Geodesic motion from the EFE

[TrickyDicky]

In order to clarify what the EFE tells us about geodesic motion, it is important to remember that by the local flatness theorem, we can at any point p introduce a coordinate system (Riemann normal coordinates) so that the first derivatives of the metric at that point vanish.
We can choose to use them at the COM of a massive body, say a neutron star in a binary system with two stars of similar mass. Now the definition of COM in GR may not be as straight forward as in Newtonian theory but there are references provided in this forum that show that it is defined in GR and it is perfectly valid to use it.
http://www.springerlink.com/content/mg846n70582873n8/
http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0456v1.pdf

So let's imagine this orbiting neutron star and that we could record its path in video and then plug it into a computer that traces the path of the COM of the star as a stream of dots, each representing a snapshot of the star's COM at a certain moment t. By the property of the vanishing of the metric first derivative and the Christophel connection at that point of the manifold we make the gravitational field vanish at that point. We can perform that operation at everypoint of the path described by the neutron star so that we define a line where every point is a momentarily comoving frame or a local lorentz frame.
Is that line not a geodesic in curved spacetime?

 

[atyy]

According to this argument, wouldn't every wordline be geodesic?

 

[Mentz114]

...local lorentz frame.

If you mean a frame that experiences no proper acceleration - then yes.

$$\nabla_\nu V_\mu V^\nu = 0$$

is the definition of a geodesic.

I'm not so sure about the COM frame of the orbiting neutron star. I suppose neutron stars are pretty hard so it is more plausible than if they were gello stars.

 

[TrickyDicky]

According to this argument, wouldn't every wordline be geodesic?

I should have added in the absence of any non-gravitational force.

 

[atyy]

If you aren't assuming that the worldline is geodesic, then the local flatness theorem doesn't show that the worldline is geodesic, since one can have Riemann normal coordinates anywhere (http://arxiv.org/abs/1102.0529, section 8, see comments after Eq 8.9). To distinguish between geodesic and non-geodesic worldlines using "special" coordinates, you could try Fermi normal coordinates. These have vanishing Christoffel symbols on the worldline only if it is geodesic (section 9.1-9.5 , see comments after Eq 9.16).

 

[TrickyDicky]

If you aren't assuming that the worldline is geodesic, then the local flatness theorem doesn't show that the worldline is geodesic, since one can have Riemann normal coordinates anywhere (http://arxiv.org/abs/1102.0529, section 8, see comments after Eq 8.9). To distinguish between geodesic and non-geodesic worldlines using "special" coordinates, you could try Fermi normal coordinates. These have vanishing Christoffel symbols on the worldline only if it is geodesic (section 9.1-9.5 , see comments after Eq 9.16).

I'm not asuming that the worldline is geodesic, the example shows a way to construct a path made of a succesion of points that are local lorentz frames momentarily (the moment the NS COM passes thru them). By what I read in your reference, my example fulfills the properties demanded to Fermi local coordinates, which are just local coordinates adapted to a geodesic so that the local metric is Euclidean (first derivatives of the metric vanish), all Christoffel symbols vanish and all this for infinitesimal lengths of time(mometarily comoving frame of the COM) . According to the EFE in this scenario the path of the NS COM is non-accelerated and therefore its proper time is the longest possible (follows extremal path by the variational principle). Under these circumstances the question was whether the path of the example's NS is geodesic.
A plausible consequence is that an extense body in this circumstance has to spin around an axis wth the COM in its center to keep the angular momentum conservation, which happens to be the case in all known planets and stars.

 

[PAllen]

I want to add that I am very interested in the answer to this question, or any relevant material a real expert here could point to on it.

I will add a slightly different slant to what trickydicky has said, because I have no doubt that gravitational waves are prediction of GR, but I think the question is still just as significant, and an hour or so of searching led me to no clue as to an accepted answer, let alone any analysis of the question.

