Definitions (continued from the cosmology forum)

[Fredrik]

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

The speed of light is constant.

By definition, distance and time are such as to make the speed of light constant.

Ergo, the speed of light is constant.

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.

 

[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.

 

[Fredrik]

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

 

[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).

 

[Dale]

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.

 

[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 ;-)

 

[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?

 

[Fredrik]

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.

 

[Chrisc]

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

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.

 

[JesseM]

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?

"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.

 

[Mentz114]

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?

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).

 

[Chrisc]

"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, ...

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.

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).

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.

 

[Dale]

It would seem they have little option but to conclude the speed of light differs between the labs when they are in motion.

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

 

[Mentz114]

Chrisc:

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.

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.

 

[Chrisc]

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

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.

 

[Chrisc]

There aren't 'three components of a measurement', only a time interval.

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

That is the empirical data. c is not a variable. In fact, what you said in the section I quoted is almost gibberish.

I can't respond to this.

 

[Dale]

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.

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.

 

[Chrisc]

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.

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?

 

[JesseM]

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.

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.

 

[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.

 

[JesseM]

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.

What do you mean "measured exactly the same rulers"? Their rulers are only the same when they come to rest relative to one another, when they were moving relative to one another and each measured the other's clock to be slowed down by 0.044710245045, they also measured each other's rulers to be shrunk by the same factor.

Suppose we are looking at things in A's frame, while B is moving at 0.999c relative to A. And suppose that when the back ends of their two rulers line up, at x=0 light-seconds and t=0 seconds in A's frame, at that moment a light flash is set off at this point. Finally, suppose that B has clocks at either end of his ruler which are synchronized in his frame, and the clock at the back end of his ruler also read t'=0 when it lined up with the back of A's ruler. Because of the relativity of simultaneity, B's two clocks will be out-of-sync in A's frame; since the two clocks are a distance of 1 light-second apart in B's frame (the length of B's ruler in his frame), and they are moving at 0.999c in A's frame, in A's frame they are out-of-sync by (1 light second)(0.999c)/c^2 = 0.999 seconds (in general if two clocks are synchronized and a distance x apart in their own rest frame, then in a frame where they're moving at speed v along the axis between them, they'll be out-of-sync by vx/c^2). So at t=0 in A's frame, when the clock at the back end of B's ruler reads t'=0 seconds, the clock at the front end of B's ruler reads t'=-0.999 seconds.

Now, in A's frame B's ruler is 0.044710245045 light seconds long, so at t=0 the front end is at x=0.044710245045 light seconds. And the front end is moving at 0.999c, so in A's frame the front end's position as a function of time is given by x(t) = 0.999c*t + 0.044710245045. Meanwhile, if in A's frame the light beam is moving at 1c, and it started at x=0 at t=0, the light beam's position as a function of time must be x(t) = 1c*t. So we can set these equal to find the time in A's frame that the light catches up with the front end of B's ruler--this gives us 1c*t = 0.999c*t + 0.044710245045 which implies 0.001c*t = 0.044710245045, so t = 0.044710245045/0.001c = 44.710245045 seconds. Naturally this happens at position x = 44.710245045 light-seconds in A's frame, as you can see if you plug the time into either of the x(t) functions above. All this was based on the assumption that B's ruler had a length of 0.044710245045 light-seconds in A's frame, that it was moving at 0.999c in A's frame, and that the light beam was moving at 1c in A's frame.

So we know in A's frame the time the light reaches the front of B's ruler is t = 44.710245045 seconds. But since B's clocks are slowed down by a factor of 0.044710245045 in A's frame, the clock at the front of B's ruler will only have advanced forward by 44.710245045 * 0.044710245045 = 1.999 seconds between the time the light flash is set off and the time the light reaches the front end of B's ruler. And since in A's frame the clock at the front end of B's ruler initially read -0.999 seconds when the light flash was set off, due to the relativity of simultaneity, that means that when the light finally reaches the front end of B's ruler the clock there will read -0.999 + 1.999 = 1 second. So you see, it makes perfect sense in A's frame that the clock at the back end of B's ruler read 0 seconds when the flash was set off there, and the clock at the front end of B's ruler read 1 second when the light reached it, in spite of the fact that in A's frame B's ruler was not 1 light-second long but only 0.044710245045 light-seconds long, and B's clocks were slowed down by a factor of 0.044710245045.

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.

I don't get what you mean by "speed of light varies". If they both agree that light moves at 1c using their own rulers and clocks (as in the above example where if we look at the event of the light flash being set off next to the back end of B's ruler, and the event of the light flash reaching the front of B's ruler, then in B's frame the distance between these events was 1 light-second and the time between them was 1 second, while in A's frame the distance between them was 44.710245045 light-seconds and the time between them was 44.710245045 seconds), doesn't that mean by definition that the speed of light is constant between their two frames, not that the speed of light varies between frames? What do you mean by "varies" anyway?

