Rotational Symmetry & Gravitational Waves: Exploring the Connection

[Dr.Brain]

Ok , during my study , I read that the bodies which are rotating spherically symmetrically do not emit GW;s as in don't dent the spacetime memberane...same is with the bodies moving with uniform velocity ...why is that??...has it something to do with conservation of angular momentum??

 

[pervect]

Ok , during my study , I read that the bodies which are rotating spherically symmetrically do not emit GW;s

correct.

as in don't dent the spacetime memberane...

Spherically symmetric bodies still distort space-time, even though they do not emit gravitational waves.

same is with the bodies moving with uniform velocity ...why is that??...has it something to do with conservation of angular momentum??

Electromagnetic waves are emitted when one has a changing dipole moment. But there is no such thing as a gravitational dipole moment (or rather it is always zero) - therefore gravitational waves are emitted only when a body has a changing quadropole moment.

See http://en.wikipedia.org/wiki/Quadrupole for a defintion of "quadropole" moment (and some discussion of general multipole expansions).

You can also try looking at https://www.physicsforums.com/showthread.php?t=145020

for a past discusion about gravitational waves (a bit advanced, though).

 

[Chris Hillman]

Attempted clarification

But there is no such thing as a gravitational dipole moment (or rather it is always zero)

Or rather, in quadrupole radiation thory we treat an isolated radiating system using the weak-field approximation to the fully nonlinear EFE, aka "linearized gtr". Then, we can choose a center-of-mass coordinate system so that the dipole moment vanishes (for all times under consideration). If we do not adopt center of mass coordinates, the dipole moment of the system, taken wrt the origin of our coordinates, will be a nonzero constant. Usually, in fact, we choose a comoving center-of-mass system so that the linear momentum of the system also vanishes.

However, the important point here is what pervect said, that in EM, the lowest order radiation is associated with a time varying dipole moment in the distribution of electrical charge, while in gtr, the lowest order gravitational radiation is associated with a time varying quadrupole moment in the distribution of mass-energy. The textbook General Relativity by Stephani offers a particularly clear introduction to quadrupole radiation theory which emphasizes these points.

 

[Dr.Brain]

ok

Ok I read the wikipedia description of Quadropole as w.r.t Gravitational Wave emission , so what I infer is that , there is no dipole existing because m>0 , which is possible in case of magnets and charges.

It says a spheroid (ellipse rotated 360) , will possesses a quadropole , how is that? ...whereas a perfect spherical planet will not ...!...so will the planet form a monopole ...what in the case of binary star systems , because they emit GW's , so do two stars orbitting each other form a quadropole?...can I have a more detailed desciption for quadropole's in case os system's emitting GWs.

thnx.:shy:

 

[Chris Hillman]

What version? What article? Why WP?

Ok I read the wikipedia description of Quadropole as w.r.t Gravitational Wave emission , so what I infer is that , there is no dipole existing because m>0 , which is possible in case of magnets and charges.

It says a spheroid (ellipse rotated 360) , will possesses a quadropole , how is that? ...whereas a perfect spherical planet will not ...!...so will the planet form a monopole ...what in the case of binary star systems , because they emit GW's , so do two stars orbitting each other form a quadropole?...can I have a more detailed desciption for quadropole's in case os system's emitting GWs.

Gosh, what version of what WP article are you reading? I can't seem to find the text you "quoted" (yes?), not even searching wikipedia.org for the words you used via Google!

Note that there are many ways in which the words "monopole, dipole, quadrupole, ... multipole" are used in statistics, math, and physics. They really are closely related but this might not be obvious. The key feature which all multipole expansions have in common is that they are describing deviations from the most symmetric form, via a kind of generalized power series analogous to $$a_0 + a_1/r + a_2/r^2 ...$$ where r is something like distance from a "center of mass". This might at least help you to understand why a perfectly spherically symmetric gravitational field has a multipole expansion which terminates with the "monopole term". Higher order multipoles, if they exist, would describe how the field deviates from being spherically symmetric.

