Is dτ Invariant Under Transformations Beyond Lorentz in Special Relativity?

[friend]

Can the metric of special relativity be derived from requiring the infinitesimal line segment, dτ, to be invariant in space and time? If we parameterize a line segment by the variable τ marked off along the line (that exists in space and time dimensions) is the length in τ of that line segment only invariant with respect to the Lorentz transformations? Or are there other coordinate Xformations for which dτ would also be invariant? Thank you.

 

[Mentz114]

This thread https://www.physicsforums.com/showthread.php?t=651640 will be of interest.

 

[friend]

This thread https://www.physicsforums.com/showthread.php?t=651640 will be of interest.

Thank you. I'm trying to keep up with that thread. There's some interesting stuff there. I tried asking some questions there. But they seem to be focused on their own goals in that thread. So I'd like to post my questions here to see if I can generate interest and get comments here without cluttering up someone else's thread.



I'm given to understand that

dτ²=dt²-dx² = dt'²-dx'²

when (t',x') are the Lorentz transformation of (t,x).

Perhaps it's instructive to consider in what circumstances dτ should want to be considered invariant wrt to coordinate changes. Maybe those requirements are the driving force behind the necessity of the Lorentz transformations.

For example, the most obvious use of dτ is in the calculation of the line integral,

\int_{{\tau _0}}^\tau {d\tau '} = \tau - {\tau _0}
which is the length of a line measured in terms of segments marked off along the length of the line. Then, of course, we can always place this line in an arbitrarily oriented coordinate system and express τ in term of those coordinates.

So the question are: 1) when do we want to use the coordinates (t,x), and 2) when would we want τ-τ₀ to be invariant wrt to those coordinates?

As far as 1) goes, usually, we specify a curve in space by parameterizing the space coordinates with an arbitrary variable, call it "t". But since the x and t coordinates are arbitrarily assigned, the length of the curve can depend on the (t,x) coordinates. But if you specify that the length of the curve is invariant, then this requires the Lorentz transformations between coordinate systems.

But for 2) what should require the length of the curve to be invariant? Perhaps if we have a more fundamental requirement like

\int_{{\tau _0}}^\tau {f(\tau - {\tau _0})d\tau } = a
this will require the length of τ-τ₀ to be invariant wrt to coordinate changes in (t,x). For example, maybe {f(\tau - {\tau _0})} might be a probability distribution along a path so that its integral along the path must be 1 in any coordinate system.

Did I get this all right? I would appreciate comments. Thank you.

 

[Mentz114]

Before I attempt to answer your questions, I should point out that the LT is the only known transformation that retains the causal structure of SR. Time-like, null and spacelike intervals retain this property under LT. Also, crucially, identifying the proper interval as the time recorded on a clock traveling on the curve in question eliminates all clock paradoxes (time-bomb paradoxes) because the clock times are invariant. I know you're most probably aware of this.

So the question are: 1) when do we want to use the coordinates (t,x), and 2) when would we want τ-τ0 to be invariant wrt to those coordinates?

It seems to me that the answer to 2) is 'always - if SR is to be consistent'. I don't understand the first question. We have to use some coordinates or other in this methodology.

<br /> \int_{{\tau _0}}^\tau {f(\tau - {\tau _0})d\tau } = a<br />
Right now I can't see any meaning in this. I'll have to think about it.

 

[PeterDonis]

I'm given to understand that

dτ²=dt²-dx² = dt'²-dx'²

when (t',x') are the Lorentz transformation of (t,x).

Yes.

Perhaps it's instructive to consider in what circumstances dτ should want to be considered invariant wrt to coordinate changes.

As Mentz114 said, the answer to this is "always". The reason is that d\tau is a direct observable; it corresponds to the elapsed time on a clock following the given worldline for an infinitesimal segment. Direct observables must have the same value in all coordinate systems, so their values have to be invariant under coordinate changes.

Maybe those requirements are the driving force behind the necessity of the Lorentz transformations.

Exactly; that's the main point of that other thread Mentz114 referenced.

For example, the most obvious use of dτ is in the calculation of the line integral,

\int_{{\tau _0}}^\tau {d\tau &#039;} = \tau - {\tau _0}
which is the length of a line measured in terms of segments marked off along the length of the line.

It's also the time elapsed on a clock following the given worldline from \tau_0 to \tau. So as I said above, it's a direct observable.

Then, of course, we can always place this line in an arbitrarily oriented coordinate system and express τ in term of those coordinates.

Yes, and the value of \tau must be the same regardless of which coordinates you express it in.

So the question are: 1) when do we want to use the coordinates (t,x)

As opposed to what other coordinates?

2) when would we want τ-τ₀ to be invariant wrt to those coordinates?

Always.

As far as 1) goes, usually, we specify a curve in space by parameterizing the space coordinates with an arbitrary variable, call it "t".

