Monday, 15 February 2016

How to decode gravitational waves from black holes

The gravitational-wave signal that we observed on September 14th last year, and announced to the world last Thursday, came from two black holes spiralling into each other. Those are the first two black holes we have ever observed colliding. The result of the collision was an even bigger black hole, and the last moments of the signal were the closest we have ever come to measuring a signal from a black hole. These are now three famous objects in the history of science. What do we know about them, and how can we be sure?

The properties of the black holes are encoded in the gravitational-wave signal. Unfortunately, the signal is very weak (that is why it took one hundred years to find gravitational waves after Einstein first predicted them), and, worse still, the features of the black holes usually have only a very subtle effect on the shape of the signal.

To make really precise measurements, we need an accurate theoretical description of the signal produced by any possible configuration of two black holes. We can use that as a key to decode the signal, and work out what sort of black holes we have seen.

That's where the work of my collaborators and I -- and our competitors! -- comes in. Black holes can be completely described by Einstein's general theory of relativity. That includes an exact (no approximations, no assumptions, nothing left out) description of two black holes orbiting each other, spiralling together, and merging, and the gravitational-wave signal produced as the black holes whip up space-time in their wake. There is only one problem: Einstein's equations can be written in what physicists loftily call an "elegant" form, but they are in fact 10 equations, and each equation has hundreds of terms, and the effects of one equation feed into the results of all of the others, and it is impossible to write down a simple solution to this whole mess, except for the most simple and uneventful situations. We can, for example, describe one black hole, sitting in space, all alone, and doing absolutely nothing. Such solutions are a theorist's adventure playground, but they are not much use if you are interested in two black holes falling into each other.

Thankfully, this is not 1687. Scribbling on a piece of paper is not the only way to solve equations. We can also feed them into a computer, and get it to do all the work. And that's what we do [1]. As usual, it turns out to be a lot harder than it sounds, and we only worked out how to do this just over ten years ago. But since then we have been working like crazy to be ready for what we found last September. There is still a long way to go to be able to decode any black-hole-binary signal we find, but fortunately we already have enough information to measure the properties of the two black holes we just observed.

Most of our work has been on what we optimistically call the "simple cases". If the two black holes are not spinning, then they orbit around each other on a fixed plane. This means that if you traced out the orbits, you would have a picture of disc; you can accurately draw the orbits on a flat sheet of paper. The figure shows this for two black holes, each with the same mass.

The motion during the last orbits of two black holes, 
each with the same mass. This system is consistent 
with the GW150914 observation.
In these binaries, each orbit produces two oscillations of the gravitational-wave signal. The strength of the oscillations depends on how fast the black holes are orbiting, and the frequency of the oscillation depends on how long it takes to complete one orbit. As the black holes get closer, they move faster, and each orbit is completed more quickly, so the signal gets stronger, and its frequency increases, i.e., the oscillations get closer together. The signal strength and frequency both ramp up faster and faster until the black holes meet and merge. Then we have one black hole, which in the beginning is a mis-shapen blob, but quickly settles down to the shape of a slightly flattened ball. The "settling down" also produces a gravitational-wave signal that very quickly dies away; this is a very characteristic "ringdown" signal, and how fast it dies away, and the frequency of the waves given off (the frequency no longer changes) depends entirely on the mass of the final black hole, and how fast it is spinning. (We shouldn't be surprised that it is spinning: two things whirling around each other came together: we would expect the final object to be spinning very fast indeed.)

We say that it is "simple" to describe these signals, because all that happens is that the strength grows more and more rapidly until the merger, and then dies down, and the frequency grows more and more rapidly until merger, and then levels off. The details depend on the masses of the two black holes: it is from slight variations in this behaviour that we can infer the black-hole masses from an observation.

We had produced models of these kinds of signals by about 2007, at least for binaries where the black holes had similar masses.

Things get more complicated if the original black holes are spinning. If we imagine the black holes as spinning tops, then the easiest spinning cases to describe are the ones where the tops are perfectly upright. In these cases, the effect of the black holes' spinning is that they spiral together more slowly or more quickly, depending on whether they are spinning clockwise or anti-clockwise.

You might think that if the masses affect how fast the black holes spiral together, and the spin also affects how fast the black holes spiral together, then the two effects would mix together, and make all of our measurements very difficult. And you would be right. If black holes could not spin, we could have measured the masses of GW150914 more accurately. Fortunately the signal was strong enough that we could still do a reasonable job, and got the values to within a bit worse than 10%.

