# Definitions (continued from the cosmology forum)

1. Feb 3, 2009

### Fredrik

Staff Emeritus
This is a reply to a comment made in the cosmology forum.

That's not correct. Do you really believe that physicists are that dumb? The real story goes something like this:

The speed of light is a constant in the real world.
Therefore, we define a mathematical model in which the speed of light is a constant.
Ergo, this model might be useful in a theory of physics.

2. Feb 3, 2009

### JesseM

I think it's a bit of an oversimplification to say "the speed of light is constant in the real world", since speed is coordinate-dependent, and non-inertial coordinate systems are not "wrong" per se. It would be more accurate to say "the laws of physics are Lorentz-symmetric, meaning they work exactly the same in the family of inertial coordinate systems related by the Lorentz transformation, and in this family of coordinate systems the speed of light is constant". As an analogy, it would be possible to construct a family of coordinate systems where the speed of a sound wave was the same in every coordinate system, but the laws of physics would not obey the same equations in all of these coordinate systems.

Once consequence of this is that if observers at rest in different Lorentzian coordinate systems use identical physical procedures to construct rulers at rest relative to themselves, they will all get the same value for the coordinate length of their own rulers in their rest frames; likewise if they use identical physical procedures to construct clocks at rest relative to themselves, they will all get the same value for the coordinate time between ticks of their own clocks in their rest frames (if these things weren't true it would mean the laws of physics don't work the same way in different Lorentzian coordinate systems, since they'd be using the same physical procedure to construct their rulers/clocks yet getting different coordinate descriptions of them). So, if they all use these identically-constructed rulers and clocks to measure the two-way speed of light, they'll all get the same answer.

3. Feb 3, 2009

### Fredrik

Staff Emeritus
I agree of course. I just wanted to keep it as simple as possible, and I was a bit lazy.

4. Feb 3, 2009

### malawi_glenn

Yes, the thing is that we can never prove the light speed to be constant and no one has ever made such claim that we have either. So the argument is a straw-man attack..

The 'light speed is constant in all frames' is one axiom in SR, and axioms can not be proven correct by deductive logic, they can only be used for derive theorems and formulas -> observables.

From the two axioms of SR, we can derive many formulas, such as time- and length contraction and see if those things exists and thus makes SR a coherent framework for describing nature. And in fact, that is what has been proven during the 20th century - that the physical relations derived from the axioms of SR makes sense. But I stress that it can never be used to prove the axioms - in no axiomatic systems can the axioms be proven (Gödel's theorem).

5. Feb 3, 2009

### Staff: Mentor

The grain of truth in hartlw's comment is that today's definition of the meter makes the constancy of c a tautology in SI units.

However, in other unit systems, including older versions of SI units, the constancy of c is not tautological. In these other unit systems you can reasonably measure the speed of light. When you do so using previous versions of SI units you find that not only is c constant, but it is constant to such a high degree of precision that the primary source of error in determining c is the uncertainty in the length of the meter.

6. Feb 3, 2009

### malawi_glenn

But the definition of 1 meter is not used in theories, choice of units = choice of gauge -> Physics is independent of those choices.

But yeah, it is fun to know that in SI units the speed of light constancy is a tautology ;-)

7. Feb 4, 2009

### Chrisc

If experimental physicist "A" constructed a one meter ruler while their lab was moving at near light speed wrt physicist "B", when brought to rest with B, A's ruler would be identical to a one meter ruler constructed by B?

If identical clocks were used by A and B to measure 1/300000s of a light signal to mark their meters, would the discrepancy in their clock times at rest after constructing their rulers, indicate light speed varied between them while marking their meters?

8. Feb 4, 2009

### Fredrik

Staff Emeritus
1. Yes.
2. I don't understand the first part of the question, but the speed of light is always the same in all inertial frames.

9. Feb 4, 2009

### Chrisc

The times marked by A and B in measuring a light signal traversing one meter is 1/300,000s in their respective labs. This time is the measure they use to construct their meter sticks. (SI units)
Assuming their clocks were synchronized before A began to move, the total time marked by clock A will be less than that marked by clock B once they are again at rest in B's lab?
If so, A and B will reason the time dilation in conjunction with the construction of identical meter sticks leaves only one variable in their methodology - the speed of light.
While the constancy of the speed of light is confirmed by the empirical evidence of identical meter sticks, the discrepancy in the total time marked by their clocks leads them to reason the speed of light differs between their labs when in motion, but is constant wrt any "measure" of length/time in either lab.

10. Feb 4, 2009

### JesseM

"Total time marked by clock A" between what two events? You can't use a single clock to measure events at different positions in your frame, like the events of light passing either end of a meter stick at rest in your frame. You could measure the two-way speed of light with a single clock by having the light emitted next to the clock and then reflected at the other end of the stick so you can note the time on the same clock when the light returns. But even if the light leaving the clock and the light returning to the clock happen at the same position in one frame, they'll happen at different positions in the frame where the clock is moving, so in that frame you'd need two synchronized clocks to measure the time between these events.

11. Feb 4, 2009

### Mentz114

Chrisc:
I think yes, by applying this argument -

1. the rulers are constructed using a clock and a light beam
2. so each maker sets a clock to measure an interval 1/c. Call this interval $\tau_0$. These inervals would transform between the frames thus $\tau_0'=\gamma\tau_0$
3. when the frames are brought to mutual rest after the rulers are made, either by giving a positive boost to one, or a negative boost to the other, the subsequent length transformation cancels the $\gamma$ leaving both ruler makers admiring each others (identical) products.

