What is the predicted location of gravity in orbiting objects?

[Xilor]

Because gravity travels at the speed of light in GR, there seem to be a few possible places where gravity could pull you towards if dealing with some massive object in some orbit, which one of these is predicted?1. To the position where you see the object (the place where the object used to be)

Or one of these options where you are pulled towards the expected present location of the object:
2. Using the Newtonian perspective of a straight line (wherein orbits are only due to acceleration), using the perspective of 'present location' in your reference frame.
3. Using the GR perspective of a straight line (wherein gravity involves moving along straight lines), using 'present location' in your reference frame.
4. Using Newtonian lines, using 'present location' as seen in some other frame, like that of the object.
5. Using GR lines, using 'present location' as seen in some other frame, like that of the object.

Or:
6. Where it 'actually' is (in some frame?), after accounting for factors like non-gravitational acceleration etc. Aka, some form of instantaneous-like position.It feels there's some potential problems with all of them, so I can't figure this one out nor find any conclusive statements elsewhere, except that it probably isn't option 1.

And followup question, has whatever answer is predicted actually been shown experimentally/through data? I imagine that with the difficulty of measuring gravitational acceleration to one particular object, and with the low velocities in our area of space the minimal differences would actually be rather hard to detect.

 

[A.T.]

Because gravity travels at the speed of light in GR,...

Gravity itself doesn't travel at all. Only disturbances of the space-time geometry propagate. Your question was discussed here many times. See also direction of Coulomb force on charges in relative motion vs. visual position of the charged objects.

 

[Xilor]

Gravity itself doesn't travel at all. Only disturbances of the space-time geometry propagate.

That sounds like a more accurate way of phrasing it, but does by itself not really give an answer.

Your question was discussed here many times.

Yes, but I could only find discussions that said it wasn't option 1. Which of the other options is accurate wasn't something I could find something conclusive on and is actually mostly what I'm interested in.

See also direction of Coulomb force on charges in relative motion vs. visual position of the charged objects.

I'll try to look for that, thanks for the suggestion.

 

[phinds]

Also, just to be sure you are clear, if you are talking about GR, then gravity does not pull you. It is in the nature of things that you travel along a geodesic is spacetime unless there is some force making you do otherwise. In the situation I believe you to be describing, there IS no other force (in fact there is no force at all) so nothing is pulling you, you are simply following a geodesic in spacetime.

 

[Xilor]

Also, just to be sure you are clear, if you are talking about GR, then gravity does not pull you. It is in the nature of things that you travel along a geodesic is spacetime unless there is some force making you do otherwise. In the situation I believe you to be describing, there IS no other force (in fact there is no force at all) so nothing is pulling you, you are simply following a geodesic in spacetime.

Yes I'm clear on that. I used language like that to describe the difference between some of the options (the difference between a Newtonian line and a GR line).

Let me rephrase the original question to be sure we're all clear on what is being asked:Spacetime has curves that influence the way straight lines work. In a space with a curvature that is only influenced by a single mass, my expected vision on standing still (as is of course normal in my frame) involves that the relative velocities of everything I see around me appear impacted by some point 'a', which coincides approximately with a mass I see. My expected view is also that this point 'a' will orbit me. And some possible orbits of 'a' around me are unfortunately so elliptical, that the mass moves along with 'a', will once it reaches a point close to its pericenter have a larger radius than distance to me. And this is the point where I die. All because some mass decided to accelerate into me, while I was just standing still, minding my own business.

However, if I happen to not die, and this mass happens to be moving relative to some other point b, perhaps along the geodesics of a second mass. Then I should expect that the point 'a' moves relative to me in a more complicated manner than a simple orbit around me. And part of this complication is how this point 'a' behaves as compared to the mass I see that mostly appears to align with it. This is the part my question is about.From what I've understood. The answer is not option 1. But I cannot tell which of the other options it is myself.
6. would seem to involve some form of superluminal communication and might involve some 'preferred frame' that decides what the present even is.
2. and 4. Seem unsatisfying as I expect all lines to be like GR lines.
3. and 5. Would appear to imply that spacetime propagations propagate along spacetime lines, which I'm sort of open to but am unsure of. It would for example make me question whether spacetime propagations could even escape a black hole.
4. and 5. Make me worried as I don't expect preferred frames and would not know which frame to pick.
2. and 3. Make me worried as it would seem to imply that someone right next to me at a different velocity, would experience a different point 'a' and could thus in extreme cases have a completely different idea about what spacetime around us is like, and we would seem to anticipate different results.

