Is the Principle of Relativity different for different theories?

[Pushoam]

The images have been taken from the section : Relativity according to Galileo and Newton, page no.66, special relativity , A.P.French,1968What I understood is:
According to the first paragraph,
Laws of transformation are needed so that a theory which describes a phenomenon w.r.t. one reference frame could be applied to explain the same phenomenon w.r.t. another reference frame.

Does it mean that Laws of transformations are different for different theories?A physical statement of what these invariants are is called a principle of relativity.
For Galilean transformation, acceleration is invariant.
So, does it mean that the following statement:
"Acceleration is invariant under Galilean transformation ."
is a principle of relativity for Newtonian dynamics.
So, for different theories , there will be different principle of relativities. Right?

Fundamental equations of theory usually defines the principle of relativity applicable to the theory.
e.g.
The equation F = ma defines the principle of relativity, i.e. force in Newtonian Dynamics.

 

[PeterDonis]

Does it mean that Laws of transformations are different for different theories?

Yes. For example, Newtonian mechanics uses Galilean transformations, whereas special relativity uses Lorentz transformations. And since the actual transformation laws obeyed in experiments can be tested, we can show experimentally that the Newtonian transformation laws are wrong and the special relativity transformation laws are correct.

for different theories , there will be different principle of relativities

Yes.

"Acceleration is invariant under Galilean transformation ."
is a principle of relativity for Newtonian dynamics

The equation F = ma defines the principle of relativity, i.e. force in Newtonian Dynamics

These I'm not sure about. I would expect the principle of relativity applicable to a given theory to be defined simply by the transformation laws themselves, not necessarily consequences of them.

Or, if one is going to define the principle of relativity for a given theory in terms of invariants, I think one would have to be able to use the invariants to uniquely determine the transformation laws. This works for special relativity, since the Lorentz transformations are the unique ones that preserve the relevant invariants (the speed of light and spatial isotropy). I'm not sure if it works for Newtonian mechanics if we take the relevant invariants to be acceleration and spatial isotropy--that is, I'm not sure the Galilean transformations are the unique ones that preserve those two invariants.

 

[vanhees71]

Assuming that the principle of inertia is valid, i.e., that there exists a class of reference frames, the inertial frames, where a free particle moves with constant velocity, for which for any observer at rest relative to an inertial frame time and space are homogeneous and space is isotropic and that the set of symmetry transformations forms a Lie group you find that there are, up to equivalence, only two spacetime manifolds fulfilling these properties, namely the Galilean spacetime and the Minkowski spacetime.

For details, see e.g.,

V. Berzi and V. Gorini, Reciprocity Principle and the Lorentz Transformations, Jour. Math. Phys. 10, 1518 (1969)
http://dx.doi.org/10.1063/1.1665000

 

[dextercioby]

The Galilean and Minkowski spacetimes are flat, hence is makes sense to ask a different question: How are two frames of references A,B connected (thus by which transformations of variables), so that the motion of object O in reference frame A which is along a straight line is mapped into the motion of the object O in reference frame B which is also along a straight line? The answer you can find here: http://www.mathpages.com/home/kmath659/kmath659.htm.