Actually, \tau itself can be used to parameterize the worldline. The coordinates (t, x) are then functions of the parameter \tau.

But since the x and t coordinates are arbitrarily assigned

No, they're not, because this...

the length of the curve can depend on the (t,x) coordinates.

...is false. The length of the curve is an observable. See above.

But if you specify that the length of the curve is invariant, then this requires the Lorentz transformations between coordinate systems.

Yes.

But for 2) what should require the length of the curve to be invariant?

Because it's a direct observable. See above.

 

[friend]

I appreciate your comments. I can see that you are knowledgeable in SR. However, I don't think you understand what I'm trying to do here. I'm trying to explain where the SR metric, with its speed of light, comes from to begin with. So I'm trying to avoid starting with the observed speed of light as in the usual development of SR. And I'm trying to find principles that would give rise to the metric of SR, with its implied speed of light.

So my thinking is that since the Dirac delta function is used in the development of QM, it might also be used to develop SR. And here's how I figure. Since we have
\int_{ - \infty }^{ + \infty } {\delta (\tau &#039; - {\tau _0})d\tau &#039;} = 1
we can embed this line integral in a background spacetime (t,x), where \tau then becomes a function of the coordinates (t,x). And then it seems that the requirement that this integral remain constant wrt changes in coordinate system requires the coordinate transformations to be Lorentzian to keep {\tau - {\tau _0}} invariant. Does this sound right to you?

 

[Mentz114]

I appreciate your comments. I can see that you are knowledgeable in SR. However, I don't think you understand what I'm trying to do here. I'm trying to explain where the SR metric, with its speed of light, comes from to begin with. So I'm trying to avoid starting with the observed speed of light as in the usual development of SR. And I'm trying to find principles that would give rise to the metric of SR, with its implied speed of light.

So my thinking is that since the Dirac delta function is used in the development of QM, it might also be used to develop SR. And here's how I figure. Since we have
\int_{ - \infty }^{ + \infty } {\delta (\tau - {\tau _0})d\tau &#039;} = 1
we can embed this line integral in a background spacetime (t,x), where \tau then becomes a function of the coordinates (t,x). And then it seems that the requirement that this integral remain constant wrt changes in coordinate system requires the coordinate transformations to be Lorentzian to keep {\tau - {\tau _0}} invariant. Does this sound right to you?

My first take on this is that there's nothing wrong with your argument in the second paragraph, but it doesn't tell us anything we didn't know. Incidentally, is the prime on the differential a typo or significant ?

The speed of light is in the metric itself because for a null curve 0 = dt²-dx² implies (dx/dt)²=1.

 

[PeterDonis]

I appreciate your comments. I can see that you are knowledgeable in SR. However, I don't think you understand what I'm trying to do here. I'm trying to explain where the SR metric, with its speed of light, comes from to begin with.

This still doesn't completely clear up what you're trying to do, at least not for me. When you say "the SR metric", it seems to me that you really mean "the Lorentz transformation", since that's what preserves the spacetime interval. So it seems to me that the question you're really trying to ask is "where does the Lorentz transformation come from?" and find an answer that doesn't depend on any assumptions about the speed of light.

If the latter is really your question, the best explanations I've seen that don't depend on any assumptions about the speed of light, start from the assumptions of translation and rotation invariance. That is enough to narrow down the possibilities to either the Lorentz transformation or the Galilei transformation, which is just the limit of the Lorentz transformation as c -> infinity. More precisely, translation and rotation invariance are enough to show that there must be some "invariant speed" c that appears in coordinate transformations, but which may be infinite (Galilei transformation) or finite (Lorentz transformation). The only way to choose between these two options is by experiments like the Michelson-Morley experiment, which show that the invariant speed is finite.

And then it seems that the requirement that this integral remain constant wrt changes in coordinate system requires the coordinate transformations to be Lorentzian to keep invariant. Does this sound right to you?

Not really. The delta function \delta ( \tau - \tau_0 ) is zero unless \tau = \tau_0. I don't see what that's supposed to mean physically; it looks like you're saying that no proper time elapses at all except at the single event \tau_0, where *all* of the proper time on the entire worldline of the object elapses. That doesn't seem reasonable to me.

 

[friend]

This still doesn't completely clear up what you're trying to do, at least not for me. When you say "the SR metric", it seems to me that you really mean "the Lorentz transformation", since that's what preserves the spacetime interval. So it seems to me that the question you're really trying to ask is "where does the Lorentz transformation come from?" and find an answer that doesn't depend on any assumptions about the speed of light.

If the latter is really your question, the best explanations I've seen that don't depend on any assumptions about the speed of light, start from the assumptions of translation and rotation invariance. That is enough to narrow down the possibilities to either the Lorentz transformation or the Galilei transformation, which is just the limit of the Lorentz transformation as c -> infinity. More precisely, translation and rotation invariance are enough to show that there must be some "invariant speed" c that appears in coordinate transformations, but which may be infinite (Galilei transformation) or finite (Lorentz transformation). The only way to choose between these two options is by experiments like the Michelson-Morley experiment, which show that the invariant speed is finite.

