What is the nature of the force causing deviation from a circular orbit in GR?

[Buzz Bloom]

I thought I understood the following, but recently I am having doubts about my understanding.

https://www.physicsforums.com/threads/is-a-2-body-elliptical-orbit-stable-in-gr.898258/
@PAllen said:
I don't believe any orbital system is stable in GR due to gravitational radiation. The time scale may be enormous, but all orbits decay.​

Consider two spherically symmetric bodies of equal mass in an otherwise empty space in orbits about the point halfway between them. Suppose the initial conditions are such that Newtonian gravity would predict circular orbits. I interpret PAllen's quote to mean that GR predicts that the orbits would decay following some kind inward spiral paths while emitting gravitational waves.

Q1: Does anyone disagree with this?

There are two possibilities:
a: The circular orbits are GR geodesics.
b: The spirals are GR geodesics.

Q2: Which is correct? My guess is (b), although I have doubts.

If (a) some force must cause the bodies to deviate from the circular geodesics as explained by @gnnmartin.

https://www.physicsforums.com/threads/is-gravity-really-a-force.917446/page-4
Once you realize that space and time are linked, and that space/time is not necessarily flat, then you need to refine Newton's definition. It is always possible to describe local smooth space/time as flat space with time the same everywhere, and if an object is not acted on by a force when space/time is so described, then it is moving along a timelike geodesic, hence we can define force as that which causes a body to deviate from a timelike geodesic. With this definition, gravity is not a force.
​

Q3: If (a) is correct, what is the nature of the force that causes the deviation from a circle? Does this force have a name?

 

[Dr Aaron]

In "real" space, even in the absence of mass there would be no absence of field effects that would alter even a circular orbit that would otherwise be a potential geodesic course. Field effects in space would always alter any orbit as an external entity, masking the inherent nature of a theoretical orbit. This effect is so powerful that one of the limits to the lifespan of any satellite NASA places in orbit is the amount of fuel required for "station-keeping" for its functional parameters, to correct deviations for perturbing forces. Whether in macro-space or micro-space, you can't find an area without field effects. Is that what you are looking for?

 

[Vitro]

@Buzz Bloom, sorry I don't have a reference to point to, but I think in GR only so called "test objects" with negligible mass really follow geodesics, objects with significant mass in curved spacetime do not exactly.

Edit: negligible when compared to the mass causing that curvature, like the mass of a satellite compared to the mass of the Earth.

 

[pervect]

I thought I understood the following, but recently I am having doubts about my understanding.

https://www.physicsforums.com/threads/is-a-2-body-elliptical-orbit-stable-in-gr.898258/
@PAllen said:
I don't believe any orbital system is stable in GR due to gravitational radiation. The time scale may be enormous, but all orbits decay.​

Consider two spherically symmetric bodies of equal mass in an otherwise empty space in orbits about the point halfway between them. Suppose the initial conditions are such that Newtonian gravity would predict circular orbits. I interpret PAllen's quote to mean that GR predicts that the orbits would decay following some kind inward spiral paths while emitting gravitational waves.

Q1: Does anyone disagree with this?

I have some concerns about the two bodies being spherically symmetrical. They'd naturally tend not to be, due to tidal effects.

Basically, the idea of spherically symmetric bodies is an idealization of rigid bodies, actual bodies are more complex, , and rigid bodies are an approximation that works in some applications, but GR isn't generally one of those applicatoins.

If one is able to find some notion of the center-of-mass of the extended bodies (this would be easy if they were spherical,but I don't think they would be), one can ask if this idealized center-of-mass follows a geodesic or not.

I haven't actually done this, and I suspect there are some thorny issues in defining a 'center of mass" at all. Part of the issue is that I'm not really familiar with modelling the motion of extended bodies in GR. This is certainly something that can be and has been done, I've even seen papers that discuss this. The author Dixon comes to mind, and google finds https://www.jstor.org/stable/2416466?seq=1#page_scan_tab_contents. But I'm not familiar with the details. It's rather an advanced topic.

