Invariance of the speed of light

[DAC]

SR section 1.7. Einstein states if a train and light beam are moving in the same direction, the speed of the light as seen from the train is c-v. ( c being the speed of light and v the speed of the train ). c-v being smaller than c is resolved by time dilation or length contraction.
My question is, in c-v, what is the train's speed relative to? Given light's speed is being measured relative to the train, shouldn't the train's speed also be measured relative to the train? As the train's speed relative to itself is zero, this means c-v , no matter what v's speed is, always gives c.

Separate question. Do objects aligned sideways, or vertically along their direction of travel contract, as objects aligned lengthways do? If so shouldn't the terminology change? e.g direction of motion contraction?

 

[ShayanJ]

Einstein is talking about a Galilean speed transformation from one inertial frame to another applied to light. Then he says Length contraction and time dilation clean up that mess that Galilean transformation caused!
About your second question, we're not supposed to pack all of the meaning of something inside its name. People hear about length contraction and if they want, they can come and learn it actually happens for dimensions parallel to the direction of motion.

 

[Nugatory]

If the speed of the train relative to some observer somewhere is $v$, then the speed of the light relative to the train according to that observer will be $c-v$. You've described the particular case in which the observer is at rest relative to the train; another interesting case is the one in which the observer is at rest relative to the ground.

 

[Nugatory]

Separate question. Do objects aligned sideways, or vertically along their direction of travel contract, as objects aligned lengthways do? If so shouldn't the terminology change? e.g direction of motion contraction?

The contraction is indeed in the direction of travel, and that's not necessarily the longest dimension of the object in question. However, the "length contraction" terminology has had better than a century to put down roots, so we're stuck with it.

 

[harrylin]

Einstein is talking about a Galilean speed transformation from one inertial frame to another applied to light. Then he says Length contraction and time dilation clean up that mess that Galilean transformation caused!

If the speed of the train relative to some observer somewhere is $v$, then the speed of the light relative to the train according to that observer will be $c-v$. You've described the particular case in which the observer is at rest relative to the train; another interesting case is the one in which the observer is at rest relative to the ground.

Just to clarify to DAC the above apparent difference of opinion: I agree with Shyan that here Einstein is casually referring to the Galilean transformation, and with "with respect to" he subtly means "as measured by". In his example, the Galilean transformation gives the answer of classical physics to the question: 'what relative speed* between the light and the train will a "train observer" measure?'.
It happens that the "Galilean" answer is the same as the answer to the question about the "relative speed" between the train and the light as measured by the "embankment observer", because in classical physics it is assumed that the result of measurements is independent of the measurement system that is used. The Lorentz transformation gives a different answer, so that according to the "train observer" the speed of light relative to the train is indeed c-0=c.

*note: in textbooks two partially incompatible definitions of "relative speed" are used, and that can be confusing. Happily Einstein, DAC, Nugatory and myself use the same definition.

 

[ghwellsjr]

SR section 1.7. Einstein states if a train and light beam are moving in the same direction, the speed of the light as seen from the train is c-v. ( c being the speed of light and v the speed of the train ). c-v being smaller than c is resolved by time dilation or length contraction.
My question is, in c-v, what is the train's speed relative to? Given light's speed is being measured relative to the train, shouldn't the train's speed also be measured relative to the train? As the train's speed relative to itself is zero, this means c-v , no matter what v's speed is, always gives c.

I presume that by SR section 1.7 you mean this passage:

http://www.bartleby.com/173/7.html

I think your confusion is that you are assuming that everything Einstein is saying up to the last paragraph is what he actually believes but in reality, he is painting the picture of what Lorentz and most other scientists believed up to 1905 and became known as the Lorentz Ether Theory (LET). In that theory, light propagates at c in all directions only in one frame, the rest state of the ether. In reality, it is impossible to measure or determine the propagation speed of light, otherwise referred to as the one-way speed of light. We can easily measure the round trip speed of light using a single clock or timer and a mirror some measured distance away. This was explained by Lorentz using length contraction and time dilation so that even an observer moving with respect to the ether would still get c as the value of the round-trip speed of light but he would not claim that the light took the same amount of time to get to the mirror as it took for the reflection to get back to him. The observer on the train is an example of such an observer and he would believe under LET that the light did not propagate at c with respect to him but with respect to the ether as symbolized by Einstein by the embankment. Therefore, the observer on the train would claim that the light propagating away from him is traveling at less than c by his speed relative to the embankment.

