B Gravity and speed of information for orbiting planets

[HansH]

With Newton gravity the orbits of planets or stars can be calculated based on one over the square of the distance giving a force pointing from one mass to the other mass. on the other hand based on the shape of the orbit the corresponding amplitude and direction of gravity (as a vector) can be derived based on F=m.dv/dt so dv/dt=F/m= g.

With Einstein we learned that gravity is not infinitely fast but propagates with the speed of information equal to the speed of light. taking this speed into account with newton however results in gravity vector pointing to a position back in time leading to instable orbits, especially then the masses orbiting eachother both move, such as by 2 stars with aspprox. same mass orbiting eachother.

With Einstein the force of gravity is replaced by a curvature of spacetime giving orbits being free falling trajectories (geodesics). however in thought experiments we could calculate an equivalent gravity vector giving a force and a direction.

Now my question is: if you do that, where is the direction of this vector than pointing to? is this the position of the light coming from the star, so basically the position back in time? or is it something different. my assumption would be that it cannot be the position back in time, because of the instable orbits not being the orbits we have in practice. And also because in daily circumstances, Newton should give almost same results as general relativity, so also based on that the calculated gravity vector must point into the direction of the other star at the current moment insted of the monment back in time when the star sent out the information. (light and gravity)

 

[Dale]

The same situation applies in electromagnetism. Basically in both cases you find the retarded time, and at the retarded time you determine the position and velocity of the source, call that the retarded position and retarded velocity.

The force does not come from either the retarded position or the current position. It comes from the retarded position plus the retarded velocity times the difference between the current time and the retarded time.

 

[HansH]

I have some difficulties in understanding what you are saying: "retarded velocity times the difference between the current time and the retarded time" that sounds like a time x speed=distance. I would expect it is the distance between the source of gravity and the mass that sees the curvature of spacetime.

We also had this discussion at the dutch forum and there I took the next thought experiment; hope that helps to further get clearification.
The sun in its frame of reference and a fast moving rocket with speed v in a straight line along the x axis at distance d from the sun.

the next picture is the situation seen from the rocket frame of reference. from the rocket you see the sun moving toe the left with speed -v

now the question is: what happens with a mass in the rocket at the moment that I let it float, basically allowing it to follow a geodesic. (the rocket goes in a straight line so is forced not to follow a geodesic)
now the question is what is the direction of free falling mass in the rocket? is this according to fz1 pointing to the position of the sun (faint drawing) where it was at the moment it sent out the information of gravity that is received now in the rocket or pointing to some different direction?

 

[Dale]

that sounds like a time x speed=distance

Yes, it is.

I would expect it is the distance between the source of gravity and the mass that sees the curvature of spacetime.

No. It is the distance that the source would travel if it continued at the retarded velocity from the retarded time until the current time.

pointing to some different direction?

It is a different direction. It is the direction from the location I described above.

Basically you start at the retarded position and extrapolate where the source will wind up by moving at the retarded velocity until the current time.

 

[HansH]

So if we talk about constant speed then if I understand you right it is the currernt position of the sun at the moment that the information of the gravity reaches the rocket? so basically what you would also expect with Newton.

 

[Dale]

So if we talk about constant speed then if I understand you right it is the currernt position of the sun at the moment that the information of the gravity reaches the rocket? so basically what you would also expect with Newton.

Yes

 

[HansH]

But if so, is then my conclusion right that spacetime curvature takes care of this effect basically extrapolating for the speed (as I could predict the current position of an object of say 10 lightminutes away, so initially sounds as not causal as I measure gravity at a different direction (in the future) then where the light seems to comes from.)

 

[Ibix]

In the weak field approximation the Einstein field equations take the same form as Maxwell's equations, and you can view the field as providing a "gravitoelectric" and "gravitomagnetic" force that behave the same as the EM forces. The gravitoelectric force points at the retarded position of the Sun (as you guessed), but when you add the gravitomagnetic force you get a resultant that points at where the Sun is now. So orbits are stable.

