# LIGOpalooza

So we found gravitational waves today. Who, what, when, where, why and how?

First, I should say that I’m a particle physicist moonlighting as a particle astrophysicist. I have a grad-school level understanding of General Relativity, but I’m not a professional relativist, and I don’t study gravitational waves for a living. My goal is just to talk a bit about what happen, what I learned, and to generally geek out because this is a big discovery and I'm pretty excited to see it happen, even just as a bystander.

So first, what are gravitational waves? As a very important side note, they aren’t gravity waves. Gravity waves are waves on the surface of a fluid: you might know some of them as “ocean waves.” These are very important, but presumably discovered by the groundbreaking scientist Lucy some time in the Pleistocene.

Gravitational waves are waves in the spacetime metric. At the quantum level, everything that we experience is a manifestation of one field or another: electrons, quarks, photons, New York, and so on. Each point in these fields can oscillate (the field strength can change, and wants to restore itself to some equilibrium value). Moreover, the field value at one location can affect the field values nearby; so if I pull the field away from the equilibrium value at one point, the field value next to it gets pulled aside as well. Think of this crudely as like a guitar string: I pull at one point, and all the string distorts, with the points further away moving less. You can see this in terms of a quantum field by looking at the electric field in the presence of a charged particle. The charge “pulls” the field in the same way that my finger pulls the guitar string (and in fact, I pull a guitar string through application of the EM fields which are part of my finger pulling the charges in the string), and the field further away from the charge is still affected, but less so than nearby.

Of course, if you let a guitar string go, you know you’ll get a wave. What happens is that each point in the string wants to restore itself to the equilibrium point, but when it reaches that point, it has kinetic energy and so overshoots, and oscillates back and forth. Since the displacement of the string at each point affects the displacement of the bits of the string nearby, these oscillations can propagate up and down the string, giving you a wave. So a wave is just a “coupled” oscillation.

Same thing with an electric charge. If you wave a charge back and forth, the electric field nearby tries to change to continue to point towards the charge. Electric field values at one location affect the electric and magnetic field values nearby, and so the information that the charge is moving can propagate outwards in an oscillatory manner: a wave. In particular, this wave is what we call light: you get light by waving charges back and forth.

So, now onto gravity. With general relativity, Einstein taught us that the way to think about gravity is as result of something called the space-time metric. The metric is how we measure distances and time: it’s the fundamental ruler that tells you how big something is and long it takes time to tick. The reason we fall down towards the center of the Earth is that the mass of the Earth affects the spacetime metric (similar to how a charged particle affects the electromagnetic field). Mass and energy causes the metric to “bend,” which is to say that lengths and times are different near a mass than they would be far away (in “flat” space). So we fall down because time is ticking slower at our feet than at our heads (not by much, but the gravitational field of the Earth is very small as these things go, it just feels big to us monkeys). So the Earth sitting in space deforms the metric of spacetime: the field of lengths and times. This is often analogized as a rubber sheet, bent by the presence of mass. But let’s not get into that now. Imagine that you suddenly removed the mass of the Earth (which is impossible on a number of levels). Then the metric would try to spring back to it’s flat-space configuration. In doing so, it would oscillate, and just like the electromagnetic field, the oscillations near the place where the Earth was would spread out along the spacetime metric. We would call these coupled oscillations waves in the spacetime metric, or gravitational waves.

Now, you can’t remove mass or energy from the Universe, so you can’t set up waves in spacetime just by blinking a planet or star out of existence. But in the same way, you can’t remove charge from the Universe, so you set up electromagnetic waves by waving charges around. So if you want to set up gravitational waves, you do it by taking a mass or some energy, and moving it around really fast: this will force the metric around the mass to change to keep up with the location of the mass itself, and get you your wave.

Problem is, charges are small, abundant, easy to move, and create big deformations in the electromagnetic field. So it’s easy to create electromagnetic waves: just turn on a light or switch on your radio. Gravity is a very very very weak force. The ratio of the force from an electron due to its charge and due to its mass is enormous:

$$F_{\rm elec}/F_{\rm grav} \sim 10^{42}.$$

So you need to move an enormous amount of mass around to see a sizable wave. Like, say a few solar masses.

There are some other differences: electromagnetic waves can come from “dipoles:” a charge and the opposite charge next to each other. This creates a particular pattern of radiation in the wave. Gravitational waves can’t come from dipoles (there are no anti-masses to stand in for negative charges for one thing), which means that gravitational waves are coming from the “quadrapole” of moving masses. Turns out this is directly related to the fact that the EM field is spin-1 and the metric is spin-2, but moving on.