To try and answer some of your questions, though - if you were able to somehow identify some particular point - a center of mass of some sort - to describe the motion of an extended body, the motion of this center-of-mass point would most likely not follow a geodesic in the case of a pair of bodies of equal mass orbiting each other. (I can't say that I'm absolutely certain of this, since I haven't performed any calculations that would be so sophisticated, however).

The rest of your question appears mostly to be an effort to fit GR into the familiar mold of rigid bodies, point masses, forces acting on these idealized point masses, and the resulting ordinary differential equations. Unfortunately, the basic answer to this is that you can't force GR into such a narrow mold, you need to learn new techniques to understand the theory.

There is some possibility of forcing the equations of GR into a model based on partial differential equations, for whatever it's worth. The heart of GR are the field equations ad those are just partial differential equations. This may not help if you've never studied partial differential equations, however.

 

[Buzz Bloom]

Is that what you are looking for?

Hi Dr Aaron:

If I understand you correctly, you are saying that (a) is rhe correct answer to (2).

Regarding (3) are you also saying that "field effects" are the source of forces that cause deviation from circular orbits?

I understand that that there are many ubiquitous EM fields in our universe which produce an EM force which could cause such deviations from circular orbits if the bodies are made of baryonic matter. Are you saying that if the bodies were made of dark matter, and no matter what the dark matter might consist of, there would always be some kind of "field effect" that would cause this kind of deviation?

Regards
Buzz

 

[Dr Aaron]

I cannot speak to what dark matter bodies respond or fail to respond to. I do wish I had that type of insight. However, while all orbits decay and produce a spiral effect, if not countered, there are so many types of field (and other) effects that affect satellite motion that we can not really ascertain what the orbits would be like without these effects. Gravitational/EM fields are generated by the Sun, Earth, the moon, Saturn, van Allen belts, anomalies in our planet's structure, solar wind effects, atmospheric drag, and so on that we have nowhere to test orbits without these influences. And that would incorrectly presume we understand all forces within our heliosphere. I would tend to believe that satellites are likely to follow geodesic orbits but only as influenced by the total forces acting on the spacecraft , along with the sum effects acting upon space-time itself. For this reason, I would tend to lean toward (b) being the more correct answer for question one. Still, we can't identify all forces acting on a spacecraft so we cannot assert this as an indisputable finding. This answer would relate more to macro objects such as physical bodies and satellites rather than the micro objects like sub-atomic particles, which I am not the best person to ask about. The external forces acting on an orbital object, however, would be unlikely to generate gravitons, if one believes these truly exist.

 

[PAllen]

I thought I understood the following, but recently I am having doubts about my understanding.

https://www.physicsforums.com/threads/is-a-2-body-elliptical-orbit-stable-in-gr.898258/
@PAllen said:
I don't believe any orbital system is stable in GR due to gravitational radiation. The time scale may be enormous, but all orbits decay.​

Consider two spherically symmetric bodies of equal mass in an otherwise empty space in orbits about the point halfway between them. Suppose the initial conditions are such that Newtonian gravity would predict circular orbits. I interpret PAllen's quote to mean that GR predicts that the orbits would decay following some kind inward spiral paths while emitting gravitational waves.

Q1: Does anyone disagree with this?

There are two possibilities:
a: The circular orbits are GR geodesics.
b: The spirals are GR geodesics.

Neither are geodesics. In GR, only test bodies exactly follow geodesics. Specifically, geodesic motion follows from the field equations in the limit of ever smaller bodies against a given background geometry, with further technical assumptions being required (e.g. the dominant energy condition). Look up a series of papers by Gralla and Wald on the latest and greatest version of derivations of geodesic motion in the limit (only) [their proofs formally use that motion must be timelike, not necessarily geodesic, to derive the geodesic limit for zero size body limit; but they footnote prior work that to prove timelike motions for bodies from the EFE, the dominant energy condition is required; thus their proofs, though more elegant, have the same axiomatic requirements as the earlier work of Geroch. ]

Q2: Which is correct? My guess is (b), although I have doubts.