In the last paragraph, Einstein points out that it is possible for the propagation speed of light (the one-way speed) to be c for both the embankment observer and the train observer which is the basis for his theory of Special Relativity.

 

[harrylin]

I presume that by SR section 1.7 you mean this passage:

http://www.bartleby.com/173/7.html

I think your confusion is that you are assuming that everything Einstein is saying up to the last paragraph is what he actually believes but in reality, he is painting the picture of what Lorentz and most other scientists believed up to 1905 [..]

Yes, nearly so: Einstein painted the classical, or "Galilean" transformation. But already in 1895 Lorentz had come up with a synchronization that he named "local time", and the corresponding time transformation differed from the Galilean transformation (and this one was specified as a first order approximation):
local time t' = t − vx/c² or, in his notation for system velocity along x:
t' = t − p_xx/V²
The development from "classical" to "relativistic" evolved in little steps. :)
- https://en.wikisource.org/wiki/Tran...odies/Section_V#Reduction_to_a_resting_system.

 

[ghwellsjr]

Yes, nearly so: Einstein painted the classical, or "Galilean" transformation. But already in 1895 Lorentz had come up with a synchronization that he named "local time", and the corresponding time transformation differed from the Galilean transformation (and this one was specified as a first order approximation):
local time t' = t − vx/c² or, in his notation for system velocity along x:
t' = t − p_xx/V²
The development from "classical" to "relativistic" evolved in little steps. :)
- https://en.wikisource.org/wiki/Tran...odies/Section_V#Reduction_to_a_resting_system.

Maybe so, but that is not what Einstein was discussing in the referenced chapter. He stated:

"Prominent theoretical physicists were therefore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which were contradictory to this principle."

If they rejected the principle of relativity, then there was no need for either the Galilean transformation or Lorentz's transformation. I'm not saying that Einstein painted the historical picture accurately or correctly but in order to answer the OP's question, we need to focus on what Einstein said, not what we think he should have said.

 

[DAC]

Getting back to my original question, which i'll reframe, is time dilation/length contraction necessary when all moving observers see their own speed as zero? Given c-v when v equals 0, equals c.

 

[Nugatory]

Getting back to my original question, which i'll reframe, is time dilation/length contraction necessary when all moving observers see their own speed as zero? Given c-v when v equals 0, equals c.

You may be thinking about it backwards. Time dilation and length contraction are logical consequences of the invariance of the speed of light, not things that are needed to explain that invariance.

We start with Einstein's postulate that the speed of light is the same for all observers. From this, we derive the Lorentz transformations, and then from the Lorentz transformations we derive all sorts of interesting things including time dilation and length contraction.

 

[harrylin]

[..] in order to answer the OP's question, we need to focus on what Einstein said, not what we think he should have said.

Exactly!

 

[ShayanJ]

Getting back to my original question, which i'll reframe, is time dilation/length contraction necessary when all moving observers see their own speed as zero? Given c-v when v equals 0, equals c.

You seem to be very confused!
Let's assume Galilean transformations are correct. Now imagine an observer trying to measure the speed of light. He finds a value which we don't know what is it but he calls it c. Then he starts to move with speed v and again tries to measure the speed of light. This time he measures the speed of light to be $c\pm v$. Then he changes speed to w and again measures the speed of light and this time he gets $c\pm w$. So he considers the first frame, the rest frame and the other two as two other inertial ones moving w.r.t. the first one and also each other.
But because we assumed nothing about the mentioned frames, he could as well called one of the moving frames, as his rest frame and so he would call the first measured speed $c'$ and then in other measurements, he would get $c'\mp v$ and $c' \mp w$.
That could be very natural about speed of ordinary things we see around us. So why strange about light? It turns out that we have reasons to say this actually is strange about light. Because its not something like other things which can be accelerated to different speeds. For light, somehow its speed is one of its defining features. Also we have Maxwell's equations that give us a very special speed for light. So we say light should have only one speed. But this fact should somehow be reconciled with the previous experiment that the guy measured different speeds for light in different frames.
The solution people proposed before SR, was to say that there is a special inertial frame which defines absolute rest and anyone at rest in that sense, measures c as the speed of light and others won't agree but others are wrong because they're in the wrong frame of reference. This theory was troublesome and complicated.
Einstein didn't like this theory and so tried to propose another solution. He said there is no special inertial frame that is the "right" frame and defines absolute rest. He somehow brought democracy between people in different inertial frames and said they all have equal right in doing physics so they should all measure c for the speed of light. This idea made things simpler than the ether theory(which had a special frame defining absolute rest).

