# Einstein postulates and the speed of light

#### MHD93

Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate).
Serway Modern Physics:
Note that postulate 2, the principle of the constancy of the speed of
light, is consistent with postulate 1: If the speed of light was not the same in
all inertial frames but was c in only one, it would be possible to distinguish
between inertial frames, and one could identify a preferred, absolute frame
Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)

Thanks

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#### ghwellsjr

Gold Member
Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate).
Serway Modern Physics:
Note that postulate 2, the principle of the constancy of the speed of light, is consistent with postulate 1: If the speed of light was not the same in all inertial frames but was c in only one, it would be possible to distinguish between inertial frames, and one could identify a preferred, absolute frame in contradiction to postulate 1.
Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)

Thanks
No, it's not true. Whoever wrote that statement doesn't understand Einstein's Special Relativity. That person thinks that it is possible to measure or observe or be aware of the speed of light that is talked about in postulate 2. But if you look at what Einstein wrote in his 1905 paper, he said in the introduction:
...light is always propagated in empty space with a definite velocity c...
And in section 2 he said:
Any ray of light moves in the “stationary” system of co-ordinates with the determined velocity c...
It is clear that he is talking about the One-Way Speed of Light which cannot be measured.

In section one, he discusses how to measure the two-way speed of light and he says:
In agreement with experience we further assume the quantity to be a universal constant—the velocity of light in empty space.
"Experience" there means experiment-something that can be measured.

If you read these first two sections of his paper, you will see that what he is doing is assigning the unknowable one-way speed of light to the measurable value of the two-way speed of light. And he can do this because he postulates without proof that light propagates in any Inertial Reference Frame at c. Prior to that, the Lorentz viewpoint was that light propagated at c only in the absolute ether rest state.

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#### Fredrik

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Gold Member
Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one
The first postulate is just a loosely stated idea, so you can't derive mathematical statements from it. But you can write down a mathematical statement that people would agree is an acceptable way to make the first postulate (and something else, like isotropy of space) mathematically precise, and then you can try to derive things from that. What you will find is that there's an invariant speed, i.e. that there are lines in spacetime that represent motion that's assigned the same speed by all inertial coordinate systems. However, you will not find the value of the invariant speed this way. It can be any positive real number, and it can also be infinite.

In this context, the significance of the second postulate is only that it tells us that the invariant speed is finite.

(or more precisely, its converse contradicts the first postulate).
Converse? Do you mean its negation?

Now, is that true? And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature (note that their argument, that is, the first postulate indicates the second, doesn't involve experiments)
Sound waves don't have the same speed in all inertial coordinate systems. Electromagnetic waves do.

It's a consequence of relativistic quantum mechanics that for any elementary particle, the value of $-E^2+\vec p^2$ (in units such that the invariant speed is 1) is the same in all inertial coordinate systems. That value is one of the things that distinguishes one particle species from another. It turns out that a theory that involves particles for which that value is 0 makes extremely accurate predictions about experiments involving light. Those particles are called photons, and it can be shown that the equality $-E^2+\vec p^2=0$ implies that they move at the invariant speed.

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#### Meir Achuz

Homework Helper
Gold Member
c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's seond postulate necessarily follows from the first.

#### Fredrik

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c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's seond postulate necessarily follows from the first.
You mean from the first and Maxwell's equations?

#### bcrowell

Staff Emeritus
Gold Member
Now, is that true?
I agree with it, but you'll find many people who wouldn't.

And if yes, what's so special about light than other object (ex, sound waves) to consider its speed a law of nature
The speed of light is predicted to have a unique value by Maxwell's equations, which are laws of physics. The speed of baseballs isn't predicted to have a unique value by Newton's laws.

I wouldn't get too hung up on Einstein's 1905 axiomatization. It's really not optimal from a modern point of view. In 1905, electromagnetism was the only known fundamental field. From a modern point of view, it doesn't make sense to give it a special role. Here's something with a more modern outlook: http://arxiv.org/abs/physics/0302045

#### Nugatory

Mentor
c is a parameter in Maxwell's equations. Electromagnetism would be different in each Lorentz frame if c were not the same. c must be constant to satisfy the first postulate.
c was measured in 1856 by the ratio of magnetic to electric phenomena with no relation to the speed of light.
It turns out that c determines the speed of light, which thus must be invariant if c is.
E's second postulate necessarily follows from the first.
I must confess that I find this to be the most natural way of developing SR: PoR plus Maxwell, doubling down on the first postulate.