If (a) some force must cause the bodies to deviate from the circular geodesics as explained by @gnnmartin.

https://www.physicsforums.com/threads/is-gravity-really-a-force.917446/page-4
Once you realize that space and time are linked, and that space/time is not necessarily flat, then you need to refine Newton's definition. It is always possible to describe local smooth space/time as flat space with time the same everywhere, and if an object is not acted on by a force when space/time is so described, then it is moving along a timelike geodesic, hence we can define force as that which causes a body to deviate from a timelike geodesic. With this definition, gravity is not a force.

​

This statement is only true within the approximation of geodesic motion.
​

Q3: If (a) is correct, what is the nature of the force that causes the deviation from a circle? Does this force have a name?

No force is needed. The orbital motion leads to a dynamic geometry, and the bodies interact with the dynamic geometry so as to radiate away momentum and angular momentum.

 

[phinds]

... dark matter bodies ...

I'm pretty sure such a concept is internally contradictory. The processes that allow/cause planets and stars to form do not apply to DM

 

[Buzz Bloom]

In GR, only test bodies exactly follow geodesics.

Hi Paul:

Thank you very much for this concept which is new to me.

Based on this concept it would seem that the quote of @gnnmartin is not correct. Or am I misinterpreting something?

It is always possible to describe local smooth space/time as flat space with time the same everywhere, and if an object is not acted on by a force when space/time is so described, then it is moving along a timelike geodesic, hence we can define force as that which causes a body to deviate from a timelike geodesic.

Regards,
Buzz

 

[Buzz Bloom]

I'm pretty sure such a concept is internally contradictory. The processes that allow/cause planets and stars to form do not apply to DM

Hi phinds:

I understand that DM cannot form "bodies" in the same manner (by radiating photons) that baryonic matter does. However, since at the present time the true nature of DM is not known, what it might be made of remains only speculations. I have read somewhere (?) some speculations about the possibility that DM might lose energy by means of weak force interactions. If it turns out some day that this is found to be true, would you then agree that it should be possible for DM to form bodies eventually, although it might take a very long time?

Regards,
Buzz

 

[Dr Aaron]

But isn't dark matter's gravitational influence thought to have provided an "assist" to the early formation of baryonic structures, including stars and galaxies?

 

[mfb]

But isn't dark matter's gravitational influence thought to have provided an "assist" to the early formation of baryonic structures, including stars and galaxies?

For galaxies. It is slow enough to clump on the scale of galaxies via gravity alone. It doesn't form smaller clumps.

If it turns out some day that this is found to be true, would you then agree that it should be possible for DM to form bodies eventually, although it might take a very long time?

It will fall into black holes on much shorter timescales.

Then there's Hubble expansion acting to increase the space between any two objects.

Not if the objects are gravitationally bound.

 

[Buzz Bloom]

Then there's Hubble expansion acting to increase the space between any two objects.

Hi Chris:

Although I am certainly not an expert, I am pretty sure from various threads in these forums that this is wrong. Objects that are gravitationally bound together remain so, and the Hubble Expansion (HE) does not change this. Thus star-planet systems and galaxies remain stable with respect to HE, and generally galaxy clusters and some super clusters do also. Most super clusters may not be stable with respect to HE, and all larger formations are likely to not be stable.

Regards,
Buzz

 

[Buzz Bloom]

It will fall into black holes on much shorter timescales.

Hi @mfb:

Thank you for this insight. I had not thought of that.

ADDED

I remember reading in another thread recently

https://www.physicsforums.com/threads/dark-matter-and-black-hole-interaction.916923/#post-5778660​

that very small fraction of DM can fall into a black hole. In order for that to happen the DM particle must actually hit the event horizon, or must be traveling very slowly.

Does this change your calculation?

Regards,
Buzz

 

[Buzz Bloom]

The rest of your question appears mostly to be an effort to fit GR into the familiar mold of rigid bodies, point masses, forces acting on these idealized point masses, and the resulting ordinary differential equations. Unfortunately, the basic answer to this is that you can't force GR into such a narrow mold, you need to learn new techniques to understand the theory.

Hi pervect:

Actually I was not trying to do fit GM into anything. I was trying to gain understanding of some posts by people who know much more about GM that I do. I used the thought experiment of the two bodies as an attempt to create a context in which I could attempt to explain my confusion.

