SR derived solely from one postulate

And are you defining "meter stick" in physical terms, or are you defining it to agree with whatever coordinate system is being used regardless of the underlying laws of physics?
It is completely arbitrary, so regardles of the underlying physics. One observer might grab a thimble and call it a meter and another might call it the length of a football field. The condition is that both of their ships must be identical, so if compared side by side, they must be nose to nose and tail to tail. So each measuring 50 meters for the length when one uses the thimble to count out meters and the other uses football fields, when the ships are compared directly, they will not have identical lengths.
 

JesseM

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It is completely arbitrary, so regardles of the underlying physics. One observer might grab a thimble and call it a meter and another might call it the length of a football field.
Not if they are both committed to defining their meter (or any other standard length) using the same physical procedure, which was what I was talking about--they can each call a standard thimble a meter, or each call a football field a meter, but they can't use different physical definitions (unless they define a meter in terms of a certain coordinate transformation rather than a standard physical procedure, which was the other option I suggested).
grav-universe said:
The condition is that both of their ships must be identical, so if compared side by side, they must be nose to nose and tail to tail. So each measuring 50 meters for the length when one uses the thimble to count out meters and the other uses football fields, when the ships are compared directly, they will not have identical lengths.
But are the observers' definitions of length supposed to match up with length in their rest frame as defined by the coordinate transformation in question, or not? If they are, then unless the laws of physics are invariant under this coordinate transformation, then the ships can have different rest lengths (i.e. length in each ship's current rest frame) when the ships are in motion relative to one another, despite the fact that they physically line up when brought to rest in the same frame.
 
I don't think you can take the two postulates in the 1905 paper as a logically rigorous axiomatization of SR.
... IMO the purpose of the postulates is just to lay out a philosophical set of criteria to apply to candidate theories.
...
Newton and Einstein were both writing for audiences who had strong preconceptions.
Yes, exactly.

Actually, if you try to take the postulates as the OP wrote:
1) the laws of physics are the same in every inertial frame
2) light is measured travelling isotropically at c in every inertial frame
and you take these too seriously/literally, problems start to arise. On the other hand, one could argue the "essence" of relativity has remained the same since Galileo's time, but that we have made it more precise and rigorous over the years.

I think I remember reading a paper once where, based solely on the idea of "relativity" and assuming space and time is described uniformly in an inertial frame (or alternatively, how they defined an inertial frame) you already can get transformations between frames with an arbitrary constant involved. If this constant is set to c, you get the Lorentz transformations, if you take the limit as the constant -> infinity, you get Galilean transformations.

Anyway, I agree with you that the 1905 paper should not be considered the end-all of SR. Like all physics, a greater understanding of it was gained over time, and there are better presentations of it as well. I would consider a modern definition as: SR postulates all physical laws have poincare symmetry.

I have seen physicists and textbooks take this as the modern statement, but this could easily devolve into what is truly the "essence" of SR. For example, one could not derive from that statement that there is no medium for light (as there is for sound for example). I will say though that my personal taste lines up well with the symmetry definition of SR (probably because I talk physics with so many particle physicists).

actually, we should be able to derive SR from the first postulate alone.
and
The way I've often presented it in the past is that there is only one postulate: the laws of physics are the same in every inertial frame. Maxwell's equations are a law of physics, so #2 follows from #1. To me, the split between #1 and #2 is a historical artifact of the 19th-century picture of physics
I would strongly disagree with this.
If you take that stance, then SR couldn't tell us whether physical laws have Galilean or Lorentz symmetry. I believe SR takes a strong stance on that distinction and says Lorentz symmetry.

Well folks, looks like the show might be over already. Although I believe the mathematics for what I have demonstrated to be solid, an observant poster in http://www.bautforum.com/against-mainstream/100435-sr-derived-solely-one-postulate.html" has already shown me that unless the first postulate is also taken into account, there is nothing that says that different materials might not contract to varying degrees in different frames unless the physics is applied in the same way for all inertial frames, and I am inclined to agree. To that end, if this thread needs to be moved elsewhere in this forum or closed altogether, then that is fine. :)
There's an even simpler/more general way to explain why your original idea wouldn't work (although it is an interested idea, and many people here have probably thought through it at some point in their education):
Since the starting statement only deals with light, at best you could prove that the propagation of electromagnetic waves has Lorentz symmetry. There is no way this can put any restriction on any other physics.

In short: SR is more than just Lorentz transformations between inertial frames.
You cannot claim to have derived SR if you derive the Lorentz transformation (otherwise indeed, Lorentz would be the founder of SR). It is the postulate that ALL physics has that symmetry. Consider muon decay. I love that example because originally SR was "discovered" through electromagnetism ... and everything known at that time was dominated by electromagnetism (the "shape" of material bodies, interactions of gas, etc. all come down to electromagnetic interactions, and while they couldn't prove it so, they already strongly suspected it). But muon decay is not electromagnetic at all. And sure enough, it has the same symmetry. Then we discovered the strong nuclear interactions, and they too had this symmetry! Lorentz's refusal to let go of a "relativistic aether", that while unmeasureable was still there ... missed the great intuitive leap Einstein was able to make. EM was teaching them about a profound symmetry of the universe/physics itself.


