Einstein simultaneity: just a convention?

[Ken G]

You seem to have a misunderstanding. There is the general theory of relativity, which simplifies to the special theory of relativity in regions of flat spacetime (i.e. special relativity means "special case" of the more general theory of relativity. Special relativity, in turn, simplifies to galilean relativity for v<<c.

You certainly can have accelerating observers and all sorts of forces in special relativity.

I am aware of that, my point was the reason for the word "special" in the title of the theory has to do with the specialness of inertial observers, not the specialness of the absence of gravity. Inertial observers are the observers for whom the postulates of special relativity apply, that's pretty special. Accelerated observers can be treated, but not directly with the postulates, one must first find an inertial frame, apply the postulates there, and convert to the accelerated frame to find the metric that applies to the accelerated frame (such as the Rindler metric for uniform acceleration). The postulates of special relativity are wrong in an accelerated frame. They are also wrong when there's gravity, so that's why gravity had to be added, but what made it "general" was the ability to treat all observers with the same formalism.

 

[Ken G]

He didn't say "the coordinate expression for the metric is invariant" -- he said "the metric is invariant".

Indeed, and he expressed that metric as an explicit function g(dX,dX). That function won't apply for, say, radio coordinates, so if that function is interpreted as "the metric" (the clear insinuation), then it is not invariant for all inertial observers in all coordinate charts. Orthonormal charts only, i.e., Lorentz transformations-- that's the problem.

Compare -- lengths and angles are invariants of Euclidean geometry, even though formulas for computing them can have varying forms between different coordinate charts.

Only if you either restrict to orthonormal coordinate charts, or redefine what you mean by "distance" to assure that it is preserved in any coordinate chart, which makes it an invariant by construction not by geometry. In other words, I do not need to define a concept of "distance" the same way the Minkowski norm does, nor do I need to agree with its answers for that quantity, to be doing the same physics-- and we should find a way to teach it that reflects that. Does the normal way?

 

[Ken G]

In general relativity, "gravity" is simply the tendency of objects to travel in a straight line through space-time; i.e. inertial travel.

Like I said, it's hard to classify as "real" or "coordinate" in nature (coordinate being in how it doesn't show up in a locally inertial coordinate system, or as you say creates travel in a straight line, until you look at tidal effects on more than one particle, and real being how it indeed exhibits tidal effects on multiple particles).

(note: force is an invariant of motion. Furthermore, the notion of force is very different from the notion of coordinate acceleration)

I believe you are distinguishing proper acceleration from coordinate acceleration. An important distinction in general, but I'm missing the specific relevance here. The reason I said it's tricky to categorize gravity is because locally it is governed by the equivalence principle, so its affect on a single free particle cannot be detected, but if you have several particles, you can detect it as a real effect even without there being the presence of any proper acceleration. Does that count as a real force, or a coordinate force?

 

[Dale]

I am aware of that, my point was the reason for the word "special" in the title of the theory has to do with the specialness of inertial observers, not the specialness of the absence of gravity.

No, it is neither. It is "specialness" of flat spacetime which is not the same thing as the absence of gravity.

In GR the worldline of an inertial observer is a geodesic, in SR the worldline of an inertial observer is a straight line. By definition, a flat spacetime is one where all the geodesics are straight lines. So SR is the special case of GR in a flat spacetime.

Inertial observers are the observers for whom the postulates of special relativity apply, that's pretty special.

You are mixing up "observers" and "reference frames". The postulates of SR apply to equally well to all observers, inertial and non-inertial, as analyzed in an inertial reference frame. You can use the postulates of SR without modification to analyze a non-inertial observer from an inertial reference frame. You cannot use the postulates of SR without modification to analyze even an inertial observer from a non-inertial reference frame.

 

[Ken G]

No, it is neither. It is "specialness" of flat spacetime which is not the same thing as the absence of gravity.

Wiki confirms my expectaton: "The theory is termed "special" because it applies the principle of relativity only to inertial frames." (http://en.wikipedia.org/wiki/Special_relativity)

You are mixing up "observers" and "reference frames".

I wouldn't say that, though your issue is largely semantic. In Einstein's approach to special relativity, which is what I am questioning, there is no difference, as he extends the inertial observer to an entire global frame, and noninertial observers are referenced to inertial ones instantaneously comoving. Of course this is not done in general relativity, but that only underscores my issue with thinking that the Einstein conventions are anything but a coordinate convenience. I don't know that I'm right, but so far there have been no successful challenges.

The postulates of SR apply to equally well to all observers, inertial and non-inertial, as analyzed in an inertial reference frame.

By invoking that frame, you are invoking an observer, to give the values in that frame scientific meaning. But it's something of a moot point, I am saying that what is "special" about it is that you cannot use its postulates to infer what a noninertial observer will measure, unless you first reference the noninertial observer to one in an inertial frame, do the calculation in that frame, and then transform back. Specifically, neither the form of the laws of physics, nor the speed of light, will be the same for any noninertial observer. In short, the postulates won't work for that observer, which is very much the spirit of special relativity.

You can use the postulates of SR without modification to analyze a non-inertial observer from an inertial reference frame.

I know that. Nevertheless, it violates the idea that "the laws of physics are the same for all observers", instead it becomes "the laws of physics become the same for all observers only once they are translated to an inertial frame". If you're going to do a translation, you might just as well pick an absolute frame and always transform to that-- it is still a violation of the spirit of relativity, and that's just what general relativity fixes via formal unification of accelerating frames with gravity using the equivalence principle.

 

[Antenna Guy]

In GR the worldline of an inertial observer is a geodesic, in SR the worldline of an inertial observer is a straight line. By definition, a flat spacetime is one where all the geodesics are straight lines. So SR is the special case of GR in a flat spacetime.

Can a geodesic in GR be a straight line?

What curves a geodesic in GR?

Regards,

Bill

 

[Dale]

Wiki confirms my expectaton: "The theory is termed "special" because it applies the principle of relativity only to inertial frames." (http://en.wikipedia.org/wiki/Special_relativity)

If you are going to try to win an argument by appeal to authority you should at least try to do better than Wikipedia. RL Faber. "Differential Geometry and Relativity Theory: An Introduction" has a whole chapter entitled "Special Relativity: the Geometry of Flat Spacetime". Or MS Parvez. "On the theory of flat spacetime" which says in the abstract "Special relativity, in essence, is a theory of four-dimensional flat spacetime".

I note that you didn't address the point I made about geodesics.

it violates the idea that "the laws of physics are the same for all observers", instead it becomes "the laws of physics become the same for all observers only once they are translated to an inertial frame".

Sorry Ken, you need to read up a little more. The first postulate is in fact closer to your second statement than the first. The first postulate is, in Einstein's words, "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good ... The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion".

 

[Dale]

Can a geodesic in GR be a straight line?

Yes, anywhere the space is not curved.

What curves a geodesic in GR?

The intrinsic curvature of the space, which is in turn caused by the presence of energy.

 

[bernhard.rothenstein]

conentionality

I'm curious about how people here view Einstein's prescription for determining simultaneity in an inertial frame, and how the extension of that approach to other inertial frames spawns the Lorentz transformation. It seems to me the competing pictures here are that this is an arbitrary way (in the sense of, not physically forced, even if convenient) to coordinatize time, and hence the Lorentz transformation is an arbitrary mapping between the coordinates of different reference frames, versus saying that the Einstein convention is fundamental to what we mean by time, and the Lorentz transformation is fundamental to what we mean by motion. I am rather of the former school, that what is physically fundamental is a deeper symmetry that allows the Einstein convention to be a particularly convenient coordinate choice, but that its physical significance comes entirely from how it simplifies the coordinatizations when we apply the laws of physics. But others might argue that the simplification is so fundamental that it would be foolish for us to imagine that "reality itself" could be doing anything different, even if just a means for recognizing equivalent possibilities.

Note, in particular, that the isotropic and constant speed of light in an inertial frame is a ramification of Einstein's coordinatization prescription, so an equivalent way to ask this is, is the isotropic speed of light a law of nature or just the proof that there exists a particularly elegant coordinate possibility? As the former is often taken as a postulate of special relativity, are we messing up the proper axiomatic structure of our art here?

Please follow the following thoughts:
1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).
2.the t(E) and t'(E) readings could be brought in a physically correct relationship with the readings of other clocks synchonized in a different way. In the case of the synchronized transformation (Mansouri and Sexl, Abreu and Homen, Guerra) or inertial transformations (Selleri) it reads
t(E)=t(v)+Vx/cc (1)
With (1) the Lorentz transformation becomes
t'(E)=t(v)/(1-VV/cc)
the concept of simultaneity having an absolute character.
3.It is considered that under such conditions the reference frame I is in absolure rest relative to the ether the motion of I' relative to it having an absolute character confering to I' some properties (anisotropy).
4.My oppinion is that those properties are merely introduced by the shift from the reading t(E) to the reading t(v). Using different physically correct relationships between t(E) and the reading t of a differently synchronized clock we obtain different transformation equations which confer different properties to I'.
I would highly appreciate your oppinion.

 

[Hurkyl]

Indeed, and he expressed that metric as an explicit function g(dX,dX). That function won't apply for, say, radio coordinates, so if that function is interpreted as "the metric" (the clear insinuation), then it is not invariant for all inertial observers in all coordinate charts. Orthonormal charts only, i.e., Lorentz transformations-- that's the problem.

Again, you're confusing "the metric" with "coordinate representation of the metric"

Compare -- lengths and angles are invariants of Euclidean geometry, even though formulas for computing them can have varying forms between different coordinate charts.

Only if you either restrict to orthonormal coordinate charts, or redefine what you mean by "distance"

It sounds like you're claiming that, in Euclidean geometry, 'distance' is a coordinate-dependent notion. If that really is what you're saying, then I posit that you need to review elementary geometry before continuing to reflect upon physics.

 

[Hurkyl]

Can a geodesic in GR be a straight line?

All geodesics are, by definition, straight lines.

What curves a geodesic in GR?

Geodesics are not curved.



