# Postulate and special relativity

1. Jan 11, 2008

### bernhard.rothenstein

Please tell me if it is correct to state that
Everithing that is a result of the two postulate can be considered as a postulate. So we can postulate the existence of the length contraction in order to derive the Lorentz-Einstein transformations?
Thanks

2. Jan 11, 2008

### malawi_glenn

the lenght contraction arises due to Lorentnz-transformation, and Lorentz transformation comes from the two postulates of special relativity.

3. Jan 11, 2008

### HallsofIvy

I'm not certain exactly what you are asking. Mathematically, a postulate is "given" and anything that "is a result" of one or more postulates is a "theorem", not a "postulate".

4. Jan 11, 2008

### bernhard.rothenstein

special relativity terminology

Thank you. So if I consider that the invariance of the space-time interval is an invariant then length contraction and time dilation which are a consequence of it are theorems. Can I say that if we derive the Lorentz-Einstein transformations using one of them, then the derivation is a result of a theorem?

5. Jan 11, 2008

### HallsofIvy

Yes, that would be theorem. Anything that is derived from some other fact is a theorem.

(Frankly, I am not comfortable using mathematical or Logical terms in Physics. The "postulates" of relativity are themselves derived from experimental results. They are not "postulates" in the exact sense.)

6. Jan 11, 2008

### Shooting Star

I may be wrong, but to me it seemed like what you wanted to say was:

if A and B are postulates, and A AND B together implies C, then can we treat C itself as a postulate? The answer is no, because C need not imply A and B individually. This is basic logic, nothing to do with Physics.

7. Jan 15, 2008

### danime

In physics does not make sense to concern with postulates and theorems. There are experimental facts valid within certain conditions and consequences that makes the body of theory.
The experimental facts may be invalid in other conditions.
They are not postulates, first because they are not postulated, but observed.

Also, the derivations of the experimental facts are not theorems, because they can contradict the original facts and be logically correct. Example: The Abraham-Lorentz force are not trully causal but can be derived from the coulomb force and the relativity which is itself causal.

8. Jan 15, 2008

### Shooting Star

> They are not postulates, first because they are not postulated, but observed.

One of the very first "postulates" I came across in school was Avogadro's Postulate, stating that equal volumes of gases contain equal number of molecules.

What had been "observed" at that time -- the molecules? They were not even universally believed in at that time.

9. Jan 15, 2008

### HallsofIvy

Very true. However, the next part of physics, after observing the experimental results, is to draw logical conclusions from those observations- that is the part of physics that uses mathematics. It is common to accept the experimental results as "postulates" in the mathematical sense in order to do that.

Notice that "postulates" is in quotes. See my previous post.

10. Jan 15, 2008

### danime

It's interesting to cite Hamilton. In the first of series of papers in which he begun to make the optical-mechanical analogy and defines his principal function, related to the action function, and the W function, related to the hamiltonian, he says that the evolution of science as the quest of understanding the nature begins with the observation of facts, a deductive process. As long as we can see related things we can use the inductive process to imagine a theory correlating the things and then make predictions and make new experiments to test the theory.

We just expand our capacity to make previsions. We never understand anything and our theories are just methods to make previsions within certain scope. Mathematics is useful as long as it's the ideal tool to deal with patterns and we only try to correlate patterned observations. Those which don't seem to fit in the patterns we are used to or cannot be promptly reproduced are ignored. So a huge mass of observed things are ignored or not useful.

11. Jan 15, 2008

### bernhard.rothenstein

Thanks to all. Would be correct to start teaching special relativity as:
Accept the following facts
1. The laws of physics are the same in all inertial reference frames
2. The speed of light is the same in all inertial reference frames,
in order to obtain results in accordance with special relativity theory
without mentioning concepts like postulate or theorem?

12. Jan 15, 2008

### Shooting Star

Pauli added a third postulate: provided that the first two do not contradict each tother.

13. Jan 16, 2008

### bernhard.rothenstein

Pauli quotation

Do they?

