Which Of These Neutron Stars Must Have Had Its Angular Momentum Changed By A Binary Companion?
Conservation of angular momentum in GR
- Thread starter TrickyDicky
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In this context, at that place is something I don't sympathize nigh an example likewise found oftentimes in GR textbooks: in the Earth-moon arrangement there is a tidal torque due to the moon's influence and as well the sunday's, that changes the earth spin angular momentum by acting on the equator burl and that slows downwards the earth's spin, nevertheless in this instance the total angular momentum is effectively conserved past correcting the orbit angular momentum thru its enlarging of about 4.v cm/yr.
What makes the total angular momentum to be conserved in this detail setting? Is this small system considered practically spherically symmetric?
Answers and Replies
Angular momentum IS conserved in whatsoever asymptotically flat spacetime, fifty-fifty 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 athwart momentum is Not conserved, for case Hobson's GR in the affiliate about the Kerr metric in folio 313: "Note, however, that the total angular momentum of a particle is not a conserved quantity, since the spacetime is not spherically symmetric almost any betoken."
On the other mitt the total athwart momentum in a spherically symmetric spacetime is conserved and therefore the quadrupole moment'due south 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 argument is not corect but in the context of the moon-earth system I'm not sure what you hateful past it anyway, are you saying that the moon-earth system can exist considered asymtotically flat? Delight explain.
TrickyDicky, I have no access to that textbook and probably couldn't make much utilise of it if I did, but but a idea that may be off the mark. Sometimes authors misfile owing to failure to specify context adequately. Maybe 'Total angular momentum is not conserved' Schutz refers to above is that of the orbiting masses just ('radiation reaction' couple slowing the bodies down)? And we are supposed to assume the balance is in the GW'southward? In the case of the World-Moon arrangement, my assumption is probably that GW's are also weak to consider - ie tidal transfer is way larger in event and taken to exist momentum conserving. I call up Clifford Will covers calcs on Earth-Moon tidal coupling in http://relativity.livingreviews.org/Articles/lrr-2006-3/ [Broken] but perchance wrong there.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 take to exist radiated away as gravitational radiation. (run across Schutz, affiliate 9: exercises 39,40 and 47).In this context, there is something I don't empathize about an example too constitute frequently in GR textbooks: in the Globe-moon system in that location is a tidal torque due to the moon's influence and also the lord's day's, that changes the world spin angular momentum by acting on the equator bulge and that slows down the world'southward spin, however in this case the full angular momentum is effectively conserved by correcting the orbit angular momentum thru its enlarging of almost 4.five cm/year.
What makes the total angular momentum to exist conserved in this detail setting? Is this small-scale organisation considered practically spherically symmetric?
In #three
...Hobson'south GR in the affiliate about the Kerr metric in folio 313: "Note, however, that the full angular momentum of a particle is non a conserved quantity, since the spacetime is non spherically symmetric nearly whatsoever point."
Just maybe another example of context not properly specified by writer - an unstated assumption the particle'due south angular momentum is coupling to that of a much larger spinning mass via it'due south Kerr metric - ie a GR variant of Earth-Moon tidal exchange?
On the other paw, would be most interested if none of my comments above hit the mark!
In this case, this seems to exist non controversial common knowledge in GR, I can assure yous my citations are not out of context, I only didn't quote the whole paragraphs.Sometimes authors misfile attributable to failure to specify context fairly. Perhaps 'Full angular momentum is not conserved' Schutz refers to in a higher place is that of the orbiting masses just ('radiation reaction' couple slowing the bodies downward)? And nosotros are supposed to presume the balance is in the GW'due south? In the case of the Earth-Moon system, my assumption is probably that GW'south are too weak to consider - ie tidal transfer is way larger in outcome and taken to be momentum conserving.In #three
Just mayhap another case 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!
See Wikipedia Angular momentum folio: "Athwart momentum in relativistic mechanics:
In modern (late 20th century) theoretical physics, athwart momentum is described using a dissimilar formalism. Under this ceremonial, athwart momentum is the 2-form Noether charge associated with rotational invariance (Every bit a consequence, angular momentum is non conserved for general curved spacetimes, unless it happens to be asymptotically rotationally invariant)."