However, the historical development of the theory is important here. If it were really as clear as all that, then shortly after Maxwell the entire physics community would have realized that they couldn't have all three of the PoR, Galilean transforms, and Maxwell's equations, and would have started looking for an alternative to the Galilean transforms. And that's not even close to what happened.... So apparently this "most natural" way of developing SR wasn't at all natural in historical context.

Thus, even if the second postulate is in some sense redundant, at the time it was essential to the argument. It granted permission to abandon the ether and Galilean transforms. Now, a century later, we're altogether comfortable without ether and Galilean transforms and the second postulate sounds to the modern ear more like "Oh, and I really mean that first postulate". If we no longer feel the need for the second postulate, that just goes to show how convincing the theory has become over the past century.

#### PAllen

You mean from the first and Maxwell's equations?
and that Maxwell's equations hold in all inertial frames, unchanged. This is not required by the principle of relativity if you believe that an exotic material aether picks a preferred frame in which Maxwell's equations hold in their standard form. It is not necessary to consider it a violation of the principle of relativity to be able to detect motion relative to (exotic) matter (any more than the ability to detect motion relative to CMB isotropy is a violation of any relativity principle).

Thus, even with the type of modern arguments Frederik (and Bcrowell) mention, you need something else, take your pick:

- that there is a finite invariant speed
- that Maxwell's equations are interpreted without reference to aether, or that aether is undetectable
- that light speed is constant in all inertial frames

you've got to pick something more.

#### Nugatory

Mentor
Thus, even with the type of modern arguments Frederik (and Bcrowell) mention, you need something else, take your pick:

- that there is a finite invariant speed
- that Maxwell's equations are interpreted without reference to aether, or that aether is undetectable
- that light speed is constant in all inertial frames
The third is a restatement of the second, it it not? And also a more specific form of the first?

But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?

That's a rhetorical question, of course, and it's why I believe that the second postulate is best understood in historical context. Einstein was developing his thinking in an era that accepted (almost without question) the existence of the aether, so "no aether" was a new and important and challenging idea. Today it's as easy to just never introduce the notion of aether in the first place, and then we don't need a postulate to get rid of it.

#### Fredrik

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Gold Member
But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?
I prefer the first option on the list: We assume that the invariant speed that's predicted by (mathematical interpretations of) the principle of relativity, translation invariance and rotation invariance, is finite rather than infinite.

When we make the second assumption about "light", it sounds like SR is just an ingredient in a theory of electrodynamics, rather than a framework in which both classical and quantum theories of particles and fields can be defined. If we write down a theory of electrodynamics in this framework, it's automatically independent of ether, angels and tortoises.

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#### PAllen

The third is a restatement of the second, it it not? And also a more specific form of the first?
Not really. The third can be taken independently of Maxwell's equations. The first says nothing about light. If you assume the first you must empirically determine that light is the invariant speed.
But it's the second that is the most interesting, because it's so odd that it's needed. Why, if we must state that Maxwell's equations are to be interpreted without reference to aether, do we not also need to state that they are to be interpreted without reference to the breath of angels, or the giant tortoise that supports the entire universe, or ...?
Maxwell's equations include wave propagation. All prior experience with waves suggested a propagation medium. It is easy from our modern standpoint to laugh at aether, but I think it was quite natural for many physicists in the 1800s to suppose all waves must have a material medium to propagate in. Once you assume aether is material, however exotic, it is not necessary to assume that Maxwell's equations, in the 'standard' form, hold only in the aether frame, is a violation of POR.
That's a rhetorical question, of course, and it's why I believe that the second postulate is best understood in historical context. Einstein was developing his thinking in an era that accepted (almost without question) the existence of the aether, so "no aether" was a new and important and challenging idea. Today it's as easy to just never introduce the notion of aether in the first place, and then we don't need a postulate to get rid of it.
If you want to treat POR as part of space and time symmetries and nothing else, you still need at least (1), as Fredrik has noted.