The geometry of space-time yields a meaning to the term "straight" (which I will henceforth call 'intrinsic-straight'). Among other things, the path traced out by a geodesic is defined to be 'intrinsic-straight'.

There is a 'usual' way to attach an affine structure to R^4, and this gives another meaning to the term "straight" (which I wlil henceforth call 'coordinate-straight'). There is no good reason to do so -- it's simply that in a different context, we often use the same set R^4 as the underlying set of an affine space, so there is a temptation to invoke that affine structure in this context.

The point is, when you select coordinates, that affine structure usually has absolutely nothing to do with the geometry of space-time; the notions of intrinsic-straightness and coordinate-straightness are different. An intrinsic-straight path will generally be coordinate-curved. Similarly, a coordinate-straight path will generally be intrinsic-curved.

 

[Antenna Guy]

The point is, when you select coordinates, that affine structure usually has absolutely nothing to do with the geometry of space-time; the notions of intrinsic-straightness and coordinate-straightness are different. An intrinsic-straight path will generally be coordinate-curved. Similarly, a coordinate-straight path will generally be intrinsic-curved.

I think I follow.

Would it be correct to say that a particle following a geodesic (intrinsic-straight path in space-time) could be mapped as a classical trajectory (coordinate-curved path in space)?

Regards,

Bill

 

[Dale]

Hi bernhard, I wondered how long it would take for you to join the conversation!

Please follow the following thoughts:
1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).

Yes, everyone already agreed with that. The underlying Minkowski geometry is present regardless of the synchronization convention.

 

[bernhard.rothenstein]

convention simultaneity

Hi bernhard, I wondered how long it would take for you to join the conversation! Yes, everyone already agreed with that. The underlying Minkowski geometry is present regardless of the synchronization convention.

what about my thoughts 2 and 3?

 

[Dale]

what about my thoughts 2 and 3?

For 3 the ether is non-physical and the absolute frame is arbitrary, so I don't care.

For 2 I don't know enough about the specific transformations you referenced to comment about them in particular. But I already gave general comments about coordinate systems and synchronization conventions in https://www.physicsforums.com/showpost.php?p=1720437&postcount=26".

 

[Ken G]

If you are going to try to win an argument by appeal to authority you should at least try to do better than Wikipedia. RL Faber. "Differential Geometry and Relativity Theory: An Introduction" has a whole chapter entitled "Special Relativity: the Geometry of Flat Spacetime". Or MS Parvez. "On the theory of flat spacetime" which says in the abstract "Special relativity, in essence, is a theory of four-dimensional flat spacetime".

Both of those chapter headings are of course true (I never said otherwise), and neither are responsive to the issue of "what is the word 'special' there to imply".

I note that you didn't address the point I made about geodesics.

It required no comment, I am aware that geodesics in general relativity become straight lines in special relativity. Again, it's simply not responsive to the question of the meaning of "special", and again I repeat that this comes from the specialness of the treatment of inertial frames. That is the key element that distinguishes the approach of general and special relativity, in any kind of spacetime.

Still, this is not a terribly important semantic question-- you are welcome to your opinion on that matter, and it may be unanswerable because both the name and the theory have evolved so much that multiple interpretations of that process may be possible.

Sorry Ken, you need to read up a little more. The first postulate is in fact closer to your second statement than the first. The first postulate is, in Einstein's words, "the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good ... The laws by which the states of physical systems undergo change are not affected, whether these changes of state be referred to the one or the other of two systems of co-ordinates in uniform translatory motion".

Again, I can't agree that those statements support your contention. When Einstein said "all frames for which the equations hold good", he obviously means "all inertial frames". And in the second sentence, he said that the changes have to be referred to the coordinates used by inertial observers, but if one is going to "refer" to coordinates willy nilly, there is no problem with simply using "the coordinates of the King", and be done. His point is that you can choose any inertial frame, i.e., any of the special frames, to refer to, and use that special frame, where the postulates apply, to translate between measurements by noninertial observers. It is implicit that the coordinates of that inertial frame correspond to the measurements of a hypothetical inertial observer, i.e. of the special class of observers that define special relativity. Whether or not that special observer actually exists is irrelevent, physics uses hypothetical observers all the time, as did Einstein.

 

[Dale]

Ken, this conversation is getting repetitive and boring.

In summary:
1) The general theory of relativity simplifies to the special theory of relativity in flat spacetime, hence SR is a special case of GR.
2) The first postulate refers to inertial reference frames, not inertial observers.
3) Einstein synchronization is a convention.
4) The fundamental and coordinate independent concept of SR is the Minkowski geometry of spacetime.

I'm done.

 

[Ken G]

Again, you're confusing "the metric" with "coordinate representation of the metric"

Normally, you start out by defining a bunch of vectors, and a metric is a way to take those vectors two at a time and associate a number with each pair in a bilinear way. Unless you plan to enumerate every such pairing, you will need a convenient way to name the vectors, such that the metric can work automatically on that naming convention. That's called a coordinatization. Then you name the metric by how it functions on that vector-naming convention, that's what is meant by the "Minkowski metric". Thus, the naming convention on the vectors is presumed in the naming of the metric. It is not I who confuses that with what a metric is-- it is the way the Minkowski metric is taught that does that, and this is very much the point of the thread.

It sounds like you're claiming that, in Euclidean geometry, 'distance' is a coordinate-dependent notion. If that really is what you're saying, then I posit that you need to review elementary geometry before continuing to reflect upon physics.

I posit you need to read my words more carefully. What I was saying is that the way we produce a concept of distance is by the use of a metric, or an inner product if you will. You said that Euclidean geometry "preserves" lengths and angles. I presumed your use of the word "preserve" meant "leaves invariant under some type of mapping of the space into itself". Then I pointed out that, if you did mean that, the statement only holds on the subclass of mappings that are "orthonormal" under the action of your metric. If you only meant that distances are by definition the same no matter how you coordinatize the space, then (1) there's no meaning to the word "preserve", as there's nothing to preserve, you've already declared by fiat what the distance is, (2) that would hold in any geometry that admits a metric, and (3) that is what I meant by "redefining what you mean by distance" (I probably should have said redefine what you mean by an inner product that magically knows what the vector was before its name got changed).

Oh, and drop the haughtiness, it's not being backed up.

 

[Ken G]

1) The general theory of relativity simplifies to the special theory of relativity in flat spacetime, hence SR is a special case of GR.

Agreed, nor did I ever express any disagreement. The reason behind this is summed up well by John Baez(http://math.ucr.edu/home/baez/physics/Relativity/SR/acceleration.html):
"The difference between general and special relativity is that in the general theory all frames of reference including spinning and accelerating frames are treated on an equal footing. In special relativity accelerating frames are different from inertial frames. Velocities are relative but acceleration is treated as absolute. In general relativity all motion is relative. To accommodate this change general relativity has to use curved space-time. In special relativity space-time is always flat."

So there it is, we see that indeed special relativity does involve the restriction to flat spacetime, and the reason for this, and where the word "special" comes from, is that inertial frames are treated differently than noninertial frames, i.e., inertial frames are special in that theory. Take it up with Dr. Baez, I can't delve any deeper than I have already.

2) The first postulate refers to inertial reference frames, not inertial observers.

Indeed, but be careful you are not implying that the explicit reference to inertial frames does not include the all-important implicit reference to inertial (possibly hypothetical) observers. How else do you plan to define an inertial frame but by using observers, possibly hypothetical, with accelerometers that read zero? (See the definition you linked-- unfortunately that same library has no entry for "inertial reference frame".)

3) Einstein synchronization is a convention.

Agreed, when we have access to the concept of the measurements of hypothetical inertial observers. Then when we add the Einstein convention, we get that the connection between those observer coordinates is the Lorentz transformation, and we find that the invariant distance is given by the Minkowski metric. If we don't use the Einstein convention, we get neither the Lorentz transformation nor invariance of the Minkowski metric as it is normally expressed (or alternatively, we need to define a new Minkowski metric commensurate with the new time coordinatization).

4) The fundamental and coordinate independent concept of SR is the Minkowski geometry of spacetime.

If that were true, there'd be no need for this thread. But it isn't. "Minkowski geometry", as it is generally used, means a geometry spawned by an inner product that deviates from Euclidean by a -1 in one of the terms, using a particular choice of basis vectors chosen from a special class that represent a particular physically-motivated ordering of events by inertial observers. That is not "coordinate independent", because that particular metric, defined in the Minkowski way, is only invariant when acting on those special coordinates generated by inertial observers, coordinates which are connected by Lorentz transformations, which are of course the orthonormal transformations under the action of the Minkowksi metric (which is why it is invariant for those transformations). This I would say is precisely the basis of Baez's remark: "In special relativity accelerating frames are different from inertial frames."

Put mathematically, it is a very basic theorem of metric spaces that g(x,y)=g(Lx,Ly) only works if L is an orthonormal transformation (indeed, that defines the orthonormal transformations under the action of g). Ergo "Minkowski geometry" is very much the geometry associated with the Lorentz transformations and the Einstein simultaneity convention. These form a subclass of linear coordinatizations, and hence "Minkowski geometry" is not "coordinate independent". Just what is coordinate independent in all this is precisely the question behind this thread.

I'm done.

We all decide how much we want to know.

 

[Ken G]

1.Einstein's transformation equations hold only with Einstein synchronized clocks t(E) and t"(E).

Yes, the coordinate form of the standard "Lorentz transformation" between inertial frames (as defined by a population of hypothetical inertial observers) requires that inertial observers use the Einstein simultaneity convention.

2.the t(E) and t'(E) readings could be brought in a physically correct relationship with the readings of other clocks synchonized in a different way.

Yes, I would say that any arbitrary prescription could be used to synchronize clocks that merely followed some very weak constraints (supportive of metric spaces), and it would merely spawn a new way to transform between the inertial observers' coordinates. The special treatment of those observers would still allow the first postulate of SR to apply, but the second postulate would be lost. In that sense, I see the second postulate as superfluous, and Einstein's simultaneity convention should be elevated to the level of a postulate if one wanted to work in the standard coordinatization. If one wanted a coordinate-free treatment, one would simply assert that the speed of light is whatever is necessary to allow the first postulate to hold.