 

[PeterDonis]

@Xilor You should look at Carlip's paper on aberration and the speed of gravity:

http://arxiv.org/abs/gr-qc/9909087

It explains how "the direction gravity appears to pull things" (in the Newtonian approximation) is almost, but not quite, the same as the direction of the source "now", and how the small differences (i.e., "aberration of gravity") arise from relativistic effects. It also explains why the amount of aberration for gravity is much smaller than the amount of aberration for light (electromagnetism), even though both interactions propagate at the speed of light.

 

[Xilor]

@Xilor You should look at Carlip's paper on aberration and the speed of gravity:

http://arxiv.org/abs/gr-qc/9909087

It explains how "the direction gravity appears to pull things" (in the Newtonian approximation) is almost, but not quite, the same as the direction of the source "now", and how the small differences (i.e., "aberration of gravity") arise from relativistic effects. It also explains why the amount of aberration for gravity is much smaller than the amount of aberration for light (electromagnetism), even though both interactions propagate at the speed of light.

Thanks for the link!

A significant part of that part of that paper is clearly above my level, so I'm not entirely sure if I'm drawing the correct conclusions out here, but what seems to be said in terms of my original question is:

Most of the 'pull' of gravity is targeted at an extrapolated 'instantaneous' position, thanks to some velocity dependent terms. But some of the retarded position still persists as not everything is canceled is out. So basically some combination that is mostly focused on one of the options 2/3/4/5 with a bit of option 1 mixed in. Unfortunately I couldn't really find anything in there that would seem to separate between 2/3/4/5 that well. It does appear to at least address the problem of having possibly different relative velocities by mentioning expected ambiguities of order v², which at least indicates some preference for 2/3.
The Newtonian vs GR line issue does not seem discussed well in the writing either, but a particle traveling along geodesics is mentioned, so perhaps that part is just hidden in some of the maths I can't follow?
From this it would seem to me that out of these options that if anything, this paper appears to be leaning towards option 3 (with some option 1 mixed in). But some of the language and the relatively recent date of this paper suggest to me that this topic might not yet have a clearly defined consensus yet. The paper admits that experimentally very few has been shown about this at least. That experimentally, not even the speed of gravity has been properly shown, so let alone something detailed like the influence of your own velocity compared to the mass.

Is that about right?

 

[PeterDonis]

Most of the 'pull' of gravity is targeted at an extrapolated 'instantaneous' position, thanks to some velocity dependent terms. But some of the retarded position still persists as not everything is canceled is out.

This is basically it, yes.

So basically some combination that is mostly focused on one of the options 2/3/4/5 with a bit of option 1 mixed in. Unfortunately I couldn't really find anything in there that would seem to separate between 2/3/4/5 that well.

I think that points to confusion in the way your options are formulated, not in GR itself. In particular, your concept of "Newtonian vs. GR lines" seems physically meaningless to me. I think you would be better served by taking some time to learn the standard math and terminology of GR so that you could reformulate your questions (if you still have them after learning those things) in standard terminology.

some of the language and the relatively recent date of this paper suggest to me that this topic might not yet have a clearly defined consensus yet.

No. The treatment in the paper is entirely standard GR and there is no dispute or lack of consensus about it.

The paper admits that experimentally very few has been shown about this at least.

