Einstein simultaneity: just a convention?

[Ken G]

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?

 

[mitesh9]

I think I can't understand what you are asking for!
As I have understood it, my answer would be, there is nothing fundamental in Einstein's definition of time (or for that matter simultaneity) or Lorentz transforms, and they are just arbitrary (and convenient, as u said) ways to understand the 'mnemonics' of 'motion'. We will keep on refining them until we have matched the ultimate way of understanding everything. However, It's just me, others may (and should, will) differ.

 

[Ich]

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?

It is the mere "coordinate possibility", which makes not only the speed of light but also the electromagnetic and mechanical laws isotropic, that let's us assume that this symmetry is a law of nature.

 

[Ken G]

Both of the responses so far seem more in keeping with "the former" answer, in that mitesh9 echoes my use of "convenience", and Ich cautiously inserts "lets us assume" that the speed of light is isotropic. Probably Ich is straddling the line a bit, and likely identifies more with "the latter" camp from the OP, but only in a kind of "Occam's Razor" sort of way. To be a full-fledged member of "the latter" camp, someone would need to interpret the one-way speed of light itself, not the symmetry that "permits the assumption", to be a constant of nature. I'm wondering if anyone else sees it more firmly in "the latter" camp, and if that's not our understanding, why do we teach it that way?

 

[mitesh9]

... why do we teach it that way?

You will notice Sir, that the teachings are not working for some reason, else, you would have found the answers with second flavor!

 

[Ken G]

I tend to agree, though I am not familiar with your colorful use of the expression "second flavor"!

 

[Hurkyl]

As the former is often taken as a postulate of special relativity, are we messing up the proper axiomatic structure of our art here?

For the record, axioms are not an intrinsic part of a theory. They are more like a spanning set for a vector space; axioms are simply a 'computationally' convenient way for working with a theory.

 

[phyti]

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.
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?

Using the 2nd postulate, c is constant..., you can derive the same results in SR, with one exception. Time dilation is physically real, length contraction is an interpretation.
The 1st postulate was a philosophical preference.

 

[Ken G]

For the record, axioms are not an intrinsic part of a theory. They are more like a spanning set for a vector space; axioms are simply a 'computationally' convenient way for working with a theory.

I'm not at all sure what you mean by that, or if you are distinguishing axioms from postulates (I should have used the latter term, as I believe postulates are chosen optionally to test their ramifications whereas some view axioms as kind of self-evident truths). But nevertheless, this is very much the question I'm asking-- is the isotropic speed of light truly a postulate of relativity, or has it been misnamed, as it is instead an assumption of convenience that is deeply related to a particular choice of coordinates?

 

[Ken G]

Using the 2nd postulate, c is constant..., you can derive the same results in SR, with one exception. Time dilation is physically real, length contraction is an interpretation.
The 1st postulate was a philosophical preference.

This is also right at the heart of what I'm asking, i.e., the difference between a physical principle and a philosophical preference. But I don't understand what you are saying-- if you had to clarify more clearly what is a physical postulate, what is a philosophical preference, and what is a coordinate choice, how would you recast the description of special relativity?

 

[Hurkyl]

I'm not at all sure what you mean by that, or if you are distinguishing axioms from postulates

In formal logic, axioms are nothing more than a means for presenting a theory. There is no intrinsic quality that distinguishes between the chosen axioms and the other statements in that theory. To wit, any mathematical theory can be axiomatized in infinitely many different ways. (including the "every statement of the theory is an axiom" axiomization)

The main point I'm trying to make is that there is no mathematical content in your question -- it's purely a question of pedagogy. (or possibly of philosophy)

 

[Ken G]

In formal logic, axioms are nothing more than a means for presenting a theory. There is no intrinsic quality that distinguishes between the chosen axioms and the other statements in that theory. To wit, any mathematical theory can be axiomatized in infinitely many different ways. (including the trivial "every statement of the theory is an axiom" axiomization)

I see what you are saying, the theory is essentially every prediction it makes of the invariants, and the paths used to arrive at the prediction is a particular choice of axiomatization. That's a helpful insight, bringing into better focus some of the things bouncing around in my head, and gibes with what I was calling "the former" presentation of the axiomatic structure of relativity. So I should not have asked if we are messing up the axiomatization itself, I should have asked are we messing up the way we describe the meaning of that axiomatization. Because the way relativity is taught is invariably "at first we thought there was an ether but then Michelson-Morely proved there wasn't", when it should be said that "at first we thought the basic symmetry was built around a preferred frame, but found the symmetry supports an elegant coordinatization that doesn't require that concept". Ironically, cosmology returns us to something closer to the former position, which is why the way special relativity gets taught could be viewed as counterproductive.

The main point I'm trying to make is that there is no mathematical content in your question -- it's purely a question of pedagogy. (or possibly of philosophy)

That's how I tried to frame it, yes. But I see that the "messing up" comment suggested otherwise-- what I meant was that we may be messing up the proper pedagogy. I'm still interested if there is anyone who doesn't see it that way, before I conclude that we are.

