Conservation of angular momentum in GR

[TrickyDicky]

Total angular momentum is not conserved due to lack of spacetime spherical symmetry, it is precisely this fact that causes the angular momentum of the quadrupole moment to have to be radiated away as gravitational radiation. (see Schutz, chapter 9: exercises 39,40 and 47).

In this context, there is something I don't understand about an example also found often in GR textbooks: in the Earth-moon system there is a tidal torque due to the moon's influence and also the sun's, that changes the earth spin angular momentum by acting on the equator bulge and that slows down the earth's spin, however in this case the total angular momentum is effectively conserved by correcting the orbit angular momentum thru its enlarging of about 4.5 cm/year.
What makes the total angular momentum to be conserved in this particular setting? Is this small system considered practically spherically symmetric?

[Bill_K]

Angular momentum IS conserved in any asymptotically flat spacetime, even if radiation is present.

[TrickyDicky]

Angular momentum IS conserved in any asymptotically flat spacetime, even if radiation is present.

Consider the Kerr spacetime, I think it is asympotically flat but it has axisymmetric metric, so it is not spherically symetric, this according to various GR textbooks means total angular momentum is NOT conserved, for instance Hobson's GR in the chapter about the Kerr metric in page 313: "Note, however, that the total angular momentum of a particle is not a conserved quantity, since the spacetime is not spherically symmetric about any point."

On the other hand the total angular momentum in a spherically symmetric spacetime is conserved and therefore the quadrupole moment's angular momentum is absent and can not produce gravitational radiation. I believe all this to be basic stuff with no much room for disagreement.

So apparently your statement is not corect but in the context of the moon-earth system I'm not sure what you mean by it anyway, are you saying that the moon-earth system can be considered asymtotically flat? Please explain.

[Q-reeus]

Total angular momentum is not conserved due to lack of spacetime spherical symmetry, it is precisely this fact that causes the angular momentum of the quadrupole moment to have to be radiated away as gravitational radiation. (see Schutz, chapter 9: exercises 39,40 and 47).

In this context, there is something I don't understand about an example also found often in GR textbooks: in the Earth-moon system there is a tidal torque due to the moon's influence and also the sun's, that changes the earth spin angular momentum by acting on the equator bulge and that slows down the earth's spin, however in this case the total angular momentum is effectively conserved by correcting the orbit angular momentum thru its enlarging of about 4.5 cm/year.
What makes the total angular momentum to be conserved in this particular setting? Is this small system considered practically spherically symmetric?
TrickyDicky, I have no access to that textbook and probably couldn't make much use of it if I did, but just a thought that may be off the mark. Sometimes authors confuse owing to failure to specify context adequately. Maybe 'Total angular momentum is not conserved' Schutz refers to above is that of the orbiting masses only ('radiation reaction' couple slowing the bodies down)? And we are supposed to assume the balance is in the GW's? In the case of the Earth-Moon system, my assumption is probably that GW's are too weak to consider - ie tidal transfer is way larger in effect and taken to be momentum conserving. I think Clifford Will covers calcs on Earth-Moon tidal coupling in http://relativity.livingreviews.org/Articles/lrr-2006-3/ but maybe wrong there.

In #3 ...Hobson's GR in the chapter about the Kerr metric in page 313: "Note, however, that the total angular momentum of a particle is not a conserved quantity, since the spacetime is not spherically symmetric about any point."

Just maybe another example of context not properly specified by author - an unstated assumption the particle's angular momentum is coupling to that of a much larger spinning mass via it's Kerr metric - ie a GR variant of Earth-Moon tidal exchange?
On the other hand, would be most interested if none of my comments above hit the mark!

[TrickyDicky]

Sometimes authors confuse owing to failure to specify context adequately. Maybe 'Total angular momentum is not conserved' Schutz refers to above is that of the orbiting masses only ('radiation reaction' couple slowing the bodies down)? And we are supposed to assume the balance is in the GW's? In the case of the Earth-Moon system, my assumption is probably that GW's are too weak to consider - ie tidal transfer is way larger in effect and taken to be momentum conserving.

In #3

Just maybe another example of context not properly specified by author - an unstated assumption the particle's angular momentum is coupling to that of a much larger spinning mass via it's Kerr metric - ie a GR variant of Earth-Moon tidal exchange?
On the other hand, would be most interested if none of my comments above hit the mark!
In this case, this seems to be not controversial common knowledge in GR, I can assure you my citations are not out of context, I just didn't quote the whole paragraphs.
See Wikipedia Angular momentum page: "Angular momentum in relativistic mechanics:
In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant)."

It is precisely the no conservation of total angular momentum in GR that allows the quadrupole moment to exist.

I grant you my own comparison with the moon-earth system might be not valid here and thus my question in post #1 but I think Bill-k made a not exact assertion AFAICS in his answer.

[Q-reeus]

In this case, this seems to be not controversial common knowledge in GR, I can assure you my citations are not out of context, I just didn't quote the whole paragraphs...
OK sorry I wasn't giving you much credit there. :blushing:
See Wikipedia Angular momentum page: "Angular momentum in relativistic mechanics:
In modern (late 20th century) theoretical physics, angular momentum is described using a different formalism. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a result, angular momentum is not conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant)."
Well it makes sense if energy is ill-defined in curved spacetime, momentum too. While asymptotically flat has for me a clear interpretation not so asymptotically rotationally invariant. This specifies whether there is some nonzero metric coupling to the spin of a test particle at infinite distance from some source of angular momentum (say a notional Kerr BH), or something else? This is new to me.
It is precisely the no conservation of total angular momentum in GR that allows the quadrupole moment to exist.
That surprises me. Two masses connected by a spring undergoing linear oscillation have a nonzero mass quadrupole moment, right? Or are we restricted to the quad moment of say two co-orbiting bodies where angular momentum is inherently present?

[Bill_K]

Consider the Kerr spacetime, I think it is asympotically flat but it has axisymmetric metric, so it is not spherically symetric
Of course Kerr is not spherically symmetric in the near zone, and that's not what I said. It is spherically symmetric in the asymptotic region, which is all you need. Asymptotically flat means it approaches Minkowski space as you get far away.

this according to various GR textbooks means total angular momentum is NOT conserved... "Note, however, that the total angular momentum of a particle is not a conserved quantity, since the spacetime is not spherically symmetric about any point."
This quote is referring to the angular momentum of a test particle, not the total angular momentum. The angular momentum of the particle is not conserved but the total angular momentum is conserved.

On the other hand the total angular momentum in a spherically symmetric spacetime is conserved and therefore the quadrupole moment's angular momentum is absent and can not produce gravitational radiation. I believe all this to be basic stuff with no much room for disagreement.
I agree completely with this statement, it is quite basic and there should be no disagreement.

I'm not sure what you mean by it anyway, are you saying that the moon-earth system can be considered asymtotically flat? Please explain.
Yes certainly, the gravitational field of the Earth-Moon system is asymptotically flat. The combined gravitational field goes down as M r-3, where M is the total mass, and the spacetime asymptotically approaches Minkowski space.

[TrickyDicky]

Well it makes sense if energy is ill-defined in curved spacetime, momentum too. While asymptotically flat has for me a clear interpretation not so asymptotically rotationally invariant. This specifies whether there is some nonzero metric coupling to the spin of a test particle at infinite distance from some source of angular momentum (say a notional Kerr BH), or something else? This is new to me.
In fact that term "asymptotically rotationally invariant" seems to be not very commonly used, normally just rotational invariance is mentioned.
That surprises me. Two masses connected by a spring undergoing linear oscillation have a nonzero mass quadrupole moment, right? Or are we restricted to the quad moment of say two co-orbiting bodies where angular momentum is inherently present?
You are right. I'm actually referring only to binary systems of co-orbiting bodies configurations.

[Cleonis]

[...] in the Earth-moon system there is a tidal torque due to the moon's influence and also the sun's, that changes the earth spin angular momentum by acting on the equator bulge and that slows down the earth's spin, however in this case the total angular momentum is effectively conserved by correcting the orbit angular momentum thru its enlarging of about 4.5 cm/year.
What makes the total angular momentum to be conserved in this particular setting?

There are two distinct effects. (Probably there a bit of cross-influence, but afaik that's negligable.)
The two have in common that they each relate to a way of the Earth not being perfectly spherical.

1. Moon and Sun act upon the Earth's equatorial bulge, giving rise to a torque upon the Earth, hence a corresponding gyroscopic precession. For the Earth that gives the precession of the equinox.

2. The moon acts upon the tidally distorted Earth. Due to the internal friction the Earth's tidal distortion lags behind. Given that lag there is a gravitational effect that increases the Moon's orbital energy, and it decreases the Earth's rotational energy. The number of 4.5 cm/year increase of Moon orbit altitude is the increase of Moon orbital energy.

(As I understand it: when the Moon first formed it was rotating somewhat faster than 1 rotation per month. The Moon was tidally distorted by the Earth, and rotating more than once a month. Internal friction makes such distortion lag behind. This lag gave the Earth opportunity to slow the Moon's rotation down. At some point in time the Moon reached a state of tidal lock. )

Anyway, the precession of the equinox and the Earth rotation slowing down arise both from gravitational distortion effects, but it's two different effects.
Neither of them are relativistic effects, which makes it unlikely that any relativity textbook author would mention them.

[TrickyDicky]

Of course Kerr is not spherically symmetric in the near zone, and that's not what I said. It is spherically symmetric in the asymptotic region, which is all you need. Asymptotically flat means it approaches Minkowski space as you get far away. Ok, that solves the doubt about asymptotically rotational invariance in the above post too.


This quote is referring to the angular momentum of a test particle, not the total angular momentum. The angular momentum of the particle is not conserved but the total angular momentum is conserved.
Ok, I see.


Yes certainly, the gravitational field of the Earth-Moon system is asymptotically flat. The combined gravitational field goes down as M r-3, where M is the total mass, and the spacetime asymptotically approaches Minkowski space.
I see, I didn't consider the fact that it could be modelled that way.

Thanks for the clarifications

[PAllen]

My understanding of some of the issues touched on here is as follow:

A pair of orbiting bodies in GR does not exactly conserve angular momentum considering just the two bodies. Neither do a mutually orbiting pair of opposite charges, considering mass to be gravitationally negligible, and only EM force significant. The deviation in the gravity case is much smaller (due to existence only of quadrupole radiation), versus dipole for the EM case.

