Could SR not be built from only one postulate?

[guitarphysics]

No, not along my line of thinking but interesting nevertheless- at least the parts that I could understand.

Nice 'signature', by the way :D.

 

[strangerep]

I think I prefer strangerep's way of looking at it best; as far as I can tell, it doesn't rely on the rest of physics, and is very precise. However, it seems like a much more mathematical way of looking at the problem and I feel there should be more physics (like in Ben's approach).

Well, the physical content (using only a very small set of intuitive assumptions), and its mathematical development proceed along the following lines:

Following Rindler [1],

Postulate 1 (Principle of Relativity):
The laws of physics are identical in all inertial frames,
or, equivalently,
the outcome of any physical experiment is the same when performed with identical initial conditions relative to any inertial frame.

This requires a definition of inertial frames. Still following Rindler, an Einsteinian inertial frame is a reference frame in which spatial relations, as determined by rigid scales at rest in the frame, are Euclidean and in which there exists a universal time in terms of which free particles remain at rest or continue to move with constant speed along straight lines (i.e., in terms of which free particles obey Newton's first law).

The boundary between the physics and the maths lies in this: an observer can reasonably possesses local length scales (a very short rigid rod), and a local clock (measuring short time intervals). That much is physical. One then imagines that the rod could be successively laid end-over-end indefinitely to create a spatial coordinate grid. Similarly, one imagines that the clock could be duplicated endlessly, with the duplicates moved to spatially remote locations.

The (abstract) space of dynamical parameters needed to describe such an arrangement is assumed to correspond to (possibly a subspace of) $R^4$, i.e., 3 space and 1 time. Similarly, velocities are assumed to correspond to (possibly a subspace of) $R^3$. Thus we imagine a 7-dimensional velocity extended phase space of parameters. (We need not extend any further to higher dimensional phase spaces involving acceleration, jerk, etc, since the requirement of inertial motion restricts acceleration to be zero.)

These imaginings are made more precise using group theory. E.g., the basic physical spatial displacement defined by the rod is expressed in terms of a transformation of these dynamical parameters. Demanding that such transformations preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group, one can derive quite strong restrictions for the possible form that such transformations may take.

It's similar for temporal displacements, and velocity boosts (preserving the origin). One takes the basic physical operation of a small temporal displacement expressed via a local clock, or a velocity difference (holding the spatiotemporal origin invariant), expresses these as transformations, and imposes the same group theoretic requirements when composing multiple such transformations.

Summarizing, the physical content consists of the concept of inertial motion of an observer, and the availability of means for measuring very local spatial displacements, very local temporal delay, and relative velocities of other such observers who are momentarily at the 1st observer's origin. One also assumes spatial isotropy: that there is no preferred spatial direction.

Then one asks for the most general transformation of the parameters in the (abstract) velocity phase space which preserve the condition of zero acceleration. That encapsulates all the ways in which 2 inertial observers could be labelling the 7-dimensional abstract phase space in "different" (though physically equivalent) ways. The parameters of the transformation between 2 such observers may then be identified as the physically meaningful relative observables (spatial displacement, temporal delay, relative velocity, spatial orientation, etc). One naturally adopts an additional principle of "physical regularity": that finite values of these relative observables must be mapped by the transformations to finite values. That restricts the allowable transformations a bit further.

Then it's just a matter of grinding through the math of Lie group theory applied to such transformations to find the most general possibility, as I described earlier. The method is firmly grounded in realistic physics, which can be expressed quite concisely. The detailed math is extensive, of course, taking many pages if one performs all calculations explicitly. But overall it's a good thing: from a small set of physical concepts based on intuitive local operations, one derives an extensive theory. The possibility of an invariant limiting speed constant, and an invariant length constant, emerge as derived consequences. Only their values need be determined experimentally.References:

[1] W. Rindler, Introduction to Special Relativity,
Oxford University Press, 1991 (2nd Ed.), ISBN 0-19-85395-2-5.

 

[Fredrik]

I will briefly describe my view on the postulates and "derivations" of the Lorentz transformation here.

