Gravitational Waves Emitted by a Binary System

[lomidrevo]

Let's assume a binary system with an inclination angle $i$ (angle between the orbital plane and line of sight). Then, according to this source - equations (128) and (129) - for the amplitudes of the tensor polarization modes ("plus": $+$ and "cross": $\times$ polarization) I could write:
$$h_{+} \propto (1 + \cos^2 i) $$
$$h_{\times} \propto (2\cos i) $$

So now, if I would take the extreme cases.. If it happens that we observe merging binaries perpendicular to their orbital plane ($i = 0$), our detector(s) would measure gravitational waves (GWs) with both of the polarization modes, with equal amplitudes. In case we see the orbital plane edge-on ($i = \pi / 2$), then we detect only GWs with "plus" polarization, and the amplitude of the measured distortion is only half when compared to the first case.

Is my understanding correct?

Note: this is not a homework. I've been just curious why most of the visualizations provided on internet shows GWs propagating in the plane of the orbit. The above suggest that radiation emitted perpendicularly to the plane is more intense. But I see, it would be quite difficult to draw these waves

 

[PeterDonis]

I've been just curious why most of the visualizations provided on internet shows GWs propagating in the plane of the orbit.

Because they're using a "rubber sheet" type of visualization, where you can only show one plane anyway, and the obvious plane to pick is the plane of the orbit. Since they're just pop science visualizations anyway, one should not expect them to be very accurate.

 

[vanhees71]

Isn't the rubber-sheet visualization of gravity in the sense of GR the worst visualization you can choose? It's coneptually completely wrong: It suggests as if gravity is described as some substance embedded in a flat (3D?) space.

Concerning to GR, gravity is described as gauging the global Lorentz symmetry of special relativity, i.e., making it a local symmetry and thus also the notion of inertial frames local notions. It then turns out that you can reinterpret this description as a dynamical pseudo-Riemannian spacetime manifold (or, if one includes also spin, a Einstein-Cartan manifold with a pseudometric and compatible connecction with torsion). No embedding in any higher-dimensional space is needed. Everything physical can be described by geometrical "objects" (tensors/spinors) "living" in this spacetime manifold.

It's of course impossible to explain this highly abstract mathematics on the popular-science level :-(.

 

[hmmm27]

Actually, the rubber-sheet visualisation just "clicked" for me a few weeks ago : at first glance it shows only a literal-ish "gravity well", but the well also illustrates the gravitic expansion of space within it (which is what I was looking for) ; 3d in a 2d representation.

 

[lomidrevo]

Agreed, the visualizations might be pretty inaccurate, especially the one with rubber-sheet. But to be fair, it is not easy to visualize curved 4 dimensional spacetime. And not even mentioning the "ripples" in such spacetime. But it would be correct if those sources described (or at least mentioned) the limitations of the provided graphics, which is not the case, generally.
Fortunately, some more credible sources provides at least partial description. For example:
https://www.ligo.caltech.edu/page/what-are-gw
In the embedded video, the GWs are being emitted in the orbital plane, but in the text they write:

Einstein's mathematics showed that massive accelerating objects (such as neutron stars or black holes orbiting each other) would disrupt space-time in such a way that 'waves' of undulating space-time would propagate in all directions away from the source. These cosmic ripples would travel at the speed of light, carrying with them information about their origins, as well as clues to the nature of gravity itself.

 

[PeterDonis]

the well also illustrates the gravitic expansion of space

What do you mean by "gravitic expansion of space"?

 

[hmmm27]

What do you mean by "gravitic expansion of space"?

Space in a gravity well is larger. Wrong word ?

 

[PeterDonis]

Space in a gravity well is larger.

Larger than what? Larger how?

 

[pervect]

The volume of a thin spherical shell for a stationary observer in the Scharzschild spacetime for a given radius r and [correctio] thickness $\Delta r$ is larger than the volume element of a thin spherical shell in flat spacetime, I think.

The necessary space-time split to define the spatial volume element is induced by the timelike congruence of worldlines of stationary observers outside the event horizon.

[add]
I should add that this comes from the fact that the measured thickness $\sqrt{g_{rr}} \Delta r$ is larger than $\Delta r$, which is essentially a coordinate thickness.

 

[hmmm27]

In regards the "rubber sheet" illustration, a dimple has more surface area... or else it wouldn't be a dimple. Any way of showing hypothetical (fanciful) negative-gravity regions ?

 

[PeterDonis]

hypothetical (fanciful) negative-gravity regions ?

What are you referring to here?

 

[hmmm27]

Nothing concrete, to my knowledge, though I'm considering posting a question if such a thing has been observed, perhaps in those massive grav waves that the thread is actually about.