3.It is considered that under such conditions the reference frame I is in absolure rest relative to the ether the motion of I' relative to it having an absolute character confering to I' some properties (anisotropy).

Yes, it is a matter of sheer preference, a la Occam's Razor, to exclude anisotropy. In other words, if we later found some new physics that required anisotropy, no previous experiments would suddenly seem strange, we would merely have to use a different simultaneity convention and/or a different status of what are the "special" reference frames.

4.My oppinion is that those properties are merely introduced by the shift from the reading t(E) to the reading t(v). Using different physically correct relationships between t(E) and the reading t of a differently synchronized clock we obtain different transformation equations which confer different properties to I'.

To me, the key unanswered issue is, "how should we think of all this so that none of the arbitrary choices matter, i.e., what possibilities are ruled out by experiment and what is just what we accept from our preference for simplicity?"

 

[Ken G]

If that really is what you're saying, then I posit that you need to review elementary geometry before continuing to reflect upon physics.

Oh, and drop the haughtiness, it's not being backed up.

Perhaps I should have instead said "I will if you will"!

 

[Hurkyl]

4) The fundamental and coordinate independent concept of SR is the Minkowski geometry of spacetime.

If that were true, there'd be no need for this thread.

So, you can see why he might have become exasperated.

But it isn't. "Minkowski geometry", as it is generally used,

I have ever only seen "Minkowski geometry" used to refer to the notion of a coordinate-independent description of spacetime. (Conversely, those who reject Minkowski geometry prefer coordinate-dependence)

means a geometry spawned by an inner product that deviates from Euclidean by a -1 in one of the terms, using a particular choice of basis vectors

Everything can be described by a coordinate-based approach: that's why coordinates are useful. But that doesn't mean everything is coordinate-dependent.

In fact, I don't remember the last time I have ever heard of Minkowski space being described in a coordinate-dependent manner -- it's usually described in terms of its ,[/URL] which an intrinsic property of a metric, and is invariant under all coordinate transformations. In fact, it is effectively the only property of a metric that is invariant under all coordinate transformations.

 

[DrGreg]

Ken,

Sorry I haven't had chance to respond for a few days.

To echo what Hurkyl has been saying, the modern "geometrical" view of spacetime uses terminology slightly differently than the way you've been using it. It might help to forget relativity for a while and go back to 2D Euclidean geometry. The metric here is given by

ds^2 = dx^2 + dy^2​

where x and y are orthonormal Cartesian coordinates. However, that equation is not the metric; it is the equation for the metric in a particular coordinate system. It turns out that the same equation works for all other orthonormal Cartesian coordinates. But it doesn't work for other coordinates. For example, in "skew" coordinates, where the axes are at an angle of \alpha to each other, the metric is given by

ds^2 = dx^2 + dy^2 - 2\,dx\,dy\,\cos \alpha​

And in polar coordinates the equation is

ds^2 = dr^2 + r^2d\theta^2​

The above three equations are not three different metrics. They all represent the same metric, viz. the 2D Euclidean metric, expressed in different coordinate systems. And the metric has a physical interpretation as "distance", which is invariant under any coordinate change.

In relativity, even though the physical interpretation of the metric is a little more complicated, the same principle applies.

Note that if you rescale rapidity to be c \log_e k then it approximates to coordinate-speed at low speeds.

True, but that's not really a speed, it's a Doppler factor. That's the thing we can measure, speed requires a coordinatization.

But I am saying, if you do the maths, you will find that for low speeds the natural logarithm of the Doppler factor, viz. c \log_e k really does approximate to coordinate speed (at "everyday" terrestial speeds the two values would be indistinguishable), so you could use rapidity as a coordinate-independent measure of motion that is fully compatible with Newtonian (non-relativistic) speed.

That had me thinking for awhile, but I don't think that would give a unique result. After all, there are infinitely many pairs of mutually stationary objects that could have one object at each event, all with different distances between them. If you further stipulate that the objects must be stationary with respect to the observer doing the measurement, it just means each such pair comes with their own observer, each finding a different "proper distance" between the events. If the events themselves don't have a concept of being "stationary", which they don't normally, then we still have no way to know which observer is getting the "proper" result.

Actually you are right here: what I said isn't enough to define the "interval" between two events. Every inertial observer can measure a different distance between events in the way I said. The "interval" is the shortest possible distance that any inertial observer might measure between those two events, assuming that minimum is not zero (otherwise your two events are timelike separated).

They are not "different"-- everyone can measure something with an accelerometer. The inertial ones are simply defined as those who measure zero.

The point I was alluding to is that to an inertial observer in GR, Special Relativity still appears to be approximately true in a small local region around himself/herself. (The phrase "approximately true" can be made precise by means of calculus.) An inertial observer, in GR, can set up a local, Einstein-synced coordinate system in such a way that ds^2 = dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2 is still true at the origin of the coordinate system (although it won't be true elsewhere). (And conversely, non-inertial observers can never set up a local Minkowski approximation.) In that sense, inertial observers are "different", even though, as you rightly say, all observers, inertial or not, can set up coordinate systems.

The inner product, or "metric" is invariant, that is you always get the same answer for g(X,Y) no matter what coordinate system you use to carry out the calculation.

Not if you use "radio coordinates". This is part of the point-- the metric space has more general properties than the form of the metric.

No, this is a terminological issue. I think you are thinking of "the metric" as being the formula for ds in terms of the coordinates. I am saying that "the metric" is an entity that exists independently of coordinates, that you can define physically in terms of proper time and proper distance, and whose mathematical properties can be formulated in terms of vector equations, not component equations. So in spherical radar coordinates the equation

ds^2 = du \, dv - \frac{(u-v)^2}{4} ( d\theta^2 + \sin^2 \theta d\phi^2)​

represents exactly the same metric as

ds^2 = ds^2 = dt^2 - dx^2/c^2 - dy^2/c^2 - dz^2/c^2​

expressed in Minkowski coordinates. Both equations are the Minkowski metric. The metric is an operator that maps a pair of vectors to a scalar.

 

[DrGreg]

To get back to the original question, is Einstein synchronisation arbitrary or is there some good reason for it? One good reason is the mathematical one that it makes the maths simpler, and it makes it easy to compare one frame against another and confirm that neither is "special" in any way.

For those that are not aware, there is another "natural" synchronisation method called "ultra slow clock transport". The obvious Newtonian way to sync 2 inertial clocks A and B at rest relative to each other is to put a 3rd clock C next to A, sync it to A, then put it next to B and sync B to C. We know that method is no good in relativity, for if you then moved C back to A you would find that C was no longer synced to A (the twin "paradox"). Syncing B to A gives a different result than syncing A to B, by this "fast clock transport method".

But what if we move C from A to B v-e-r-y s-l-o-w-l-y? The twin paradox discrepancy gets less the slower you go. Although you could never achieve zero speed in practice, you can consider, mathematically, what would happen in the limit. It turns out, when you do the maths, that this method of "ultra slow clock transport" synchronisation gives exactly the same result as Einstein synchronisation (and experiments have confirmed this).

 

[Histspec]

For those that are not aware, there is another "natural" synchronisation method called "ultra slow clock transport". The obvious Newtonian way to sync 2 inertial clocks A and B at rest relative to each other is to put a 3rd clock C next to A, sync it to A, then put it next to B and sync B to C. We know that method is no good in relativity, for if you then moved C back to A you would find that C was no longer synced to A (the twin "paradox"). Syncing B to A gives a different result than syncing A to B, by this "fast clock transport method".

But what if we move C from A to B v-e-r-y s-l-o-w-l-y? The twin paradox discrepancy gets less the slower you go. Although you could never achieve zero speed in practice, you can consider, mathematically, what would happen in the limit. It turns out, when you do the maths, that this method of "ultra slow clock transport" synchronisation gives exactly the same result as Einstein synchronisation (and experiments have confirmed this).

A nice description and comparison of Einstein synchronization and "slow clock transport" can be found in:

Mansouri R., Sexl R.U.: A test theory of special relativity. I: Simultaneity and clock synchronization. In: General. Relat. Gravit.. 8, Nr. 7, 1977, pp. 497–513.​

Experiments, which confirmed the equivalence between those methods, were made by:

Wolf P. and Petit G., Satellite test of special relativity using the global positioning system, Phys. Rev. A56, 6, 4405, (1997).​

See also:
en.wikipedia.org/wiki/Einstein_synchronisation

 

[Aether]

It turns out, when you do the maths, that this method of "ultra slow clock transport" synchronisation gives exactly the same result as Einstein synchronisation (and experiments have confirmed this).

Wouldn't this method of "ultra slow clock transport" give exactly the same result as any other synchronization method if one assumes the same conventional isotropy/anisotropy of speeds as assumed for the other method? There is no unique connection between Einstein synchronization and slow clock transport.

 

[Ken G]

I have ever only seen "Minkowski geometry" used to refer to the notion of a coordinate-independent description of spacetime. (Conversely, those who reject Minkowski geometry prefer coordinate-dependence).

The key point I was making is, a metric is only invariant on mappings of the vector space into itself that constitute the "orthonormal transformations" under that metric. Ergo, one cannot say the "Minkowski metric is invariant" and "Minkowski geometry is coordinate independent" in the same breath, they are contradictory. They both have their separate meanings, it is true, but the meanings are different. If you want to count the latter as true, as is the conventional choice, then the former statement is not coordinate independent. That basic confusion is at the heart of what we are trying to get to the bottom of-- the contradiction between imagining that "Minkowski geometry" is coordinate independent, but it is generated by a "Minkowksi metric" (as in any textbook) that is not in general an invariant. If the latter requires assumptions not required in the former, then will the real "Minkowki metric" please stand up?

In fact, I don't remember the last time I have ever heard of Minkowski space being described in a coordinate-dependent manner -- it's usually described in terms of its ,[/URL] which an intrinsic property of a metric, and is invariant under all coordinate transformations. In fact, it is effectively the only property of a metric that is invariant under all coordinate transformations.

