Could SR not be built from only one postulate?

[WannabeNewton]

@atyy and PAllen, the distinction between Galilean space-time and Minkowski space-time is the latter presumes a space-like foliation with separate Euclidean temporal and spatial metrics whereas the latter contains no such foliation and presumes the Minkowski metric. From here one can demand that equations (or "the laws of physics") be Lorentz (or Poincare) covariant and get SR. Why are inertial frames required for any of this? Equations will be Lorentz covariant as long as they are written in terms of spinor or tensor representations of the Lorentz group which is a completely frame-independent condition. After the dust settles we can simply define an inertial frame in terms of zero rotation and acceleration. I honestly don't see any need to talk about inertial frames before the dust settles.

 

[PAllen]

@atyy and PAllen, the distinction between Galilean space-time and Minkowski space-time is the latter presumes a space-like foliation with separate Euclidean temporal and spatial metrics whereas the latter contains no such foliation and presumes the Minkowski metric. From here one can demand that equations (or "the laws of physics") be Lorentz (or Poincare) covariant and get SR. Why are inertial frames required for any of this? Equations will be Lorentz covariant as long as they are written in terms of spinor or tensor representations of the Lorentz group which is a completely frame-independent condition. After the dust settles we can simply define an inertial frame in terms of zero rotation and acceleration. I honestly don't see any need to talk about inertial frames before the dust settles.

The discussion was do you need anything other than a physical principle of relativity+homgeneity+isotropy, to get SR. To apply POR, you need a non-circular physical definition of inertial frames. Then you need something physical (experiment or law) to select SR vs. GR (Galilean relativity, not General relativity).

 

[atyy]

@atyy and PAllen, the distinction between Galilean space-time and Minkowski space-time is the latter presumes a space-like foliation with separate Euclidean temporal and spatial metrics whereas the latter contains no such foliation and presumes the Minkowski metric. From here one can demand that equations (or "the laws of physics") be Lorentz (or Poincare) covariant and get SR. Why are inertial frames required for any of this? Equations will be Lorentz covariant as long as they are written in terms of spinor or tensor representations of the Lorentz group which is a completely frame-independent condition. After the dust settles we can simply define an inertial frame in terms of zero rotation and acceleration. I honestly don't see any need to talk about inertial frames before the dust settles.

Yes, for defining SR, that's the modern way. But the old way using the Principle of Relativity and the speed of light still works.

Incidentally, were you actually commenting on Sugdub's question whether an inertial frame can be determined without reference to the laws of physics, assuming SR is true? In theory, yes. In practice, no, since one has to build some instruments to measure acceleration, rotation etc. And in calibrating them, the laws of physics will be used.

 

[Sugdub]

I'm still at a complete loss as to why we need to talk about the "laws of physics" or any of the other verbiage.

An inertial frame is simply one which has both zero 4-rotation and 4-acceleration; every excellent textbook on relativity I have ever seen defines it in this way, in terms of kinematical concepts. What's the issue?

In which way can a mathematical concept such as a coordinate system be physically zero-accelerated? May be you assume its origin remains collocated with a zero-accelerated physical body? Then how can one characterize a zero-accelerated body unless a postulate states that its accelerated or non-accelerated state of motion is an objective property of this body?

Apart from setting a postulate as suggested above, one necessarily comes back to invoking physical laws, leading to circular statements as per the Wikipedia dedicated article:

Within the realm of Newtonian mechanics, an inertial frame of reference, or inertial reference frame, is one in which Newton's first law of motion is valid. However, the principle of special relativity generalizes the notion of inertial frame to include all physical laws, not simply Newton's first law... According to the first postulate of special relativity, all physical laws take their simplest form in an inertial frame, ...

 

[WannabeNewton]

In which way can a mathematical concept such as a coordinate system be physically zero-accelerated?

Who said anything about a coordinate system having zero acceleration? All I said was the frame has zero acceleration. All this means is the object of interest at rest in the frame has zero acceleration.

Then how can one characterize a zero-accelerated body unless a postulate states that its accelerated or non-accelerated state of motion is an objective property of this body?

There is no need for such a postulate. No such postulate exists in SR. It is simply a consequence of the definition in both Newtonian and relativistic mechanics.

Apart from setting a postulate as suggested above, one necessarily comes back to invoking physical laws, leading to circular statements as per the Wikipedia dedicated article

http://articles.adsabs.harvard.edu//full/1967QJRAS...8..252D/0000252.000.html

Also just because inertial frames are defined in a certain way in Newtonian mechanics doesn't mean we need to follow the same tired route in relativity. As atyy mentioned there is a much more coherent and fundamental way to approach SR, as opposed to the antiquated approach taken by Einstein and some of his contemporaries.

 

[WannabeNewton]

The discussion was do you need anything other than a physical principle of relativity+homgeneity+isotropy, to get SR. To apply POR, you need a non-circular physical definition of inertial frames. Then you need something physical (experiment or law) to select SR vs. GR (Galilean relativity, not General relativity).

Ah I see; I probably should have read the entire discourse.

In theory, yes. In practice, no, since one has to build some instruments to measure acceleration, rotation etc. And in calibrating them, the laws of physics will be used.

I don't disagree there.

 

[strangerep]

(Not sure whether I should stay involved with this, but... maybe one more post...)

First, let's replace the phrase:

"The laws of physics are identical in all inertial frames."

by the equivalent:

"The outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame."

Then we can seek a physical definition of "inertial frame"...

The task is not to express an observer's local experiences without referring to "laws of physics" (or, equivalently, the "outcome of any physical experiment [...]"). Rather, the task is to relate one observer's experiences to those of others. That's why it's called "relativity".

Of course each observer already possesses some physical concepts such as (local) position, time, and devices for measuring such things locally, and hence also a concept of differential ratios thereof (velocity, acceleration, etc). An observer equipped with suitable accelerometers, gyroscopes, etc, can tell whether heshe is accelerating or not. If heshe detects no acceleration, then heshe is an inertial observer. (In this sense, non-acceleration is indeed a property that an observer can ascribe to hisherself.)