A quick aside to show why the improved precision of the modern statements of SR are preferred/necessary, consider this in terms of the "two postulate" statement:
If applying a parity transformation to an inertial coordinate system yields another inertial frame, does this mean special relativity predicts parity invariance?

The historical "two postulate" way is a great intro to SR. The precision of modern restatements are necessary though.
 
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Not if they are both committed to defining their meter (or any other standard length) using the same physical procedure, which was what I was talking about--they can each call a standard thimble a meter, or each call a football field a meter, but they can't use different physical definitions (unless they define a meter in terms of a certain coordinate transformation rather than a standard physical procedure, which was the other option I suggested).
I am steering away from the physics being the same in every frame, so no physical properties are considered other than the length alone by a direct comparison of lengths to determine that they are identical or by measuring both to be the same by using a third ruler of arbitrary length.

But are the observers' definitions of length supposed to match up with length in their rest frame as defined by the coordinate transformation in question, or not? If they are, then unless the laws of physics are invariant under this coordinate transformation, then the ships can have different rest lengths (i.e. length in each ship's current rest frame) when the ships are in motion relative to one another, despite the fact that they physically line up when brought to rest in the same frame.
In this case, a third observer is placed in a frame directly between the other two, where the other two rulers or ships might be travelling in opposite directions at the same relative speed to the third observer. If the two rulers were to travel in the same direction at v to the third observer, then a direct comparison of their lengths can be made by the third observer. If the lengths are equal, then the rulers are identical. Now, assuming that observations are homogeneous in any direction, then if one of the rulers travelled to the left instead of the right of the third observer, it will still have the same length, so having one ruler travel to the left at v and the other to the right at v, if they are still observed to have the same length, then they would have the same length as travelling in the same direction to the right at v in order to gain a direct comparison in the same way, so they are identical. One can always find a frame that exists midway between two other frames in order to make such a comparison.
 

JesseM

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In this case, a third observer is placed in a frame directly between the other two, where the other two rulers or ships might be travelling in opposite directions at the same relative speed to the third observer. If the two rulers were to travel in the same direction at v to the third observer, then a direct comparison of their lengths can be made by the third observer. If the lengths are equal, then the rulers are identical.
What do you mean by "identical", though? Are you concluding that because they have equal length in the third observer's own rest frame, they must have identical rest lengths in their own respective rest frames? If so that would not be a valid conclusion, one can find coordinate transformations where this is not true. If that's not what you're concluding, can you explain what you meant by "then the rulers are identical"? Presumably you don't just mean they have the same length in the frame of the third observer, because in that case "If the lengths are equal, then the rulers are identical" would just be repeating the same thing twice using different phrasing.

edit: Maybe I answered my own question below? When you say the rulers are identical, do you mean that if brought to rest relative to one another, they would line up?
grav-universe said:
Now, assuming that observations are homogeneous in any direction, then if one of the rulers travelled to the left instead of the right of the third observer, it will still have the same length
But does the second postulate justify the assumption that "observations are homogenous in any direction"? How so? The fact that the third observer measures the objects going in opposite directions to have identical lengths in his frame is a statement about coordinate length, while the idea that the two rulers would line up if brought next to each other is a statement about physical length which depends on the form of the laws of physics. For example, if we assumed the laws of physics were Newtonian but used the Lorentz transformation to define different coordinate systems, I don't think it would in fact be true that just because two objects going at equal and opposite velocities in some frame had the same length in that frame, they would necessarily have the same physical length when brought next to each other in the same frame.

Anyway, even if your statement were true, how is this supposed to prove that the Lorentz transformation follows from the second postulate? You never addressed my request for a violation of the second postulate in the non-Lorentzian coordinate transformation I mentioned in post #13. Do you or do you not think that it is possible to find a numerical example where a signal is moving at c in one of the frames given by that transformation, but not moving at c in another?
 
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The two postulates of SR are:

1) the laws of physics are the same in every inertial frame
2) light is measured travelling isotropically at c in every inertial frame

I intend to derive SR by applying only the second postulate alone, ...
It is clear that it is totally obvious to derive SR from the first postulate alone.
You can easily come to the general transformation with one unkown parameter, that you can label 'c' if you want.
The Galilean Relativity appears then as a very special case, almost unlikely.
If Galileo had been able of such an approach, he would have wondered why the special case would prevail and he would have tried to find out an experimental value for 'c'.