HOWEVER, the total angular momentum of bodies plus gravitational radiation; or charges plus EM radiation *does* conserve angular momentum exactly.

Issues like the rotational symmetry or assymptotic flatness of the universe relate to whether you can define a universally conserved angular momentum. As long as, within some large region, you have sufficient overall flatness, there is conservation of angular momentum within the large region (e.g. the milkyway's mass, energy, plus gravitational radiation conserve angular momentum).

For me, a good source on some of these issues is (in passing):

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

[pervect]

I can't find the reference in Schutz that says that angular momentum isn't conserved. Is this "A First course in General Relativity?" Perhaps there's some change with editions.

There is mention in Held, "General Relativity and Gravitation", about various ways to define angular momentum in an asymptotically flat space-time, in a chapter by Winicour. My version of Schutz also cites Held:


The measurement of the mass and angular momentum of a source by looking at its distant gravitational field is discussed in Misner et al(1973), Ashtekar(1980), and Winicour (1980)

the Winicour reference is found in Held's collection of papers, "General Relativity and Gravitation, 100 yeras after the birth of Einstein".

[add]
I"m also reasonably certain that there is a conserved orbital angular momentum for a particle orbiting a spining black hole. As I said, I couldn't find the section of Schutz that you were referring to - it's not my favorite textbook.

Wiki seems to back me up on this:

http://en.wikipedia.org/w/index.php?title=Carter_constant&oldid=420908353


The Carter constant is a conserved quantity for motion around black holes in the general relativistic formulation of gravity. Carter's constant was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the energy, axial angular momentum, and particle rest mass provide the four conserved quantities necessary to uniquely determine all orbits in the Kerr-Newman spacetime (even those of charged particles).


As none of the metric coefficients is a function of \phi, I think you should have rotational symmetry as is required by Noether's theorem. The space-time will be "stationary" rather than static because of the dt \, d\phi term, but that shouldn't spoil the conservation laws - it does prevent you from having "hypersurface orthogonality" though.

[Q-reeus]

Here's something I've hastily put together as purely hand-waving argument (no numbers, no equations) you folks might like to pick over while I get to a very late sleep.
Take an ideally 'incompressible' hollow right circular cylinder, spinning at constant angular velocity about it's major vertical axis and supported on frictionless bearings. Now apply gradually pressure to the cylinder ends, placing the sides under purely axial compression. Unlike the case of an ideally incompressible fluid so compressed, there is no resulting induced circumferential stress. A relativistic transformation into the frame of a moving element in the cylinder wall will therefore notice no circumferential stress gradient during the time changing squeeze phase, and therefore no relativistic force in that direction. So, to the extent elastic energy is absent (and we have idealized to an 'incompressible' solid), our cylinder undergoes no change in either spacial dimensions or angular velocity as a result of the stressing process. Ho hum you might say, but here's the rub. Pressure/stress is a source of gravity (active mass) in GR, and by the equivalence principle that means equally also a source of both passive and inertial mass. The latter is key here. Stressing the cylinder has increased it's inertial mass 'for free' and therefore it's angular momentum and rotational kinetic energy. So in principle we can endlessly cycle this process and generate angular momentum and energy - spin up in unstressed state, spin down in stressed state, over and over. Of course in the real world case friction and finite elastic energy contributions and cross couplings to the 'pure pressure' effect will be overwhelmingly greater. But these will be material dependent factors and hence not fundamental. Can it be shown there is no such effect at all though? Cheers!

[pervect]

Wincour's article in Held (pg 74, eq 2-10) gives the expression for the total angular momentum as a volume intergal as:


\int_{\Sigma} \xi^{b} \left(T^{a}{}_{b} - \frac{1}{2} \, \delta^{a}{}_{b} T \right) dS_{a}


Here \xi^{b} is the appropriate killing vector - for angular momentum, it would be a phi-translation vector.

I *think* dS_a is just a unit future (i.e. normal to the volume element sigma) - and I *think* this implies that pressure doesn't contribute to angulalr momentum (though you are correct that it contributes to energy) - because the kronecker delta factor vanishes when a is not equal to b, so the dependence on T is present only when you have a time-like killing vector and are computing energy, it doesn't contribute when you have a space-like killing vector and are computing linear or angular momentum.


I'm only 50/50 on this...

[Q-reeus]

Wincour's article in Held (pg 74, eq 2-10) gives the expression for the total angular momentum as a volume intergal as:


\int_{\Sigma} \xi^{b} \left(T^{a}{}_{b} - \frac{1}{2} \, \delta^{a}{}_{b} T \right) dS_{a}


Here \xi^{b} is the appropriate killing vector - for angular momentum, it would be a phi-translation vector.

I *think* dS_a is just a unit future (i.e. normal to the volume element sigma) - and I *think* this implies that pressure doesn't contribute to angulalr momentum (though you are correct that it contributes to energy) - because the kronecker delta factor vanishes when a is not equal to b, so the dependence on T is present only when you have a time-like killing vector and are computing energy, it doesn't contribute when you have a space-like killing vector and are computing linear or angular momentum.
I'm only 50/50 on this...
I'm assuming pervect your last entry refers to mine in #13, in which case thanks for your interest in the problem, but it was posed as a potential counterexample to the kind of generalized theorems employing the style of high end maths you have used. Sorry but I can't really follow it at that level; could you please translate it back to the specifics of the setup? What I expected was an argument that because the cylinder axial stress is normal to the motion, v.sigma = 0 (v the local velocity, sigma the axial stress) for any given moving element and thus no actual 'boost' to momentum and KE. My objection in turn would be that implies a breakdown in the principle of equivalence - if the pressure generated inertial mass is directional in nature, why not the same for the active and passive mass? But no-one would argue the gravitational properties of the latter two would be anything but isotropic, surely. Nearly all cases of astrophysical interest seem to be limited to fluids under pressure, and here isotropy hides the issue - ie you just consider rho + 3p, the 3 factor signifying that all three orthogonal components of pressure in a fluid are equally contributing to the effective mass density. But is it consistent? If ma = mp = mi here, it automatically implies that for any given acceleration a, there will be a full contribution F = mia = 3pa owing to fluid pressure inertial mass contribution. Which necessarily implies a full contribution from the two components of p orthogonal to a. Which in turn brings us back to the case of our spinning cylinder - either the principle of equivalence fails for pressure, or we have a genuine perpetuum mobile (we should use v|sigma|, not v.sigma). What's wrong with this line of reasoning?

[TrickyDicky]

Which in turn brings us back to the case of our spinning cylinder - either the principle of equivalence fails for pressure, or we have a genuine perpetuum mobile What's wrong with this line of reasoning?
Hi, Q-reeus
Probably I don't understand your thought experiment very well but I would say that if in a experiment about conservation of momentum-energy you leave out friction and compressibility you can get all kind of perpetuum mobiles.

[Q-reeus]

Hi, Q-reeus
Probably I don't understand your thought experiment very well but I would say that if in a experiment about conservation of momentum-energy you leave out friction and compressibility you can get all kind of perpetuum mobiles.
Firstly TrickyDicky I feel a little guilty now, having jumped in with a different angle altogether - even though it has kinda stuck to the title, didn't really stop to think if this was really a 'highjacking' of your thread. If you feel that I will put a stop to this now - not that there's much to stop mind you!:redface:
On the matters you have raised, I agree the natural instinct is to assume this has got to be wrong if one stipulates physically impossible idealizations. But in this context it follows a long line of similar examples. To isolate the essential features, remove any facors that merely complicate but are not germane. Friction is a fairly obvious one - not claiming a perfect flywheel or anything like that. Finite compressibility is less obvious but is standard procedure for what seems like an endless succession of papers in AJP etc that keep rolling out on 'hidden momentum' and 'stored field momentum' for instance, and many of these are actually quite close to this scenario. As explained in #13, these two factors are both material dependent and thus not of fundamental significance to the principle in question. One caveat there though; SR dictates an upper theoretical limit to rigidity (Born Rigid), but I believe even there it does not fundamentally touch on the principles in question. And I have not really claimed a PMM as such - merely pointed out this is implied imho if stress as source of inertial mass has the properties I think is so. So if you would like to add anything, here or by my starting a new thread, be my guest. Same old problem here though, an almost deathly silence!:smile:

[TrickyDicky]

.... If you feel that I will put a stop to this now
No problem at all.

So if you would like to add anything, here or by my starting a new thread, be my guest. Same old problem here though, an almost deathly silence!:smile:
After reading pervect's post I would say his answer makes sense, in GR the conservation laws are encoded in symmetries of the system, one way to look at these symmetries uses Killing vectors, where energy conservation is related to the time-like Killing vector and angular momentum is related to space-like kiling vector, if the formula pervect writes applies here it means (I think, if not please pervect explain) that the cylinder might gain energy (in a static setting), but not angular momentum and therefore no spin cycle and no free energy.

[bcrowell]

I can't find the reference in Schutz that says that angular momentum isn't conserved. Is this "A First course in General Relativity?" Perhaps there's some change with editions.

The argument is pretty elementary: angular momentum isn't a scalar. No non-scalar quantity can be globally conserved for all spacetimes in GR, because parallel transport is path-dependent, but you can't add up the quantity without parallel-transporting to a single point first. (A related but slightly different way of putting it is that GR doesn't have global frames of reference, so there is no frame of reference that could be used in order to express the vector representing the total.)

[pervect]

Q-reeus. I thought your experiment was interesting. My tentative conclusion is simply that the angular momentum does not change when you compress the cylinder, as one would expect.

It is true that if you have a cylinder with a bolt through it, and that if you tighten up the bolt to put the cylinder under pressure, that the Komar mass of the cylinder increases. While the Komar mass, of the cylinder increases because of the pressure, the Komar mass of the bolt decreases, because it's under tension. Unless something actually deforms, doing work, the whole closed system doesn't gain or lose mass, it's just distributed differently.

However, (assuming my understanding is correct) even though rotating cylinder is more massive, it appears to be the case that it doesn't have any more angular momentum. (Again, using the Komar formula, which can be generalized to handle angular and linear momentum.) The pressure terms, which do contribute to the Komar mass, simply don't matter to the angular momentum.

That's assuming I've understood the equations properly. It all appears to make sense, but I don't have a lot of "worked problems" in this area to alert me to potential confusion on my part, and I've learned to be cautious about that situation.