The "inertial reference frame" imagined by an inertial observer is simply an intuitively natural extrapolation of locally performable operations, e.g., moving 1 step to the right, waiting until 1 minute has elapsed according to hisher clock, etc. To be an "inertial motion", such operations must be non-accelerative once completed, meaning that (e.g.,) after the spatial translation of moving 1 step to the right heshe still detects no acceleration.

Then we assume that (the mathematical expressions of) these operations form a Lie group, since that seems to be the case for strictly local operations, at least as far as they can reach.

Such local experience can be extended a bit further by the "radar" method, if the observer has a light emitting source (e.g., a torch) and a device for receiving light and noting its direction (e.g., a pair of eyes that implement binocular vision). Such parallax methods allow an observer to relate remote events to hisher imagined reference frame.

(I'll skip the additional complications/ambiguities that arise beyond the useful range of the radar method or more sophisticated parallax techniques.)

 

[strangerep]

I probably should have read the entire discourse.

Tsk, tsk.

 

[PAllen]

Such local experience can be extended a bit further by the "radar" method, if the observer has a light emitting source (e.g., a torch) and a device for receiving light and noting its direction (e.g., a pair of eyes that implement binocular vision). Such parallax methods allow an observer to relate remote events to hisher imagined reference frame.

Well, if you are talking about drawing conclusions from the POR + experiments, before using the Radar method you first have to establish the constancy of light speed (no need to worry about one way / two way if we are assuming isotropy). Having done such an experiment, you already find SR selected rather than Galilean relativity.

 

[strangerep]

Well, if you are talking about drawing conclusions from the POR + experiments, [...]

Actually, I was trying to describe how one might reach the concept of an inertial frame, beginning at a physically plausible starting point. Probably, I should have ditched the radar stuff in my previous post, since it confuses the logic -- as you pointed out.

[Edit: ... and thank you for pointing it out, btw. ]

 

[Sugdub]

An observer equipped with suitable accelerometers, gyroscopes, etc, can tell whether heshe is accelerating or not.

Yes, however one cannot demonstrate that the devices called “accelerometer”and “gyroscope” actually measure “accelerations” and “changes in orientation” without invoking somehow the laws of physics. Such devices can provide a more accurate assessment of our state of motion than our senses, once it has been demonstrated that they are fit to purpose, but beforehand the need remains for an independent definition of “inertial” (see below).

I agree with many of your statements which I find better than mine, in particular the need to refer to a consensus between observers. However I think a more logic presentation can be proposed if the postulates leading to SR are set at a deeper level, as follows.

… non-acceleration is indeed a property that an observer can ascribe to hisherself.

This is not derived from your assumptions, it is a postulate, the first postulate from which everything will flow: my sensation reveals an objective property, a qualification upon which all observers will agree. Then, assuming I do not sense any acceleration or rotation in my body, my state of motion can be defined as “inertial”. Hence a 4-coordinate system attached to my body and reflecting my time flow provides an inertial frame of reference. I think this definition is immune from contradictions whilst removing the indirect reference to laws of physics (via "accelerometers").

A second postulate is required in order to derive the Lorentz transformation (assuming no further constraint will reduce the generality of the development alongside Rindler's approach), whereby the difference between representing oneself at rest or in constant motion is non-objective, conventional.

Thanks to the first postulate, the equivalence relationship “to be in constant relative motion” structures the family of all possible frames of reference in such a way that the associated transformation will map an inertial frame onto an inertial frame. Thanks to the arbitrariness set by the second postulate, one may conclude that:

"The outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame."

In conclusion two independent postulates are required for developing SR, which are of a more general nature that those often proposed.

 

[Fredrik]

the first postulate from which everything will flow: my sensation reveals an objective property, a qualification upon which all observers will agree. Then, assuming I do not sense any acceleration or rotation in my body, my state of motion can be defined as “inertial”. Hence a 4-coordinate system attached to my body and reflecting my time flow provides an inertial frame of reference. I think this definition is immune from contradictions whilst removing the indirect reference to laws of physics (via "accelerometers").

There's no significant difference between specifying that human senses should be used and specifying that an accelerometer (defined by instructions on how to build one) should be used. All you have done is to use a different device to detect acceleration.

 

[PAllen]

In conclusion two independent postulates are required for developing SR, which are of a more general nature that those often proposed.

That's fine so far. But to conclude SR is true rather than Galilean relativity, you must do some experiments that distinguish them. Your distinguishing experiment need not have anything to do with light, which you may consider a virtue of this approach (e.g. it could be time dilation).

 

[Sugdub]

That's fine so far. But to conclude SR is true rather than Galilean relativity, you must do some experiments that distinguish them. Your distinguishing experiment need not have anything to do with light, which you may consider a virtue of this approach (e.g. it could be time dilation).

I'm not sure how to interpret your statement. In the following I will assume you mean that a reference to actual experimental results is necessary before one can state that the postulates we are dealing with lead to the Lorentz transformation of SR as opposed to the Galilean transformation of the Newtonian mechanics.

If this is the case I disagree. I already explained why in a previous input but I welcome comments.

The Lorentz- or the galilean- transformation are the only possible outcomes but they are exclusive. So the choice between both must be sorted out before reaching any conclusion. On the one hand, the Lorentz transformation necessarily deals with a 4-dimensions space-time coordinate system since it entails a dependency between space coordinates (typically x) and the time coordinate. This dependency is constitutive of SR: it cannot be eliminated without abandoning the theory.

On the other hand, the Lorentz transformation gets squeezed down to the galilean transformation by negating this dependency between space and time so that only a 3-dimensions space coordinate system is required, complemented with a separate, independent, invariant 1-dimension time coordinate system. This reduction from 4 to 3+1 dimensions corresponds to a loss of generality. This loss can be triggered by various hypotheses or constraints acting on top of the postulates, for example by imposing that the time coordinate is left invariant.