The slant I would take is as follows: You have two compact bodies of similar mass co-orbiting. I completely agree that COM of each should be well defined in this case (and maybe more generally). As each moves, its influence on the metric 'moves', and this change takes finite time to propagate to the other body (no instant action at a distance in GR). From this alone (finite propagation time), it is possible to prove that the two bodies (by themselves) don't exactly conserve angular momentum (they come very close because of velocity *and* acceleration dependent 'potential' effects, as described in Steve Carlip's paper on aberration and the speed of gravity). Also, conservation of momentum and energy fail (of the two bodies considered in isolation). Trickydicky's proposal that the rotation of the bodies can compensate doesn't work. The solution provided by GR is gravitational radiation, which carries away the lost conserved quantities, leading to complete conservation when the GW is included in the system. Note, on an even more elementary level, you have a mutual orbit, with each source moving periodically, its effects on the metric propagating with finite speed, thus finite speed propagation of periodic changes to the metric. What to call these if not GW?

Now, it is obvious that this state motion cannot be described in terms of geodesic motion in some *background* geometry (which, of course, would exclude GW). However, trickydicky was never really claiming this, there was just mutual mis-understandings and lack of precision in expression.

However, admitting GW, the question still remains: there is some complete spacetime geometry that is an exact solution of the similar mass two body problem in GR (though we can't specify it analytically). Do the COM's of the co-orbiting bodies follow geodesics of this exact solution?

A key, relevant point, is that a paper linked a different thread by physicsmonkey stated that the answer for the extreme mass ratio situation, where all the GW can be attributed to the smaller body (which inspirals into the larger), is *yes*, it does follow a geodesic of the total geometry (that includes the GW)). This is at least suggestive that the answer could be yes for the similar mass case (possibly with simplifications for compact bodies that can sufficiently characterized by their COM and mass and angular momentum).

 

[TrickyDicky]

The slant I would take is as follows: You have two compact bodies of similar mass co-orbiting. I completely agree that COM of each should be well defined in this case (and maybe more generally). As each moves, its influence on the metric 'moves', and this change takes finite time to propagate to the other body (no instant action at a distance in GR). From this alone (finite propagation time), it is possible to prove that the two bodies (by themselves) don't exactly conserve angular momentum (they come very close because of velocity *and* acceleration dependent 'potential' effects, as described in Steve Carlip's paper on aberration and the speed of gravity). Also, conservation of momentum and energy fail (of the two bodies considered in isolation).Trickydicky's proposal that the rotation of the bodies can compensate doesn't work. The solution provided by GR is gravitational radiation, which carries away the lost conserved quantities, leading to complete conservation when the GW is included in the system. Note, on an even more elementary level, you have a mutual orbit, with each source moving periodically, its effects on the metric propagating with finite speed, thus finite speed propagation of periodic changes to the metric. What to call these if not GW?

Let's remember that if we understand (as it is usually demanded in GR) matter and energy and their interactions and gravitational influences, in a geometrical way, since indeed there is no action at a distance, perhaps is not correct either to talk about "propagation" of something that is inserted in the geometry( however changing it may be), because expressing in terms of propagation seems to be reminiscent of "action at a distance", when in GR what changes is the dynamical geometry, dictated by the gravitational sources. Unlike the case of the propagation of EM waves where the sources of radiation do not affect the geometry so that they act on a fixed background.
It's worth remembering also that GW are a theoretical consequence of the linear approximation of GR, not the non-linear regime we are considering here.

Now, it is obvious that this state motion cannot be described in terms of geodesic motion in some *background* geometry (which, of course, would exclude GW). However, trickydicky was never really claiming this, there was just mutual mis-understandings and lack of precision in expression.

Right, I always referred to the spacetime of the full exact nonlinear GR solution for the sources, not the background of the linear approximation.