[Sam Gralla]

Some of the confusion in this thread is a result of the fact that angular momentum can be defined either at spatial infinity or at null infinity. Angular momentum defined at spatial infinity is conserved, for every asymptotically flat spacetime, period. This is because even if you radiate angular momentum the radiation never makes it out to spatial infinity, and the angular momentum of the radiation is included when you calculate the angular momentum, no matter what time (slice) you choose. Angular momentum defined at null infinity is not conserved, precisely due to the radiation loss that Schutz talks about.

[Q-reeus]

No problem at all.
Thanks - quite a relief!
After reading pervect's post I would say his answer makes sense, in GR the conservation laws are encoded in symmetries of the system, one way to look at these symmetries uses Killing vectors, where energy conservation is related to the time-like Killing vector and angular momentum is related to space-like kiling vector, if the formula pervect writes applies here it means (I think, if not please pervect explain) that the cylinder might gain energy (in a static setting), but not angular momentum and therefore no spin cycle and no free energy.
Fair enough but my rather simple approach is to test general theorems by way of a specific setup, and just see if all situations have been fully and accurately accounted for by said theorem(s). See my comments to pervect's post in #20.

[Q-reeus]

Q-reeus. I thought your experiment was interesting. My tentative conclusion is simply that the angular momentum does not change when you compress the cylinder, as one would expect.
It is true that if you have a cylinder with a bolt through it, and that if you tighten up the bolt to put the cylinder under pressure, that the Komar mass of the cylinder increases. While the Komar mass, of the cylinder increases because of the pressure, the Komar mass of the bolt decreases, because it's under tension. Unless something actually deforms, doing work, the whole closed system doesn't gain or lose mass, it's just distributed differently.
Agreed, no argument there. However it matters much that the mass is distributed differently when a rotating system is under consideration. If for instance the bolt as assumed lies on the axis of rotation, it's contribution to angular momentum will be relatively tiny, even though it cancels the overall system mass change from the cylinder. I had actually envisaged a stationary external agent in applying pressure (eg. G-clamp) which could be quite compressible itself but an irrelevant factor - only changes to the rotating mass is important imo.
However, (assuming my understanding is correct) even though rotating cylinder is more massive, it appears to be the case that it doesn't have any more angular momentum. (Again, using the Komar formula, which can be generalized to handle angular and linear momentum.) The pressure terms, which do contribute to the Komar mass, simply don't matter to the angular momentum.
Well that would be extremely problematic from my pov given points raised in #15. How could this be explained? I argued in #13 there would be no circumferential forces of relativistic origin, owing to the purely axial nature of the induced stress. Therefore no physical mechanism to reduce angular velocity. Now if cylinder Komar mass increase is 'really real', how on earth can there not be an increase in angular momentum? My suspicion is that conservation of energy/momentum is a priori built into the Komar 'proof' of conservative behavior. Hence the specific counterexample. I have argued in #15 one cannot have it both ways - if ma = mp = mi (WEP) holds for pressure induced mass, transverse components of pressure must contribute to inertial mass mi just as for the component in line with any acceleration a. That leaves no way around the conclusion angular momentum must increase for the setup of #13. The other alternative is there is indeed a directional nature to pressure generated mi (but not for ma or mp!). Hence WEP must fail here, which would still be news imo. My bet is though WEP holds. Thoughts?

[TrickyDicky]

Some of the confusion in this thread is a result of the fact that angular momentum can be defined either at spatial infinity or at null infinity. Angular momentum defined at spatial infinity is conserved, for every asymptotically flat spacetime, period. This is because even if you radiate angular momentum the radiation never makes it out to spatial infinity, and the angular momentum of the radiation is included when you calculate the angular momentum, no matter what time (slice) you choose. Angular momentum defined at null infinity is not conserved, precisely due to the radiation loss that Schutz talks about.

So I guess in the Hulse-Taylor binary pulsar system angular momentum is defined at null infinity, not asymptotically flat situation, and thus radiates GW, and angular momentum is not conserved.
However in the earth-moon orbit system angular momentum is conserved because it is defined at spatial infinity, and it is an asymptotically flat modeled situation.

Is this right according to your explanation?
How do we choose to define angular momentum either at null infinity or at spatial infinity for a given orbital system?

[PAllen]

So I guess in the Hulse-Taylor binary pulsar system angular momentum is defined at null infinity, not asymptotically flat situation, and thus radiates GW, and angular momentum is not conserved.
However in the earth-moon orbit system angular momentum is conserved because it is defined at spatial infinity, and it is an asymptotically flat modeled situation.

Is this right according to your explanation?
How do we choose to define angular momentum either at null infinity or at spatial infinity for a given orbital system?

No, I don't think this is right. As I explained earlier, the Hulse Taylor system *including its gravitational radiation* conserves angular momentum. The fact that the system excluding radiation does not is completely trivial and non-relativistic and is analagous to the loss of angular momentum in a pair of co-orbiting charges in EM + SR; including the EM radiation, angular momentum is conserved.

In a radiating system, it seems not very sensible to use a null-infinity definition, because you want to include the radiation at all times.

I think the Hulse-Taylor system is normally analyzed with an asymptotic flatness condition. It is not static, or spherically symmetric, but can be asymptotically flat.

[TrickyDicky]

No, I don't think this is right. As I explained earlier, the Hulse Taylor system *including its gravitational radiation* conserves angular momentum. The fact that the system excluding radiation does not is completely trivial and non-relativistic and is analagous to the loss of angular momentum in a pair of co-orbiting charges in EM + SR; including the EM radiation, angular momentum is conserved.

In a radiating system, it seems not very sensible to use a null-infinity definition, because you want to include the radiation at all times.
This doesn't agree with Sam Gralia's statement "Angular momentum defined at null infinity is not conserved, precisely due to the radiation loss...".
The fact is we include the radiation precisely because angular momentum is not conserved.
Precisely the reason dipole moment is not radiated as GW is because linear momentum is conserved.

I think the Hulse-Taylor system is normally analyzed with an asymptotic flatness condition. It is not static, or spherically symmetric, but can be asymptotically flat. This I could agree with. Edit: In this case angular momentum is defined at spatial infinity and therefore conserved because the GW can't reach spatial infinity, is this right?

[Q-reeus]

TrickyDicky: In the other recent thread, I mentioned an Angelo Loinger, who is a strictly GR theorist (of sorts), and claims full GR does not admit to GW's - if true then coupled to binary pulsar data, automatically means failure of angular momentum conservation in toto. For sure 'controversial' but anyway the paper is relatively short, and if you do a search at arXiv.org will not be hard to find - you could no doubt make more of than I could. Just a thought.

[Sam Gralla]

So I guess in the Hulse-Taylor binary pulsar system angular momentum is defined at null infinity, not asymptotically flat situation, and thus radiates GW, and angular momentum is not conserved.
However in the earth-moon orbit system angular momentum is conserved because it is defined at spatial infinity, and it is an asymptotically flat modeled situation.

Is this right according to your explanation?
How do we choose to define angular momentum either at null infinity or at spatial infinity for a given orbital system?

I think probably all that is going on here is that the earth-moon system analysis you read about is in a non-relativistic (i.e., Newtonian) limit, where there is no gravitational radiation. In reality it will radiate a bit of angular momentum, but my guess is this is tiny and simply being neglected in whatever you read about. In a binary pular, on the other hand, there is a lot of gravitational radiation.

Whether to use angular momentum defined at null or spatial infinity is just a choice. If you want to see how much angular momentum escapes as radiation, you use null infinity. If you want to see that angular momentum is always conserved, as long as the angular momentum in the radiation is included, you use spatial infinity. Both are well defined for any asymptotically flat spacetime (although the definitions of asymptotic flatness people give in the respective cases can be a little bit different). I didn't really make this comment in response to your original question; it's just I noticed some people were saying "angular momentum is conserved" and some people were saying "no, you can radiate it away", and the different definitions of angular momentum are probably the cause there.

[PAllen]

TrickyDicky: In the other recent thread, I mentioned an Angelo Loinger, who is a strictly GR theorist (of sorts), and claims full GR does not admit to GW's - if true then coupled to binary pulsar data, automatically means failure of angular momentum conservation in toto. For sure 'controversial' but anyway the paper is relatively short, and if you do a search at arXiv.org will not be hard to find - you could no doubt make more of than I could. Just a thought.

I don't know his credentials, but he says many things all other GR researcher's disagree with. For example, in the intro to one of his papers 'refuting' gravitational waves he says:

"The exact (non-approximate) formulation of general relativity (GR)
does not allow the existence of physical gravitational waves (GW’s). I have
given several proofs of this fact [1]. Quite simply, we can observe, e.g.,
that bodies which interact only gravitationally describe geodesic lines, and
therefore – as it is very easy to see – they do not generate any GW."

Except that going all the way back to Einstein and Infeld (in the 1940s), and everyone else who derives equations of motion directly from the field equations, finds that bodies follow geodesics only in the limit of point particles. Real, massive bodies closely approximate geodesics, but the difference cannot be ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole body of work.

[Q-reeus]

...Except that going all the way back to Einstein and Infeld (in the 1940s), and everyone else who derives equations of motion directly from the field equations, finds that bodies follow geodesics only in the limit of point particles. Real, massive bodies closely approximate geodesics, but the difference cannot be ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole body of work.
I'm in no position to argue that - you are probably quite right. I never noticed him referring his findings to the binary pulsar results which seemed odd, but owing to how short his arguments were, I figured someone with a good grasp of GR (not me) could sort out his logic without too much sweat

[bcrowell]

Loinger is a kook with academic credentials. In addition to the kookery discussed by PAllen, Loinger and his coauthor Marsico believe that there is something wrong with the standard analysis of black holes: http://arxiv.org/abs/1011.2600 Basically they make a big deal out of the fact that an observer at infinity sees infalling matter take an infinite amount of time to reach the event horizon. They treat this as having Dramatic Physical Implications, and they criticize other authors: "Now, if we take into account the decisive role of the Hilbertian gravitational repulsion, which is neglected by [Broderick et al.][...]" This is stupid and misleading, since Broderick hasn't made a mistake by ignoring some relevant effect. The "repulsion" isn't really a repulsion; it's just a *coordinate* acceleration with no intrinsic physical significance. An observer at infinity sees infalling material as taking infinite time to reach the event horizon, and since the Schwarzschild t coordinate is the time measured by an observer at infinity, the coordinate acceleration d2r/dt2 has an outward direction.