The Lorentz transformation appears to be the most general solution for the required transformation, the only one which deals with a 4-dimensions integrated coordinate system. Since both possible outcomes are exclusive, a conclusion in favor of the galilean transformation could only be reached following the conscious acceptance of a reduction of generality, e.g. by adding that the time coordinate remains invariant or equivalently by adding the possibility for a signal propagating at an infinite speed, or by adding that simultaneity at a distance is a reality. The addition of any of the above constraints on top of the postulates will trigger the derivation of the galilean transformation. Otherwise, the Lorentz transformation will be arrived at.

 

[PAllen]

I'm not sure how to interpret your statement. In the following I will assume you mean that a reference to actual experimental results is necessary before one can state that the postulates we are dealing with lead to the Lorentz transformation of SR as opposed to the Galilean transformation of the Newtonian mechanics.

If this is the case I disagree. I already explained why in a previous input but I welcome comments.

The Lorentz- or the galilean- transformation are the only possible outcomes but they are exclusive. So the choice between both must be sorted out before reaching any conclusion. On the one hand, the Lorentz transformation necessarily deals with a 4-dimensions space-time coordinate system since it entails a dependency between space coordinates (typically x) and the time coordinate. This dependency is constitutive of SR: it cannot be eliminated without abandoning the theory.

On the other hand, the Lorentz transformation gets squeezed down to the galilean transformation by negating this dependency between space and time so that only a 3-dimensions space coordinate system is required, complemented with a separate, independent, invariant 1-dimension time coordinate system. This reduction from 4 to 3+1 dimensions corresponds to a loss of generality. This loss can be triggered by various hypotheses or constraints acting on top of the postulates, for example by imposing that the time coordinate is left invariant.

The Lorentz transformation appears to be the most general solution for the required transformation, the only one which deals with a 4-dimensions integrated coordinate system. Since both possible outcomes are exclusive, a conclusion in favor of the galilean transformation could only be reached following the conscious acceptance of a reduction of generality, e.g. by adding that the time coordinate remains invariant or equivalently by adding the possibility for a signal propagating at an infinite speed, or by adding that simultaneity at a distance is a reality. The addition of any of the above constraints on top of the postulates will trigger the derivation of the galilean transformation. Otherwise, the Lorentz transformation will be arrived at.

That just becomes another assumption. In addition to POR, isotropy, homogeneity you add: of the two remaining possibilities, pick the one you like. Any 'natural philosopher' of circa 1800, brought up to speed on the math, would say you obviously want to add a postulate of simultaneity to rule out the nonsensical alternative (what we call SR). In fact, Newton had such postulate: time flows equably and consistently for all observers. Unlike for symmetries, I don't see any convincing argument for one conclusion over another except experiment.

 

[Sugdub]

...In fact, Newton had such postulate: time flows equably and consistently for all observers.

Indeed it was this a priori assumption which prevented physicists looking for a transformation of 4-coordinate events. Once Einstein understood that his postulate on the invariance of the speed of light was incompatible with this a priori assumption, he could set more general 4-dimensions equations in view of producing a genuine transformation of space-time events, … and later on he understood that a further generalization could encompass the non-uniform gravity field... This is the way science is progressing.

Unlike for symmetries, I don't see any convincing argument for one conclusion over another except experiment.

Imposing the invariance of the time coordinate also acts as an asymmetry since it restricts the transformation to 3 out of 4 coordinates of any event. There is no fundamental difference.

 

[PAllen]

Imposing the invariance of the time coordinate also acts as an asymmetry since it restricts the transformation to 3 out of 4 coordinates of any event. There is no fundamental difference.

How is "no time invariance" fundamentally more natural "time invariance"? Note, time is distinguishable in SR (the geometry is pseudo-Riemannian, not Riemannian). Or you can say, manifolds are preferable to fiber bundles (?).

Also, note that the value of c only chooses units not physics. Thus, there exactly two physically distinguishable choices to make (there is or is not an invariant speed), not infinite versus one. You can choose "time invariance" or "c invariance". You are not making headway convincing me that anything other than experiment chooses SR over Galilean relativity.

 

[strangerep]

You [Sugdub] are not making headway convincing me that anything other than experiment chooses SR over Galilean relativity.

 

[WannabeNewton]

Tsk, tsk.

 

[Sugdub]

How is "no time invariance" fundamentally more natural "time invariance"? Note, time is distinguishable in SR (the geometry is pseudo-Riemannian, not Riemannian). Or you can say, manifolds are preferable to fiber bundles (?).

Also, note that the value of c only chooses units not physics. Thus, there exactly two physically distinguishable choices to make (there is or is not an invariant speed), not infinite versus one. You can choose "time invariance" or "c invariance". You are not making headway convincing me that anything other than experiment chooses SR over Galilean relativity.

Yes, I agree that in order to derive the Lorentz transformation a finite constant (c) with the dimension of a speed must somehow be injected according to which “instantaneous actions at a distance” get excluded. Obviously my previous inputs overlooked it. Thanks for this lesson.

Still I wish to challenge the rationale for invoking either a “law of physics” (such as Maxwell's equations or the “invariance of the speed of light”) or some experimental results (such as the Michelson and Morley experiment) as a valid foundation for the formal derivation of the Lorentz transformation. In my view, statements about the world, how it is, how it works, what happens there,... are just inappropriate. We should not accept any a priori statements in this range.

SR provides a formal framework into which a model / description / simulation of the world and associated phenomena will get hosted. The purpose of that framework is to specify how our formal description of the same phenomena should be evolved when we change the perspective from which this description is proposed. The SR framework should be physically neutral, it should provide an empty structure, in the same way as in GR the actual curvature of space-time relies upon the effective presence of energy or mass.

The two postulates I have proposed for SR are not about the world, they are about us: we sense accelerations and rotations whereas we do not sense speed or rest. The symmetries we have discussed are not about the world itself, they deal with our a priori concepts of space and time: we only grasp differences in position, in orientation, in time, not their absolute values ... But the addition of c as an external constraint, somehow linked to a belief in the existence of a “law of nature”, in order to complement this set of postulates and symmetry rules does not fit well. It has no bearing to the meta-rules which the SR formal framework must comply with.