BTW, whenever you cite a WP article, you should first hit the "permanent link" button in the WP sidebar (look to the left) and then copy and paste the complete url this actions gives you. This ensures that even years later, someone reading PF will know what version of what article you were discussing. This is important because WP articles are so unstable.

In a very recent thread, I pointed out in response to a similar question that recent versions of the WP article titled "Gravitational radiation", in the section titled "Source of gravititational waves", are seriously misleading. This was frustating for me because months ago I noticed that this section contained many serious errors and was very badly written, so I rewrote that section. Unfortunately, more recent (and probably well intentioned) editors have once again munged it, so that recent versions are again misleading, which leads (apparently) to confusion on the part of readers like yourself.

If this is the WP article you are quoting from, try the earlier version listed at http://en.wikipedia.org/wiki/User:Hillman/Archive (scroll down to "Gravitational radiation" and click on the link).

(Just took a look and the current versions (22 Dec 2006) of the WP articles on "Multipole moments", "Dipole", "Quadrupole", "Gravitational radiation" all contain multiple errors, ranging from silly mistatements to seriously misleading misinformation.)

Even better, I hope you will consider checking out of your local university library a copy of one of the excellent gtr textbooks listed at http://math.ucr.edu/home/baez/RelWWW/reading.html#gtr

 

[pervect]

Let's look again at the defintions. And do think "weak fields".

monople moment = $\int$ dM

Anytime you have a mass, you have a "monopole moment". But this doesn't cause gravitational radiation, i.e. you need more than the existence of mass to create gravitational waves

dipole moment - component of this in the 'r' direction = $\int r dM$

The dipole moment just describes the motion of the center of mass of the system. This will be a straight line in the weak field Newtonian approximation.

A complete description of the dipoole moment is three numbers, one component each direction (x direction, y direction, z direction). All of the components are constant in the weak field (and zero in the center of mass frame). It's easiest to think about the center of mass frame when analyzing a system. Hence I said that the dipole moment is zero, but I probably should have said "zero in the center of mass frame in the weak field approximation".

quadropole moment - component in the r direction $\int r^2 dM$.

This may be familiar to you as the "moment of inertia" tensor for a rotating body. What this means: a symmetrical top doesn't emit any gravitational radiation when it spins. An assymetrical top will emit gravitational radiation. Look at the "moment of inertia" ellipsoid of a rotating object. If it is spherical, no gravitational radiation. If the ellipsoid is symmetrical around the axis of rotation, no gravitational radiation. If the ellispoid is not symmetrical around the axis of rotation, you have gravitational radiation.

Examples: a spinning sphere will not emit gravitational radiation (symmetric about axis of rotation)

A spinning dumbell shape (two spheres joined by a short rod) will emit gravitational radiation if it's rotating like thisx-----x

where the leftomst mass is moving "up" and the rightmost mass is moving "down", i.e. if the spin axis points up out of the page. But it won't if you spin the dumbell shape so that it stays in the above orientation with a spin axis lying in the horizontal directionl

Orbiting binary stars have roughly this configuration - they do emit gravitational radiation.

 

[Dr.Brain]

Let's look again at the defintions. And do think "weak fields".

monople moment = $\int$ dM

Anytime you have a mass, you have a "monopole moment". But this doesn't cause gravitational radiation, i.e. you need more than the existence of mass to create gravitational waves

dipole moment - component of this in the 'r' direction = $\int r dM$

The dipole moment just describes the motion of the center of mass of the system. This will be a straight line in the weak field Newtonian approximation.

A complete description of the dipoole moment is three numbers, one component each direction (x direction, y direction, z direction). All of the components are constant in the weak field (and zero in the center of mass frame). It's easiest to think about the center of mass frame when analyzing a system. Hence I said that the dipole moment is zero, but I probably should have said "zero in the center of mass frame in the weak field approximation".

quadropole moment - component in the r direction $\int r^2 dM$.