 

[Chrisc]

What do you mean "measured exactly the same rulers"? Their rulers are only the same when they come to rest relative to one another, when they were moving relative to one another and each measured the other's clock to be slowed down by 0.044710245045, they also measured each other's rulers to be shrunk by the same factor.

I said "have" measured (past tense)- when they are back at rest together in B, they find they "have" measured exactly the same length.

I don't get what you mean by "speed of light varies". ... doesn't that mean by definition that the speed of light is constant between their two frames, not that the speed of light varies between frames? What do you mean by "varies" anyway?

Their method of construction is to "measure" the speed of light. They both claimed to have measured the speed of light to be 300,000k/s(i.e. it is independent of the motion of the observer and source). To prove this they must confirm that their clocks run at identical rates and the meter sticks each constructed are of identical length. This can only be confirmed by comparison when they are back at rest in B's lab. They make this comparison and find both (rate and length) are identical.

If this were all the evidence they had they would have to either claim it was some kind of illusion or they would have to "fail" the laws of mechanics.
But they have one more critical piece of evidence that must be added to their conclusion.
The clocks in A, the lab that accelerated and maintained .999c for 1 second, have marked less time than the clocks in B.
This change in total time marked by identically constructed clocks is empirical evidence of time dilation.

From this they must conclude, based on their construction of identical meters, that A's measure of length was, while in motion, also a real physical change, for they have the physical evidence in the meter stick A constructed.
All of the above and your previous post, indicates any direct "measure" of the speed of light will show it to be constant.

What we seem to be arguing is not the constancy of a measure, but what constancy means with respect to different measures.
The only reason A and B have any inclination to question the rate of their clocks and the length of their meters is because they think (could they observe the other while in motion) the other has measured something different.
As I mentioned above, they indeed have measured something different. They find the same ratio of length/time in the motion of light we call c, but unless they fail the laws of physics they must conclude that the meter A measured while in motion, was NOT identical to the meter B measured and the time marked by A while in motion was NOT identical to the time marked by B.
So while the ratio of length and time remained constant in their measures WHAT they measured was not identical while in motion. Since what they measured was the speed of light, then the speed of light is not identical while in motion, but is always a constant measure.
Don't take this to mean that the speed of light varies. I am not, and have not said that.
I am saying the first postulate of Einstein's theory (SR), the principle of relativity, maintains the laws of mechanics NOT because A and B find all things are identical between them anywhere or at any time - they don't. It upholds the laws because the "same equations hold good" for each of them. The same equations find the speed of light constant for each of them, but not identical between them while in motion.
The point of the principle is that the whole universe may change when A observes B, but as long as the equations hold good for the measurements of each (scientific method), the laws are perfectly valid working tools.

 

[JesseM]

I said "have" measured (past tense)- when they are back at rest together in B, they find they "have" measured exactly the same length.

I still don't get it. Do you agree that it only makes sense to talk about "length" relative to a particular choice of coordinate system? If so, please be specific about what coordinate systems you are talking about in the above statement, and what pair of events you want to measure the length between. Obviously when they are at rest with respect to one another they share the same rest frame and thus agree about length measurements, but are you claiming this means they did agree on the length traveled by the light beam when they were moving relative to one another? That wouldn't make sense--in my previous example, you can see that in B's rest frame the light traveled a length of one light-second between the event of it being emitted and the event of it reaching the right end of B's ruler, but in A's rest frame the length it traveled between these same two events was 44.710245045 light-seconds. Do you disagree with those numbers?

Their method of construction is to "measure" the speed of light. They both claimed to have measured the speed of light to be 300,000k/s(i.e. it is independent of the motion of the observer and source). To prove this they must confirm that their clocks run at identical rates and the meter sticks each constructed are of identical length. This can only be confirmed by comparison when they are back at rest in B's lab. They make this comparison and find both (rate and length) are identical.

They "both claimed to have measured the speed of light to be 300,000k/s" in their rest frames when they were moving relative to one another--how is the length of B's ruler when he comes to rest relative to A relevant to the length he measured when he was in motion relative to A? These are two different frames, and again, "length" can only be defined relative to a particular choice of frame.

If this were all the evidence they had they would have to either claim it was some kind of illusion or they would have to "fail" the laws of mechanics.