The postulates are nowhere near as significant as most texts will make you think. It should be emphasized that they're not even part of the theory. SR is defined by (purely mathematical) definitions of terms like "Minkowski spacetime" and "proper time", and a few correspondence rules that tell us how to interpret the mathematics as predictions about results of experiments.

The "derivations" of the Lorentz transformation that start with the postulates are certainly interesting and fun, but they shouldn't be viewed as proofs. They should be viewed as ways to guess how to define a new theory, or rather, a new framework in which to define theories. Once we have defined the mathematics of the theory properly, we can prove theorems that resemble the postulates.

The proper way to turn the "derivations" into actual derivations (i.e. proofs) is to first interpret the postulates as mathematical statements. Then you can take those statements as the starting point of a proof. The question is, what are we really proving? There's no obvious answer to the question of what mathematical statement best corresponds to the principle of relativity. So we still won't be able to say that we have (rigorously) derived the Lorentz transformation from the postulates. We have derived it from one mathematical interpretation of the postulates.

Because of this, I prefer to do those "derivations" in a way that's not completely rigorous, and to use language that indicates what parts of the argument are really just clever guesses. See e.g. this post. (Start reading at the line that starts with "The explicit". The "numbered statements" that I'm referring to in that post are the postulates).

I also think that some of the theorems (with rigorous proofs) that take a mathematical interpretation of the principle of relativity as a starting point are very interesting. The ones I've looked at can be interpreted as saying that SR and Newtonian mechanics are the only possible theories of physics in which $\mathbb R^4$ is the underlying set of spacetime (the mathematical structure that represents real-world space and time), and inertial coordinate systems can be defined on that set.

 

[Sugdub]

... Then one asks for the most general transformation of the parameters in the (abstract) velocity phase space which preserve the condition of zero acceleration. That encapsulates all the ways in which 2 inertial observers could be labelling the 7-dimensional abstract phase space in "different" (though physically equivalent) ways. The parameters of the transformation between 2 such observers may then be identified as the physically meaningful relative observables (spatial displacement, temporal delay, relative velocity, spatial orientation, etc). One naturally adopts an additional principle of "physical regularity": that finite values of these relative observables must be mapped by the transformations to finite values. That restricts the allowable transformations a bit further...

It would seem that the physical world can be split into two classes of objects: those which have an inertial state of motion and those which have a non-inertial state of motion. Neglecting gravitational effects, the Lorentz transformation of SR could be essentially derived from two postulates:

1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

2) all inertial frames of reference are physically equivalent, so that the laws of physics (i.e. the internal rules belonging to a physics theory) are invariant through a change of the inertial frame of reference.

Is this correct?

 

[PAllen]

It would seem that the physical world can be split into two classes of objects: those which have an inertial state of motion and those which have a non-inertial state of motion. Neglecting gravitational effects, the Lorentz transformation of SR could be essentially derived from two postulates:

1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

2) all inertial frames of reference are physically equivalent, so that the laws of physics (i.e. the internal rules belonging to a physics theory) are invariant through a change of the inertial frame of reference.

Is this correct?

For (2), the result you get depends on what laws of physics you include. If you use Newtonian machanics, you get the Galilean transform from these assumptions. If you use Maxwell's equations, you get the Lorentz transform. Note, part of the genesis of SR was working out what dynamical laws of mechanics replace Newtonian mechanics. If you include as the laws of physics Newtonian mechanics + Maxwell, you have a self contradictory system.

 

[Meir Achuz]

If you include as the laws of physics Newtonian mechanics + Maxwell, you have a self contradictory system.

...or Newtonian mechanics is wrong, and not a law of physics. That is why it is better to say something like 'all experimental results'.

 

[Sugdub]

For (2), the result you get depends on what laws of physics you include. If you use Newtonian machanics, you get the Galilean transform from these assumptions...

I'm trying to elaborate on Rindler's approach (I assume it is valid and recognized) as presented by strangerep in #32 according to which it seems possible to derive the Lorentz transformation of SR, referring neither to the invariance of the speed of light nor to Maxwell's equations, from constraints applying to the transformation between inertial frames of reference, in particular the need for transformations to “preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group”. I find this approach interesting and the purpose of my contribution is to challenge my own understanding of Rindler's approach. Actually I assume it entails as well that finite accelerations map onto finite (non-zero) accelerations, hence the first postulate I suggest:

First postulate: the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”.

