Einstein's 1905 section 2: light speed in moving frames

[vanhees71]

What is the "Galilean metrics"?

In the standard formulation of Newtonian mechanics, there's time as an oriented 1D "real line" with the standard topology and space, which is a 3D Euclidean affine manifold. There's not (pseudo-)metrical meaning that would justify to introduce spacetime vectors as is very natural in special relativity leading to Minkowski space, where four-vectors have a very convenient meaning.

 

[robphy]

What is the "Galilean metrics"?

Galilean spacetime is a special case of a Newton-Cartan spacetime
( https://en.wikipedia.org/wiki/Newton–Cartan_theory )
which has

In Newton–Cartan theory, one starts with a smooth four-dimensional manifold M and defines two (degenerate) metrics.
A temporal metric $t_{ab}$ with signature (1,0,0,0) used to assign temporal lengths to vectors on M
and a spatial metric $h^{ab}$ with signature (0,0,0,1) [ used to assign spatial lengths to vectors on M].

 

[vanhees71]

This doesn't look very natural though.

 

[robphy]

This doesn't look very natural though.

Your opinion has been noted.
It's an established approach in relativity research,

introduced by Élie Cartan[1][2] and Kurt Friedrichs[3] and later developed by Dautcourt,[4] Dixon,[5] Dombrowski and Horneffer, Ehlers, Havas,[6] Künzle,[7] Lottermoser, Trautman,[8] and others.

The point is...
can we formulate it to explain things and to make predictions that can be tested by experiment?

 

[David Lewis]

We have a wagon of length L, as measured in a frame in which it is traveling with speed v to the right.

Is L the rest length of the wagon?

 

[PeroK]

Is L the rest length of the wagon?

No.