I don't see what you mean by "some kind of illusion". A can measure the fact that B's clock is slowed down in A's frame without B needing to decelerate--if B passes one of A's clocks when it reads 0 seconds, then passes another one of A's clocks (which is synchronized with the first according to A's definition) when it reads 1 second, but meanwhile B's clock has only advanced forward by 0.044710245045 seconds between these events, why isn't this sufficient to show B's clock is running slow in A's frame? Likewise, B can show A's clock is running slow in B's frame by using two clocks which are at rest in B's frame and synchronized according to B, and noting how much time has passed on A's clock between passing them. And if one of them accelerates, then whichever one accelerates (A or B) will be the one whose clock has elapsed less time when they reunite, so this doesn't tell you anything about whose clock was "really" running slower before either accelerated, not in any objective frame-independent sense anyway.

From this they must conclude, based on their construction of identical meters, that A's measure of length was, while in motion, also a real physical change, for they have the physical evidence in the meter stick A constructed.

Just as with time, B can measure the fact that A's ruler is shrunk in B's rest frame without A having to change velocities--B can note that the back end of A's ruler was next to one clock when it read T, and the front end of A's ruler was next to a different clock when it read T, so the length of A's ruler in B's frame must just be the distance between these clocks in B's frame.

What we seem to be arguing is not the constancy of a measure, but what constancy means with respect to different measures.

I'm not really arguing, I just don't understand what you mean by the words "speed of light varies", along with a lot of other phrases you're using.

The only reason A and B have any inclination to question the rate of their clocks and the length of their meters is because they think (could they observe the other while in motion) the other has measured something different.
As I mentioned above, they indeed have measured something different. They find the same ratio of length/time in the motion of light we call c, but unless they fail the laws of physics they must conclude that the meter A measured while in motion, was NOT identical to the meter B measured and the time marked by A while in motion was NOT identical to the time marked by B.

Sure.

So while the ratio of length and time remained constant in their measures WHAT they measured was not identical while in motion. Since what they measured was the speed of light, then the speed of light is not identical while in motion, but is always a constant measure.

"The speed of light is not identical" to what? You're continuing to speak in extremely vague english, it would help if you could give some mathematical example or definitions.

Don't take this to mean that the speed of light varies. I am not, and have not said that.

But in a previous post you said "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."

I am saying the first postulate of Einstein's theory (SR), the principle of relativity, maintains the laws of mechanics NOT because A and B find all things are identical between them anywhere or at any time - they don't. It upholds the laws because the "same equations hold good" for each of them. The same equations find the speed of light constant for each of them, but not identical between them while in motion.

What is "not identical between them"? Again, can you speak in terms of frames here, since the first postulate is explicitly about the laws of physics in different inertial frames? The equations governing light are identical when expressed in one inertial frame's coordinates as they are when expressed in a different inertial frame's coordinates, do you agree? If so, what is it exactly that you're saying is "not identical between them"?

The point of the principle is that the whole universe may change when A observes B,

Huh? How does the universe change at all when A observes B?

but as long as the equations hold good for the measurements of each (scientific method), the laws are perfectly valid working tools.

You really aren't expressing yourself in a way I can make any sense of, the above sentence is completely meaningless to me. Again, it would really help if you would try to express your ideas using well-defined mathematical terms or even a numerical example, not just vague english phrases.

 

[Dale]

Thanks for the effort Chrisc, that was not quite a rigorous derivation, but it was helpful. Here is a key problem.

They confirm this by laying their rulers side by side and noting they are exactly the same length.

If they place their clocks side by side they also note that they measure exactly the same duration too.

I said "have" measured (past tense)- when they are back at rest together in B, they find they "have" measured exactly the same length.

When they are back at rest together they find they "have" measured exactly the same time also.

 

[Chrisc]

Thanks for the effort Chrisc, that was not quite a rigorous derivation, but it was helpful. Here is a key problem.If they place their clocks side by side they also note that they measure exactly the same duration too.

That is exactly my point!
They do at rest in B, but during the "critical second" when they constructed their meter they didn't.

A left B and traveled at .999c for one second "while constructing their meter stick".
Upon returning to rest in B their meter stick is identical to B's, and their clocks run at exactly the same rate.
As they constructed the sticks by marking light signals, the sticks represent their measure of length and time.
As they are identical when back together at rest in B, we say they marked identical length and time.
As the length and time they marked was the speed of light, the speed of light is independent of the motion of the source and observer.
Now how do they reason all of this perfectly logical methodology against the fact that A's clocks have marked
less total time than B's?
A's clocks WERE running slower than B's during the critical second "while they constructed their meter stick".
If so why is their meter stick identical when back at rest in B?
Because the speed of light they marked was, while a constant measurement, not (while in motion) the same as marked by B.

The rigorous proof you're looking for is Lorentz transformation. There is no new or unique formula I need to show you.
I need to make you understand the Lorentz transformation represent a physically "real" phenomenon -
time really does change and length really does change therefore the speed of light
really does change, but will never be measure to do so, it will always be measured a constant.

