The discussion centers around the concept of congruent worldlines in a static gravitational field as presented in Carroll's General Relativity book. Participants explore the implications of light propagation in a gravitational field, the nature of static fields, and the relationship between gravity and spacetime curvature. The conversation includes theoretical reasoning and interpretations of gravitational effects on light.
Participants express differing interpretations of the implications of static gravitational fields and the nature of light's interaction with gravity. There is no consensus on whether the assumptions made in the argument are valid or if they lead to circular reasoning.
Participants note that the discussion involves nuanced definitions of static fields and spacetime geometry, as well as the implications of the equivalence principle. Some statements reflect assumptions that may not be universally accepted within the context of the discussion.
I would like to point out that when you see a map on a piece of paper you have to know how that map is actually built (i.e. what map it is). In the case of post#1 it is the inertial/Minkowski map for Minkowski/flat spacetime.Ibix said:No. You can always draw straight axes on a map of the nastiest twistiest spacetime you can imagine (as long as you avoid coordinate singularities and the like). The axes won't necessarily be straight in the real world!
Indeed.cianfa72 said:would like to point out that when you see a map on a piece of paper you have to know how that map is actually built (i.e. what map it is).
No, it's in an arbitrary static spacetime in coordinates that reflect that symmetry. In context, it's probably Schwarzschild spacetime in Schwarzschild coordinates, but the argument is more general.cianfa72 said:In the case of post#1 it is the inertial/Minkowski map for Minkowski/flat spacetime.
Why ? If you assume Schwarzschild spacetime in Schwarzschild coordinates then spacetime is curved and that's actually what that argument is supposed to show (i.e. give a proof).Ibix said:No, it's in an arbitrary static spacetime in coordinates that reflect that symmetry. In context, it's probably Schwarzschild spacetime in Schwarzschild coordinates, but the argument is more general.
I was stating the eventual conclusion there. The only assumption being made is that observers at any fixed ##z## always see the same result of any gravitational experiment (so the lines of constant ##z## are what we will eventually call integral curves of the timelike Killing vector field). That's why the light paths are the same. But we aren't assuming anything about what those paths are, which is why they are drawn as arbitrary squiggles.cianfa72 said:If you assume Schwarzschild spacetime in Schwarzschild coordinates then spacetime is curved and that's actually what that argument is supposed to show (i.e. give a proof).
Ok, so the lines of constant ##z## in the map represent (in the map) the integral curves of the timelike KVF -- a 'static' spacetime also requires that the timelike KVF is hypersurface orthogonal. Basically the values of ##z## "label/ identify" each of those timelike KVF's integral curves.Ibix said:The only assumption being made is that observers at any fixed ##z## always see the same result of any gravitational experiment (so the lines of constant ##z## are what we will eventually call integral curves of the timelike Killing vector field). That's why the light paths are the same.
I'm stuck with this point: observers at any fixed ##z## always see the same result only for local gravitational experiment. Is that true?Ibix said:The only assumption being made is that observers at any fixed ##z## always see the same result of any gravitational experiment.
I'm not sure what "local" is meant to mean in this context, honestly.cianfa72 said:I'm stuck with this point: observers at any fixed ##z## always see the same result only for local gravitational experiment. Is that true?
How does the observer at fixed ##z## measure the time along the path the object is falling from rest ?Ibix said:An observer at fixed ##z## in a static field who always releases the ball from rest and allows it to fall the same distance will always measure the same time
Bounce a radar pulse off it. The nature of a static spacetime means that the radar pulse takes equal times on both legs of its journey, so you may not be able to measure distance accurately that way without further work, but you can determine at what time (by your clock) the echo happens.cianfa72 said:How does the observer at fixed ##z## measure the time along the path the object is falling from rest ?
I believe that's true only in the coordinate chart adapted to the hypersurface orthogonal's timelike KVF (i.e. in the coordinate chart in which the coordinate time worldlines are orbits of timelike KVF).Ibix said:The nature of a static spacetime means that the radar pulse takes equal times on both legs of its journey,
Sure, but that's fine, as long as you use the same coordinates for each run. If you want something coordinate free, go with the "record the time you see it land".cianfa72 said:I believe that's true only in the coordinate chart adapted to the hypersurface orthogonal's timelike KVF (i.e. in the coordinate chart in which the coordinate time worldlines are orbits of timelike KVF).
Do you mean record the time on the observer's clock at fixed ##z## when the radar pulse bouncing from the land event come back to the observer at fixed ##z## ?Ibix said:If you want something coordinate free, go with the "record the time you see it land".