That's because it's extremely difficult to do direct experiments to measure the speed of gravity. There are two main reasons for this. First, the source of gravity, stress-energy, obeys a local conservation law that is much more general than, say, conservation of charge for electromagnetism. Plenty of objects have no charge, but everything has stress-energy. So simple thought experiments like "what if we suddenly removed the Sun from the solar system?" don't work--you can't just remove the Sun, it has a huge amount of stress-energy that can't just disappear.

The second reason is that gravity is so weak that you need a huge object, like the Earth or the Sun, to be a significant enough source to measure its effects--but huge objects like that take a huge amount of effort to move. We can't just arbitrarily wiggle the Sun and see what effect that has on the Earth's orbit. And wiggling the masses we can arbitrarily wiggle just doesn't produce enough effect to measure.

So our best bet is indirect measurement, of the sort the paper describes--look for small effects that are due to the fact that, as you say above, not everything cancels out. One such effect is the perihelion precession of planets like Mercury; that would not be there if the speed of gravity were instantaneous. There are others as well, as the paper describes. And that indirect evidence is quite clear: gravity propagates at the speed of light.

 

[Dale]

Unfortunately I couldn't really find anything in there that would seem to separate between 2/3/4/5 that well.

I don't think that your question or the answer options are well formulated in GR. I read your explanation and tried to mentally translate it into GR, but I couldn't.

 

[Xilor]

I think that points to confusion in the way your options are formulated, not in GR itself. In particular, your concept of "Newtonian vs. GR lines" seems physically meaningless to me. I think you would be better served by taking some time to learn the standard math and terminology of GR so that you could reformulate your questions (if you still have them after learning those things) in standard terminology.

I don't think that your question or the answer options are well formulated in GR. I read your explanation and tried to mentally translate it into GR, but I couldn't.

Hmm, interesting. I wouldn't have thought these questions would cause any form of confusion. I must be missing something big somewhere.

Perhaps this makes it more sense, or at least make it easier to see why my thoughts are incompatible with GR thinking:

On the Newtonian vs GR lines:
Because we have to look at an extrapolated 'instantaneous' position, it's nice to know how to extrapolate this. I imagine one option is to simply extrapolate where the mass will be, and take that point. That would naturally be along normal GR lines, so along the orbit of the object.
I imagine another option to be that instead of following the mass, maybe should really be following some center of a propagating spacetime distortion. Since at least some part of the distortion is traveling and must get space-separated from the mass to reach wherever I'm standing, there is already some separation between distortion and mass expected. That the extrapolated center of all these distortions could also end up separated from the actual position of the mass makes sense to me, especially considering the possibility of some acceleration of the mass by some other effect, which I'm assuming has no apparent gravitational effect until you get to see it. All the effects of gravity and GR, would to you then seem positioned around this center, instead of the mass.
So how does this extrapolated point really move? It by itself would not appear to have any mass and is of an entirely different type of 'thing' than the mass. So it would not surprise me if it's movement moves along something more similar to Newtonian-like lines, rather than neatly following the orbit of its origin mass. Basically, does the extrapolation follow orbit lines, or does it do something else?

On the reference frame to be used:
Because finding the extrapolated 'instantaneous' position involves a relative velocity of the mass, and because the nature of what spacetime events are in the 'present' depends on your reference frame/velocity. It would make sense to discuss how SR reference frames play in. I would imagine that with high enough velocity differences, two observers right next to each other could come to different answers about the position of the center of distortion in the present. And if you have different answers, you must apparently expect different spacetime curvatures. Which would impact the expected straight lines of these objects among things. And so I'd imagine the two observers would disagree on how these two observers will move in the future, how much time dilation there is, etc. That seems sort of odd, so I'm wondering if this is in fact the correct interpretation, or if perhaps some other frame should be used, or if there's some other option here. I had previously understood spacetime curvature to be basically invariant, but if velocities and the 'present' become terms, how can this be when these are frame dependent?

 

[Dale]

Perhaps this makes it more sense, or at least make it easier to see why my thoughts are incompatible with GR thinking:

On the Newtonian vs GR lines:
Because we have to look at an extrapolated 'instantaneous' position

In GR there is no physical meaning to this position. The extrapolated instantaneous position can be anything simply by change of coordinates.