When they are back at rest together they find they "have" measured exactly the same time also.

Yes, exactly the same time was measure by clocks running at different rates.

 

[matheinste]

Hello Chrisc.

Sorry to break in as this question is really for my benefit to check my understanding of the situation.

Quote:-
----I need to make you understand the Lorentz transformation represent a physically "real" phenomenon - time really does change and length really does change therefore the speed of light really does change, but will never be measure to do so, it will always be measured a constant.-----

I was under the impression, but i may be wrong, that these real changes in time rate and length were "required" to be such that the speed of light remains constant. In fact i thought that they were consequences of this constancy.

Also, to me it seems that saying that the speed of light changes but we can never measure that change is analogous to saying that there is an ether but we can never detect it. However, i would not like to get into an argument over that.

Matheinste.

 

[Dale]

The rigorous proof you're looking for is Lorentz transformation. There is no new or unique formula I need to show you.

No, the Lorentz transformation is not a rigorous proof. If you were to use the Lorentz transformation to derive your conclusion then that would be a rigorous proof. This is what you have not done. Even in post 20 where you did very well and did a lot of good work you did not derive your conclusion.

I need to make you understand the Lorentz transformation represent a physically "real" phenomenon -
time really does change and length really does change therefore the speed of light
really does change, but will never be measure to do so, it will always be measured a constant.

This is the interpretation of Lorentz in his aether theory. It is experimentally indistinguishable from Einstein's formulation. Most scientists (including myself) prefer Einstein's formulation because of Occham's razor, but I honestly don't care which interpretation you prefer since they are experimentally indistinquishable.

 

[Chrisc]

Hello Chrisc.

Sorry to break in as this question is really for my benefit to check my understanding of the situation.

Quote:-
----I need to make you understand the Lorentz transformation represent a physically "real" phenomenon - time really does change and length really does change therefore the speed of light really does change, but will never be measure to do so, it will always be measured a constant.-----

I was under the impression, but i may be wrong, that these real changes in time rate and length were "required" to be such that the speed of light remains constant. In fact i thought that they were consequences of this constancy.

These real changes in time rate and length are why clocks and rulers measure the speed of light as a constant.
The constant measure of the speed of light is a consequence of their changes.

Also, to me it seems that saying that the speed of light changes but we can never measure that change is analogous to saying that there is an ether but we can never detect it. However, i would not like to get into an argument over that.

Matheinste.

Not at all, its a matter of deduction.
Any direct measure the speed of light will remain constant with any other direct measure of the speed of light.
But under certain conditions of measuring, a comparison between two direct measurements shows us something has changed.
Unless we throw out our definition of speed, length/time, the same measure of a speed under differing time and length leaves
us no option but to deduce time and length changed.
This change in time has been well documented.
That the speed of light is a constant means this also documents a change in length.

 

[Chrisc]

No, the Lorentz transformation is not a rigorous proof. If you were to use the Lorentz transformation to derive your conclusion then that would be a rigorous proof. This is what you have not done. Even in post 20 where you did very well and did a lot of good work you did not derive your conclusion.

You seem to be asking me to restate Einstein's derivation and claim it is my own.
The conclusion was derived by Einstein.
I will refer you to the pertinent conclusions in his paper:
On the Electrodynamics of Moving Bodies
Section 4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks.

"whence it follows that the time marked by the clock (viewed in the stationary system) is slow by 1-sqrt(1-v^2/c^2) seconds per second,
or--neglecting magnitudes of fourth and higher order--by1/2v^2/c^2."

"If we assume that the result proved for a polygonal line is also valid for a continuously curved line, we arrive at this result:
If one of two synchronous clocks at A is moved in a closed curve with constant velocity until it returns to A, the journey lasting t seconds,
then by the clock which has remained at rest the traveled clock on its arrival at A will be 1/2 tv^2/c^2 second slow."

This is the interpretation of Lorentz in his aether theory.
It is experimentally indistinguishable from Einstein's formulation.

No, I am not stating Lorentz's aether theory. I am not suggesting any theory nor I am suggesting there is any absolute nature to space
and time against which dilation and contraction are measured.
I am pointing out the fact that time and length do change as Einstein said (quoted above) and there must be reason behind it.

Most scientists (including myself) prefer Einstein's formulation because of Occham's razor, but I honestly don't care which interpretation you prefer since they are experimentally indistinquishable.

Einstein's formulation is a mathematical convention not an explanation or reason.
Occham's razor is invoked to distinguish the simplicity of competing theory not to distinguish theory from mathematical convention.

 

[JesseM]

No, I am not stating Lorentz's aether theory. I am not suggesting any theory nor I am not suggesting there is any absolute nature to space
and time against which dilation and contraction are measured.
I am pointing out the fact that time and length do change as Einstein said (quoted above) and there must be reason behind it.