There is some possibility of forcing the equations of GR into a model based on partial differential equations, for whatever it's worth. The heart of GR are the field equations ad those are just partial differential equations. This may not help if you've never studied partial differential equations, however.

I have studied PDEs, and in general I am fairly comfortable with working with them, although non-linear equations get to be a bit tricky. I have also worked with numerical solutions of such equations. My difficulty with the GM equations is a very limited exposure to tensors. I find that that I am overwhelmed by being unable to properly chose a suitable coordinate system that will enable me to work with fewer then too many simultaneous equations. I also have conceptual difficulties with setting up the components of the stress-energy tensor.

Regards,
Buzz

 

[PAllen]

Hi Paul:

Thank you very much for this concept which is new to me.

Based on this concept it would seem that the quote of @gnnmartin is not correct. Or am I misinterpreting something?Regards,
Buzz

It is only true in approximation. However, the approximation covers a lot of territory, in practice.

 

[Buzz Bloom]

Look up a series of papers by Gralla and Wald on the latest and greatest version of derivations of geodesic motion in the limit

Hi Paul:

I took a look at the abstract for

https://arxiv.org/abs/0806.3293​

and found it unintelligible, and that is before getting into the PDEs in the article's body.

There is general agreement that the MiSaTaQuWa equations should describe the motion of a "small body" in general relativity, taking into account the leading order self-force effects. However, previous derivations of these equations have made a number of ad hoc assumptions and/or contain a number of unsatisfactory features. For example, all previous derivations have invoked, without proper justification, the step of "Lorenz gauge relaxation", wherein the linearized Einstein equation is written down in the form appropriate to the Lorenz gauge, but the Lorenz gauge condition is then not imposed--thereby making the resulting equations for the metric perturbation inequivalent to the linearized Einstein equations.​

I think I have learned a lot of new to me concepts from the discussion in his thread, and I would like to thank you and the other posters for their contributions.
Note: I inserted some explanatory context in brackets in the quotes.

In "real" space, even in the absence of mass there would be no absence of field effects that would alter even a circular orbit that would otherwise be a potential geodesic course. Field effects in space would always alter any orbit as an external entity, masking the inherent nature of a theoretical orbit.

I think in GR only so called "test objects" with negligible mass really follow geodesics, objects with significant mass in curved spacetime do not exactly.

Neither [circular orbits or spirals] are geodesics. In GR, only test bodies exactly follow geodesics. Specifically, geodesic motion follows from the field equations in the limit of ever smaller bodies against a given background geometry, with further technical assumptions being required (e.g. the dominant energy condition).

No force is needed. The orbital motion leads to a dynamic geometry, and the bodies interact with the dynamic geometry so as to radiate away momentum and angular momentum.

It [DM] will fall into black holes on much shorter timescales.

There remains one point that still confuses me. There seems to be three different reasons/explanations why a non-test particle body fails to follow a geodesic.

1. From @PAllen and @Vitro: In GR, only test bodies exactly follow geodesics.
2. From @Dr Aaron: Field effects in space would always alter any orbit as an external entity.
3. From PAllen: The orbital motion leads to a dynamic geometry, and the bodies interact with the dynamic geometry so as to radiate away momentum and angular momentum.​

(1) is not an explanation, but rather a general statement of a GM truth.

Question: Are (2) and (3) two independent mechanisms which explain the truth of (1), or are they two different interpretations of the same mechanism?

Regards,
Buzz

 

[mfb]

In an idealized setup, you have only GR. "Field effects" is not a very clear concept anyway. In actual planetary systems, you have various things influencing the orbits more than gravitational waves, but that doesn't matter for your idealized setup. For pulsars, emission of gravitational waves can be the dominant effect of orbital changes.
To emit gravitational waves, your objects both have to have mass - they won't follow geodesics. They will spiral in.

 

[PeterDonis]

To emit gravitational waves, your objects both have to have mass - they won't follow geodesics. They will spiral in.

It's not actually clear that the inspiraling objects (e.g., binary pulsars) won't follow geodesics, just from the fact that they are spiraling in; the spacetime geometry is not stationary, so free-fall geodesic orbits will not be slowly precessing ellipses, as they would be in a stationary situation.