 

[Ich]

I'm not sure whether our pedagogy is fundamentally flawed in this aspect. Do people actually say that the ether ist disproved? There are other things to worry about, like the teaching of SR in ether terms, which is quite common.
But I think it is right that it is not common knowledge that there are infinitely many theories which are experimentally indistinguishable from SR. It is also quite hard to explain at an introductory level why these theories are nevertheless unacceptable.

 

[mitesh9]

I tend to agree, though I am not familiar with your colorful use of the expression "second flavor"!

"Second flavor" meant to depict the second option you gave, amongst the two to choose between... You see, from the given two, we (me and Ich) chose to go with the first one (or first flavor?)!

 

[Dale]

I think the pedagogy is fundamentally flawed. Students are famously unsuccessful at learning the relativity of simultaneity, and based on my own experience I would strongly favor introducing 4-vectors and the Minkowski norm as early as possible.

However, as to the OP's Q: There is a physical significance to the Einstein synchronization convention, namely the isotropy of the one-way speed of light. There is also a mathematical significance, namely that synchronization gives an orthogonal basis set. That said, the physical significance does seem somehow "less" since there seems to be no physical significance to the fact that two events are simultaneous in some frame since they cannot be causally related.

 

[Ken G]

I'm not sure whether our pedagogy is fundamentally flawed in this aspect. Do people actually say that the ether ist disproved?

In my experience, yes.

There are other things to worry about, like the teaching of SR in ether terms, which is quite common.

I haven't seen that. Perhaps there is more than one problem going around.

But I think it is right that it is not common knowledge that there are infinitely many theories which are experimentally indistinguishable from SR. It is also quite hard to explain at an introductory level why these theories are nevertheless unacceptable.

Why are they unacceptable? Also, stimulating that kind of question may be as important to a student of science as relativity itself.

 

[Ken G]

"Second flavor" meant to depict the second option you gave, amongst the two to choose between... You see, from the given two, we (me and Ich) chose to go with the first one (or first flavor?)!

Ah, I see. Frankly it surprised me how preferred that flavor is-- among the scientists I know, that flavor comes close to being blasphemous.

 

[Ken G]

However, as to the OP's Q: There is a physical significance to the Einstein synchronization convention, namely the isotropy of the one-way speed of light.

It sounds like you are saying that because people have problems with the physical significance of the Einstein simultaneity convention, it is best to move right to its ramifications (the Minkowski norm) and avoid confusion in applying it. But are we not covering our tracks a bit too much with that approach? In other areas of physics, we teach that multiple coordinate systems are equally valid, it's just that some conform better to the symmetries so are more convenient.

There is also a mathematical significance, namely that synchronization gives an orthogonal basis set.

But not a unique one, correct?

That said, the physical significance does seem somehow "less" since there seems to be no physical significance to the fact that two events are simultaneous in some frame since they cannot be causally related.

That's very much a key point, I would say.

 

[mitesh9]

Ah, I see. Frankly it surprised me how preferred that flavor is-- among the scientists I know, that flavor comes close to being blasphemous.

Sure! But why only the scientists? It is equally "blasphemous" here on PF as well!
Ironically enough, if you do not accept SR and GR, you are not fit to be scientist. Logic is no absolute either, It is relative indeed (i.e. if it matches with SR, it's true, else not)!
But yes, It surely (fortunately and thankfully) disqualifies me to be a scientists.

 

[Ken G]

Perhaps it would be helpful to distinguish two separate issues, one being whether or not the observational evidence has made the case as to the value of special relativity, and the other is whether we are being true to the lessons of relativity as to what we accept as the most general way to understand that theory.

 

[DrGreg]

Einstein simultaneity is indeed "just a convention". In the same way that when we describe 3D Euclidean space using xyz coordinates, it is conventional to choose the 3 space axes to be orthogonal to each other. There is no necessity to do so, it just makes the maths a hell of a lot easier.

When you are learning relativity, having to cope with lots of different coordinate systems (not necessarily orthogonal) would be an added complication to a subject that is already difficult for many to grasp. (And trying to cope without any coordinates seems nigh on impossible to me!)

The way that the postulates of relativity are usually phrased and interpreted implies that they are more than just physical assumptions; they are also assumptions about what coordinate systems we will use. I can't see any way of coming up with totally coordinate-free postulates. The best we can do is say that if we choose to measure things in a particular way, we will not be able to distinguish one observer's frame from another.

 

[dx]

Ken G,

I think you will find William L. Burke's book "Spacetime, Geometry, Cosmology" very much in your spirit. He emphasizes very clearly that simultaneity has no physical significance and is just a convention, and he also says that he considered not mentioning simultaneity at all since the physical predictions of relativity are completely independent of the way we choose to define simultaneity.

 

[Ken G]

When you are learning relativity, having to cope with lots of different coordinate systems (not necessarily orthogonal) would be an added complication to a subject that is already difficult for many to grasp. (And trying to cope without any coordinates seems nigh on impossible to me!)

I can certainly agree with that, but my concern is the way this practice can often merely replace one set of questions with another. In other words, everyone learning relativity has the usual "ten questions" or so, and by using a standard (best, even) coordinatization, we can find nice neat (though difficult) answers to those questions. Then we send them home.