So in your thought-experiment where A and B are moving relative to one another and then A decelerates to come to rest relative to B, you're not claiming there's any objective truth about whether A's ruler was longer or shorter than B's before A decelerated?

 

[Dale]

You seem to be asking me to restate Einstein's derivation and claim it is my own.

Certainly not. You don't need to do another derivation of the Lorentz transform.

What you need to do is to use the Lorentz transform to derive this questionable assertion of yours:

they have little option but to conclude the speed of light differs between the labs when they are in motion.

 

[Chrisc]

So in your thought-experiment where A and B are moving relative to one another and then A decelerates to come to rest relative to B, you're not claiming there's any objective truth about whether A's ruler was longer or shorter than B's before A decelerated?

I am claiming the objective measures of A and B during construction of their meters meets with the criteria of scientific method.
Likewise the objectiveness of identical clocks cannot be questioned.
When back at rest in B they again use objective methodology to discover their clocks have marked different total time, hence objective truth is not absolute truth, it is relative truth.
As such the speed of light is a relative measure, not an absolute measure.

 

[Chrisc]

Certainly not. You don't need to do another derivation of the Lorentz transform.

What you need to do is to use the Lorentz transform to derive this questionable assertion of yours:

Again the derivation is Einstein's. I am not suggesting anything different than the derivation of time dilation and length contraction as set out by Einstein in his paper.

I am suggesting the significance of these equations, what they tell us about length and time, are often set aside or ignored outright in favour of the blind application of his formalism.
It is as if the study of relativity has fallen prey to the "shut up and calculate" approach of QT.

I won't copy all his equations here as I'm sure you are very familiar with them.
I will quote in Einstein's own words the significance these equations.

"...For velocities greater than that of light our deliberations become meaningless; we shall, however, find in what follows, that the velocity of light in our theory plays the part, physically, of an infinitely great velocity."

Note he says "physically". The speed of light is a "physical" constant of relative measure that changes between frames by the infinite resolution of the Lorentz factor.

 

[Dale]

I am suggesting the significance of these equations, what they tell us about length and time, are often set aside or ignored outright in favour of the blind application of his formalism.

You still have not established that they "tell us" what you say they do. Specifically, you have still failed to substantiate your claim that the Lorentz transform leads to a situation where:

they have little option but to conclude the speed of light differs between the labs when they are in motion.

You claim that we blindly use the formulas and then suggest that the solution is for us to blindly trust you to tell us what the formulas really say with no better derivation than your repeated assertions.

 

[Mentz114]

Chrisc:

hence objective truth is not absolute truth, it is relative truth.
As such the speed of light is a relative measure, not an absolute measure.

You are saying there are two kinds of truth. This is woolly logic at best and meaningless at worst. What is your definition of 'truth' ?

How does a 'relative' measure differ from an 'absolute' measure ? Do you mean 'observer dependent' vs 'covariant' ?

 

[Chrisc]

You still have not established that they "tell us" what you say they do. Specifically, you have still failed to substantiate your claim that the Lorentz transform leads to a situation where:

Is it necessary to substantiate empirical evidence?
Their meters sticks are identical length, their clocks have measured different total time.
I cannot substantiate the reason for following the only path available
if you have no intention to follow any path.

You claim that we blindly use the formulas and then suggest that the solution is for us to blindly trust you to tell us what the formulas really say with no better derivation than your repeated assertions.

I said "are often", not necessarily by you.
I have not asked you to blindly trust me.
I have stated what appears, according to the evidence, to be the only option.
If there are more please let me know, if there are none, then whether you consider
it mine or Einstein's, or anyone else's, it is the only option.
As I said above, the best derivation is that offered by Einstein, if my assertions
seem repetitive it is because you repeatedly ask me for a derivation other than Einstein's.

 

[Chrisc]

Chrisc:

You are saying there are two kinds of truth. This is woolly logic at best and meaningless at worst. What is your definition of 'truth' ?

Truth in as much as you've used it, is observer dependent.
It is true A measured one second.
It is true B measured one second.
It is true their clocks have marked different total time.
When both observers, observe the "truth" of the different times marked by their clocks,
the truth that they both measured the same quantity becomes relative.

How does a 'relative' measure differ from an 'absolute' measure ? Do you mean 'observer dependent' vs 'covariant' ?

A relative measure is a measure determined by relative mechanics.
An absolute measure is classical concept independent of relativity.
I'm not sure what you mean by "vs"

 

[JesseM]

Is it necessary to substantiate empirical evidence?
Their meters sticks are identical length, their clocks have measured different total time.

Are you assuming that because their meter sticks are identical length when they come to rest relative to one another, they must have been identical length when they were in motion relative to one another? If so, why?