Ideally, I don't want to start with any assumptions about what kind of transformations apply. I'd like the transformation to be derived from some more fundamental requirement not based on any measurement of anything physical. I have reason to believe that the Dirac delta function is a mathematical representation of causality. At least in linear systems a system is considered causal depending on how it responds to an impulse = dirac delta. So if there is some logic for starting with the integral of the Dirac delta, then perhaps the constant speed of light and the Lorentz transformations can be derived from that logic.

Not really. The delta function \delta ( \tau - \tau_0 ) is zero unless \tau = \tau_0. I don't see what that's supposed to mean physically; it looks like you're saying that no proper time elapses at all except at the single event \tau_0, where *all* of the proper time on the entire worldline of the object elapses. That doesn't seem reasonable to me.

I'm not entirely confident that I want to start with the Dirac delta function. For if we change the coordinate on the curve to \tau &#039; = a\tau , then d\tau &#039; = ad\tau , and \tau = \tau &#039;/a , and d\tau = d\tau &#039;/a . Then we have \tau - {\tau _0} = \tau &#039;/a - \tau {&#039;_0}/a = (\tau &#039; - \tau {&#039;_0})/a and \delta (\tau - {\tau _0}) = \delta ((\tau &#039; - \tau {&#039;_0})/a) . And this gives us by the scaling property of the Dirac delta,
\int_{ - \infty }^{ + \infty } {\delta (\tau - {\tau _0})d\tau } = \int_{ - \infty }^{ + \infty } {\delta ((\tau &#039; - \tau {&#039;_0})/a)\frac{{d\tau }}{a}} = \int_{ - \infty }^{ + \infty } {a \cdot \delta (\tau &#039; - \tau {&#039;_0})\frac{{d\tau &#039;}}{a}} = \int_{ - \infty }^{ + \infty } {\delta (\tau &#039; - \tau {&#039;_0})d\tau &#039;} = 1
So it seems this integral is insensitive to changes in the \tau coordinate. Does this mean that ANY changes in the backgound (t,x) could change the \tau coordinate in arbitrary ways that still results in the integral being 1? In other words, the integral no longer depends on the \tau - {\tau _0} not changing. Wouldn't that mean that this integral has no power to restrict the background transformations to the Lorentz transformations? Or does this scaling property allow only changes in \tau by a constant factor? Can I still get the Lorentz Xformation from this?

Perhaps it would be better to integrate a different function other than the Dirac delta. Then I'd not be dealing with the scaling property and the integral could restrict the background transformation of (t,x) to the Lorentz type.

 

[PeterDonis]

Ideally, I don't want to start with any assumptions about what kind of transformations apply.

The assumptions of translation and rotation invariance aren't assumptions about what kinds of transformations apply. They're very general assumptions about the physics, that happen to have as a consequence the restriction on possible transformations that I gave.

I'd like the transformation to be derived from some more fundamental requirement

I'm not sure why translation and rotation invariance wouldn't count as "fundamental requirements".

not based on any measurement of anything physical.

This seems like a strange thing to want. In order to test whether *any* fundamental assumption actually holds, you're going to have to make physical measurements and compare them with what you would expect to see if the assumption were true.

I have reason to believe that the Dirac delta function is a mathematical representation of causality. At least in linear systems a system is considered causal depending on how it responds to an impulse = dirac delta.

That's not the same as the delta function itself being a representation of causality. The representation of causality is in the *response* function, not the function describing the initial impulse.

So if there is some logic for starting with the integral of the Dirac delta

Why are you starting with integrals in the first place? If you're looking for fundamental requirements, it seems to me that you would want to start with something local, i.e., to only look at physical quantities and their derivatives (or infinitesimals) at a single point. The integral representation of physics over a finite length or in a finite region is built up from the local representation; it's not fundamental in itself.

 

[friend]

Why are you starting with integrals in the first place? If you're looking for fundamental requirements, it seems to me that you would want to start with something local, i.e., to only look at physical quantities and their derivatives (or infinitesimals) at a single point. The integral representation of physics over a finite length or in a finite region is built up from the local representation; it's not fundamental in itself.

My question is: what logical necessity could there possibly be for the invariance of d\tau that could give rise to the Lorentz metric with its constant speed of light. So in effect I'm trying to explain the LT and c. I don't think an explanation of such can start by measurements of c, that would be a circuilar argument. Yes, of course, whatever theory one derives must be confirmed by experiment. But I don't think experiment is an explanation for the experiment. That would be finding some math that fits the data without an explanation for the data or the math, IMO. If you are not comfortable with this approach to the problem, you don't have to contribute.

