Third postulate of special relativity?

[zonde]

Hi!
I was trying to think over how one can arrive at Lorentz transformations starting from absolute reference frame (with only first postulate) and came to conclusion that there must be third postulate hidden somewhere in special relativity. And I found it here:
"The length of the moving rod [placed perpendicularly to direction of motion] measured in the stationary system does not change, therefore, if v and -v are interchanged. Hence follows that l/φ(v)=l/φ(-v), or
φ(v)=φ(-v)."

 

[CompuChip]

Hi!
I was trying to think over how one can arrive at Lorentz transformations starting from absolute reference frame (with only first postulate) and came to conclusion that there must be third postulate hidden somewhere in special relativity.

I don't know what you call the first and second postulate, but if I'm not mistaken one of them contradicts the existence of an absolute reference frame. So either you didn't express yourself carefully enough, or you're drawing a conclusion from false premises. Not to mention the fact that there seems to be no need whatsoever to add extra postulates to SR.

And I found it here:
"The length of the moving rod [placed perpendicularly to direction of motion] measured in the stationary system does not change, therefore, if v and -v are interchanged. Hence follows that l/φ(v)=l/φ(-v), or
φ(v)=φ(-v)."

Lengths are only contracted along the direction of relative motion. In perpendicular directions they do not change. You don't need this as a postulate, it already follows from the first two.

 

[zonde]

I don't know what you call the first and second postulate, but if I'm not mistaken one of them contradicts the existence of an absolute reference frame. So either you didn't express yourself carefully enough, or you're drawing a conclusion from false premises. Not to mention the fact that there seems to be no need whatsoever to add extra postulates to SR.

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
And that about an absolute reference frame was just an intro but not the point in question.

Lengths are only contracted along the direction of relative motion. In perpendicular directions they do not change.

You are using circular argument here if you base this statement on formulas of Lorentz transformations. This quoted part appears in process of derivation of Lorentz transformations.

You don't need this as a postulate, it already follows from the first two.

How?

 

[HallsofIvy]

Because velocity is a vector. Working in only one direction you can use the component of velocity in that direction. Since the Lorentz formulas apply to components of vectors, perpendicular to the direction of motion, it is exactly as if there were no relative motion.

 

[robphy]

Here is an old post
https://www.physicsforums.com/showthread.php?p=850688#post850688

There is a thought experiment arguing that there is no contraction (or expansion) in the spatial directions perpendicular to the motion. Essentially, it involves two identical winged objects (call them A and B) with a nail or other marking pen at the tip of each wing.
Suppose A and B travel inertially toward each other with different speeds along a line.
Suppose there is contraction in the perpendicular direction.
Then, in A's frame, B's wing contraction will mark up A's wing.
Invoking the relativity principle, in B's frame, A's wing contraction will mark up B's wing.
Logically, both can't happen. Therefore, there can be no contraction in the perpendicular direction.

 

[CompuChip]

You are using circular argument here if you base this statement on formulas of Lorentz transformations. This quoted part appears in process of derivation of Lorentz transformations.

No and yes
The Lorentz transformations can be derived from the postulates (see any standard reference on SR). From the formulas thus derived, it follows that there is no contraction in the perpendicular directions. I don't see how this is a circular reasoning.

And yes, the quoted part appears in the process of deriving the Lorentz transformations. They can be derived from the two postulates. Therefore, both are a consequence of the postulates and no third one is needed, nor an "absolute" reference frame.

 

[Ich]

There are indeed two further postulates, which Einstein did not express explicitly.
3. Homogeneity of Space and Time, which makes the solution to his differential equation linear,
4. Isotropy of Space and Time, which allows the symmetry argument you quoted.
Working on GR, Einstein concluded that both postulates above cannot be globally valid. The equivalence principle states that they are still valid locally, which is the starting point of GR.
Hence, local Lorentz Invariance is the basis of relativity, and SR is said to be valid only locally in the presence of gravitation.

 

[zonde]

Because velocity is a vector. Working in only one direction you can use the component of velocity in that direction. Since the Lorentz formulas apply to components of vectors, perpendicular to the direction of motion, it is exactly as if there were no relative motion.

That is not correct.
In formula (from Einstein's paper about SR)
η=φ(v)y
v is relative motion along X-axis but not vector component of v along Y-axis
or similary in
y'= φ(v)φ(-v)y
v is relative motion along X-axis
And φ(v)φ(-v) it is not discarded based on something like this but exactly based on statement that "The length of the moving rod measured in the stationary system does not change".