Actually we all know why c is necessary, why “no signal can travel at an infinite speed”, why we must impose this constraint on our formal framework. It is not a postulate about the world and neither an external constraint derived from experiments. It reflects the causal structure we impose to any abstract construction deserving to be labelled as a “physics theory”. We can't accept that our theories claim “explaining” phenomena through “instantaneous actions at a distance”. Causes and effects must be ordered in time otherwise they can't be distinguished from each other. As long as our physics theories abide to some concept of causality, they must fit within a formal framework imposing a maximum limit for the speed of any signal invoked in a causal explanation.

My conclusion is that we impose the existence of c as a consequence of our own internal mental structure, it is not imposed to us by external experiments and neither by some miraculous knowledge about the world. Comments are, of course, welcome.

 

[strangerep]

[...] But the addition of c as an external constraint, somehow linked to a belief in the existence of a “law of nature”, in order to complement this set of postulates and symmetry rules does not fit well.

It is not imposed as an "external constraint, linked to a belief [...]". Only its value is determined by experiment.

The Galilean case is simply an approximation of what happens in the Poincare case as $|v|/c$ becomes small. (Most people just say "as $c$ becomes large", but it's better to have a dimensionless quantity when taking limits.)

In this sense there are not 2 separate cases, but only one -- and it comes with a universal invariant speed $c$, whose value must be determined by experiment.

My conclusion is that we impose the existence of c as a consequence of our own internal mental structure, [...]

Rubbish. If someone puts an axe through your skull, the causal consequence (your death) is not dependent on your mental structure (i.e., it doesn't depend on whether you're awake, asleep, or in a coma).

 

[PAllen]

It is not imposed as an "external constraint, linked to a belief [...]". Only its value is determined by experiment.

The Galilean case is simply an approximation of what happens in the Poincare case as $|v|/c$ becomes small. (Most people just say "as $c$ becomes large", but it's better to have a dimensionless quantity when taking limits.)

In this sense there are not 2 separate cases, but only one -- and it comes with a universal invariant speed $c$, whose value must be determined by experiment.

Rubbish. If someone puts an axe through your skull, the causal consequence (your death) is not dependent on your mental structure (i.e., it doesn't depend on whether you're awake, asleep, or in a coma).

I don't quite agree with this. Infinity is not a value, so the Galilean case is more properly viewed as the alternative where there is no invariant speed. As for different values of c, IMO this is just units. There is no way to physically distinguish between different values of c without fixing many other things. Thus, for me, the two choices are between a pseudo-riemannian metric of (1,-1,-1,-1) on a 4-manifold vs. a fiber bundle with invariant t and invariant distance (in the Euclidean 3-manifold). These two (no more) physically distinguishable options fall out of the derivation.

 

[Fredrik]

I don't quite agree with this. Infinity is not a value, so the Galilean case is more properly viewed as the alternative where there is no invariant speed.

I think of what we're doing simply as finding all groups of permutations of $\mathbb R^4$ that take straight lines to straight lines, so to me it makes the most sense to acknowledge that for each such group, there's a set of lines that aren't just taken to straight lines, but are invariant under transformations that preserve the origin. In the case of Galilean transformations, these are the lines that are drawn horizontally in a spacetime diagram. It makes sense to think of them as representing motion with infinite speed.

As for different values of c, IMO this is just units. There is no way to physically distinguish between different values of c without fixing many other things. Thus, for me, the two choices are between a pseudo-riemannian metric of (1,-1,-1,-1) on a 4-manifold vs. a fiber bundle with invariant t and invariant distance (in the Euclidean 3-manifold). These two (no more) physically distinguishable options fall out of the derivation.

That's the physical way of looking at it. (Nothing wrong with that of course ). The mathematical way is that the groups with different positive values of $c^2$ are all isomorphic. The ones with negative values of $c^2$ have to be ruled out by other methods. In the 1+1-dimensional case, it's sufficient to assume that 0 is an interior point of the set of velocities associated with the elements of the group, i.e. that there's an open interval containing 0 such that for each v in that interval, there's a transformation with velocity v.

 

[strangerep]

I don't quite agree with this. Infinity is not a value, so the Galilean case is more properly viewed as the alternative where there is no invariant speed.

In the derivation, the constant emerges most naturally as having dimension of inverse speed squared. So one could just as easily think of it as taking a limit as an inverse speed constant approaches 0, (which is certainly a number).

As for different values of c, IMO this is just units. There is no way to physically distinguish between different values of c without fixing many other things. Thus, for me, the two choices are between a pseudo-riemannian metric of (1,-1,-1,-1) on a 4-manifold vs. a fiber bundle with invariant t and invariant distance (in the Euclidean 3-manifold). These two (no more) physically distinguishable options fall out of the derivation.

Physical experiments can only give results within error bounds applicable for the apparatus. E.g., an apparatus with insufficient accuracy to probe relativistic scales could only say that the value of "c" is greater than <whatever> (expressed in terms of local length and time standards, of course).

(Strictly speaking, there are certainly cases in mathematics where a sequence of elements of an abstract space $S$ converges, yet the limit is not in $S$. But in our case, $S$ is a 1-parameter space of groups of linear transformations of the solution manifold of $d^2x/dt^2=0$ (with $c$ being the "parameter" in $S$). I don't think such subtleties affect this case unless one invokes some more subtle topological issues. But I could be wrong about that.)

 

[PAllen]

In the derivation, the constant emerges most naturally as having dimension of inverse speed squared. So one could just as easily think of it as taking a limit as an inverse speed constant approaches 0, (which is certainly a number).

Physical experiments can only give results within error bounds applicable for the apparatus. E.g., an apparatus with insufficient accuracy to probe relativistic scales could only say that the value of "c" is greater than <whatever> (expressed in terms of local length and time standards, of course).