I'd like to mention some objections made in other threads to the geodesic motion of objects with very diferent masses: that initially parallel geodesics at the same point are the same geodesic (because the geodesic equation is a second-order differential equation, and there are uniqueness theorems for the solutions of such equations) and also that the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics.
The answer to these is easy, to the unique solutions of the second-order differential geodesic equation, it is obvious that the uniqueness comes from providing both a unique initial position, which it is given here and also a unique initial direction vector u, that is not the case here since for two different masses we would have two different initial directions from the common point. So this is one reason they don't have to be the same geodesic, but the main reason is that in GR we are dealing obviously with non-euclidean geometry and in such geometries the fifth postulate of Euclid is no longer valid and therefore it is possible to have more than one parallel geodesic with a common point, nothing prevents two geodesics parallel at the same point to diverge and both be geodesics.

 

[Sam Gralla]

I want to add that I am very interested in the answer to this question, or any relevant material a real expert here could point to on it.

I believe I qualify as an expert on the motion of bodies in general relativity, but I'm having trouble discerning a well-defined question from the posts in this thread. I will say from the outset that the "center of mass" of a body (if you succeed in defining it) will not follow a geodesic of the exact metric. The motion of a finite size body will depend in detail on what the body is made of, and there is no reason to expect some simple property like geodesic motion. What is true is that in the limit where the body is infinitesimally small it will follow a geodesic of the "metric without the body". This was established under some restrictive assumptions by Geroch and Jang, and later established firmly by Geroch and Ehlers (the latter is available on the arXiv). It has also been shown in one of my own papers.

You can't expect geodesic motion (or any simple motion) except in the limit of small size.

 

[bcrowell]

I believe the paper Sam Gralla is referring to is this: http://arxiv.org/abs/gr-qc/0309074v1

 

[TrickyDicky]

I believe I qualify as an expert on the motion of bodies in general relativity, but I'm having trouble discerning a well-defined question from the posts in this thread.

That is quite puzzling, specific questions in quite well-defined settings have been formulated in several posts in this thread. You can say you have chosen to ignore them, which is a valid position too.

I will say from the outset that the "center of mass" of a body (if you succeed in defining it) will not follow a geodesic of the exact metric. The motion of a finite size body will depend in detail on what the body is made of, and there is no reason to expect some simple property like geodesic motion.

Maybe the problem is that the notion of geodesic motion is being restricted to the inertial paths of flat spacetime, in that sense it is of course true that in a curved spacetime, absolutely no body of finite mass and size describes geodesic motion "sensu stricto" but that it can be approximated in the linearized GR when there is a big difference of mass wrt the second body of reference. But then again this thread refers to the non-linear exact GR proper. In this case the geodesic motion of the linearized approach loses its meaning and might be a self defeating concept for real finite mass bodies. Maybe I should then specify and say that my example in the OP refers to the geodesic equation of GR, and according to the way it is set up it certainly fulfills the equation in the context of curved of spacetime, even if it doesn't follow geodesic motion in the context of a perturbation of the flat background metric of the linearized approach to GR.

What is true is that in the limit where the body is infinitesimally small it will follow a geodesic of the "metric without the body".

This is trivially true, of course.

 

[TrickyDicky]

One of the important differences between geodesics in curved spacetime and geodesics in Minkowski spacetime as a background approximate metric, is that in flat spacetime both the initial conditions (initial point and initial velocity vector) are fixed, but this is a property of Minkowski spacetime, in curved spacetime there is more than one geodesic path between any two events, which one of them is a certain body going to follow is certainly going to depend on its particular conditions of mass-energy and may be different for different bodies with different masses. In the linearized approach there is only one geodesic, and of course requires test particles in the limit of low mass. That is how linearized GR works. And it gives good approximations for many problems.
So the specific question remains, does the COM of the NS of the OP follow a geodesic in curved spacetime in non-linear GR?

 

[bcrowell]

I will say from the outset that the "center of mass" of a body (if you succeed in defining it) will not follow a geodesic of the exact metric.

As a concrete way of seeing this, here is an equation for the rate at which a spinning body's momentum deviates from geodesic motion:
$$dP^a/ds = -(1/2)u^bS^{cd}R^a_{bcd}$$
Papapetrou, Proc. Royal Soc. London A 209 (1951) 248. The relevant result is summarized in MTW, p. 1121.