You also have to measure the wave. This presents a problem. If I’m sitting in the middle of a gravitational wave, that means the spacetime metric is oscillating. Time ticks slower, then faster, than slower again. Lengths get longer, than shorter, than longer. But here’s the problem: it’s not that I will see my clock ticking at a different rate: EVERYTHING ticks at a different rate, including my cognitive processes. I won’t see a meter getting shorter, since the meter stick I’m using gets shorter as well. How do I measure this?

Deflection of light of stars due to the mass of the sun, measured during a solar eclipse (Campbell 1922).

I use light. Light waves will also get fundamental changes in length and time as the metric oscillates. However, general relativity tells us that spacetime cares about energy in all of its forms: so mass (since $E = mc^2$) changes the metric, but so does kinetic energy. Things that are moving very fast have significant kinetic and rest energy, and react to bending spacetime in a slightly different way than slow moving things (here, fast moving means “near the speed of light”). Light, being light, is massless, and so has all of its energy in the form of kinetic energy, and so is the ultimate relativistic particle. We’ve already seen the difference in the bending of light and matter from spacetime: in one of the earliest confirmations of Einstein’s theory when the bending of light from distant stars passing near the Sun (observed during a total Solar eclipse) was seen to be twice what the Newtonian (non-general relativistic) prediction would say it was. We now use this effect in gravitational lensing.

So light will bend differently than the non-relativistic matter that we and the Earth are made out of. So if a gravitational wave passes through the Earth, the length of a meter will change, but the length of time it takes light to travel that meter will change in a slightly different way. So you can see that change.

LIGO set up (from the LIGO Physical Review Letter). The upper left shows the two locations of the LIGO interferometers in America. The center is a sketch of the interferometer: light is split, sent down the two 4 km arms, and then recombined to see a interference patter. Upper right shows the minimum strain they can measure as a function of frequency.

What LIGO does is sent laser light down two arms, each 4 km in length, and set at 90 degrees away from each other. The laser light is split, sent down each arm, and then returns. LIGO then recombine the light, and if one arm was slightly longer than the other, the two waves won’t exactly match up. So tiny changes in length can be measured. This is called interferometry, and is the standard way to measure differences in length. LIGO is just the most precise interferometer ever.

Of course, gravity is very very weak. So gravitational waves cause only exceeding small changes in length. We measure the size of the wave signal in terms of the “strain” $\Delta h/h$, the relative change in length $\Delta h$ divided by the length $h$. The strain of the signal LIGO saw today, of two 30 solar mass black holes moving at half the speed of light, is only about $10^{-21}$, which is saying that LIGO can measure the length of their “meterstick” (a 4 kilometer path) down to 0.004 of a femtometer (an atomic nucleus is about a femtometer across).

The technology here is just bonkers. I’m completely unqualified to speak about how they do it. EVERYTHING will cause oscillations in length which completely outstrip the signal. So they need to cancel those spurious noises. This is one reason LIGO has two locations: one in Louisiana and another in Washington state: someone downshifting their truck on a highway in Louisiana will cause a signal in that detector, but not in Washington.

But now let’s look at the result.

Best fit regions for the mass and total spin of the final black hole after the merger (LIGO)

As I said, gravitational waves need a huge amount of mass to move around very quickly in order to generate a reasonable signal. What LIGO saw was the signal of two black holes colliding, with masses of $36^{+5}_{-4}$ and $29^{+4}_{-4}$ solar masses. After the collision, there was a single black hole with $62^{+4}_{-4}$ solar masses. During the collision 3 solar masses of energy was radiated out. That’s astounding. A supernova sends out $10^{51}$ ergs of energy in light, and $10^{52}$ ergs in neutrinos. This is $10^{54}$ ergs of energy, 100 times more than the energy sent out in the previous benchmark for the Biggest Booms Around. There’s some degeneracy between the measurement of the spin of the black holes and their masses, but they have enough handles to get some information on that.

The black holes were located $410^{+160}_{180}$ Mpc (or 1.6 billion light years) from Earth, resulting in a strain of $\sim 10^{-21}$ at Earth. Again, the fact they can measure this is completely bonkers.

If there was no such thing as gravitational radiation, two masses orbiting would orbit each other stably forever. So the black holes in question would never have collided without gravitational radiation, just kept on keeping on. However, they were bleeding off orbital energy into these waves in the metric, which caused them to spiral inwards. As they got closer, the rate of energy loss increased, accelerating the process. We’d already indirectly seen evidence of gravitational waves in neutron star pairs. The 1993 Nobel Prize went to Hulse and Taylor, who saw the change in binary pulsar timing, due to this gravitational radiation.