Why the integral? As I said in post 3, the most obvious use of dτ is in the calculation of the line integral,


∫dτ′=τ−τ0

And when would τ−τ0 be invariant. Well you could have τ−τ0=constant. But then you ask why. So you notice that this is just another way of saying f(τ−τ0)= constant. And I suppose that this could always be put in the form

∫g(τ'−τ0)dτ′=constant

And one thing to consider is the integral of the Dirac delta equal to one, always, in any coordinate system, right?

 

[PeterDonis]

My question is: what logical necessity could there possibly be for the invariance of d\tau

Because d\tau represents a direct physical observable, the time elapsed on a clock over a very small portion of its worldline. It seems reasonable to want your mathematical model of physics to represent direct physical observables by quantities that are invariant under coordinate transformations. Whether that counts as "logical necessity" seems to me to be beside the point; every physical theory has to be based on some assumptions that can't be proved, but just have to be taken as given.

that could give rise to the Lorentz metric with its constant speed of light.

That's where translation and rotation invariance comes in; and I also should have included invariance under boosts, i.e., under changes of inertial frame, because of the principle of relativity: physics should look the same to two observers who are inertial and are moving at a constant velocity relative to each other.

I don't think an explanation of such can start by measurements of c, that would be a circuilar argument.

You don't have to start with measurements of c if you start with translation, rotation, and boost invariance; those requirements are enough to narrow down the possible transformations to Lorentz or Galilei transformations, as I said before. But you do have to know about measurements of c if you want to determine whether Lorentz or Galilei transformations are the right ones. I don't know of any other way to make that determination.

Part of the problem here may be the issue of what counts as an "explanation". If I say that physics has to be translation, rotation, and boost invariant, does that count as an explanation of why only Galilei or Lorentz transformations are allowed? If I need experiments to tell me that the invariant speed is finite instead of infinite, does that count as an explanation of why Lorentz transformations, not Galilei transformations, are the right ones to use? To me these are questions about words, not about physics.

Of course you could continue to ask questions, like "why does the invariant speed have to be finite rather than infinite?". AFAIK nobody has a good answer to that question, other than "that's what we find experimentally". Does that mean we don't have an explanation? Again, that seems to me to be a question about words, not about physics.

 

[TrickyDicky]

My question is: what logical necessity could there possibly be for the invariance of d\tau that could give rise to the Lorentz metric with its constant speed of light. So in effect I'm trying to explain the LT and c.

The LT (together with the Galilean transformation when not including the second postulate of SR) as commented by Peter Donis is based first of all on the reasonable assumption of isotropy and homogeneity, geometrically this is a feature of constant sectional curvature spaces, this assumption introduces the choice of a metric and since the motion that is dealt with in this case is inertial motion the logical constant curvature metric to choose (also remember this is 1905 for Einstein's SR and 1907 for Minkowski coming up with the natural space for SR, Einstein hadn't introduced the idea of curvature in physics yet) was the flat one, this leads to the linearity of both Galilean and Lorentz transformations.

The invariance of a finite c was Einstein second postulate of SR, there is no logical necessity for a postulate, it just happens to be backed up by experiments.

Also invariance of dτ is not equal to SR since it is shared by GR for instance.

EDIT: I see Peter has already answered, sorry about the overlapping.

 

[friend]

The invariance of a finite c was Einstein second postulate of SR, there is no logical necessity for a postulate, it just happens to be backed up by experiments.

Also invariance of dτ is not equal to SR since it is shared by GR for instance.

OK, so what I'm beginning to understand is that the invariance of d\tau only implies the flat metric locally. However, does this mean that the invariance of \int_{{\tau _0}}^\tau {d\tau &#039; = } \tau - {\tau _0} would specify a more global flat space, with a Lorentzian or Galilean transformation? If c is not infinite, then the background metric is Lorentzian and the same along the trajectory, is this right?

Perhaps the only reason that there must be a finite c is because if it were infinite, then everything would happen all at once and there would be no distinction between cause and effect. How's that sound? It's a little more basic than measuring c.

 

[PeterDonis]

OK, so what I'm beginning to understand is that the invariance of d\tau only implies the flat metric locally.

Yes.

However, does this mean that the invariance of \int_{{\tau _0}}^\tau {d\tau &#039; = } \tau - {\tau _0} would specify a more global flat space, with a Lorentzian or Galilean transformation?

Yes. It also assumes that the curve parametrized by \tau is a straight line in that global flat spacetime (I use that word in preference to "space" since time is one of the dimensions in this "space".)

If c is not infinite, then the background metric is Lorentzian and the same along the trajectory, is this right?

In a flat spacetime, yes.

Perhaps the only reason that there must be a finite c is because if it were infinite, then everything would happen all at once and there would be no distinction between cause and effect.

I'm not sure that works, because Newtonian physics has an "infinite c", and everything doesn't happen all at once in Newtonian physics.

 

[TrickyDicky]

I'm not sure that works, because Newtonian physics has an "infinite c", and everything doesn't happen all at once in Newtonian physics.