One more evidence that mankind (on an historic time scale) is still unable to free itself from empiricism and still has hard times with abstraction!

see: http://adsabs.harvard.edu/abs/1994AmJPh..62..157S
 
It is clear that it is totally obvious to derive SR from the first postulate alone.
I commented on the derivation of general transformations between inertial frames above. I won't repeat everything, but I'll say again:

SR is more than just Lorentz transformations between inertial frames.
You cannot claim to have derived SR if you derive the Lorentz transformation (otherwise indeed, Lorentz would be the founder of SR). It is the postulate that ALL physics has that symmetry.

Also, (again as noted in my previous post), if you take that first postulate as literal, then you would be claiming SR predicts parity invariance and is therefore proven wrong experimentally. The historical "two postulate" way is a great intro to SR. The precision of modern restatements are necessary though.
 

JesseM

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I commented on the derivation of general transformations between inertial frames above. I won't repeat everything, but I'll say again:

SR is more than just Lorentz transformations between inertial frames.
You cannot claim to have derived SR if you derive the Lorentz transformation (otherwise indeed, Lorentz would be the founder of SR). It is the postulate that ALL physics has that symmetry.
Yeah, but if you derive the Lorentz transformation from the first postulate as lalbatros suggested, then since you're assuming the validity of the first postulate that automatically implies the claim that all the laws of physics behave the same way in all the coordinate systems that this transformation gives you. Of course as both you and lalbatros mentioned earlier, the first postulate cannot actually be used to uniquely derive the Lorentz transformation, instead it gives you something similar to the Lorentz transformation but with a parameter in place of c that can take any value, including infinity, in which case you get the Galilei transformation.
 

Fredrik

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(Even though the first part of this post was inspired by JesseM's post above, this isn't really a reply to him. I just wanted to say a few things that I think everyone should keep in mind when they discuss "derivations" of SR).

Right. If we start with the assumption that transition functions between inertial frames are smooth bijections [itex]\phi:\mathbb R^3\rightarrow\mathbb R^3[/itex] that take straight lines to straight lines, and form a group G, then we find that G is either the Galilei group or isomorphic to the Poincaré group.

It doesn't make sense to use "the first postulate" as the starting point, since it's not even a mathematical statement. We have to translate it to a mathematical statement first. I did that above, but there are plenty of other ways to do it. We could e.g. define Minkowski spacetime first, define an inertial frame to be any of the coordinate systems that are associated with the isometries of the metric in a specific way (which I won't describe right now), and then interpret the first postulate to be the idea that the "laws" of all the interesting theories of physics that we can define in this framework can be expressed as relationships between tensor fields on Minkowski spacetime.

There's certainly no obvious way to interpret Einstein's postulates as mathematical axioms, and people tend to interpret them them in a way that lets them prove what they want to prove, and it's rather silly to say that the results obtained this way have been derived from Einstein's postulates.

The way I've often presented it in the past is that there is only one postulate: the laws of physics are the same in every inertial frame. Maxwell's equations are a law of physics, so #2 follows from #1.
I'm not a big fan of this view. What you're describing isn't a derivation of SR from the first postulate. It can at best be described as a derivation of SR from a mathematical reformulation of the first postulate and Maxwell's equations (which of course are already mathematical statements). But Maxwell's equations are absurdly complicated compared to the axiom that the invariant speed is finite, so what you're suggesting isn't a very elegant solution. It also seems to be wildly inconsistent with your own point (b):
(b) ...and there's clearly no reason to describe c as the speed of light. c is a geometrical property of spacetime, the maximum speed of causality.
 
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Using only the 2nd postulate, the measured speed of
light is constant in space for all inertial frames,
(and independent of its source), you can derive
time dilation, addition of velocities, and
variation of length measurements.

Mapping constant linear motion in 2 or 3 dimensions
to the perception space of the observer,
demonstrates the hyperbolic form (gamma).
Time is linear only in radial directions for the observer,
as noted in the popular and simplistic one dimensional expositions.
It's been done already!

My suggestion for grav-u, try to simplify your examples with fewer observers.

My constant reminder: SR is a theory of transformation of coordinates.
 

JesseM

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Using only the 2nd postulate, the measured speed of
light is constant in space for all inertial frames,
(and independent of its source), you can derive
time dilation, addition of velocities, and
variation of length measurements.

Mapping constant linear motion in 2 or 3 dimensions
to the perception space of the observer,
demonstrates the hyperbolic form (gamma).
Time is linear only in radial directions for the observer,
as noted in the popular and simplistic one dimensional expositions.
It's been done already!
Are you disagreeing with my posts #13 and #15 which attempt to show that a coordinate transformation with an arbitrary constant A in place of gamma will still result in a constant speed of light in all frames, satisfying the second postulate?
 
