A Understanding gravitational waves (GR)

Orodruin

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In order to have monopole radiation, the time derivative of the monopole moment must be non-zero. Since the monopole moment is conserved, this is not the case.

The same argument goes for the dipole moment but with the second time derivative. By momentum conservation, this second derivative is zero.

The quadrupole moment on the other hand can undergo periodic oscillation and therefore give rise to a radiation field (proportional to the third time derivative of the quadrupole moment squared).
 
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The monopole moment (total mass), and dipole moment (essentially displacement of the center of mass).
OK so the monopole moment represents the total mass and the dipole moment represents the displacement of the center of mass.

What does the quadrupole moment represent though?
 

pervect

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Mmm I see, thanks.



Could someone please recommend a book/paper which explains why this is the case (with the Math of course)?
MTW (amazon link) talks about it in section $15.2, but I'm not sure that's necessarily the best place to start.

Certainly, I'd expect that if you open the book to this section, without reading the prerequistes, you'd most likely wind up confused. And it's midway through a very long, very dense book, so to get the context for the brief section that talks about it, a small number of pages, would be a large amount of reading.

The title of $15.2 might provide a clue as to how intelligible it will be without reading the previous 15 chapters.

"Bianchi identity d##\mathcal{R}##=0 as a manifestation of "Boundary of boundry = 0"

Basically, the main issue I see is that there's a a lot of background information that you'll need to understand the explanations. I rather suspect you don't have that background from your questions, but of course it's hard to be really sure of what background you might have.

The reason I'm not sure that MTW is the best book is because it's notation is a bit antiquated. But it's very good about explaining things, and I usually rather like the informal presentation. I'm not really a fan of the comma and semicolon component notation, though. MTW is not too expensive - I've seen some really low-priced ebook versions, but the formatting of the diagrams and long equations strike me as probably being likely to be illegible without the printed verison.

There are more modern books out there, I'm not sure which to recommend. Schutz, for instance is very popular, but most people I talk to on PF who try to learn GR from his book wind up very confused. I don't own it, when I read pieces and snippets of it, though, I usually wind up agreeing with the posters who are asking about said snippets that it's confusing :(.

There are always Carroll's online lecture notes, <<web link>>, which are free, though I don't know if or where Carroll would talk about the particular issue you are concerned at at the moment.

I think I already mention Landau and Lifschitz, "Classical Theory of fields". They had a pretty good derivation of the quadrupole formula as I recall, but I'm not too familiar with them overall. They cover a lot more than just gravity, which is both good and bad. The quadrupole formula is very similar to the "retarded potentials" method for electromagnetism, <<wiki link>>.

I'm pretty sure there is a thread about GR books in general, and I see the task ahead of you as one of picking up a whole lot of background to get to the small piece that you're interested in.
 

vanhees71

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I am aimed at understanding why the changing of the quadrupole moment means that the orbiting black holes emit gravitational radiation.

But first let's delve into the calculation of the quadrupole moment ##Q_{ij}## (it's shown in the paper, page 10).

I don't see why the paper defines the quadrupole moment as:

$$Q_{ij} = \int d^3 x \rho (x) (x_i x_j - \frac{r^2\delta_{ij}}{3})$$

Particularly the usage of the factor:

$$x_i x_j - \frac{r^2\delta_{ij}}{3}$$

I guess it is setting the system's center of mass in the origin of the Cartesian coordinates but that is not enough for me.
In Cartesian coordinates the multipole moments are defined as the irreducible moments of the mass distribution. Irreducible moments are defined by the irreducible polynomials of the ##x_i##. Irreducible means that the moments are symmetric and traceless.

For the 2nd irreducible moment you start from ##x_i x_j##, which is already symmetric of course, but you have to get rid of the trace part. The only invariant tensor you can form from the ##x_i## alone is ##x_k x_k \delta_{ij}##. Thus the right moment is given by the factor you quote, ##x_i x_j-x_k x_k \delta_{ij}/3##, and indeed this gives you the quadrupole moment.

This construction gets soon pretty tedious. You can try it for the octopole, starting from ##x_i x_j x_k##.

The general construction is much simpler in terms of spherical coordinates leading to the spherical harmonics ##\text{Y}_{lm}(\vartheta,\varphi)##, which can all be determined fully algebraically using the orbital angular-momentum operator from wave mechanics, ##\hat{\vec{L}}=-\mathrm{i} \vec{x} \times \vec{\nabla}##.
 

bob012345

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No, they aren't. They are in free-fall orbits about their common center of mass. "Acceleration" in GR means proper acceleration; objects in free fall have zero proper
While I have no doubt you are completely correct, that still makes no sense to me. Two object that are attracted to each other are moving closer and if they are closer they are more attracted. In Newtonian terms, they are accelerating towards each other and the acceleration is not constant. As we observe them, are not those two black holes moving faster as time progresses until they collide? Thanks.
 
