# Recent observation of gravitational waves

It was exciting to hear that LIGO detected the merging of two black holes one billion light-years away. One of the black holes had 36 times the mass of the sun, and the other 29. After the merging the mass was that of 62 suns, with the rest converted to gravitational radiation. Could someone explain, without assuming a deep knowledge of general relativity, how these conclusions were reached?

Short version: LIGO matches their data onto waveforms calculated in numerical relativity. The mathematical study of black hole solutions plays a significant role in this; we couldn't trust our inferences if we didn't know a priori that black holes rapidly stabilize into a handful of low-parametric stationary configurations.

Classical black holes are solutions to the vacuum Einstein's equations for the Lorentzian metric. The stationary ones are relatively simple things: Bunting & Mazur proved that the only static, axisymmetric solution of the coupled Einstein & Maxwell equations is the Kerr-Newman-et-al solution, characterized by mass $M$, angular momentum $J$, and charge $Q$ ($\simeq 0$ in practice). (For a review of the development of these ideas, see Robinson's Four decades of black hole uniqueness theorems and P. Mazur's Black Hole Uniqueness Theorems .)

Non-stationary black hole solutions to Einstein's equations also exist, but they are more difficult to study, being analytically intractable. Our understanding of them relies on a mix of general considerations and numerical simulations. The basic picture is straightforward, however. Two black holes caught in each other's reach will rapidly collapse into a single black hole, and emit a spherically expanding pulse of gravitational radiation. By the time this radiation reaches Earth, only the leading spherical harmonic (the quadrupole moment) is potentially observable. Everything else dies off in powers of $r$.

After gauge fixing & choosing coordinates, the quadrupole moment can be treated as a perturbation to the 3d Euclidean signature metric on a space-like hypersurface in our local Minkowski-like patch of spacetime. It has the form $$h_{ij}(r, t) \simeq \mbox{ transverse traceless part of}\frac{2G}{rc^4} \frac{d^2}{dt^2} I_{ij}(t-r)$$ where $I_{ij}$ is the quadrupole moment of the source.

Numerical simulations allow you to calculate the shape of these waveforms as a function of the initial and final parameters of a black hole merger. This is done via a modified ADM formalism; they gauge fix and time-evolve data from one spacelike hypersurface to another. I'm not expert enough to say anything interesting; it's all hard detail. (To learn more, I'd start at Living Reviews in Relativity, which is a remarkable journal.)

LIGO's laser interferometry is sampling local perturbations to the spatial metric. These are more or less continuous measurements and (after the terrestrial backgrounds and detector noise are subtracted) fairly clean ones. You get a high resolution look at passing waveforms in the band of sensitivity.

When LIGO sees a signal, they match it onto a database of pre-calculated waveforms, which encodes their priors about what observations are possible in their detectors, given what we known about likely gravitational radiation sources. The data is high enough resolution that you can match pretty accurately and read off the parameters of the merging binary black holes.

If you want to get into the fun details, LIGO has released their data for the big collision, along with a Python-based tutorial in signal processing.

https://losc.ligo.org/s/events/GW150914/GW150914_tutorial.html

• @AlexanderChervov It's a purely experimental matter: We were not able to produce sufficiently sensitive detectors until a few years ago (after years of calibration on the LIGO sensors). If you're asking about what exactly the experimental breakthroughs were, I don't know. – Danu Feb 14 '16 at 13:24
• This answer currently contains pretty much zero mathematics. I think that, given the fact that this is a research level mathematics site, some quantitative discussion would be appropriate. – Danu Feb 14 '16 at 21:00
• @AlexanderChervov Without modern computers, it's difficult to study the time evolution of a merger through the period which produces the peak radiation intensity. You can use the Post Newtonian approximation to look at the initial approach and the final ringdown, but you can't see what's going on as the event horizons merge. – user1504 Feb 15 '16 at 4:24
• @user1504 I see that your original response was "Fine. Bah." (edited away now for something slightly less angry-sounding). You seem to think my suggestion is not very useful. If you disagree with me you are by no means obliged to indulge me, as I'm sure you know. In any case, I think the answer has now significantly improved. – Danu Feb 15 '16 at 8:21
• @AlexanderChervov - An exhaustive answer to your question is here: books.google.com/books?id=HdP3Tscyr3AC. The book is actually quite good IMO. – Steve Huntsman Feb 16 '16 at 17:20

The numbers 36,29,62 were obtained as the best match between the received signals and the output of computer simulations. The 90% confidence intervals on these numbers are about $\pm 4$. The details (largely beyond my understanding) are in the technical paper here.