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Guest post – Jeff Wyss: The Relativistic Train April 30, 2008

Posted by dorigo in Blogroll, mathematics, physics, science.

Jeff is a physics professor at the University of Cassino, and a long-time colleague and friend of mine. He worked in the SLD and CDF collaborations as a particle physicist, but later moved on to study radiation damage on silicon detectors for particle and astroparticle applications.

Besides admiring him for his wicked sense of humor, which he uses to make the workplace around him always a pleasant place, I have the highest esteem of Jeff as a professor, because he is quite skilled in explaining physics concepts in simple terms. He always looks for the most intuitive way to understand things, as you might appreciate in the contribution he offers below.

The following describes a very elegant and simple derivation of the relativistic formula for the addition of velocities, w = (u+v)/(1 + uv/c^2).

It is due to David Mermin. I fell in love with it and have been telling it for the past four years now to the students of my general physics course. The students are first year telecommunications and electrical engineering students. Before sitting in on my course all of them have heard about Einstein and most of them heard the expression “the velocity of light is constant”. I do not have the time to discuss special relativity in detail. My course is quite traditional. I discuss reference frames, inertial frames, Galilean transformations and covariance of Newton’s laws. I then point out that when describing mechanical waves the frame that is stationary respect to the medium is a special reference! In particular the wave motion can be made to disappear by moving respect to the medium with a velocity equal to that of the wave. It is clear at this point that the constancy of the velocity of light cannot be understood by assuming Newton’s laws and then modeling light as a mechanical wave in a medium (the ether). I then restate the constancy of the velocity of light and begin Mermin’s derivation.

The derivation uses:

  • only one reference frame (no use of Lorentz transformations),
  • simple kinematics (always good to brush up on),
  • the constancy of the velocity of light (something that every telecommunications and electrical engineering student should know),
  • the idea that some things are invariant; i.e. while many quantities are relative, observers will agree on some absolutes.

Consider a train of length L moving along the x-axis at a constant velocity v respect to an inertial frame of reference (the observer watching the events unfold). At the trailing end of the train a loaded gun is aimed in the forward direction and fired at time t=0: the bullet and flash of light emerge and travel in the forward direction with different speeds: w the velocity of the bullet, c the velocity of light. A mirror at the front end of the train reflects the light back towards the advancing bullet. Let f be the fraction of the length of train that the reflected light travels before meeting up with the bullet. The constancy of light (Einstein’s dictum) tells us that the velocity of light in the forward direction is equal to the velocity of light in the backward direction; i.e. c_F = c_B = c.

The space-time plot looks like this:

Let t_F be the time for the light flash to reach the forward-going mirror and t_B be the time the reflected light needs to return from the mirror and meet up with the forward-moving bullet. Simple kinematics allows us to label the space-time plot:

Simple algebra:

It is important to note that the expression for f we just obtained is valid if the velocity of light in the forward and backward direction are equal. Note:

  • A classical pre-Einstein physicist would say this expression is valid only if the observer is stationary respect to the ether frame.
  • On the other hand Einstein says that any inertial observer would use the same velocity of light; i.e. Einstein tells us that this expression is valid for any observer (generic inertial frame).

Following Einstein we consider a particular observer (frame), one that is moving along with the train. For this observer the velocity of the train is v = 0. For clarity let us use the symbol u to indicate the velocity of the bullet with respect to this observer; i.e. with respect to the train.

Suppose the train has 10 windows and the reflected light and the bullet meet up at the third window from the front (f=0.3). It is important to realize that all observers will agree on the value of f. The fraction f is an invariant!

The constancy of the velocity of light allows us to impose the invariance of f the following way:

Q.E.D. !


1. goffredo - April 30, 2008

I read Mermin’s discussion about 10 years ago in an appendix of a book of his that I checked out from a library. I think it was entitled “Einstein’s mirror”. I have a good memory of the book. There were a few more original contributions to explaining relativity. Three cheers for Mermin. HE was good!

I did not add anything. I think I even used his notation. I suggested to Tommaso to post it as EVERY physics colleague I asked, and many teach (that is why I asked), did not know it. My students like it very much as they find the reasoning easy to follow and can all pin-point to the role of Einstein’s dictum. The spin-off discussions (Newton’s limit; special cases such as u=c and v=c; … make them interested to know more. Only at this point do I mention briefly Lorentz trasformations as it allows me to recap on Galileo trasformations and the Newton way of adding velocities. I’ve asked the demonstration several times in oral exams because I think it is a healthy excercise to reason out the kinematics from the simple space-time diagram.