A better argument is that, as the objects emit gravitational waves, they experience a "back reaction" due to conservation of momentum--basically the waves are a (very weak!) form of "rocket exhaust" that apply a (very small!) thrust to the objects. That means the objects will experience a (very small!) proper acceleration, which means they cannot be moving on geodesics. I don't know if this has been explicitly calculated for, e.g., a binary pulsar, however; the actual effects might be so small that the observed orbits would not be measurably different from geodesics of the non-stationary spacetime geometry.

 

[pervect]

My not-very-good understanding, based mostly on Baez's "Insight" article on "problems with the continuum" is that there are various ad-hoc methods of calculating or at least estimating back-reaction, but there isn't any completely general and totally satisfactory theoretical method to deal with it.

 

[PeterDonis]

there are various ad-hoc methods of calculating or at least estimating back-reaction, but there isn't any completely general and totally satisfactory theoretical method to deal with it.

That's my understanding as well. AFAIK there are no exact or even approximate closed form solutions known for this, so all we have are numerical simulations. Those simulations give good results for, e.g., predicting the change in orbital parameters over time, but I don't know if they're fine-grained enough to make predictions about, e.g., exactly what proper acceleration each pulsar in a binary pulsar system will experience.

 

[Buzz Bloom]

as the objects emit gravitational waves, they experience a "back reaction" due to conservation of momentum--basically the waves are a (very weak!) form of "rocket exhaust" that apply a (very small!) thrust to the objects.

Hi Peter:

There seems to be no end to the new concepts which I am being taught. I did not previously understand that that the gravitational waves (GWs) were emitted from each of the two bodies as the moved relatively to their common center of gravity. I had been conceptualizing that the GWs were being created by an interactions between each body and something else. The something else things were unclear, and at different points in my recent education from this thread were (1) EM fields and (2) time varying irregularities in the space-time distortions due to the fact that the bodies are not test particles.

New Question: What causes a body to radiate GWs? Could the cause be the tidal influences of each body on the other?

Regards,
Buzz

 

[Buzz Bloom]

Hi pervect:

Can you provide a link to the Baez's "Insight" article on "problems with the continuum"? I tried a search but failed to find it.

Regards,
Buzz

 

[PeterDonis]

I did not previously understand that that the gravitational waves (GWs) were emitted from each of the two bodies as the moved relatively to their common center of gravity.

This is an approximation, not an exact statement. The problem is that, unlike with the case of EM waves, there is no way to localize the energy and momentum carried by GWs. I should have clarified that the "rocket exhaust" argument I was giving was heuristic. A more exact calculation would, in principle, show a (very small) nonzero proper acceleration for each of the pulsars in the binary, but you would not be able to point to a specific "piece" of gravitational radiation being emitted from a given pulsar and say that it was "pushing" on the pulsar, the way you could with an ordinary rocket exhaust or an emission of EM waves.

What causes a body to radiate GWs?

The GWs are not being radiated from the pulsars individually; see above. The general "source" of gravitational radiation is the third time derivative of the quadrupole moment of the system as a whole.

 

[Buzz Bloom]

The GWs are not being radiated from the pulsars individually; see above. The general "source" of gravitational radiation is the third time derivative of the quadrupole moment of the system as a whole.

Hi Peter:

I confess I do not understand how the quadrupole moment is calculated, but I get the impression from the above quote that someone with sufficient math skill and understanding of the GR equations could analyze the behavior of a binary system with given initial conditions and approximately determine the consequent spiral paths as well as well the time dependent magnitude of the generated GWs. Is this correct?

If so, would the accuracy of this approximation likely be somewhat similar (within a few orders of magnItude) to the accuracy of the GR calculation of the previously unexplained (using Newtonian calculations) precession of Mercury's elliptical orbit?

Regards,
Buzz

 

[PeterDonis]

I do not understand how the quadrupole moment is calculated

https://en.wikipedia.org/wiki/Quadrupole#Gravitational_quadrupole

someone with sufficient math skill and understanding of the GR equations could analyze the behavior of a binary system with given initial conditions and approximately determine the consequent spiral paths as well as well the time dependent magnitude of the generated GWs. Is this correct?