#### ghwellsjr

Gold Member
I wouldn't get too hung up on Einstein's 1905 axiomatization. It's really not optimal from a modern point of view. In 1905, electromagnetism was the only known fundamental field. From a modern point of view, it doesn't make sense to give it a special role. Here's something with a more modern outlook: http://arxiv.org/abs/physics/0302045
When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?

#### PAllen

When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?
I think that is a valid way of looking at it.

#### Fredrik

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Gold Member
When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light?
What makes you say that? The assumption they're making is that for each inertial coordinate system S, there's an inertial coordinate system T with the property that for all events p, if S assigns it the coordinates (t(p),x(p)), then T assigns it the coordinates (t(p),-x(p)).

This doesn't seem to have any obvious connection to a statement about an invariant speed.

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#### Fredrik

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I have a couple of other issues with Pal's argument. To get from (1) and (2) to (3) and (4), he's using that for all transformations f, if the velocity of f is v, then the velocity of f-1 is -v. This doesn't follow from the principle of relativity alone.

An informal argument for it would look something like this: Consider two guns built according to identical specifications, in gun factories that are identical except for their velocities and orientation in space. (They are oriented in opposite directions, so the guns will be aimed in opposite directions). Now let's get rid of the factories and keep only the guns. Suppose that they meet at some event, which is assigned coordinates (0,0) by their comoving inertial coordinate systems. Suppose also that they're both fired at (or near) that event, and that the specifications are such that the bullet from gun A will end up comoving with gun B. Then the principle of relativity and principle of isotropy (which in 1+1 dimensions means reflection invariance, not rotation invariance) demand that the bullet from gun B will end up comoving with gun A. To be more precise, the principle of relativity suggests that guns according to identical specifications must fire bullets at the same speed relative to the gun, and the principle of isotropy suggests that the speed of the bullets won't depend on how the gun factory was oriented.

I don't see an informal argument that doesn't rely on reflection invariance. An alternative to this is to introduce a function f that takes the velocity of S' in S to the velocity of S in S', and make a technical assumption about its properties. The principle of relativity strongly suggests that $f\circ f$ is the identity map, but this doesn't imply that f(v)=f(-v) for all v unless we assume continuity or something. The assumptions must of course also imply that f is not itself the identity map. This point was (I think) first argued by Berzi & Gorini (pdf). The argument can be found in Giulini's The rich structure of Minkowski space as well. I actually haven't studied the details myself, because I was trying to find an approach that doesn't require ugly technical assumptions.

Another issue with the paper is that the argument that rules out K<0 is pretty weak. He uses strong words like "not self-consistent", but the results he derives from the assumption K<0 are just "unexpected". There's no clear statement of what mathematical statement it contradicts. He says that "we want Av to reduce to unity when v=0", but desire is of course irrelevant. This could be interpreted as an assumption that we're dealing with a connected topological group, but then why doesn't he say that he's making an assumption like that?

I recently tried to work out my own version of this argument. I was able to derive a genuine contradiction from the assumption that K<0, with only one simple technical assumption: 0 is an interior point in the set of velocities. Unfortunately I didn't realize until after I was done that, just like Pal and many others before him, I too had used reflection invariance right at the beginning.

I now think that there is no "nothing but relativity" argument of the sort I was hoping to find (no ugly technical assumptions, no use of reflection invariance), at least not for the 1+1-dimensional case. There is however a theorem for the 3+1-dimensional case that looks really awesome. Unfortunately, you have to go to a university library to read the proof, and it's a rather horrible exercise in matrix multiplication. Gorini's theorem can be stated like this: Let G be a subgroup of GL(ℝ4) such that the subgroup of G that takes the 0 axis to itself is
$$\left\{\left.\begin{pmatrix}1 & 0^T\\ 0 & R\end{pmatrix}\right| \,R\in\operatorname{SO}(3) \right\}.$$ Then either G is the group of Galilean boosts (which has invariant speed +∞), or there's a c>0 such that
$$G=\left\{\Lambda\in\operatorname{GL}(\mathbb R^4)\left| \Lambda^T\eta_c\Lambda=\eta_c\right.\right\},$$ where
$$\eta_c=\begin{pmatrix}-c & 0 & 0 & 0\\ 0 & 1 & 0 & 0\\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1\end{pmatrix}.$$ This is of course the "Lorentz" group with invariant speed c. (The actual "Lorentz group" has invariant speed 1, but this doesn't have any deeper significance. It's just a choice of units).

