The Light Speed Postulate and its Interpretation in Derivations of the Lorentz Transformation

Physics isn't concerned about appearances, and it is not philosophy. The derivation is sufficient.

 

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I was not thinking that x1 and x2 are independent, I defined them as independent . x1 is the fixed location where experimenter 1 has placed his detector in his system, x2 is the fixed location where experimenter 2 has placed (independently) his detector in his system. There is by definition no connection between the two. Still the speed of light postulate must hold

(1) x1=c*t1
(2) x2=c*t2

but (x1,t1) and (x2,t2) do not describe the same event as they relate to different detectors. And if you want the event 1 to coincide with event 2 in frame 2, you would need the additional constraint

(3) x2=x1'

 

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No it wouldn't. In a derivation, it is made clear what is assumed and what then follows and readers can follow the logic for themselves. You almost certainly want to mention the thing you intend to show can be derived, as that is what makes your work worth reading.

 

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I don't see where you're going with this.

If it needs to be said, then an implicit assumption* in relativity is that there's just one spacetime in which events occur, so that different reference frames are just different ways of using coordinates to describe where and when things happen. In particular, if a light pulse reaches experimenter 2's detector at position x2' at time t2'**, then this event occurs in experimenter 1's reference frame somewhere and at some time, describable with some coordinates x2 and t2, whether experimenter 1 makes an effort to independently observe/measure this for himself or not.

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* It's usually implicit in introductions to special relativity. It's made very explicit in general relativity, where spacetime is modelled as a four-dimensional pseudo-Riemannian manifold and special relativity is recovered as the special case where this manifold has zero curvature everywhere.

** I'm following the convention of putting primes on experimenter 2's coordinates and no primes on experimenter 1's coordinates, since I find this clearer.

 

I think tsmid envisions two different light signals being detected by two different detectors. The two different detection events are completely unrelated:
(1) x1 = c*t1
(2) x2 = c*t2

Then he chooses the special case where x1/x2=t1/t2=q=1 at which point he is also choosing the special case where x1=x2 and t1=t2:
(3) x1/x2 = t1/t2 = q = 1
(4) t1 = t2 and x1 = x2

Then he jumps to the following conclusion which is incorrect under SR:
(7) x1' = x2

Where x1' is the transformed coordinate of x1.

 

No, there is only one light signal/flash being emitted from the origin of both reference frame (with the light source at rest in frame 1 but moving in frame 2,) but this flash is detected by two different detectors, one at the fixed distance x1 in frame 1, the other independently at the fixed distance x2 in frame 2. And if you apply the postulate invariance of the speed of light to this scenario, Eqs.(1) and (2) must hold (otherwise Eq.(2) would depend on v, the velocity of detector 2 relative to the light source).

The crucial point is that in this case x1 and x2 are not coordinate transforms of each other, as they are not related to the same but different detection events. They are both independent variables (the distance at which each experimenter chooses to place his detector in his reference frame).

The crucial point is that in this case x1 and x2 are not coordinate transforms of each other, as they are not related to the same but different detection events. They are both independent variables (the distance at which each experimenter chooses to place his detector in his reference frame).

 

That seems okay, for what it's worth. But both the Galilean and Lorentz transformation equations deal with coordinates of events, and yet you seem to be only concerned with two proper distances.

You might as well define rod 1 to be at rest in system 1 with its left endpoint fixed to the origin of system 1, and with a proper length of x1. Then define rod 2 to be at rest in system 2 with its left endpoint fixed to the origin of system 2, and with a proper length of x2. Then, when you choose q=1, you are essentially choosing x1=x2 which simply means that the rods would be identical in length when held stationary with respect to one another.

In that case, allowing for the possibility of length contraction (as SR has) then the length of rod 1 as measured by system 2 could be length-contracted to a length of x1/gamma where gamma>1. Assuming the light was emitted at t'=0 in system 2, at that time, the location on the x' axis where the right endpoint of rod 1 would be located would be x'=x1/gamma.

But it will take some time for the light to travel along the moving, length contracted rod. Why don't you calculate that amount of time as t'=(x1/gamma)/(c-v) for the case where the light and the rod are moving in the same direction, or t'=(x1/gamma)/(c+v) for the case where the light and the rod are moving in opposite directions?

Then you could calculate the location on the x' axis where the right endpoint of rod 1 would be located at that time using x1'=(vt')+(x1/gamma). Instead you just conclude x1'=x2 without any justification.

 

Yes.

The length of rod 1 as measured by system 2 is irrelevant for the determination of the speed as defined above. It is given by the time it takes for the light signal to get from one endpoint of rod 2 to the other. Rod 1 doesn't enter into this definition at all. But IF you want to define the speed of light in system 2 by the time it takes the light signal to get from one endpoint of rod 1 to the other (the question is why would you), then this definition must be consistent with the first one.

 

Right, but then any transformation equation would also be irrelevant, because rod 2 is stationary in system 2. Only rod 1 is moving at velocity v through system 2. So what exactly do you mean by your equation (7) x1'=x2 if not to introduce a transformation of some sort? And if it is supposed to be a transformation, then how do you justify that you derived it by ignoring the moving rod 1 altogether, and relying only on the stationary rod 2?