 

[Dale]

I have stated what appears, according to the evidence, to be the only option.

You have not even shown that it is an option, let alone the only option. It is hard to take you seriously when you so obviously don't even understand what it means to derive your claim.

 

[Mentz114]

Chrisc,
thank you for trying to answer my questions. I suspect you have this idea that there's something going on that isn't described by physics. Something absolute perhaps. This is really a question for philosophers, because physics is constrained by what is measured, or can in principle be measured. Things that don't fall into this category are not the stuff of physics.

 

[Chrisc]

Sorry for the delay, it's been a busy week.

Chrisc,
thank you for trying to answer my questions. I suspect you have this idea that there's something going on that isn't described by physics. Something absolute perhaps. This is really a question for philosophers, because physics is constrained by what is measured, or can in principle be measured. Things that don't fall into this category are not the stuff of physics.

Far from philosophy, the fact that A's clocks marked less total time during the trip than B's clocks
is empirical evidence - the stuff of physics. Likewise the fact that A's ruler is identical to B's ruler is empirical evidence.
And finally the fact that both A and B measured light speed to attain the length of their rulers leaves the responsibility
of explaining this evidence of "physical change" firmly in the lap of physics - not philosophy.

You have not even shown that it is an option, let alone the only option. It is hard to take you seriously when you so obviously don't even understand what it means to derive your claim.

I understand what it means to derive my claim, but as you mentioned earlier you expect I should use Lorentz transformation to justify my claim.
My claim is not derivation requiring the use of Lorentz transformation, it is deduction, deduction made possible by the derivations of Einstein using Lorentz transfromations- i.e. bodies in constant linear motion experience time dilation and length contraction.
With this in mind, I will not derive time dilation or length contraction, but take them as viably derived and deduce a further consequence of their derivation.

A executes a round trip starting from and returning to B.
At the apogee of this trip A maintains a constant speed of .999c relative to B,
during which A constructs a 300,000km meter ruler by marking the emission point of a light signal
at a pre-arranged time and the extent of the signal along a stick coinciding with A's clocks marking one second.
This is accomplished by two physicists in A each with identical, synchronized clocks.
A's clocks are also identical to and synchronized with two clocks left at B where the same experiment is performed.
The physicists in A discover their light signal has traveled 300,000km when their clocks have marked one second after emission time.
The physicists in B discover their experiment produces the same results.
On their way back to B, the physicist in A phone ahead to B and exchange their results.
Both A and B conclude the constancy of the speed of light, i.e. it is independent of the speed of the source and observer.

Upon returning to B, the physicists lay their rulers side by side and confirm they are indeed exactly the same length.
But when they set their clocks side by side they notice that A's clocks are no longer synchronized with B's clocks.
A's clocks have lost 1-sqrt(1-v^2/c^2) seconds for every second marked by B's clocks while A was in motion.(where v is the speed of A relative to B)
They also notice that A's clocks are now, as they were before the trip, marking time at exactly the same rate as B's clocks.
All of the above has been derived by Einstein.

The evidence is -
1. - identical ruler lengths
2. - the speed of light in both labs was measured to be 300,000km/s
3. - A's clocks are no longer synchronized with B's clocks.

The Deductions are:
As the clocks in A were running at the same rate as the clocks in B before and after the trip, but have marked less total time than B's clocks,
the only point at which the clocks in A could have lost time was "during" the trip.
As it was "during" the trip that the physicists in A constructed their ruler, there now exists a logical conflict between the evidence
of the measures of A and B and the definition of speed, for two identical measures of speed attained over identical
measures of length, require each measure mark identical durations of time - a criteria that cannot be met when
the clocks in A run at different rates than the clocks in B.Taken together, the evidence requires we deduce that the magnitude of motion measured by A and expressed as a ratio of Length/Time
was less, due to the equality of length and inequality of time marked by A wrt that marked by B, than the magnitude of motion likewise measured by B.
The magnitude of motion measured by each was the motion of light, therefore the magnitude of the motion of light differs
between A and B while each is in motion wrt the other.
Because both length and time changed for A wrt B "WHILE" they made their measurements, (also derived by Einstein)the ratio of
Length/Time expressed by the measures of A and B remains the same or constant, therefore the ratio of Length/Time
that is the speed of light remains constant.

Are you assuming that because their meter sticks are identical length when they come to rest relative to one another, they must have been identical length when they were in motion relative to one another? If so, why?

No, I am deducing they were not.
I think my answer to DaleSpam above will explain what I am saying about the evidence of the length of their rulers.

 

[atyy]

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 A has accelerated and decelerated relative to an inertial frame, then A's clock is not an inertial clock.

 

[Mentz114]

Chrisc:
Are you saying that the speed of light was 'actually different in the two frames' , and this is the only way to explain the clock and ruler evidence ?