It is precisely the no conservation of total athwart momentum in GR that allows the quadrupole moment to exist.
I grant you lot my own comparison with the moon-earth arrangement might be not valid hither and thus my question in post #1 but I think Beak-1000 fabricated a not exact assertion AFAICS in his answer.
OK lamentable I wasn't giving you much credit there.In this instance, this seems to be not controversial common noesis in GR, I tin can clinch you my citations are non out of context, I just didn't quote the whole paragraphs...
Well information technology makes sense if energy is ill-defined in curved spacetime, momentum likewise. While asymptotically flat has for me a clear interpretation not then asymptotically rotationally invariant. This specifies whether there is some nonzero metric coupling to the spin of a test particle at space distance from some source of angular momentum (say a notional Kerr BH), or something else? This is new to me.See Wikipedia Angular momentum page: "Angular momentum in relativistic mechanics:
In modern (late 20th century) theoretical physics, angular momentum is described using a dissimilar ceremonial. Under this formalism, angular momentum is the 2-form Noether charge associated with rotational invariance (As a effect, athwart momentum is not conserved for full general curved spacetimes, unless information technology happens to be asymptotically rotationally invariant)."
That surprises me. Two masses connected past a spring undergoing linear oscillation have a nonzero mass quadrupole moment, right? Or are nosotros restricted to the quad moment of say two co-orbiting bodies where angular momentum is inherently present?It is precisely the no conservation of total angular momentum in GR that allows the quadrupole moment to be.
Of class Kerr is not spherically symmetric in the well-nigh zone, and that'south not what I said. It is spherically symmetric in the asymptotic region, which is all yous need. Asymptotically flat ways it approaches Minkowski space as you get far away.Consider the Kerr spacetime, I think it is asympotically flat only it has axisymmetric metric, so it is not spherically symetric
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.this according to diverse GR textbooks means total angular momentum is Not conserved... "Note, however, that the total angular momentum of a particle is non a conserved quantity, since the spacetime is not spherically symmetric about any point."
I agree completely with this statement, it is quite basic and there should be no disagreement.On the other hand the total angular momentum in a spherically symmetric spacetime is conserved and therefore the quadrupole moment's athwart momentum is absent-minded and tin can not produce gravitational radiations. I believe all this to exist basic stuff with no much room for disagreement.
Yes certainly, the gravitational field of the World-Moon organization is asymptotically apartment. The combined gravitational field goes downwardly equally M r-3, where Thou is the total mass, and the spacetime asymptotically approaches Minkowski space.I'g not sure what you mean by it anyway, are you saying that the moon-earth organisation can be considered asymtotically flat? Please explain.
In fact that term "asymptotically rotationally invariant" seems to be non very commonly used, usually just rotational invariance is mentioned.Well it makes sense if energy is ill-defined in curved spacetime, momentum too. While asymptotically apartment has for me a clear interpretation non and then asymptotically rotationally invariant. This specifies whether in that location 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.
Yous are right. I'm actually referring only to binary systems of co-orbiting bodies configurations.That surprises me. Two masses connected by a jump undergoing linear oscillation take a nonzero mass quadrupole moment, correct? Or are we restricted to the quad moment of say two co-orbiting bodies where angular momentum is inherently present?
[...] in the Earth-moon system there is a tidal torque due to the moon'due south influence and too the sun's, that changes the globe spin athwart momentum by acting on the equator bulge and that slows down the earth's spin, however in this case the full angular momentum is effectively conserved by correcting the orbit angular momentum thru its enlarging of nigh four.5 cm/year.
What makes the full angular momentum to be conserved in this particular setting?
At that place are two singled-out furnishings. (Probably in that location a bit of cross-influence, but afaik that's negligable.)
The two have in mutual that they each relate to a style of the Globe not being perfectly spherical.
1. Moon and Sun act upon the Earth's equatorial burl, 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 World's tidal distortion lags behind. Given that lag there is a gravitational effect that increases the Moon's orbital free energy, and it decreases the Earth'due south rotational free energy. The number of four.5 cm/year increment of Moon orbit distance is the increase of Moon orbital free energy.