Right, what actually happens in Newtonian physics rather than "happenning all at once" which would imply a block-like interpretation is that there is no relativity of simultaneity. Time and space are not related in the same way they are in SR.

 

[friend]

If c were infinite, you could cause and emission over there by forcing an absorbsion of a photon over here. My argument is that it is not logical for nature to ever be ambiguous about cause and effect. And one way to do this (if not the only way) is to ensure that c is not infinite. Some people even try to derive the Lorentzian metric from the requirement of causality alone. Although I think they are presupposing causality and not arguing for its necessity to begin with.

EDIT: To continue with these thoughts... Suppose a particle could travel with infinite speed, it would be impossible to say where the particle was at that instant. It could be anywhere from the edge of the universe to your present location. It would turn a point particle at one instant into an 13.75ly line an infinitesimal instant later, which would be a totally different thing than a particle. Further, there would be no frame of reference in which the particle is at rest. Would this violate the differential structure of the universe? Doesn't causality require a different position for each value of time?

 

[friend]

I guess my question is whether diffeomorphism invariance requires the choice of Lorentz transformations above Galileans transformations. Is diffeomorphism invariance between coordinate transformations a more basic necessity to physics than measuring the speed of light? I mean, if the laws of physics necessarily require diffeomorphisms between frames of reference, does that necessarily exclude the infinite speeds used in the Galilean transformations? Can there be a diffeomorphism between two frames of reference when one of them has and infinite speed? And if not, does that make diffeomorphism invariance more basic than the speed of light? Thanks.

 

[friend]

This is probably an easy question for some:

I've seen where if we start with the Lorentz transforms, we can show that dτ and even τ-τ₀ is not changed. But I wonder if we can go the other way. If we start with the requirement that dτ or τ-τ₀ should not vary, can we then derive the Lorentz transformations? I'd appreciate any help on this. Thanks.

 

[friend]

I think I found a paper that does what I'm asking. It seems to derive the Lorentz transformations from the properties of spacetime alone. I'd appreciate comments on this paper, 7 pages, relatively easy math. Thank you.

One more derivation of the Lorentz transformation

 

[PeterDonis]

I'd appreciate comments on this paper

This is the sort of thing I was referring to in posts #8 and #12, when I said that translation, rotation, and boost invariance are enough to narrow down the possibilities to Lorentz or Galilei transformations. I think there have been other published papers taking a similar approach, but I haven't been able to find links.

 

[friend]

This is the sort of thing I was referring to in posts #8 and #12, when I said that translation, rotation, and boost invariance are enough to narrow down the possibilities to Lorentz or Galilei transformations. I think there have been other published papers taking a similar approach, but I haven't been able to find links.

Correct me if I'm wrong. But I think he is using the causality condition to choose between Lorentz and Galilei transformations. (see Hypothesis 4: causality, page 276, second to last page)

One more derivation of the Lorentz transformation

 

[PeterDonis]

Correct me if I'm wrong. But I think he is using the causality condition to choose between Lorentz and Galilei transformations.

He explicitly says that his causality requirement (that the sign of the time interval be unchanged by a transformation) is satisfied by Galilei transformations for all time intervals; so he can't be using that requirement to rule out Galilei transformations in favor of Lorentz transformations. He only uses causality to rule out a third possibility, which he gives as case 9i) on page 275. I don't know if this third possibility has ever been given a name.

As far as I can see, this paper doesn't rule out Galilei transformations, and doesn't claim to. The only hint it gives as to why Lorentz transformations might be preferred is that it several times refers to Galilei transformations as the "singular limit" of Lorentz transformations ("singular" presumably refers to the fact that Galilei transformations are what you get when you take the limit of Lorentz transformations as c -> infinity). But the paper doesn't elaborate on that.

 

[Mentz114]

I think I found a paper that does what I'm asking. It seems to derive the Lorentz transformations from the properties of spacetime alone. I'd appreciate comments on this paper, 7 pages, relatively easy math. Thank you.

One more derivation of the Lorentz transformation

Interesting paper. The four hypotheses do seem the minimum to end up with a group of transformations that preserve causality. The bad news is that the group is non-commutative, so making two boosts in different directions becomes a problem.

Did you see the 'note added ...'. Apparently this was all known in the 1920's.

 

[friend]

I think I found a paper that does what I'm asking. It seems to derive the Lorentz transformations from the properties of spacetime alone. I'd appreciate comments on this paper, 7 pages, relatively easy math. Thank you.

One more derivation of the Lorentz transformation

I still have questions about his math. Is this a good thread to ask such questions? Or should I start a new thread to discuss this paper specifically?

 

[strangerep]

The bad news is that the group is non-commutative, so making two boosts in different directions becomes a problem.

Why is that a problem? Boosts in different directions don't commute, so in deriving the full group of (linear) inertial motion transformations for 1+3D spacetime, one must take account of ordinary rotations...

