Gravitational Wave: Energy & Amplitude Changes

[Meerio]

I was wondering how the energy and amplitude change over a distance

 

[PeterDonis]

What sources have you looked at to try to answer this question? There are plenty of online discussions of gravitational waves, including Sean Carroll's online lecture notes on GR:

https://arxiv.org/abs/gr-qc/9712019

 

[Meerio]

I honestly cannot filter the answer out of that paper, my physics skills aren't good enough for that yet. I talked to someone about it but he said the amplitude didn't change. Also I read somewhere that energy is conserved until (part of) the gravitational wave is absorbed. The second part is pretty logical but I just want to confirm the thing about the amplitude is right

 

[PeterDonis]

I talked to someone about it but he said the amplitude didn't change.

"I talked to someone" isn't a valid reference. I need a textbook or peer-reviewed paper. If the person you talked to can give you such a reference, that would help.

On its face, this statement seems obviously false; the "amplitude" of GWs (at least with the usual meaning of that term) is what a detector like LIGO detects, and obviously such a detector will detect a stronger signal the closer it is to the source. But without seeing some actual reference, I can't be sure what your source means by "amplitude".

I read somewhere that energy is conserved until (part of) the gravitational wave is absorbed.

"I read somewhere" isn't a valid reference either. Where did you read it?

On ifs face, this statement is too vague to know whether it's true or false. What "energy" is being talked about? That term doesn't have a single meaning.

 

[Meerio]

Yes, I agree about the energy question, but I talked to a PHD student about the subject, by amplitude I mean the amount space is stretched/squeezed relative to itself, I have also read in the link you've sent me it consists of 2 components with an angle of 45 deg relative to each other. But what is the relation between the distance to the source and the amplitude?

 

[pervect]

In the context of linearized weak-field gravity, the amplitude (more formally, the "peak gravitational wave strain h") from a compact source (such as a binary inspiral measured by Ligo) falls off as 1/r, i.e. $h \propto 1/r$.

There are two polarizations of the GW, generally written as $h_+$ and $h_x$, one can consider $h = \sqrt{(h_+)^2 + (h_x)^2}$. But for detectors such as Ligo, only one component is measured, thus the Ligo measurement of amplitude will under-represent the "real" amplitude.

See for instance the wiki article https://en.wikipedia.org/wiki/Gravitational_wave. The Ligo detection paper http://journals.aps.org/prl/abstract/10.1103/PhysRevLett.116.061102 might also be of some interest.

Energy is tricky to define in GR, but in the context of linearized gravity the average energy flux carried by a GW is proportional to the square of the rate of change of h, i.e. $E \propto \dot{h}^2$, where the over-dot represents taking the time derivative. The reference I have online, http://www.tat.physik.uni-tuebingen.de/~kokkotas/Teaching/NS.BH.GW_files/GW_Physics.pdf, actually writes this as $E \propto \omega^2 h^2$, which applies for sinusoidal GW's. Since non-sinusoidal waves are important in the Ligo case, I'll give the more general formula. I'd give a better reference on this if I had one, this is what I have available.

If one is not interested in the effects of frequency on GW's, only about the effects of distance, one might say that $h \propto 1/r$ and $E \propto 1/r^2$

It may be worthwhile reviewing the definition of "Energy Flux" - wiki gives "Energy flux is the rate of transfer of energy through a surface. The quantity is defined in two different ways, depending on the context." The right context in this situation is that the average energy flux is the average rate of energy transfer per unit area.

The part about "averaging" the energy is important for technically reasons. It's an error to assume that GW energy can be localized in any traditional manner. It's rather difficult to explain these technical aspects, the general approach I've seen in popularizations is to ignore the issue entirely. I'm not sure how much harm ignoring this issue does for a broad, non-specialist understanding, so I won't spend a lot of words emphasizing the issue, I will just mention that it exists.

 

[PeterDonis]

by amplitude I mean the amount space is stretched/squeezed relative to itself

Then that is indeed what a detector like LIGO detects, and it does fall off with distance. Pervect's post is a good summary of what happens.

 

[Meerio]

Thank you!