(Strictly speaking, there are certainly cases in mathematics where a sequence of elements of an abstract space $S$ converges, yet the limit is not in $S$. But in our case, $S$ is a 1-parameter space of groups of linear transformations of the solution manifold of $d^2x/dt^2=0$ (with $c$ being the "parameter" in $S$). I don't think such subtleties affect this case unless one invokes some more subtle topological issues. But I could be wrong about that.)

But once you go from looking just at transforms to mathematical structure, there are just two possibilities. There either is or isn't a pseudo-riemanian manifold, and if there is, it can always be given the metric with determinant -1. Here we do have the case that the limit is not a Minkowski space, while all the other elements are.

As for 1/speed squared, that is 1, in natural units. Then , the limit is 1.

 

[strangerep]

But once you go from looking just at transforms to mathematical structure, there are just two possibilities. There either is or isn't a pseudo-riemanian manifold, and if there is, it can always be given the metric with determinant -1. Here we do have the case that the limit is not a Minkowski space, while all the other elements are.

Indeed.

This reminds me of an observation that Gerry Kaiser made a long time ago. If one performs the Inonu--Wigner contraction while remaining in a Poincare irrep, one gets instead the centrally-extended Galilei group, including canonical (position--momentum) commutation relations like those of QM. This is very different from what we get from a "naive" contraction to ordinary Galilei. Yet, (afaik), the centrally-extended Galilei group doesn't show up if we start from the POR alone, but only if we go to Poincare and then contract within an irrep. [IIUC]

Evidently, one gets different answers depending on whether the group's homogeneous space is considered more (or less) important physically than its irreps. (QFT would suggest that the irreps are more important.)

I recall that there can be subtleties in the distinction between representation--contraction vs abstract group contraction, but I'm no expert on that.

As for 1/speed squared, that is 1, in natural units. Then , the limit is 1.

Yes, but that just shows that one cannot usefully perform the Inonu--Wigner group contraction (i.e., Poincare->Galilei) in that way.

 

[dmummert]

I will confess that I skipped to the end, I have another lecture this evening. If I missed this point being introduced, I apologize. All of this hinges on one other assumption as yet unclaimed - that of invariant physical space. If in fact any of the dimensional constants change - as they do, I have been led to believe - then the two postulates cannot be combined into one. The proposal suffers from one other deficit, in that they are co-dependent. SR was ever meant to measure conditions in local groups, and in fact could never state with any authority that these conditions held at every point. Those ideas are left over from Newtonian Era thinking - which admittedly still serves for all kinds of ballistics problems.

During my skim, I remember seeing a statement to the effect that all infinities are equal. In fact, they are not. It matters a great deal HOW you got there, and how fast. Physics hates infinities, and except as a concept, there is no proof they exist (a geometry proof notwithstanding). You can calculate with them, and you can avoid calculating with them. If you have performed a division by zero, umm, look for an error somewhere. I remember as a lark once in high school physics I ran some calculations with infinities, but labeled them. I got some interesting results. Some of those equations showed up in college texts later. No notes, sorry. The same went for imaginary numbers, which I was told I could safely ingnore, since it 'didn't make sense to carry it through'. Twistor theory.

The bottom line, I think, is to carefully examine your base assumptions, or at least state ALL of them.

 

[strangerep]

I will confess that I skipped to the end,

That's unwise -- it's too easy to make yourself look silly.

[...] All of this hinges on one other assumption as yet unclaimed - that of invariant physical space.

If you had studied earlier posts in this thread you might have seen some where I tried to make a distinction between an abstract space of certain parameters (chosen by reference to an inertial observer's experience) used to express the free equations of motion, and finding the maximal symmetry group thereof. The more concrete notions of relative physical quantities emerge as parameters in those symmetry transformations.

So assumptions about the enveloping parameter space were not "unclaimed" -- but merely implicit in the use of the theory of dynamical symmetry groups.

If in fact any of the dimensional constants change - as they do, I have been led to believe -

Really? I bet you don't have experimental evidence of variability of the local speed of light in vacuum.

[...] The proposal suffers from one other deficit, in that they are co-dependent. SR was ever meant to measure conditions in local groups, and in fact could never state with any authority that these conditions held at every point.

Again, if you had studied the whole thread you might have seen that issues surrounding spacetime homogeneity were not ignored. Without spacetime homogeneity, one gets a more general theory (known as "projective" or "de Sitter" relativity). Adding the assumption of homogeneity reduces this to the usual Poincare relativity.

During my skim, I remember seeing a statement to the effect that all infinities are equal.

Really? I don't remember that. If you see silly statements like that you should reference them in a quote box. Anyone who has done some university--level pure math knows that there is more than one kind of infinity.

In fact, they are not. It matters a great deal HOW you got there, and how fast.

These are just sweeping statements, conveying little meaning. How one "got to infinity"?? What are you talking about? (And if you answer, make sure it remains on-topic for this thread.)

The same went for imaginary numbers, which I was told I could safely ingnore, since it 'didn't make sense to carry it through.

No idea what you're talking about here. One certainly can't "ignore" imaginary numbers in modern theoretical physics.

Twistor theory.

Is this just a random phrase or did you have a point about twistor theory?

The bottom line, I think, is to carefully examine your base assumptions, or at least state ALL of them.

Actually, the "bottom line" is that you should study nontrivial threads properly before relieving yourself in them by ill-informed brain--f*rts.

 

[Fredrik]

During my skim, I remember seeing a statement to the effect that all infinities are equal. In fact, they are not.

While it's true that there's more than one infinity in math, I don't think it's relevant here. A line in a spacetime diagram is either horizontal or not.

 

[dmummert]

That's unwise -- it's too easy to make yourself look silly.

I agree. However, I also agreed to be neutral and productive. In that spirit, that is the only thing I'm going to say about the matter of the rest of the of the response.

If you had studied earlier posts in this thread you might have seen some where I tried to make a distinction between an abstract space of certain parameters (chosen by reference to an inertial observer's experience) used to express the free equations of motion, and finding the maximal symmetry group thereof. The more concrete notions of relative physical quantities emerge as parameters in those symmetry transformations.