Try thinking in terms of invariants, not coordinates. Both "instantaneous" and "position" are coordinate concepts. I would recommend going back to the Carlip paper. If you don't understand it then read Sean Carroll's lecture notes on general relativity. Then go back to the Carlip paper.

 

[Xilor]

In GR there is no physical meaning to this position. The extrapolated instantaneous position can be anything simply by change of coordinates.

Try thinking in terms of invariants, not coordinates. Both "instantaneous" and "position" are coordinate concepts. I would recommend going back to the Carlip paper. If you don't understand it then read Sean Carroll's lecture notes on general relativity. Then go back to the Carlip paper.

Your comment here is exactly the nature of my question! People (including the author of that paper) speak of an 'extrapolated instantaneous position', but I don't know what they mean by that, exactly because of the possibility to have different coordinates. If there's some invariant point, then what is it? The paper uses velocity in some of its formulas, which is not invariant. And it keeps referring to an "instantaneous" position, the paper does also use quotes around that word, to show it isn't really instantaneous, but makes no effort to express in language what exactly is being talked about here. The formulas go over my head, especially since many of the variables remain unexplained, presumably because they are GR jargon unfamiliar to me. I suppose I'll go read those lecture notes and see if that helps.

 

[PeterDonis]

People (including the author of that paper) speak of an 'extrapolated instantaneous position', but I don't know what they mean by that, exactly because of the possibility to have different coordinates.

The "extrapolated instantaneous position" is a heuristic way of describing what's going on, and you are correct that it assumes a particular choice of coordinates (the ones that Carlip used in his paper), as do other terms used in the heuristic description. There is no exact ordinary language description of what is going on; that's why we express physical theories in math, not ordinary language.

I would imagine that with high enough velocity differences, two observers right next to each other could come to different answers about the position of the center of distortion in the present. And if you have different answers, you must apparently expect different spacetime curvatures.

No, just different ways of slicing up spacetime into space and time (because each observer is using the "natural" way of doing that for his motion). Spacetime curvature is independent of how you do that.

One way of trying to shift viewpoints on the "speed of gravity" issue is by analogy with how the "speed of light" issue gets dealt with in GR. In SR the rule is simple: no causal influence can propagate faster than light. But in GR that simple rule doesn't work as it's stated, because there is no way to define a relative velocity between objects that are not co-located, so "faster than light" has no global meaning. Instead, the rule becomes: no causal influence can propagate outside the light cone. Light cones--null surfaces generated by all of the null curves passing through a given event--are an invariant geometric feature of spacetime, so they can serve as an invariant causal boundary.

The "speed of gravity" works similarly: if you look at a particular event in spacetime, all of the effects of "gravity" at that event can only be caused by sources of gravity (i.e., stress-energy) in the past light cone of that event, and whatever is happening to stress-energy at that event can only affect "gravity" in the future light cone of that event. So, for example, if we look at a particular event on the worldline of the planet Mercury, instead of asking "where gravity appears to point", we ask "what sources of gravity are in the past light cone of that event", and of course the main one will be the Sun--more precisely the intersection of the Sun's worldline with the past light cone of that event. The "retarded position" of the Sun in Carlip's paper is basically what that intersection looks like in the coordinates he is using.

The complicating factor, if you are used to Newtonian gravity, is that what is propagating from the Sun in the past light cone of that event on Mercury's worldline is not just a Newtonian force vector--it's more complicated than that. All the talk about "extrapolated position" is one way of trying to describe (some of) the complications, as modeled in a particular coordinate system, in reasonably intuitive language. But ultimately, what Mercury "sees" at a given event on its worldline is a local spacetime geometry which is due to the stress-energy present in the past light cone of that event, and it moves so as to keep its worldline a geodesic in that local spacetime geometry. That is the closest I can come to trying to describe the invariants involved in ordinary language.