 

[Ibix]

But if so, is then my conclusion right that spacetime curvature takes care of this effect basically extrapolating for the speed (as I could predict the current position of an object of say 10 lightminutes away, so initially sounds as not causal as I measure gravity at a different direction (in the future) then where the light seems to comes from.)

It remains causal. The force points at where the Sun is now assuming the Sun is in freefall. If I were to accelerate the Sun with a sufficiently powerful rocket, the Earth would continue in its current orbit for eight minutes.

In a sense, the maths works out as pointing at a forecast of where the Sun is now. The forecast can be incorrect if the Sun does something unexpected.

 

[Ibix]

This paper by Carlip does the maths. It's pretty far above B-level, I'm afraid.

 

[Dale]

so initially sounds as not causal as I measure gravity at a different direction

I am not sure how you can possibly think from what I said that there is anything not causal. The direction of the force is determined entirely from the retarded position and the retarded velocity. Please identify exactly where the not causal possibly enters.

 

[HansH]

I am not sure how you can possibly think from what I said that there is anything not causal. The direction of the force is determined entirely from the retarded position and the retarded velocity. Please identify exactly where the not causal possibly enters.

I think as Ibix explains, it remains causal: "pointing at a forecast of where the Sun is now" so based on the behavior of the sun it seems the curvature knows how to curve spacetime at the position of my rocket in the picture such that the force remains pointing towards the extrapolated position of the sun at the moment in the past being the best quess of the position of the sun at the momnent that the rocket receives the information. .

 

[HansH]

The forecast can be incorrect if the Sun does something unexpected.

do you think it is possible at all that the sun does something unexpected. For example suppose the sun is pushed away due to a collapse with a very fast moving other sun then I assume we can exactly predict what happens next, while curvature is also based on that second sun approaching, so I assume curvature of spacetime then also behaves based on that prediction.

And if the sun would instantly explode and all mass being converted into radiation then still nothing would change as the expanding sphere of radiation effectively behaves the same at the mass it was build off until the sphere reachers the earth (or the rocket in my example)

or am I wrong?

 

[HansH]

This paper by Carlip does the maths. It's pretty far above B-level, I'm afraid.

no problem to be asbove the B level. akthough i cannot follow the details I think page 5 shows an interesting result about the basic behavior. as I understand gravity information progresses but toward its position extrapolated from this retarded data. so to my understanding this means nature knows how to progress with its gravity towards my rocket such that I measure a direction as if gravity progresses with infinite speed. But as I cannot follow the details I still don't understand why.

 

[Ibix]

do you think it is possible at all that the sun does something unexpected. For example suppose the sun is pushed away due to a collapse with a very fast moving other sun then I assume we can exactly predict what happens next,

In the weak field approximation the equations are linear, and you could just add up the fields of the two stars. I would expect that you could decompose the movement of the rocket into the effects of the two stars, and that it would act as if the stars had passed through each other until information about the collision travelled to the rocket. I haven't done any maths to check that, though.

Note that I think the weak field approximation assumes that pressure is not a significant contribution to the stress-energy and this may well not be the case during a collision between stars.

And if the sun would instantly explode and all mass being converted into radiation then still nothing would change as the expanding sphere of radiation effectively behaves the same at the mass it was build off until the sphere reachers the earth (or the rocket in my example)

Birkhoff's theorem says that the spacetime outside a spherically symmetric stress-energy distribution is Schwarzschild, so as long as the explosion is spherically symmetric the Earth is unaffected until some of the matter or radiation expands beyond its orbit.

 

[PeterDonis]

So if we talk about constant speed then if I understand you right it is the currernt position of the sun at the moment that the information of the gravity reaches the rocket? so basically what you would also expect with Newton.

Not quite. @Dale included the word "velocity" in his posts for a reason. In the weak field approximation of GR, where we can, with an appropriate choice of reference frame, think of gravity as a "force", it's still not a force that depends solely on position, as with Newton. It also depends on velocity. The velocity-dependent terms in the force compensate for most of the effects of aberration--i.e., of the fact that the interaction travels with a finite speed.