[TrickyDicky]

I think probably all that is going on here is that the earth-moon system analysis you read about is in a non-relativistic (i.e., Newtonian) limit, where there is no gravitational radiation. In reality it will radiate a bit of angular momentum, but my guess is this is tiny and simply being neglected in whatever you read about. In a binary pular, on the other hand, there is a lot of gravitational radiation.
Yes, I also think that's all.
I guess in a Newtonian-limit non-relativistic analysis of the binary pulsar, you do have total angular momentum conservation with no gravitational radiation, with the orbit angular momentum (shortening of the orbital period) compensated by the spin angular moment of the pulsar (acceleration of spin) just like in the earth-moon system newtonian treatment the slowing of the earth spin is compensated by the enlarging of the orbit and therefore total angular momentum is conserved.

Whether to use angular momentum defined at null or spatial infinity is just a choice. If you want to see how much angular momentum escapes as radiation, you use null infinity. If you want to see that angular momentum is always conserved, as long as the angular momentum in the radiation is included, you use spatial infinity. Both are well defined for any asymptotically flat spacetime (although the definitions of asymptotic flatness people give in the respective cases can be a little bit different). I didn't really make this comment in response to your original question; it's just I noticed some people were saying "angular momentum is conserved" and some people were saying "no, you can radiate it away", and the different definitions of angular momentum are probably the cause there.
Yeah, there is some confusion about asymptotically flat spacetimes, it is not an easy theme for non-experts at least. Part is derived from the fact that this and many other common themes in GR have a lot to do with the concepts of "conformal infinity" and "null infinity" that are not usually treated with depth in introductory texts but that are vital to give a sound theoretical basis to GW, blackholes etc.
Here is a pretty informative site about this: http://relativity.livingreviews.org/Articles/lrr-2004-1/

Also as I read angular momentum in GR is a thorny subject, or it has been until relatively recently.

[TrickyDicky]

Real, massive bodies closely approximate geodesics, but the difference cannot be ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole body of work.
Most likely this guy's work is flawed, but not for this reason.
If we are limiting the analysis to bodies trajectories is perfectly licit to talk about geodesics for any body trajectory, the fact that in the surface of a massive body test particles at rest are not following geodesic motion is not relevant in this case. The fact is curvature of spacetime (gravity) manifests thru the convergence-divergence of geodesic trajectories.

[bcrowell]

Real, massive bodies closely approximate geodesics, but the difference cannot be ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole body of work.
Most likely this guy's work is flawed, but not for this reason.
If we are limiting the analysis to bodies trajectories is perfectly licit to talk about geodesics for any body trajectory, the fact that in the surface of a massive body test particles at rest are not following geodesic motion is not relevant in this case. The fact is curvature of spacetime (gravity) manifests thru the convergence-divergence of geodesic trajectories.
No, PAllen was correct, and he wasn't saying anything about surfaces. Bodies with nonnegligible mass don't follow geodesics because they radiate gravitational waves.

[Q-reeus]

If we are limiting the analysis to bodies trajectories is perfectly licit to talk about geodesics for any body trajectory, the fact that in the surface of a massive body test particles at rest are not following geodesic motion is not relevant in this case. The fact is curvature of spacetime (gravity) manifests thru the convergence-divergence of geodesic trajectories.
Can't argue the specifics, but agree 100% it's not proper to dismiss anyone who makes a serious argument without presenting their entire rationale first. While no doubt PAllen in #29 meant only the best, it would have been a bit kinder to A.Loinger to have quoted the remainder of the intro "..If we add non-gravitational forces, the conclusion remains the same, because the new trajectories do not possess kinematical elements (velocity, acceleration,
time derivative of the acceleration, etc.) different from those of the geodesic motions."
Not saying the final conclusion in the full course of time would be different, but, inadvertently or not, excising aspects of his pov paints a particularly simplistic 'straw man' picture. Likewise, bcrowell's costic assessment in #31 to my mind doesn't address the fact that while 'Hilbertian repulsion' may be a poor way of labelling things, is it truly 'of no physical significance' that in Schwarzschild coords a distant observer finds infalling matter will decelerate to zero velocity on approach to the nominal EH of a nominal stationary BH? Isn't this a legitimate 'physically meaningful' perspective, even if the terminology 'repulsion' is somewhat misguided? I'm not the only one to have pointed out that 'infinite redshift' is more than mere 'optical illusion' in this setting -it has a genuine physical meaning, imho at least. And honestly, it may be true Loinger is ultimately off the mark, but is it right to label such folks with terms like 'kook', 'nutter'. 'fringe', 'weirdo', [insert your own pejorative and demonizing term(s) here..]? More reasonable surely to just say 'mistaken', 'minority view' etc. Can't help but see parallels with a particularly intolerant and 'huge vote of hands' religious movement that has just one word for dissenters - 'infidel'.
One last thing here. Notwithstanding TrickyDicky's accepting stance, fact is I have created a 2-stream thread here, and am officially 'pulling' my part of that at this point. Will be in due time be reformulating and starting as new thread - who knows, I might even get some decent feedback (exempting two participants from that observation!).

[PAllen]

Can't argue the specifics, but agree 100% it's not proper to dismiss anyone who makes a serious argument without presenting their entire rationale first. While no doubt PAllen in #29 meant only the best, it would have been a bit kinder to A.Loinger to have quoted the remainder of the intro "..If we add non-gravitational forces, the conclusion remains the same, because the new trajectories do not possess kinematical elements (velocity, acceleration,
time derivative of the acceleration, etc.) different from those of the geodesic motions."


I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

Deriving motion from the field equations is a complex prodedure, but it has been done now dozens of different ways by many researcher's over decades, reaching common conclusions. There is no need to bring in non-gravitational forces to dispute the erroneous starting point.

[Q-reeus]

I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

Deriving motion from the field equations is a complex prodedure, but it has been done now dozens of different ways by many researcher's over decades, reaching common conclusions.
Thanks for that clarification. As I say I can merely comment on the sidelines at that level. If I recall it right though Einstein himself was a doubter of GW's some twenty odd years after publishing the final version of GR. Lucky I believe to have his reputation saved by an anonymous journal referee who knocked back his 'rebuttal' of GW's as physically real. Subsequently someone persuaded him otherwise no doubt. And to his dying day refused to believe BH's were possible - maybe a sad case of pupil's supplanting teacher; dunno myself.

[PAllen]

Thanks for that clarification. As I say I can merely comment on the sidelines at that level. If I recall it right though Einstein himself was a doubter of GW's some twenty odd years after publishing the final version of GR. Lucky I believe to have his reputation saved by an anonymous journal referee who knocked back his 'rebuttal' of GW's as physically real. Subsequently someone persuaded him otherwise no doubt. And to his dying day refused to believe BH's were possible - maybe a sad case of pupil's supplanting teacher; dunno myself.

Yes, it is well known that Einstein never accepted that Black holes could form by realisitic physical processes. However, he didn't live to see the work of Chandresekhar, Penrose, and Hawking. I expect he would have been swayed.

[TrickyDicky]

I stopped quoting where I did for a very specific reason - the point at which a statement was made that differed with 70 years of virtually all other researcher's conclusions going back to Einstein and Infeld. Further, it is, in fact, pretty obvious that if all massive bodies follow geodesics exactly, there is no GW. So this is the fundamental starting point which must be disputed. And the way it is disputed is as I described: derive motion from the field equations rather than a-priori assuming the geodesic hypothesis.

So would you argue that the adimensional point that is the center of gravity of a massive body is not following a geodesic motion no matter how massive, is it not following an inertial path (in GR terms)?

[PAllen]

So would you argue that the adimensional point that is the center of gravity of a massive body is not following a geodesic motion no matter how massive, is it not following an inertial path (in GR terms)?

Yes, massive, non-singular bodies do not exactly follow geodesics. Out of curiosity, I researched a little when this was first convincingly shown, and it is even earlier than I thought. Vladimir Fock showed in 1939 that non-singular bodies that are massive enough to have self gravitation (again, no need for any non-gravitational forces to be considered) do not exactly follow geodesics.

[EDIT: And, continuing the historic trend of putting messy calculations on a more rigorous footing, here is a paper by a recent contributor to these forums verifying the main conclusions of Fock:

http://arxiv.org/abs/0806.3293
]

[TrickyDicky]

Yes, massive, non-singular bodies do not exactly follow geodesics.


This is IMO an important point worth to clarify. I would say that by the Equivalence principle, the adimensional center of mass of any massive body is considered "a local inertial frame" and must be following exactly a geodesic path. Is this statement not true? Or is the Equivalence principle not stating this? Or is the equivalence principle not valid here?

[PAllen]

This is IMO an important point worth to clarify. I would say that by the Equivalence principle, the adimensional center of mass of any massive body is considered "a local inertial frame" and must be following exactly a geodesic path. Is this statement not true? Or is the Equivalence principle not stating this? Or is the equivalence principle not valid here?

Why would the inside of massive body be expected to be (precisely) a local inertial frame? I would expect the opposite, in the sense the metric cannot be transformed to Minkowski form over the scale of interest (the massive body).

The principle of equivalence is problematic as a precise rule, rather than a useful guide, as Bcrowell and others here have helped me understand with pointers to various papers.

However, Clifford Will discusses his take on its usage for precisely this scenario of massive self gravitating bodies:

http://relativity.livingreviews.org/Articles/lrr-2006-3/

especially sections 3.1.2, 4.1.1 and 4.1.2

[TrickyDicky]

Why would the inside of massive body be expected to be (precisely) a local inertial frame? I would expect the opposite, in the sense the metric cannot be transformed to Minkowski form over the scale of interest (the massive body).

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?




The principle of equivalence is problematic as a precise rule, rather than a useful guide
But it's still valid, right? at the very least in its weak form.

[PAllen]

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?




But it's still valid, right? at the very least in its weak form.

I don't know of any simple justification. When a body must be treated as both a source of gravitation and responding to gravitation, The EFE are are complex, non-linear. I have read through a couple of such analyses of motion from the EFE, most carefully an old one by Synge (1960), and followed that different methods keep coming to the same conclusion. I was under the belief that center of mass is ill defined in GR, but have not really thought much about it. Birkhoff's theorem is only defined for non-rotating spherical bodies, which is what would lead me to think that the ability to treat any other body as equivalent to mass at a point is suspect.

As for the equivalence principle, in its most classic form (indistinguishability of an accelerating lab from a lab sitting on a planet) , it is true only locally, when all tidal effects can be ignored. Will describes 3 other formulations (all different from Einstein's original formulation). He argues that GR is the only known theory, consistent with experiment, that satisfies all of his 3 forms of equivalence principle. His strong one is specifically designed to be tested in the presence of massive self gravitating bodies that radiate and don't follow geodesics exactly.