Those zeroes in my notation for the rotation subgroup denote the 3×1 matrix with all zeroes. The notation should be interpreted as a short way of writing a 4×4 matrix whose components are numbers, not as a 2×2 matrix whose components are matrices. I like to number coordinates and the rows and columns of these matrices from 0 to 3, so a transformation that takes the 0 axis to itself is a $\Lambda\in G$ with $\Lambda_{i0}=0$ for all $i\in\{1,2,3\}$.

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#### harrylin

Hello

Some authors claim that Einstein's second postulate (constant speed of light) simply emerges from the first one (or more precisely, its converse contradicts the first postulate). [..]
Thanks
Hi. It's quite the contrary: the second postulate is (at first sight) "apparently irreconcilable" with the first postulate.
- http://www.fourmilab.ch/etexts/einstein/specrel/www/

#### PAllen

What makes you say that? The assumption they're making is that for each inertial coordinate system S, there's an inertial coordinate system T with the property that for all events p, if S assigns it the coordinates (t(p),x(p)), then T assigns it the coordinates (t(p),-x(p)).

This doesn't seem to have any obvious connection to a statement about an invariant speed.
I think the point is that while two way speed is independently measurable, one way speed is not; thus the assumption of invariant one way speed is equivalent to assuming isotropy for light (Edwards frames are anisotropic). The newer derivations can be interpreted as replacing this special assumption about light with a generally assumed symmetry of space/time.

#### bcrowell

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When that paper invokes the "isotropy of space" (between equations 6 and 7), aren't they invoking Einstein's second postulate, except they aren't limiting it to the speed of light? Is that your point, not that Einstein's second postulate was included in the first but that it should be stated in a more general context that also includes light?
No, isotropy of space simply means that space (not spacetime) is the same in all directions, i.e., that the laws of physics are invariant under spatial rotations.

The type of axiomatization used by Pal actually doesn't refer, even implicitly, to any dynamical laws of physics such as Newton's laws or Maxwell's equations. Unlike Einstein's 1905 axiomatization, it's purely geometrical. Geometrically, the laws imply that either (a) spacetime is Galilean, or (b) spacetime is Lorentzian with some frame-invariant velocity c. In a universe where electromagnetism didn't exist, this axiomatization would still make sense, and we would still have a frame-invariant velocity c.

#### bcrowell

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Another issue with the paper is that the argument that rules out K<0 is pretty weak. He uses strong words like "not self-consistent", but the results he derives from the assumption K<0 are just "unexpected". There's no clear statement of what mathematical statement it contradicts.
I agree that this is a flaw in his presentation. The way I prefer to present this is that there is a separate axiom asserting causality, i.e., there exist events P and Q such that Q is later than P in all frames of reference. Since K<0 describes a rotation in the x-t plane, it allows us to do a 180-degree rotation that reverses the time-ordering of any two events, and that violates the axiom.

I now think that there is no "nothing but relativity" argument of the sort I was hoping to find (no ugly technical assumptions, no use of reflection invariance), at least not for the 1+1-dimensional case.
I don't see what's objectionable about those as assumptions. If you're interested in seeing other presentations of this flavor, try:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

P. Frank, H. Rothe, Ann Phys (Lepizig) 34 (1911) 825.

L.A. Pars, Philos. Mag., 42 (1921) 249

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

David Mermin, American Journal of Physics, 52 (1984) 119

Gannett, "Nothing but Relativity, Redux," http://arxiv.org/abs/1005.2062

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I

I don't think Pal's is the best treatment in this style, but I do think that this general approach is the only sensible one in the 21st century. It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view. The reason I always point people to Pal's treatment is that it's conveniently available on arxiv.