The advantage I see is that this postulate caters for a definition of an “inertial frame of reference” which avoids the circularities of usual definitions: on the one hand they refer to Newton's first law which refers to the absence of forces which refers to inertial motion (first circularity), and on the other hand they refer to a law of physics (Newton's first law) in order to define the framework into which laws of physics should be formulated (second circularity).

However your comment shows that my second postulate is inappropriate. So let me replace it with something less ambiguous, as follows:

Second postulate: the difference between the rest state and an inertial state of motion is arbitrary, so that all inertial frames of reference (as defined above) are physically equivalent. Therefore the laws of physics (i.e. the internal rules belonging to a physics theory) which hold in a given inertial frame of reference should also be valid in any other inertial frame of reference. This in turn implies that their mathematical formulation shall remain invariant through the transformation between inertial frames of reference.

Hopefully the above is only a rewording of the constraints presented in #32. Assuming this can lead to the Lorentz transformation (as per my limited understanding of Rindler's approach), I doubt it can equally lead to the galilean transformation unless an additional constraint is imposed whereby the time coordinate remains invariant through the transformation.

 

[PAllen]

I'm trying to elaborate on Rindler's approach (I assume it is valid and recognized) as presented by strangerep in #32 according to which it seems possible to derive the Lorentz transformation of SR, referring neither to the invariance of the speed of light nor to Maxwell's equations, from constraints applying to the transformation between inertial frames of reference, in particular the need for transformations to “preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group”. I find this approach interesting and the purpose of my contribution is to challenge my own understanding of Rindler's approach. Actually I assume it entails as well that finite accelerations map onto finite (non-zero) accelerations, hence the first postulate I suggest:

First postulate: the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”.

The advantage I see is that this postulate caters for a definition of an “inertial frame of reference” which avoids the circularities of usual definitions: on the one hand they refer to Newton's first law which refers to the absence of forces which refers to inertial motion (first circularity), and on the other hand they refer to a law of physics (Newton's first law) in order to define the framework into which laws of physics should be formulated (second circularity).

However your comment shows that my second postulate is inappropriate. So let me replace it with something less ambiguous, as follows:

Second postulate: the difference between the rest state and an inertial state of motion is arbitrary, so that all inertial frames of reference (as defined above) are physically equivalent. Therefore the laws of physics (i.e. the internal rules belonging to a physics theory) which hold in a given inertial frame of reference should also be valid in any other inertial frame of reference. This in turn implies that their mathematical formulation shall remain invariant through the transformation between inertial frames of reference.

Hopefully the above is only a rewording of the constraints presented in #32. Assuming this can lead to the Lorentz transformation (as per my limited understanding of Rindler's approach), I doubt it can equally lead to the galilean transformation unless an additional constraint is imposed whereby the time coordinate remains invariant through the transformation.

No, the reasoning in #32 only establishes the either Galilean transform or Lorentz transform with some invariant speed TBD are the only possibilities. It is then either choice of what you consider to be laws (if you approach this axiomatically) or what experimental data you have that selects between Galilean and Lorentz, and also determines what the invariant speed is.

[edit: I think you fail to grasp the last sentence of #32:

"The possibility of an invariant limiting speed constant, and an invariant length constant, emerge as derived consequences. Only their values need be determined experimentally."

This says, based on the POR and the additional plausible assumptions and much math, you get some invariant speed as a possibility. The other possibility is no invariant speed = Galilean relativity.]

 

[strangerep]

1) the difference between the inertial and non-inertial states of motion is absolute, so that any "7-dimensional abstract phase space" which matches the boundary between both classes of objects (i.e. which correctly assigns its inertial or non-inertial state of motion to any object) can be said providing an “inertial frame of reference”;

That doesn't really make sense.

For present purposes, one only needs the physical input that an observer can distinguish between his own accelerated/unaccelerated state -- hence that an equation like $$\frac{d^2 x}{dt^2} ~=~ 0$$is applicable to all unaccelerated observers. One then seeks the maximal set of symmetries of that equation.