The group law assumption in all these types of derivations boils down to the hypothesis that boosts along a fixed direction should form a 1-parameter Lie group.

Did you see the 'note added ...'. Apparently this was all known in the 1920's.

Maybe some of it was known even earlier. The standard references are Ignatowsky and also Frank+Rothe (though I've been unable to source them, and they appear to be in German anyway -- I haven't found a translation).

 

[strangerep]

I still have questions about his math. Is this a good thread to ask such questions? Or should I start a new thread to discuss this paper specifically?

Well, it's your thread so it seems reasonable that you get to choose what's on-topic...

BTW, such derivations become a little clearer (imho) if the step involving the group composition law is split into two steps. The result that a v-independent constant emerges can be found by using the fact that elements of a 1-parameter Lie group must commute. After establishing this, the full composition law can be used more easily to derive the velocity addition rule.

 

[Fredrik]

I guess my question is whether diffeomorphism invariance requires the choice of Lorentz transformations above Galileans transformations.

It does not. Diffeomorphisms do not even involve a metric.

Can there be a diffeomorphism between two frames of reference when one of them has and infinite speed?

There are no inertial frames such that the speed of one of them relative to the other is infinite.

 

[friend]

For example, he writes on the bottom left of page 272 of,

One more derivation of the Lorentz transformation,

"..., we consider (3) for given (x,t) and (x',t') as a set of two equations in n unknowns {a₁,a₂,...,a_n}. It is clear that, if n≥2, these equations will, in general, have solutions; an interval between two physical events might then have arbitrary coordinates... On the other hand if n=0, there would be no other inertial transformations than space and time translations, and no proper theory of relativity. From this argument we may conclude that n=1,..."

I want to make sure I understand him here. Is he saying that if you have two equations in more than two unknowns, then you have an underdetermined set of equations, with not enough equations to solve for the unknows? But I thought for n=2, there is two equations in two unknowns, so that you do have enough info to solve for the two unknown a's. What am I missing?

Also, he seems to conclude n=1 on the basis of a physical argument ("no proper theory of relativity") which I thought he was trying to avoid. Thanks for any help in advance.

 

[strangerep]

Is he saying that if you have two equations in more than two unknowns, then you have an underdetermined set of equations, with not enough equations to solve for the unknows? But I thought for n=2, there is two equations in two unknowns, so that you do have enough info to solve for the two unknown a's. What am I missing?
Also, he seems to conclude n=1 on the basis of a physical argument ("no proper theory of relativity") which I thought he was trying to avoid.

[Original reply replaced by...]

I think this part of his argument is questionable, since it ignores uniform dilations of the form $(t',x') = (e^\alpha x, e^\alpha t)$, where $\alpha$ is independent of $x$ and $t$. Including such dilations would lead to 2-parameter transformations. He dispenses with dilations implicitly a bit later in the paragraph after eq(9) where he describes the case $K(a)=0$ as pathological, and in the related footnote 10 he mentions the "static group" or "Carroll" group. So he's implicitly assuming a common scale in both frames.

Another way of looking at this is that for the case $v=0$, the old and new frames are one and the same, apart from a possible scale difference in their respective units of time and length -- which are promptly assumed to be standardized uniformly when the origins coincide.

 

[friend]

[Original reply replaced by...]

I think this part of his argument is questionable, since it ignores uniform dilations of the form $(t',x') = (e^\alpha x, e^\alpha t)$, where $\alpha$ is independent of $x$ and $t$. Including such dilations would lead to 2-parameter transformations. He dispenses with dilations implicitly a bit later in the paragraph after eq(9) where he describes the case $K(a)=0$ as pathological, and in the related footnote 10 he mentions the "static group" or "Carroll" group. So he's implicitly assuming a common scale in both frames.

Another way of looking at this is that for the case $v=0$, the old and new frames are one and the same, apart from a possible scale difference in their respective units of time and length -- which are promptly assumed to be standardized uniformly when the origins coincide.

It's easy for me to view equation (1) as a Taylor series expansion of the transformation functions between the primed and unprimed coordinates, expanded about (x,t)=(0,0). Then he seems to be trying to derive the coefficients of that expansion using his hypothesis. He thinks there is no loss of generality if he sets the first coefficients of those expansions to 0 so that equation (4) holds. But I'm not even seeing that.

He starts out with a good idea and an interesting approach to the problem. But he's not communicating it very well as far as I'm concerned.

 

[strangerep]

It's easy for me to view equation (1) as a Taylor series expansion of the transformation functions between the primed and unprimed coordinates, expanded about (x,t)=(0,0).

Eq(1) is not a Taylor series. He's simply writing down the most general form for a coordinate transformation in 1+1D spacetime that depends on various parameters $a_i$ .
The new (primed) coordinates are simply functions of the old (unprimed) coordinates and the parameters $a_i$. Nothing deeper than that at this stage.

