Is there a simple way of deriving Lorentz transformation?

1. Mar 21, 2013

dailin223

Is there a simple way of deriving Lorentz transformation?
I don't find the typical derivations in text book so convincing, which seems to use too many intuitive postulations...

2. Mar 21, 2013

bobc2

Last edited: Mar 21, 2013
3. Mar 21, 2013

strangerep

SR can be derived without the Light Postulate (i.e., without assuming the speed light is a universal constant in any inertial frame). But a lot more math ability is needed to grind through the derivation.

IOW, we merely introduce the concept of an inertial observer (who feels no acceleration, and can set up a Euclidean coordinate system). Then the Relativity Postulate -- that the laws of physics are identical in all inertial frames -- implies the Lorentz transformations (together with translations in space and time, and uniform dilations), together with the (possible) existence of a constant limiting speed, whose value must be determined experimentally.

But there's some serious math to grind through to get there.

In contrast, the Light Postulate simplifies the math considerably, hence tends to appear more often in textbooks.

So it's a bit of a tradeoff between the nonintuitive Light Postulate vs some heavy math using only the (imho, quite intuitive) Relativity Postulate.

Last edited: Mar 21, 2013
4. Mar 21, 2013

bcrowell

Staff Emeritus
I disagree. Two counterexamples:

http://arxiv.org/abs/physics/0302045
http://www.lightandmatter.com/html_books/lm/ch23/ch23.html#Section23.1 [Broken]

The latter is my own presentation. Some straightforward algebra details are relegated to a homework problem.

Last edited by a moderator: May 6, 2017
5. Mar 21, 2013

strangerep

Umm, you quoted 3 of my sentences. Which one(s) do you disagree with?
That paper by Pal has a few problems/deficiencies. He relies early on rigid rods, etc, to get linear transformations. Such a motivation is only intuitive at ordinary lab scales, but less so on the very small and very large scales. Later, his argument about the sign of $K = \pm 1/c^2$ is rather weak. Also, for the argument to be complete, one should thoroughly analyze the properties of a body in the limit as its velocity relative to an observer approaches $c$, and show that they coincide with observed properties of light.
I haven't read that before, so I'll do that before replying. (BTW, has that been refereed? -- I'm wondering about the PF rules...)

Edit: Ok, I've read it. I see you rely on a principle that time depends on the observer's state of motion. This is a nonintuitive postulate -- which you justify by appeal to experiment. OK, that's another way of reaching the result, I guess. Personal tastes differ.

Last edited by a moderator: May 6, 2017
6. Mar 21, 2013

Staff: Mentor

It might help if you could tell us which postulates/assumptions you find troublesome. It would be a shame if we were to point you at one the derivations that you don't like; better to identify what you don't like so that we can avoid it or justify it.

7. Mar 22, 2013

dailin223

I read several derivations, and every one of them involves using light signals find the coefficients of Lorentz Transformation, is there a derivation not referring to light signals...

8. Mar 22, 2013

strangerep

What level of math are you comfortable with?

E.g., are you comfortable with the math in the paper by Palash Pal that Ben referenced earlier? I.e., http://arxiv.org/abs/physics/0302045

9. Mar 22, 2013

Staff: Mentor

There are, but in many ways they are more complex, less intuitive, and less directly connected to the physical situation that we're analyzing. Light signals show up in the physical derivations of the Lorentz transform because of Einstein's second postulate about the constancy of the speed of light.

So I think your question may be: Why are we so willing to accept this postulate?
There are two reasons:
1) It is supported by experiment and observation; it seems the world really does work this way.
2) The simplest interpretation of Maxwell's laws of electricity and magnetism suggest that electromagnetic radiation, aka light, should behave this way. (There are other ways of interpreting these equations, but they end up making additional assumptions and get more and more unwieldy as we try to reconcile them with experimental observations).

10. Mar 22, 2013

Histspec

One can also derive it directly from the following experimental results, as shown by HP Robertson and others:
http://en.wikipedia.org/wiki/Test_theories_of_special_relativity

*The Michelson-Morley experiment demonstrates the direction independence of the speed of light with respect to a preferred frame.

*The Kennedy-Thorndike experiment demonstrates the independence of the speed of light on the velocity of the apparatus with respect to a preferred frame.

*The Ives-Stilwell experiment demonstrates the relativistic Doppler effect, and thus the relativistic time dilation.

If you combine those results, you arrive at effects like time dilation and length contraction, as well as the complete Lorentz transformation.

11. Mar 22, 2013

bcrowell

Staff Emeritus
I gave two counterexamples in #4.

12. Mar 22, 2013

dailin223

Thank you guys so much, I really appreciate Ben's ref. of Pal's paper!

13. Mar 23, 2013

haael

The one I like most is:
$\Lambda \Lambda g = g$
Where 'g' is the Minkowski metric. This basically states: "Lorentz transformation is an isometry of a 4-dimensional flat space with 3 spatial dimensions and 1 time dimension."

14. Mar 23, 2013

strangerep

And you declined to respond to my post #5. (A discussion has to be 2-way in order to be a discussion.)