Coordinate Systems: GR vs Newtonian

[Jonnyb42]

I just want to ask a simple question:

Is it true that Newtonian/Classical Mechanics does not hold true for all coordinate systems, while General Relativity does?

 

[haushofer]

You mean the form of the equations of motion? Yes, that's true.

The Einstein equations are covariant with respect to general coordinate transformations. But for instance, the Poisson equation is only invariant with respect to the Galilei group (plus some extra transformations involving accelerations).

A simple example is given by the EOM of a free Newtonian particle:

$$\frac{d^2 x^i}{dt^2} = \ddot{x}^i = 0$$

Going to a rotating frame of reference means

$$x^i \rightarrow R^i_{\ j}(t)x^j$$

where R is an element of SO(3). Plugging this into the EOM gives rise to a centrifugal and Coriolis force. A similar game can be played for arbitrary accelerations.

 

[Russell E]

Is it true that Newtonian/Classical Mechanics does not hold true for all coordinate systems, while General Relativity does?

Depending on what you mean, the answer is that Newtonian mechanics holds in some coordinate systems, all coordinate systems, or no coordinate systems.

If you are just referring to the fact that inertial acceleration equals the second derivative of the space coordinates with respect to the time coordinate only for a special class of coordinate systems, then that's true. If you (wrongly but not uncommonly) regard the identification of inertial acceleration with that second derivative as being part of "Newtonian mechanics", then you would say Newtonian mechanics "is true" only when phenomena are described in terms of a certain class of coordinate systems.

On the other hand, the conventional identification of d2x/dt2 with inertial acceleration is really just that, i.e., a convention or convenience, tending to make us prefer certain coordinate systems because the equations of Newtonian mechanics are simplest when expressed in terms of such coordinates. This does not imply that Newtonian mechanics is violated when we simply translate the description of phenomena from one system of coordinates to another. So from this standpoint we would say Newtonian mechanics "is true" for all coordinate systems.

On the third hand, we obviously have to say that, in fact, Newtonian mechanics is not exactly "true" with respect to ANY system of coordinates, first because it doesn't account for the effects of Lorentz invariance (e.g., the change in inertia with speed), and second because it doesn't account for the possibility of curved spacetime.

The same three answers apply to general relativity, but for different reasons.

I just want to ask a simple question:

There are no simple questions, there are only simple answers.

 

[bcrowell]

GR can be done in a coordinate-dependent formalism or a coordinate-independent one. Newtonian mechanics can be also be done in a coordinate-dependent formalism or a coordinate-independent one.

Some relevant papers:
http://arxiv.org/abs/gr-qc/0603087
http://arxiv.org/abs/gr-qc/0506065
http://arxiv.org/abs/gr-qc/9506077
http://th-www.if.uj.edu.pl/~acta/vol21/pdf/v21p0755.pdf