# Putting numbers in Einstein's thought problem

1. Jan 26, 2010

### DanRay

The thought problem in section 7 of “Relativity” is the beginning of Einstein’s rationale for using his version of Lorentz Transformations. He uses a railroad embankment as one “rigid reference-body (frame)”, and a train car on the tracks moving at a constant speed parallel to the embankment as a second (inertial)“rigid reference body”. He has removed the air above the embankment so his “ray of light” will be propagated in a vacuum. He does this is to demonstrate why simple addition of velocities won’t work the same for the light in this thought problem as it did for a man walking along the railroad car in the same direction as the train’s motion in section 6 of “Relativity” where he demonstrated the “theorem of the addition of velocities” in regard to the speed of the man relative the embankment. (W = v + w)

The question posed by Einstein in his section 7 problem is this: “If a ray of light is sent along the embankment, we see from the above that the tip of the ray will be transmitted with the velocity c relative to the embankment. ---” The unknown of the problem is the speed of the tip of the ray relative to a train carriage moving at speed v (relative to the embankment) in the same direction as the light. He sets up an equation using the standard equation for an addition (or in this case, subtraction) of velocities problem with the resultant velocity w as the unknown: w=c-v.

It is Einstein’s next comment that is the beginning of his justification for using the Lorentz Transformations: “The velocity of propagation of a ray of light with respect to the carriage thus comes out smaller than c.” That of course violates the second postulate of Special Relativity and leads him into his famous statement at the end of this same section. “As a result of an analysis of time and space, it became evident that in reality there is not the least incompatibility between the principle of relativity and the law of propagation of light,----”

And we’re off Galilean/Newtonian Universe out; Time dilation, rod shortening, speed limits and eventually the curving space-time continuum in.

Some time ago I began to wonder what would happen if I put actual numbers into Einstein’s section 7 problem. This is what I found: a light flash begins from a source situated 100 meters away from a point (A) on the embankment at the moment a mirror on the back of the carriage aligns with point A. The carriage is moving 100km/h. It will take the “tip of the ray” 1/3rd of a microsecond (333ns)to reach point A on the embankment but the train will only have moved .00925 millimeter in that time. The “tip of the ray” will cover that extra distance in .030833ns. Obviously these are distances and intervals far too small for human observers to perceive.

But what if the speed of the carriage is 200,000km/sec. (2/3 c). As before the tip of the ray of light will reach point A on the embankment in 1/3rd microsecond in which time the train will have moved a very perceptible 66.6 meters. The light will catch up to the observer on the train in just 1 microsecond from its starting point at a point 200 meters from the observer on the embankment. Though a human observing from Einstein’s omnipotent point of view would clearly see the 200 meter difference, he would not likely be able to perceive a time difference of only 1 microsecond. (Light traveling at c moves 300m/microsecond.). So it would appear to the observer that the light arrived at both the point on the embankment and the mirror on the train 200 meters down the track simultaneously. If the reflected light from the mirror is accurately timed it will be back at the source exactly 2 microseconds from when it began and if you could substitute an accurate measuring device for the mirror it would be measured as exactly c.

I don’t think there is a problem here for either of Einstein’s Special Relativity postulates but for me it puts up a big question mark about the rationale for his use of the Lorentz Transformations and all of the phenomena that derive from. That doesn’t necessarily prove that none of those are true, it just seems to negate Einstein’s pathway to them. The Lorentz Transformations are the only math Einstein used in relation to Special Relativity.

Am I missing something here? Please tell me the flaw in my reasoning.

Last edited: Jan 27, 2010
2. Jan 27, 2010

### JesseM

The mirror aligns with point A, or the mirror aligns with the source? I assume from the later comments you probably mean that the mirror is aligned with A...
1 microsecond in the embankment frame, yes. But you don't appear to be considering the time it takes the light to go from the source to the back of the train in the train's own rest frame, or the distance the light travels in this time in the train frame, and wasn't that the whole point of Einstein's thought-experiment, to compare the speed of a ray of light in two different frames?
"Observation" does not literally mean what you see with your unaided eyes, it refers to what you measure using equipment which is at rest in your frame. You're allowed to use cameras and precise clocks and the like.
Again, you're sort of missing the point of the thought-experiment if you don't figure out what the speed would be when measured by equipment at rest relative to the train. Obviously since you assume in your calculations that the light moves at c in the embankment frame, it's necessarily going to be true that the light takes 1 microsecond to travel 100 + 200 meters to catch up with the back of the train, and another microsecond to travel 300 meters back, since all these measurements are themselves specific to the embankment frame.