If one admits non-zero acceleration, then one must begin with a much larger abstract phase space. E.g., for constant acceleration, the phase must be enlarged to 10 dimensions, and the group of possible transformations is much larger than fractional-linear -- since one is now preserving an equation of motion like $$\frac{d^3 x}{dt^3} ~=~ 0 ~.$$The maximal symmetry group of this equation has been (partially) investigated, but I'm not sure if anything physically-useful has come out of that.

For non-constant accelerations, the phase space becomes ever larger, maybe even infinite-dimensional for truly arbitrary acceleration.

To understand this setting properly, one needs to know about continuous symmetries of differential equations. There's a significant body of theory about this -- I found the textbook of Stephani [1] quite useful. (The case of inertial motion symmetries is actually an exercise in one of his early chapters: the generators derived in that exercise correspond to fractional-linear transformations when integrated).

I'm trying to elaborate on Rindler's approach (I assume it is valid and recognized) as presented by strangerep in #32 according to which it seems possible to derive the Lorentz transformation of SR, referring neither to the invariance of the speed of light nor to Maxwell's equations, from constraints applying to the transformation between inertial frames of reference, in particular the need for transformations to “preserve inertial motion (i.e., that zero acceleration is mapped to zero acceleration), and that they form a Lie group”.

Note that I only quoted Rindler insofar as his 1st postulate and definition of inertial frame. Although he does perform a group-theoretic derivation of the Lorentz transformation, he adds an extra postulate about homogeneity, which restricts fractional-linear transformations down to linear transformations. The more general case is discussed in the other references I gave.

Assuming [#32] can lead to the Lorentz transformation (as per my limited understanding of Rindler's approach), I doubt it can equally lead to the galilean transformation unless an additional constraint is imposed whereby the time coordinate remains invariant through the transformation.

Actually, all these possibilities are contained in the most general (fractional-linear) approach. In the limit where spatial distances are small compared to the universal length constant, one recovers the Poincare transformations, i.e., standard special relativity. Then, in the limit where relative speed is much smaller than $c$, one recovers Galilean transformations.

---------------
References:

[1] H. Stephani,
Differential Equations -- Their solution using symmetries,
Cambridge University Press, 1989C, ISBN 0-521-36689-5.

 

[atyy]

...or Newtonian mechanics is wrong, and not a law of physics. That is why it is better to say something like 'all experimental results'.

I prefer to state specifically that Maxwell's equations are a "law of physics", or to assume that Poincare symmetry is a symmetry of the "laws of physics". If one says all experimental results (to date), that includes GR with accelerated expansion of the universe, in which case SR does not hold.

 

[Meir Achuz]

If one says all experimental results (to date), that includes GR with accelerated expansion of the universe, in which case SR does not hold.

That is excluded by the word "constant in the postulate:
"any two observers moving at constant speed and direction with respect to one another will obtain the same results for all experiments"
If gravity were included the phrase "at the same location" would have to be included, but I think that is implied in the postulate, which gives only relative velocity in the difference beteen the two observers.

 

[Fredrik]

...or Newtonian mechanics is wrong, and not a law of physics.

All theories are wrong. Some are just less wrong than others. So if a statement that's part of a "wrong" theory of physics can't be a law of physics, there are no laws of physics.

 

[Sugdub]

That doesn't really make sense.

For present purposes, one only needs the physical input that an observer can distinguish between his own accelerated/unaccelerated state ... One then seeks the maximal set of symmetries of that equation. ...If one admits non-zero acceleration, then one must begin with a much larger abstract phase space.

Thanks for your answer. I have the strong feeling that your rejection of my first postulate is due to a misunderstanding since it deals with physical objects, not with observers.

Although I agree with the statements quoted above, I'm trying to eliminate any direct reference to “observers” performing measurements or experiments. Yes, only an “observer” who feels unaccelerated can imagine being attached to an inertial frame of reference and the transformation between inertial frames of reference will map zero-accelerated observers to other zero-accelerated observers. But assuming one of these non-accelerated observers observes a non-zero-accelerated object, the said transformation between inertial frames of reference will map this non-zero-accelerated object onto a non-zero-accelerated object: if an object is accelerated when represented in one inertial frame of reference, it must have an accelerated motion in any other inertial frame of reference, irrespective of the presence of any “observer”. This is what I tried to express in my first postulate which deals with objects and not with observers. Therefore there is no need to involve complex maths.