Since the binary pulsar as a whole has angular momentum, the center of mass of the entire system will not follow a geodesic. The centers of mass of the individual stars do not follow geodesics either. (They don't follow geodesics since they're spinning themselves, and even if you could stop their spin, I don't see how they could both follow geodesics if their common center of mass didn't.)

A key, relevant point, is that a paper linked a different thread by physicsmonkey stated that the answer for the extreme mass ratio situation, where all the GW can be attributed to the smaller body (which inspirals into the larger), is *yes*, it does follow a geodesic of the total geometry (that includes the GW)). This is at least suggestive that the answer could be yes for the similar mass case (possibly with simplifications for compact bodies that can sufficiently characterized by their COM and mass and angular momentum).

As you scale down the mass m of the satellite, the power radiated in gravitational waves drops like $m^3$. In the limit of very small m, there are no gravitational waves, so the issue disappears. This is essentially the reason why the Ehlers-Geroch theorem is true.

 

[TrickyDicky]

As a concrete way of seeing this, here is an equation for the rate at which a spinning body's momentum deviates from geodesic motion:
$$dP^a/ds = -(1/2)u^bS^{cd}R^a_{bcd}$$
Papapetrou, Proc. Royal Soc. London A 209 (1951) 248. The relevant result is summarized in MTW, p. 1121.

As I've explained if you restrict the concept of geodesic path to that of the flat background metric of linearized GR, it is straightforward that no massive body can follow geodesic motion, and obviously a test particle at the low mass size and mass limit can't spin so it is obvious that according to your restricted definition a spinning body has to deviate from geodesic motion.

Since the binary pulsar as a whole has angular momentum, the center of mass of the entire system will not follow a geodesic.

This is not even being discussed here.

The centers of mass of the individual stars do not follow geodesics either. (They don't follow geodesics since they're spinning themselves, and even if you could stop their spin, I don't see how they could both follow geodesics if their common center of mass didn't.)

Independently of whether or not the COM of the star follows your idea of a geodesic, it certainly is true that a massive rigid body (like all known planets and stars) will either spin around its COM in order to keep angular momentum conservation or fracture itself due to the tidal forces (i.e. shoemaker-levy comet), or deform if it's not so rigid. In the case of the neutron stars they're prety massive and compact bodies so they spin really fast.

 

[Samshorn]

I'd like to mention some objections made in other threads to the geodesic motion of objects with very diferent masses: that initially parallel geodesics at the same point are the same geodesic... The answer to these is easy... it is obvious that the uniqueness comes from providing both a unique initial position... and also a unique initial direction vector u, that is not the case here since for two different masses we would have two different initial directions from the common point. So this is one reason they don't have to be the same geodesic...

That's a non-sequitur, because the objection (as you stated it) stipulated "initially parallel geodesics at the same point", which signifies that they have the same initial position AND the same initial direction - that's what "parallel" means. (Hopefully it goes without saying that we are talking about directions in spacetime, not in space.)

...but the main reason is that in GR we are dealing obviously with non-euclidean geometry and in such geometries the fifth postulate of Euclid is no longer valid and therefore it is possible to have more than one parallel geodesic with a common point, nothing prevents two geodesics parallel at the same point to diverge and both be geodesics.

No, that's incorrect. In any smooth manifold with a well-defined metric there is a unique geodesic path through a given point in a given direction. This is just as true in a curved manifold as it is in a flat manifold. Perhaps what you are (mis)remembering is the fact that, in hyperbolic geometry (for example) there can be infinitely many distinct geodesics through a given point that never intersect with a given "line". But those geodesics are not parallel to each other at that common point.

 

[TrickyDicky]

That's a non-sequitur, because the objection (as you stated it) stipulated "initially parallel geodesics at the same point", which signifies that they have the same initial position AND the same initial direction - that's what "parallel" means. (Hopefully it goes without saying that we are talking about directions in spacetime, not in space.)