[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]

[bcrowell]

The center of mass is the single point on a structure which characterizes the motion of the object if the object shrinks to a point mass.
In a curved spacetime, obviously the center of mass is not following an SR inertial path, but a geodesic, or the straightes path in a curved space, but now you are saying that a massive enough body's trajectory,(wich can be characterized by that of its center of mass, as if it were a test particle)is not a geodesic, and I just don't understand what the reason is, all books I consult say in the absence of non-gravitational forces a body follows geodesic motion, can you explain with your own words why this is not the case?

Amplifying on PAllen's response to this question: --

If the books you are referring to are carefully written, they should not say that "in the absence of non-gravitational forces a body follows geodesic motion." They should say that "in the absence of non-gravitational forces a body with negligible mass follows geodesic motion."

The basic reason is the one I gave in #34.

-Ben

[bcrowell]

[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]

Another rigorous paper is this one: Ehlers and Geroch, http://arxiv.org/abs/gr-qc/0309074v1

It is nontrivial to even formulate what is meant by geodesic motion of a test body, and a lot of the Ehlers and Geroch paper is taken up with that.

Another thing to note is that if energy conditions are violated, you can't necessarily prove geodesic motion of test bodies.

I made an attempt here http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3) to explain the Ehlers and Geroch argument in a way that would be accessible to people who are not big-time GR technicians. It's hard to say whether I succeeded, since I'm not a big-time GR technician myself :-)

[TrickyDicky]

I don't know of any simple justification. When a body must be treated as both a source of gravitation and responding to gravitation, The EFE are are complex, non-linear. I have read through a couple of such analyses of motion from the EFE, most carefully an old one by Synge (1960), and followed that different methods keep coming to the same conclusion. I was under the belief that center of mass is ill defined in GR, but have not really thought much about it. Birkhoff's theorem is only defined for non-rotating spherical bodies, which is what would lead me to think that the ability to treat any other body as equivalent to mass at a point is suspect.
It is true that the center of mass is ill defined in GR, according to Krasinski "Introduction to general relativity": " In relativity, so far there is not even a generally accepted definition of the centre of mass, although work on this problem is being done. Thus, when we consider the orbits of point bodies in relativity, we are in fact extending the theory into the domain in which it has not been worked out yet. Nevertheless, these results agree with observational tests."
In practice orbits are considered geodesic paths and I haven't found any reason to think this shouldn't apply to the binary pulsar orbit. A reference where this is stated explicitly would help.

As for the equivalence principle, in its most classic form (indistinguishability of an accelerating lab from a lab sitting on a planet) , it is true only locally, when all tidal effects can be ignored. Will describes 3 other formulations (all different from Einstein's original formulation). He argues that GR is the only known theory, consistent with experiment, that satisfies all of his 3 forms of equivalence principle. His strong one is specifically designed to be tested in the presence of massive self gravitating bodies that radiate and don't follow geodesics exactly.
That is a valuable opinion but it's just C. Will's, not necessarily the consensus in GR.
[Edit: If you're willing to tackle it, I really got the impression the Sam Gralla paper I linked to is much more careful than any prior presentation I've read. Since you seem particularly sensitive to 'ignored details' (not a bad thing at all), this paper might be more satisfying than other treatments. Repeating the link:
http://arxiv.org/abs/0806.3293 ]
The notion of self-gravitation is not without its own conceptual problems , since the gravitational field can be made to vanish for a free-falling body following a geodesic path or just by choosing the appropriate coordinates.

The bottom-line is much of this seems to be not fully worked out yet, and no generally accepted conclusion has been drawn, in this context, leaning on recent papers not yet fully validated to reject completely a perhaps valid premise is not appropriate IMO.
It just seems unfair to anybody's work, no matter how flawed it seems to dismiss it just by saying that his premise is clearly wrong because it is obvious that GW exist when his purpose is precisely to prove that they don't exist.

[TrickyDicky]

I made an attempt here http://www.lightandmatter.com/html_books/genrel/ch08/ch08.html#Section8.1 (subsection 8.1.3) to explain the Ehlers and Geroch argument in a way that would be accessible to people who are not big-time GR technicians. It's hard to say whether I succeeded, since I'm not a big-time GR technician myself :-)
There seems to be some conflating between geodesic in curved spacetime and inertial paths in flat spacetime here. For instance when you assert: "The world-line of a such a body therefore depends on its mass, and this shows that its world-line cannot be an exact geodesic, since the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics."
First, in flat space parallel inertial paths remain paralle but in curved space all initially parallel geodesics diverge.
Second, I thought all bodies, no matter their mass are affected in the same way by gravity (curvature), isn't that the principle of equivalence between inertial mass and gravitational mass? or the reason a hammer and a feather fall at the same time on the moon's floor.
Also, why can't both be geodesics?, in a sphere longitude lines departing from the poles diverge and they are all geodesics. The divergence of the masses is derived from the fact that in a real setting gravitational fields are not uniform, orbiting bodies have a near spherical shape so when acting as sources of curvature they produce different geodesics for different giving rise to tidal forces, in fact that is the reason we feel the curvature, and what geodesic deviation measures.

Also it is obvious that not all points in a massive body follow a geodesic path,(maybe this is what you and Pallen mean) the further from the center of mass the more so, at the surface of the body, that is gonna be rotating, a test particle at rest is not allowed to follow geodesic motion, that is felt as an acceleration that is commonly described "as gravity" but that in fact is due to non-gravitational forces, like the EM forces that keep matter united or the Pauli exclusion principle affecting fermions.

The cited Geroch paper seems to be in agreement with what I'm saying.


On the other hand if all massive bodies, (which is the same that saying all bodies) fail to follow geodesic paths that would mean the only type of motion is non-geodesic motion and this doesn't seem to be right in GR.


Perhaps it would be interesting to discuss this in a new thread since it is not directly related to the OP.

[bcrowell]

There seems to be some conflating between geodesic in curved spacetime and inertial paths in flat spacetime here. For instance when you assert: "The world-line of a such a body therefore depends on its mass, and this shows that its world-line cannot be an exact geodesic, since the initially tangent world-lines of two different masses diverge from one another, and these two world-lines can't both be geodesics."
First, in flat space parallel inertial paths remain paralle but in curved space all initially parallel geodesics diverge.
Initially parallel geodesics at a distance from one another diverge. Initially parallel geodesics at the same point are the same geodesic. (This follows from the fact that the geodesic equation is a second-order differential equation, and there are uniqueness theorems for the solutions of such equations.)


Second, I thought all bodies, no matter their mass are affected in the same way by gravity (curvature), isn't that the principle of equivalence between inertial mass and gravitational mass? or the reason a hammer and a feather fall at the same time on the moon's floor.
This is the same issue we keep coming back to. That is only an approximation for small masses.


Also, why can't both be geodesics?, in a sphere longitude lines departing from the poles diverge and they are all geodesics.
They aren't parallel at the pole.


The cited Geroch paper seems to be in agreement with what I'm saying.
No, you've misunderstood the paper.


On the other hand if all massive bodies, (which is the same that saying all bodies) fail to follow geodesic paths that would mean the only type of motion is non-geodesic motion and this doesn't seem to be right in GR.
Geodesic motion is the limiting case where the body's mass is small. This is standard stuff that you can find in any GR textbook. If you tell us what GR textbook(s) you own, we can point you to the relevant material in them.

[bcrowell]

The bottom-line is much of this seems to be not fully worked out yet, and no generally accepted conclusion has been drawn, in this context, leaning on recent papers not yet fully validated to reject completely a perhaps valid premise is not appropriate IMO.
No, it has been fully worked out, there is a generally accepted conclusion, and it's been a generally accepted conclusion since roughly 1940 (some history here http://en.wikipedia.org/wiki/Sticky_bead_argument#Einstein.27s_double_reversal ).


It just seems unfair to anybody's work, no matter how flawed it seems to dismiss it just by saying that his premise is clearly wrong because it is obvious that GW exist when his purpose is precisely to prove that they don't exist.
You've misread what PAllen wrote in #29. He didn't assume gravitational waves. He described calculations that assume only the validity of the Einstein field equations.

[TrickyDicky]

This is the same issue we keep coming back to. That is only an approximation for small masses.

The Weak Equivalence principle is an approximation for small masses?


They aren't parallel at the pole.

Ok change diverge by converge, are they not parallel at the equator?

No, you've misunderstood the paper.

Maybe so, but is it not trying to prove that small bodies follow geodesic motion?, how does that disagree with my trying to shed light about why shouldn't any size body's trajectory, which is a unidimensionalline regardless its size, be considered a geodesic motion?.

Geodesic motion is the limiting case where the body's mass is small.I know the standard reason given to assert this is that massive bodies have a self-gravitation that interferes with the background curvature, but this seem counterintuitive when it is also asserted that freefalling bodies gravitational field vanishes at the appropriate coordinates, and that their inertial mass is equivalent to their gravitational mass.

[TrickyDicky]

No, it has been fully worked out, there is a generally accepted conclusion, and it's been a generally accepted conclusion since roughly 1940 (some history here http://en.wikipedia.org/wiki/Sticky_bead_argument#Einstein.27s_double_reversal ).

I was referring to the center of mass in GR, what does that have to do with the GW and Einstein story?


You've misread what PAllen wrote in #29. He didn't assume gravitational waves. He described calculations that assume only the validity of the Einstein field equations.

In fact I was thinking of your post #34 when I wrote that. I misquoted Pallen.

[PAllen]

The Weak Equivalence principle is an approximation for small masses?

Here is a reasonably precise statement, with appropriate caveats, from Clifford Will:
"An alternative statement of WEP is that the trajectory of a freely falling “test” body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition "


Ok change diverge by converge, are they not parallel at the equator?

Bcrowell was clear that he was talking about parallel at the same point. An easy way to see that this must be true (though Bcrowell already gave you a perfectly clear one) is to use the parallel transport definition of geodesic: a curve that parallel transports its tangent vector. There is one curve for each tangent vector at a given point.


I know the standard reason given to assert this is that massive bodies have a self-gravitation that interferes with the background curvature, but this seem counterintuitive when it is also asserted that freefalling bodies gravitational field vanishes at the appropriate coordinates, and that their inertial mass is equivalent to their gravitational mass.

Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.

[TrickyDicky]

Here is a reasonably precise statement, with appropriate caveats, from Clifford Will:
"An alternative statement of WEP is that the trajectory of a freely falling “test” body (one not acted upon by such forces as electromagnetism and too small to be affected by tidal gravitational forces) is independent of its internal structure and composition "
So once again, are you saying that the WEP is not valid for massive bodies? only for idealized test particles? just asking...



Bcrowell was clear that he was talking about parallel at the same point. An easy way to see that this must be true (though Bcrowell already gave you a perfectly clear one) is to use the parallel transport definition of geodesic: a curve that parallel transports its tangent vector. There is one curve for each tangent vector at a given point.
He also said if they are at the same point they are the same geodesic.


Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.
We are talking about its path, and I'm saying the earth's path is a geodesic. The earth is of course a source of curvature too.

[Q-reeus]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?

[PAllen]

He also said if they are at the same point they are the same geodesic.



No, this is what Bcrowell said, that I was justifying a different way:

"Initially parallel geodesics at a distance from one another diverge. Initially parallel geodesics at the same point are the same geodesic. (This follows from the fact that the geodesic equation is a second-order differential equation, and there are uniqueness theorems for the solutions of such equations.)"

I was saying the parallel transport definition of a geodesic makes it obvious that this must be so.

[Mentz114]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

[cosmik debris]

Are you saying that the earth, which is free falling around the sun, may be treated as if it has no gravitational field? I've never heard such a claim. Can you justify it or give a reference for what this could mean.

I'm guessing he means that the connection can be made to vanish for a body in geodesic motion.

[Q-reeus]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum...
OK thanks, but it's all been centering here about small departures from geodesic motion tying in or not with GW emission, and if spatial extent was a prerequisite, seemed very unlikely there would be even close to a match to the linearized GR quadrupole formula. Guessing the Hulse-Taylor system was basically treated as two orbiting point masses re orbital period decline at least - maybe extended bodies approach for more refined calcs. :zzz:

[bcrowell]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

I don't think this is true. The body radiates gravitational waves at a rate that is proportional to the square of its mass. Therefore its trajectory depends on its mass, and can't be a geodesic.

[TrickyDicky]

So in the context of full GR treatment, when talking of 'small body' vs 'large body' re any departure from geodesic motion of co-orbiting masses, is this basically referring to mass as the only important determinant, or spatial extent? If the latter, wouldn't this imply it all hinging on 2nd and higher order extended body correction terms, making the rationale for GW generation quite different from the linearized theory where point masses are assumed?
Yes, it seems contradictory that GW are derived from linearized , Newtonian limit theory where point masses are assumed and the center of mass is completely accepted representation the body motion no matter how massive. and at the same time we are told that GW can't exist if massive bodies follow geodesic motion.
I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.Good point too.

I'm guessing he means that the connection can be made to vanish for a body in geodesic motion.
Yes that is exactly wha I mean, thanks. Connections can be derived from the vanishing of the covariant derivative of the metric tensor and we can introduce a special coordinate system, called a geodesic coordinate system, in which the connection vanishes for a body in geodesic motion.

[PAllen]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

Do you have a reference to a GR definition of COM? Spurred by discussion on this thread, I reviewed some of my GR books on this. J. L. Synge claims to prove that for every foliation of spacetime in to spacelike hypersurfaces, there is a *different* COM for a given world tube; therefore there is no such thing as an invariant COM in GR. His philosophy GR is that all physical quantities must have some invariant definition, therefore there is no physically well deffined COM in GR. Admittedly, his work is circa 1960, and the field evolves. Thus, if you have a more recent reference I would be interested.

[Mentz114]

Do you have a reference to a GR definition of COM? Spurred by discussion on this thread, I reviewed some of my GR books on this. J. L. Synge claims to prove that for every foliation of spacetime in to spacelike hypersurfaces, there is a *different* COM for a given world tube; therefore there is no such thing as an invariant COM in GR. His philosophy GR is that all physical quantities must have some invariant definition, therefore there is no physically well deffined COM in GR. Admittedly, his work is circa 1960, and the field evolves. Thus, if you have a more recent reference I would be interested.
I haven't got a reference but I'll do some resarch. It was probably proved by Frottmann in 1742.

I have a lot of respect for J. L. Synge, so I'll have to invoke a local frame to define the COM with some normal coordinates. If the extension is smaller than the radius curvature, but not too small, it might work.

I've sketched a proof, but it needs thinking about.

[Edit]After 5 minutes I have found a paper where the conservation of angular momentum is used to define a 'centre-of-mass' line inside the world tube. So I got it backwards.

Schattner, (1978)
http://www.springerlink.com/content/mg846n70582873n8/


This recent survey

http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0456v1.pdf

concludes

4. Equivalence of center of mass

The constant mean curvature foliation constructed in the previous section gives
a geometric center of mass. By our construction, it is easy to see that the geometric
center of mass (3.8) is equal to the classical Hamiltonian notion (1.2) because each
constant mean curvature surface is roughly round centered at p = C + O(R1−2q).

[PAllen]

I suspect that for an incompressible solid extended body falling in the presence of tidal forces, the COM must follow a geodesic or there will be a torque around the COM, which would violate conservation of angular momentum.

Maybe it's possible to prove this, but I haven't come across it.

Another thought on this, related to Bcrowell's comment is that conservation of angular momentum is achieved by taking into account the body plus its radiation (which can carry angular momentum). Thus, it is possible that the body alone does not conserve angular momentum.

[Mentz114]

Another thought on this, related to Bcrowell's comment is that conservation of angular momentum is achieved by taking into account the body plus its radiation (which can carry angular momentum). Thus, it is possible that the body alone does not conserve angular momentum.

Sorry, I was editing while you posted this.

So, can a body in free-fall slow down by radiating GW ?

[pervect]

I'd say yes, the binary-pulsar being an example.

[Mentz114]

I'd say yes, the binary-pulsar being an example.

Are they not speeding up ? But this is radiation from geodesics that have non-zero proper acceleration.
(I'm basing that on the Hagihara frame where there is acceleration towards the centre).

I was thinking about falling bodies, and whether they could have a non-zero quadrupole moment. My guess is not. So a falling extended body must conserve angular momentum, no ?

[PAllen]

Are they not speeding up ? But this is radiation from geodesics that have non-zero proper acceleration.

I was thinking about falling bodies, and whether they could have a non-zero quadrupole moment. My guess is not. So a falling extended body must conserve angular momentum, no ?

How can a geodesic have non-zero proper acceleration? They are in free fall, but not exactly following geodesics, and radiating GW.

I wouldn't be surprised if you could demonstrate that two isolated objects falling directly toward each other do not radiate and must follow a geodesic. However, that is a very specialized case.

[Mentz114]

How can a geodesic have non-zero proper acceleration? They are in free fall, but not exactly following geodesics, and radiating GW.


Darn, I just checked the Hagiahara frame and the equatorial frame has no acceleration ( as you say, it's a geodesic ).

OK, but I asked if a falling body can slow down by radiating and the circular orbits are not relevant to that.

I wouldn't be surprised if you could demonstrate that two isolated objects falling directly toward each other do not radiate and must follow a geodesic. However, that is a very specialized case.

Still not an answer. And so what if it's a 'specialized' case. We have to consider everything.

Anyhow, if a falling body begins to rotate, conservation of angular momentum is lost whether or not it radiates.

[PAllen]

Darn, I just checked the Hagiahara frame and the equatorial frame has no acceleration ( as you say, it's a geodesic ).

OK, but I asked if a falling body can slow down by radiating and the circular orbits are not relevant to that.



Still not an answer. And so what if it's a 'specialized' case. We have to consider everything.

One of the discussions on this thread is whether real massive bodies exactly follow geodesics, in general, if there are no forces on them. While interesting, a finding that idealized massive bodies in one unique state of relative modtion exactly follow geodesics does little to answer the general issue.

[PAllen]

I haven't got a reference but I'll do some resarch. It was probably proved by Frottmann in 1742.

I have a lot of respect for J. L. Synge, so I'll have to invoke a local frame to define the COM with some normal coordinates. If the extension is smaller than the radius curvature, but not too small, it might work.

I've sketched a proof, but it needs thinking about.

[Edit]After 5 minutes I have found a paper where the conservation of angular momentum is used to define a 'centre-of-mass' line inside the world tube. So I got it backwards.

Schattner, (1978)
http://www.springerlink.com/content/mg846n70582873n8/


This recent survey

http://arxiv.org/PS_cache/arxiv/pdf/1101/1101.0456v1.pdf

concludes

Thanks a lot for the references. Very interesting. (I don't think there is any contradiction with Synge; his definitions were different, but presumably, for some reasonable coordinates, his would match the modern definition(s) [several of them shown to be equivalent in the cited paper]. )

[Mentz114]

One of the discussions on this thread is whether real massive bodies exactly follow geodesics, in general, if there are no forces on them. While interesting, a finding that idealized massive bodies in one unique state of relative modtion exactly follow geodesics does little to answer the general issue.

Granted.

My original assertion is a cinch to prove in Newtonian gravity with a spherically symmetric field and might extend to a local coordinate system on a geodesic. But I have other things to do. The motion of extended bodies in GR is obviously a big and difficult topic.

[bcrowell]

I'm pretty sure that if two masses are free-falling toward one another, they radiate. The quadrupole moment is changing. Actually purely radial motion is one of the easiest cases to use if you want to convince someone without a lot of math that gravitational radiation exists. Taylor and Wheeler have an argument to this effect in Spacetime Physics (where they have Atlas lifting a gigantic weight).

I'm not sure it makes sense to discuss the proper acceleration of a massive self-gravitating body in GR. If you attach an accelerometer to the earth, all you measure is the earth's field. An accelerometer attached to a radiating body acts like part of that body, radiating coherently along with it.

[EDIT] Deleted my original version of the second paragraph, which I decided was wrong.

[TrickyDicky]

This seems to be a case when the principles, the foundations of a theory like GR can not be refuted by supposed consequences of something, like GW that is said to be derived from the theory, without incurring in contradiction or falacy.