14. Jan 16, 2008

### HallsofIvy

I agree. That would hardly be a "postulate" but a "proviso". Either they do or they don't.

15. Jan 16, 2008

### Shooting Star

As of today, the majority of scientists believe they do not.

"Proviso" is a much better term. I don't remember who mentioned the word "postulate", but he was a big shot. Pauli had said something to the effect that behind the two postulates, there lies the "tacit assumption of the third postulate."

16. Jan 16, 2008

### Shooting Star

First of all, the link cannot be followed, but there's a way to fix it, which I don't know. Perhaps a mentor will help.

Last edited by a moderator: May 3, 2017
17. Jan 17, 2008

### Fredrik

Staff Emeritus
I disagree with the attitude towards postulates in physics that some posters have expressed in this thread. Every theory of physics needs a set of unambiguous statements that defines the theory, just as "group theory" (mathematics) needs the definition of a group. Those statements are the theory's postulates. I don't care if we call them "postulates" or "definitions" or something else, but it's ridiculous to say that we shouldn't concern ourselves with these things. No matter what we call them, they are the starting point of the theory, and everything that the theory says can be derived from them. So of course it's important to make sure that they really are unambiguous and consistent.

Einstein's postulates were most certainly not experimental facts. Yes, they had some experimental support, but it's not like someone had proved that they were true. They are assumptions that attempt to define the theory. From a mathematical point of view, they do a terrible job, because they contain hidden assumptions and use terms that need to be rigorously defined to make sense (in particular "inertial frame").

Because of that, there may be a way to interpret them as contradictory statements. I don't know if there is, but I do know that there's also a way to interpret them as statements that certainly aren't. Those statements (together) are equivalent to this one:

"Space and time can be represented mathematically by Minkowski space"

This is the only "postulate" (or whatever you prefer to call it) that's needed. It's the statment that defines what special relativity is.

Now, about the "proviso" that the postulates must not contradict each other... I have spent some time thinking about what the hidden assumptions are and how to define an inertial frame without explicitly mentioning the Minkowski metric, and I believe that the definition of an inertial frame must look something like this:

"A inertial frame is a member of the only subgroup of the group of all global coordinate systems on the set of space-time events M, that map straight lines to straight lines and is consistent with Einstein's postulates"

I guess if we try to represent M with something other than $\mathbb{R}^4$, this group might not exist or not be unique. So the "proviso" can be thought of as a guideline to follow when we attempt to make the postulates rigorous. However, it's so absurdly obvious that it makes no sense to even mention it.

18. Jan 17, 2008

### Fredrik

Staff Emeritus
I would call them postulates and use them to derive theorems. But I would also explain the problems, and show how to turn them into unambigous mathematical statements. If you are unable to do that, there's another perfectly valid approach, which is probably more appropriate for less advanced students anyway:

1. Show them Einstein's postulates. Explain how the first one was motivated by Newtonian mechanics and the second by the Michelson-Morley experiment and Maxwell's equations. Also admit that they aren't very good from a mathematical point of view.

2. Use them to "derive" the Lorentz transformation non-rigorously. Skip every step that is difficult, if you can motivate it by saying that "it seems reasonable to guess that this is correct". (In particular, you don't try to prove that homogeneous Lorentz transformations must be linear. Just guess that they are).

3. When you finally arrive at the Minkowski metric, you explain that now we are able to guess that the entire content of the theory can be stated like this: "Space and time can be represented mathematically by Minkowski space".

4. Explain that it's completely irrelevant that we used sloppy proofs to get to this point, because we were just trying to find a good way to define the theory properly. Now that we have a definition, i.e. now that we actually have a theory (which we didn't before), it's up to the experimentalists to determine how well it approximates nature.