 

[Xilor]

The "extrapolated instantaneous position" is a heuristic way of describing what's going on, and you are correct that it assumes a particular choice of coordinates (the ones that Carlip used in his paper), as do other terms used in the heuristic description. There is no exact ordinary language description of what is going on; that's why we express physical theories in math, not ordinary language.

..

All the talk about "extrapolated position" is one way of trying to describe (some of) the complications, as modeled in a particular coordinate system, in reasonably intuitive language.

Alright, I suppose I'll just have to accept that it's just one of those things that make way more sense mathematically than intuitively. Thanks for your explanation.
What it sounds like now: There's not really a 'point' like a spacetime event, as that would be coordinate dependent. Instead we have some complicated invariant effects that show up in the maths, which end up changing things in a way that sort of makes the effects resemble them coming from a different point, but they don't really.
So then say formula (2.3) in the paper, the gammas present there are only there to help identify the 'point' in its local coordinate system, but this point gets canceled out somehow so that in another coordinate system, the effects would still appear in the same invariant way?Funnily enough, it seems someone made a popular reddit thread yesterday about the exact same thing, and the top comment ended up quoting the same paper you did.
https://www.reddit.com/r/askscience/comments/4zwl8e/is_the_earth_pulled_toward_where_the_sun_is_now
Unfortunately, no one seemed to have come up with a great intuitive explanation there either. But it might still be of a little help if someone lands here after googling.

 

[Battlemage!]

Would imagining spacetime as an uncut loaf of bread and individual frames of reference as different angles of cutting the bread be a decent intuitive analogy?

 

[Xilor]

Would imagining spacetime as an uncut loaf of bread and individual frames of reference as different angles of cutting the bread be a decent intuitive analogy?

Oh I have no problems intuitively understanding reference frames and cones of light and all that. The lack of intuitive understanding is just about this particular behavior of GR, which intuitively would seem to depend on the velocity and reference frame of an observer, yet apparently is invariant.

 

[Battlemage!]

Oh I have no problems intuitively understanding reference frames and cones of light and all that. The lack of intuitive understanding is just about this particular behavior of GR, which intuitively would seem to depend on the velocity and reference frame of an observer, yet apparently is invariant.

Ahaha I just meant for my own understanding. GR is above my head.

 

[Xilor]

Ahaha I just meant for my own understanding. GR is above my head.

Ah alright!
I suppose it kind of would be a decent analogy using 4 dimensional bread. The bread would represent spacetime, and a cut could represent a certain time as seen by an observer. The angle of the cut would depend on the velocity of an observer. And the cut that goes right through that observer could be seen as the expected 'present' of that observer. The past and future would be 'cuts' at the same angle that are above/below present.
But make sure to remember that there is not just one objective 'bread'. The combination of all those planes at different times makes its own 'bread'. With two observers traveling at some relative velocity, both would interpret their own bread as the normal way of seeing things, and would think of the other observer as having a cut at some angle.
But that's not perfect either. Two observers right next to each other at different velocities would in fact be seeing the exact same events simultaneously. But if you had just used the bread analogy and angled the 'present', you might've expected in the bread analogy that the 'cone of light' would have neatly angled along with the present, thereby making the observers see something completely different. This is not true, the cone remains the same for two observers at the same place regardless of velocity. Both observers will agree that anything inside of the 'past' cone is in the past and anything inside the 'future' cone must be in the future, but they will not agree by how much these events are in the past/future and can disagree on simultaneity of two events. On anything that's outside of the cones, the observers may even disagree whether something is past/future.

 

[Nugatory]

I suppose it kind of would be a decent analogy using 4 dimensional bread. The bread would represent spacetime, and a cut could represent a certain time as seen by an observer. The angle of the cut would depend on the velocity of an observer. And the cut that goes right through that observer could be seen as the expected 'present' of that observer. The past and future would be 'cuts' at the same angle that are above/below present.

This analogy works pretty well for special relativity (although if you take it too seriously you can get caught up in the sterile block world debate). The geometry of flat spacetime is Minkowski, not Euclidean, but that doesn't mess up the analogy as long as you don't go measuring distances between points inside the loaf of bread.