 

[zonde]

There is a thought experiment arguing that there is no contraction (or expansion) in the spatial directions perpendicular to the motion. Essentially, it involves two identical winged objects (call them A and B) with a nail or other marking pen at the tip of each wing.
Suppose A and B travel inertially toward each other with different speeds along a line.
Suppose there is contraction in the perpendicular direction.
Then, in A's frame, B's wing contraction will mark up A's wing.
Invoking the relativity principle, in B's frame, A's wing contraction will mark up B's wing.
Logically, both can't happen. Therefore, there can be no contraction in the perpendicular direction.

This seems like circular argument too because you can not use relativity principle (if I understand correctly what do you mean by that) in derivation of relativity principle e.g. Lorentz transforms.

 

[zonde]

No and yes
The Lorentz transformations can be derived from the postulates (see any standard reference on SR). From the formulas thus derived, it follows that there is no contraction in the perpendicular directions. I don't see how this is a circular reasoning.

Even if I am talking about that derivation (taken from some standard reference on SR for example this one
http://www.fourmilab.ch/etexts/einstein/specrel/www/" )?

And yes, the quoted part appears in the process of deriving the Lorentz transformations. They can be derived from the two postulates. Therefore, both are a consequence of the postulates and no third one is needed, nor an "absolute" reference frame.

 

[robphy]

This seems like circular argument too because you can not use relativity principle (if I understand correctly what do you mean by that) in derivation of relativity principle e.g. Lorentz transforms.

The point is:
since these inertial observers are symmetrical in this situation described earlier,
neither will have their wings contract in the direction perpendicular to the motion.

No postulate of the "constancy of the speed of light" is used.
This argument also holds for the Galilean principle of relativity... so, the Lorentz transformations are not invoked. In fact, there is no explicit use of any transformations.

 

[Fredrik]

There's been a bunch of threads about Einstein's postulates before. This is a summary of what I've been saying in some of the more recent threads:

1. Einstein's "postulates" aren't well-defined enough to be the axioms of a mathematical theory. The biggest problem with them is that they use the concept "inertial frame" without a definition.

2. Any definition of "inertial frame" that's appropriate for special relativity will actually include both of Einstein's postulates in some form.

3. This means that Einstein's postulates are useless both as a definition of special relativity and as a starting point of rigorous derivations of theorems.

4. The postulates are just a list of properties that Einstein thought that a good theory of space and time should have. He wanted to find a new theory and at the time he was only willing to consider theories that had those specific properties.

5. This means that there's nothing wrong with using very non-rigorous methods when you're trying to derive something from Einstein's postulates. (In fact it wouldn't make any sense at all to try to do things rigorously). For example, if you can't prove that Lorentz transformations must be linear maps, then don't. Just guess that they are and move on.

6. When you do the sort of things I'm talking about in 5, you will end up with something that looks a lot like Minkowski space. This suggests that you can take Minkowski space to be the mathematical model of space and time in the new theory.

7. Minkowski space is mathematically well-defined, and therefore an excellent starting point for derivations of mathematical theorems about the properties of space and time. It also includes a version of the second postulate explicitly and suggests a very natural way to include the first postulate. (It also includes the additional postulates that Ich mentioned).

8. No mathematical model can make predictions about the outcome of any experiment all by itself. We also need to make some identifications between things in the model and things in the real world. (I recently started this thread about that, but it didn't really go anywhere). Those identifications can't be derived from anything else, so they must be postulated.

9. What I'm saying in 8 is that we need to specify what sort of gizmos in the real world we think will be able to measure real-world quantities that correspond to things in the mathematical model. Every such specification is a postulate of the theory. For example, this is a statement that (unlike Einstein's postulates) deserves to be called a postulate of both special and general relativity: "What a clock measures corresponds to the proper time along the curve in Minkowski space that represents the clock's motion".

It seems to me that what Zonde is going for in #1 is another example of an identification between something in the real world and something in the model. It's necessary to specify in some way what sort of thing in the model corresponds to "what we measure with a meter stick" in the real world. So I agree with Zonde that it's definitely necessary to postulate something like that, although I might do it a bit differently.

 

[zonde]

To all
It seems that I have introduced some confusion by not explicitly stating that the quoted part in first post I took from Einstein's 1905y paper "On the Electrodynamics of Moving Bodies"
Sorry for that.

 

[zonde]

There's been a bunch of threads about Einstein's postulates before. This is a summary of what I've been saying in some of the more recent threads:
...

Thank you for your answer.
There are interesting points in your post. The point about problems with definition of inertial frame I didn't understand however. Well maybe this concept is not defined but as I see it is described and in axiomatic systems there can be primitive terms that are described rather than defined. But of course I am not familiar with arguments about that problem.

It is interesting how you propose to establish relations between mathematical models and real world. However I am interested in slightly different direction of findings. I would rather forsake constancy of light speed for Euclidean space and time (global, historical and in different scales). It allows to view mainstream theories from different side and make interesting conclusions.