(Every bit I understand it: when the Moon first formed information technology was rotating somewhat faster than i rotation per month. The Moon was tidally distorted past the World, and rotating more than than once a calendar month. Internal friction makes such distortion lag behind. This lag gave the World opportunity to irksome the Moon's rotation downwardly. At some point in time the Moon reached a state of tidal lock. )
Anyway, the precession of the equinox and the Earth rotation slowing downwards arise both from gravitational distortion effects, but it's ii different effects.
Neither of them are relativistic effects, which makes it unlikely that any relativity textbook writer would mention them.
Ok, that solves the doubt most asymptotically rotational invariance in the above mail also.Of class Kerr is non spherically symmetric in the most zone, and that'due south not what I said. It is spherically symmetric in the asymptotic region, which is all you need. Asymptotically flat means information technology approaches Minkowski infinite every bit y'all get far abroad.
Ok, I run across.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 just the full athwart momentum is conserved.
I meet, I didn't consider the fact that it could be modelled that way.Yes certainly, the gravitational field of the World-Moon system is asymptotically flat. The combined gravitational field goes down every bit Chiliad r-3, where M is the total mass, and the spacetime asymptotically approaches Minkowski infinite.
Thank you for the clarifications
A pair of orbiting bodies in GR does non exactly conserve angular momentum considering just the two bodies. Neither exercise a mutually orbiting pair of opposite charges, considering mass to be gravitationally negligible, and only EM strength significant. The difference in the gravity example is much smaller (due to being but of quadrupole radiation), versus dipole for the EM example.
HOWEVER, the total angular momentum of bodies plus gravitational radiations; or charges plus EM radiation *does* conserve athwart momentum exactly.
Issues like the rotational symmetry or assymptotic flatness of the universe relate to whether you can define a universally conserved athwart momentum. Equally long every bit, within some big region, you accept sufficient overall flatness, there is conservation of angular momentum within the large region (e.g. the milkyway'due south mass, free energy, plus gravitational radiations conserve angular momentum).
For me, a expert source on some of these issues is (in passing):
http://arxiv.org/abs/gr-qc/9909087
There is mention in Held, "General Relativity and Gravitation", almost various ways to define angular momentum in an asymptotically flat space-time, in a chapter by Winicour. My version of Schutz besides 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 nativity of Einstein".
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I"yard as well 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 non my favorite textbook.
Wiki seems to back me up on this:
http://en.wikipedia.org/w/index.php?championship=Carter_constant&oldid=420908353
The Carter abiding is a conserved quantity for motion around black holes in the general relativistic conception of gravity. Carter'south abiding was derived for a spinning, charged black hole by Australian theoretical physicist Brandon Carter in 1968. Carter's constant along with the free energy, axial angular momentum, and particle rest mass provide the four conserved quantities necessary to uniquely make up one's mind all orbits in the Kerr-Newman spacetime (even those of charged particles).
As none of the metric coefficients is a function of [itex]\phi[/itex], I think you lot should have rotational symmetry as is required by Noether's theorem. The infinite-time will exist "stationary" rather than static considering of the [itex]dt \, d\phi[/itex] term, but that shouldn't spoil the conservation laws - it does prevent yous from having "hypersurface orthogonality" though.
Take an ideally 'incompressible' hollow right circular cylinder, spinning at constant angular velocity about it'due south major vertical axis and supported on frictionless bearings. Now utilize gradually pressure to the cylinder ends, placing the sides under purely centric compression. Dissimilar the instance 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 fourth dimension irresolute squeeze stage, and therefore no relativistic force in that management. And so, to the extent elastic energy is absent (and we take idealized to an 'incompressible' solid), our cylinder undergoes no change in either spacial dimensions or angular velocity as a event of the stressing process. Ho hum you might say, simply here's the rub. Pressure/stress is a source of gravity (agile mass) in GR, and by the equivalence principle that means equally also a source of both passive and inertial mass. The latter is central here. Stressing the cylinder has increased information technology's inertial mass 'for free' and therefore it's angular momentum and rotational kinetic energy. And then in principle we can incessantly cycle this process and generate angular momentum and energy - spin upwards in unstressed state, spin downwards in stressed state, over and over. Of grade in the existent globe example friction and finite rubberband energy contributions and cross couplings to the 'pure pressure level' result will be overwhelmingly greater. Simply these will be material dependent factors and hence non cardinal. Can it be shown at that place is no such effect at all though? Cheers!