Note that I only quoted Rindler insofar as his 1st postulate and definition of inertial frame. Although he does perform a group-theoretic derivation of the Lorentz transformation, he adds an extra postulate about homogeneity, which restricts fractional-linear transformations down to linear transformations...

Yes, homogeneity and isotropy of space, as well as homogeneity of time merely reflect the absence of good reasons to inject asymmetries in our representations of space and time. Any alternative would require a justification counteracting the empirical evidence. The two postulates I formulated are very general in their nature: since we do not sense any difference between velocity and rest, it would be irrational to assume a priori an absolute difference between uniform motion and rest. Conversely, our sense of acceleration suggests the opposite a priori assumption about the difference between inertial and non-inertial state of motion.

It is noticeable that the second postulate proposed by Einstein in 1905 about the invariance of the speed of light is of a less general nature, so that the set of postulates and assumptions from which he derived the Lorentz transformation lacks homogeneity. However the main effect of his second postulate is to inject a dependency between space and time quantities, a parameter (c) homogeneous to a velocity which is left invariant through changes of the inertial frame of reference: this imposes correlated changes in the transformation between space (x coordinate) and time physical quantities. Therefore the most general transformation compatible with all constraints cannot be squeezed down to a mere transformation of space coordinates. De facto it deals with space-time and this rules out the galilean transformation.

No doubt, injecting Maxwell's equations as a constraint has the same effect: the transformation will induce correlated changes in space (x coordinate) and time physical quantities and this also rules out the galilean transformation. However, injecting Maxwell's equations leads to the same pattern as Einstein's second postulate insofar the set of conditions which leads to the Lorentz transformation lacks homogeneity: Maxwell's equations relate to a specific range of phenomena whereas the postulate on relativity of motion and the homogeneity / isotropy symmetries encompass all phenomena.

On the other hand, one may decide to inject laws of the Newtonian mechanics as a constraint, which are incompatible with the perspective of a parameter homogeneous to a velocity remaining invariant. This would also lack homogeneity but more importantly it rules out the perspective of an invariant correlation between changes in space (x coordinate) and changes in time. The most general transformation between inertial frames of reference falls down to a mere transformation of space coordinates, not of space-time. This will lead to the galilean transformation.

Actually, all these possibilities are contained in the most general (fractional-linear) approach. ..

As a conclusion, I think the above shows that:

i) the two postulates I proposed, complemented with space and time symmetries established empirically lead to either the Lorentz transformation or the galilean transformation, which are exclusive;

ii) the Lorentz transformation which embraces space-time is more general than the galilean transformation insofar the former reduces to the latter proviso the addition of one constraint, e.g. imposing that time is not affected by the transformation between inertial frames of reference, or imposing that c is infinite, or imposing that simultaneity at a distance makes sense, etc... in which case the Lorentz transformation is ruled out, leaving the galilean transformation as the only possible outcome.

Overall, I believe that the Lorentz transformation is the most general solution that can be derived from the two postulates I proposed, maximising the impact of empirical symmetries without injecting any additional constraint. This should not come as a surprise since the universality of time is somehow a hidden postulate of the Newtonian mechanics.

 

[strangerep]

[...] I'm trying to eliminate any direct reference to “observers” performing measurements or experiments.

Then you're not doing physics. A frame of reference is an abstract construction of a particular observer.

[...] a parameter (c) homogeneous to a velocity [...]

That phrase doesn't make sense. I guess you mean "a parameter (c) with dimensions of speed".

You also seem to have used the word "homogeneous" in a distorted way elsewhere in your post. Have you actually studied the Rindler textbook reference I gave? He explains it reasonable detail. (I sense that you are uncomfortable with math, but without math it's all just hand-waving.)

 

[Sugdub]

Then you're not doing physics. A frame of reference is an abstract construction of a particular observer.