Right-- that's why it would normally be considered true that the Minkowski signature is really the heart of special relativity-- not invariants of the Minkowski metric. Yet I will bet you that if you pick up virtually any physics textbook, you will quickly find a confusion between what "coordinate independent" means and what "invariance under Lorentz transformations" means. They get enmeshed as if they were saying the same thing, and untangling that confusion is the progress we are making.

What you are now saying is that the sole "physical" aspect of the Minkowksi metric is that it is symmetric (in the sense <x_i,x_j> = <x_j,x_i>) and gives three positive and one negative norm on an any orthogonal basis. It will not give that on arbitrary bases, however, as if the basis vectors are strange combinations of observables, or if they are the observables of an accelerated observer over a finite time period. Note in particular what happens to the "postulates of special relativity" in the latter cases-- we find they make coordinate assumptions. What's more, any metric with that signature would successfully generate special relativity with the appropriate definitions (i.e., not using the Einstein simultaneity convention, or not requiring inertial observers for finite-time calculations). Thus if the "real heart of special relativity" does not make those assumptions, then the "postulates of special relativity", as they are normally taught, are not in fact the real heart of special relativity. Now we're getting somewhere.

 

[Ken G]

The above three equations are not three different metrics. They all represent the same metric, viz. the 2D Euclidean metric, expressed in different coordinate systems.

Actually, I think your "skew" metric is indeed a different metric. You have just changed the metric when you changed the basis vectors, to make the new basis an orthonormal one under the new metric. I'm pretty sure that for a single metric, saying <x,y> = <Ox,Oy> is the definition of O being an orthonormal coordinate transformation under that metric.

I think part of the problem here is that metrics normally work on a single vector space, from which you select two vectors, but for them to transform in an invariant way you actually have to take one vector from the vector space and the other from the "dual space", so if you want the first vector space to be covariant vectors, you have to select a contravariant vector from its dual space. If you do that, you obtain complete coordinate independence, but that is not the normal way that metrics operate. Maybe we shouldn't be using a Minkowski "metric" at all.

But I am saying, if you do the maths, you will find that for low speeds the natural logarithm of the Doppler factor, viz. c \log_e k really does approximate to coordinate speed (at "everyday" terrestial speeds the two values would be indistinguishable), so you could use rapidity as a coordinate-independent measure of motion that is fully compatible with Newtonian (non-relativistic) speed.

It seems to me that "coordinate speed" and "coordinate-independent measure of motion" are having a little fight in that sentence.

The point I was alluding to is that to an inertial observer in GR, Special Relativity still appears to be approximately true in a small local region around himself/herself...In that sense, inertial observers are "different", even though, as you rightly say, all observers, inertial or not, can set up coordinate systems.

But if there is no difference between an observer whose accelerometer reads zero, and one whose reads something else (which is true on the scales you are describing), then there is still nothing "special" about the one who is inertial. The "specialness" in special relativity appears on finite times, where the physics comes in, and the accelerometer reading becomes important.

I am saying that "the metric" is an entity that exists independently of coordinates, that you can define physically in terms of proper time and proper distance, and whose mathematical properties can be formulated in terms of vector equations, not component equations.

It is my impression that your remark here would only be true if the vectors that the metric acts on were selected from dual spaces (one covariant and one contravariant), but normally metrics are defined with both vectors from the same space. When the latter is used, metrics are only invariant when acting with respect to orthonormal bases, so that is a coordinate constraint that does single out inertial observers observing vectors of finite (i.e., not infinitesmal) length. Nevertheless, as in the above exchange with Hurkyl, it does not appear that the invariance of the metric is a terribly crucial property, as it is actually its signature that determines the physics within any particular coordinate system.

Both equations are the Minkowski metric. The metric is an operator that maps a pair of vectors to a scalar.

I would say that both equations share the signature of the Minkowski metric, and generate Minkowski geometry, but they are not the same metric. Moreover, the way the Minkowski metric is usually taught is as a single metric, not as a class of metrics that all spawn the same geometry but differ in the values of the norms.

 

[Ken G]

To get back to the original question, is Einstein synchronisation arbitrary or is there some good reason for it? One good reason is the mathematical one that it makes the maths simpler, and it makes it easy to compare one frame against another and confirm that neither is "special" in any way.

I agree, there's certainly plenty of motivation from Occam to set up special relativity the way it is done. My issue, however, is when we "cover our tracks" and assert statements of our own choosing, to make the math simple, as though they were "truths about reality" (that's often how the postulates of relativity are taught, I've seen very few counterexamples). The place you'd see the difference is if we ever found evidence that those postulates were wrong, would we say "hey, but I thought we had observations to back them", and the answer would be "no, the observations only backed more general postulates, we added additional elements for no reason other than to simplify the math. We did the same thing with Newton's laws and look where that got us".

 

[Hurkyl]

The key point I was making is, a metric is only invariant on mappings of the vector space into itself that constitute the "orthonormal transformations" under that metric. Ergo, one cannot say the "Minkowski metric is invariant" and "Minkowski geometry is coordinate independent" in the same breath, they are contradictory.

Change-of-basis transformations are not "mappings of the vector space into itself". (although, they are equivalent to "mappings of the {coordinate-representation of the vector space} into itself")

From this, and subsequent comments, it looks like you're still confusing "the metric" with "the coordinate representation of the metric". I'm quite serious when I say you should reconsider Euclidean geometry before you continue thinking about Minkowski geometry. (Since I assume you understand the Euclidean case)

The invariance of the metric under local Lorentz transformations means that if you change which direction you look, physics remains the same.

The coordinate-independence of the metric means that lengths and angles remain the same, no matter what chart you use to compute them.

 

[Ken G]

Change-of-basis transformations are not "mappings of the vector space into itself".

Changing to a different observer is, and that's what we are ultimately talking about here.

I'm quite serious when I say you should reconsider Euclidean geometry before you continue thinking about Minkowski geometry.

The issue was never about Euclidean geometry vs. Minkowski geometry (we agree that is the crucial geometric difference). The issue was about the invariance of a metric, what the "Minksowki metric" means, and how that is different from "Minkowski geometry" (in virtually any textbook). Those are not the same questions, that's the point.

The invariance of the metric under local Lorentz transformations means that if you change which direction you look, physics remains the same.

That's incorrect, it means that if you change from one inertial observer to another, physics remains the same.

The coordinate-independence of the metric means that lengths and angles remain the same, no matter what chart you use to compute them.

Let's define a metric, and denote it by < , >. Now I tell you that <x,y> = <Ox,Oy> for some transformation O. We may imagine that O is how things look different when I change from one observer to another, and we are asserting that the metric remains invariant under that change. Question: what can we say about O?

 

[DrGreg]

Wouldn't this method of "ultra slow clock transport" give exactly the same result as any other synchronization method if one assumes the same conventional isotropy/anisotropy of speeds as assumed for the other method? There is no unique connection between Einstein synchronization and slow clock transport.

Well I'm not exactly sure what you mean by "the same conventional isotropy/anisotropy of speeds". If you mean one-way coordinate-speed of light isotropy, then you are assuming Einstein synchronization.

Mansouri & Sexl (mentioned in this post) make some homogeneity and "Lorentzian" assumptions which amount to assuming Einstein's postulates are true when expressed in a suitable sync-convention-independent way.

You can also prove the equivalence of ultra slow clock transport and Einstein synchronization using Bondi's k-calculus and radar coordinates, which do not depend on any sync convention.

Assuming SR is true, as we can prove ultra slow clock transport and Einstein synchronization are equivalent, then ultra slow clock transport cannot be equivalent to anything that is not equivalent Einstein synchronization.

Any experimentally confirmed difference between ultra slow clock transport and Einstein synchronization would amount to a disproof of relativity. It hasn't happened yet.

 

[Hurkyl]

Changing to a different observer is, and that's what we are ultimately talking about here.

If, by that, you don't mean a change of coordinates, then you need to explain.

The issue was never about Euclidean geometry vs. Minkowski geometry (we agree that is the crucial geometric difference).

Euclidean and Minkowski geometry are identical in all aspects relevant to this discussion. For example, they are both affine spaces equipped with a symmetric, nondegenerate bilinear form (that is compatable with the affine structure). I assume we both consider Euclidean geometry 'simpler', and so there is much to gain reviewing that case first.

 

[Ken G]

If, by that, you don't mean a change of coordinates, then you need to explain.

When you change the observer, you will change the basis vectors used to label the events of spacetime in terms of the physical measurables "distance" and "time". That is both a change in coordinates, in that the labels are changing in a particular way, and a transformation of the vector space into itself, as all the events are now seen from a different perspective-- that of a new observer. The events are the same, but the vectors are different (indeed, nonlinear transformations would make them no longer even members of a vector space at all). If the transformation was Lorentzian, which happens when you are changing between inertial observers and are using the Einstein simultaneity convention, then the "Minkowski metric" connecting any two events, as it is normally defined, will be invariant. But in general, it will not. If you want to get something that is invariant to all linear transformations, you must choose one vector from the vector space and the other from its dual space, as I dimly understand the situation.

Euclidean and Minkowski geometry are identical in all aspects relevant to this discussion. For example, they are both affine spaces equipped with a symmetric, nondegenerate bilinear form (that is compatable with the affine structure). I assume we both consider Euclidean geometry 'simpler', and so there is much to gain reviewing that case first.

Well, if by "all aspects" you mean "in terms of the meaning of a transformation of a vector space, a coordinatization, and a dual space", then I suppose you are right, and those are all interesting and important but quite mathematical issues that I think we all have much to learn about.

But what I have in mind is a much more interesting physical issue, namely, "what are the minimal postulates required to describe the physics of special relativity." When that is the goal, then the different signatures (of the Euclidean and Minkowski metrics) are indeed important, and I am beginning to suspect that the minimal postulates say that the geometry of spacetime is described by a symmetric metric with a signature with three positive and one negative eigenvalue. Note this requires no simultaneity convention, nor the statement that the speed of light is isotropic, nor the requirement that physics look the same from all inertial reference frames. Interesting, is it not, that those are basically the "three pillars of special relativity" as it is normally taught, and my goal is to get to the bottom of this apparent flaw in the standard architecture.