3. Jan 27, 2010

### DanRay

Dear Jesse M,
Thanks for your reply. However I must take exception to all that you say. Section 7 of Relativity is Einstein's 1916 explanation of the reasoning path he took to discovering all of the things you are talking about. You are wanting me to accept that Einstein should have used time dilation to prove time dilation. You have to accept that at some point he didn't know what he would discover and if you read Section 7 you will see that the whole point of it and this problem is to explain what led him to use the Lorentz Transformations. Sections 8 through 10 are also part of this reasoning journey. He introduces Lorentz and time and space variations in Section 11. My point about my findings is that there is nothing in this problem that justifys the introduction of the transfprmations in the first place and although I didn't mention it before if you look at the two examples I gave and any others you care to examine the difference that satisfies the addition of velocities problem is always in the different arrival time of the "tip of the ray" first to point A then the carriage.

Simply you can't use a conclusion to prove itself!

And by the way Einsteins observer in this case and I think in all cases in this book is the human mind he doesn't invoke any instrumentation at all.

4. Jan 27, 2010

### Ich

DanRay,

I can't make sense of you first post. As JesseM put it, you seem to be completely missing the point.
Your answer to JesseM also seems to have nothing to do with what he said.
And this is simply wrong:
Maybe you can formulate a specific question or claim, which can then be discussed.

5. Jan 27, 2010

### yuiop

I do not have the book you mention, but I assume Einstein demonstrated time dilation is a consequence of the constant speed of light postulate and the light clock thought experiment before he gets to the train thought experiment. In the latter experiment, he demonstrates that what appears simultaneous to one observer, does not appear simultaneous to anotyhe observer with relative motion to the first observer.

The justification is that without the Lorentz transformations the postulates of the constant speed of light and the laws of physics being equal in all inertial reference frames are violated.

6. Jan 27, 2010

### DanRay

Dear Ich,

Thanks for your concern. Everything I put in quotes is from Einsteins 1916 book "Relativity" and the problem I am dealing with is Einstein's own. I simply put numbers in where he had none and kept it true to his original. In this book Einstein is explaining the reasoning he used to develop his Special Relativity conclusions all of which are based on Lorentz Transformations (other than the two postulates themselves). My specific claim is that I can't see a reason to apply Lorentz Transformations to this problem without using the conclusions he (Einstein) seeks to prove by using them. This problem was meant to establish Einstien's logic for using the Transformations in the first place.

I would hope that anyone who wants to understand my point would actually read this book. The section on Special Relativity is only 64 pages long in the copy I have so it is an easy read if you quickly comprehend what he wrote.

7. Jan 27, 2010

### Mentz114

DanRay
1. Is the light source on the embankment or the train ?
2. see bold - whose frame ? There is no absolute simultaneity between spatially separated events.

Your obsession with this is puzzling. Putting numbers into the formula cannot say anything different from what the formulae say.

Also, we're discussing principles here, so it's absurd to think that 1 sec interval is different from a 1 nano-sec interval in relativity.

8. Jan 27, 2010

### Ich

There is nothing based on Lorentz Transformations. Einstein uses the postulates to derive the transformation.
Transformations are used to get from one coordinate system to another. As JesseM rightly said, all your quotes and numbers are dealing with only one coordinate system. So it is absolutely impossible that the reasoning so far could be based on a certain transformation. Therefore your answer to JesseM doesn't make sense.
Then, I don't see at all what you're getting at with the numbers. They seem correct, and irrelevant.
Further, your comment I quoted could be read as a claim that all of SR is based on the inability of a human observer to resolve short timescales. I can't imagine anyone seriously claiming such things, so there must be a misunderstanding.

Again, could you provide us with a clue as to what you're trying to convey? If you believe to have spotted an error in Einsteins popular account, could you point us to it?

9. Jan 27, 2010

### sylas

You can get the book online, in a pdf of an English translation, at relativitybook.com.

