vanhees71 said:
So what are you disputing then with
@PeterDonis ?
Well, disputing...let's call it discussing :P : How the Newtonian potential transforms if you go
beyond the Galilei group of transformations.
Let me formulate how I see things after the (enlightening!) discussion with Peter.
In the textbook Newtonian limit of GR, the Einstein field equations become the Poisson equation, and the geodesic equation becomes Newton's 2nd law. The gradient of the Newton potential shows up in the Christoffel connection,
<br />
\Gamma^i_{00} = \partial^i \phi<br />
So one can easily deduce that under a transformation, which we can call an acceleration,
<br />
x^{'i} = x^i + \xi^i (t)<br />
we have in the Newtonian limit of GR that
<br />
\Gamma^{'i}_{00} = \Gamma^{i}_{00} - \ddot{\xi}^i \ \ \rightarrow \ \ \partial^{'i} \phi' = \partial^i \phi - \ddot{\xi}^i<br />
Such a transformation leaves both the geodesic equation and Einstein's field equations in the Newtonian limit covariant. This transformation can be physically understood as the fact that locally in spacetime, where the Newtonian force is approximately constant, one can choose a coordinate frame such that
<br />
\Gamma^{'i}_{00} = 0<br />
by choosing the acceleration appropriately (##\ddot{x}^i = (0,0,g)## if the gravitational field is point in the z-direction). This is the interpretation of the statement that "a freely falling observer feels itself being weightless".
So, the correspondence principle suggests that this is how ##\partial^i \phi## should transform under accelerations in the Newtonian theory. However, in the Newtonian theory, ##\phi## transforms as a scalar. The induced transformation on ##\partial^i \phi## by accelerating does not reproduce the transformation that the GR-limit gives. In GR, ##\partial^i \phi## shows up in the connection, and we know how connection coefficients transform. In Newtonian theory, ##\partial^i \phi## is just the gradient of a scalar.
But still, the action of a Newtonian point particle coupled to the Newton potential (and it's equation of motion, Newton's 2nd law) is covariant under the transformation
<br />
\partial^{'i} \phi ' = \partial^{i} \phi - \ddot{\xi}^i, \ \ \ \ \ \ x^{'i}(t) = x^i + \xi^i (t)<br />
This transformation of ##\partial^i \phi## is not "induced by the coordinate transformation". But there is a formalism in which we can interpret this kind of transformations: we call it the formalism of pseudosymmetries, which has its origin in sigma-models. The transformation of ##x^i(t)## does not induce the transformation on ##\partial^i \phi##; instead the transformation of ##\partial^i \phi## can be regarded as a
redefinition of ##\partial^i \phi##. This "redefinition" (which for PeterDonis was the "magic"-part) is still a symmetry of the action, but because this redefinition is not induced by the transformation of ##x^i(t)##, the corresponding symmetry does not have an accompanying Noether charge. That's why we call it a "
pseudosymmetry". It's the price to pay if you want to connect the Newtonian limit of GR to the usual Newton theory on the level of symmetries.