The reason that's important, once again, is that when observations some day show that special relativity breaks down even in the absence of gravity (say, in quantum mechanics), we'll want to know what are the postulates that our observations really did back up, and what ones did we just imagine they backed up, in the process of mistaking Occam "simplicity" for something more akin to "computational convenience".

 

[Hurkyl]

But what I have in mind is a much more interesting physical issue, namely, "what are the minimal postulates required to describe the physics of special relativity."
...
The reason that's important, once again, is that when observations some day show that special relativity breaks down even in the absence of gravity (say, in quantum mechanics), we'll want to know what are the postulates that our observations really did back up, and what ones did we just imagine they backed up

I have more to say, but no time this morning. But I did want to make one quick comment:
Mathematically speaking, at least, "minimal postulates" are not unique. There are many many different ways of formulating any theory.

In terms of your long-term goal, I think that guessing at the 'one true formulation of special relativity' is the wrong approach -- if you instead learn many different ways of formulating special relativity, you're much more likely to know one that can be tweaked to accommodate the new data.

 

[Ken G]

Mathematically speaking, at least, "minimal postulates" are not unique. There are many many different ways of formulating any theory.

Right, you made that point earlier and that is a very valid one. It is not really the "minimum postulates" that count here, it is the minimal theory. By that I mean, the theory that unifies all the observations, without making unique predictions about what is outside the intended realm of explanation of the measurement set. Newtonian mechanics should have been done that way too, it would have saved us a lot of false surprise (surprise we had no real business being surprised about).

In terms of your long-term goal, I think that guessing at the 'one true formulation of special relativity' is the wrong approach -- if you instead learn many different ways of formulating special relativity, you're much more likely to know one that can be tweaked to accommodate the new data.

That's not the issue, the goal is not to find an equivalent formulation, but a less restrictive one. For example, uniting all metrics with the same signature is already a less restrictive form of dynamics than requiring invariance of a particular one.

 

[Hurkyl]

By that I mean, the theory that unifies all the observations, without making unique predictions about what is outside the intended realm of explanation of the measurement set.

With the description you've given thus far, it appears that a database of all experimental data is precisely the "minimal theory" you seek. But it is not useful scientifically (it cannot be falsified), nor practically (it cannot make predictions).

That's not the issue, the goal is not to find an equivalent formulation, but a less restrictive one.

What exactly do you mean by "less restrictive"? My initial reaction is that that's a disadvantageous trait for a scientific theory -- the less restrictive a theory's predictions, the less the possibility for failure, and thus the less confidence we get by empirically testing it. Conversely, we gain a lot of confidence when a theory passes a test in which it makes very specific predictions.

e.g. if we are considering "space is globally Minkowski" versus "space is locally Minkowski" -- the former assertion is very specific. Every piece of experimental data consistent with the former assertion is, of course, also consistent with the latter assertion.

So, according to Bayesian statistical inference, given lots of experimental data confirming both of these assertions, it is correct to favor the stronger assertion, and so we conclude "space is globally Minkowski". (And, of course, being good statisticians, we are willing to drop that conclusion if later evidence contradicts it)

For example, uniting all metrics with the same signature is already a less restrictive form of dynamics than requiring invariance of a particular one.

They look equivalent to me. Every metric of signature +--- determines a unique class of coordinate charts (related by Poincaré transformations) in which the coordinate representation of the metric is given by d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2.

The "affine 4-space equipped with a compatable metric of signature +---" formulation does have pedagogical value due to its manifest coordinate-independence, but it is describing exactly the same theory as "affine 4-space equipped with a distinguished class of coordinate charts, and a metric whose coordinate representation in any coordinate chart is d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2."

 

[Ken G]

With the description you've given thus far, it appears that a database of all experimental data is precisely the "minimal theory" you seek.

No, a "database" is not a theory at all because it is not unified. That's what an "explanation" means-- a way to see all that data as a consequence of a single theory.

But it is not useful scientifically (it cannot be falsified), nor practically (it cannot make predictions).

Correct, the minimal theory cannot be falsified, that is why it is such a useful springboard to making the kinds of "extensions" I specifically mentioned above. It is the extensions that make predictions, and are falsifiable. That way, you know what you are doing, and avoid the "scattershot" approach by which Newtonian mechanics was replaced by special relativity, and that same scattershot approach is how special relativity is still taught today.

You see, there is no point in making predictions of experiments you cannot do, so it makes more sense to look at the experiments you can, and tailor a theory that starts with fitting all the experiments you have done, and simply extends to make a prediction for the new experiment, without being weighed down with a host of other predictions that are not being tested and probably aren't right. That way, we avoid the continual mistake of "believing" in aspects of our theories that we failed to identify as being purely out of convenience. We could also avoid this annoying illusion that science undergoes "revolutions", rather than simply learns new stuff.

What exactly do you mean by "less restrictive"?

I mean the theory would come with fewer requirements on how we picture reality, and a broader understanding of the possibilities that work equally well. For special relativity, that means that inertial observers would not be singled out as special in any way, the speed of light would not need to be isotropic, and no one would need to claim "experiments show there is no ether". We would simply set up the mathematical machinery we need to get the dynamics right, and not bother to make claims about reality that we have no way to test. Because, when we later figure out a way to test them, more often than not we discover we were wrong, and science historians will make a big deal about the shocking revolution, when in fact we were simply pretending to know something we did not know.

Ironically, this is exactly what happened with the Michelson-Morely experiment, but we missed the full lesson there. The lesson was not "M-M showed us we made the wrong assumptions", as it is normally taught, but instead, "M-M showed us the danger in making assumptions that we simply don't need to unify the observations we have on hand". We should have simply gone into M-M with an open mind, realizing that we were entering a new regime and anything could happen. We could have come equipped with several possible extensions of our current theory, and used the experiment to distinguish them, but no one needed to act the least bit surprised when one extension worked better than another.

My initial reaction is that that's a disadvantageous trait for a scientific theory -- the less restrictive a theory's predictions, the less the possibility for failure, and thus the less confidence we get by empirically testing it. Conversely, we gain a lot of confidence when a theory passes a test in which it makes very specific predictions.

But what "confidence" do you mean? Confidence that the theory is indeed working for unifying a particular measurement set, and other measurements that fit into the same overall framework, or confidence that the theory will work when applied to some completely new measurement? The former kind of confidence is the confidence that builds bridges-- the latter is the one that makes fools of the best thinkers of all time.

e.g. if we are considering "space is globally Minkowski" versus "space is locally Minkowski" -- the former assertion is very specific. Every piece of experimental data consistent with the former assertion is, of course, also consistent with the latter assertion.

The former is the more restrictive theory, because it makes more assertions about reality, and has more ways to be false. So this is a good example of just what I'm talking about-- the latter unifies our current observations, the former is false (it breaks down either if there is gravity, or if the observer accelerates). The latter requires extensions to expand its usefulness into those realms, but that's just what it should need.

So, according to Bayesian statistical inference, given lots of experimental data confirming both of these assertions, it is correct to favor the stronger assertion, and so we conclude "space is globally Minkowski". (And, of course, being good statisticians, we are willing to drop that conclusion if later evidence contradicts it)

We already know that is false.

They look equivalent to me. Every metric of signature +--- determines a unique class of coordinate charts (related by Poincaré transformations) in which the coordinate representation of the metric is given by d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2.

I agree that +--- is equivalent to -+++, it didn't matter because we are comparing to +++ with absolute time.

The "affine 4-space equipped with a compatable metric of signature +---" formulation does have pedagogical value due to its manifest coordinate-independence, but it is describing exactly the same theory as "affine 4-space equipped with a distinguished class of coordinate charts, and a metric whose coordinate representation in any coordinate chart is d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2."

Again, that is not the coordinate representation of the Minkowski metric in any coordinate chart-- only any orthogonal coordinate chart. Furthermore, we need a finite concept of distance-- an infinitesmal one does not suffice to determine the dynamics, so the latter requires a special treatment of inertial observers, un awkward and unnecessary aspect of the theory that is often mistaken for a physical statement of some kind.

 

[Hurkyl]

A trivial theory is still a theory -- aesthetic grounds are not sufficient justification for rejecting it. And besides the 'database theory' is the only theory (up to equivalence) that makes no assertions beyond the experimental data. This is fairly easy to see: if you have a theory that is not equivalent to the database theory, then either it deals with things that are not experimental results, or it makes assertions that cannot be proven by the data.

And, of course, it is a trivial exercise to show that each piece of experimental data is a theorem of the database theory.

For special relativity, that means that Inertial Observers would not be singled out as special in any way,

But they can be singled out: an observer is inertial if and only if his worldline is straight,

the speed of light would not need to be isotropic

and it's an easy theorem that null vectors have 'speed' one in any orthonormal affine coordinate chart.

The theory of special relativity, like any other theory, is formulation independent: you get the same theory no matter how you formulate it. e.g. if you formualte it in terms of inertial observers and Poincaré-invariant coordinate metrics, you get exactly the same theory as if you formulate it in terms of a coordinate-independent metric with a specified signature.

Even Lorentz relativity is effectively the same as special relativity. LR includes an extra constant symbol denoting an orthonormal coordinate frame, but is otherwise exactly the same theory as special relativity. (mathematically speaking, at least)

We should have simply gone into M-M with an open mind, realizing that we were entering a new regime and anything could happen.

Tomorrow is a new regime too. Yes, a closed mind is bad for science... but so is naïeveté. Scientific theories have been well-supported by empirical evidence, and that affords us confidence that they will continue to be correct. When going into a new experiment, we should have exactly as much confidence in our theories as they deserve... no more, and no less.

But what "confidence" do you mean?

The confidence afforded to us by the scientific method.

The former is the more restrictive theory, because it makes more assertions about reality, and has more ways to be false. So this is a good example of just what I'm talking about-- the latter unifies our current observations, the former is false (it breaks down either if there is gravity, or if the observer accelerates).
...
We already know that is false.

The point is, before we had evidence contradicting the former, it was scientifically correct to favor the "globally Minkowski" hypothesis over the "locally Minkowski" hypothesis. Why was that scientifically correct? Because the "globally Minkowski" hypothesis had stronger empirical support.