Meaning no offense, but your points are incoherent; and in so far as they are comprehensible, they are incorrect and show that you have not understood the book.

As others have said, the issue is not about measurement accuracies. Einstein shows in the book a popular exposition of how to get the Lorentz transformations, using simple language and the basic assumption of constancy of the speed of light for all observers. You can assume all times and distances are known to complete accuracy.

Cheers -- sylas

10. Jan 27, 2010

### JesseM

I didn't say that he would have "used time dilation to prove time dilation", I was just saying that the point of the thought-experiment in section 7 was to compare the distance/time measured in the embankment frame with the distance/time measured in the train frame, so if your own problem doesn't even refer to the train frame it kind of misses the point. Looking on section 7 of this 1920 edition of the book online, you can see that Einstein doesn't derive any specifics about time dilation or length contraction in section 7, his point in section 7 is just that if you assume rulers and clocks work the same way they do in classical Newtonian physics (no time dilation, length contraction, or disagreements about simultaneity) then you'd reach the conclusion that the train observer would measure the light to travel at w = c - v, and he points out that this is in contradiction with the assumptions of relativity. So, he's using this section just to show the negative conclusion that rulers and clocks cannot work the same way in relativity that they do in Newtonian physics, but he doesn't derive how they actually do behave until later. Your own numbers do not even show what speed the train-observer would measure for the light using Newtonian assumptions, so like I said, missing the point.
That's wrong, in early sections of the book he makes clear that whenever he talks about "systems of coordinates" or "rigid reference bodies", they are ideally to be defined by noting positions on a system of rigid rods (see section 2) and clocks (see the last paragraph of section 3). So, for example, when he says "People travelling in this train will with advantage use the train as a rigid reference-body (co-ordinate system)" in the opening of section 9 where he talks about the relativity of simultaneity, he is referring to measurements made on such a system of rigid rods and clocks. Also note that in section 8 when introducing simultaneity, he talks very specifically about measuring the time of events on a system of clocks, and defining the time of an event in terms of the reading on the clock that was in the event's immediate vicinity when it happened:
The importance of defining time in terms of readings on clocks which have been "synchronized" using the assumption that light travels at c in the clock's own rest frame (this is known as the Einstein synchronization convention) is also given central importance in Einstein's 1905 paper where he first introduced special relativity--take a look at section 1 of that paper, "Definition of Simultaneity".

Last edited: Jan 27, 2010
11. Jan 28, 2010

### DanRay

Most of you seem to think I don't understand what I'm talking about or you think everything I have said is obvious. I do know that Einstein had already developed and had many discussions about time dilation and foreshorting etc. What I also know is that in these early sections of his book "Relativity" he is trying to demonstrate his reasoning path to using the Lorentz Transformations and all that stems from them. And I am also well aware that Einstein could have put these or any numbers into this thought problem and determined the correct answers. My original thought was that it would be obvious from my demonstration that the only thing that would perceptively vary would be the arrival distance differences and the time of arrival for the "tip of the ray" (Einstein's own term) to the point on the embankment and then the very fast moving carriage. We all agree and of course Einstein already knew that the speed of light would be measured the same in both places. I believe that it was invalid for him to put c into a standard addition of velocities problem that he knew requires the possibility of variance for any value in the problem. The differences I refer to for time of arrival and position relative to the embankment at arrival of the tip of the ray can only be observed once for any continuous ray of light. From then on the speed (and perhaps a red shift) are the only thing that can be measured.

Here is a side issue that I think is related to this thought problem. Did anyone ever wonder why only one of the changes indicated by the Lorentz Transformations can happen with each permutation and which phenomena (time dilation, forshortening, apparent speed) is indicated is dependent on which value is the unknown. Can anyone explain why only one is true at a time within the same set of facts?

Last edited: Jan 28, 2010
12. Jan 28, 2010

### DanRay

By the way, that 1920 on line version of relativity I'm sure reads the same as these sections in my 15th addition version which is still in print, copyright 1961 by the Estate of Albert Einstein. My copy is a paperback Published by Three Rivers Press (\$8.95 US). The 15th edition first appeared in 1952 and has a fifth appendix titled "Relativity and the Problem of Space" which was new in this edition. If you haven't read that I think you would all find it very interesting. I don't know which appedicies appear in the 1920 version, I have yet to check that.