Precisely , have you heard about parallel transport of the velocity vector in GR and how it is not possible to keep the initial direction in a unique way? Well, that amounts to saying you can't really introduce the initial direction in a curved spacetime setting the same way you can do it in Minkowskian spacetime of SR and this is what I meant,that you can't introduce a unique initial direction, because for two bodies with different masses you can't assure that after a certain amount of time they will keep the same vectorial direction in a curved spacetime.

No, that's incorrect. In any smooth manifold with a well-defined metric there is a unique geodesic path through a given point in a given direction. This is just as true in a curved manifold as it is in a flat manifold. Perhaps what you are (mis)remembering is the fact that, in hyperbolic geometry (for example) there can be infinitely many distinct geodesics through a given point that never intersect with a given "line". But those geodesics are not parallel to each other at that common point.

You are again conflating a flat manifold (like minkowski spacetime) with the fact that the definition of manifold demands that it must be locally Euclidean, but as soon as you consider a finite size entity you get deviation from Euclidean geometry , and of course in a non-euclidean geometry (however in elliptic geometry there are no parallel lines at all) such as hyperbolic geometry you can have infinite parallels intersecting at a point. Look it up in any non-euclidean geometry book.

 

[TrickyDicky]

Summarizing, there seems to be a distinction between geodesics in the weak field GR approximation background flat spacetime and geodesics in curved spacetime that led to some confusion and mutual misunderstanding.
It is obvious that geodesic motion defined from the linearized setting only admits one geodesic between two given events and is always referred to test particles understood as particles in the low mass (and size) limit, and described in the papers linked in this and related threads.

It is also true that geodesic motion in curved spacetime as understood in GR contemplates the fact that there can be more than one geodesic path between two given distant events (variational principle approach to the EFE) , each one of them could be a local extremal path (proper time path-dependence of a each particular massive body).

Under this last meaning it would be licit IMO to consider that the COM of the orbiting neutron star of the OP (BTW the COM is not spinning) follows one possible particular geodesic path between some arbitrary spacetime events (local extremal path), certainly not the one it would follow if the body was a test particle or had different mass and not necessarily the maximum global geodesic path for the particular global spacetime curvature of the manifold.
This can be extended to the worldlines of any object not acted upon by external forces (non-gravitational), i.e. not measuring any acceleration in an accelerometer located at its COM.
I think this is a specific enough statement so that either non-experts or relativity experts can give their opinion. (I include in this last group the SR permanent team (JesseM, DaleSpam, etc )

 

[bcrowell]

That's a non-sequitur, because the objection (as you stated it) stipulated "initially parallel geodesics at the same point", which signifies that they have the same initial position AND the same initial direction - that's what "parallel" means. (Hopefully it goes without saying that we are talking about directions in spacetime, not in space.)
[...]
No, that's incorrect. In any smooth manifold with a well-defined metric there is a unique geodesic path through a given point in a given direction. This is just as true in a curved manifold as it is in a flat manifold. Perhaps what you are (mis)remembering is the fact that, in hyperbolic geometry (for example) there can be infinitely many distinct geodesics through a given point that never intersect with a given "line". But those geodesics are not parallel to each other at that common point.

And all of this has already been explained to TrickyDicky recently in another thread.

 

[TrickyDicky]

And all of this has already been explained to TrickyDicky recently in another thread.

And clarified to Samshorn in post #16, I guess you haven't read it.

 

[Samshorn]

And clarified to Samshorn in post #16, I guess you haven't read it.

The path-dependence of parallel transport in curved manifolds quite obviously does NOT imply non-uniqueness of geodesics from a given point in a given direction. Your beliefs are completely bonkers.

 

[TrickyDicky]

it is possible to have more than one parallel geodesic with a common point, nothing prevents two geodesics parallel at the same point to diverge and both be geodesics.

In any smooth manifold with a well-defined metric there is a unique geodesic path through a given point in a given direction. This is just as true in a curved manifold as it is in a flat manifold. Perhaps what you are (mis)remembering is the fact that, in hyperbolic geometry (for example) there can be infinitely many distinct geodesics through a given point that never intersect with a given "line". But those geodesics are not parallel to each other at that common point.