#### Fredrik

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I agree that this is a flaw in his presentation. The way I prefer to present this is that there is a separate axiom asserting causality, i.e., there exist events P and Q such that Q is later than P in all frames of reference. Since K<0 describes a rotation in the x-t plane, it allows us to do a 180-degree rotation that reverses the time-ordering of any two events, and that violates the axiom.
The approach I used was roughly this: I called the group of coordinate transformations G. I assumed that there's an ε>0 such that for all v in (-ε,ε), there's a $\Lambda\in G$ with velocity v. This seemed like the "smallest" technical assumption one can possibly make that leads to good things. Then I proved that if K<0, the following statements are true.
1. There's no proper $\Lambda\in G$ with velocity c. (If there is, then the 00 component of $\Lambda^2$ would be 0, and this means that the velocity of $(\Lambda^2)^{-1}$ is infinite, contradicting my assumption that every transformation has a velocity in ℝ).
2. For each v in (-ε,ε), there's a proper $\Lambda\in G$ with velocity v.
3. There's a proper $\Lambda\in G$ and a positive integer n such that the velocity of $\Lambda^n$ is c. (Since $\Lambda^n$ is proper too, this contradicts item 1 on this list).

I don't see what's objectionable about those as assumptions.
I find assumptions that can be motivated by physical principles more appealing than assumptions that are introduced just to ensure that the theory is simple enough to work with. But I guess there's no logical reason for this. It's just a matter of taste. I also find assumptions that involve advanced math uglier than assumptions that involve simple math.

If our goal is only to find out what a transformation between inertial coordinate systems looks like, then there is nothing objectionable about assuming reflection invariance. My goal was however more ambitious. I wanted to show that if G is the group of transformations, then there's a K≥0 such that G is equal to one of exactly five groups. The number is five because there are exactly five inequivalent answers to the question of which of these three matrices are members of G:
$$P=\begin{pmatrix}1 & 0\\ 0 & -1\end{pmatrix},\qquad T=\begin{pmatrix}-1 & 0\\ 0 & 1\end{pmatrix},\qquad -I=\begin{pmatrix}-1 & 0\\ 0 & -1\end{pmatrix}.$$ After a lot of work, I was finally able to prove this. This result would (in my opinion) have been really beautiful if we could interpret the statements $P\in G$ and $P\notin G$ respectively as "space is reflection invariant" and "space is not reflection invariant". I find this interpretation objectionable because the best way to justify my starting assumptions is to use reflection invariance. I was really frustrated when I figured this out. This is the reason why I haven't said that everyone should read my proof instead of Pal's. If anyone wants to see it, there's a pdf attached to this post.

If you're interested in seeing other presentations of this flavor, try:

W.v.Ignatowsky, Phys. Zeits. 11 (1911) 972

P. Frank, H. Rothe, Ann Phys (Lepizig) 34 (1911) 825.

L.A. Pars, Philos. Mag., 42 (1921) 249

Rindler, Essential Relativity: Special, General, and Cosmological, 1979, p. 51

David Mermin, American Journal of Physics, 52 (1984) 119

Gannett, "Nothing but Relativity, Redux," http://arxiv.org/abs/1005.2062

Morin, Introduction to Classical Mechanics, Cambridge, 1st ed., 2008, Appendix I
Thanks for the tips. I have heard of most of those, but I haven't studied any of them. Most of these are probably difficult to access for most of us.

I don't think Pal's is the best treatment in this style, but I do think that this general approach is the only sensible one in the 21st century. It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view.
I agree, but "wrong" may be too strong a word. If we remove the references to "light" and just say that there's a finite invariant speed, we have made the assumptions pretty without changing the mathematics, and now the only problem is that the assumptions are stronger than they need to be. This may be undesirable, but it has the advantage that it makes the calculations significantly easier.

The reason I always point people to Pal's treatment is that it's conveniently available on arxiv.
It's a good article, so it makes sense to link to it.

#### harrylin

[...] It's ridiculous that people are still slavishly following Einstein's 1905 axiomatization, which is just philosophically wrong from a modern point of view. [..]
I suspect that that "ridiculous" thing is non-existent, as I don't know people who "slavishly" follow such things. For many people it will be as it is for me, a matter of precise definitions of what the axioms are of which theory. "SR" happens to be a definition that was made by Einstein. Also, it is rare in modern physics to subject a theory to philosophical arguments. One merely subjects a theory to experimental verification, as philosophy is a matter of personal opinion.