This may be familiar to you as the "moment of inertia" tensor for a rotating body. What this means: a symmetrical top doesn't emit any gravitational radiation when it spins. An assymetrical top will emit gravitational radiation. Look at the "moment of inertia" ellipsoid of a rotating object. If it is spherical, no gravitational radiation. If the ellipsoid is symmetrical around the axis of rotation, no gravitational radiation. If the ellispoid is not symmetrical around the axis of rotation, you have gravitational radiation.

Examples: a spinning sphere will not emit gravitational radiation (symmetric about axis of rotation)

A spinning dumbell shape (two spheres joined by a short rod) will emit gravitational radiation if it's rotating like this


x-----x

where the leftomst mass is moving "up" and the rightmost mass is moving "down", i.e. if the spin axis points up out of the page. But it won't if you spin the dumbell shape so that it stays in the above orientation with a spin axis lying in the horizontal directionl

Orbiting binary stars have roughly this configuration - they do emit gravitational radiation.

Right.So I think changing quadropole moment (change in $\int r^2 dM$ ), can be thought of a change in Inertia , or to say , change in distribution of mass through an imaginary plane , containing the axis of rotation as a line in its plane . I hope that sums it up . !??

 

[Dr.Brain]

If this is the WP article you are quoting from, try the earlier version listed at http://en.wikipedia.org/wiki/User:Hillman/Archive (scroll down to "Gravitational radiation" and click on the link).

(Just took a look and the current versions (22 Dec 2006) of the WP articles on "Multipole moments", "Dipole", "Quadrupole", "Gravitational radiation" all contain multiple errors, ranging from silly mistatements to seriously misleading misinformation.)

I have been reading the same text that you're referring to , and if its misleading , please tell if there are any better links out there.

 

[Chris Hillman]

Dr. Brain, I'll try to help if I can, but you still have not specified what WP article you are looking at, much less which version! All I know right now is that you are apparently looking at some version of some WP article and are confused by something you read. Since you didn't paste an exact quote, I have no idea if you simply misunderstood something, or if what you read is wrong, or both. Based upon your initial post in this thread, I do suspect that you are confused about something involving multipole moments and gravitational radiation.

Please tell us which article you are examining, in particular, where exactly you found the information (misinformation?) you are asked about in your initial post, and if possible which version (I'll settle for "current version" if you are looking at the current version of some article).

 

[Dr.Brain]

This is the extract I was referring to:
"There are gravitational monopoles; they are very commonly represented by ideal, stationary, spherically symmetric suns, planets, and so on. A gravitational quadrupole can be represented by two massive balls (say, lead) on opposite ends of a light rod, or, more simply, just as a long massive rod or a thin massive disk. A prolate (American football-shaped) or oblate spheroidal mass has a quadrupole moment. For example, the Earth is flattened at the poles, so it has a quadrupole moment"

It has been taken from http://en.wikipedia.org/wiki/Quadrupole [/URL]
under the heading Gravitational Multipoles.

 

[MeJennifer]

So can we univocally conclude that: "No, there is absolutely no gravitational radiation in case of a rotating mass with an equatorial bulge"?

And "no" should not mean: "well just a little bit" or "of course we only consider weak field approximations", or "well we assume it is not rotating too fast" or "well of course only in this special, super non-physical, coordinate system" etc.

 

[Chris Hillman]

(rofl)

This is the extract I was referring to:
"There are gravitational monopoles; they are very commonly represented by ideal, stationary, spherically symmetric suns, planets, and so on. A gravitational quadrupole can be represented by two massive balls (say, lead) on opposite ends of a light rod, or, more simply, just as a long massive rod or a thin massive disk. A prolate (American football-shaped) or oblate spheroidal mass has a quadrupole moment. For example, the Earth is flattened at the poles, so it has a quadrupole moment"

It has been taken from http://en.wikipedia.org/wiki/Quadrupole [/URL]
under the heading Gravitational Multipoles.[/QUOTE]

This is a hilarious example of the kind of thing I am talking about, Wikipediawise! The phrases "thin massive disk" and "American football-shaped object" suggest that the original source of this article was some material I inserted many months ago into another WP article. It looks like some person or persons unknown came along and took part of what I wrote, but completely altered the meaning. In particular, this person or persons appears to have seen an example which I used, in which I compared the effect of adding an equatorial bump with the effect of adding an equatorial bugle to a spherically symmetric mass distribution. Unfortunately, whoever did this completely misunderstood what I had written, so that this article now reads like total nonsense, because the version you quote [I]is[/I] total nonsense! OK, I guess you can quit worrying now. You are not crazy, it is just that some editors of WP are very bad writers, or have no idea what they are talking about, or both.