Maybe here is a source of misunderstanding, when I responded to this in post #16, I was only referring to the first two sentences of the paragraph, I should have quoted only those.
I thought it was obvious that when I spoke about parallels (both in the original post-first quoted here and in post 16) intersecting at a point I meant parallels to another line, not to each other. The point I was trying to make was clearly about geodesics, not about parallels themselves, but to make it it was necessary to remember the non-validity of Euclid's fifth postulate in hyperbolic geometry.

The path-dependence of parallel transport in curved manifolds quite obviously does NOT imply non-uniqueness of geodesics from a given point in a given direction. Your beliefs are completely bonkers.

Only I have never put it in those terms, I always specified that if the direction is fixed, as in flat spacetime, there is only one geodesic. It is only when you compute the path between two events, as in giving boundary conditions instead of fixing completely the initial conditions that you can have more than one geodesic in curved spacetime. We are talking about a scenario in which different bodies may clearly have different momenta with different velocity vectors at a given location.

So clearly Samhorns is trying to troll this thread by misinterpreting me, besides gratuitous insulting remarks not allowed in this site.

 

[Samshorn]

I always specified that if the direction is fixed... there is only one geodesic... We are talking about a scenario in which different bodies may clearly have... different velocity vectors at a given location.

You contradict yourself. The direction of a path in spacetime is determined not just by the spatial direction but also by the velocity. When people say the geodesic path through a given point in a given direction is unique, they aren't just talking about the direction in space, they are talking about the direction of the path in spaceTIME, which signifies that both the velocity and the spatial direction are given. That's why I said in my first message "hopefully it goes without saying that we are talking about directions in spacetime, not just in space".

 

[TrickyDicky]

You contradict yourself. The direction of a path in spacetime is determined not just by the spatial direction but also by the velocity.

There's no contradiciton, you are just still missing my point. I'm pointing out the difference between straight lines in flat spaceTIME and geodesics in curved spaceTIME.

When people say the geodesic path through a given point in a given direction is unique, they aren't just talking about the direction in space, they are talking about the direction of the path in spaceTIME, which signifies that both the velocity and the spatial direction are given.

So what? I'm not disagreeing with this.

 

[Samshorn]

Originally Posted by Samshorn: "When people say the geodesic path through a given point in a given direction is unique, they aren't just talking about the direction in space, they are talking about the direction of the path in spaceTIME, which signifies that both the velocity and the spatial direction are given."

So what? I'm not disagreeing with this.

You are both agreeing and disagreeing. First you say that "if the direction is fixed... there is only one geodesic..", which is true, but then you say that two geodesics passing through a given point in a given direction may subsequently diverge. Those two claims are mutually self-contradictory. When this is pointed out, you answer that the two paths can diverge because they may have different velocities, but then we explain to you that this is a misconception, because if they have different velocities they are not moving in the same direction. You have stipulated that they are moving in the same direction, but then you say they are not moving in the same direction. But then you say they are moving in the same direction, and then you say they are not... Etc. Etc. And then you say you are not contradicting yourself.

If you agree that there is a unique geodesic path through any given point in any given direction, then you agree with what everyone has been telling you.

 

[TrickyDicky]

I never said two geodesics with the same direction may diverge, I never stipulated that, it was put in those terms in the objections I referred first.
Please , go troll elsewhere.

 

[Samshorn]

I never said two geodesics with the same direction may diverge...

Here's a verbatim quote: " ...nothing prevents two geodesics parallel at the same point to diverge and both be geodesics." But now you agree that your statement was nonsense, and my correction was accurate. You're welcome.

 

[TrickyDicky]

Here's a verbatim quote: " ...nothing prevents two geodesics parallel at the same point to diverge and both be geodesics." But now you agree that your statement was nonsense, and my correction was accurate. You're welcome.

Yeah, sure, you're right, thanks, bonkers is the word, right?