13. Jan 28, 2010

### sylas

The things that depend on the observer -- that vary depending on who is measuring -- include
• The distance from the location where the tip of ray starts out to the the location where it is observed.
• The time it takes from when the tip of the ray starts out to the time when it is observed.
• The time showing on the passenger's watch at the instant when the ray starts out.

These things are respectively
• The relativity of distances.
• The relativity of elapsed times.
• The relativity of simultaneity.

These differences are nothing to do with measurement accuracies. They are all a consequence of the Lorentz transformations. They are explained in the chapters 8, 9 and 10, after the chapter 7 which first shows that the conventional Galliean transformations are inadequate. In chapter 11 the Lorentz transformations themselves are introduced.

As Einstein shows, the Lorentz transformations be shown to follow by accepting that the speed of light is a constant, and that there is a linear transformation between times and distances in two frames. The book basically shows how this works, without using too much maths, because it is intended for teaching relativity to interested novices. The mathematical derivations of the Lorentz transformations is in appendix 1.

Cheers -- sylas

Last edited: Jan 28, 2010
14. Jan 28, 2010

### JesseM

But in section 7, he is not actually giving the reasoning that leads to the specific equations of the Lorentz Transformation, he is just making the negative argument that if we assumed that the train observer's rulers and clocks bore the same relation to the embankment observer's rulers and clocks as they do in standard Newtonian physics, then if the embankment observer measured the light to move at c in his frame, the train observer would measure it to move at w = c - v in his own frame, which would contradict one of the basic assumptions of relativity which Einstein had mentioned earlier. Do you disagree that he was just making this type of negative argument (a sort of proof-by-contradiction that Newtonian assumptions must be wrong if the basic assumptions of relativity are correct) in section 7?
c appears in classical electromagnetism as the speed of all electromagnetic waves, it already had a known value. But classical physicists assumed that the equations of classical electromagnetism could only work precisely in one preferred frame (the rest frame of the luminiferous aether), and that anyone moving at velocity v relative to this preferred frame would measure electromagnetic waves to travel at c+v or c-v relative to themselves. Einstein was exploring what conclusions would hypothetically follow if we instead started out with the assumption that the laws of classical electromagnetism must work precisely in every inertial frame, so that every frame would measure an electromagnetic wave to move at c (then once he had found out what conclusions would hypothetically follow from this starting assumption, it would of course be an experimental matter to see whether the conclusions match up with what is actually seen in nature). Since this is his basic starting assumption, it doesn't make much sense to say that he should have considered "the possibility of variance" in c.

15. Jan 28, 2010

### DanRay

Sylas, you are alway kind to my misteps and I appreciate that. I am keenly aware of everything that you say, but you are attempting to introduce the conclusion as the reason to head toward it. All three of the things you mentioned are as you have said yourself derived from the results of the Lorentz Transformations and I think therefore cannot be used as an argument for employing them before thay are introduced. That is where we are with Einstein's section 7 problem. Please think about what I said about putting a constant into an equation that is designed for variables. In this old standard equation and it's variations any one of the values can be the unknown and each one varies acording to the combined values of the other two. We know Einstein already knew that c was invariable so I think he should have known that it didn't belong in the addition of velocities problem as a presumed variable.

Do you have any thoughts about why you can only derive each of the things you mentioned from one variation or other of the Transformations. In other words if the unknown is the speed there is no dilation or forshortening,you use the measured values for those two and end up with your "Relativistic speed". If the unknown is the distance you use the measured speed and and time and the result is "relativistic distance". If the unknown is the time you use the measured values for the other two and you have "relativistic time". And this can all be for the same event in the same frame with three different observers with the different combinations of information mentioned.

Cheer to you too. DanRay

Last edited: Jan 28, 2010
16. Jan 28, 2010

### Mentz114

DanRay,
have you read 'On the Electrodynmaics of Moving Bodies' ? Length contraction and time dilation are shown to be a consequence of the invariance of the speed of light.

I have to say I find it difficult to interpret this

17. Jan 28, 2010

### sylas

That's actually backwards. All the three things mentions are consequences of the invariance of the speed of light for all observers.

