# Galilean relativity & the road to special relativity

Firstly, I just want to make sure that I've understood the notions of relative and absolute quantities correctly.

Elementary analysis shows that position and velocity are relative quantities. Indeed, position is clearly relative as two inertial frames $S$ and $S'$ displaced by a constant displacement vector $\mathbb{r}_{0}$ will measure the position of an object to be at $\mathbb{r}$ and $\mathbb{r}'$ respectively, the two positions related by $\mathbb{r}=\mathbb{r}'+\mathbb{r}_{0}$. As these two frames are arbitrary and neither can be distinguished from the other as a preferred absolute rest frame, it must be that position is relative. This argument also holds if the two frames $S$ and $S'$ are in relative motion to one another, related by $\mathbb{r}=\mathbb{r}'+\mathbb{v}t$, where $\mathbb{v}$ is the relative velocity between the two frames. Clearly it follows from this (by differentiating with respect to time) that velocity is also relative.

Now, if I understand it correctly, Newton introduced the notion of absolute space, and thus defining the absolute position and velocity of a given object as their position and velocity measured relative to this frame. Thus these relative quantities defined in the previous paragraph are all related to absolute quantities (that in principal will be the same for all observers at rest relative to absolute space, regardless of where they are located in this space). However, as a result of Galileo's principal of relativity ruling out the existence of a frame at absolute rest, i.e. absence of absolute space, it follows that the concepts of absolute position and velocity do not exist and therefore are truly relative quantities, dependent on the frame that they are measured in.

Secondly, if we consider Maxwell's equations, which are not invariant under Galilean transformations, but we require them to hold in all inertial frames, doesn't it immediately follow that the speed of light has the same constant value in all inertial frames from this assumption (given that Maxwell's equations imply a constant speed of light). Why is it given as an axiom of special relativity?

• Newton introduced an absolute time, that all observers would agree on, however nothing in Newton's work requires an absolute space. Newton understood full well that positions and velocities were relative - after all, he was familiar with Galileo's work. I don't understand your last paragraph. What exactly are you asking? Actually I'm not what the first part of your question is asking either. – John Rennie Jun 19 '15 at 11:07
• @JohnRennie I looked at this earlier, thinking the same as you that only absolute time was involved. There is a quote on Wiki from Newton directly: Absolute space, in its own nature, without regard to anything external, remains always similar and immovable. The rest of the quote agrees with your comments regarding his views on relative motion. I'm not silly enough to argue with yourself on GR, I just wondered how relevant this quote is. Probably of no relevance if he didn't use the absolute space concept anyway, as you say above. – user81619 Jun 19 '15 at 11:52
• @JohnRennie sorry, I realise it's perhaps not best worded. The first part is really a check of whether I've understood the notions of relative position and velocity correctly? And the second part is asking why the speed of light being constant and frame independent is taken as an axiom of SR when it appears to follow if one accepts that Maxwell's theory is valid in all in all inertial frames?! – Will Jun 19 '15 at 12:02
• @Will "when it appears to follow if one accepts that Maxwell's theory is valid in all in all inertial frames?!", This fact is incompatible with Galilean relativity, since Galilean relativity necessarily puts no upper bound on speed, whereas your assumption is equivalent to assuming the existence of a finite upper bound on speed. Actually, one way to formulate SR is to state the following axiom: the laws of physics including that of electromagnetism are invariant in all inertial frames of reference. – Omar Nagib Jun 19 '15 at 13:08
• Are you asking if Newton was wrong? Yes. Newton was wrong. He was wrong about attempting to introduce absolute coordinates. That, however, is not the difference between Newtonian mechanics and special relativity, both of which have the same notion of relativity. The difference is that the equations of electrodynamics are correct at large velocities and the equations of Newtonian mechanics are not and that this has been sufficiently tested experimentally. Was Newton wrong about his equations, then? No, not really, he simply couldn't know better given the experimental limits of his time. – CuriousOne Jun 19 '15 at 19:51

## 2 Answers

Why is it given as an axiom of special relativity?