 

[harrylin]

Getting back to my original question, which i'll reframe, is time dilation/length contraction necessary when all moving observers see their own speed as zero? Given c-v when v equals 0, equals c.

While I also largely agree with the other answers, I see no reason to think that you "have it backwards" or that you are maybe "very confused".
By chance my post #7 (which only was intended as a historical footnote), can be elaborated into an answer to your question about time dilation/length contraction. As I stressed, the 1895 transformations were admitted to be only good for first order approximations, giving a simple "fix" for apparent simultaneity only. The remaining issue can be more easily understood when you let light in the train bounce back with a mirror. If the light bounces back parallel to the motion, this will take longer than in rest according to the reckoning with the railway track's reference system. This will also be the case for light that is bounced back laterally, but not by the same amount. In order to fix the transformations to second order, Lorentz therefore added length contraction as well as frequency change; it is indeed necessary for the relativity principle to hold, while still assuming the validity of Maxwell's laws. Einstein showed next how the same can be derived directly from two fitting postulates, in the way he explains in the book that you are reading.

 

[Chronos]

Harrylin, I think you [as many in his time did] missed the fundamental concept Einstein was attempting to express - there is no fundamental reference frame.

 

[harrylin]

Harrylin, I think you [as many in his time did] missed the fundamental concept Einstein was attempting to express - there is no fundamental reference frame.

Hi Chronos, I am well aware of Einstein's concepts at different times in his life; and his philosophy has little to do with the topic here. It appears therefore that you misunderstand some of the clarifications in this thread. We all agree - necessarily - on the essential points, so that all of the answers may be helpful for DAC. For clarity it is useful to explain the same thing in different ways.

 

[ghwellsjr]

Getting back to my original question, which i'll reframe, is time dilation/length contraction necessary when all moving observers see their own speed as zero? Given c-v when v equals 0, equals c.

According to the reference, Einstein defined v as that speed of the train relative to the embankment. If you want to make v equal to 0 then the train is stationary on the embankment and no time dilation/length contraction are necessary to explain anything.

But you are missing the point. As I stated in post #6, it is impossible to measure the one-way speed of light which is what Einstein was talking about. Even on the embankment or a train stationary on the embankment, it is not possible to measure how long it takes for light to travel a given distance. What could be measured was the average round-trip speed of light using a mirror as I also stated in post #6 and it always comes out to be c for any inertial observer which would include an observer stationary on the embankment where v=0 or an observer traveling at any constant speed v relative to the embankment. That's a fact.

But that fact is explained differently by Lorentz and by Einstein.

Lorentz says that the light is propagating (one-way) at c only relative to the embankment and that the reason the moving train observer still measures c for his round-trip measurement is because his ruler has been physically length contracted and his clock has been physically time dilated. Note that since this measurement necessarily involves a round trip, the actual speed is c-v for the first part of the trip and c+v for the second part of the trip. Note that it takes a long time for the light to get to the mirror at c-v and a very short time for the light to get back at c+v but the total time is such that the measurement comes out to be c.

Einstein says that there is no conflict if the train observer assumes that the light propagates at c and his ruler is not length contracted and his clock/timer is not time dilated. But this means that in the moving train's rest frame, it is the embankment observer whose ruler is length contracted and whose clock is time dilated.

So I think the answer you are looking for is "yes". Length Contraction and Time Dilation are always involved both for Lorentz and Einstein whenever there are relatively moving observers--it's just a question of which one it applies to.

 

[Fantasist]

Time dilation and length contraction are logical consequences of the invariance of the speed of light.

Only if you require that the measurements of the coordinates of a light signal in two reference frames should be connectable by means of a velocity dependent transformation.

 

[Nugatory]

Only if you require that the measurements of the coordinates of a light signal in two reference frames should be connectable by means of a velocity dependent transformation.

I suppose so... But I'm finding it hard to imagine an alternative to that assumption. Consider a photodetector connected to an explosive device. We can assign all sorts of different and velocity-dependent coordinates to the location of the photodetector, but the bomb will either be detonated if the flash of light hits the detector or not detonated if the flash of light does not hit the detector, no matter what coordinates we use. That implies the existence of a transformation between the various choices of coordinates.

 

[Dale]

Only if you require that the measurements of the coordinates of a light signal in two reference frames should be connectable by means of a velocity dependent transformation.

That is not an assumption. That is a definition of what we mean by a transformation between frames. Inertial coordinate systems are also connectible by translations and rotations, but when we refer to "different frames" we are specifically referring to coordinate systems that are connected by a boost.