Of course, with the evidence we now have, "locally Minkowski" has stronger empirical support.

They look equivalent to me. Every metric of signature +--- determines a unique class of coordinate charts (related by Poincaré transformations) in which the coordinate representation of the metric is given by d\tau^2 = dt^2 - dx^2 - dy^2 - dz^2.

I agree that +--- is equivalent to -+++, it didn't matter because we are comparing to +++ with absolute time.

Huh? That has absolutely nothing to do with what I said in that quote.

Again, that is not the coordinate representation of the Minkowski metric in any coordinate chart-- only any orthogonal coordinate chart.

That was a typo, sorry. It was supposed to say "affine 4-space equipped with a distinguished class of coordinate charts, and a metric whose coordinate representation in any distinguished coordinate chart..."

Furthermore, we need a finite concept of distance-- an infinitesmal one does not suffice to determine the dynamics

That's what calculus is for.

 

[Hurkyl]

When you change the observer, you will change the basis vectors used to label the events of spacetime in terms of the physical measurables "distance" and "time". That is both a change in coordinates, in that the labels are changing in a particular way, and a transformation of the vector space into itself, as all the events are now seen from a different perspective-- that of a new observer.

Looking at the same events from a different perspective -- that sounds exactly like you're leaving Minkowski space unchanged, but changing the coordinate chart you're using.

For a vivid (but Euclidean) example -- put a sheet of paper on the floor and look at it. Now, walk somewhere else and look at the paper again. Did the paper change?

The events are the same,

And since, physically speaking, events in 'reality' correspond to points in Minkowski space, we see that the operation you propose doesn't transform Minkowski space. (In fact, there is nothing physical that enacts a transformation of Minkowski space)

but the vectors are different (indeed, nonlinear transformations would make them no longer even members of a vector space at all).

Minkowski space is not a vector space; it is an affine space. You can view an affine space as a vector space by choosing an 'origin' and corresponding each point of the affine space with the vector given by subtracting off the origin. If you change the origin, then yes, that correspondence will change.

It looks like you're trying to make your observer correspond to an origin but that doesn't make sense -- the origin is a single point, whereas the observer occupies an entire worldline. (actually an entire 3+1-dimensional region -- we only get a worldline if we assume zero spatial extent)

If you want to get something that is invariant to all linear transformations, you must choose one vector from the vector space and the other from its dual space, as I dimly understand the situation.

If all linear transformations of interest act trivially, you get invariance automatically. That's what happens with a coordinate change -- the change-of-coordinates transformation doesn't do anything to Minkowski space; it only changes the coordinate functions, and the coordinate spaces.


Now, the fact that the symmetry group of Minkowski space is Poincaré group is interesting... and I suspect the thing you're really interested in; coordinate changes are just a red herring. And the key point is that Minkowski space is not symmetric under skew transformations, or a rescaling along a single axis; only Poincaré transformations preserve the Minkowski structure.

 

[Hurkyl]

In fact, studying geometry by its symmetry group is the topic of the Erlangen program.

 

[Ken G]

A trivial theory is still a theory -- aesthetic grounds are not sufficient justification for rejecting it.

Not aesthetic grounds-- the grounds would be the definition of what a theory is.

And besides the 'database theory' is the only theory (up to equivalence) that makes no assertions beyond the experimental data.

There have to be some defining assumptions that theories make, such as objectivity and repeatability. These can never be proven, only falsified. That is the important kind, the "bridge-building" kind, of predictions that theories must make. These are of the "weather prediction" kind, and are the useful predictions, the bridge-building predictions, that science makes. To make predictions of that nature, there is no need to pretend theories are things that they are not.

Nevertheless, that kind of prediction is often (unfortunately) viewed as a trivial aspect of a theory-- people sometimes treat theories as if their value (erroneously) is their ability to predict outside the box of the core assumptions that define what a theory is. Those latter kinds of "predictions" are really just guesses, a way to extend a theory that, once tested, form a means to create new theories, i.e., they become predictions of the important kind. One doesn't need a theory to form a hypothesis, though they can be a helpful guide if we need one. Unfortunately, the latter gets all the attention, despite being extraneous to the value of science, and results in all kinds of misconceptions about what science is and what you can use it for (not to mention a list of "revolutions" in scientific thinking-- rather than just big discoveries, which is all they really are).

But they can be singled out: an observer is inertial if and only if his worldline is straight,

That is circular reasoning, you simply define straight that way. All we can say is their accelerometers read zero, if we want to think of that as special that's up to us-- there's no need to go and build physics around it.

and it's an easy theorem that null vectors have 'speed' one in any orthonormal affine coordinate chart.

At last we see the appearance of the word "orthonormal", which I've been hammering for awhile now.

The theory of special relativity, like any other theory, is formulation independent: you get the same theory no matter how you formulate it. e.g. if you formualte it in terms of inertial observers and Poincaré-invariant coordinate metrics, you get exactly the same theory as if you formulate it in terms of a coordinate-independent metric with a specified signature.

I remain unconvinced of that, and this is an important purpose of the thread. The key thing I have maintained is not that SR makes false predictions for quantitative measurements within the regime where it has been tested, nor that it is unable to predict the dynamics of any particle with a known proper acceleration that satisfies certain other assumptions (as are necessary in either classical physics or Dirac's formulation of quantum mechanics). Rather, its problems are pedagogical, in that it may make unnecessary guesses that could prove to be false in future experiments outside the realm where it has been tested. Such false "predictions" are not an important part of any theory, just as it was not an important part of Newton's laws that they work to arbitrary speeds (and the fact that they don't has in no way compromised their use in situations where they are warranted).

The pedagogical problems of special relativity include the fact that its postulates cannot be applied from the reference frame of an accelerated observer. Also, they imply choices about how we picture reality that are not supported, they are merely assumed. As such, it generates explanations for "why things happen the way they do" that are inconsistent between observers. A classic example is, what is the cause of a blueshift between two rockets in free space. If we take Einstein's convention for "stationary" meaning the frame of any inertial observer describing their universe, then the cause of blueshift observed by an inertial observer is always the squeezing of the wavelength due to the motion of the source, coupled with time dilation of the source. However, a more flexible interpretation of the "cause" of that phenomenon is that the wave period simply depends on the proper time of any receiver on any path that connects the path between the absorption of the prior wavecrest and the following wavecrest (calculus could make that even more precise). That accounts for everything, we do not need either of the two "postulates of special relativity" to perform that calculation, we need only the signature of the metric and the conventions by which the observer measures time (i.e., they will ultimately ratio the period of a wave to the period of a clock).

The rest is pure language and arbitrary picture/coordinates, and does not belong as part of the postulates of a theory. Once again, where you will see the problem with the latter is when some observation contradicts those postulates, and we'll ask, "but why did we expect the postulates to hold, based on the database we already had?" The answer to that will be, "there was no reason, we were deluding ourselves".

Tomorrow is a new regime too.

Yes, but all that goes right into the definition of a theory, as I alluded to above. We do not need to add special postulates to handle that, it is in all scientific theories from the start. This is my point, the importance of understanding what aspects of our theory are there because that's how we define scientific theories, what aspects are there because they unify existing observations, what parts are extensions that we are curious about testing and have no idea if they will work or not (like Newton and arbitrary speed), and what parts are just pure fantasy (like MWI) that we have no reason whatsoever to ever pass a falsifiable test.

The confidence afforded to us by the scientific method.

But I still don't know which of the two versions of "confidence" you mean. I would say the confidence afforded to us by the scientific method is of the first kind I listed, but you seem to be talking about the second situation.

The point is, before we had evidence contradicting the former, it was scientifically correct to favor the "globally Minkowski" hypothesis over the "locally Minkowski" hypothesis. Why was that scientifically correct?

It wasn't, any more than it was "scientifically correct" to think Newton's laws would extend to arbitrary speed, or that Ptolemy's model would hold up to more precise observations. The only things that are scientifically correct are to expect predictions "within the box" of the current dataset to work, that's like predicting the weather or building a bridge. Other types of predictions are called "guesses", and are not scientifically correct to expect to work (a point history has been rather clear on, especially once you bear in mind that "the winners write the history").

Because the "globally Minkowski" hypothesis had stronger empirical support.

No, it had no empirical support (even in the absence of gravity), as it was only formulated and tested for inertial observers. Indeed, it breaks down when you leave that observational regime, as is not untypical of phyical theories.

Of course, with the evidence we now have, "locally Minkowski" has stronger empirical support.

If by that you mean that "global Minkowski is known to be wrong", I agree.

Huh? That has absolutely nothing to do with what I said in that quote.

I thought you were pointing out that -+++ is the same as +---. What is written is merely a re-affirmation of what I've been saying all along-- that the Minkowski metric is invariant only under the transformations of the Poincare group (and is not invariant under arbitrary coordinate transformations or changes of observer, though its signature is).

That's what calculus is for.

If you want to use calculus to integrate the metric between events from the perspective of a constantly accelerated observer, you need to integrate the Rindler metric, not the Minkowski metric. The latter gives you the wrong answer, that's the point.

 

[Ken G]

Looking at the same events from a different perspective -- that sounds exactly like you're leaving Minkowski space unchanged, but changing the coordinate chart you're using.

That depends on what you mean by "Minkowski space", this is very much the point here. What many (most) mean by that phrase is, "a metric space ruled by the Minkowksi metric", but if you take that meaning, your statement is wrong. It is only right if you take the more general meaning of a "space governed by Minkowski geometry, constrained by metrics of Minkowski signature and the tensorial transformation rules between them." More to the point, you seem to be imagining that events themselves are members of a vector space, but they are not-- that stucture must be imposed on them by choosing basis vectors, i.e., by the coordinates. Hence, changing observer/coordinates is indeed a mapping of the vector space into itself, a mapping that if it leaves the events unchanged it changes the vectors that are associated with them, or if it leaves the vectors unchanged then it changes the events associated with them. I think one is the covariant picture, the other the contravariant. I believe the "dual space" is what you get if you make the opposite choice.