 

Yes, there is obviously no transformation of the endpoint coordinates of rod 1 necessary if you use the endpoints of rod 2 to determine the speed of light in the latter's reference frame. But Einstein needs the transformation to derive the Lorentz transformation, so he uses instead the endpoints of rod 1 to define the speed of light in the reference frame of rod 2. Now assuming the light signal starts at the origin in both reference frames, this means that when it reaches the endpoint of rod 2, this location must also coincide with the endpoint of rod 1 i.e. x2=x1', because otherwise you would get a different speed of light from both methods (as the distance the light covers within the same time t2 would be different)

 

Okay, but I don't think Einstein would have any problem if system 2 wanted to use rod 2 to measure the speed of light. There would be no transformation required, but it would work just fine.

No, that does not follow. As long as x1'/t1'=x2/t2 then the speed of light is still c because x2/t2=c. It does not matter if x1'=100*x2 (for example) as long as t1'=100*t2.

Your error is assuming the same time t2 for light traveling two different distances.

 

Here is a simple diagram that does not even consider length contraction:

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In the top diagram, the rod is stationary, and the light reaches both the left and right endpoints of the rod at the same time, t=2.

In the bottom diagram, the rod is moving at some velocity v in the rightward direction. Note that the light reaches the left endpoint of the rod some time before t=2, and the light reaches the right endpoint of the rod some time after t=2.

The speed of light is still the same in both cases, but the times of the light detection events are not the same.

 

That, like your diagram above, is what you are tacitly implying on the basis of our everyday experience (i.e. the Galilei transformation), where we have the equations

x1 =ut
x2 =(u-v)*t = x1-vt = x1'

But here we have instead

x1=ct
x2=ct

and x2 does in no way, shape or form depend on v, and thus can not be represented as a function of x1' (whether it is x1' or x1'/100)

 

Frame 2 measures the speed of rod1 as v, and frame 2 also measures the speed of the light as c, and therefore, frame 2 must measure the "closing speed" between the light and the moving rod1 as c-v (for the case where the light travels in the same direction that the rod is moving), or c+v (for the case where the light travels in the opposite direction that the rod is moving). This remains so under both Galilean relativity and SR.

Right, x2 does not depend on v. But x1' does depend on v, assuming your notation represents the place on the x axis where frame 2 measures the light reaching the end of the moving rod1. Since one variable depends on v and the other variable does not, you cannot just write x1'=x2 without any justification. That is my objection to your approach, and it still stands.

 

In the Galilean case, the appearance of the 'closing velocity' u-v in x2=(u-v)*t expresses the fact that the speed of an object is not invariant. If you fire a bullet with speed u from a car that is itself moving with speed v on the ground (against the direction of the bullet), then ground based detectors will measure the speed of this bullet as u-v (i.e. the bullet will reach the detector location x2=(u-v)*t after time t). This can be verified without any reference to events related to separate detectors co-moving with the car. And if you replace the bullet by a light signal, it can be analogously verified that x2=ct (independent of v).

I am not requiring x2=x1', Einstein is. He defines the propagation speed of the signal in frame 2 by events related to detectors stationary in frame 1 rather than frame 2 itself. As explained above, experience shows that you can do this in Galilean relativity, but not if the speed is supposed to be invariant, because if we have x1=ct and x2=ct, it follows that x1'=x1, which obviously is not possible (unless v=0).

 

Because the point of the exercise was to consider two different scheme for describing the geometry of phenomena that happen in reality. Thus you need two systems for describing every event and a means of converting back and forth between the two systems. Here Einstein limited himself to Cartesian coordinates so that the equations of motion that described inertial trajectories were simple.

 

I think we all might be talking about different things. Please look at the top half of post #48 where I say to tsmid, "You might as well define rod 1 to be at rest in system 1 with its left endpoint fixed to the origin of system 1, and with a proper length of x1. Then define rod 2 to be at rest in system 2 with its left endpoint fixed to the origin of system 2, and with a proper length of x2. Then, when you choose q=1, you are essentially choosing x1=x2 which simply means that the rods would be identical in length when held stationary with respect to one another." Note that tsmid agrees to this by saying, "Yes."

In that case, x2 would not be dependent upon v at all. But of course x1' must depend on v, because (x1', t1') represents the transformation from system 1 to system 2 of the coordinates of an event which happens at (x1, t1) in system 1. In that case, x1' would not be equal to x2 at all, unless v=0.

At least that is what I thought this was supposed to be about, but I could very well be wrong. It does not help that tsmid uses notations with 1 and 2 instead of the usual primed and unprimed coordinates, and it also does not help that he considers x1 and x2 to be distances rather than coordinates, etc.

 

You can obviously always convert between two systems back and forth to describe an event (assuming that there is a one-to-one correspondence between the two), but this is only half the story here. The other half is how you define the speed of an object/signal in the respective system. And the point is that the latter can be done without using such a conversion at all, simply by using different events for this in the two systems.

 

1 and 2 relate here to different events (independent measurements), the prime (by common convention) to the same event. So x1' and x1 relate to the same event, x2 and x1 not (even if x2=x1').

If the distance starts at the origin (which is the convention I chose for simplicity), then it is equal to the coordinate.

 

Yes, I figured as much. But I don't think rpernner realized that your (x2,t2) event is not the same event as your (x1,t1) event. The only reason he agrees with your claim that x1'=x2 is on the grounds that he assumes one would want to describe a single event from both systems, and therefore he thinks you are saying that system 2 measures the single event at x1'=x2 and t1'=t2 (even though that would actually be unnecessarily redundant notation).

So when you claim x1'=x2 for the case where q=1 and therefore x1=x2, you now clearly admit that the x1' and x2 in that equation do not pertain to one single event. But the whole purpose of a transformation equation is to describe one single event as measured by both systems. Therefore you cannot claim either x1'=x2 or x1'=x1 as transformation equations.