# Could SR not be built from only one postulate?

1. May 31, 2014

### 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.

2. May 31, 2014

### PAllen

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).

Last edited: May 31, 2014
3. May 31, 2014

### atyy

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.

4. May 31, 2014

### Sugdub

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, ...

5. Jun 1, 2014

### WannabeNewton

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.

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.

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.

6. Jun 1, 2014

### WannabeNewton

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

I don't disagree there.

7. Jun 1, 2014

### 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.)

Last edited: Jun 1, 2014
8. Jun 1, 2014

Tsk, tsk.

9. Jun 1, 2014

### PAllen

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.

10. Jun 1, 2014

### strangerep

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. ]

Last edited: Jun 1, 2014
11. Jun 2, 2014

### Sugdub

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.

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:

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

12. Jun 2, 2014

### Fredrik

Staff Emeritus
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.

13. Jun 2, 2014

### PAllen

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).

14. Jun 2, 2014

### Sugdub

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.

15. Jun 2, 2014

### PAllen

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.

Last edited: Jun 2, 2014
16. Jun 5, 2014

### Sugdub

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.

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.

17. Jun 5, 2014

### PAllen

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.

18. Jun 5, 2014

### strangerep

19. Jun 5, 2014

### WannabeNewton

20. Jun 10, 2014

### Sugdub

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.

21. Jun 10, 2014

### strangerep

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).

22. Jun 10, 2014

### PAllen

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.

23. Jun 10, 2014

### Fredrik

Staff Emeritus
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.

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.

24. Jun 10, 2014

### strangerep

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.)

Last edited: Jun 10, 2014
25. Jun 11, 2014

### PAllen

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.