# Nonlinear combination of velocities implies no absolute time?

Landau 1961 begins with a brief presentation of special relativity. This question is about the validity of a certain argument that they use in building up the foundations of the subject from scratch, so I think I should first sketch what they consider foundational.

They define an inertial frame as one in which Newton's first law holds. They telescope Einstein's two 1905 postulates into a single principle of relativity, which is that the laws of physics are form-invariant w.r.t. changes of inertial frame. They assert (based on unspecified experiments) that forces are propagated at a finite speed. They also seem to assume (presumably also based on experiment) that this speed is a maximum and universal for all interactions.

Given a maximum velocity, they argue that time is not absolute. The argument is that in Galilean relativity, velocities combine linearly, which is incompatible with a maximum velocity and with the Michelson-Morley experiment. Also, Galilean relativity posits absolute time. Therefore absolute time must be wrong. This seems like a clear logical fallacy to me. (A implies B, and A also implies C. B is false, so C must be false.)

[EDIT] Oops, crucial mistake in my original statement of the question. The parenthetical originally ended with "C must be true." Should have ended with "C must be false."

Is there any way of salvaging this argument? Am I misunderstanding it?

Landau and Lifshitz, The classical theory of fields, 2nd ed., 1961

## 1 Answer

The gap in the argument, if there is one, may be bridged by this: Nothing but Relativity

We deduce the most general space-time transformation laws consistent with the principle of relativity. Thus, our result contains the results of both Galilean and Einsteinian relativity. The velocity addition law comes as a bi-product of this analysis. We also argue why Galilean and Einsteinian versions are the only possible embodiments of the principle of relativity.

Essentially, the most general spacetime transformation consistent with the relativity principle includes an undetermined invariant speed parameter.

A universal time coordinate and a linear velocity addition formula only exist in the limit as the invariant speed goes to infinity which yields the Galilean transformation.

Using this form in Eq. (23), we thus obtain that the most general transformation equations consistent with the principle of relativity are of the form

Given the form of the function A from Eq. (26), it is now easy to deduce that

which is the velocity addition law.

Specific theories of relativity, of course, have to make extra assumptions in order to determine the value of K. In the case of Galilean relativity, this extra assumption shows up in the form of the universality of time, which means $t′ = t$ for any $v$.

Obviously, this requires $K = 0$. The extra assumption for Einstein’s theory of relativity is the constancy of the speed of light in vacuum. From Eq. (29), it is easy to see that $K^{ −1/2}$ is an invariant speed, independent of the frame of reference. Thus, $K = 1/c^2 > 0$ in this case.

• In general I like Pal's approach (which dates back to Ignatowsky in 1911), and it's more in harmony with the modern way of thinking about SR than Landau's somewhat archaic presentation. +1 for a possible way of fixing the argument, by giving an approach that clearly establishes that the only possibilities are Galilean relativity and SR. However, this seems way beyond anything that could have been implied by Landau, especially since Landau has not yet established that time is relative in SR. – user4552 Jun 6 '13 at 3:30
• @+1 for the reference. Homogeneity giving linearity of the transformation is very interesting (Linearity is absolutely fundamental, for instance, if one inertial observer sees these 2 systems as independent, another observer must see also these 2 systems as independent, and this implies linearity - if we consider momentum transformation). Note that there is an supplementary axiom in the Pal paper: $x' = 0 <=> x = vt$. And, yes, isotropy will give some particular 1-dimensional subgroup of $SL(2,R)$. [$SO(1,1)$ for relativity transformation] – Trimok Jun 6 '13 at 11:02