So, what keywords would I use to google for what I'm talking about ? ... which could be codeified as "local expansion of space due to local gravity".

 

[PeterDonis]

what keywords would I use to google for what I'm talking about ? ... which could be codeified as "local expansion of space due to local gravity".

There aren't any keywords for this because it's not a known concept, as far as I know.

In post #9 @pervect described a property of the geometry of spacetime around a spherically symmetric massive body that might be what you are thinking of; but "local expansion of space" is not a good description of that property.

 

[hmmm27]

There aren't any keywords for this because it's not a known concept, as far as I know.

Ah, well in that case the rubber-sheet model is pretty stupid, if all it shows is simply attraction.

In post #9 @pervect described a property of the geometry of spacetime around a spherically symmetric massive body that might be what you are thinking of; but "local expansion of space" is not a good description of that property.

Perhaps : I did get the idea out of a black hole thread, though I don't think it was specific to an event horizon... I'll see if I can dig it up. Thanks, both.

 

[PeterDonis]

in that case the rubber-sheet model is pretty stupid

I agree that the rubber sheet model is not a good model to use if you actually want to understand GR; we have had many, many threads here at PF in the past about this.

if all it shows is simply attraction.

It doesn't show "attraction". It shows one particular aspect of the spacetime geometry around a massive body. And that aspect is not the same as the aspect that produces "attraction"--the rubber sheet is not showing you "gravitational potential".

 

[hmmm27]

It doesn't show "attraction". It shows one particular aspect of the spacetime geometry around a massive body. And that aspect is not the same as the aspect that produces "attraction"--the rubber sheet is not showing you "gravitational potential".

If an object traverses a gravity well, the rubber sheet model makes it look like it's a greater distance... rather than what I now assume one is supposed to do, ie: look at it from above, where the gridlines don't look stretched.

 

[PeterDonis]

If an object traverses a gravity well, the rubber sheet model makes it look like it's a greater distance

Greater distance than what?

what I now assume one is supposed to do, ie: look at it from above, where the gridlines don't look stretched

That doesn't mean the spacetime geometry is flat. It isn't.

The "gridlines" in the rubber sheet model represent Schwarzschild coordinates--specifically the radial and one of the angular coordinates. The radial coordinate is an "areal radius"--in the rubber sheet diagram, where the second angular coordinate is suppressed, the radial coordinate $r$ labels a circle in the angular direction whose circumference is $2 \pi r$. (If we added back the second angular coordinate, the radial coordinate $r$ would label a 2-sphere whose surface area was $4 \pi r^2$.)

It is possible to come up with an answer to the "greater distance than what" question I asked above by making use of what I have just stated in the previous paragraph. Post #9 by @pervect describes part of how to do that. And you have to do that in order to give any meaning to claims like the one of yours that I quoted at the start of this post.

 

[pervect]

Ah, well in that case the rubber-sheet model is pretty stupid, if all it shows is simply attraction. Perhaps : I did get the idea out of a black hole thread, though I don't think it was specific to an event horizon... I'll see if I can dig it up. Thanks, both.

Yes, the rubber sheet analogy as is usually presented is very misleading.

One can start to get somewhere if one draws space-time diagrams on a curved surface (such as the surface of a sphere), but unfortunately the popularizations do not do this. One of our posters, AT , has a number of graphics of this sort.

Most popularizations depict a curved space, not curved space-time. The fault for the misunderstandings are in my mind the fault of the presentation, not in the reader.

Being able to draw and understand space-time diagrams is one of the important gateway to special relativity, and then onto General relativity, which needs special relativity as a foundation.

 

[vanhees71]

Another problem is that already Minkowski diagrams in SR are not as easy to read as many (most?) people think. At least for me, it's hard to really forget about "Euclidean thinking", because I'm very much used to interpret the plane on the paper as a Euclidean plane. But if I draw a Minkowski diagram, depicting e.g., the world lines of particles in 1D motion, I must completely forget about Euclidean geometry and have to substitute it with Minkowskian/Lorentzian geometry. If I start drawing for one inertial frame, I use what looks like a Cartesian coordinates, and I can choose the unit grid arbitrarily. Now drawing another inertial frame, I have to draw the worldline of an observer at rest in the origin in the new frame, which is a straight line in the forward light-cone. This is the temporal axis of the new frame. Then I can construct the spatial axis of the new frame by mirroring it at the light cone. The crucial difference to Euclidean geometry then is that I get the temporal and spatial unit lengths on the new axes by drawing the time-like and space-like hyperbola rather than unit circles in Euclidean geometry. So I must forget to think about "length" in a Euclidean way and substitute it by Lorentzian geometry and must forget to think about "angles" at all.