So as the black holes spiral in, the deformations in the metric get bigger and bigger. That causes bigger and bigger waves of gravity to be sent out. When they reach the Earth a billion plus years later, LIGO measured an increase in the strain, as well as a change in the frequency of the wave: the black holes were orbiting faster and faster, sending out peaks of the waves more often. Here’s what the signal looks like, combined with when those waves were sent out from the collision.

Measurement of merger event, from the LIGO Physical Review Letter. Top is a sketch of the location of the two black holes as time goes on, center is the actual measurement (strain over time), and bottom is the inferred velocity and separation between the two black holes

This is just beautiful data. Really amazing and so cool to look at. Let’s look at it some more.

More LIGO data (again from the PRL). The two separate measurements, in the two LIGO locations are shown.

The frequencies are in the audio band, so you can convert this actual collision of black holes into sound you can hear, without any real post-processing. Just play loud noise when the strain was large, soft when it was small. You get a characteristic “chirp.” So black holes go chirp in the night. Something of a letdown, perhaps, but hey, they outshone the Universe for 0.2 seconds, so I’ll let them chirp.

So what do we learn? Well, from the rate at which the signal amplitude and frequency change during the inspiral, they can get the masses of the black holes, and the mass of the final resulting black hole (the numbers I wrote above). I wonder how much degeneracy there is from the possible alignment of the spins of the black holes, which apparently can increase or decrease the amount of gravitational wave radiation. But I only know that from reading some old papers of Kip Thorne’s, so I’m going to guess the man himself thought of it during the analysis.

Immediately after the collision, you have a black hole that isn’t exactly spherical. Black holes would like to be spherical, so the spacetime deformation that makes up a black hole will rearrange itself to be a spherical spinning black hole (a Kerr black hole). This will, of course, radiate gravitational waves. This is the “ringdown” at the end, and so we’re really seeing the formation of a black hole from two merging objects. Which is again, pretty incredible.

During the actual collision, in principle you can learn about how gravity acts when the deformations of the metric are large. However, I gather that there’s some lack of theoretical control here, so it’s not clear that we’ve learned anything yet. Broadly, everything matches with Einstein’s predictions. In particular, LIGO finds that the signal they saw was compatible at the 96% level with the predictions of General Relativity.

From the fact that the high and low frequency signals both arrived in times that are consistent with the predictions of the computer simulations of the merger, LIGO learned that there’s no measurable frequency-dependence on the speed of gravity. That can be interpreted as a measurement of the mass of the graviton. We expect that gravity has massless force carriers: the graviton should be massless and move at the speed of light, just like the photon. It turns out that is a necessary condition for the force of gravity to drop off like $1/r^2$. However, we like data, and this measurement allows us to place a limit on the propagation of gravitons: they move at or at least very near the speed of light, so they’re massless or very close to it.

LIGO's limit on the wavelength of the graviton, related to the inverse mass of the graviton.

From the fact that LIGO saw this event almost immediately, they learn that there’s a population of 30 solar mass black holes out there. Unless we got unbelievably lucky, there have to be more. That tells us something about how black holes form, which tells us something about how stars form. From what I can tell, LIGO’s result already is suggesting evidence for particular models of stellar formation in the earliest stars (what are called Pop III stars). However, you can’t hang too much on one event, so we’ll need to see more, and once we have a statistical sample, we’ll have a much better idea of what’s going on. But still: we have a new way of looking at the Universe, and that will tell us a lot about how things work.

So what’s next?

Predictions for gravitational wave signals from Cutler and Thorne (gr-qc/0204090)

Well, as I just said, one event is amazing, but more is better. This event was very bright, and very clear, so it was seen immediately. There should be smaller black hole mergers out there, many more, we hope. LIGO will start seeing those soon (stay tuned). That will start giving us real statistical information about the population of black holes in the Universe.

We’ll start seeing neutron star mergers too, we hope. Neutron stars are less massive than black holes, but in the collision we’ll get information about how the pressure waves of the collision propagate through the incredibly dense matter that make up the neutron star. We have a lot of questions about how that form of matter works, and the gravitational wave data might speak to that. This is in addition to the information we’ll gain just on the population of neutron stars out there as well.

There are other possible signals, though we may not be able to see them at LIGO.

If a supernova isn’t perfectly symmetrical, then the motion of all that gas being thrown around in the death of the star will release gravitational radiation. That will tell us interesting information about how supernovae occur.

If we build something like LIGO, but in space (called LISA), we can do even better, and start seeing gravitational waves from the collision of bubbles of particles forming in the early Universe. We have no direct probe of that epoch as of yet.

Plus, every time we have looked in the Universe in a new way, we find things we didn’t expect. We now have a new observatory for the Universe: gravitational waves. There will be something surprising found, something we had no idea was possible until it was seen. What that will be, I have no idea. But I can’t wait.