But if they think more deeply about it, they now find the next ten questions, that are raised by that approach (like, is the speed of light "really" constant?, or is the universe built to have the laws of physics be the same in all frames, or did we build physics to make that true?). So we need to be careful that, in giving good answers to the first ten questions, we are not promulgating bad answers to the next ten. It's a bit of a quandary, perhaps a kind of "two-pass" approach, as is used in mechanics for example, is a good way to go for advanced students.

The way that the postulates of relativity are usually phrased and interpreted implies that they are more than just physical assumptions; they are also assumptions about what coordinate systems we will use.

I would actually call that less than purely physical assumptions, as in, physical assumptions plus a few crutches that compromise the physical structure to keep us from having to think about the tougher questions about what the structure of the theory really is. I don't mind crutches-- as long as we recognize we are doing it.

I can't see any way of coming up with totally coordinate-free postulates.

Neither can I, but I'll bet someone can!

The best we can do is say that if we choose to measure things in a particular way, we will not be able to distinguish one observer's frame from another.

I agree, that would be an excellent place to start-- and is not the usual approach.

 

[mitesh9]

Perhaps it would be helpful to distinguish two separate issues, one being whether or not the observational evidence has made the case as to the value of special relativity, and the other is whether we are being true to the lessons of relativity as to what we accept as the most general way to understand that theory.

Well Sir, Now the things are more clearer from a few more responses. I am equally surprised to see that there are many of us who think that the einstein simultaneity is just a convention, and no more physically significant outcomes are extracted from that.

I think what you say about distinguishing two issues makes sense to me. the first one being the issue with observational evidences of SR, which have, though initially improved the position of the theory amongst others, have largely hurt the theory later. The problem with the evidences is that they are so indirect, that it becomes almost impossible to accept them as proofs of SR, instead, alternative explanations are sometimes so strong (may not be acceptable in the domain of SR), that they tend to prove the evidences against the SR. In case there would have been a single direct evidence of SR, people would have not any problem accepting teachers lessons regarding the theory. We do not question Quantum mechanics, though it is more complex mathematically then SR, yet we question SR. And this may precisely be the reason, we are taught to accept relativity, yet we refrain from saying that it is "Physical reality".

 

[dx]

I am equally surprised to see that there are many of us who think that the einstein simultaneity is just a convention, and no more physically significant outcomes are extracted from that.

Do you mean to say that einsteinian simultaneity is more than a convention?

 

[Dale]

It sounds like you are saying that because people have problems with the physical significance of the Einstein simultaneity convention, it is best to move right to its ramifications (the Minkowski norm) and avoid confusion in applying it. But are we not covering our tracks a bit too much with that approach? In other areas of physics, we teach that multiple coordinate systems are equally valid, it's just that some conform better to the symmetries so are more convenient.

Pedagogically I think that is actually not correct. Some brief mention of alternative coordinate systems may be made by a particularly thourogh professor, but most students can go through their entire undergraduate physics coursework without ever actually working a problem in a non-cartesian or non-inertial coordinate system. If we don't use a complicated multi-coordinate pedagogical approach when analyzing everyday situations, where experience and intuition serve us well, then what would be the benefit of further complicating an already difficult teaching situation in extra-ordinary relativitistic situations?

When I am looking at choosing a coordinate system here are my three desires (in order of importance):
1) simplified math
2) orthogonality
3) physical significance

The Einstein convention has 1 and 2, and is light on 3. Most other synchronization conventions lose 1 and 2 for little if any improvement in 3.

If I were to choose any other coordinate system it would be radar coordinates. There the physical significance of the axes is clear, and they are orthogonal. Having never done any actual work with them I cannot speak about the simplified math, which is really the most important IMO, particularly pedagogically, but they may turn out to be good for simplicity as well.

 

[Ken G]

Some brief mention of alternative coordinate systems may be made by a particularly thourogh professor, but most students can go through their entire undergraduate physics coursework without ever actually working a problem in a non-cartesian or non-inertial coordinate system.

That is partially true-- we would never study the gravity of a point source in cylindrical coordinates, for example. But there are several non-inertial coordinate systems that are very common indeed, such as rotating systems for analyzing a Foucalt pendulum or Roche lobe overflow in a binary star. And globally non-Cartesian systems are also common. Still, I think the valid point you are making is that we generally do select coodinates in a sensible way, never making work for ourselves simply to prove that it would have been possible to do it some other way.

What I think distinguishes relativity, pedagogically, is that it is so close to physics-as-philosophy that it becomes more important to make these distinctions clear. Most first-year physics majors are taught that centrifugal forces, for example, are "ficticious", i.e., they are coodinate forces that don't obey Newton's third law. So we say, in effect, "we are choosing coordinates to fit a symmetry but don't think this is something real". If we feel it is so important to make that distinction in elementary physics, why not for the more advanced physics? Is it not even more important to get our "ontological ducks" in order for advanced students than for introductory ones?

 

[Dale]

the first one being the issue with observational evidences of SR, which have, though initially improved the position of the theory amongst others, have largely hurt the theory later. The problem with the evidences is that they are so indirect, that it becomes almost impossible to accept them as proofs of SR, instead, alternative explanations are sometimes so strong (may not be acceptable in the domain of SR), that they tend to prove the evidences against the SR.