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While I have no doubt you are completely correct, that still makes no sense to me. Two object that are attracted to each other are moving closer and if they are closer they are more attracted. In Newtonian terms, they are accelerating towards each other and the acceleration is not constant. As we observe them, are not those two black holes moving faster as time progresses until they collide? Thanks.
Yes, the attraction changes and in Newtonian physics that means they are applying a gravitational force to each other? So what? This thread is about GR, not Newtonian physics. Things just naturally travel along geodesics (ie are in "free fall")
 

Ibix

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In Newtonian terms,
Unfortunately, GR uses rather different terms.
As we observe them, are not those two black holes moving faster as time progresses until they collide?
You can certainly find a way to state that. The problem is that it's not coordinate independent, so it always leads to a thousand "ifs" and "excepts" and "depends on your point of views". For example, the distance between two things whose "surfaces" can't rigorously be described as places in space and don't really have centers in any meaningful sense is a surprisingly difficult thing to pin down. So we use things like quadropole moments which are actually invariants.
 

FactChecker

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While I have no doubt you are completely correct, that still makes no sense to me. Two object that are attracted to each other are moving closer and if they are closer they are more attracted. In Newtonian terms, they are accelerating towards each other and the acceleration is not constant. As we observe them, are not those two black holes moving faster as time progresses until they collide? Thanks.
You can think that way (and many great physicists did before relativity) as long as you can accept some incredible coincidences and till some calculations are wrong. Why does the "mass" that resists acceleration EXACTLY match the mass that attracts other objects? Why does a large mass "attract" something like light that has no mass and seems to only travel in straight lines through space? How does gravitational attraction work with no intermediate entity to transmit the force? Why is the orbit of Mercury a little different from the Newtonian prediction? Relativity clarified all that beautifully. It redefined what a "straight line" means and what an acceleration away from a "straight" path means.
 

Orodruin

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More precisely, proper acceleration is measured by an accelerometer, and nothing else.

There is such a concept as coordinate acceleration, and it can be measured if a suitable reference frame is set up, so I don't think we can just say "acceleration" without qualification is measured by an accelerometer. Particularly not in a thread where the distinction between proper and coordinate acceleration is now a topic of discussion.
I think in general "acceleration" would be taken to mean proper acceleration, but I guess you are right about it being relevant for this discussion. [Moderator's note: That discussion has now been moved to a separate thread.]

However, this thread is not about the distinction between proper and coordinate acceleration ... it is about understanding gravitational waves.
 
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this thread is not about the distinction between proper and coordinate acceleration ... it is about understanding gravitational waves.
Yes, fair point. I'll look at spinning off the "proper vs. coordinate acceleration" subthread into a separate thread.
 
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I'll look at spinning off the "proper vs. coordinate acceleration" subthread into a separate thread.
Done. The proper vs. coordinate acceleration discussion is now in this thread:

 

bob012345

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I'm going to chew on these answers for a while as well as do some reading overnight about the subject. Thanks for all the answers. I do appreciate them! If it's spun off to a new thread, I'll respond there then. Thanks again!
 
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If you are trying to understand gravitational waves, I'd suggest you look for papers that model them rather than looking at data.
Do you guys have in mind any paper/book which nicely models gravitational waves?

I’ve been reading chapter 7 of Carroll’s book but I’d like to have more references.
 
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Ryder Lewis Introductory to General Relativity has a nice intro level into GW waves. The book will step you into all the main aspects of SR and GR as well. It's one of the textbooks I used to step into GR. Very useful in its details particularly when learning the mathematics
 

George Jones

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Do you guys have in mind any paper/book which nicely models gravitational waves?

Ryder Lewis
Eats, Shoots & Leaves, i.e., "Ryder, Lewis". :oldbiggrin:

I though about recommending this book. I love its final paragraph:
"In this chapter we have considered a few of the topics that have appeared on the agenda since Einstein's day, as a consequence of his great theory. There are, of course other questions raised by by General Relativity, perhaps the most famous of which is quantum gravity. How should a quantum theory of gravity be constructed? There is as yet no generally agreed answer to this question, but many clever people have devoted many years to thinking about it. At the end of an introduction to Einstein's theory, however, it is best not immediately to start thinking about the next challenge. Like a climber who has arrived at the top of his mountain, we should simply sit down and admire the view. Is it not absolutely remarkable that Einstein was able to create a new theory of gravity in which the geometry of space itself became a part of physics? What would Euclid have thought?"

Other, perhaps less abstract references: "A General Relativity Workbook" by Thomas Moore; "Modern General Relativity: Black Holes, Gravitational Waves, and Cosmology" by Mike Guidry.

A book devoted to the topic: "Gravitational Waves Volume 1: Theory and Experiments" by Michele Maggiore.
 
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I liked that passage as well lol
 

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