I anyone wishes to have the *.ppt file and/or the *.doc files just send me an email (goffredov@yahoo.it)

2. Doug - April 30, 2008

Hi Jeff Wyss,

Isn’t this true only if the speed of light can exceed c as in variable speed of light [VSL].

If MAX(speed of light)=c, then c+v=c, for any magnitude of v.

3. Per - April 30, 2008

Ha! Nice derivation. Thanks for sharing!

4. Nick - May 1, 2008

The key factor of this derivation is the fact that even for the observer, the speed of light is the same off the train as it is if you were riding on the train, correct?

Good stuff! I just got Einstein’s book “Relativity” and now I’m even more excited to read it. 🙂


5. goffredo - May 1, 2008

Yes Nick. That is the key factor. It stands out so clearly and that is why I like the derivation.

By the way as I have a bad memory I decided it best to look on the web. Indeed the book where I found Mermin’ derivation is NOT by Mermin.
Here it is

6. goffredo - May 1, 2008

Hi Nick
yes the key factor is the constancy of the speed of light namely that the speed is independent of the motion of the observer and the light source. The velocity “addition” formula encodes this; e.g.
U “plus” C = C; (no boost will make light go any faster).
Indeed even C “plus” C = C.

By the way as I have a very bad memory I decided to check on the web and indeed the book where I found Mermin’s derivation was NOT written by him!
Here it is

7. DB - May 1, 2008

Very nice, Jeff. And great diagrams.

Looking at Einstein’s 1905 paper, it’s interesting that he deliberately takes a complicated approach to deriving the Lorentz transformation. In a footnote he even refers to a short cut, but one which would not have so clearly shown the compatibility between the principle of relativity and the constancy of the velocity of light.
Of course, once he gets the Lorentz transformation, his derivation of the addition of velocities just relies on a neat application of Pythagoras’s theorem. He also points out that it is an example of how Lorentz transformations form part of a group (what we now call the Poincare group, the foundation of quantum field theory) – little did he realise at the time just how significant this group property would become!.
It’s interesting to compare this with his neat derivation of the addition of velocities in his book “The meaning of relativity” written seventeen years later. Once he introduces spatial rotations as a special example of a Lorentz transformation, he derives the addition of velocities in just one line!

8. goffredo - May 1, 2008


I VERY WARMLY (hotly) suggest “SPACETIME PHYSICS” by Taylor and Wheeler. The analogy, both formal and fertile (useful in describing reality), between ordinary (circular) trigonometric functions and hyperbolic “trigonometric” functions is easy and every student should know it. But most do not.

In particular a physics student should know it. He would appreciate how the simple trigonometric law of “addition of slope” can be seen to be related to the law of “addition of velocity”. This is discussed in great detail for several pages in SPACETIME PHYSICS because the book is elementary (but not easy). The final sensation is that if one assumes knowledge of hyperbolic functions then the connection between euclidean rotational transformations and psuedo-eucliean space-time (lorentz) trasformations can truely be made in just one or two lines maximum, and the law of addition of velocities is immediate.

Ah the power of mathematics!!! People react to its power in two different ways. Some get excited and breathless when they see how great physics concepts, made at great expense in terms of experimental and cognitive effort, can be expressed elegantly and very succinctly. Others, mainly professional mathematicians and an annoying number of theoretical “physicists”, do not appreciate real Physics because they see the “physics” as trivial.

9. Nick - May 1, 2008

One thing I am wondering now is how we can detect a red or blue shift in the light? I’m assuming it must still travel at the speed of light, but it’s wavelength must contract or expand.

Here’s the book I have: http://www.amazon.com/Relativity-Special-General-Albert-Einstein/dp/0517884410

In the back it shows his derivation of the Lorentz Transformation, pretty cool.

That IS pretty interesting, It’s amazing how things change as we figure out new things.


10. dorigo - May 2, 2008

hi nick,

this is jeff’s job here, but in short, we know the wavelength of ionized hydrogen and other elements, so we measure shifts from the standard values very accurately.


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