In a nutshell: because of the Michelson-Morley experiment. This ought to have detected a variation in the speed of light caused by our motion through space. But it didn't. So Einstein reasoned that speed = distance / time, and that if the speed didn't change, your the time did. And then if your time changed, your measurement of distance also had to change. Something like that, Maybe somebody else can express it better or give a reference.

Note that IMHO the postulate "works" because of the wave nature of matter. See The Other Meaning of Special Relativity by Robert Close for what I think is a well-argued article. Again in a nutshell: when you're made of waves along with your rods and clocks, you always measure wave speed to be the same. Because you calibrate your rods and clocks using the motion of waves, then you use them to measure the motion of waves! It's a tautology, see http://arxiv.org/abs/0705.4507. Also note that Einstein abandoned the postulate when he developed general relativity. See for example this and this. The speed of light varies in the room you're in. If it didn't, light wouldn't curve and your pencil wouldn't fall down.

• Ah, so is the point that either Maxwell's equations were valid in one special frame of reference, the ether, and as such the speed of light was only constant in this frame, or as they discovered, the ether doesn't exist and as Maxwell's equations appear to accurately describe electromagnetic phenomena the only possible why our is to postulate that the speed of light is the same in all inertial frames. Also, is what I've put about relative position and velocity correct? – Will Jun 19 '15 at 13:41
• The ether thing doesn't sound right Will. Einstein is said to have done away with the aether, but he described space as the aether of general relativity, see his 1920 Leyden Address and the arXiv. What you said about position & velocity sounds off too. The CMB dipole anisotropy lets us gauge our speed through the universe, which is absolute as it gets. – John Duffield Jun 20 '15 at 8:46
• Position and velocity are definitely relative quantities though (they very much depend on our frame of reference), even in classical mechanics (c.f. Tong's notes damtp.cam.ac.uk/user/tong/relativity/dynrel.pdf pages 5 and 6). The dipole anisotropy that we observe is due to the peculiar velocity of the Milky Way with respect to the rest of our local galaxy cluster and so is a relative quantity (it's not absolute as people at different points in the universe would measure a different anisotropy and so would measure our velocity to be different). – Will Jun 20 '15 at 10:09
• The thing to note is that a special-relativity reference frame is little more than a state of motion. When you use that to demonstrate that motion is "merely relative", your argument is tautological. Also note that your absence of absolute space goes against the grain of what Einstein said about space being a something rather than a nothing, wherein a field is a state of space, see the Expanding the theory section here. And do note that that every observer can use the CMB reference frame to gauge their motion through the universe. – John Duffield Jun 20 '15 at 10:49

Secondly, if we consider Maxwell's equations, which are not invariant under Galilean transformations, but we require them to hold in all inertial frames, doesn't it immediately follow that the speed of light has the same constant value in all inertial frames from this assumption (given that Maxwell's equations imply a constant speed of light). Why is it given as an axiom of special relativity?

In all physics frameworks, by which I mean classical mechanics, electrodynamics, thermodynamics and quantum mechanics , one finds similar mathematics, usually differential equations. What separates the frameworks is the region of validity where the model holds.

So Maxwell's equations hold for electromagnetic quantities and the Lorenz transformations ( note Lorenz) are developed within the framework.

The genius of Einstein was in thinking outside the box: He took the proven constant velocity of electromagnetic waves and the kinematics that the Lorenz transformations imposed ,and applied them to massive particles. That is his contribution, that no massive particle can have a speed larger than the velocity of light. The Lorenz transformations for massive particles gave what we now call special relativity .

Not everybody can think outside the box, and many who want to be the new Einstein offer outside the box proposals that are rejected as crank proposals. Einstein had at his fingertips the mathematical formulations of generations of mathematicians and physicists. The combination , background knowledge and out of the box thinking, gave the successful theories he has proposed: his theories/models fitted the data.