These spin effects are also "easy" to describe in a theoretical model: it is the same general behaviour we had already modelled, and required only a few adjustments.That said, it took several years to add these effects into our models. By 2010 several models had been produced, and if we had observed a binary black hole then, we could have used these models to decode the signal and measure how massive the two black holes were, and how fast they were spinning.

(It turns out that the effect on the signal of the spins is so subtle that, unless the signal is very strong, we can only measure the total amount of spin from both black holes. If both black holes are spinning clockwise, that has a strong effect on the signal, and we can measure it. If one is clockwise and the other anti-clockwise, their effects on the signal can cancel out, and we might not see any sign of spinning at all. This is a very important effect, and I've no idea why I put it in parentheses.)

There is another problem: if we had used these models, we might have been fooled. If we assume that the signal behaves this way, but is in fact quite different, then we will incorrectly decode our observation and all of our measurements might be wrong.

Could this happen?

Yes, indeed.

There is no reason to expect that binaries will form in these very special configurations. In general the black holes could be spinning in any direction. These are not tops spinning on a table, they are objects floating in space, and if we spun a top in space we could set the nose pointing in any direction we wished -- there is no reason for it to be straight up and down. (Whatever up and down might mean in empty space!)

Now the black holes are not necessarily spinning clockwise or anti-clockwise to the orbit, but in some arbitrary direction. All of this spinning affects the surrounding space-time in such a way that the binary's orbit tips. Think of the binary like a plate set spinning on a smooth table. Previously the plate was set spinning flat on the table. Now the plate is thrown haphazardly onto the table, and wobbles as it spins. Black-hole binaries that do this are said to be "precessing".
A precessing binary. The dynamics are very different to the non-precessing case above,  
but this configuration is also consistent with the GW150914 observation. 

The wobbling plays havoc with our simple picture of how the gravitational-wave signal strength and frequency behave; the wobbling of the orbit means that the signal wobbles as well. Depending on the setup of the binary, it can wobble a lot. Most importantly, the wobbling we see in the signal will depend on the direction we look at it.

Think of the plate again. If you look at it from above, and far away, and it is a perfectly smooth plain white plate, then it may be hard to tell that it is wobbling at all. All we can see is a round plate. Maybe the sides of the plate are going up and down, but if we are far away and cannot see it very clearly, we might not be sure.

If we look at it from the side, it's a completely different story. All we can see is the edge of the plate, and now the wobble is the most distinctive feature we see.

From far away, we would also expect that the rapidly wobbling plate will be much harder to see at all from the edge, compared to the sight of almost all of the plate from above.

The same happens with gravitational-wave signals from precessing binaries. If we see them from above (or below), it is difficult to tell if they are precessing. On the other hand, the signal is much stronger, so we are much more likely to see signals like this. If the binary is "edge-on" to us, then we can see the precession effects very clearly, but the signal is going to be weaker. In this edge-on case, the signal will also behave very differently to the non-precessing cases I described earlier -- so if we try to decode the signal assuming that it is a non-precessing binary, we might do a very bad job of working out anything about the two black holes.

Here are the signals from two binaries. On the left is a binary where the black holes have the same mass, and there is no spin (the first case shown above). The binary is viewed from above. On the right is the signal from the precessing binary shown earlier, also seen from directly above. The two signals are very difficult to distinguish.


Now, here are the two same sources, but seen from the side. The equal-mass non-spinning binary signal looks the same; the change in orientation only affects the strength of the signal, not its shape. The precessing signal, on the other hand, is completely different. If we try to match this signal to a model that looks like the non-precessing-binary signal, it's quite possible that our estimates of the black-hole masses and spins will be badly skewed [2].


That is the problem we might have had if we had detected a binary black hole in 2010. It was a disturbing problem, because, if you think that all of this wobbling and procession and so on is very confusing, you are not alone. In 2010 we did not have theoretical models for gravitational-wave signals from mergers of precessing binaries, and we had no idea how to produce them. Lots of people claimed to have clever ideas, but no-one was especially convinced.

I was one of those people with extravagant claims. I scored the job I have right now on the basis of a grant proposal in which I bragged that (a) the problem of precessing binaries was "the biggest theoretical challenge facing our field", and (b), I would solve it. It was all a bluff. If you look carefully at all of the technical talk, the precise bullet-point descriptions of planned methodology, and month-by-month timelines describing exactly how I and my collaborators would nail this problem, the one-line summary was, "We'll think about it."