So assumptions about the enveloping parameter space were not "unclaimed" -- but merely implicit in the use of the theory of dynamical symmetry groups.

Sorry - as you say - I do look silly by not reading all of the intervening text. Did, at some point, someone come to the conclusion that SR could not be built from one postulate? Because there seem to be a great number of constraints.

Really? I bet you don't have experimental evidence of variability of the local speed of light in vacuum.

My original statement;

-->If in fact any of the dimensional constants change - as they do, I have been led to believe -

And my response; no, I have not performed any experiments that show the variability of the speed of light under any conditions. In a vacuum is deliberately left out. In fact, I will go out on a limb and say that it has shown itself to be remarkably steady in the brief period that we have been measuring it. Some other conditions may support a longer (unobserved) flat value. But, I would not have made the interjection without at least some knowledge of the opinions of others who had performed experiments and come to the conclusion that some of the constants were 'drifting'. I have no knowledge of their equipment nor experimental method. I do, however, have a thought experiment whereby the speed of light could change and you could not measure it.

--->During my skim, I remember seeing a statement to the effect that all infinities are equal. In fact, they are not.

Really? I don't remember that. If you see silly statements like that you should reference them in a quote box. Anyone who has done some university--level pure math knows that there is more than one kind of infinity.

These are just sweeping statements, conveying little meaning. How one "got to infinity"?? What are you talking about? (And if you answer, make sure it remains on-topic for this thread.)

I have a little bit of trouble with that for at least two reasons. The first, and primary one is the process of arriving at one's answer is often non-commutative. Another is citing university math in support of a statement and then not connecting the two. The third is sniping and then admonishing.

 

[strangerep]

Did, at some point, someone come to the conclusion that SR could not be built from one postulate? Because there seem to be a great number of constraints.

I'll answer that only after you've studied the whole thread, and references given therein. Otherwise, I'm just rehashing things that have gone before.

The rest of your post is either speculative, or too far off-topic for this thread, so I won't pursue it.

 

[fhenryco]

I would not mind too much if SR could be derived from many more than one postulate: the real issue is the status of these postulates: are they symmetry principles (homogenity , isotropy, a relativiy principle) or are they something else that nobody would have expected and that looks just ugly such as the principle taht there exists an universal invariant speed , which then will be the speed of any staff with zero mass hence probably the speed of light if photons have indeed zero mass.

The reason why we like symmetry principles is not an arbitrary one : for instance the principle of invariance under spacetime translations just means that the laws are the same averywhere and any time and this has to do with the intuition that science itself is possible.

So I'm indeed interested in trying to avoid the second principle of SR: the existence of a universal invariant speed but not necessarily by trying to deduce everything from a single principle but rather by replacing this second principle (invariance of c) by something that more looks like a symmetry principle and would exclude the galilean group.

One usually agrees that eventually SR really unifies space and time : it mixes x and ct in just the same way i.e
in the transformation you can make a permutation of x and ct and at the same time x' and ct' and this will leave the transformation laws invariant : in the galilean case where c is infinite the previous sentence would make no sense at all and this is a reason why i believe galilean transformation cannot just be considered as a special case of Lorentz transformation with c infinite.

Now if i wanted to derive the lorentz transformation i would start by just a transformation transforming a given (x,ct) into another (x',c't') where c and c' are not necessarily equal but just needed from the beginning becaus one cannot mix in a vector compnents which are not expressed in the same units , so the speeds c and c' are needed. Then i would impose a perturmation symmetry i.e invariance of the transformation laws under the simultaneous exchange of x and ct and at the sametime x' and c't' ... and from this i would demonstrate that necessarily c=c' if the transformation we are speaking about represents the physical transformation between two frames at constant relative speed. Actually it seems that it works and is quite straightforward and would allow to replace the speed of light constancy by a permutation principle...

 

[Dale]

So I'm indeed interested in trying to avoid the second principle of SR: the existence of a universal invariant speed but not necessarily by trying to deduce everything from a single principle but rather by replacing this second principle (invariance of c) by something that more looks like a symmetry principle

Huh? The invariance of c is a symmetry principle.

 

[fhenryco]

Huh? The invariance of c is a symmetry principle.

Not in the same sense as isotropy , homogeneity or relativity principle : those symmetry principles are based on the idea that there is no privileged staff : all positions, speeds, directions must be treated in the same way. The permutation symmetry as i think about would have to be a kind of generalisation of isotropy to 4d : the transformation must treat on the same footing the fourth coordinate (ct) and the three others for instance x and ct in a boost along x.

You could notice that for instance even in simple rotations for instance about x, there is no such permutation symmetry under the exchange y <-> z and y' <-> z' but of course this is due to the fact that in this case the permutation symmetry must be accompanied by the angle reversal because rotations are physically oriented... so the invariance under permutation is actually also satisfied in the case of rotations which i believe is a consequence of isotropy.

Eventually the minkowskian vs the euclidian signature might be a consequence of a difference between oriented (boost) and non oriented transformation (rotation).

I would summarize in that way: suppose you already have 3 coordinates x, y, z then you add the fourth coordinate x4 ; there are two ways for x4 to satisfy a permutation symmetry when performing a transformation which mixes x4 with any of the 3 others : either a rotation (oriented) and then x4 is a spacelike coordinates as x,y,z , or a boost (non oriented) transformation and then x4 is a timelike coordinate with flipped signature.

Does all this makes sense ;-) i hope so, i'd like! waiting for other helpful critical comments

 

[Dale]

Personally, I still don't understand your objection to the invariance of c. You said "not in the same sense" and then wandered off into permutations.

When x doesn't change under y then x is a symmetry of y. The speed c is therefore a symmetry of inertial transforms.

I am fine if you don't want to accept the invariance of c as a postulate, but your stated dissatisfaction with it just seems odd to me.