[tex]
\int_{\Sigma} \xi^{b} \left(T^{a}{}_{b} - \frac{1}{two} \, \delta^{a}{}_{b} T \right) dS_{a}
[/tex]
Hither [itex]\11^{b}[/itex] is the appropriate killing vector - for angular momentum, information technology would exist a phi-translation vector.
I *think* dS_a is just a unit future (i.e. normal to the book element sigma) - and I *think* this implies that pressure doesn't contribute to angulalr momentum (though you are right 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 merely when you have a time-similar 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'chiliad only 50/50 on this...
I'm assuming pervect your terminal entry refers to mine in #13, in which case thanks for your interest in the problem, merely information technology was posed as a potential counterexample to the kind of generalized theorems employing the style of high finish maths you take used. Sorry merely I can't really follow it at that level; could you please translate it dorsum to the specifics of the setup? What I expected was an argument that considering the cylinder axial stress is normal to the move, v.sigma = 0 (five the local velocity, sigma the centric stress) for any given moving element and thus no actual 'boost' to momentum and KE. My objection in plow would exist 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 contend the gravitational properties of the latter two would be annihilation but isotropic, surely. Nearly all cases of astrophysical involvement seem to exist limited to fluids under pressure, and here isotropy hides the outcome - ie yous just consider rho + 3p, the iii factor signifying that all three orthogonal components of pressure in a fluid are every bit contributing to the effective mass density. But is it consistent? If ma = one thousandp = thoui hither, information technology automatically implies that for whatsoever given acceleration a, there will be a full contribution F = one thousandi a = 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 u.s.a. dorsum to the case of our spinning cylinder - either the principle of equivalence fails for pressure, or we take a genuine perpetuum mobile (we should apply v|sigma|, not five.sigma). What's incorrect with this line of reasoning?Wincour'south article in Held (pg 74, eq ii-x) gives the expression for the total angular momentum as a book intergal as:[tex]
\int_{\Sigma} \xi^{b} \left(T^{a}{}_{b} - \frac{i}{2} \, \delta^{a}{}_{b} T \right) dS_{a}
[/tex]Hither [itex]\xi^{b}[/itex] is the advisable killing vector - for athwart momentum, it would exist a phi-translation vector.
I *think* dS_a is just a unit time to come (i.e. normal to the volume chemical 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, and then the dependence on T is nowadays only when you have a time-similar killing vector and are computing energy, information technology doesn't contribute when you have a space-similar killing vector and are computing linear or angular momentum.
I'm only 50/50 on this...
Hello, Q-reeusWhich in plough brings us dorsum to the case of our spinning cylinder - either the principle of equivalence fails for pressure, or nosotros have a genuine perpetuum mobile What'southward wrong with this line of reasoning?
Probably I don't sympathize your thought experiment very well but I would say that if in a experiment almost conservation of momentum-energy you go out out friction and compressibility y'all can get all kind of perpetuum mobiles.
Firstly TrickyDicky I feel a little guilty now, having jumped in with a dissimilar angle altogether - even though it has kinda stuck to the title, didn't really stop to think if this was actually a 'highjacking' of your thread. If you feel that I volition put a finish to this now - not that there's much to stop mind you!Hi, Q-reeus
Probably I don't sympathize 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 go all kind of perpetuum mobiles.
On the matters you have raised, I hold the natural instinct is to presume this has got to be incorrect if i 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 annihilation like that. Finite compressibility is less obvious only is standard procedure for what seems like an countless succession of papers in AJP etc that go along rolling out on 'hidden momentum' and 'stored field momentum' for instance, and many of these are actually quite close to this scenario. Equally explained in #13, these 2 factors are both cloth dependent and thus not of fundamental significance to the principle in question. One caveat at that place though; SR dictates an upper theoretical limit to rigidity (Born Rigid), merely I believe even there it does non fundamentally bear on the principles in question. And I have not really claimed a PMM as such - only pointed out this is unsaid imho if stress as source of inertial mass has the properties I think is so. And so if you would similar to add together anything, here or by my starting a new thread, be my guest. Same old problem here though, an almost deathly silence!