For a vivid (but Euclidean) example -- put a sheet of paper on the floor and look at it. Now, walk somewhere else and look at the paper again. Did the paper change?

As I said, if the events are taken to be invariant, then the vectors have changed. As in your example.

And since, physically speaking, events in 'reality' correspond to points in Minkowski space, we see that the operation you propose doesn't transform Minkowski space.

I thought this is what you are imagining, but I think you are incorrect. The events are one thing, the points in Minkowksi space are another, and the connection is made via the coordinatization. You are imagining that they both stay the same when we change oberver/coordinates, but they do not-- one or the other must change.

Minkowski space is not a vector space; it is an affine space.

Translations of the origin are of no interest to me, we are talking about changing reference frame. In other words, we talking about the vectors that connect events, not the vectors that connect events to an origin.

It looks like you're trying to make your observer correspond to an origin but that doesn't make sense -- the origin is a single point, whereas the observer occupies an entire worldline.

Rest assured, that is not what I'm trying to do. The fact that an observer is on a worldline is very much my concern with the standard formulation of special relativity-- it not only requires the worldline be inertial, it further require the worldline has always been inertial and always will be. That's terribly over-restrictive, and there's just no need for it.

That's what happens with a coordinate change -- the change-of-coordinates transformation doesn't do anything to Minkowski space; it only changes the coordinate functions, and the coordinate spaces.

Again, this is not the standard formulation, involving the Minkowski metric.

Now, the fact that the symmetry group of Minkowski space is Poincaré group is interesting... and I suspect the thing you're really interested in; coordinate changes are just a red herring.

Yes, this is what I'm saying, except I'm saying that the red herring is the Minkowski metric. We don't need a symmetry group that limits our postulates, indeed general relativity figures out how to do it with no such limitation.

And the key point is that Minkowski space is not symmetric under skew transformations, or a rescaling along a single axis; only Poincaré transformations preserve the Minkowski structure.

That's what I've been saying, as I recall right about the time I was "boring" and "exasperating" DaleSpam right out of the thread.

 

[Hurkyl]

Not aesthetic grounds-- the grounds would be the definition of what a theory is.

Mathematically, that definition is: (given an ambient formal logic and formal language)

A theory is a collection of statements made in the given formal language that is closed under logical deduction.​

(And given any set S of statements in a formal language, they generate a theory -- in particular, the theory consisting of all logical statements that are provable from S)

We don't need a symmetry group that limits our postulates

There is always a symmetry group, whether you state it explicitly or not.

general relativity figures out how to do it with no such limitation.

e.g. general relativity is symmetric under any isometry of differential manifolds, and the frame bundle is symmetric under global and local Lorentz transformations.

I thought this is what you are imagining, but I think you are incorrect. The events are one thing, the points in Minkowksi space are another, and the connection is made via the coordinatization.

*shrug* I guess there's nothing left to say but "you're wrong". (And similarly for many of the points in your previous posts)

Translations of the origin are of no interest to me

Minkowski space does not have an origin. It is not a vector space. (Just like Euclidean space)

The fact that an observer is on a worldline is very much my concern with the standard formulation of special relativity-- it not only requires the worldline be inertial, it further require the worldline has always been inertial and always will be.

It wasn't, any more than it was "scientifically correct" to think Newton's laws would extend to arbitrary speed, or that Ptolemy's model would hold up to more precise observations.

If you don't have empirical evidence that Newton's laws shouldn't extend to arbitrary speeds, then you don't have any scientific grounds for expecting them to fail for arbitrary speeds. (I'm assuming you meant to make a sensical statement -- we knew even before special relativity that notions of absolute velocity had no physical meaning)

 

[Ken G]

Mathematically, that definition is: (given an ambient formal logic and formal language)

A theory is a collection of statements made in the given formal language that is closed under logical deduction.​

As this is a physics forum, I would have preferred the scientific definition. For that, I think Wiki (http://en.wikipedia.org/wiki/Theory) does fine:
"In science a theory is a testable model of the manner of interaction of a set of natural phenomena, capable of predicting future occurrences or observations of the same kind, and capable of being tested through experiment or otherwise verified through empirical observation. "

The key words there are "model" and "observations of the same kind". I would say Wiki is right on target here, even so far as underscoring the imporance of "inside the box" predictions over "outside" ones.

There is always a symmetry group, whether you state it explicitly or not.

My point is that the symmetry group is all there is-- there is nothing special about inertial observers simply because they exhibit that symmetry. Arbitrarily accelerated observers exhibit other symmetries, what we need is not a way to flag the inertial ones, but rather a prescription for matching the symmetry to the observer.

A good concrete example of this is the twin paradox. With the standard formulation of SR, the reduced aging of the traveling twin is explained as "due to time dilation", as if time dilation itself was something other than an arbitrary coordinatization. Or we can choose the accelerated frame, and then the standard SR formulation with the Einstein simultaneity convention allows us to shift between inertial frames, account for time dilation, and tack on the simultaneity shift due to switching frames. Then the "reason" for the younger twin comes out sounding like "it's all due to the simultaneity convention when you accelerate", as if that was something physical rather than yet another arbitrary coordinatization. Neither of those are decent physical explanations in my view, they are both simply mistaking a coordinate convenience for a statement about how reality works. The better statement of why the difference in ages occurs is simply that different time elapses on different spacetime paths connecting two events-- but that's not the explanation that stems directly from the postulates of SR (even if we know it is in fact correct).

*shrug* I guess there's nothing left to say but "you're wrong". (And similarly for many of the points in your previous posts)

You said that when you defended the idea that the Minkowski metric was invariant to any coordinate change, but now you are recognizing orthonormal transformations, and when you claimed that changing an observer was "just a coordinate change, not a mapping from spacetime into itself", a claim you have also apparently backed off on.

Minkowski space does not have an origin. It is not a vector space. (Just like Euclidean space)

Metrics apply to vector spaces, so once again, if "Minkowski space" takes on its usual meaning as "the spacetime vector space with the Minkowski metric", then it is a vector space.

I'm confused what confuses you about this perfectly natural statement about the normal formulation of special relativity (and I remind you of the John Baez quote I posted earlier in the thread).

If you don't have empirical evidence that Newton's laws shouldn't extend to arbitrary speeds, then you don't have any scientific grounds for expecting them to fail for arbitrary speeds. (I'm assuming you meant to make a sensical statement -- we knew even before special relativity that notions of absolute velocity had no physical meaning)

You may indeed assume I was making a sensible statement there. Furthermore, if you have no reason to expect they do extend to arbitrary (relative-- obviously) speeds, then why do you think it is "scientifically correct" to expect they will? This is precisely what I am saying is not scientifically correct, as history has shown many times. It is scientifically correct to form no opinion in advance of the observation.

 

[Hurkyl]

The key words there are "model" and "observations of the same kind".

Be aware that many people (even technical people!) often make statements with an implicit assumption of nontriviality -- I would be extremely hesitant to take such an common-language heuristic explanation as being accurate on such a detail. (in fact, I would even expect experts to disagree on that detail)

That said, the database theory only needs a slight tweak to predict "observations of the same kind" -- it takes 'same kind' perfectly strictly, and only makes predictions of new experiments that are identical to previous ones.

I would say Wiki is right on target here, even so far as underscoring the imporance of "inside the box" predictions over "outside" ones.

You seem to read that statement much differently -- I would read it as, for example, a theory of fluid motion is not expected to make predictions about photonics.

That said, you are being very dogmatic about what's "inside the box". Every experiment ever performed was performed "before today". Expecting any of that data to give us information about "tomorrow" is clearly just an extrapolation -- no different in principle than extrapolating kinematics to high relative velocities, or assuming that the sun's core obeys same nuclear physics as we observe in the laboratory.

A good concrete example of this is the twin paradox.

The twin (pseudo)paradox is, by definition of 'paradox', an example of fallacious reasoning. The resolutions of the twin paradox are meant specifically to identify the flaw in the reasoning and explain why it is a flaw.

You said that when you defended the idea that the Minkowski metric was invariant to any coordinate change, but now you are recognizing orthonormal transformations, and when you claimed that changing an observer was "just a coordinate change, not a mapping from spacetime into itself", a claim you have also apparently backed off on.

Yes, the metric on Minkowski space is invariant under coordinate change. Yes, what you described as 'changing an observer' appeared to be nothing more than a coordinate change. Yes, for an actual (affine) transformation of Minkowski space itself to respect the metric, it must be Poincaré.

Metrics apply to vector spaces, so once again, if "Minkowski space" takes on its usual meaning as "the spacetime vector space with the Minkowski metric", then it is a vector space.

Yes, vector spaces may have metrics¹. So can affine spaces. And pseudo-Riemannian manifolds by definition have a metric. Minkowski space, like Euclidean space, is not a vector space.

1: The kind of metric we're talking about here. The metric on a metric space is another concept, and the two notions are not compatable in the case of interest here.

I'm confused what confuses you about this perfectly natural statement about the normal formulation of special relativity (and I remind you of the John Baez quote I posted earlier in the thread).

Your assertion that all worldlines are inertial is patently false. And note that the Baez quote doesn't say anything about observers or worldlines.

Furthermore, if you have no reason to expect they do extend to arbitrary (relative-- obviously) speeds, then why do you think it is "scientifically correct" to expect they will?

I wouldn't.

But we do have reasons to expect Newtonian mechanics to work for arbitrary relative speed: all of that pesky empirical evidence supporting Newtonian mechanics. In fact those reasons are still applicable today.

 

[Ken G]

In fact, studying geometry by its symmetry group is the topic of the Erlangen program.

Indeed, and I am suspecting that the right way to do special relatilvity is similar. By "right", I mean the way that carries no unnecessary concepts that are included purely for convenience and familiarity, but which ultimately replace the actual point of what has been discovered about reality with pictures (like the isotropic speed of light or the Einstein simultaneity convention) that are useful in practice but of deceptive physical content. They are fine for doing calculations, but may not be the best way to unify speical relativity with other advances in physics.

 

[Ken G]

That said, the database theory only needs a slight tweak to predict "observations of the same kind" -- it takes 'same kind' perfectly strictly, and only makes predictions of new experiments that are identical to previous ones.