I'm often inclined to think that a Minkowski diagram is less easy to understand than the algebra of four-vectors in the affine Lorentzian manifold.

 

[pervect]

Another problem is that already Minkowski diagrams in SR are not as easy to read as many (most?) people think.

I suppose I have to agree. Drawing the diagrams is the first step. The actual drawing part should be trivial, though it's difficult to get people to do even that much, I find :(.

Understanding that the diagrams represent a non-Euclidean geometry needs to occur at some point, usually after one has learned the basics of just drawing them. This is related to other important topics that all mesh together - the relativity of simultaneity, the fact that space-time really needs to be treated as a 4d continuum. The space-time diagram can be a gateway to understanding all of these topics, but it is often resisted.

On the topic of affine geometry, I was recently re-reading "The Parable of the Surveyor", from Taylor & Wheeler's "Space-time physics", which I still recommend (both the book and the section of it which is the parable) . The moral of the parable is that it explains why space-time in SR is a 4d continuum. I think that talking about the space-time continuum is something that people learn by rote but don't really appreciate the need for. Taylor & Wheeler talk about a hypothetical universe in which north-south distances and east-west distances were measured by different methodologies using different units, and how one might come to understand that north-south and east-west are part of a larger structure, the 2 dimensional plane. They then use this understanding to motivate why we would treat space-time as a larger structure, rather than separating it into space and time.

Re-reading the "parable of the surveyor", I came to the realization that the geometry where north-south and east-west were incommeasurable was an affine geometry. So affine geometry could be a gateway to some important insight into SR, if one somehow happened to learn about it before one learned about SR. But I suspect this will be rare.

 

[romsofia]

I don't have time to type up how I visualize all this, but I can provide a cool paper on the topic, and their accompanying youtube videos as the links in the paper are broken!
The paper: https://arxiv.org/pdf/1012.4869.pdf
The youtube channel: https://www.youtube.com/c/SXSCollaboration/videos

 

[pervect]

I don't have time to type up how I visualize all this, but I can provide a cool paper on the topic, and their accompanying youtube videos as the links in the paper are broken!
The paper: https://arxiv.org/pdf/1012.4869.pdf
The youtube channel: https://www.youtube.com/c/SXSCollaboration/videos

I took a quick look at just the introduction to the paper.

I never worried too much about the magnetic part of the Riemann. (Or in this case, the Weyl - I may be using a different but similar decomposition in my own thinking). Does the vorticity of the magnetic part have some physical significance that you can explain? It's interesting that it's invariant as the paper explains - that leads me to believe it's important, but I don't have an intuitive feel for it's physical significance.

This is a bit off-topic of the original poster, I think, but I find it interesting. Perhaps a moderator could split the thread off?

 

[lomidrevo]

I don't have time to type up how I visualize all this, but I can provide a cool paper on the topic, and their accompanying youtube videos as the links in the paper are broken!
The paper: https://arxiv.org/pdf/1012.4869.pdf
The youtube channel: https://www.youtube.com/c/SXSCollaboration/videos

Thanks for sharing this! Some of their visualizations are pretty easy to understand, but others seem to be more sophisticated. I will read the paper when I have more time.

 

[lomidrevo]

This is a bit off-topic of the original poster, I think, but I find it interesting. Perhaps a moderator could split the thread off?

I don't mind to keep it here, it is an interesting question :-) But of course, this is up to moderator to decide

 

[lomidrevo]

The youtube channel: https://www.youtube.com/c/SXSCollaboration/videos

One of those videos: Simulation of GW170104, is very helpful to me. It simulates a merger of two BHs, and in the video description they write this:

The lower part of the movie shows the two distinct gravitational waves (called 'polarizations') that the merger is emitting into the direction of the camera. The modulations of the polarizations depend sensitively on the orientation of the orbital plane, and thus encode information about the orientation of the orbital plane and its change during the inspiral. Presently, LIGO can only measure one of the polarizations and therefore obtains only limited information about the orientation of the binary. This disadvantage will be remedied with the advent of additional gravitational wave detectors in Italy, Japan and India.

The resolution is quite low to properly read the labels of the vertical axes, but I believe the first graph is for "plus" polarization, and the second one for "cross" polarization. Generally, higher amplitude for "plus" mode would be consistent with my understanding described in post #1. The ration of the amplitudes of the two modes can be used to infer the inclination angle.
I can also see that phase difference of the modes is $45^\circ$, as expected. And finally, as they say, we need second detector to be able to decompose the signal in two modes. E.g. Ligo + Virgo oriented at different angle (ideally about $45^\circ$).
So that is my "takeaway" from this :-)