That is not a physically meaningful statement because it is not falsifiable. Nor does it have any physical consequences. It makes no difference what the 'actual' value was. In fact, there's no such thing except to absolutists who never accept relativity and keep plugging on and on and on with their vague meaningless arguments.

 

[Dale]

You just don't get how to establish your claim. I'm sorry, but this is pointless.

The evidence is -

You should be able to take your scenario and use the Lorentz transform to do a step-by-step derivation of these measurements.

The Deductions are:

You should then be able to do a step-by-step derivation of your deductions from your "evidence".

Since I am now bored with this conversation I will go ahead and tell you where you are making your mistake. The key error is that you are treating time and space differently and thereby effectively mixing frames in order to reach your deduction. You note that the clock is running slowly when it is in motion, but fail to note that ruler is contracted when it is in motion: when you take both into account you get c. You also note that the ruler is the same length when at rest and then deliberately ignore that the clock is the same rate when at rest: when you take both into account you get c. In all of these ways you are treating time different from distance, so it is not surprising that you get a "change" in speed.

Finally, you worry about the fact that the clock has "accumulated" less time, but the accumulated time is a measurement of the total spacetime interval along the worldline of the clock. Since the clocks took different paths through spacetime your expectation that they show the same accumulated time seems strange to me. It is like expecting that the odometer of two cars show the same mileage on a trip from New York to Miami despite the fact that one went by way of Washington DC and the other went by way of Denver.

 

[Al68]

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

Many of the laws of physics are this way even going back to Newton. As one example of many, F=ma in an inertial frame, and an inertial frame by definition is a frame in which F=ma.

I think a lot of people would be surprised by how many of our "laws of physics" are true by definition or convention.

Al

 

[Dale]

Many of the laws of physics are this way even going back to Newton. As one example of many, F=ma in an inertial frame, and an inertial frame by definition is a frame in which F=ma.

I think a lot of people would be surprised by how many of our "laws of physics" are true by definition or convention.

You could also define an inertial frame as one in which Newton's 3rd law holds (thus eliminating frames with fictitious forces). Then Newton's 2nd law is not a tautology. Or you could formulate your mechanics in terms of frame-independent geometric quantities as in GR.

My point is that although under one particular set of definitions something may be tautological does not mean it is so in general. F=ma is one example, as is the constancy of c. They are not just true by definition, but they are so well established experimentally that they are often used in definitions of other things like the meter and inertial frames.

 

[Chrisc]

Since I am now bored with this conversation I will go ahead and tell you where you are making your mistake. The key error is that you are treating time and space differently and thereby effectively mixing frames in order to reach your deduction.

That is the key, but it is the mistake you are making not me.
You are mixing the evidence gathered between frames to reach your conclusions.

You note that the clock is running slowly when it is in motion, but fail to note that ruler is contracted when it is in motion:when you take both into account you get c.

No I did not fail to note the ruler is contracted when in motion I stated that quite clearly, and yes taking both into account you get c - while A is constructing their ruler - Not while A and B are at rest in B.

You also note that the ruler is the same length when at rest and then deliberately ignore that the clock is the same rate when at rest:

No, I did not deliberately ignore that the clocks run at the same rate at rest, again I stated that fact quite clearly.

when you take both into account you get c.

Yes when both are taken into account you get c, and this is where you are mixing the evidence of frames.
The length of A's ruler was "constructed" while A's clock WAS running slow, NOT while A is at rest in B where A's clocks
now run at the same rate as B's.

Finally, you worry about the fact that the clock has "accumulated" less time, but the accumulated time is a measurement of the total spacetime interval along the worldline of the clock.

Once again you are stating the formalism that let's you calculate the measures, but does not explain
why the measures are contradictory.

Since the clocks took different paths through spacetime your expectation that they show the same accumulated time seems strange to me.

It seems strange to you because you think I expect them to show the same accumulated time, I do not.
That time dilates and length contracts is not something I expect is wrong.
I expect those, who upon recognizing the power of the formalism to predict this very real change between frames, then realize it signifies something more than the formalism itself.
If you carried around a ruler that performed the transformations of space and time for you, would you do physics by ignoring
that you need the ruler to do physics? You seem to be of the opinion that as long as you can build this ruler of transformation
you should not concern yourself with the fact that you need to build it. Physics is not the act of measurement in and of itself.
Physics is also responsible for explaining the physical significance of those measures.

It is like expecting that the odometer of two cars show the same mileage on a trip from New York to Miami despite the fact that one went by way of Washington DC and the other went by way of Denver.

No it is like expecting the evidence of the odometers will not be contradicted by the drivers claiming they both maintained the same speed and arrived at the same time.
A and B both claim the same speed of light and both show the same length of ruler which is contradicted by
the time accumulated on A's clocks.