I can't see any way to adapt my “first postulate” and the subsequent definition of an “inertial frame of reference” in order to replace “physical objects” with “observers”. However nothing prevents attaching an hypothetical “observer” to each inertial frame of reference as I defined it, but what he/she will actually “observe” will be “distorted” by the Doppler effect over the signals transporting the information about remote events. It depends on what you wish to represent... But you're right, I'm not a physicist.

That phrase doesn't make sense. I guess you mean "a parameter (c) with dimensions of speed".

Yes, my command of English is rather limited. Does my statement make sense once properly worded?

You also seem to have used the word "homogeneous" in a distorted way elsewhere in your post.

I meant that some assumptions whereby Maxwell's equations are valid, or whereby Newton's first law is valid, are less general than other assumptions like the principle of relativity of motion or the isotropy and homogeneity of space. May be non- “homogeneous” was inappropriate to qualify the association of different categories ... But my main reservation in respect to such assumptions is that they appear to bring circularity: they invoke some specific “laws of physics” in order to set / define the mathematical framework into which “laws of physics” should be stated.

Have you actually studied the Rindler textbook reference I gave? He explains it reasonable detail. (I sense that you are uncomfortable with math, but without math it's all just hand-waving.)

No, I'm quite convinced that the Rindler textbook will fly far above my head, for both maths and physics. But I found your presentation interesting, because it seems to indicate that the above “circular” references can be overcome, the only one which remains according to your post #32 being the definition of the “Einsteinian inertial frame” which still refers to Newton's first law, so that I suggested a different approach for that specific definition through a “first postulate” leading to a new definition of an inertial frame of reference. That's all, I'm afraid.

 

[strangerep]

I meant that some assumptions whereby Maxwell's equations are valid, or whereby Newton's first law is valid, are less general than other assumptions like the principle of relativity of motion or the isotropy and homogeneity of space.

[...] my main reservation in respect to such assumptions is that they appear to bring circularity: they invoke some specific “laws of physics” in order to set / define the mathematical framework into which “laws of physics” should be stated.

I found your presentation interesting, because it seems to indicate that the above “circular” references can be overcome, the only one which remains according to your post #32 being the definition of the “Einsteinian inertial frame” which still refers to Newton's first law,

Yes, in the most general treatment, one need not appeal to other laws of physics to restrict the possibilities. A remarkable amount can be derived just from the concept of preservation of "zero acceleration".

BTW, the Rindler definition of "Einsteinian inertial frame" I quoted previously mentions "universal time". This is sufficient for ordinary special relativity, but in fact this is relaxed even further in the more general fractional--linear approach. Instead, one works with a weaker assumption: that spacetime (in the absence of gravitation) seems the same to inertial observers anywhere and anywhen.

[...] I'm quite convinced that the Rindler textbook will fly far above my head, for both maths and physics.

Well, I would encourage you to at least look at some of his early chapters before adopting such a defeatist attitude.

 

[Sugdub]

Yes, in the most general treatment, one need not appeal to other laws of physics to restrict the possibilities. A remarkable amount can be derived just from the concept of preservation of "zero acceleration".

BTW, the Rindler definition of "Einsteinian inertial frame" I quoted previously mentions "universal time". This is sufficient for ordinary special relativity, but in fact this is relaxed even further in the more general fractional--linear approach. Instead, one works with a weaker assumption: that spacetime (in the absence of gravitation) seems the same to inertial observers anywhere and anywhen...

As long as the definition of an "Einsteinian inertial frame" refers to “inertial observers”, one still needs to provide an acceptable definition for the word “inertial”. The issue at stake is whether this can be done without invoking “laws of physics” such as "Newton's first law" or circular definitions like the "absence of forces".

 

[WannabeNewton]

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?

 

[atyy]

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?

Couldn't it be possible that spacetime is Lorentzian and flat, but the laws of physics do not have Poincare symmetry?

 

[PAllen]

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?

Because the topic is to derive SR, not assume it. The concept of 4-rotation and 4-acceleration presumes it. A physical definition of inertial frame needs some further physical assumption or experimental finding to select between Galilean spacetime and Minkowski spacetime.

 

[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. ]