 

[PAllen]

I still think the best approach for a postulate besides POR, isotropy, and homogeneity (which leaves exactly 2 choices: SR or Galilean relativity) is that you must pick some additional invariance:

t invariance -> Galilean relativity
c invariance -> SR

This is leaner than "laws of physics". The other major approach is you don't postulate at all. You do any experiment which will distinguish. For example, SR predicts you can keep muons in a storage ring for for some time if they are very energetic. Galilean relativity says you would never succeed in this given the low energy decay time of 2 microseconds. Of course, you wouldn't get very far building your accelerator or storage ring without using laws that already imply c invariance.

 

[Dale]

I like that approach. I also like Robertson's approach of just making a general theory and letting experiment determine the parameters.

 

[Sugdub]

... So I'm indeed interested in trying to avoid the second principle of SR: the existence of a universal invariant speed but not necessarily by trying to deduce everything from a single principle but rather by replacing this second principle (invariance of c) by something that more looks like a symmetry principle and would exclude the galilean group.

The second principle deals with the invariance of the speed of light. Since in the absence of gravitation SR must encompass all phenomena, this specific reference to the propagation of light in empty space is problematic insofar it artificially injects an asymmetry in SR foundations. What is missing is a different, more general, justification for the existence of an invariant speed c. I suggested a way forward in #70 : a physics theory which excludes instantaneous actions at a distance necessarily imposes the existence of a finite maximum speed. True or False?

One usually agrees that eventually SR really unifies space and time : it mixes x and ct in just the same way ...

What is your proposed justification for injecting a constant c in your equations? What role does it play?

 

[robphy]

Here an old post of mine from a now-closed thread on Galilean relativity:

https://www.physicsforums.com/showthread.php?p=4112167#post4112167

Essentially copied from my earlier self...

I describe Galilean relativity by saying that
it has a maximum signal speed that is infinite.
Light speed is still finite, but not invariant under Galilean boosts.
An infinite speed is invariant under Galilean boosts.

Special relativity has a maximal signal speed that is finite,
and that light's speed is equal to that maximal signal speed.

In their respective geometries/relativities,
these maximum signal speeds correspond to eigenvectors of the boosts.
A useful way to encode this [relation between Galilean and Minkowskian]
is to define a dimensionless quantity (I call the indicator)
<br /> \epsilon^2=\frac{c_{light}}{c_{max\ signal}}<br />
where c_{light}=3×10^8\ m/s plays the role of a conversion constant.
The physics is in c_{max\ signal}, determined by experiment.

\epsilon^2 has the value 0 for the Galilean spacetime, and 1 for Minkowski space.

With this, one can formulate special relativity with this indicator
so that one can clearly obtain the Galilean limits by having this indicator tend to zero.

A maximum signal speed corresponds to a null-directed (in that metric) eigenvector.
If that signal speed is infinite, then spacelike[=orthogonal to timelike] coincides with null,
which leads to a "t-invariance".

This is in the spirit of the Cayley-Klein formulation of Euclidean and the constant-curvature non-euclidean geometries.


As I mentioned earlier in this thread, there is another approach which could be regarded as more primitive: using a causality axiom (the causal structure) to obtain the Minkowski spacetime
https://www.physicsforums.com/showpost.php?p=4751681&postcount=30

 

[PAllen]

The second principle deals with the invariance of the speed of light. Since in the absence of gravitation SR must encompass all phenomena, this specific reference to the propagation of light in empty space is problematic insofar it artificially injects an asymmetry in SR foundations. What is missing is a different, more general, justification for the existence of an invariant speed c. I suggested a way forward in #70 : a physics theory which excludes instantaneous actions at a distance necessarily imposes the existence of a finite maximum speed. True or False?

False. There is no contradiction between every interaction having some speed of propagation (different for different types of interaction, or perhaps different for different energy domains), and t invariance. To conclude a finite maximum speed, you must assume an upper bound on such speeds. Further, you must conclude no relative speed can exceed this upper bound (rather than assuming that objects with very large relative speed cannot interact).

 

[Dale]

it artificially injects an asymmetry in SR foundations.

What?!? You are certainly free to consider the invariance of c to be artificially injected, but it is patently a symmetry, not an asymmetry.

 

[Fredrik]

Since in the absence of gravitation SR must encompass all phenomena, this specific reference to the propagation of light in empty space is problematic insofar it artificially injects an asymmetry in SR foundations.

Not specifically into the SR foundations, but into the common framework in which both SR and pre-relativistic classical mechanics are defined. That framework admits two different groups of functions that describe a coordinate change from one inertial coordinate system to another. To choose the group is to choose the theory.

Edit: I wouldn't use the term "asymmetry" either. It's simply a choice of which symmetries to include in the theory.

What is missing is a different, more general, justification for the existence of an invariant speed c.

I don't think it can be some idea that can be arrived at purely by intellectual means. In the future, we may be able to think of this as a prediction made by some future theory. At the moment, I think we have to rely on experiments.

I suggested a way forward in #70 : a physics theory which excludes instantaneous actions at a distance necessarily imposes the existence of a finite maximum speed. True or False?

You will certainly have to limit your attention to some class of theories to make such an argument. If we focus on theories that use $\mathbb R^4$ as a model of space and time, and is consistent with (mathematical statements corresponding to) the principle of relativity, I suppose that your statement is true. But I don't think it's an improvement over the simple idea that we can use experiments to distinguish between the two possibilities.

What is your proposed justification for injecting a constant c in your equations? What role does it play?

The thing is, when we set out to find the group of functions that "translate" between inertial coordinate systems on $\mathbb R^4$, that constant shows up without being "injected". You start out with several undetermined parameters, and then you find that you can get rid of all but one by using the principle of relativity and symmetry principles.

 

[Fredrik]

When x doesn't change under y then x is a symmetry of y. The speed c is therefore a symmetry of inertial transforms.

Isn't it the other way around? When the Lagrangian doesn't change under translations, translations are a symmetry of the Lagrangian. I can't think of a reason to call the invariant lines (or the speed they represent) a symmetry.

Edit: I think it's OK to call the group of functions that change coordinates from one inertial coordinate system to another a symmetry group, since the presence of that group in the theory reflects invariance properties of spacetime. So each of those coordinate change functions (Galilean or Poincaré transformations) can be considered a "symmetry".