World lines define a very real changes between observers.
You cannot lay the world lines of A and B upon each other without transforming their time and space accordingly. That you must transform the time and space should tell you that time and space are transformed between A and B while they are in motion.
When time and space are transformed between them the definition of speed holds the speed of light
constant for both in their transformed dimensions of time and space, which makes it clear that the c of A is not the c of B
while they are in motion, unless and until and because their measures are transformed between them.

 

[matheinste]

Hello chrisc.

Like Dalespam and others, i am quite at a loss as to understanding your argument, but i add the proviso that that in itself does not make it incorrect, although i believe it is. In your last post#47 you say that "world lines define very real changes between observers". These changes are brought about as a consequence of the postulates of SR, one of them being, either implici or implied, that c remains the same in all inertial frames. Given this fact, you cannot then use it to prove the inequality of c in the two frames. In the "real" world what we measure as c is c. We cannot measure the value c as being the same in two inertial frames and then say it is not the real value one of them.

Matheinste.

 

[Dale]

Chrisc, at this point until you post a formal proof I am done because until you do so I apparently cannot show you your error. I have already told you where it is, but because you haven't actually done the math you don't realize that you have it backwards. So without a rigorous derivation it is pointless to continue.

 

[Chrisc]

The experiment consists of the construction of a ruler in A and in B, the length of which is to be determined by
marking the distance a light signal has traveled after one second from emission in each frame.
A and B are inertial frames equipped with identical, synchronized clocks.
Each frame is equipped with the equation of speed
L/T

(1)​

where L is a measure of dimension Length and T a measure of dimension Time.
Let A and B agree the equations of mechanics must hold good in their frames for the laws to be valid.
A moves away from B and maintains a speed of .999c for a period of time sufficient to prepare and perform the experiment.

Labeling the coordinate system of B (x,y,z,t) and that of A (x1,y1,z1,t1)
we find via Einstein's derivations the following transformations of measures
between A and B hold the speed of light constant as measures of L/T in each frame.

while in motion (i.e. conducting the experiment) -
for every measure of B's ruler in the direction of motion (x), A's ruler measures:
¥(x-vt)

(2)​

Where ¥ is the Lorentz factor = 1/sqrt(1-v^2/c^2), x is the length measured by B, v is the velocity of A,
t the time measured by B and c the speed of light measured by B in kilometers per second.

while in motion (i.e. conducting the experiment) and for ever after evidenced by A's clocks -
for every measure of time on B's clocks, A's clocks measure:
¥(t-vx/c^2)

(3)​

A measure of speed in B is:
L/T = x/t

(4)​

A measure of speed in A is:
L/T = x1/t1

(5)​

substituting x1 from (2) and t1 from (3) in (4) a measure of speed in A as calculated by B is:
L/T = ¥(x-vt)/¥(t-vx/c^2)

(6)​

The ratios:
L/T = x/t = ¥(x-vt)/¥(t-vx/c^2)

(7)​

Therefore the ratio of L/T that is c is constant measure by A in A and by B in B.

but in the magnitudes of each dimension measured that comprise this constant ratio L/T
we find a variation between measures by B in B and by B in A where:
(I cannot find the html keystroke for "not equal" so please read n= as "not equal")
x/t n= x1/t1

(8)​

We can find this by entering the magnitude of B's measures of L/T in the units of B:
L/T = x/t = 300,000km/1s

(9)​

and the magnitude of B's measure of L/T in A in the units of B:
L/T = ¥(x-vt)/¥(t-vx/c^2) = 13,413.0735135km/0.044710245045s

(10)​

Since:
300,000 n= 13413.0735135

(11)​

and
1 n= 0.044710245045

(12)​

then although the ratios:
L/T = 300,000km/1s = 13,413.0735135km/0.044710245045s = c

(13)​

c n= 300,000km/0.044710245045s

(14)​

and
c n= 13,413.0735135km/1s

(15)​

Therefore in the magnitude of dimensions L and T measured by B and A respectively, B will find:
x/t n= ¥(x-vt)/¥(t-vx/c^2)

(16)​

Hence the different magnitudes of the dimensions L and T between A and B, result in a different magnitude of constant ratio
of their measures of L/T that is c. In other words, although a constant ratio of motion, the magnitude of motion that is speed of light changes
between A and B while they are in motion.

This is not a radical observation, it is why the physicists in A will age less than those in B. The mechanics of aging are not unique to physicists,
all mechanics are subject to the same relative change between frames in motion (assuming the acceleration of one wrt to the other).

For those who feel inclined to claim that this difference is irrelevant as physics can only deal with real measures, remember the dilation
of A's clocks is a real measure.