 2 in a moving FOR. Lets put these clocks on a train and a platform. One on left side of train one on right, one on left side of platform, one on right. In the platform's FOR, the train and the platform are the same length, the clock distance is also the same. So in the platform's FOR when the left two clocks are at the same point, the right two clocks are also at the same point. Yet in the train's frame the distance between it's clocks and the platforms clocks is different. So in the train's FOR, when the left pair of clocks are in the same place the right pair of clocks can not be in the same place.

So if the left clocks read 0 when they line up, and the right clocks also read 0 when they align. In the train's frame of reference when the left clocks both read 0, the right clocks can't also read 0 since they aren't aligned. Since the math alone isn't convincing you, draw out an actual example of what it would look like in reality, take time dilation, length contraction, and that both frame needs to agree on events. Draw the clocks, draw them moving towards other clocks, look at how time must behave, and what "now" would have to be in different frames.

19. Apr 9, 2014

### andromeda

It may seem that I am a bit picky and going off-side from my main challenge, but I have to disagree with your low rating of simultaneity although such opinion is shared by many.
If I was to rate simultaneity and causality I would agree the later seems a little bit more important, but to disregard simultaneity as a non-issue?
Simultaneity in descriptive terms is the boundary that separates before from after and cause from effect(locally). How is that the boundary is not important?
My words may not be convincing however we should take note that there are quite different views:
According to M. Jammer [1] Einstein[2] stated that relativity of simultaneity “is the most important, and also the most controversial theorem of the new theory of relativity. It is impossible to enter here into an indepth discussion of the epistemological and ‘naturphilosophischen’ assumptions and consequences which evolve from this basic principle.”

In another place Jammer [1] states: "Reichenbach had already recognized that the notion of simultaneity plays an important role in the metrical geometry of special relativity when he defined the length of a moving line segment as the distance between simultaneous positions of its end points".

I can rest on Lorentz transformation to calculate what happens in the primed system but I cannot understand how “at once” becomes before and after. Being in this thread I want to understand just that rather than move the controversial issue away from me.

[1] Jammer, M. Concepts of Simultaneity: From Antiquity to Einstein and Beyond . Baltimore : Johns Hopkins University Press, 2006.

[2] A. Einstein, “Vom Relativitäts-Prinzip,” Vossische Zeitung, 26 April 1914 (no. 209), pp. 1–2;
The Collected Papers (1996), vol. 6, p. 4; English translation (1997), vol. 6, pp. 3–5.

20. Apr 9, 2014

### andromeda

Thanks. I will look at this example as well. It is good to have different angles on the same problem