That is why they are introduced where they are introduced, to motivate the subsequent derivation of the Lorentz transformations. The argument for relativity of time, or distance and of simultaneity stands quite apart from the transformations, and that's partly why you will have problems putting numbers to chapters 7 to 10.

The way Einstein develops the explanation here doesn't depend on specific numbers. You can put them in, of course, but you don't have all the tools to give numbers for the train passenger given numbers for the observer on the embankment.

The book itself shows that you derive each of the three points without using the transformations at all. You only need that the speed of light is constant for all observers.

Given this, you can show that time must be relative, and that distance must be relative, and that simultaneity must be relative, simply by showing that they cannot be equal for the two observers. In these initial chapters, that is done without transformations and without calculating actually HOW MUCH difference there is.

Felicitations -- sylas

18. Jan 29, 2010

### DanRay

Yes I read it a long time ago but I don't currently have it. It may be online, I haven"t checked. I am currently reading Poincare's "Science and Method" first published in 1908. He also used Lorentz Transformations in ways simular to Einstein but he never did give up the ether concept. However he did deal with the changes derived from the transformations with conclusions much like Lorentz's own.

By the way I believe if Einstein were still with us he would have a sense of humor about my approach even if he thought it was comical and futile. He welcomed debate and opposing theories and never debased those who disagreed with him. And if you think my thoughts are an attack on Einstein and Relativity, think again. As far as I am concerned his 1905 achievements still rank as the single most amazing thing that any human has ever acomplished and probably will stand for a long time to come if not forever.

19. Jan 29, 2010

### chronon

There are two things I would mention
1) Science works along the lines of:
Devise a theory (somehow), make predictions from the theory, see whether these agree with experiment. The fact that the derivation of the predictions may depend on the theory doesn't matter.

However, if you try to derive the theory from observations or from basic principles, then you are likely to get into arguments about whether the derivation depends on the theory.

2) The train thought experiment is one dimensional, and I doubt whether it is possible to untangle the effects of length contraction from those of time dilation - to do that you also need a light beam going perpendicular to the direction of motion

20. Jan 29, 2010

### sylas

You seem unwilling to accept that you haven't yet actually given any reason to think you have some new insight that everyone here has missed.

Your latest post adds absolutely nothing to the substance of discussion; only lots of private reasons why we should all take you seriously, and unjustified false assumptions about everyone being refusing to consider new ideas. But I am taking you seriously. I do that by dealing with the actual substance of your points. But your latest post doesn't say anything of interest to the actual point.

The fundamental premise of special relativity is that light moves in straight lines at the same speed, for all observers. You appear to accept that. The Lorentz transformations follow from this.

Real science is perfectly allowed to question the hypothesis that the speed of light is constant for all observers. But you don't seem to be doing that. Is that right?

If so, there's a problem; because the Lorentz transformations follow as a mathematical consequence of the invariance of lightspeed for observers in Euclidean 3D space.

And a second post...
The various simple thought experiments with the train go into all three spatial dimensions, and the derivation I cited derives the full Lorentz transformations in 3D space from these thought experiments.

Cheers -- sylas

21. Jan 29, 2010

### JesseM

Did you read my previous post to you? (you never responded to it...) I gave you a link to the 1905 paper online at the end.
And what is that reason?? You certainly haven't given any clear summary of what your objection is. Do you understand that in the book you refer to Einstein was not trying to prove that relativity must actually be true in the real world--instead he was just exploring the logical consequences that would follow if the basic postulates of relativity (namely, the postulate that all inertial frames see the same laws of physics, and the postulate that all inertial frames measure light to move at c) were true? Once he has shown what consequences would follow if that were true, then of course it becomes a matter of experiment to check if those consequences actually match up to what we see in the real world.

22. Jan 29, 2010

### DanRay

I apologize for the personal digression about my Daughter. I spent the day today at the funeral of a dear friend and and at such times my daughter's untimly death and thoughts of her loom heavy in my mind.

I do not agree that the Lorentz Transformations follow in the manner you say. I am well aware that they were formed by Lorentz for a different purpose than Einstein adapted them for use with special Relativity. Lorentz and Irish Physicist Fitzgerald both tried to explain away the ether experiment of Michelson and Morely by postulating that "each body of light foreshortens when it moves against the stationary ether" when the instrument is set up to measure light parallel to the earths movement in orbit. Einstein with his new insights into light rejected the concept of an ether and Lorentz's idea of foreshortening and adapted the equations to a different purpose. Poincare did some of that too but he actually believed in the ether to the end of his life and had different reasoning for both relativity and the Lorentz Transformations.