#### Fredrik

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I suspect that that "ridiculous" thing is non-existent, as I don't know people who "slavishly" follow such things. For many people it will be as it is for me, a matter of precise definitions of what the axioms are of which theory. "SR" happens to be a definition that was made by Einstein. Also, it is rare in modern physics to subject a theory to philosophical arguments. One merely subjects a theory to experimental verification, as philosophy is a matter of personal opinion.
I don't consider Einstein's postulates to be the definition of special relativity. Every theory is defined by its mathematics and its correspondence rules (i.e. assumptions about how to interpret the mathematics as predictions about results of experiments). The postulates aren't even mathematical statements. They can be interpreted as mathematical statements, but this isn't even a good way to specify the mathematics.

I would say that the mathematics of SR is defined by the definitions of mathematical terms like "Minkowski spacetime" and "proper time". Together with the correspondence rule for time measurements, this defines a framework in which both classical and quantum theories of matter can be defined. I would define "special relativity" as this framework, not as a specific theory of physics. When we e.g. explain the twin paradox by drawing a bunch of lines in a spacetime diagram, what we're using is a special relativistic theory of a finite number of non-interacting classical particles. But it's much easier to just say that we're using "special relativity".

In my opinion, a "derivation" of the Lorentz transformation based on (a mathematical interpretation of) Einstein's postulates is only of historical interest. This derivation is a good way for someone who knows Newtonian mechanics and Maxwell's equations (or the result of the Michelson-Morley experiment) to find a mathematical structure that might be appropriate to use in a new theory of space, time and motion, or a theory of electrodynamics without ether. But once we've found it, it makes more sense to build the definition around that structure than around the postulates.

Even Gorini's theorem, which I find really awesome, doesn't really tell us anything about SR. What it tells us is how Galilean spacetime and Minkowski spacetime fit into a larger hierarchy of mathematical models of space and time.

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#### rjbeery

C is defined in terms of distance by time, yet all manner of spatial measurements ultimately rely upon c. If you look at the current NIST definition of a meter it explicitly does so, but prior to this it is easy to see that even our concept of any arbitrary measuring stick would depend on the propagation rate of force carriers. It's a self-referential system. IOW, unless the rate of clocking changed locally (which would contradict SR's first postulate), we have no choice but to find c locally invariant, therefore SR's second postulate is redundant.

#### harrylin

I don't consider Einstein's postulates to be the definition of special relativity. Every theory is defined by its mathematics and its correspondence rules (i.e. assumptions about how to interpret the mathematics as predictions about results of experiments). The postulates aren't even mathematical statements. They can be interpreted as mathematical statements, but this isn't even a good way to specify the mathematics.
It should be a good way to specify the physics...
[..]I would define "special relativity" as this framework, not as a specific theory of physics.
It appears to me that the framework that you call SR is in fact known as Minkowski spacetime. That is of course a very suited framework for applying SR, as it was conceived for that very purpose.
[..] In my opinion, a "derivation" of the Lorentz transformation based on (a mathematical interpretation of) Einstein's postulates is only of historical interest. This derivation is a good way for someone who knows Newtonian mechanics and Maxwell's equations (or the result of the Michelson-Morley experiment) to find a mathematical structure that might be appropriate to use in a new theory of space, time and motion, or a theory of electrodynamics without ether. But once we've found it, it makes more sense to build the definition around that structure than around the postulates. [..]
Well of course we don't need to relate to the postulates when we use the LT. However, they follow from each other and "historical interest" can be very useful for a good and logical understanding of the foundations of a theory. In many discussions here I found that people who asked questions about SR did so because they didn't understand how to correctly apply the transformations; while if they understood the postulates, they probably would understand how to correctly apply the LT. Regretfully Serway's book as cited in the OP implies a misunderstanding of the second postulate and I find that particularly unhelpful.

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#### Fredrik

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It should be a good way to specify the physics...
The physics is in the correspondence rules, and the postulates don't say anything about those.

It appears to me that the framework that you call SR is in fact known as Minkowski spacetime.
I would be OK with that definition too, but I chose to also include the correspondence rule that all theories in this framework have in common. This is the one that says that clocks measure proper time. To be more precise: For all events A,B such that both are on the world line of a clock, and B is in the chronological future of A, the number displayed by the clock at B, minus the number displayed by the clock at A, is the proper time of the segment of the clock's world line from A to B.

...if they understood the postulates, they probably would understand how to correctly apply the LT.
I'm somewhat skeptical of this.