- #36
PeterDonis
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vanhees71 said:You don't need to make a closed loop of geodesics.
Yes, I've already recognized that. Please see post #33.
vanhees71 said:You don't need to make a closed loop of geodesics.
pervect said:I suspect the OP want's a more general description of Fermi-Walker transport than that, one that could work along space-like curves.
PeterDonis said:You can apply the Fermi-Walker transport law along any spacelike curve; you just pick one of the spacelike vectors in the orthonormal tetrad to be the tangent vector to the spacelike curve. This case is not much discussed in the literature, but it works fine mathematically, since the Fermi-Walker transport law does not depend on the curve being timelike.
The case I'm not sure about is applying the Fermi-Walker transport law along a null curve. Obviously you can't choose an orthonormal tetrad in that case if you want to include the curve's tangent vector.
In that case (flat space-time in global Lorentzian coordinates) space-time is basically an affine space in affine coordinates thus parallel transporting a tangent vector (by Levi-Civita trivial connection) simply preserves its components.pervect said:In flat space-time, parallel transport in any inertial frame is just a matter of translating the vector in said frame. The curved space-time case is more interesting, but I don't think I need to rehash what others wrote there.
pervect said:I think it probably matters that it be of unit length.
vanhees71 said:even for a null vector line you have
$$
u_{\mu} u^{\mu}=0=\text{const}
$$
and thus
$$
\dot{u}_{\mu} u^{\mu}=0
$$
where the dot means a derivative wrt. an arbitrary parameter
vanhees71 said:Maybe my notation was not well enough explained
vanhees71 said:can be spacelike
vanhees71 said:depending on the specific light-like curve
Quoting first item in that post:PAllen said:You might want to study post #177 in that thread for a summary of the accumulated understanding developed int the thread.
What do you mean with 'observable' Minkowski metric ? Basically that distance and time measurement are consistent with a Minkowski metric in which the (timelike) paths of observers at fixed height ##z## from ground are assumed to be geodesics of the underlying space-time ?PAllen said:1) They posit a theory where the Minkowski metric is the observable metric for distance and time measurements.This is pretty clearly stated."
Yes.cianfa72 said:Quoting first item in that post:
What do you mean with 'observable' Minkowski metric ? Basically that distance and time measurement are consistent with a Minkowski metric in which the (timelike) paths of observers at fixed height ##z## from ground are assumed to be geodesics of the underlying space-time ?
Which is the procedure they posit to be used to setup that global Lorentz frame ?PAllen said:3) They posit it is possible to set up global Lorentz frame physically using a described procedure.
cianfa72 said:Which is the procedure they posit to be used to setup that global Lorentz frame ?
ok, but I'm confused about the following:PeterDonis said:Start with a family of observers at rest at infinity. Then extend to all finite distances ##r## from the origin by adding observers at each ##r## who are at rest relative to the observers at infinity (as they can confirm by exchanging round-trip light signals). The worldlines of all these observers provide the timelike "grid lines" of the global Lorentz frame, and the spacelike "grid lines" are then just mutually orthogonal spatial geodesics that are also orthogonal to those timelike worldlines.
cianfa72 said:The global frame being built this way is a Lorentz one (the observed metric due to space and time measurements results in ##ds^2 = dz^2 - dt^2##)
cianfa72 said:that is not the case because to gravitational time dilatation between obsevers at fixed height
Trying to sum up (sorry but I'm a non-expert )PeterDonis said:the hypothetical theory of gravity that he is describing (basically, a theory in which gravity is treated as a "force field" like electromagnetism on a flat spacetime background) predicts we should be able to do. But Schild's argument then amounts to the observation that we can't actually do it, because of gravitational time dilation--the actual observed metric doesn't match the predicted metric that we should observe according to the theory.
cianfa72 said:the predicted metric should be Minkowski using that (global) coordinate system
cianfa72 said:In both cases light ray worldlines of Schild's argument at first and second emission are nevertheless 'congruent' -- they are obtained one from the other by a Lie transport along the fixed height observer woldlines (global Lorentz frame timelike 'grid lines') that are actually orbits of ##\partial_t## KVF in those coordinates.
cianfa72 said:In both cases gravitational time dilatation for lower and upper observers (as turn out from experiment) gets a contradiction
Regarding spacelike "grid lines" I'm a bit confused about their actual physical significance. I'm aware of timelike worldlines physical significance as paths taken in spacetime by massive particles, but what about spacelike worldlines ?PeterDonis said:The worldlines of all these observers provide the timelike "grid lines" of the global Lorentz frame, and the spacelike "grid lines" are then just mutually orthogonal spatial geodesics that are also orthogonal to those timelike worldlines.
cianfa72 said:Regarding spacelike "grid lines" I'm a bit confused about their actual physical significance.
cianfa72 said:ok, but I'm confused about the following:
The global frame being built this way is a Lorentz one (the observed metric due to space and time measurements results in ##ds^2 = dz^2 - dt^2##) thus coordinate clocks 'wearing' by observers providing timelike 'grid lines' must result 'Einstein synchronized' upon exchanging repetuted round-trip light signals. However that is not the case because to gravitational time dilatation between obsevers at fixed height as discussed in this thread
pervect said:From a metric standpoint ##ds^2 = dz^2 - dt^2## gets replaces with ##ds^2 = dz^2 - z^2 dt^2## - or something equivalent. That's the Rindler metric.