This is completely wrong. SR is one of the most well-tested theories ever, with http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html" . For you to not accept the verdict of such strong experimental evidence is for you to reject science and the scientific method.

 

[Dale]

What I think distinguishes relativity, pedagogically, is that it is so close to physics-as-philosophy that it becomes more important to make these distinctions clear.

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor, or, as Einstein said, "Everything should be made as simple as possible, but not simpler."

Relativity was not developed and accepted because of some philosophical crusade in the scientific community at the time. It was developed and accepted on the exact same basis as all other successful scientific theories: it was the simplest theory that fit the observed experimental data. Classical physics couldn't explain the data, and other theories that could explain the data (like Lorentz's ether) were more complicated. That is pure science, and other than Occham's Razor I really see very little philosophical in it.

 

[Antenna Guy]

If I were to choose any other coordinate system it would be radar coordinates. There the physical significance of the axes is clear, and they are orthogonal. Having never done any actual work with them I cannot speak about the simplified math, which is really the most important IMO

In cartesian, if you set one ordinate to a constant, you get a plane.

In spherical (I assume that is what you meant by "radar"), if you set one ordinate to a constant, you get either a sphere, cone, or plane depending upon which ordinate you choose to hold constant. Spheres are quite handy for defining closed surfaces about things that radiate (speaking from the antenna perspective, of course ).

Regards,

Bill

 

[DrGreg]

If I were to choose any other coordinate system it would be radar coordinates. There the physical significance of the axes is clear, and they are orthogonal. Having never done any actual work with them I cannot speak about the simplified math, which is really the most important IMO, particularly pedagogically, but they may turn out to be good for simplicity as well.

Yes, I'd momentarily forgotten about radar coordinates. If you restrict yourself to one space dimension (which is commonplace when learning the theory), they work really well, they have an easy-to-grasp physical significance, there is no clock synchronisation to worry about, and many of the equations are actually simpler than their (t,x) counterparts! You measure motion using the physically measurable doppler factor k (or rapidity = log_ek, which is additive) instead of velocity.

For example, see this post where I give Bondi's proof, from first principles (i.e. directly from the postulates), that, in radar coordinates (u,v), the Lorentz transform becomes

u' = k u
v' = k^{-1} v,​

the metric is

ds^2 = du \, dv,​

and how these transform into standard Einstein-synced (t,x) coordinates.

Unfortunately, radar coordinates are less convenient if you want to work in 2 or 3 space dimensions. You can either work with (u,v,y,z), which is OK if there's no motion in the y or z directions, or else use spherical polar coords (u,v,\phi,\theta), but then you have non-linear coords which are generally more painful, and usually considered only in GR rather than SR.

 

[Ken G]

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor, or, as Einstein said, "Everything should be made as simple as possible, but not simpler."

It seems to me the difficulty students have with relativity stems primarily from how highly it disagrees with their intuition about time and space. I'd call that a more significant philosophical impact than Occam's razor. It is the first introduction for most people of how much different reality can behave than we think, and that's largely the pedagogical importance of the theory.

Relativity was not developed and accepted because of some philosophical crusade in the scientific community at the time.

Nor is that required for a theory to have important philosophical content. Quantum mechanics wasn't either.

 

[mitesh9]

This is completely wrong. SR is one of the most well-tested theories ever, with http://www.edu-observatory.org/physics-faq/Relativity/SR/experiments.html" . For you to not accept the verdict of such strong experimental evidence is for you to reject science and the scientific method.

I think I should make it clear that I'm not against SR (or Einstein for that matter, nor am I gifted enough to ever expect this), instead, being a chemist, the best fit for my status in PF can be as a hobbyist relativist. The point I raised was that the "overwhelming" evidences has not stopped the scientific community to conspire about SR, instead, these evidences are the only points the anti-relativists target to prove SR wrong (or at least inherently inconsistent).

Why do you think NASA and Stanford Uni. sent Gravity probe, if SR has been established by so called "overwhelming evidences"? Just to shut up anti-relativists, which are not even considered the part of scientific community, and treated in ever-increasingly harshest possible manner world-over? Of course, it is not just to satisfy the curiosity, I suppose!

 

[Ken G]

For example, see this post where I give Bondi's proof, from first principles (i.e. directly from the postulates), that, in radar coordinates (u,v), the Lorentz transform becomes

u' = k u
v' = k^{-1} v,​

the metric is

ds^2 = du \, dv,​

and how these transform into standard Einstein-synced (t,x) coordinates.

Right, so although it's common to state that the Minkowski norm is "coordinate independent", that's only true within a coordinate subclass. What we need to know is, what is the core principle that unites the Minkoswki norm with the radio norm? A mathematician could probably say it in one line, but I wouldn't understand a single word-- I want the physical statement, and I feel that we should teach relativity to reflect that, rather than asserting a constant speed of light as if it were a physical fact (that is very much what is normally done).

 

[dx]

Why do you think NASA and Stanford Uni. sent Gravity probe, if SR has been established by so called "overwhelming evidences"? Just to shut up anti-relativists, which are not even considered the part of scientific community, and treated in ever-increasingly harshest possible manner world-over? Of course, it is not just to satisfy the curiosity, I suppose!

