# B Derivation of time dilation without light clocks

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1. Sep 3, 2016

### Tazerfish

In the way I was taught about special relativity, time dilation is like the fundamental building block from which you derive things like relativistic mass and length contraction.

So it has always struck me as quite odd, that the derivation of time dilation (in some sense the basis of special relativity) uses something as abstract as light bouncing up and down in a box to measure time.
Something that has never been built in reality to my knowlege.
I don't have any problems with this proof.It is very elegant and simple.
Yet there rarely is just one nice derivation or proof for such things
and I have never seen any alternetative for this case.

Have you ?

Is there a way to derive it by considering some other process ? :)

2. Sep 3, 2016

### A.T.

That's a confusing way to think about it. Time dilation and length contraction are on the same footing, as consequences of the Lorentz transformation.

Derive from what? From the two postulates? The key thing in them is the source independence of light's propagation, so I don't see how you can avoid using that.

3. Sep 3, 2016

### Tazerfish

I thought about proving that some other process would slow down as you approached the speed of light.
Independant of the consideration of light clocks.Like an oscillation of some sort.
But the highlighted part in your post has been key.The derivation seems much more intuitive now.
Thanks

4. Sep 3, 2016

### Ibix

Pal (https://arxiv.org/abs/physics/0302045) derives transforms using the principle of relativity only (and an assumption of linearity in the transforms, and assumptions of homogeneity and isotropy of space). There's no mention of light anywhere. He ends up with two options - Galilean relativity and Einsteinian relativity. You can then eliminate Galilean relativity by experiment.

Or one can postulate that spacetime is a 4d entity obeying Minkowski geometry with c as a scale factor between spatial and temporal directions; everything falls out of that, and once again you can verify by experiment.

In practice, no one uses light clocks as there are perfectly good (precise and reliable) atomic clocks. The beauty of the light clock for thought experiments is that one doesn't need to know how velocities transform in general. One simply has to postulate that the speed of light is frame invariant and that, if I see you doing +v, you see me doing -v. You can, in principle, do thought experiments with a pendulum clock, but you need to know how to transform the velocity of the pendulum - and the full velocity transforms are a bit much to pull out of thin air.

5. Sep 3, 2016

### Orodruin

Staff Emeritus
This is not a very pedagogical nor modern way to teach SR. I doubt time dilation was ever a fundamental building block in any reasonable approach.

Also note that relativistic mass is an antiquated concept that is generally not used in modern treatments.

This is just one of the mote heuristic ways of deriving time dilation. The more mathematical approach is to just derive it straight out of the Lorentz transformations.

6. Sep 3, 2016

### A.T.

...involving light.

7. Sep 3, 2016

### Ibix

Not necessarily, I think. Cosmic ray muons are tough to explain without time dilation - no light needed in that one.

8. Sep 3, 2016

### Tazerfish

I might have to add that we never went into it in great detail.
We didn't have a lot of time for the topic and our mathematical knowlege was quite limited at the time.
I don't blame my teacher for going the quick and ditry route, we simply didn't have the time to do it right.
Sometimes it is better to teach something that is not entirely correct and to just clear up the misconceptions afterwards.

9. Sep 3, 2016

### A.T.

Proving based on what?

10. Sep 3, 2016

### robphy

Try this [apparently] not well known method called the k-calculus by Bondi, which I recently mentioned in another thread.

https://www.physicsforums.com/threa...n-these-approaches-to-sr.883403/#post-5554866
"There the metric and the Lorentz transformation are not in the foreground of the discussion.
It is the principles of relativity, with focus on the radar method and the Doppler effect.
(Secretly, the approach is using the eigenbasis of the Lorentz Transformation.)

https://archive.org/details/RelativityCommonSense
https://en.wikipedia.org/wiki/Bondi_k-calculus
"
(Bondi)
https://books.google.com/books?id=hxYqGQUGXewC&pg=PA88&lpg=PA88&dq=bondi+"value+of+k"+common+sense
You probably have to go back to see the development of the diagram,
and ahead to see how k is related to v.
k is the Doppler factor

(D'Inverno)
https://books.google.com/books?id=hQdh3SVgZ8MC&pg=PA24&dq="bondi+factor"

Here's an ancient post of mine on it:
https://www.physicsforums.com/threads/time-difference-light-emitted-vs-observed.113915/#post-934989