Then he seems to be trying to derive the coefficients of that expansion using his hypothesis. He thinks there is no loss of generality if he sets the first coefficients of those expansions to 0 so that equation (4) holds. [...]

He's simply restricting the transformation equations to the class of "inertial frames with common spacetime origins", which can be achieved by applying spacetime translations to either the old or new coordinates. That's how he gets eq(4) -- from the restriction that the old and new origins coincide (achieved by using up the translational freedom).

He starts out with a good idea and an interesting approach to the problem. But he's not communicating it very well as far as I'm concerned.

It's hard to write a paper suitable for Am. J. Phys. which is also pedagogical, but I've seen far more obscure papers than this one. As with most research literature, one must persevere and try to understand and reproduce the steps in detail.

 

[friend]

I think I found a paper that does what I'm asking. It seems to derive the Lorentz transformations from the properties of spacetime alone. I'd appreciate comments on this paper, 7 pages, relatively easy math. Thank you.

One more derivation of the Lorentz transformation

He writes,

"I will take as a starting point the statement of the principle of relativity in a very general form: there exists an infinite continuous class of reference frames in spacetime which are physically equivalent."

Instead of accepting on faith this principle of relativity in a very general form, I'd like to offer a reason that would give rise to this "infinite continuous class of reference frames". It might be that whatever gives rise to this requirement might also help in understanding the other symmetries of spacetime that allow us to fully specify the Lorentz nature of the transformations in the reference frames.

It seems the dirac delta function specifies a diffeomorphism between coordinate transformations that constitute the reference frames.

Consider the defining property of the Dirac delta function,
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = 1
If we change coordinates to y = y(x) so that x = x(y), then dx = \frac{{dx}}{{dy}}dy.
Then we can use the notation
\frac{{dx}}{{dy}} = \sqrt {g(y)} = \sqrt {\frac{{dx}}{{dy}} \cdot \frac{{dx}}{{dy}}}
in order to be consistent with higher dimensional versions. We also have
x - {x_0} = \int_{{x_0}}^x {dx&#039;}
And by using the transformation, y = y(x), so that {y_0} = y({x_0}), this integral becomes,
x - {x_0} = \int_{{x_0}}^x {dx&#039;} = \int_{{y_0}}^y {\sqrt {g(y&#039;)} dy&#039;}
And the original integral can be transformed to,
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = \int_{ - \infty }^{ + \infty } {\delta (\int_{{x_0}}^x {dx&#039;} )dx} = \int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (\int_{{y_0}}^y {\sqrt {g(y&#039;)} \,dy&#039;} )\,\,\sqrt {g(y)} \,\,dy} = 1
where {y^{ + \infty }} = y(x = + \infty ) and {y^{ - \infty }} = y(x = - \infty ).

Then using the composition rule for the Dirac delta, we have
\int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (\int_{{y_0}}^y {\sqrt {g(y&#039;)} \,dy&#039;} )\,\,\sqrt {g(y)} \,\,dy} = \int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\left( {\frac{{\delta (y - {y_0})}}{{|\sqrt {g({y_0})} \,|}}} \right)\,\,\sqrt {g(y)} \,\,dy = } \frac{1}{{|\sqrt {g({y_0})} \,|}}\int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (y - {y_0})\,\,\sqrt {g(y)} \,\,dy}
where {{y_0}} is where \int_{{y_0}}^y {\sqrt {g(y&#039;)} \,dy&#039;} = 0.

And then using the sifting property of the Dira delta we have
\frac{1}{{|\sqrt {g({y_0})} \,|}}\int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (y - {y_0})\,\,\sqrt {g(y)} \,\,dy} = \frac{1}{{|\sqrt {g({y_0})} \,|}}\left( {\,\sqrt {g({y_0})} \,\int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (y - {y_0})\,\,dy} } \right) = \int_{{y^{ - \infty }}}^{{y^{ + \infty }}} {\delta (y - {y_0})\,\,dy} = 1
So that finally,
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = \int_{ - \infty }^{ + \infty } {\delta (y - {y_0})\,dy} = 1
where I assume that using an integration interval of - \infty \le y \le + \infty changes nothing.

So it appears here that any invertible coordinate transformation (diffeomorphism) leaves the integration of the Dirac delta unchanged in form, meaning it is diffeomorphism invariant.

The properties of the Dirac delta function are necessary in any reference frame or any space. They are true on principle alone. So if the Dirac delta is required by some logic, then that logic would appear to also specify an "infinite continuous class of reference frames in spacetime".

 

[strangerep]

[...]
So that finally,
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} = \int_{ - \infty }^{ + \infty } {\delta (y - {y_0})\,dy} = 1
where I assume that using an integration interval of - \infty \le y \le + \infty changes nothing.