Gravity Probe tests GR, not SR.

 

[DrGreg]

Right, so although it's common to state that the Minkowski norm is "coordinate independent", that's only true within a coordinate subclass. What we need to know is, what is the core principle that unites the Minkoswki norm with the radio norm? A mathematician could probably say it in one line, but I wouldn't understand a single word-- I want the physical statement, and I feel that we should teach relativity to reflect that, rather than asserting a constant speed of light as if it were a physical fact (that is very much what is normally done).

The interval "ds²" is invariant -- the same value between a given pair of nearby events according to every observer. If the observer is using standard Einstein-synced Minkowski coords, the interval is always given by the formula

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

(the c might be in a different place or the signs might be opposite according to what your metric sign convention is, but once that's decided, all observers use the same formula).

The physical significance of the interval (as I chose to write it) is:

- if ds² > 0, ds is the proper time taken by an inertial observer to travel between the events (and it is also the longest proper time that anyone, inertial or not, could take to travel between the events) "Proper time" means time measured by your own clock between events that occur at zero distance from yourself (so no sync required).

- if ds² < 0, \sqrt{-c^2ds^2} is the proper distance between the events measured by an inertial observer who considers them to be Einstein-simultaneous

- if ds² = 0, it is possible for a photon of light to pass through both events.

If you use coordinates other than the standard orthogonal Einstein-synced coords, you will get a different formula for ds².

For example, even with Einstein-synced time but spherical polar spatial coords, you get

ds^2 = dt^2 - dr^2 / c^2 - r^2 ( d\theta^2 + \sin^2 \theta d\phi^2) / c^2​

In Special Relativity (SR), you never use coordinates like this, but in General Relativity (GR), you have no choice but to do so. Special Relativists almost always use Einstein-synced orthogonal Minkowski coords, but General Relativists are happy to use any coordinate system you like. (But the maths of GR is a whole lot more complicated than SR.)

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction. The fact that all inertial observers using Einstein-synced clocks agree on the value of the coordinate speed of light is then really a consequence of the first postulate (because otherwise you could distinguish one frame from another). (See "Two myths about special relativity", Ralph Baierlein, http://link.aip.org/link/?AJPIAS/74/193/1 , section III.)

 

[shoehorn]

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor

Really? What about the substantivalism/relationism debate? Spacetime pointillisme? The relation of simultaneity to the automorphism group of Minkowski space?

Those are three massive philosophical questions which arise out special relativity. The substantivalist/relationist debate in particular is (or at least should be) encountered by most everyone who studies the philosophy of science at university.

 

[Ken G]

The interval "ds²" is invariant -- the same value between a given pair of nearby events according to every observer. If the observer is using standard Einstein-synced Minkowski coords, the interval is always given by the formula

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

Yes, that is the standard coordinatization, and I realize what all that gives rise to. My point is that this is usually the starting point for relativity, so there is no recognition that any other choice is even possible. I want to understand what must be true before we make this choice, what have we learned about reality, not about what a good coordinate is.

If you use coordinates other than the standard orthogonal Einstein-synced coords, you will get a different formula for ds².

Right, so the question is, what really is coordinate independent?

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction.

But that is a much weaker version, because different sources could still emit photons with the same properties but not isotropic speeds. And if the postulate is "space plays no special role in the propagation, it's always the same", then note this only applies if we treat inertial observers as special. In short, the "special" in special relativity is somewhat oxymoronic.

 

[Hurkyl]

What we need to know is, what is the core principle that unites the Minkoswki norm with the radio norm?

Right, so the question is, what really is coordinate independent?

I'll try anyways -- the answer is "lengths and angles".

Of course, the more precise version of that answer includes:
The spacelike / lightlike / timelike classification of (tangent) vectors
Proper length of spacelike paths
Proper duration of timelike paths
Circular angle in a spatial plane
Hyperbolic angle in a mixed temporal/spatial plane

There are other coordinate-independent things too, of course, such as the topology of space-time, or the mere fact that global Einstein synchronization is possible.

 

[Ken G]

The spacelike / lightlike / timelike classification of (tangent) vectors

This one seems pretty important, with its connection to causality.

Proper length of spacelike paths

I'm not so sure about this one, spacelike paths sound like pure conceptualization to me so might well be coordinate dependent, or even not invoked at all.

Proper duration of timelike paths

This is clearly a key invariant, as we can directly measure it. I think this is the crucial invariant around which a theory should be built, and that's the main advantage of radio coordinates.

Circular angle in a spatial plane
Hyperbolic angle in a mixed temporal/spatial plane

These two sound like they are connected to something important, the spacetime curvature that becomes so important for gravity, but by themselves they sound coordinate dependent to me. I'm not sure how you measure a circular angle, or an angle in spacetime, and adding angles in a triangle requires assumptions about the vertices, so I can't say for sure if these are dependent on how we conceptualize spacetime or not.

There are other coordinate-independent things too, of course, such as the topology of space-time, or the mere fact that global Einstein synchronization is possible.

Yes, topology must be fundamental. The fact that Einstein synchronization is possible might not be so fundamental, there might always be a way to do it for objects with more general properties than our reality. So it might not actually be saying anything about reality, more so than about simultaneity conventions. We need a complete mathematical understanding of what the possibilities are.