 

[Chris Hillman]

Of course there IS gravitational radiation in the case of...

Hi, MeJennifer,

So can we univocally conclude that: "No, there is absolutely no gravitational radiation in case of a rotating mass with an equatorial bulge"?

And "no" should not mean: "well just a little bit" or "of course we only consider weak field approximations", or "well we assume it is not rotating too fast" or "well of course only in this special, super non-physical, coordinate system" etc.

I take it you found this assertion someplace at Wikipedia? It's also completely the opposite of what I said in some earlier versions of various WP articles.

Let me try to briefly clarify: imagine a globe with a piece of of putty stuck over uhm... Panama City (approximately on the equator). This equatorial bump (not a "bulge") changes the mass distribution from almost perfectly spherical to a non-axisymmetric distribution with nonzero quadrupole moment. Now set the modified globe spinning. In principle, a scaled up and more massive version would product gravitational radiation because its quadrupole moment tensor picks out a preferred direction at each moment, and this preferred direction is rotating as the thing spins. This means the quadrupole moment tensor is varying with time, and gravitational radiation will be produced, in an amount which can be estimated from the quadrupole moment formula.

On the other hand, if we had instead added an axisymmetric equatorial bulge to the original sphere, and set this thing spinning about its axis of symmetry, the preferred direction (the axis of symmetry) coincides with the axis of rotation, sot it is not changing in time, so does not lead to quadrupole radiation.

This probably shows why I quit trying to help improve the Wikipedia some time ago: this kind of episode in which something correct is munged is very common, and existing social mechanisms for correcting supposed "improvements" which are actually degradations are incredibly clumsly, in part because the Wikipedia creed (at least, in some of the more extreme statements I have see) holds that articles spontaneously improve monotonically, approaching a state of perfection. Which of course is bunk, as this episode illustrates.

As for "absolutely no gravitational radiation" in the case of the sphere with a bulge, in everything I wrote I was always careful to point out that I was talking only about the radiation incorporated in the quadrupole approximation. Properly speaking, there can be other sources of gravitational radiation in addition to the quadrupole mass moment. For example, a source with constant quadrupole mass moment but time varying octupole mass moment will radiate, but this "octupole radiation" is far weaker than quadrupole radiation. And in addition to quadrupole mass moment there is a current quadrupole moment.

 

[MeJennifer]

This probably shows why I quit trying to help improve the Wikipedia some time ago: this kind of episode in which something correct is munged is very common, and existing social mechanisms for correcting supposed "improvements" which are actually degradations are incredibly clumsly, in part because the Wikipedia creed (at least, in some of the more extreme statements I have see) holds that articles spontaneously improve monotonically, approaching a state of perfection. Which of course is bunk, as this episode illustrates.

I can imagine your frustration.
Anyway I am quite happy you are on PF and I appreciate all your help.

 

[Chris Hillman]

Thanks, MeJennifer! With the passage of time I seem more inclined to laugh than cry when I discover how badly something I tried to do at WP has been munged by later edits.

 

[Dr.Brain]

Well, I appreciate your attempts chris! , even if the output is not as you wanted it to be nywayz...any links for mathematics regarding the power generated by a GW due to changing quadrupole? , and yes I am IInd year engg student , who hasnt studied tensors yet , I want to do indepth study of Gravitational Waves , any advices?