It doesn't work nearly as well for GR, because in GR the spacetime isn't flat so you can't trust any straight lines that you draw. Even a geodesic, the "straight line" of GR will appear straight when you draw it using some coordinates but not others.

 

[PeterDonis]

There's not really a 'point' like a spacetime event, as that would be coordinate dependent.

Not quite. If you are at a given event on a particular planet's worldline, which event on the Sun's worldline intersects with the past light cone of that given event is invariant, independent of coordinates. The only thing that depends on the coordinates is the coordinates . That is, what coordinates are assigned to that event on the Sun's worldline. That is just a manifestation of the GR version of length contraction and time dilation--how far a particular light ray appears to travel between two given events, and how long it takes (since those are always related by $c$ for a light ray), depends on your choice of coordinates. But it's the same light ray between the same pair of events.

 

[Xilor]

Not quite. If you are at a given event on a particular planet's worldline, which event on the Sun's worldline intersects with the past light cone of that given event is invariant, independent of coordinates. The only thing that depends on the coordinates is the coordinates . That is, what coordinates are assigned to that event on the Sun's worldline. That is just a manifestation of the GR version of length contraction and time dilation--how far a particular light ray appears to travel between two given events, and how long it takes (since those are always related by $c$ for a light ray), depends on your choice of coordinates. But it's the same light ray between the same pair of events.

You're right of course! I should have specified:
If we take 'extrapolated events' to be points in spacetime, then I assume all observers would agree on which spacetime points these hit and would also agree on the spacetime intervals between them. (although they may disagree on coordinates)
They then would be able to draw a sort of 'extrapolated event line' between them on which they could all agree.
However, they would then disagree on what point of this line intersects with the 'present'/'some moment wherein light from the mass is seen'. So which of these events is relevant would change among observers. And on top of that, there would have even been disagreement on the distance between the same event. So there would be coordinate dependency in that way.

 

[PeterDonis]

If we take 'extrapolated events' to be points in spacetime, then I assume all observers would agree on which spacetime points these hit

No, they woudn't. The "extrapolated events" are the "intersections with the present" that you describe further on:

they would then disagree on what point of this line intersects with the 'present'/'some moment wherein light from the mass is seen'.

So different choices of coordinates will result in different "extrapolated events". However, these events, as you define them, play no role in the physical predictions being made. If you look at Carlip's equations, the coordinates of the "extrapolated event" don't appear anywhere. The only event whose coordinates appear is the "retarded event", the event where the Sun's worldline intersects the past light cone, and that event is invariant. When Carlip talks about "extrapolated position", once again, he is talking about a particular way of interpreting the math; but the math itself only depends on invariants. Note that even though Carlip defines "retarded position" and "retarded velocity" 3-vectors, which are coordinate-dependent, the actual math for the "field strength" in general (equation 1.7 for EM and 2.2 for gravity) is expressed in terms of 4-vectors, 4-tensors, etc.; the 3-vector expressions are only extracted to help with interpretation in a particular set of coordinates.

 

[Xilor]

So different choices of coordinates will result in different "extrapolated events". However, these events, as you define them, play no role in the physical predictions being made. If you look at Carlip's equations, the coordinates of the "extrapolated event" don't appear anywhere. The only event whose coordinates appear is the "retarded event", the event where the Sun's worldline intersects the past light cone, and that event is invariant. When Carlip talks about "extrapolated position", once again, he is talking about a particular way of interpreting the math; but the math itself only depends on invariants. Note that even though Carlip defines "retarded position" and "retarded velocity" 3-vectors, which are coordinate-dependent, the actual math for the "field strength" in general (equation 1.7 for EM and 2.2 for gravity) is expressed in terms of 4-vectors, 4-tensors, etc.; the 3-vector expressions are only extracted to help with interpretation in a particular set of coordinates.

Alright, got it! I noticed coordinate-dependent factors in (2.3), and thought that was the more relevant one, but if that's just an extracted version that makes sense. Thank you for your help.