In any event, if some or even all of the above list are fundamental properties of any successful description of reality, we still need to express those fundamental properties in the most general way, and yet also the way that incorporates everything that the observations show. In short we don't want to imagine we need to assume anything, just to make our life simpler, that is not required to fit the observations, nor do we want to leave anything out that observations require we include. I do not see at the moment why the standard postulates of relativity accomplish that, so that's more or less what I'm asking.

 

[Hurkyl]

I'm not so sure about this one, spacelike paths sound like pure conceptualization to me so might well be coordinate dependent, or even not invoked at all.

I agree it's a little harder to imagine, but I do think it's still important. For example, it's lurking behind the scenes when we talk about (coordinate) length -- if we choose an inertial coordinate chart and ask for the length of a piece of string relative to that chart, we are actually asking for the proper length of the spacelike path defined by the string and a hyperplane of simultaneity. Now, if we choose to work with different coordinates, if we are still able to identify that spacelike path, we can compute its proper length and get the same answer as before. In fact, I think it's a very good exercise to derive the length contraction formula using just this idea. (And it might help with understanding the barn-and-pole pseudoparadox)

Another point is that, over 'infinitessimal' distances, each observer has a (spacelike) hyperplane of simultaneity, which can be useful for defining spacelike paths.

These two sound like they are connected to something important,

The 'circular' angle is just ordinary Euclidean angles. ('circular' because angles are based on the circle, and specifies which trigonometry is appropriate)

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other. (Sorry, my brain was firmly in 'geometry' mode) Angle measure in such a plane is based on the hyperbola, and uses the hyperbolic trig functions. I think 'rapidity' is the term physicists use instead of 'angle'.

The fact that Einstein synchronization is possible might not be so fundamental, there might always be a way to do it for objects with more general properties than our reality.

Nonetheless, asserting it's possibility is still a very strong assertion, and one which reality is known to violate over non-'infinitessimal' length and time scales. (I suspect it's very nearly equivalent to the special relativistic requirements on space-time, but I haven't tried working out the detail)

 

[Ken G]

Now, if we choose to work with different coordinates, if we are still able to identify that spacelike path, we can compute its proper length and get the same answer as before. In fact, I think it's a very good exercise to derive the length contraction formula using just this idea.

But length contraction is part of my issue with proper length. When we change inertial frames (by changing our velocity with respect to the rod), we infer a new length when we intercept the simultaneity hyperplane with the rod as you mention. When we back that out into the length in the frame of the rod, it returns to the correct length. But if we accelerate the rod, we have to achieve the same shrinking manually, with varying proper accelerations, to keep the rod the same length in its own frame. So that time we did do something physical to achieve the length contraction. This feels rigged to me, I sense too much of our own fingerprints "at the crime scene".

Another point is that, over 'infinitessimal' distances, each observer has a (spacelike) hyperplane of simultaneity, which can be useful for defining spacelike paths.

Yes, I've wondered about this too, it doesn't seem like we can do away with spacelike separation completely, we seem to need it in a kind of tangent space we carry around with us (rulers and whatnot). But integrating that to get finite proper distances is what doesn't make a lot of sense to me, physically that seems like something arbitrary, unlike integrating proper time, which shows on a clock.

The 'circular' angle is just ordinary Euclidean angles. ('circular' because angles are based on the circle, and specifies which trigonometry is appropriate)

But how do we measure it? You can't really measure the angle around a point, because you don't know if your protractor is warped. And if you add the angles in a triangle, you need to define the locations of the vertices. You'll probably pick a single inertial frame, but why-- why does a triangle in such a frame define something about angles? Are we learning something fundamental about reality, or just something about our biases toward frames with no forces in them? General relativity tells us that gravity can mess with those angles even in an inertial system, but choosing inertial systems is still just choosing a coordinatization, it seems to me. It seems like there's something underneath that which is more fundamental than a bias toward conceptualizing spacetime using inertial frames.

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other.

Relative velocity is not directly measurable, but redshift/blueshift is, so we can certainly pin our theory on a need to get the right answer for that. Certainly when there is gravity around, we need a more general concept than relative velocity, but I realize we are talking special relativity here. So is there really a physical concept of relative velocity even without any gravity? I'm not so sure that's a physical concept, it seems like yet another coordinate choice.

Nonetheless, asserting it's possibility is still a very strong assertion, and one which reality is known to violate over non-'infinitessimal' length and time scales. (I suspect it's very nearly equivalent to the special relativistic requirements on space-time, but I haven't tried working out the detail)

I agree with that in the way special relativity is normally constructed, and the fact that it works (in the absence of gravity) definitely restricts the universe in some important way. But I think what we tend to do is to effectively invert the Einstein prescription into a picture of how the universe works. If the mapping from all the universes that admit that description is not one-to-one onto the observation set we have at our disposal, that inverse mapping does not necessarily describe the universe correctly.