What do you mean by "identical", though? Are you concluding that because they have equal length in the third observer's own rest frame, they must have identical rest lengths in their own respective rest frames? If so that would not be a valid conclusion, one can find coordinate transformations where this is not true. If that's not what you're concluding, can you explain what you meant by "then the rulers are identical"? Presumably you don't just mean they have the same length in the frame of the third observer, because in that case "If the lengths are equal, then the rulers are identical" would just be repeating the same thing twice using different phrasing.

edit: Maybe I answered my own question below? When you say the rulers are identical, do you mean that if brought to rest relative to one another, they would line up?
Right, the rulers have identical lengths if when compared side by side in the same frame, they line up end to end. The physical properties don't matter, as they may be composed of different materials.

But does the second postulate justify the assumption that "observations are homogenous in any direction"? How so?
No, that space is homogeneous is an assumption made, or extra postulate required.

The fact that the third observer measures the objects going in opposite directions to have identical lengths in his frame is a statement about coordinate length, while the idea that the two rulers would line up if brought next to each other is a statement about physical length which depends on the form of the laws of physics. For example, if we assumed the laws of physics were Newtonian but used the Lorentz transformation to define different coordinate systems, I don't think it would in fact be true that just because two objects going at equal and opposite velocities in some frame had the same length in that frame, they would necessarily have the same physical length when brought next to each other in the same frame.
If space is homogeneous, then there shouldn't be any difference in the observation of Bob travelling to the right of Alice at v and Bob travelling to the left of Alice at v, so would be the equivalent of Alice just turning around 180 degrees to face the other way in that case, the observations being the same regardless of direction.

Anyway, even if your statement were true, how is this supposed to prove that the Lorentz transformation follows from the second postulate? You never addressed my request for a violation of the second postulate in the non-Lorentzian coordinate transformation I mentioned in post #13. Do you or do you not think that it is possible to find a numerical example where a signal is moving at c in one of the frames given by that transformation, but not moving at c in another?
I addressed it in post #24. You are right about that of course, but working through the exercise helped me to realize what extra assumption I was making in order to derive the values precisely, which was the assumption that space is homogeneous.
 
It is clear that it is totally obvious to derive SR from the first postulate alone.
You can easily come to the general transformation with one unkown parameter, that you can label 'c' if you want.
The Galilean Relativity appears then as a very special case, almost unlikely.
If Galileo had been able of such an approach, he would have wondered why the special case would prevail and he would have tried to find out an experimental value for 'c'.

One more evidence that mankind (on an historic time scale) is still unable to free itself from empiricism and still has hard times with abstraction!

see: http://adsabs.harvard.edu/abs/1994AmJPh..62..157S
That is not necessarily true, since that would be assuming that the speed of something must still be measured istropically. If nothing is measured isotropically but light had travelled ballistically with the source instead, for instance, then the physics would still be the same in every inertial frame, although we would not have derived SR from the first postulate, but should have instead derived ballistic theory in this case.
 

JesseM

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No, that space is homogeneous is an assumption made, or extra postulate required.
But what do you mean by that phrase, exactly? When Einstein talked about the homogeneity of space, I assumed he was talking about one or more spacetime symmetries like translation invariance or rotation invariance. Newtonian physics certainly respects all these symmetries, but as I pointed out above, if you assume Newtonian laws but use the Lorentz transformation to define your coordinate systems, you can have a situation where in one coordinate system two rigid objects have equal and opposite velocities and equal lengths, but if the two objects are brought to rest relative to one another, their lengths are not equal.
 

bcrowell

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bcrowell said:
The way I've often presented it in the past is that there is only one postulate: the laws of physics are the same in every inertial frame. Maxwell's equations are a law of physics, so #2 follows from #1.
I'm not a big fan of this view. What you're describing isn't a derivation of SR from the first postulate. It can at best be described as a derivation of SR from a mathematical reformulation of the first postulate and Maxwell's equations (which of course are already mathematical statements). But Maxwell's equations are absurdly complicated compared to the axiom that the invariant speed is finite, so what you're suggesting isn't a very elegant solution. It also seems to be wildly inconsistent with your own point (b):
bcrowell said:
(b) ...and there's clearly no reason to describe c as the speed of light. c is a geometrical property of spacetime, the maximum speed of causality.
There is certainly an inconsistency there. That's essentially why the first quote begins with "The way I've often presented it in the past is..." That was the way I used to present it until I understood it more deeply.

Einstein presented the structure of relativity as being closely tied to the theory of electromagnetic waves. With the benefit of another century's worth of hindsight, we can see that that's not really the right way to look at the foundations of relativity. It's just an accident of history that the only fundamental field known in 1905 was the EM field. The more modern point of view is that c is a property of spacetime, and massless particles just happen to propagate at that speed. Before I understood that, I used to teach SR from a point of view that followed Einstein's 1905 presentation, except that I telescoped the two postulates into one.
 
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But what do you mean by that phrase, exactly? When Einstein talked about the homogeneity of space, I assumed he was talking about one or more spacetime symmetries like translation invariance or rotation invariance. Newtonian physics certainly respects all these symmetries, but as I pointed out above, if you assume Newtonian laws but use the Lorentz transformation to define your coordinate systems, you can have a situation where in one coordinate system two rigid objects have equal and opposite velocities and equal lengths, but if the two objects are brought to rest relative to one another, their lengths are not equal.
Rotational invariance sounds about right for what I am describing. In other words, the physics is observed the same in a particular frame in every direction. That is not the same as the physics being the same in every frame, since potentially the physics could still be observed differently in differrent frames, but still the same in every direction in the same frame if space is homogeneous.
 

JesseM

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Rotational invariance sounds about right for what I am describing. In other words, the physics is observed the same in a particular frame in every direction. That is not the same as the physics being the same in every frame, since potentially the physics could still be observed differently in differrent frames, but still the same in every direction in the same frame if space is homogeneous.
But without the first postulate, I don't see why rotational invariance in the laws of physics themselves should imply rotational invariance in the set of coordinate systems you happen to be using. As I said, the laws of Newtonian physics themselves exhibit rotational invariance, but if you use the coordinate systems given by the Lorentz transformation to describe a Newtonian universe, you will find that in some frames physically identical objects are shrunk more when going in one direction than the other. Likewise, in our own relativistic universe, if you choose to use some non-inertial coordinate system you may find that in that coordinate system objects behave differently in different directions, that doesn't mean that the laws of physics aren't rotationally invariant.
 
But without the first postulate, I don't see why rotational invariance in the laws of physics themselves should imply rotational invariance in the set of coordinate systems you happen to be using. As I said, the laws of Newtonian physics themselves exhibit rotational invariance, but if you use the coordinate systems given by the Lorentz transformation to describe a Newtonian universe, you will find that in some frames physically identical objects are shrunk more when going in one direction than the other. Likewise, in our own relativistic universe, if you choose to use some non-inertial coordinate system you may find that in that coordinate system objects behave differently in different directions, that doesn't mean that the laws of physics aren't rotationally invariant.
It is not required with the first postulate. For instance, if an object is moved from a first frame to a second, the physics might say that the object is physically contracted to 1/2 the length in the process. Then, if the physics is the same in all frames, if one were to move the object back from the second frame to the first in the same way, then the object will again contract by 1/2, making it 1/4 the length when coming back to rest in the first frame, at least potentially that could happen although it still doesn't with SR of course because the other frame is always measured as contracted anyway. However, if space is homogeneous, then if one moves the object to a second frame and back again without the actual physics changing the length at all but only in reference to the coordinization of what is observed in space, the coordinization in the first frame is still the same as it was before, and if the physical length of the object hasn't changed as part of our definition of identical, it will be measured at the same original length.
 
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It is clear that it is totally obvious to derive SR from the first postulate alone.
If only the first postulate describes SR, what is preventing us from deriving SR as "observers in all inertial frames measure the speed of light as a multiple of the number of coins they have in their left pocket".

(I'm not joking, I'm new to SR and having trouble imagining SR without a second postulate, be it the constancy of c, Maxwell's equations or some other way to put it)
 
I thought the second postulate actually were: "the speed of light is independent on the speed of the source of light", which is not exactly the same
I've always seen the second postulate stated in terms of light moving at c in every inertial frame--saying only that it's independent of the source's speed doesn't rule out the possibility that the speed of light rays could vary with time or spatial region, for example. Einstein did state the postulate in terms of light moving at c in section 2 of his original 1905 paper: 2. Any ray of light moves in the "stationary'' system of co-ordinates with the determined velocity c, whether the ray be emitted by a stationary or by a moving body. (technically it seems he is saying here only that all light moves at c in one inertial frame, the one he has labeled the "stationary" system, but if you combine that with the first postulate it of course implies that light must move at c in every inertial frame)
lalbatros said:
It is clear that it is totally obvious to derive SR from the first postulate alone.
If only the first postulate describes SR, what is preventing us from deriving SR as "observers in all inertial frames measure the speed of light as a multiple of the number of coins they have in their left pocket".

(I'm not joking, I'm new to SR and having trouble imagining SR without a second postulate, be it the constancy of c, Maxwell's equations or some other way to put it)
Actually, what lightarrow and JesseM stated hits the nail on the head right there, and I should have extended the second postulate accordingly. If we were to try to derive SR from the first postulate alone, then since ballistic theory also includes the same physics in all frames, we would still need some generalized form of the second postulate, something on the order of "the speed of some (massless) particle exists that will always be measured to travel isotropically and not ballistically with the source, so at the same speed whether the source is stationary or moving", but of course light has been determined to be such a massless particle that travels at this isotropic speed. That postulate makes all the difference for what is derived.
 
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JesseM

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It is not required with the first postulate. For instance, if an object is moved from a first frame to a second, the physics might say that the object is physically contracted to 1/2 the length in the process.
What does "physically contracted" mean? You can't really compare the lengths of objects moving at different velocities in a coordinate-independent way.
grav-universe said:
However, if space is homogeneous,
You haven't explained what you mean by that phrase. Is it a physical statement, or a coordinate-dependent one? As I said, with Newtonian physics we normally say that the laws of physics have translation and rotation symmetries (which I guess just means that it's possible to come up with a coordinate system such that moving the origin or rotating the axes doesn't change the equations of the laws of physics, not that this would be true in all coordinate systems), yet at the same time it's also true that if you use the Lorentz transformation to define your family of coordinate systems in a universe with Newtonian laws, there will be some frames where objects which have the same length when moving at equal and opposite velocities would not have the same length if brought to rest relative to one another. Do you disagree about that, or not fully understand it? If you do understand and agree, then would you say "space is homogeneous" is true of the Newtonian universe in general, or is it a statement that's meant to be specific to your choice of coordinate system so that you might say that in a Newtonian universe, space is homogeneous in some coordinate systems but not others?
grav-universe said:
then if one moves the object to a second frame and back again without the actual physics changing the length at all but only in reference to the coordinization of what is observed in space
Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).
 
What does "physically contracted" mean? You can't really compare the lengths of objects moving at different velocities in a coordinate-independent way.
By "physically contracted", I am assuming I mean the same thing that you are referring to for a Newtonian universe, where objects physically contract rather than being a coordinatization effect. If it were Newtonian, then lengths could physically change when brought back to the initial frame, but not if the contraction is a coordinate effect.

You haven't explained what you mean by that phrase. Is it a physical statement, or a coordinate-dependent one? As I said, with Newtonian physics we normally say that the laws of physics have translation and rotation symmetries (which I guess just means that it's possible to come up with a coordinate system such that moving the origin or rotating the axes doesn't change the equations of the laws of physics, not that this would be true in all coordinate systems), yet at the same time it's also true that if you use the Lorentz transformation to define your family of coordinate systems in a universe with Newtonian laws, there will be some frames where objects which have the same length when moving at equal and opposite velocities would not have the same length if brought to rest relative to one another. Do you disagree about that, or not fully understand it? If you do understand and agree, then would you say "space is homogeneous" is true of the Newtonian universe in general, or is it a statement that's meant to be specific to your choice of coordinate system so that you might say that in a Newtonian universe, space is homogeneous in some coordinate systems but not others?
Right, coordinate only. I suppose that has to be included with the homogeneity of space, that the observations are coordinate effects, not physical.

Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).
Right, two clocks cannot compare rates when passing each other, but a frame set between the two frames of the clocks can compare rates for the clocks travelling at the same relative speed in opposite directions when viewing the time dilation as a coordinate effect within the homogeneity of space.
 

JesseM

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By "physically contracted", I am assuming I mean the same thing that you are referring to for a Newtonian universe, where objects physically contract rather than being a coordinatization effect.
What does "physically contract" mean? Consider that when we say rigid objects don't contract in a Newtonian universe, we just mean that in any of the inertial reference frames given by the Galilei transformation, the object's coordinate length remains constant regardless of change in velocity. So, for an object to contract in Newtonian physics would presumably just mean that all Galilean frames would agree the length contracted. It's still a "coordinatization effect" in the sense that all statements comparing the length of objects in relative motion depend on your choice of coordinate system, as far as I can tell.
grav-universe said:
If it were Newtonian, then lengths could physically change when brought back to the initial frame, but not if the contraction is a coordinate effect.
I was talking about rigid objects in a Newtonian universe, though. In the set of inertial frames given by the Galilei transformation, these objects never change length regardless of velocity. I take it you agree that even for such rigid objects in a Newtonian universe, if we instead use the frames given by the Lorentz transformation, then we can find cases where one frame measures two objects traveling at equal and opposite velocities to have equal lengths, yet when these rigid objects are brought to rest next to each other they are found to have unequal lengths?
grav-universe said:
Right, coordinate only. I suppose that has to be included with the homogeneity of space, that the observations are coordinate effects, not physical.
But if you're imposing the requirement that laws of physics exhibit rotational invariance in the particular coordinate systems you're using--as opposed to just saying there has to be some coordinate system where they exhibit rotational invariance--then this is basically a special case of the first postulate. Instead of requiring that the laws of physics are the same in all the coordinate systems allowed by the transformation, you're requiring that the laws of physics are the same in each subset of coordinate systems which can be related to one another by a spatial rotation transformation.
JesseM said:
Again, I don't see what it means to talk about "actual physics changing the length" as opposed to just a coordinate-dependent change in length, since there doesn't seem to be any coordinate-independent way to compare length for objects moving at different velocities (contrast with time dilation, where even though there is no coordinate-independent way to say which of two clocks is ticking slower at a given instant, if two clocks cross paths twice we can say which elapsed less total time between the crossings, since proper time is coordinate-independent).
grav-universe said:
Right, two clocks cannot compare rates when passing each other, but a frame set between the two frames of the clocks can compare rates for the clocks travelling at the same relative speed in opposite directions when viewing the time dilation as a coordinate effect within the homogeneity of space.
Huh? Of course each clock can compare their own rate to the rate of the other clock, in terms of their own frame. Why shouldn't they be able to? The answer will be coordinate-dependent (and thus the two frames will have different answers about which clock is ticking at a slower rate), but then so is the answer found by the frame "set between the two frames of the clocks" (I don't understand the phrase 'when viewing the time dilation as a coordinate effect within the homogeneity of space'--instantaneous time dilation is always coordinate-dependent, and the phrase 'within the homogeneity of space' is unclear). My point was that unlike these comparisons of instantaneous rates of ticking which are always frame-dependent, if you compare total elapsed time on two clocks between two local meetings of these clocks then there will be an objective answer that does not depend on your choice of frame (since this is just the proper time between two events on a given clock's worldline, and proper time along a worldline is frame-invariant)
 
What does "physically contract" mean? Consider that when we say rigid objects don't contract in a Newtonian universe, we just mean that in any of the inertial reference frames given by the Galilei transformation, the object's coordinate length remains constant regardless of change in velocity. So, for an object to contract in Newtonian physics would presumably just mean that all Galilean frames would agree the length contracted. It's still a "coordinatization effect" in the sense that all statements comparing the length of objects in relative motion depend on your choice of coordinate system, as far as I can tell.
That was the definition of what I referring to with "physically contract", but maybe it would be easier to define what I mean by a coordinate effect. With a coordinate effect, if two objects are at rest with identical lengths and then one accelerated to some v, then turned around and came back to rest again, the lengths would still be identical.

I was talking about rigid objects in a Newtonian universe, though. In the set of inertial frames given by the Galilei transformation, these objects never change length regardless of velocity. I take it you agree that even for such rigid objects in a Newtonian universe, if we instead use the frames given by the Lorentz transformation, then we can find cases where one frame measures two objects traveling at equal and opposite velocities to have equal lengths, yet when these rigid objects are brought to rest next to each other they are found to have unequal lengths?
I don't see that with the coordinate effects of SR, no. Times, sure, since different time dilations will give different readings on clocks brought back together, but if an observer reads identical time dilations or identical lengths for two clocks or rulers with the same relative speed and observations of each, regardless of their directions, then if brought to rest in the same frame next to each other, they should continue to have an equal tick rate and the same lengths.

But if you're imposing the requirement that laws of physics exhibit rotational invariance in the particular coordinate systems you're using--as opposed to just saying there has to be some coordinate system where they exhibit rotational invariance--then this is basically a special case of the first postulate. Instead of requiring that the laws of physics are the same in all the coordinate systems allowed by the transformation, you're requiring that the laws of physics are the same in each subset of coordinate systems which can be related to one another by a spatial rotation transformation.
I agree with that to a point, but the observations in different inertial frames for the same relative speed to those frames can still potentially be different, and the physics can also be different overall. However, since applying homogeneous space as I did still worked out to having the same physics in each frame, then that must be the natural result.


Huh? Of course each clock can compare their own rate to the rate of the other clock, in terms of their own frame. Why shouldn't they be able to? The answer will be coordinate-dependent (and thus the two frames will have different answers about which clock is ticking at a slower rate), but then so is the answer found by the frame "set between the two frames of the clocks" (I don't understand the phrase 'when viewing the time dilation as a coordinate effect within the homogeneity of space'--instantaneous time dilation is always coordinate-dependent, and the phrase 'within the homogeneity of space' is unclear). My point was that unlike these comparisons of instantaneous rates of ticking which are always frame-dependent, if you compare total elapsed time on two clocks between two local meetings of these clocks then there will be an objective answer that does not depend on your choice of frame (since this is just the proper time between two events on a given clock's worldline, and proper time along a worldline is frame-invariant)
I mean that two clocks with a relative speed between them cannot directly compare tick rates any more than two frames can directly compare rulers in order to determine that they are identical. It would require a third frame set between these two speeds in homogeneous space that can directly compare them this way.
 

JesseM

Science Advisor
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That was the definition of what I referring to with "physically contract", but maybe it would be easier to define what I mean by a coordinate effect. With a coordinate effect, if two objects are at rest with identical lengths and then one accelerated to some v, then turned around and came back to rest again, the lengths would still be identical.
So even in a Lorentz ether theory, which imagines that things "really contract" due to their velocity relative to the ether, you'd say it's just a coordinate effect since if the objects are brought to rest relative to each other (and thus have identical velocities relative to the ether) they will be the same length again?
grav-universe said:
I don't see that with the coordinate effects of SR, no.
I wasn't exactly talking about SR, I was talking about applying the Lorentz transformation in a Newtonian universe where the laws of physics are not Lorentz-invariant (physicists would usually take the Lorentz-invariance of the laws of physics to be the definition of SR). You're saying that even in this case, you don't see why rigid bodies could be measured to have equal lengths in a frame where they have equal and opposite velocities, but not have equal lengths when brought to rest relative to one another?

Imagine that in a Newtonian universe, we define a single unprimed frame so that it's a standard Newtonian inertial frame where Newton's laws apply. Then we define a family of other frames using the Lorentz transformation on the unprimed coordinates:

x' = gamma*(x - vt)
t' = gamma*(t - vx/c^2)

Now suppose that in the unprimed frame, we have a rigid measuring-rod that's at rest and 10 light-seconds long, and another rigid measuring rod that's moving in the +x direction at 0.8c and is 6 light-seconds long. Now consider a coordinate system, given by the transformation above, that is moving at 0.5c in the +x direction. In this coordinate system, the first rod is moving in the -x' direction at 0.5c, while the second rod is moving in the +x' direction at 0.5c (I can prove this if you like, but consider the relativistic velocity addition formula, which says that if the unprimed frame observes the primed frame to be moving at 0.5c and the primed frame observes the second measuring-rod to be moving at 0.5c, then the unprimed frame will observe it to be moving at (0.5c + 0.5c)/(1 + 0.5*0.5) = 0.8c).

Let's say that in the unprimed frame, both measuring-rods start with their left end at x=0 at t=0. Since rod #1 is at rest in the unprimed frame, rod #1's left end will have position as a function of time given by:
x(t) = 0 light-seconds
And rod #1's right end will have position as a function of time given by
x(t) = 10 l.s.

Meanwhile since rod #2 is moving at 0.8c, rod #2's left end will have:
x(t) = 0.8c*t
And since rod #2 is 6 light-seconds long in the unprimed frame, rod #2's right end will have:
x(t) = 0.8c*t + 6

Now consider two events in the unprimed frame: (x=0, t=0) and (x=10, t=5). Obviously the first event lies on the worldline of both the left end of rod #1 and the left end of rod #2 (i.e. it's the event of the left ends of both rods lining up), since we established that both their left ends started at x=0 at t=0. But the second event happens to lie on the worldline of both the right end of rod #1 and the right end of rod #2 (so it's the event of the right ends of both rods lining up), since the right end of rod #1 remains fixed at x=10, and since plugging in t=5 into the function x(t) = 0.8c*t + 6 gives x = 0.8*5 + 6 = 4 + 6 = 10.

Finally, consider what happens when you use the coordinate transformation to find the coordinates of these two events in the primed frame. The first event will become (x'=0, t'=0) while the second event will become (x'=8.66, t'=0). So the key here is that these two events are simultaneous in the primed frame--the left ends of both rods line up at x'=0 at t'=0, while the right ends of both rods line up at x'=8.66 at t'=0. Since "length" in a given frame is just the distance between two ends of an object at a single moment in that frame, both rods must have equal lengths of 8.66 light-seconds in the primed frame. And as I said before, they also have equal and opposite velocities of 0.5c in the primed frame.

But remember that in the unprimed frame, rod #1 is 10 light-seconds long while rod #2 is 6 light-seconds long. And the unprimed frame is just an ordinary Newtonian inertial frame, where according to Newtonian laws objects will not change length when they change velocities. So if rod #2 is brought to rest next to rod #1, they will still be different lengths.
grav-universe said:
I mean that two clocks with a relative speed between them cannot directly compare tick rates any more than two frames can directly compare rulers in order to determine that they are identical.
Still doesn't make any sense to me. In each frame you can compare the tick rates of the two clocks--for example, if the two clocks are moving at 0.8c relative to one another, then in the rest frame of clock #1 it'll be true that clock #2 is ticking at 0.6 the rate of clock #1, while in the rest frame of clock #2 it'll be true that clock #1 is ticking at 0.6 the rate of clock #2. So here each frame is comparing the tick rates of the two clocks in terms of their own coordinates. I don't know what you mean when you say talk about two clocks comparing tick rates, as opposed to using a single frame to compare the tick rates of two clocks.
grav-universe said:
It would require a third frame set between these two speeds in homogeneous space that can directly compare them this way.
The third frame would just give a third answer about the relative tick rates of the two clocks in terms of its own coordinates, no better or no worse than the answers found in either of the two clock rest frames. So like I said, I really don't understand what point you are trying to make here.
 

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