You've proved nothing that is not already contained in the definition of the Dirac delta itself.
I can just as well write down:
$$
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} ~=~ 1
~=~ \int_{ - \infty }^{ + \infty } {\delta (x - y_0)\,dx}
~=~ \int_{ - \infty }^{ + \infty } {\delta (y - y_0)\,dy}
$$
where, in the last step, I simply renamed the dummy integration variable.

So it appears here that any invertible coordinate transformation (diffeomorphism) leaves the integration of the Dirac delta unchanged in form, meaning it is diffeomorphism invariant.

The properties of the Dirac delta function are necessary in any reference frame or any space. They are true on principle alone. So if the Dirac delta is required by some logic, then that logic would appear to also specify an "infinite continuous class of reference frames in spacetime".

The "infinite continuous class of reference frames in spacetime" is simply a consequence of the fact that the group of coordinate transformations which preserve the equation of inertial motion, i.e.,
$$
\frac{d^2 {\mathbf x}}{dt^2} ~=~ 0
$$
turns out to be a Lie group.

This does not have to be "accepted on faith". An unaccelerated observer enclosed in an opaque box cannot tell whether he is stationary or moving relative to some external point.

 

[friend]

You've proved nothing that is not already contained in the definition of the Dirac delta itself.
I can just as well write down:
$$
\int_{ - \infty }^{ + \infty } {\delta (x - {x_0})dx} ~=~ 1
~=~ \int_{ - \infty }^{ + \infty } {\delta (x - y_0)\,dx}
~=~ \int_{ - \infty }^{ + \infty } {\delta (y - y_0)\,dy}
$$
where, in the last step, I simply renamed the dummy integration variable.

Of course you can always change the dummy variable in any equation and get the same answer. But then you're stuck having to interpret the new dummy variable as if it were the exact same thing as the old dummy variable.

I don't think you can change x₀ to y₀ without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral. But if you do, then it is no longer a dummy variable change; you're actually specifying a function from x to y.

What I think I've shown is when there is a relationship between x and y, namely, y=y(x), you get the same equation back again. Then we can interpret x and y as different coordinates. And x and y can not have any arbitrary relationship but must be a diffeomorphism. I think this means that the Dirac delta function is diffeomorphism invariant, right? So if the dirac delta is required for some reason (TBD), then that requirement also specifies the existence of some properties of the space on which it resides, namely, that it admitts a diffeomorphic coordinate patches, right?

The "infinite continuous class of reference frames in spacetime" is simply a consequence of the fact that the group of coordinate transformations which preserve the equation of inertial motion, i.e.,
$$
\frac{d^2 {\mathbf x}}{dt^2} ~=~ 0
$$
turns out to be a Lie group.

This does not have to be "accepted on faith". An unaccelerated observer enclosed in an opaque box cannot tell whether he is stationary or moving relative to some external point.

All this does is render a mathematical description of the observations. It does not explain it. What I'm trying to understand is WHY space would have the properties it does that makes necessary these kinds of observations.

Even the author of the paper is trying to derive the Lorentz transformations from properties of spacetime. He starts with the a priori existence of diffeomorphic coordinate transformations on patches of an underlying manifold (paraphrased). I'd like to go even further back and explain the need of for this property of spacetime to begin with.

 

[strangerep]

I don't think you can change x₀ to y₀ without specifying a function y(x). So I don't think you can go from the integral on the left to the middle integral.

Yes I can. Both steps are valid, provided both $x_0$ and $y_0$ are somewhere between $-\infty$ and $+\infty$.

Anyway...

Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.

 

[friend]

Yes I can. Both steps are valid, provided both $x_0$ and $y_0$ are somewhere between $-\infty$ and $+\infty$.

Anyway...

Although I'm happy to help you understand the content of published papers, I'm not going to follow you down a rabbit hole into crackpot land.

I certainly hope it's not a "rabit hole". I was trying to find the simplest construction that relates a metric to a field. Then maybe that could be used in the derivation of both the fields of QFT and curvature of GR. And it occurs to me that fields consist of individual values at each point in a space. And the minimum section of space is a infinitesimally small flat portion at the point of interest. A metric is inherently needed to do integration. So what integration process uses the least portion of space and picks out individual field values? That would be the Dirac delta function. Intuitively that seems to me like a good place to start when trying to relate space and fields in terms of their smallest constituents. One would think that the integration of the dirac delta being one is a true statement independent of any physical reason to use it. So if it does prove useful in deriving physics, then we will have succeeded in deriving physics from inherent truth.

As I recall, the integration of the dirac delta is suppose to be one no matter how small the integration interval, as long as the interval contains the zero of the dirac delta's argument. And if the interval of integration is not infinite but is small instead, then my objection to your comments holds.

But even with an infinite interval of integration, I wonder if there must be a diffeomorphism between the x and y coordinates. For it seems that the field in both coordinates needs to be sufficiently well behaved in order to do the integration. Does that mean we should be able to construct a smooth and invertible function between x and y, a diffeomorphism?