That's what I'm asking for here-- a set of postulates that not only correctly describe all the observations, but are also the minimal set that do so, so can be inverted into the full set of possible universes we are constraining. One must be cautious about inverting projections. Take the idea that all inertial observers are created equal in "the eyes of the law", if you will. That is usually framed as a fundamental statement about reality, but when one realizes that "inertial observer" just means "observer who can account for everything that is happening in terms of forces on observed objects", should we be surprised that such observers can indeed account for everything using one unified prescription? Haven't we simply excluded the observers who are going "what the heck...?"

 

[phyti]

I disagree strongly with this statement. The only philosophical aspect of special relativity of which I am aware is Occham's Razor, or, as Einstein said, "Everything should be made as simple as possible, but not simpler."

Relativity was not developed and accepted because of some philosophical crusade in the scientific community at the time. It was developed and accepted on the exact same basis as all other successful scientific theories: it was the simplest theory that fit the observed experimental data. Classical physics couldn't explain the data, and other theories that could explain the data (like Lorentz's ether) were more complicated. That is pure science, and other than Occham's Razor I really see very little philosophical in it.

The Michelson-Morley 1887 experiment supported the constant 'measured' speed of light. The body of scientific knowledge in 1900 was, by today's standard, very limited in scope and area of application. The rules of physics were derived from experiments confined to Earth (except astronomical observations), and there was never a concerted effort to prove their universality. It was a gigantic extrapolation to state "the rules are universal for all inertial frames".
Einstein preferred a deterministic behavior of the world, a common view then. This is emphasized by his objection to the randomness of quantum theory. Human nature likes a secure world with no surprises or strange behavior.
This is why I say the 1st postulate was a philosophical preference.
If the speed of light is "constant and independent of its source", then it should be possible to derive the effects of uniform motion on measurements using this postulate alone.

 

[Antenna Guy]

Yes, I've wondered about this too, it doesn't seem like we can do away with spacelike separation completely, we seem to need it in a kind of tangent space we carry around with us (rulers and whatnot). But integrating that to get finite proper distances is what doesn't make a lot of sense to me, physically that seems like something arbitrary, unlike integrating proper time, which shows on a clock.

Consider looking into what a Fresnel (pronounced fra-nel) region is.

Regards,

Bill

 

[DrGreg]

The angle in a mixed spatial-temporal plane corresponds to relative velocity -- something that is presumably physically observable when two objects pass by each other. (Sorry, my brain was firmly in 'geometry' mode) Angle measure in such a plane is based on the hyperbola, and uses the hyperbolic trig functions. I think 'rapidity' is the term physicists use instead of 'angle'.

Relative velocity is not directly measurable, but redshift/blueshift is, so we can certainly pin our theory on a need to get the right answer for that. Certainly when there is gravity around, we need a more general concept than relative velocity, but I realize we are talking special relativity here. So is there really a physical concept of relative velocity even without any gravity? I'm not so sure that's a physical concept, it seems like yet another coordinate choice.

Yes, the "hyperbolic angle" between two timelike vectors is called "rapidity" and it is equal to \log_e k, where k is the doppler factor (emitted frequency)/(observed frequency), which can be measured using only proper time. (The two timelike vectors are the 4-velocities of emitter and observer.) Note that if you rescale rapidity to be c \log_e k then it approximates to coordinate-speed at low speeds.

In terms of general relativity, it only makes unambiguous sense to measure rapidity "locally" i.e. for two observers passing by each other, so that gravitational doppler shift is excluded from consideration. In G.R. only local measurements have physical significance; "remote" measurements get distorted by the curvature of spacetime and tend to be dependent on non-physical coordinates.


"Proper distance" between two objects that are stationary relative to the observer requires no definition of simultaneity as you can take as long as you like to compare your objects against a ruler. It's only the measurement of moving objects that requires a clock synchronisation convention. The distance between two events is the proper distance between two stationary objects each of which experiences one of the events.

So the interval ds can be defined in terms of proper time (if timelike) or proper distance (if spacelike), neither requiring clock synchronisation.

Note that if you have a definition of "spatial distance" for stationary objects then you can define spatial angle via the Cosine Rule dc^2 = da^2 + db^2 - 2 \, da \, db \, \cos C.

On a final note, I believe the second postulate should really be interpreted as "the motion of a photon is independent of whatever emitted it", so that it is impossible for one photon to overtake another traveling in the same direction.

But that is a much weaker version, because different sources could still emit photons with the same properties but not isotropic speeds.

Yes, as I stated it, my 2nd postulate is weaker than the common interpretation, because I demand no coordinate system. "Isotropic speed" implies a coordinate system to measure speed.

And if the postulate is "space plays no special role in the propagation, it's always the same", then note this only applies if we treat inertial observers as special. In short, the "special" in special relativity is somewhat oxymoronic.

"Special" means "ignoring gravity", rather than the status of inertial observers.

Inertial observers are different to all other observers, in a physically measurable way: they do not experience proper acceleration, i.e. "G-forces", something they can determine using an appropriate accelerometer device, without a coordinate system. (And this definition works in GR as well as SR. Inertial observers still have special status in GR, but they no longer travel at constant velocity relative to each other.)