It is important to remark that a geodesic itself is inevitably a somewhat idealized concept, like most habitually used concepts in physics without loss of generality and validity of the empirical results obtained from them. Since the trajectory a body describes is also a set of points, necessarily a massive body, no matter its size is idealized to a massive point when speaking about its location in the trajectory it draws, a timelike geodesic path or trajectory is a set of points at which the Chrystophel symbols can be made to vanish, irrespective of the size or mass of the body that describes that path, that path is a one-dimensional curved line in curved spacetime. For reference see:
http://en.wikipedia.org/wiki/Normal_coordinates#Geodesic_normal_coordinates
This is notwithstanding the fact that in a body in motion for instance rotating there will be points that are obviously not describing the same path than the object to wich they belong because they have motion relative to the center of mass of the object.


Here is a paragraph taken from the GR textbook of a relevant relativist that summarizes perfectly what I'm trying to get across
So going to the principles of GR one finds that:

"In Einstein’s General Relativity, gravity manifests itself by a tensor field and, in the absence of other forces, the motion of a particle is determined by this tensor field; it does not depend on the mass of the particle. Einstein’s revolutionary idea is that the arena where the gravity tensor lives is itself determined by gravity, both are united in a 4-dimensional Lorentzian manifold (V, g), called the spacetime. The trajectories of particles are geodesics of the metric g. Einstein’s weak equivalence principle corresponds to the fact that in a general Lorentzian spacetime the geodesic equations are formally the same as in Minkowski spacetime with arbitrary coordinates where the non-vanishing of the Christoffel symbols signals the presence of inertial forces due to the noninertialframe of reference.

In a general spacetime (V, g), it is always possible at one given point, to choose local coordinates such that the Christoffel symbols vanish at that point; gravity and relative acceleration are then, at that point, exactly balanced. It is even possible to choose local coordinates such that the Christoffel symbols vanish along one given geodesic; astronauts in spacecraft have made popular the fact that in free fall one feels neither acceleration nor gravity; in a small enough neighbourhood of a geodesic the relative accelerations of objects in free fall are approximately zero. Massive pointlike objects in free fall follow a timelike geodesic"
General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation.


From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.
The consequences this may have for GW are not for me to spell out with authority (I'd rather let everyone draw their own conclusions), since I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.

[Physics Monkey]

From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.
The consequences this may have for GW are not for me to spell out with authority (I'd rather let everyone draw their own conclusions), since I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.

I can't agree. Let us take the simple limit of a binary where one star is much larger than the other. This allows us to study the motion of the smaller star in the bigger star's background. Now it simply must be true that the smaller star does not behave exactly as a point mass following a geodesic in the background of the big star because energy, angular momentum, etc. will be radiated in the form of gravitational waves. This conclusion is well supported by analytical, numerical, and experimental evidence.

Confusion may arise because:
1) the effect can be small, thus considerations of astronauts or moons may not be sufficient
2) calculation of the effect requires one to regulate certain divergences, thus it is technically a bit onerous

Nevertheless, the issue is well studied as the many references in the thread indicate. There is no contradiction and no need to assault the foundations of the theory. Perhaps the strongest complaint one might make is that various pedagogical sources obfuscate the physics in the name of simplicity or otherwise make overly general introductory claims. I am very sympathetic to this claim, but I don't think we should let it distract us now.

Here are two more references that I find useful:
1) a comprehensive review http://arxiv.org/abs/grqc/0306052
2) a nice conceptual framework http://arxiv.org/abs/hep-th/0409156

[TrickyDicky]

I can't agree. Let us take the simple limit of a binary where one star is much larger than the other. This allows us to study the motion of the smaller star in the bigger star's background. Now it simply must be true that the smaller star does not behave exactly as a point mass following a geodesic in the background of the big star because energy, angular momentum, etc. will be radiated in the form of gravitational waves.
But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.
Besides your example happens to go against your thesis, even in your own view, the bigger the difference in mass between two bodies in a binary orbit the less GW radiation. In fact for systems like the sun and planets the radiation is usually neglected for all practical calculations.


This conclusion is well supported by analytical, numerical, and experimental evidence.

To this date no direct experimental evidence of GW has been shown.

...various pedagogical sources obfuscate the physics in the name of simplicity or otherwise make overly general introductory claims. I am very sympathetic to this claim, but I don't think we should let it distract us now.
I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

[PAllen]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.

WEP was a motivating principle for the theory. It is not an axiom. Some respected authors (e.g. J. L. Synge strongly argue that it shouldn't even be taught anymore because, mathematically speaking, it is simply false for GR. The more consensus view is that it is valid heuristically, and can be made true in the limit, though there are numerous papers (Bcrowell has provided links) that show it is basically impossible to formulate fully precise, mathematically true, formulation of it).

The geodesic hypothesis was initially introduced as a separate element of GR from the field equations. However, after overcoming his flip flopping on GW, Einstein (with Infeld and Hoffman) became a strong proponent of the idea that the geodesic hypothesis should be deleted from the theory as if it never exised; that all motion follows from the field equations - which directly lead to tiny deviations from geodesic motion, while also showing (in the limit) that geodesic motion follows from the field equations.



To this date no direct experimental evidence of GW has been shown.

True, only strong indirect evidence. However, there are other predictions of GR that have also not been verified yet. One can even say length contraction in SR has not been directly verified. So what?


I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

Your reference is propounding the pedogogical value of the equivalence principle and the geodesic hypothesis. I am sure, if you asked the author, he/she would agree that they were glossing over the details.

[bcrowell]

From all this. I think it is a closed case that the neutron stars in a binary pulsar system describe with their orbits time-like geodesic paths.

Nope, you're wrong. The quote you gave from the textbook doesn't explicitly say so, but it's implicitly assuming that the objects have low enough masses so that radiation is negligible. This has been well understood and noncontroversial since ca. 1940.


I'm no expert, and as bcrowell admitted he is no expert either, he can't make authority claims in this line, much less dismiss the very foundations of the theory to justify a pretended derived result.
Let's get straight what would really be meant by an appeal to authority versus an appeal to logic. An appeal to authority would be if A invokes a statement by an authority figure B without fully understanding the evidence and logic behind that statement. This is what is happening when you quote from the text by "General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation." An appeal to logic would be what I did by linking to my exposition, which you didn't seem to understand, of the ideas in the Geroch proof.

The reason we keep telling you that there has been a universal consensus about this since 1940 is not because we are trying to win an argument by an appeal to authority. It's in response to your repeated erroneous claims that it is controversial or unknown, based on your failure to understand the physics.

[bcrowell]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.
No, nobody is reasoning backward. We are starting from the Einstein field equations, and as a result we find that trajectories are not geodesic and gravitational waves exist. The equivalence principle is a statement about the limiting case where the mass of the test object is small, so there is no contradiction.

[Physics Monkey]

But once again you are going from the derived conclusion to the premise, you are reasoning backwards, if you take for granted GW which is a derived prediction of the theory that might or not be confirmed, to discard a basic tenet of the theory like the WEP and geodesic motion in differential geometry, you can only get contradiction.

I merely stated that assuming gravitational waves are emitted then the orbit must not be a geodesic. Gravitational waves are derived from the theory and this derivation in no way conflicts with the basic premise that particles follow geodesics in the limit of vanishing size and mass. Are you challenging the claim the a small body orbiting a larger body in GR will emit gravitational waves?

Also, you are conflating two issues. Issue 1 is whether GR is a consistent theory that predicts gravitational waves and deviations from geodesic motion for bodies of finite size and mass. Both of these facts are well established consequences of GR. You need only go through the derivations yourself (I provided a comprehensive reference). Note that observations in our universe have nothing directly to do with issue 1. Issue 2 is whether GR is a good description for the low energy physics of gravity in our universe. This is obviously an experimental/observational question. The title of this thread and the subsequent discussion suggest to me that we are primarily discussing issue 1.


Besides your example happens to go against your thesis, even in your own view, the bigger the difference in mass between two bodies in a binary orbit the less GW radiation. In fact for systems like the sun and planets the radiation is usually neglected for all practical calculations.


I fail to see how this follows. Whether the radiation output is large or small by some measure is irrelevant for the conceptual question. I used the large mass difference example because it is a clean theoretical laboratory, but I explicitly said that the effect may turn out to be quite small. Certainly the effect exists in GR and can be made large by varying parameters. In fact, the system I described may be applicable for direct detection efforts e.g. for neutron stars orbiting supermassive black holes.


To this date no direct experimental evidence of GW has been shown.


By direct presumably you mean no one has seen a mirror wobble due to a gravitational wave. Of course I agree. However, as you surely know the effects of gravitational waves have been indirectly shown in orbital decay of binary pulsars. The observed decay agrees well with theoretical calculations using GR. Given the many other tests of GR, I'd be happy to bet a large sum that gravitational waves are a part of the physical universe. The real question now is not do gravitational waves exist (they do), but instead what we can learn from them about cosmology, black holes, etc.

Of course, this is again irrelevant for the question of geodesic motion within the theory of GR. As I said, there is convincing analytic and numerical evidence that gravitational waves exist in GR, that they are emitted in the system I described, and that relatedly bodies of finite mass do not exactly follow geodesics.


I don't think the reference I've presented obfuscate the physics in any way, maybe you conflate apparent simplicity with lack of rigour, but certainly this is not the case.
But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

You misunderstand me. I was not attempting to impugn your references but rather to encourage you by noting that this would not be first time simplified statements lead to confusion.

[Q-reeus]

Admittedly as a GR layman seems to me TrtickyDicky (and I assume still Mentz114) have logic on their side here. Are not the EFE's themselves simply a mathematical expression of 'matter tells spacetime how to curve; spacetime tells matter how to move'? So given that in vacuuo two orbiting masses can only experience metric curvature effects, what principle in the EFE's allows effective COM motion other than geodesic? My own simple reasoning here is that finite propagation delay guarantees each co-orbiting mass will 'fall slightly behind' the other - but only as determined by a distant observer. And that this is quite consistent with the COM of each mass locally following a perfect geodesic - but such perfect geodesic motion is an inspiralling non-conservative path. Maybe that's where Loinger is both right and wrong. Or is it a matter of definition that a pure geodesic path must be completely conservative in nature (again - as determined by a distant reference frame)? If so that seems imho to be a sure recipe for paradox.
EDIT: BTW the first bit above is saying I'm the laymen - not TrickyDicky - sorry for any ambiguity there.

[bcrowell]

But these are vague non-arguments , can you give some real argument to refute the references shown in my previous post?