That's exactly why I would not count it a theory, nor the goal of science.

You seem to read that statement much differently -- I would read it as, for example, a theory of fluid motion is not expected to make predictions about photonics.

Correct, I read it differently. I read it that, for example, a theory of particle dynamics that explains ideal gases would not be expected to describe the motions of those same particles when confined to the scales inside atoms.

That said, you are being very dogmatic about what's "inside the box". Every experiment ever performed was performed "before today". Expecting any of that data to give us information about "tomorrow" is clearly just an extrapolation -- no different in principle than extrapolating kinematics to high relative velocities, or assuming that the sun's core obeys same nuclear physics as we observe in the laboratory.

I agree that it is quite difficult to say categorically what is a difference "in principle", but nevertheless this is the charge that is put to science-- when you are building a bridge, for example, you face that charge all the time.

The twin (pseudo)paradox is, by definition of 'paradox', an example of fallacious reasoning. The resolutions of the twin paradox are meant specifically to identify the flaw in the reasoning and explain why it is a flaw.

The only paradox there stems from the different sounding explanations. That problem would be avoided in the approach I'm advocating.

Yes, the metric on Minkowski space is invariant under coordinate change.

(shrug)-- that is simply wrong, what more can I say. We were making progress when we established that the invariant was only the signature of the metric, and that the manifold was Lorentzian as a result. Don't backslide now.

The metric on a metric space is another concept, and the two notions are not compatable in the case of interest here.

Equivocation. We have always been talking about the standard way special relativity is described and axiomatized, right from the start of the thread. As such, "the metric on a metric space" is just what we've been talking about. What we discovered of importance, in my view, is that it is not the metric at all that generates gravity-free dynamics, it is the Lorentzian geometry of the manifold, which is connected to the signature of the metric. If you want a true coordinate-free invariant, you must work with the covariant/contravariant dual spaces, as neither the covariant metric tensor, nor the contravariant metric tensor, is by itself invariant to the kinds of transformations of spacetime into itself that we need to do physics from the perspective of different observers.

Your assertion that all worldlines are inertial is patently false.

A pretty good indicator that I never made, or even thought, any such assertion.

And note that the Baez quote doesn't say anything about observers or worldlines.

Nevertheless, what it did say is something you have simply ignored.

But we do have reasons to expect Newtonian mechanics to work for arbitrary relative speed: all of that pesky empirical evidence supporting Newtonian mechanics. In fact those reasons are still applicable today.

I haven't the vaguest idea what you are trying to say here, because taking the literal meaning is obviously "patently false".

 

[Hurkyl]

The only paradox there stems from the different sounding explanations. That problem would be avoided in the approach I'm advocating.

Students have to learn coordinates -- they are so computationally useful that it would be harmful to deny them that knowledge. And while students learn coordinates, many will make mistakes, and some will rediscover the twin pseudoparadox. Showing them an unrelated derivation of the same quantity does not help -- it does not fix their misunderstanding of coordinates.

If we were considering it as a putative paradox, your approach is entirely useless: a paradox consists of two separate arguments that lead to contradictory results. Offering yet another argument does not repair the theory.



(shrug)-- that is simply wrong, what more can I say. We were making progress when we established that the invariant was only the signature of the metric, and that the manifold was Lorentzian as a result. Don't backslide now.[/quote]
You're equivocating. The metric is invariant under all coordinate changes. Amongst symmetries of Minkowski space, it is invariant only under Poincaré transformations. You need to stop confusing those two ideas.

Equivocation. We have always been talking about the standard way special relativity is described and axiomatized, right from the start of the thread. As such, "the metric on a metric space" is just what we've been talking about.

Wrong. See:
(pseudo)Metric tensor
metric (as in a metric space)

A pretty good indicator that I never made, or even thought, any such assertion.
Nevertheless, what it did say is something you have simply ignored.

So, why did you say:

The fact that an observer is on a worldline is very much my concern with the standard formulation of special relativity-- it not only requires the worldline be inertial, it further require the worldline has always been inertial and always will be.​

?

I haven't the vaguest idea what you are trying to say here, because taking the literal meaning is obviously "patently false".

I meant exactly what I said -- this very day, we have reasons to believe Newtonian mechanics works for arbitrary relative velocities, and that is perfectly consistent with the scientific conclusion that Newtonian mechanics doesn't work for arbitrary relative velocities. I'm trying to provoke thought about how empiricism works.

 

[Ken G]

Students have to learn coordinates -- they are so computationally useful that it would be harmful to deny them that knowledge. And while students learn coordinates, many will make mistakes, and some will rediscover the twin pseudoparadox. Showing them an unrelated derivation of the same quantity does not help -- it does not fix their misunderstanding of coordinates.

But I'm not talking about the students who have made a mistake, I'm talking about those who did everything correctly and can't figure out why the two observers cannot agree on the reason that one is younger. That's a problem. The standard answer is "the reason is itself coordinate dependent", and I used to accept that answer. Now, I see it as flawed-- it misses the whole point of relativity. If the point of relativity is that different observers can use the same physics, then they should also be able to do it in a way that finds the same answer to the same question. It really isn't that hard to do relativity that way, it would just use coordinates as a convenience rather than as something seen as an integral part of the physics.

Perhaps an example would help-- the ficticious forces of a rotating reference frame. The standard way we teach that is to first show students why such forces are merely illusions, as they don't fit the postulates of Newton's laws and they emerge from the coordinates not the physics. Later, after we have convinced students such forces don't exist, we give them problems where they are terribly convenient to use, and most students end up not really caring if those forces are real or not, they work if you know how to use them. So we are in effect saying, in reality Newton's laws are true, but in practice, we can relax them if we know what we are doing. I'd say this story is a classic example of losing track of what Newton's laws are for, i.e., how to solve problems, not how to tell reality what is real.

We make the same mistake again in relativity, even as we pretend we are "fixing" Newton's laws by not making the mistake of thinking in absolute terms. We were on the right track, but should have gone all the way-- we should have found the way Newton's laws should have been formulated in the first place, where they would have made all the same predictions at low relative speeds and simply not known what they would do at high speeds nor even how high the speed needed to get to discover the breakdown. That would be the "honest" way to do science, and would have avoided any need for a "revolution" in 1905.

If we were considering it as a putative paradox, your approach is entirely useless: a paradox consists of two separate arguments that lead to contradictory results. Offering yet another argument does not repair the theory.

No, my approach is not at all useless. Let's look at how my approach applies to the centrifugal force example. I say, "why do I weigh less at the equator", and someone else says "because centrifugal force counteracts some of gravity". Someone else says "no, that's wrong, there's no such thing as centrifugal force to counteract gravity, it is because I simply require less force from the scale to move in a circle at the equator". Two different answers to the "why" question, which is the only sense to which the twin problem is a "paradox" (once we've recognized failures of our daily intuition does not count as paradoxical). In first-year physics, the first answer is "wrong" and the second is "right", but later when you learn general relativity, either answer is just as good.

What I'm saying is, let's not have either explanation, if we really want our best answer to the "why" question, let's penetrate a step deeper and find the most unifying answer, phrased in a coordinate-independent way that cares about neither the presence nor the absence of "centrifugal force". We can still look at the other answers, and the concept of centrifugal force if we want it, and tailor to the audience or the situation, but we don't imagine we are saying something physically true, we are just selecting a convenient picture to use because we know it gets the right answer.

You're equivocating. The metric is invariant under all coordinate changes.

I do not know what you mean by that statement. To me, if someone says "the metric is invariant", they mean the action of the metric on all vector pairs v_1,v_2, selected from a vector space, is invariant under any coordinate change (i.e, homeomorphisms) of the vector space. Since the action of the metric is not invariant under any homeomorphism that is not orthonormal, that contradicts your statement. Obviously you mean something different by it, but the real issue here is, if you pick up a relativity textbook and start reading about the "Minkowski metric", are you reading about the actual thing that defines the structure of Lorentzian manifolds, or are you reading about some specially coordinatized version of that true structure? I say the latter, and that's the problem with it.

Amongst symmetries of Minkowski space, it is invariant only under Poincaré transformations.

Then it is not invariant under arbitrary transformations. You have not established why there is anything to "confuse" here. Note that we are not discussing whether or not mathematicians know what a Minkowski space is, they invented it, nor are we discussing whether or not dynamics on a pseudo-Riemannian manifold are locally that of a Lorentzian manifold, we know that they are. We are talking about whether or not we should treat the Minkowski metric as something special, or as just one from a whole class of metrics with the same signature that are all equally physically "real" and equally selected from the class of metrics important for understanding spacetime from the perspective of any observer.

Wrong. See:
(pseudo)Metric tensor
metric (as in a metric space)
[/quote]There does appear to be a difference between a metric tensor and a metric space, the former being coordinate independent but taking on different forms in different coordinates. I'm still a bit confused on this point, as for example the "Rindler metric" and the "Minkowski metric" both apply to flat spacetime but for different observers. Whether or not that makes them different metrics or not is the confusing part. I was wrong about using them with the dual space-- the metric tensor is a way to choose both vectors from the same vector space without using the dual space. The connection between the mathematics and the physics, and the "specialness" of inertial coordinates, is murky yet.

So, why did you say:

The fact that an observer is on a worldline is very much my concern with the standard formulation of special relativity-- it not only requires the worldline be inertial, it further require the worldline has always been inertial and always will be.​

That quote does not claim worldlines are inertial, it says that noninertial observers also have worldlines and should be able to coordinatize spacetime using constantly changing basis vectors. For example, Einstein's simultaneity convention is a lot different for such an observer (allowing time to be perceived as going backward, for example).

I meant exactly what I said -- this very day, we have reasons to believe Newtonian mechanics works for arbitrary relative velocities, and that is perfectly consistent with the scientific conclusion that Newtonian mechanics doesn't work for arbitrary relative velocities. I'm trying to provoke thought about how empiricism works.

I'm not getting the connection, it just sounds like a contradiction. Why would we care if we have reasons to believe Newtonian physics works for arbitrary v if we know it doesn't?