Forgive me if I'm explaining something you already know, Ken, but the mathematical description of spacetime makes a distinction between a 4D vector X and its components (t,x,y,z). You can switch between lots of different coordinate representations, but they all represent the same vector which exists independently of its coordinates. Spacetime is equipped with an scalar "inner product" g(X,Y) which is analogous to the "dot product" of 3D Euclidean vectors x.y. 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. The properties of spacetime can described in terms of the properties of the metric (e.g. g(X, Y+Z) = g(X,Y) + g(X,Z) etc etc). And then ds² = g(dX,dX).

So, mathematically, spacetime is defined as a four dimensional vector space equipped with a metric g that satisfies certain conditions (which can all be expressed in a coordinate-free vector notation). I provide this as background information, as I know you are really looking for a physical rather than mathematical model.

 

[Ken G]

Note that if you rescale rapidity to be 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.

In terms of general relativity, it only makes unambiguous sense to measure rapidity "locally" i.e. for two observers passing by each other, so that gravitational doppler shift is excluded from consideration. In G.R. only local measurements have physical significance; "remote" measurements get distorted by the curvature of spacetime and tend to be dependent on non-physical coordinates.

Indeed, and my question is, are we really sure this is a gravitational effect? Maybe that is just as true in special relativity, only we have chosen to pretend otherwise because there exists a transparent globalization (based on inertial frames) in the absence of gravity.

"Proper distance" between two objects that are stationary relative to the observer requires no definition of simultaneity as you can take as long as you like to compare your objects against a ruler. It's only the measurement of moving objects that requires a clock synchronisation convention. The distance between two events is the proper distance between two stationary objects each of which experiences one of the events.

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.

That's the problem with using objects to witness events, that's really something observers should be doing, and using pairs of observers, instead of a single observer, seems to introduce ambiguities. That's why I never understood the concept of proper distance, and still don't. It seems purely coordinate dependent.

Yes, as I stated it, my 2nd postulate is weaker than the common interpretation, because I demand no coordinate system. "Isotropic speed" implies a coordinate system to measure speed.

I agree-- if we substitute your way of stating the second postulate, it would be interesting to see what possibilities would still be considered admissible ways of looking at reality. Ironically, that way of stating the postulate is normally associated with the presence of a wave medium, not the absence of one.

"Special" means "ignoring gravity", rather than the status of inertial observers.

I don't agree there, to me "special", as it is normally used, means "elevate the importance of inertial observers" (in things like simultaneity conventions, etc.) as being the ones for whom the laws of physics are the same. That seems prejudicial toward Galileo's principle of inertia, which is a kind of circular reasoning-- if we make one of the laws be that there is no acceleration without forces, then of course we are going to think the laws have a special relationship with acceleration-free frames. In "general" relativity, all observers, even the accelerated ones, are on an equal footing, because we treat "real" and "coordinate" forces in a unified way (gravity itself being hard to categorize as one or the other).

Inertial observers are different to all other observers, in a physically measurable way: they do not experience proper acceleration, i.e. "G-forces", something they can determine using an appropriate accelerometer device, without a coordinate system.

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

Forgive me if I'm explaining something you already know, Ken, but the mathematical description of spacetime makes a distinction between a 4D vector X and its components (t,x,y,z).

Yes, that's an important issue, how we obtain those components.

You can switch between lots of different coordinate representations, but they all represent the same vector which exists independently of its coordinates. Spacetime is equipped with an scalar "inner product" g(X,Y) which is analogous to the "dot product" of 3D Euclidean vectors x.y. 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.

So, mathematically, spacetime is defined as a four dimensional vector space equipped with a metric g that satisfies certain conditions (which can all be expressed in a coordinate-free vector notation).

Right-- but that by itself won't get you the Minkowski norm that we teach as if it was an inherent part of the metric space. If we use different coordinates, we get a different form for the metric, but the physics is identical. So what's the real physics here?

 

[Dale]

I don't agree there, to me "special", as it is normally used, means "elevate the importance of inertial observers" (in things like simultaneity conventions, etc.) as being the ones for whom the laws of physics are the same.

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.

 

[Dale]

Really? What about the substantivalism/relationism debate? Spacetime pointillisme? The relation of simultaneity to the automorphism group of Minkowski space?

Those are three massive philosophical questions which arise out special relativity. The substantivalist/relationist debate in particular is (or at least should be) encountered by most everyone who studies the philosophy of science at university.

People can and will debate about anything. And although the debate may even be very important, it is not an essential part of the theory itself. As a case in point I have used SR for years (not professionally) and I have no idea what you are talking about with any of those debates. It isn't that the debates are unimportant, they are just not essential to the theory.

 

[Hurkyl]

Yes, that's an important issue, how we obtain those components.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.
Right-- but that by itself won't get you the Minkowski norm that we teach as if it was an inherent part of the metric space. If we use different coordinates, we get a different form for the metric, but the physics is identical. So what's the real physics here?

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

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

 

[Hurkyl]

In "general" relativity, all observers, even the accelerated ones, are on an equal footing, because we treat "real" and "coordinate" forces in a unified way (gravity itself being hard to categorize as one or the other).

In general relativity, "gravity" is simply the tendency of objects to travel in a straight line through space-time; i.e. inertial travel. An object under the sole influence of gravity travels in a straight-line path (a geodesic), and experiences a net force of zero.

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