Steve Carlip's classic paper "Aberration and the Speed of Gravity" (I don't have a link handy, but it should be easy to find with a search) discusses this exact question in some detail, including the very helpful comparison with the case of electromagnetism.

 

[Dale]

the force remains pointing towards the extrapolated position of the sun at the moment in the past being the best quess of the position of the sun at the momnent that the rocket receives the information

Yes, and there is nothing that is non-causal in that extrapolation. It is all based on information at the retarded time.

 

[HansH]

The direction of the force is determined entirely from the retarded position and the retarded velocity.

That means in case of a constant velocity such as in my rocket example that the determination fully complies with the current position of the sun. But is it only the position and speed? Or is it also more such as derivatives of the speed as al this is predetermined by the mass and speed of the mass curving spacetime? if that is true it would mean that the force at the position of one mass always points to the current position of the other mass. Also what happens if I suddenly speedup my rocket? that means from the reference frame of the rocket I see the sum instantly move at a different speed to the left. So does the gravity of the sun as felt by the rocket then also instantly points to the new retarded position of the sun? (which is at that moment still the same position of the sun, but 5 minutes later it is further to the left based on the new rocket speed)

 

[Dale]

Or is it also more such as derivatives of the speed as al this is predetermined by the mass and speed of the mass curving spacetime?

I know that for electromagnetism it is just the retarded position and velocity. I cannot remember for gravity if it is also the retarded acceleration. I don't think so, but I am not confident about that.

what happens if I suddenly speedup my rocket?

Then the reference frame is non-inertial and it is not even clear that the question of direction of a force in a non-inertial frame is physically meaningful. At least you would need to be very explicit in what you mean by such words. I.e. you would need to write down some math just to ask the question.

 

[Ibix]

I know that for electromagnetism it is just the retarded position and velocity. I cannot remember for gravity if it is also the retarded acceleration. I don't think so, but I am not confident about that.

The EFEs reduce to the same form as Maxwell's equations in the weak field, so the behaviour in that approximation is obviously identical to the EM case. On the other hand, in the full theory you can't talk about forces or accelerations due to gravity anyway so the question doesn't make sense. Sometimes you can translate Newtonian terms into GR ones (e.g. the Newtonian acceleration due to gravity translates to the proper acceleration of a Killing observer), but I'm struggling to see how to do that in this case, beyond a straightforward assertion of general covariance: switching to a different coordinate system can't change anything physical.

I don't know if there's a slightly-less-than-linearised gravity approximation where you could talk about forces and not be reduced to Maxwell. Others (@haushofer?) might know.

 

[PeterDonis]

is it only the position and speed?

Yes. But heuristically, for electromagnetism, there are only terms linear in the speed, while in the case of gravity, there are terms quadratic in the speed as well, so for gravity the "extrapolation" that is done from the retarded information is more accurate and therefore there is much less aberration.

 

[haushofer]

The EFEs reduce to the same form as Maxwell's equations in the weak field, so the behaviour in that approximation is obviously identical to the EM case. On the other hand, in the full theory you can't talk about forces or accelerations due to gravity anyway so the question doesn't make sense. Sometimes you can translate Newtonian terms into GR ones (e.g. the Newtonian acceleration due to gravity translates to the proper acceleration of a Killing observer), but I'm struggling to see how to do that in this case, beyond a straightforward assertion of general covariance: switching to a different coordinate system can't change anything physical.

I don't know if there's a slightly-less-than-linearised gravity approximation where you could talk about forces and not be reduced to Maxwell. Others (@haushofer?) might know.

I guess one should dive into post Newtonian expansions for this, but that's not my expertise; my world is (was) strictly Newtonian :P

 

[haushofer]

I just bumped into this short note "Newtonian Gravity with a Retarded Potential",

http://kirkmcd.princeton.edu/examples/retarded.pdf

but haven't read it myself yet. It writes down the potential a la Lienard Wiechert and derives the resulting force from it. I'm not sure exactly how this expression (2) relates to the usual Newtonian limit, but maybe it's useful for @HansH :)