Here are six previous posts linking to six papers giving detailed discussions of this topic:

http://physicsforums.com/showpost.php?p=3261324&postcount=11
http://physicsforums.com/showpost.php?p=3266404&postcount=40
http://physicsforums.com/showpost.php?p=3266574&postcount=42
http://physicsforums.com/showpost.php?p=3266793&postcount=44
http://physicsforums.com/showpost.php?p=3266989&postcount=46
http://physicsforums.com/showpost.php?p=3270075&postcount=75

These are not vague non-arguments.

If you want to, you can make the effort to understand all the detailed mathematical arguments in these papers. If you don't want to do that, then it's ridiculous that you're still complaining that the rest of us are arguing based on appeals to authority, engaging in circular reasoning, violating widely accepted principles, or arriving at contradictions. If, for example, you think there is circular reasoning going on, please point out where that is happening in the Ehlers and Geroch paper.

[Mentz114]

Admittedly as a GR layman seems to me TrickyDicky (and I assume still Mentz114) have logic on their side here...

I'm agnostic, and I hope still learning about angular momentum in GR. My views can change hourly as I discover more. Questions I've asked here have been answered sufficiently to make me think, so ascribing any position to me on these issues would be a risk.

[Q-reeus]

I'm agnostic, and I hope still learning about these things. My views can change hourly as I discover more.
Fair comment - and sorry if I have made you out to be an antagonist of one color here - did hesitate slightly before typing that bit!:tongue:

[bcrowell]

So given that in vacuuo two orbiting masses can only experience metric curvature effects, what principle in the EFE's allows effective COM motion other than geodesic?
This is a good question. One way to see that this argument doesn't quite work is to see that it can be adapted to make it into an argument about electromagnetism, which then predicts the wrong thing about electromagnetism. Suppose that we have an electromagnetic field in a certain region. We have two test particles. Particle 1 has mass m and charge q. Particle 2 has mass 2m and charge 2q. Based on Newton's laws, we expect that 1 and 2 should follow identical trajectories if they are given the same initial motion. But in fact they both radiate electromagnetic waves, and the Larmour formula shows that the radiated power is (initially) equal for 1 and 2. Since the radiated power is equal, the decelerations of the particles are unequal, so they follow different trajectories. Now we could ask, "what principle in Maxwell's equations allows motion other than that predicted by Newton's laws?" The thing to realize here is that Maxwell's equations are fundamental, and Newton's laws are not. Newton's laws don't apply to electromagnetic waves. Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation.

Completing the analogy with GR, what is fundamental is the Einstein field equations(~Maxwell's equations), and what is not fundamental is geodesic motion(~Newton's laws). Newton's laws are not a good approximation for an electromagnetically radiating system. Geodesic motion is not a good approximation for a gravitationally radiating system.


My own simple reasoning here is that finite propagation delay guarantees each co-orbiting mass will 'fall slightly behind' the other - but only as determined by a distant observer. And that this is quite consistent with the COM of each mass locally following a perfect geodesic - but such perfect geodesic motion is an inspiralling non-conservative path.
To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1, even if their initial conditions are identical. Since geodesics that start out parallel at the same point are the same geodesic, it follows that the trajectories of 1 and 2 cannot both be geodesic -- in fact, neither is.

[TrickyDicky]

Let's get straight what would really be meant by an appeal to authority versus an appeal to logic. An appeal to authority would be if A invokes a statement by an authority figure B without fully understanding the evidence and logic behind that statement. This is what is happening when you quote from the text by "General Relativity and Einstein’s Equations by Yvonne Choquet-Bruhat, former president of the International committee on general relativity and gravitation." An appeal to logic would be what I did by linking to my exposition, which you didn't seem to understand, of the ideas in the Geroch proof.

What exposition?, all you did was link papers, but I guess when you do it is not an appeal to authority, that only happens when I quote pertinent paragraphs because obviously I'm very confused and wrong about all I write, I don't think I'll ever achieve your perfect understanding about GR.
You certainly don't seem to be a teacher, kind of feel sorry for your students.

[Q-reeus]

This is a good question. One way to see that this argument doesn't quite work is to see that it can be adapted to make it into an argument about electromagnetism, which then predicts the wrong thing about electromagnetism. Suppose that we have an electromagnetic field in a certain region. We have two test particles. Particle 1 has mass m and charge q. Particle 2 has mass 2m and charge 2q. Based on Newton's laws, we expect that 1 and 2 should follow identical trajectories if they are given the same initial motion. But in fact they both radiate electromagnetic waves, the Larmour formula shows that the radiated power is (initially) equal for 1 and 2. Since the radiated power is equal, the decelerations of the particles are unequal, so they follow different trajectories. Now we could ask, "what principle in Maxwell's equations allows motion other than that predicted by Newton's laws?" The thing to realize here is that Maxwell's equations are fundamental, and Newton's laws are not. Newton's laws don't apply to electromagnetic waves. Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation.

Completing the analogy with GR, what is fundamental is the Einstein field equations(~Maxwell's equations), and what is not fundamental is geodesic motion(~Newton's laws). Newton's laws are not a good approximation for an electromagnetically radiating system. Geodesic motion is not a good approximation for a gravitationally radiating system.
Thanks for your input here, but some aspects are leaving me baffled. The analogy with Newton's laws vs Maxwell's equations is apt to a point but of course represents the way Newton would have understood it without knowing about radiation, but I think we agree including radiation ('photons flitting off') Newton's laws retain their validity in a more generalized setting - all played out on a flat Minkowski backdrop.
My understanding has been that GR being a metric theory expresses all gravitationally induced effects as owing to metric curvature - including any 'radiation reaction' resulting in in-spiral. Now if that has become a bit old-hat all I can think is GR has quietly evolved into a kind of field theory. If that's even remotely correct then my earlier position clearly becomes no longer relevant. But such a transformation of the very foundations of GR would be big news to me and I think many others here, even if not to GR insiders. If not a now de facto field theory, the rationale for non-geodesic behavior still eludes me. I don't consider it fundamental in that respect that unequal masses would follow different geodesic paths - only that each mass COM is locally geodesic - ie perfectly free-fall. Again, maybe a matter of defining the term appropriately.

..To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1.

Much the same comments as above. Not trying to be picky here, but wouldn't the smaller mass, having to move that much further and faster (almost perfectly Newton's laws in action!, ha ha) be experiencing the greater power loss? EDIT: Was thinking the two masses were meant to be a binary system - but you were meanng each are separately gravitationally coupled to a much larger mass (planet or whatever)?
FURTHER EDIT: Some more thought on your last point (taking the last viewpoint is correct). I agree that is a strong consideration. Would be convinced if the sums have been done showing the net curvature of 'planet' + test mass is definitely not locally geodesic - something needing a proper explanation surely!

[TrickyDicky]

WEP was a motivating principle for the theory. It is not an axiom. Some respected authors (e.g. J. L. Synge strongly argue that it shouldn't even be taught anymore because, mathematically speaking, it is simply false for GR. The more consensus view is that it is valid heuristically, and can be made true in the limit, though there are numerous papers (Bcrowell has provided links) that show it is basically impossible to formulate fully precise, mathematically true, formulation of it).

The geodesic hypothesis was initially introduced as a separate element of GR from the field equations. However, after overcoming his flip flopping on GW, Einstein (with Infeld and Hoffman) became a strong proponent of the idea that the geodesic hypothesis should be deleted from the theory as if it never exised; that all motion follows from the field equations - which directly lead to tiny deviations from geodesic motion, while also showing (in the limit) that geodesic motion follows from the field equations.
I have to disagree with your view on relativity history,but then this is a highly subjective and opinable matter.

True, only strong indirect evidence. However, there are other predictions of GR that have also not been verified yet. One can even say length contraction in SR has not been directly verified. So what?
So what? My phrase was merely answering a previous statement saying otherwise. This question is gratuitously argumentative, don't you think?


Your reference is propounding the pedogogical value of the equivalence principle and the geodesic hypothesis. I am sure, if you asked the author, he/she would agree that they were glossing over the details.
The wikipedia article linked had all needed details. But if there is any specific detail you may wan to discuss just ask.

[bcrowell]

Thanks for your input here, but some aspects are leaving me baffled. The analogy with Newton's laws vs Maxwell's equations is apt to a point but of course represents the way Newton would have understood it without knowing about radiation, but I think we agree including radiation ('photons flitting off') Newton's laws retain their validity in a more generalized setting - all played out on a flat Minkowski backdrop.
No, there is no way to patch up Newton's laws to handle photons.

Now if that has become a bit old-hat all I can think is GR has quietly evolved into a kind of field theory.
GR is and always has been a classical field theory.



..To see that this is incorrect, consider test masses 1 and 2, with masses m and 2m. Mass 2 radiates gravitational waves with four times the power of mass 1. Therefore by conservation of energy, mass 2 in-spirals at a different rate than mass 1.
you were meanng each are separately gravitationally coupled to a much larger mass (planet or whatever)?
Yes.


FURTHER EDIT: Some more thought on your last point (taking the last viewpoint is correct). I agree that is a strong consideration. Would be convinced if the sums have been done showing the net curvature of 'planet' + test mass is definitely not locally geodesic - something needing a proper explanation surely!
Sorry, I don't follow you here.

[Q-reeus]

No, there is no way to patch up Newton's laws to handle photons.
But as you have stated earlier "Maxwell's equations conserve energy and momentum, and they predict that the test particles radiate electromagnetic waves that contain energy and momentum; therefore they predict that there is a radiation reaction on the charge emitting the radiation."
Is not conservaton of momentum by inclusion of radiation a manifestation of Newton's Third Law - equal and opposite reaction? EM radiation may be massless but there is a well defined momentum density.
GR is and always has been a classical field theory.
OK then what I meant was having taken on at least some non-metric character (seemingly implied if as has been stated WEP and geodesic motion are no longer considered fundamental to GR) - where the notion of field directly acting on matter generates a force. Without that, I cannot understand the concept of non-geodesic motion being possible, GW's or not. There is imo either free-fall, or force, or both. Can you explain then how departure from free-fall in GR is possible without introducing the notion of force. If one says GW back-reaction provides such, this to my mind only begs the question - what part of a GW is non-metric!? If one admits it's all metric, how, in a conceptual way is free-fall condition violated - assuming for simplicity an orbiting 'point' test mass?
Sorry, I don't follow you here.
Was referring to your argument about two test masses starting from the same point and following different geodesics. This indeed seems to prove non-geodesic motion in general. But I conjecture this may be untrue once the combined curvature owing to both central mass plus test mass are fully accounted for. Hence the call for some literature having proven otherwise. If you know of any, would appreciate a link. :zzz: