For a good discussion on curvature, try here:
Is Spacetime Curvature Absolute
Spacetime Curvature Observer and/or Coordinate Dependent?
https://www.physicsforums.com/showthread.php?t=596224
In that discussion :
My attempt at a Summary:
[I don’t think I’ve ever seen one, so I thought I’d give it a try..]
[#19] Is spacetime curvature observer dependent ?
None of the answers to the OP are precise…none can be unless an agreement is reached on exactly what component of 'curvature' measurement will be used...and that would provide only a partial answer; the reason is given at the end of this post with some of the great insights along the way summarized here first.…
Originally Posted by Mentz114
If we take 'spacetime curvature' to mean gravity then the answer must be 'yes'.
PeterDonis adds:
If by "gravity" you mean "particular effects of gravity", then yes. As you point out, particular effects of gravity on particular observers will always be dependent on the observer's 4-velocity…. this is a question of terminology, not physics. Whether or not "spacetime curvature" is observer-dependent depends on what you define "spacetime curvature" to mean.
passionflower asked several questions that I really liked:
[1]
So then tell me how does an observer observe the effects of
A) spatial curvature
B) temporal curvature
and [2]
All I want is to address statements that bodies are affected differently by space and time curvature depending on their velocity. These statements I consider wrong and very misleading.
I’ll try some explanations; these, too will be imprecise, but hopefully again illustrate
The problem; feel free of course pick them apart if you don’t like them:
[1] Locally, everything stays the same...spacetime stays flat….but as a differential instead of infinitesimal observation is made, curvature becomes apparent. Time dilation enables the observer to make a comparison with a ‘distant’ observer to see their clocks run at different speeds; such an observer may also see a distant inertial observer apparently moving in an arc...with some acceleration, that is, some ‘curvature’ is apparent resulting from the acceleration of either. docAl might say: "yes, that's the effect of different grid lines but drawn on the original curved graph paper.…the original gravitational field. [This is another imprecise explanation.]
[2] Jonathan Scott provided insights to this one in posts #13,14; I want to try a complementary approach: I think he said different curvatures, accelerations and velocities are all related in curved spacetime.
I am reminded that ‘light follows null geodesics’ is an APPROXIMATION; a good one, but what really follow null geodesics are test particles...ideals with no mass, no energy, no disturbance of the initial gravitational field. So we see that photons of different energy follow slightly different paths. [I got this idea elsewhere from pervect.] So these ‘bodies’ are affected by energy….and you [OP] were worried about just speed affecting curvature’!
Next, suppose we have a particle with some mass at some velocity relative to an otherwise stationary gravitational field. Send a second particle at the same initial trajectory but, say, double the velocity of the first; or impart it with some spin [angular momentum] but at the original velocity: All the particles will follow different trajectories. In docAl's language, the original gravitational field maintains it's 'curved graph paper' shape, unless it’s evolving, but there is an overlay of different grid lines [a/w each particle] on that curved graph paper representing ‘spacetime curvature…...due to different energies; hence different paths result. The ‘gravitational curvature’ may remain invariant [depending on how we define that] but the effect of the now differently curved grid lines results in a different overall ‘curvature’ and hence different worldline paths.
We seem to have agreed that: An invariant would be something that has exactly the same value in whatever frame it is calculated; And Dalespam and Mentz seem to agree that: Invariant refers to scalars (tensors of rank 0), but Tensors are 'covariant' not invariant.” And Dalespam notes:
It also depends on what means by "observer dependent". Certainly the numerical values of the components of a tensor can vary when you change frames….[
but the observers WILL agree on the total..] No controversy nor obstacles here.
Is that progress?? It sure helps me, but I don’t know about the OP…where is he??
Here comes the bad news:
So even if you all were to agree with all those descriptions, which would be a miracle, and the ones Pallen described in post #23, we are still faced with the problem of ascribing those descriptions of ‘grid lines’ and ’curved graph paper’ [or some other equivalents] to some mathematical component
of the EFE.
That seems impossible because if my understanding is correct MTW lays out an absolute impass: [from my post #27, quote courtesy of pervect from elsewhere]
MTW:
… nowhere has a precise definition of the term “gravitational field” been
given --- nor will one be given. Many different mathematical entities are
associated with gravitation; the metric, the Riemann curvature tensor, the
curvature scalar … Each of these plays an important role in gravitation
theory, and none is so much more central than the others that it deserves the name “gravitational field.”