Last edited: Sep 3, 2016
11. Sep 5, 2016

### stevendaryl

Staff Emeritus
I do not particularly prefer the abstract way of deriving the Lorentz transformations starting with homogeneity, isotropy, etc. I think it's one of those things where if you already understand relativity to a certain level of competence, then you can appreciate such a derivation, but I think if that were the first derivation I had seen, my eyes would glaze over, and I would have a hard time understanding why anyone would be interested in the topic. I found, as a young man (I think I was about 13 or 14 when I first went through it) the derivation in terms of rods and clocks and light signals really captured my imagination, and made me want to learn physics. I doubt very seriously whether I would have been similarly inspired by an abstract derivation. Sort of similarly, I find an introduction to quantum mechanics that starts with more-or-less concrete experiments such as the two-slit experiment to be a lot more inspiring than a derivation of Schrodinger's equation from principles of Galilean invariance and expectation values for observables. I'm not at all disparaging the more rigorous mathematical treatments, but in my opinion, it's useful to see both the rigorous development and a nonrigorous, but intuitive development. The worry of many physicists is that the intuitive derivations are always misleading or else have hidden (and often false or unrealistic) assumptions, and that students will get the wrong impression. I don't think that's a problem, as long as the student is told that they are only getting an incomplete picture, and that there is more to learn. If there were no student misconceptions, then there would be no need for a PhysicsForums.

Last edited: Sep 5, 2016
12. Sep 5, 2016

### vanhees71

Sure, I'd not use the abstract derivation for an introductory physics course or even at high school.

13. Sep 5, 2016

### Mister T

That's just one way of teaching it. One has to start somewhere. It's actually much better to start with the relativity of simultaneity and never mention relativistic mass. But that's just my teaching opinion.

You're taking the next step in thinking about it, rather than just leaving the lesson behind.

There may not have been any wrong doing. Teachers can, at best, begin the learning process. The student must go beyond the teacher's lesson if any real understanding is to occur. Unfortunately, for the last 15 years or so, the American primary and especially secondary systems of education have been focused on holding teachers accountable for student learning. A well-intentioned but misguided approach because it focuses on the teacher rather than the student. Some parents have taken this to mean that if their offspring are not learning, it must be the teacher's fault. (The more obstinate among them will maintain that learning did occur despite the poor teaching, but the substandard grades are due to the teacher's flawed evaluation methods).

It is not a thing to be proven. It is a thing to be demonstrated by observing the way real clocks behave. The light clock is just a teaching tool. Many if not most physicists dismissed it as flawed until its validity was demonstrated.

The phrase "approach the speed of light" can be misleading. First, it's the fact that you're observing something else move relative to you. And yes, the faster the speed the greater the effect, but the effect is there at all speeds. And it must be taken into account in cases where it makes a difference, such as the GPS satellites that are moving at only 0.001% of the speed of light.

14. Sep 6, 2016

### Battlemage!

I learned to derive it using light clocks too, but not in a class (in class they just said "this is the Lorentz transformation equations") . I just noticed the Lorentz factor looking like a side of a right triangle and went from there. But later on I learned a better way that really only assumes that the formula distance = rate x time is the same regardless of inertial coordinate system and requires only algebra (and probably some characteristics of space and time that I just took for granted).

I'll post it and you tell me if this might be better than the light clock way. Skipping no steps so that any mistake I make will be clear for someone to fix.

You have two reference frames, S and S' moving at some speed with respect to each other with their x and x' axes coinciding.

In S, you can shoot a beam of light in either direction along the x-axis. When you do, it's distance x is given by x = ct or x = -ct (depending on the direction). Likewise in S' you can do the same: x' = ct' or x' = -ct'. Setting each equal to zero gives: x - ct = 0, x + ct = 0 and x'-ct' = 0, x'-ct' = 0. Pretty straight forward so far. Then you can write them the other way: ct - x =0, ct + x = 0, and ct'-x' =0 and ct' + x' = 0.

Next I just assumed there was some functions A and B such that (1) x - ct = A(x' - ct'), (2) x + ct = B(x'+ct') and (3) ct - x = A(ct' - x'), (4) ct' + x' = B(ct + x).
Then add (1) and (2) together to get:

$$x = \frac{A+B}{2} x' + \frac{B-A}{2} ct'$$

And add (3) and (4) to get:

$$ct = \frac{A+B}{2} ct' + \frac{B-A}{2} x'$$

Then to make it easier to read, let

$$\frac{A+B}{2} = γ$$
and
$$\frac{B-A}{2} = ξ$$

leaving
(5)
$$x = γx' + ξct'$$
$$ct = γct' + ξx'$$

Then just remember that you have the inverse transformations, which would involve just swapping the sign and replacing the prime and unprimed coordinates:
(6)
$$x' = γx - ξct$$
$$ct' = γct - ξx$$

At that point all you have to do is find ξ, which is easy if you realize any object moving with uniform velocity is at rest in it's own frame, so there is always going to be a case where x' = 0. Which means that x/t in this case = v and is the speed at which S' is moving relative to S. Which then means you can solve for ξ by letting x' = 0 in the third equation:
$$x' = γx - ξct$$ with x'=0 gives
$$0 = γx - ξct$$
$$γx = ξct$$
$$γv = ξc$$
$$γ\frac{v}{c} = ξ$$