There are a lot of assumptions in this ( total symmetry) and a possible tautology, because they would need to have measured c not using a light beam.

I could be wrong - this is coffee break seminar reasoning ( no blackboard).

12. Feb 4, 2009

### Chrisc

The total time I was referring to was between A beginning to move and A returning to rest with B.
To be perfectly clear, you are right. The construction would require two clocks per lab and two physicists per lab. All four clocks would require synchronization before A begins to move.
The physicists in each lab would agree the first physicist would initiate the light signal at t0 on their clock, and the second physicist would mark the position of the light after 1/300,000s on their clock to determine the meter.
My question was not specifically about the measurement.
My question pertained to the reasoning of the physicists via scientific method when faced with three components of a measurement, two of which are empirical evidence: the "identical" meter sticks, and the time differential between the clocks on A and the clocks on B. The third component - the speed of light - is the variable they must deduce from the previous via the equation of speed Length/Time
It would seem they have little option but to conclude the speed of light differs between the labs when they are in motion.

Last edited: Feb 4, 2009
13. Feb 4, 2009

### Staff: Mentor

I would have to see a rigorous derivation of this. I don't buy the "hand waving" reasoning above.

Last edited: Feb 4, 2009
14. Feb 4, 2009

### Mentz114

Chrisc:
I don't conclude that. I have lots of options and that one isn't among them. Nothing you've said leads to this conclusion.

There aren't 'three components of a measurement', only a time interval. That is the empirical data. c is not a variable. In fact, what you said in the section I quoted is almost gibberish.

15. Feb 4, 2009

### Chrisc

Their meter sticks are identical by physical comparison at rest in B, that is empirical data, not hand waving.

From the time A began to move until the time A is back at rest with B, the total time marked by the clocks
in A differs from the total time marked by the clocks in B.
Unless you think the success of GPS is hand waving, I think you will agree such time dilation is well documented, empirical evidence.

If I have misinterpreted either of these facts, please explain.

If not, then I don't understand how you can claim identical length over differing time is identical speed.

16. Feb 4, 2009

### Chrisc

Length/time=speed are the three components I refered to.

I can't respond to this.

17. Feb 4, 2009

### Staff: Mentor

That is just more hand waving, not a rigorous derivation.

You are claiming that SR (which postulates constant c) would result in a situation where c is measured to be variable. That certainly requires a careful, step-by-step, rigorous derivation to support the claim, not mere unsupported assertions.

18. Feb 4, 2009

### Chrisc

No, you're putting words in my mouth.
I have not claimed the speed of light will be "measured" to be variable.

I said in #12
"the speed of light - is the variable they must deduce" in that they have two knowns, the length of their meters and the time of their clocks therefore the unknown, the only parameter that is variable between them, is the speed of light while they are in motion.

in #7
"light speed varied between them while marking their meters?"

in #9
"the speed of light differs between their labs when in motion, but is constant wrt any "measure" of length/time in either lab."

in #12
"the speed of light differs between the labs when they are in motion."

I have never claimed that the speed of light is not a constant measure.
I have based all of what I've said on the fact that the speed of light is always measured to be constant - that is empirical evidence.
I assume everyone agrees the length of the meters constructed while in motion are, upon comparison at rest in B, of identical lengths.
Each meter was determined as the distance traversed by light during a pre-designated interval of time, as marked by identical, synchronized clocks that run at differing rates while in motion wrt each other i.e. while constructing their meter sticks.
What more rigorous derivation do you need than: equal length/differing time=differing speed?

19. Feb 4, 2009

### JesseM

Can you give some kind of numerical example of what you're talking about? I don't really understand how measuring the time A accelerates and decelerates would relate to measuring the speed of light, nor do I understand why you think they'd be forced to conclude the speed of light differs.

20. Feb 5, 2009

### Chrisc

Let's use an example where the relative motion between A and B is .999c
This sets the time dilation to a factor of 0.044710245045, which is to say for every second marked by the clocks of each A and B, the other will mark 0.044710245045 seconds.
Let A construct a 300,000 kilometer ruler by marking (with two clocks) the initiation of the light signal in A at t= 0s and the distance the light signal has traveled at t=1s.

B constructs a ruler in the same manner using their two clocks.

B might initially think that since A's clocks are running at a rate of 1/0.044710245045 of B's clocks, and because the speed of light is constant, A will construct a ruler that is 22.366238408971 times longer than B's ruler.
A might think the same of B.
When A and B come to rest at B and compare notes, they both claim the speed of light each measured was 300,000k/s (i.e. the "measure" of the speed of light is always constant) They confirm this by laying their rulers side by side and noting they are exactly the same length.
But they are at a loss to figure out how if each of their clocks ran at 1/044710245045 the rate of the other's how they could possibly have measured exactly the same rulers when the speed of light is constant.
They read Einstein's SR and realize, while the speed of light is a physical constant which is to say their measurements of length and time will always result in the physical ratio of length to time that is the constant "c", they also realize if the laws are to be upheld in face of such a phenomenon, their measures of the dimensions length and time must change BETWEEN them WHILE they are in motion with respect to each other just as Einstein predicted. Which is to say if wrt B, time in A runs slower and length in A is contracted, then A will measure the speed of light as numerically consistent with B using units of lessor magnitude wrt B. Therefore BETWEEN A and B, and WHILE they are in motion, the speed of light varies according to the dilation and contraction of time and length respectively.