Yes, This has been done for binary pulsar systems in order to check the observed change in orbital parameters due to GW emission against the GR prediction. Hulse and Taylor won a Nobel Prize for doing this for one binary pulsar system.

would the accuracy of this approximation likely be somewhat similar (within a few orders of magnItude) to the accuracy of the GR calculation of the previously unexplained (using Newtonian calculations) precession of Mercury's elliptical orbit?

Roughly, yes, I think they're about the same order.

 

[Buzz Bloom]

Hi @PeterDonis:

Thank you for the Wikipedia reference for the quadrupole moment. The math looks like something I will be able to understand after a careful reading.

I should have thought to do this before, but I have started to explore

https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity.​

I now get that the Mercury precession prediction and the GW prediction are both solutions to the GR 2-body problem, and for this class of problems, each of the two bodies are considered to be a point mass. Is this correct? Is it also correct that a"point mass" is a "test body" in the sense of the following: "In GR, only test bodies exactly follow geodesics."

https://www.physicsforums.com/threads/qs-re-gravitational-waves.918313/#post-5788965​

If this is so, then it seems reasonable that the solution to the Mercury problem, in addition to the precession result, should also show that Mercury will follow a geodesic that will eventually fall into the sun based on the GR 2-body dynamics. Is this correct?

If so, do you know of any published calculation that gives the time frame for this result?
BTW: I found

https://www.physicsforums.com/threads/why-dont-planets-fall-into-the-sun.595442/​

which is no help.

Regards,
Buzz

 

[PeterDonis]

I now get that the Mercury precession prediction and the GW prediction are both solutions to the GR 2-body problem

The GW prediction is, but the Mercury prediction is not as the term "2-body problem" is usually used in GR. (This is one of those frequent cases where Wikipedia is not a good guide.) The Mercury prediction treats Mercury as a test body; only the Sun is a source of gravity. The 2-body problem in GR involves two sources of gravity, not one.

for this class of problems, each of the two bodies are considered to be a point mass

In Newtonian gravity, yes, this is true. Not in GR.

Is it also correct that a"point mass" is a "test body" in the sense of the following: "In GR, only test bodies exactly follow geodesics."

The term "point mass" doesn't have a well-defined meaning in GR; that's why GR textbooks don't use it.

the solution to the Mercury problem, in addition to the precession result, should also show that Mercury will follow a geodesic that will eventually fall into the sun based on the GR 2-body dynamics. Is this correct?

No. The Mercury perihelion precession does not involve any emission of gravitational waves. To predict GW emission from the solar system you would have to model the planets as sources of gravity, as is done for each of the pulsars in the binary pulsar calculations. I don't know if full numerical simulations have been done of this.

do you know of any published calculation that gives the time frame for this result?

No, I don't. You could make a rough estimate for a single planet by plugging the planet's mass and orbital parameters and the Sun's mass into formulas similar to the ones used for rough estimates of binary pulsar systems. My guess would be that you would get a time frame many orders of magnitude larger than the age of the solar system.

 

[PAllen]

The following link covers orbital decay from GW. As an exercise, they compute that Earth would spiral into the sun in 10²³ years, only 10 trillion times the current age of the universe.http://www.tapir.caltech.edu/~chirata/ph236/2011-12/lec15.pdf

 

[Buzz Bloom]

You could make a rough estimate for a single planet by plugging the planet's mass and orbital parameters and the Sun's mass into formulas similar to the ones used for rough estimates of binary pulsar systems.

Hi Peter:
I found the formulas below at

https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity​

in the section "Gravitational Radiation".

Do you know off hand whether the eccentricity of the orbit remains the same during the loss of energy/momentum, or if it a function of time?
If e is a constant, then calculating the time for the two bodies to meet is not too difficult. It become much more difficult if e varies. If e varies, do you know of a source that shows how it varies?

ADDED
I see that the Earth spiral assumes the circular orbit Eq 44 in the Hirata paper you cited. This is the same as that given in
https://en.wikipedia.org/wiki/Gravitational_wave#Binaries.

Regards,
Buzz