The amazing thing is that we actually did solve the problem. It wasn't just me, of course. Along with my students and postdocs in Cardiff, pieces of the puzzle came from research groups all over the world -- principally Spain, the US, Canada, Germany and India. Now, to our great relief, we do have theoretical models of general configurations of precessing black holes.

It was a close thing. Our most recent model was installed in the LIGO analysis codes only a few weeks before the first signal was detected. It was being painstakingly checked and double-checked in parallel to the analysis of the detection. Thankfully all was well -- in fact, the final numbers that were signed off in January had barely changed from the first preliminary results back in September.

So, what did we find? As you've probably heard, the black holes were 36 and 29 times the mass of our sun (give or take a few suns). How fast were they spinning? As noted in my parenthetical remark earlier, we can only really measure the total spin of the two black holes, and that is close to zero. It's possible that both black holes are spinning very fast, but in opposite directions, but we cannot tell for sure.

And precession?

It seems that we're looking at the binary from either above or below (most likely below). That means we cannot tell whether or not it is precessing. We think it is most likely precessing, just because it is more likely that any binary is precessing rather than not -- the chances that the spins just happen to line up perfectly straight up and down are very small -- but we cannot tell for certain from the data.

That means that we would have got pretty much the same results if we had used our old models of non-precessing binaries, and indeed a measurement was done with non-precessing models, and the results were consistent with those from the precessing model. But without a precessing model we would not have known for sure.

It would have been nice to see a more edge-on binary, where we could tell more clearly whether or not it was precessing, and maybe extract a bit more information about the black holes. But considering how much luck we had already -- we got to discover the first-ever binary black hole! -- we are not really in a position to complain.

In fact, we got really lucky. Our theoretical models are reliable if the black holes have roughly the same mass, and they are not spinning too fast. That's just the kind of system we saw. One of the black holes could have been ten times more massive than the other, or spinning incredibly fast in an inconvenient direction. Then we would have been much less certain of what we saw.

Now we have a lot to do. Next time we might not be so lucky. On Thursday and Friday we celebrated, but now it's back to work.


We detected gravitational waves! (Science)

What it feels like to detect gravitational waves. (Personal)


Why bother trying to explain gravitational waves? 
Is spacetime really curved?


1. We do use pen-and-paper calculations to estimate the paths of the two black holes, and their gravitational-wave signal, for the thousands or millions of orbits they follow before they merge. The calculations rely on approximations that become increasingly inaccurate as the binary approaches merger, and that is why we need to solve the full Einstein equations on a computer to understand what happens in the final moments of the binary's life. Many of our computer calculations cover roughly the same number of orbits as we detected on September 14th.

2. For aficionados, the precessing signal depends strongly on the source's polarisation angle, and, yes, I did choose a value to accentuate as much as possible the effects of precession.


  1. > It seems that we're looking at the binary from either above or below (most likely below).

    Cheeky bastard :D

  2. Sorry to go a bit off topic, but what sort of effect would these waves have had on matter close to the collision, 1 LY away say, or 1 AU away?

    1. The source was approximately a billion light years away. If we were only 1 light-year away, then the effect would be the same, but a billion times stronger. Since the fractional change in the length of the interferometer arms (on Earth) was ~10^(-18), it would still be ridiculously weak even if it were a billion times stronger.

    2. If we'd only been one light-year away I imagine we'd have bigger problems than detecting gravitational waves...

    3. "Since the fractional change in the length of the interferometer arms (on Earth) was ~10^(-18), it would still be ridiculously weak even if it were a billion times stronger."

      But wouldn't the strength go as 1/r^2, like EM waves? That would make the effect ~(10^9)^2*10^-18 i.e. 10^18*10-18 i.e. order unity at 1 LY, which sounds pretty substantial, and ~4*10^9 at 1 AU, which sounds pretty harrowing indeed.

    4. No, the strength falls off as 1/r.

    5. Very interesting. That does make sense, given that GR in 2D doesn't have gravitational waves at all.

  3. Really? If I was a good little science outreach person, I would do some calculations now. But since I'm lazy, I'll only note that the total mass of our favourite black-hole binary was 65 times the mass of our sun. A light-year is a hell of a lot more than 65 times the Earth's distance from the sun. (We are eight light-minutes from the sun, if someone wants to work out the factors.) So why would it have been such a problem?


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