Why absoluteness of time implies galilean transformations?

Question

In Landau course, vol.1 Mechanics, one finds the statement:

...the absoluteness of time necessarily implies that the ordinary law of composition of velocities is applicable to all phenomena.

I don't see this implication clearly.

Answer 1

We have to be careful in stating exactly what we're going to allow ourselves to assume here. We need some sort of principle of relativity -- that the laws are the same for both observers. But we don't want to assume anything else a priori, right? For instance, we don't want to assume at first that rulers have the same length for both observers -- we need to prove that.

Let's work in one dimension for simplicity.

Suppose that observer B is moving at constant velocity v relative to observer A. Suppose some object is moving along with some speed $u_B$ as measured by B. We want to show that the speed as measured by observer A is $u_A=u_B+v$.

Consider the position of the object at two different times, separated by a small amount $dt$. Since time is absolute (all observers use the same $dt$), what we want to show is equivalent to $$ dx_A = dx_B+v\,dt $$ (multiplying the original equation through by $dt$). Here $dx_A$ means $x_A(t+dt)-x_A(t)$, that is, the change in the position of the object at the two times, as measured by A, and similarly for B.

Here's a useful fact: If both observers measure the distance between two points at an instant of time $t$, they must get the same answer. The reason is symmetry. If the two disagreed, then one would have to get a bigger answer than the other. But for a measurement of this sort, there's nothing to break the symmetry between A and B -- that is, we can just change the sign of $v$, and consider B to be stationary and A to be moving, and that shouldn't affect the answer.

I think that's enough to get us there. Suppose that observer B sets of a firecracker at his location at time $t$, and another at time $t+dt$. The two observers must agree on the distance from B to the object at the time the first firecracker went off, and they must agree on the distance from B to the observer at the time the second firecracker went off. The difference between these these two numbers is $dx_B$. But the difference between these two numbers is also $dx_A-v\,dt$, since observer A knows that observer B traveled a distance $v\,dt$ during that time interval. The conclusion follows.

Answer 2

Q: absoluteness of time implies galilean transformations?
A: No.

What implies galilean transformations (ordinary law of composition of velocities) is is the non-recognition of a speed limit of propagation on the physical environment.

The long text that follows is just to illustrate the above statement.

Galileo could have known this? (lets change the course of past events with a fiction ;-)
Perhaps he had noticed that a fast boat and a slow boat leaves waves that propagate without difference between them and concluded: The speeds are not additive. As in his time all objects were slow he did not concluded that in sound and light the velocities are not additive.

But assuming the contrary, let's see what he could have thought:
A very fast plane sounds a tone and an observer, armed with an absolute ruler and a clock common to the pilot of the plane, with instant communications, measures a constant and finite speed for the sound and concludes: I can not add the speed of the plane to the speed of the sound, and observes that the sound must propagate in waves similar to the ones of the boats. These speed limits are characteristic of the propagation medium.

Galileo had read the work of Robert Hooke 'the wave theory of light' (in 1660) and Ole Roemer in 1675 had informed that the speed of light is 200,000 km/s (only in 1728, 53 years later, James Bradley measured 301,000 km/s). Since I (Galileu) have no immediate vision I will have to measure the speed of sound with the aid of a flash of light and make the appropriate corrections to have a more accurate measure of the speed of sound.

And after Galileo had known Einstein's work, which he aproved, he thought: it makes sense that the speed of light is also a constant feature of the medium. I measured its value in a closed circuit and Einstein confirmed to him: the 'c' is the average value of the round trip of a light ray, as it is written in his 1905 paper. To put an end to the confusion with the different observer referencials Galileo decided to change the perspective: I will adopt the uniform referential of the medium because it is common to all observers.