No trouble at all..... If you feel that I will put a stop to this now
After reading pervect's post I would say his answer makes sense, in GR the conservation laws are encoded in symmetries of the system, 1 fashion to look at these symmetries uses Killing vectors, where energy conservation is related to the time-like Killing vector and athwart momentum is related to space-like kiling vector, if the formula pervect writes applies here it means (I remember, if non please pervect explain) that the cylinder might gain energy (in a static setting), merely not angular momentum and therefore no spin cycle and no free free energy.So if you would like to add annihilation, here or by my starting a new thread, be my guest. Aforementioned quondam problem here though, an almost deathly silence!
I can't discover the reference in Schutz that says that angular momentum isn't conserved. Is this "A First grade in Full 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, considering parallel transport is path-dependent, but yous can't add up the quantity without parallel-transporting to a single point first. (A related just slightly different fashion 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 gild to limited the vector representing the total.)
It is true that if you have a cylinder with a commodities through it, and that if you tighten up the bolt to put the cylinder nether pressure, that the Komar mass of the cylinder increases. While the Komar mass, of the cylinder increases because of the force per unit area, the Komar mass of the bolt decreases, because it's under tension. Unless something really deforms, doing work, the whole closed system doesn't gain or lose mass, it'due south only distributed differently.
Notwithstanding, (assuming my agreement 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, only don't matter to the angular momentum.
That'due south bold I've understood the equations properly. It all appears to brand 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 nearly that situation.
Thanks - quite a relief!
Fair enough simply my rather simple approach is to exam general theorems by fashion of a specific setup, and just encounter if all situations take been fully and accurately accounted for by said theorem(s). See my comments to pervect's post in #20.Subsequently reading pervect'due south post I would say his answer makes sense, in GR the conservation laws are encoded in symmetries of the system, 1 way to look at these symmetries uses Killing vectors, where free energy conservation is related to the fourth dimension-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 proceeds free energy (in a static setting), but non angular momentum and therefore no spin wheel and no gratis energy.
Agreed, no argument there. However it matters much that the mass is distributed differently when a rotating system is under consideration. If for example the bolt as assumed lies on the axis of rotation, it's contribution to angular momentum volition be relatively tiny, even though it cancels the overall system mass change from the cylinder. I had really envisaged a stationary external agent in applying pressure (eg. Chiliad-clamp) which could be quite compressible itself but an irrelevant gene - only changes to the rotating mass is important imo.Q-reeus. I thought your experiment was interesting. My tentative conclusion is only that the angular momentum does not change when you compress the cylinder, as one would look.
Information technology is true that if y'all 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 considering of the pressure level, the Komar mass of the bolt decreases, because it's under tension. Unless something really deforms, doing work, the whole airtight organisation doesn't gain or lose mass, it's just distributed differently.
Well that would be extremely problematic from my pov given points raised in #15. How could this exist explained? I argued in #xiii 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 existent', 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 #fifteen one cannot take it both means - if ma = thoup = mi (WEP) holds for pressure induced mass, transverse components of pressure must contribute to inertial mass mi simply every bit for the component in line with any acceleration a. That leaves no way around the determination athwart momentum must increase for the setup of #13. The other culling is there is indeed a directional nature to pressure generated thousandi (just not for ma or yardp!). Hence WEP must fail here, which would still be news imo. My bet is though WEP holds. Thoughts?Even so, (assuming my understanding is correct) fifty-fifty though rotating cylinder is more than massive, it appears to exist the case that information technology doesn't take any more athwart momentum. (Again, using the Komar formula, which tin 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.
Some of the confusion in this thread is a outcome of the fact that angular momentum can be defined either at spatial infinity or at cypher infinity. Athwart momentum divers at spatial infinity is conserved, for every asymptotically apartment spacetime, period. This is considering even if you radiate angular momentum the radiations never makes it out to spatial infinity, and the athwart momentum of the radiation is included when you calculate the angular momentum, no matter what time (slice) you choose. Athwart momentum defined at zero infinity is non conserved, precisely due to the radiation loss that Schutz talks about.
So I guess in the Hulse-Taylor binary pulsar organisation 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 correct according to your explanation?
How do we cull to define athwart momentum either at null infinity or at spatial infinity for a given orbital system?