 

[Hans de Vries]

Ok , during my study , I read that the bodies which are rotating spherically symmetrically do not emit GW;s as in don't dent the spacetime memberane...same is with the bodies moving with uniform velocity ...why is that??...has it something to do with conservation of angular momentum??

A rotating spherical mass doesn't emit gravitational radiation. A rotating
spherical charge doesn't emit EM radiation either.

The reason, in both cases, is that the source is static. In the EM case each
point x,y,z within the sphere has a constant charge and a constant current.
(Their contributions to the electric and magnetic fields are propagated with
the speed c of light). The resulting total electric and magnetic fields are
therefor constant (static) as well, and thus non-radiating.

In Gravitation a Tensor is propagated from each source point rather than a
potential Vector as is the case with charges but the form of the propagator
is the same.

Regards, Hans

 

[pervect]

Well, I appreciate your attempts chris! , even if the output is not as you wanted it to be nywayz...any links for mathematics regarding the power generated by a GW due to changing quadrupole? , and yes I am IInd year engg student , who hasnt studied tensors yet , I want to do indepth study of Gravitational Waves , any advices?

Try https://www.physicsforums.com/showpost.php?p=1168838&postcount=7

where I try to summarize some of the textbook ("Gravitation" by MTW) results. Unfortunately it looks like, on checking, I totally screwed up the conversion from geometric units to standard units in that post. I'll be posting a correction shortly.


You'll basically find that the the power is insignificant in anything but an astronomical situation, and even in astronomy one needs extreme conditions to have significant gravitational wave emission.

 

[Dr.Brain]

Yes in my paper presentation I tried to put in values for the Earth-Sun Binary System , and the power came out to be approx 313 watt , which is very small , as compared to the EM radiation we reciev frm sun which is of thr order of 10^26 Watts ...

 

[Chris Hillman]

References

Hi, Dr. Brain, thanks for the thanks, and

any links for mathematics regarding the power generated by a GW due to changing quadrupole? , and yes I am IInd year engg student , who hasnt studied tensors yet , I want to do indepth study of Gravitational Waves , any advices?

The power output is given (in the quadrupole theory) by the quadrupole radiation formula, which is discussed in most gtr textbooks. Given your background, I guess the best book for you might be Ohanian and Ruffini, Gravitation and Spacetime, 2nd ed., Norton, 1994. (A textbook is always a much better source than internet! And I'd recommend that after a cursory examination of gtr textbooks in your local university library, you buy a copy of the one you like best if you want to study this in depth. See http://www.math.ucr.edu/home/baez/RelWWW/reading.html#gtr for some suggestions which include some remarks on topics covered by a selection of the best textbooks. Hartle's book came out after I wrote that, so it gets short schrift.)

 

[pervect]

I'd like to mention a very simple, though not very accurate, approximation mentioned in MTW for the amount of gravitational radiation emitted by a rotating body.

The good news is that the formula is fairly easy to understand, and it's more difficult to screw up the calculation than some of the other methods I've talked about. (I'd still have to recommend double checking the formulas I posted earlier, though I think I got the errors fixed this go-around).

The bad news is that this simple formula only gives a very rough estimate of the amount of gravitational radiation, within a few orders of magnitude only.

You compute the total non-spherical power flow in any body. If the body is grossly nonspherical (as it is in the rotating bar example), you can use the total rotational energy .5*I*w^2 divided by T, or .5*I*w^2*f to get the non-spheical power flow.

If the body is almost spherical (say a sphere with a small bump), you'd use the moment of inertia of only the bump in isolation to compute the energy and hence the power.

Note that some reason that I don't understand, MTW actually takes (.5*I*w^2)*w, which is different by a factor of 2*pi from the expression above. (see pg 979). But being off by a factor of 2*pi is not significant compared to the inaccuracies in the approximation.

Call this internal power flow Lint, i.e. Lint = (.5 I w^2)*w or (.5*I*w^2)*f. The emitted gravitational radiation will be on the ordier of (Lint^2)/(c^5/G)

c^5/G is a conversion factor for the geometric units, it's a constant equal to 3.63*10^52 watts.