Then you just plug that in to the four equations in (5) and (6). Start with the first one in (5):
$$x = γx' + γ\frac{v}{c}ct'$$
Then substitute in x' and ct' from (6):
$$x = γx' + γ\frac{v}{c}ct'$$
$$x = γ(γx - γ\frac{v}{c}ct) + γ\frac{v}{c}γ(ct - γ\frac{v}{c} x)$$

Then just clean it up and solve for the last unknown.
$$x = γ(γx - γ\frac{v}{c}ct) + γ\frac{v}{c}γ(ct - γ\frac{v}{c} x)$$
$$x = γ^2([x - vt] + \frac{v}{c}[ct - \frac{v}{c} x])$$
$$x = γ^2(x - vt + vt - \frac{v^2}{c^2} x)$$
$$x = γ^2(x - \frac{v^2}{c^2} x)$$
$$x = γ^2x(1 - \frac{v^2}{c^2})$$
$$1 = γ^2(1 - \frac{v^2}{c^2})$$
$$γ^2= \frac{1}{(1 - \frac{v^2}{c^2})}$$
$$γ= \frac{1}{\sqrt{1 - \frac{v^2}{c^2}}}$$

And then you have the Lorentz factor, so just plug it in to (5) and (6). To get time dilation, just assume that x'=0 because the person looking at their clock will be at the location of their clock if their time is proper time:
$$ct = γ(ct' + \frac{v}{c}x')$$ at x'=0 is
$$ct = γct'$$
$$t = γt'$$

Anyway, how is that derivation? Obviously if you orient your axes a certain way, y = y' and z = z', so no issue there. You can pretty much derive all of it once you get to this point.

15. Sep 6, 2016

### robphy

I didn't look at all of the steps...
But you've basically used light cone coordinates... (the eigenbasis of the Lorentz Transformations). A and B are the eigenvalues, which are the Doppler factor and its reciprocal.
Bondi's method (from post in #10), which also uses this feature, is cleaner.
My recent Insight uses Bondi's ideas to reshape causal diamonds [and thus light-clock diamonds] (which visualizes the Lorentz Transformations).

[if you know about rapidities (the Minkowskisn analogue of angle), then with A is $\exp(\theta)$ and B is $\exp(-\theta)$, half the sum gives $\cosh\theta$ and half the difference is $\sinh\theta$ where $v=\tanh\theta$.]

Last edited: Sep 6, 2016
16. Sep 6, 2016

### Tazerfish

I just wanted to thank all the people who answered this thread. It has helped me a lot
MisterT brought up an important point: If you want to learn you have to think.
The easiest derivation may not teach you as much as a harder one just because you think you understood it all.
I agree with stevendarly that I wouldn't really have appreciated a rigorous mathematic derivation when I was first introduced to special relativity.
The light clock proof is truly very easy and quick but doesn't teach you as much as the other proofs in my opinion.
You usually jump into it with your "Galilean" intuition and then realize on this purely hypothetical construct that your intuition must be wrong for the speed of light to be constant in all reference frames.
And as a little extra the formula for time dilation/length contraction/(the Lorentz factor) pops out of this consideration.
That leaves behind a lot of confusion.I felt like we just stumbled across it.

The other proofs are in a way actually more comprehensible to me, since I really doubt I would have gotten the idea for the light clock one.
To first hear of the experiments at that time that proved the Galilean view was incorrect and then try to figure out about the properties of space and time and transformations you can do to get from one reference system to another is a far more intuitive though also more complex way to derive it.
But the extra bit of work is worth it (or at least was for me).

Three more things: 1 I am fascinated that you can supposedly derive the Lorentz transformation without considering light.(Sadly, I didn't really understand the link posted by Ibix)
2 I find it cool that while the derivations that have been brought up may look very different at first glance they are actually fairly similar.
3 I have to say I am no big fan of Bondi k-calculus for introducing stuff like time dilation.
The fact that you throw around k (the doppler factor) all this time without even really knowing how it changes with velocity or why it should have the inverse value when moving in the opposite direction made me quite upset.
(But I like the diamond-diagrams robphy made in this insight article.The method seems pretty useful)

17. Sep 9, 2016

### Battlemage!

That makes sense. To be honest though, I just find this the least mathematically difficult way to do it. Only algebra is required, and even more to the point, only the basic operations of addition, multiplication, and exponents are all that is needed. I am currently trying to expand my knowledge with SR, particularly in a mathematical way of looking at it, but I find there are subtle differences in how the geometry works that make it require a bit more thought than these high school Lorentz factor derivations.

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