I have no argument with Einsteins two Special Relativity postulates I think they are among the most brilliant of scientific insights ever. But I don't think you can get to the three main consequences by reasoning alone. I believe Einstein derived them from the Lorentz Transformations applied to problems just like this one. It is easy to see from the equations themselves that if they are valid then all of these things are also true.

My original point with this problem was that if you could demonstrate with a different approach to it that the speed of light is actually measurabe as c (the mirror gives you a two way trip from the source to the carriage and back to the source) and from that and from the
known distance to the back of the train derived from its speed, you can by simply timing when it gets back to the source calculate that it did the round trip at exactly c. You then have the answer to Einstein's question with that addition of velocities problem whether you accept my statement that Einstein misused it or not by putting what he knew was a constant into a problem that required it to be variable. If there is any doubt that you should know where the mirror would be when "the tip of the ray" hits it then you are ignoring that the constancy of the speed of light is one of the most accurate measuring devices for distance known to man. The timing itself will verify that your expectations of where that will be are correct.

23. Jan 29, 2010

### JesseM

But Einstein's point was not about experimental confirmation, it was about the logical consequences if it is true that light moves at c in every frame (once we have derived these logical consequences, only then do we turn to the experimental question of whether the derived predictions actually match with experimental data). Actually performing the experiment is irrelevant here. What's more, you once again ignore the issue of what would be measured for the speed of the same light ray in the train's rest frame, and the whole point of section 7 was that if the train observer's rulers and clocks were unchanged as observed by the embankment observer (as in Newtonian physics), and if the embankment observer measured the light to move at c and the train to move at v, then this would logically imply that the train observer would measure the light to move at c-v.

Do you understand what a proof by contradiction is? If so, you can see that this is a proof by contradiction that the assumption of the train observer's rulers and clocks being unchanged as measured by the embankment observer is not compatible with the postulate that light moves at c in every frame. That was the entire point of the train example in section 7, and your example ignores the train observer's measurements entirely, so you miss the point.

24. Jan 29, 2010

### DanRay

Sorry JesseM, I tried to reply to you earlier and had problems with my internet service and today I have been otherwise occpied. I haven't yet tried to use your link but I have read all of it before. I think Einstein gives a very good accounting of all of his Relativity thinking in this book. I do realize that he always called it a theory until the day he died but he defended it passionately as well. It only gained inflexible status in the hands of later proponents. Einstein loved to debate and did it often with humor and grace along with his brilliance and insight. If you read the answer I posted earlier to Sylas you will get a better idea of where I'm coming from.

So to be clear I don't have a bit of doubt about the validity Einstein's postulates. I do have serious doubt about the validity of the Lorentz Transformations and I have reasons that I will attempt to demonstate for anyone who is interested that apply to several other thought problems that use them but I can't do it until later. I must get some sleep now and I will be busy all weekend so it might be next week before I can post much more. Regards; DanRay

25. Jan 30, 2010

### sylas

That's why everyone is sure you haven't understood this yet. This is not a matter for informed disagreement; it stands as a mathematical theorem.

The scientific questions are still open, but the mathematical ones are not. You can never prove a scientific theory is correct; you can prove a mathematical theorem. And it has all the status of a theorem that the Lorentz transformations follow from the postulates.

The very book you are citing shows the relativity of time, of distance and of simultaneity follow from the postulates by reasoning alone; and without even using the Lorentz transformations. That is the whole points of chapters 7 through 10, which establish that the postulates require the relativity of time, distance and simultaneity. THEN you get to chapter 11, which introduces the Lorentz transformations.

This still appears to miss the whole point of the chapter, which is to compare measurements of the two different observers, and show that they must give different results for durations, and distances and simultaneity. It also still appears to mix up the notion of accuracy of measurement with the derivation of qualities being measured. Measurement accuracy has nothing to do with it; this is a distraction. The point is the thing being measured, which we can assume is measured as accurately required... just like we can assume trains moving at 60% light speed.

Cheers -- sylas