Last edited: Jan 17, 2008
19. Jan 17, 2008

### danime

My position is that if you think you can do physics by postulating and deriving logical conclusions as theorems and make this a theory you have never done physics.
There are so much possibilities that are cut by the experimental facts you cannot create nothing just by bare logic. Einstein already known what he was trying to do. The postulates are just a synthetic expression of a lot of things he learned.
It's interesting to point that the great majority of Einstein's though experiments leaded him to wrong conclusions. The ones those became famous due to its correctness were based on experimental facts.
One could say that in mathematics you have postulates and try to find the consequences and in physics you have the consequences and try to find the postulates. But the reality is that even in mathematics the majority of theorems had begun with the both the postulates and the conclusions lefting the work of create the connections. If you don't know this you have never done mathematics too.

20. Jan 17, 2008

### Shooting Star

I think the postulates of Physics can be treated exactly like the axioms of Mathematics, with the terms to be used defined as clearly as possible, until the predictions run into trouble. Then new ones have to be made. The making of a postulate takes a long, long time and several years of experiments or thinking.

(Sorry, don't have time now. Will discuss later.)

I would have loved to see you saying that to Pauli's face. He used shred up other physicists for things like this. Mincemeat man, mincemeat.

21. Jan 17, 2008

### danime

It's not a matter of agreeing or disagreeing. It's a matter of doing real work and seeing that to create an entire theory you can use already existent theorems or make yours as demanded, some are even physics theorems. But they are just small steps, like a post-it to prevent you of making errors.
Even theorems that are rather abstracts and are real theories like Noether theorem are based on observations or long lasted general assumptions based on observations. It's common to redefine some quantity when some theorem fails too. :)

22. Jan 18, 2008

### Fredrik

Staff Emeritus
This is a bit incoherent, so I'm not sure what your point is. Maybe you were just trying to say what Shooting star said here:

What you're describing here is the process of finding a useful theory. I was talking about the fact that you can't claim to have a theory until you actually have a set of statements that defines it.

My post was in response to the claim that terms such as "postulates" have no place in physics, and in SR in particular. My point was that they do because SR wouldn't be a theory without a rigorous foundation.

I wouldn't have minded saying it to his face, because I know I'm right. I doubt it would have been much of a discussion though.

23. Jan 18, 2008

### danime

I think I really missed what you were saying. In principle you are right, though sometimes you cannot define a theory, but just have useful tools that make the body of the theory, once everybody knows what is the theory but cannot put it to the paper as postulates. Very simple theories as SR can be reduced to a few postulates and theorems, but try to make it with condensed matter or hydrodynamics. Even SR is contradictory when it is used as a basis to electrodynamics.

Don't underestimate the capacity of Pauli to destroy your arguments. People like Bohr, Heisenberg, Einstein, Kramers, Feynman, to cite a few have done this and became disarmed in their arguments by Pauli. Pauli was something like an oracle where people shoot their theories to see if it survives. Most of the time the response was a crude "It's foolish".

24. Jan 18, 2008

### yuiop

Interestingly the first postulate was formulated by Galileo in 1639 (over 200 years earlier than Einstein). This postulate could have been rephrased (IMHO) as "It is impossible to carry out any physical measurement that would distinguish one inertial frame from another." In Galileo's time they did not have the technology to measure the speed of light and they were uncertain if the speed of light was finite or infinite. If they had conjectured that the speed of light was finite then they could conceivably have come up with the special theory of relativity using these two postulates:

1. It is impossible to carry out any physical measurement that would distinguish one inertial frame from another.
2. The speed of light is finite.

Having worked out that for the speed of light to be finite, that objects with relative motion would have to length contract, time would dilate and that there is no universal concept of simultaneous time they probably would have decided it was all too bizarre and concluded that the speed of light must be infinite.

25. Jan 19, 2008

### Fredrik

Staff Emeritus
That version of #2 isn't sufficient. You have to postulate that it's the same in all inertial frames, not just finite. (The speed of...anything really, is also finite, but you can't build a theory of relativity around the speed of a flying duck).

Also, that version of #1 has it's difficulties. My computer is stationary in a certain group of inertial frames, and moving in another set of inertial frames. I can obviously do an experiment to determine if my computer is moving or not.