So I guess in the Hulse-Taylor binary pulsar system angular momentum is divers at goose egg infinity, not asymptotically apartment situation, and thus radiates GW, and angular momentum is non conserved.
However in the earth-moon orbit system angular momentum is conserved because information technology is divers at spatial infinity, and it is an asymptotically apartment modeled situation.Is this correct according to your explanation?
How do nosotros choose to define athwart momentum either at cipher infinity or at spatial infinity for a given orbital organisation?
No, I don't think this is right. Equally I explained before, the Hulse Taylor organisation *including its gravitational radiations* conserves athwart 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, athwart momentum is conserved.
In a radiating system, it seems not very sensible to use a null-infinity definition, considering you want to include the radiations at all times.
I think the Hulse-Taylor system is normally analyzed with an asymptotic flatness condition. It is not static, or spherically symmetric, simply can be asymptotically flat.
This doesn't hold with Sam Gralia's statement "Angular momentum divers at null infinity is not conserved, precisely due to the radiation loss...".No, I don't think this is right. As I explained before, the Hulse Taylor system *including its gravitational radiations* conserves athwart momentum. The fact that the system excluding radiation does non is completely fiddling 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 utilize a nil-infinity definition, because you lot desire to include the radiation at all times.
The fact is nosotros 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.
This I could agree with. Edit: In this case angular momentum is defined at spatial infinity and therefore conserved because the GW tin can't achieve spatial infinity, is this right?I think the Hulse-Taylor system is normally analyzed with an asymptotic flatness status. It is not static, or spherically symmetric, but tin be asymptotically flat.
So I guess in the Hulse-Taylor binary pulsar organization angular momentum is defined at null infinity, non asymptotically flat situation, and thus radiates GW, and angular momentum is not conserved.
However in the globe-moon orbit system angular momentum is conserved because information technology is defined at spatial infinity, and it is an asymptotically flat modeled state of affairs.Is this right according to your caption?
How exercise nosotros cull to define angular momentum either at null infinity or at spatial infinity for a given orbital organization?
I retrieve 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 in that location is no gravitational radiation. In reality it will radiate a bit of angular momentum, but my guess is this is tiny and merely being neglected in whatsoever you read about. In a binary pular, on the other hand, in that location is a lot of gravitational radiation.
Whether to use angular momentum defined at null or spatial infinity is just a choice. If you lot want to see how much angular momentum escapes as radiation, you use goose egg infinity. If you want to see that angular momentum is e'er conserved, as long as the angular momentum in the radiation is included, you lot use spatial infinity. Both are well defined for any asymptotically flat spacetime (although the definitions of asymptotic flatness people give in the corresponding cases can exist a piddling scrap 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.
TrickyDicky: In the other contempo thread, I mentioned an Angelo Loinger, who is a strictly GR theorist (of sorts), and claims total GR does not admit to GW's - if true then coupled to binary pulsar data, automatically means failure of athwart momentum conservation in toto. For sure 'controversial' but anyway the paper is relatively brusque, and if y'all exercise a search at arXiv.org volition non be hard to find - y'all could no doubt brand more of than I could. Just a idea.
I don't know his credentials, only 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 (not-approximate) formulation of general relativity (GR)
does non allow the existence of physical gravitational waves (GW's). I have
given several proofs of this fact [1]. Quite simply, we tin can notice, e.g.,
that bodies which interact just gravitationally describe geodesic lines, and
therefore – equally it is very like shooting fish in a barrel to see – they practice not generate any GW."
Except that going all the manner back to Einstein and Infeld (in the 1940s), and everyone else who derives equations of motion direct from the field equations, finds that bodies follow geodesics only in the limit of indicate particles. Real, massive bodies closely approximate geodesics, only the deviation cannot be ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole trunk of work.
I'm in no position to debate that - yous are probably quite right. I never noticed him referring his findings to the binary pulsar results which seemed odd, just owing to how short his arguments were, I figured someone with a proficient grasp of GR (not me) could sort out his logic without too much sweat...Except that going all the style dorsum 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. Existent, massive bodies closely approximate geodesics, just the deviation cannot exist ignored for a system like two orbiting pulsars. Thus I immediately suspect the whole body of work.
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