 

[fhenryco]

Personally, I still don't understand your objection to the invariance of c. You said "not in the same sense" and then wandered off into permutations.

When x doesn't change under y then x is a symmetry of y. The speed c is therefore a symmetry of inertial transforms.

I am fine if you don't want to accept the invariance of c as a postulate, but your stated dissatisfaction with it just seems odd to me.

As fas as i know, it never works that way: from a methematical point of view a genuine symmetry is a transformation that let's the laws of physics unchanged even though it might either transform or not various objects that enter the equations. (BTW c is not a scalar field, it is just a fundamental constant, so the invariance of c is also not the same kind of staff as the invariance of a scalar field). For instance once you have admitted the existence of a constant c and thanks to that you have your lorentz transformation, then you can ask the question : are my equations of physics going to be invariant under this transformation, if yes, this is a symmetry.

But of course behind the maths there is also the physical meaning of this invariance : it means that there is no privileged speed in the universe, in the sense that if you make an experiment on a table at rest in frame A and do the same experiment on a table at rest in frame B you will get the same results whatever the relative speed of A and B .

And the physical requirement is even more fundamental than the mathematical one which is just one of its translations in the language of maths: for instance the physical requirement of isotropy is not only translated in the invariance mathematical requirement according to which the laws of physics should be invariant under rotations but is also used to establish the Lorentz tranformations themselves for instance when one demands that the lorentz contraction should be the same for a boost at speed -u and for a boost at speed u (may be this is not used in all derivations of the Lorentz transformations but in the books i have studied it was). In both mathematical requirements there is behind the same and unic physical idea that should remain clear : we don't want any privileged direction so we must treat all of them in the same way.

The permutation symmetry is a kind of generalisation of this : i want to treat x and ct in the same way in the transformation: the physical idea is thus the same as the idea behind any other well known symmetry though the mathematical translation (invariance under permutation) is here a bit unusual.

 

[fhenryco]

What is your proposed justification for injecting a constant c in your equations? What role does it play?

c is not a priori a constant and it might be transformed into another c' in a boost: thus (x,ct) transforms (x',c't'). At the beginning we don't know what is the meaning of c and c' but we know these should be there because one cannot add time and space coordinates, thus c and c' at the beginning should merely be considered as conversion factors. But if i demand an invariance under permutation then the most general form of the boost that was:

x' =b_11 x + b_12 ct
c't' = b_21 x + b_22 ct

now must be

x' =b_1 x + b_2 ct
c't' = b_2 x + b_1 ct

and then i can derive c=c' and at the same time the usual values of b_1 and b_2 it's quite straightforward if i didnt make any mistake.

 

[Dale]

Isn't it the other way around?

I certainly could have it backwards. In any case, you have an operation and a thing which remains unchanged under the operation. Whether you use the word "symmetry" to refer to the thing or the operation or both together doesn't change the facts.

I can't think of a reason to call the invariant lines (or the speed they represent) a symmetry.

For geometric figures we speak of lines of symmetry quite often.

 

[Dale]

As fas as i know, it never works that way: from a methematical point of view a genuine symmetry is a transformation that let's the laws of physics unchanged

That is way overly restrictive. I don't know why a "mathematical point of view" would reference "the laws of physics" at all.

Mathematically, a circle has rotational symmetry regardless of any laws of physics. Physically, a disk is axisymmetric even though none of the laws of physics are.

In any case, I have no objection to your rejection of the second postulate, even though I think your stated reason is odd. In my opinion, you don't even need a reason, you can reject it on a whim or a dare if you like.

 

[fhenryco]

That is way overly restrictive. I don't know why a "mathematical point of view" would reference "the laws of physics" at all.

Mathematically, a circle has rotational symmetry regardless of any laws of physics. Physically, a disk is axisymmetric even though none of the laws of physics are.

But , i was trying to explain exactly the same thing, please read again my previous message where i explain that a symmetry is actually more a physical idea which can have many possible translations in maths.

The real issue is that the constancy of c as a principle has nothing to do with a symmetry principle, physically speaking, though mathematically we might be illusioned by the fact that indeed there is a transformation and some thing invariant (c) under this transformation.

Again, since we don't see behind the constancy of c, the physical idea that is the genuine characteristic of a symmetry (no privileged staff) , it's not , physically speaking, a symmetry!

I's not a symmetry but it can be derived from a symmetry : permutation invariance because the requirement is then to treat x and ct in the same way (no privileged staff)

 

[Dale]

The real issue is that the constancy of c as a principle has nothing to do with a symmetry principle, physically speaking, though mathematically we might be illusioned by the fact that indeed there is a transformation and some thing invariant (c) under this transformation.

Frankly, I think this is nonsense. We use math in physics in order to make sure that our theory is logical. It is not illusion, it is logic. There is a transformation and something is invariant under it. Logically, that is a symmetry.

If you don't like the second postulate, that is fine, but saying that it isn't a symmetry is absurd as is saying that mathematical conclusions are illusion.

 

[fhenryco]

Frankly, I think this is nonsense. We use math in physics in order to make sure that our theory is logical. It is not illusion, it is logic. There is a transformation and something is invariant under it. Logically, that is a symmetry.

If you don't like the second postulate, that is fine, but saying that it isn't a symmetry is absurd as is saying that mathematical conclusions are illusion.

ok, if you are not impressed by my rationnal arguments , i might just remind you that you will hardly find any textbook in relativity saying that the constancy of c is a symmetry principle. And most of the time this is the reason why many people have tried to axiomatized SR just in the hope of avoiding the arbitrariness of demanding from the beginning a constant c: some of them will make use of causality principles (to avoid action at a distance) for instance and exclude the galilean option ... it remains that i don't know many physicist which feel really confortable with the second principle of SR: most of them will just say: ok it's disturbing but the principles of physics are totally arbitrary after all: it's just experimental results that imposes us the principles, and we have to accept them wether we like them or not... (this is not my way of thinking as you may have understood)