Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Einstein postulates and the speed of light

  1. Feb 8, 2013 #1
    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:
    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
     
    Last edited: Feb 8, 2013
  2. jcsd
  3. Feb 8, 2013 #2

    ghwellsjr

    User Avatar
    Science Advisor
    Gold Member

    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:
    And in section 2 he said:
    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:
    "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.
     
    Last edited: Feb 8, 2013
  4. Feb 8, 2013 #3

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.

    Converse? Do you mean its negation?

    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.
     
    Last edited: Feb 8, 2013
  5. Feb 10, 2013 #4

    Meir Achuz

    User Avatar
    Science Advisor
    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.
     
  6. Feb 10, 2013 #5

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    You mean from the first and Maxwell's equations?
     
  7. Feb 10, 2013 #6

    bcrowell

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    I agree with it, but you'll find many people who wouldn't.

    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
     
  8. Feb 10, 2013 #7

    Nugatory

    User Avatar

    Staff: Mentor

    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.
     
  9. Feb 10, 2013 #8

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  10. Feb 11, 2013 #9

    Nugatory

    User Avatar

    Staff: Mentor

    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.
     
  11. Feb 11, 2013 #10

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
    Last edited: Feb 11, 2013
  12. Feb 11, 2013 #11

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    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.
    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.
    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.
     
  13. Feb 12, 2013 #12

    ghwellsjr

    User Avatar
    Science Advisor
    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? 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?
     
  14. Feb 12, 2013 #13

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    I think that is a valid way of looking at it.
     
  15. Feb 12, 2013 #14

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
    Last edited: Feb 12, 2013
  16. Feb 12, 2013 #15

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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\}##.
     
    Last edited: Feb 12, 2013
  17. Feb 12, 2013 #16
    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/
     
  18. Feb 12, 2013 #17

    PAllen

    User Avatar
    Science Advisor
    Gold Member

    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.
     
  19. Feb 12, 2013 #18

    bcrowell

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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.
     
  20. Feb 12, 2013 #19

    bcrowell

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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 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.
     
  21. Feb 12, 2013 #20

    Fredrik

    User Avatar
    Staff Emeritus
    Science Advisor
    Gold Member

    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 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.


    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 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.

    It's a good article, so it makes sense to link to it.
     
Know someone interested in this topic? Share this thread via Reddit, Google+, Twitter, or Facebook