[WhoBeI]

According to Einstein there is no gravity that pulls us to the center of Earth. Instead mass curves spacetime in such a way to make you experience traveling along the curvature as gravity. If the mass is rotating the curvature will also drag on the surrounding spacetime. Imagine something like a whirlpool and then consider what happens if two of those merge. While merging some of their energy will cause waves in the surrounding water that eventually fades away with range. Those waves would be the gravitational waves - ripples in spacetime.

[lomnakkus]

Incidentally, I also find the "geometrical" approach a great way to look at the whole "light vs. black holes" thing: It's not that black holes directly "trap" light or "prevent light from escaping" per se, it's more that spacetime gets curved so much that regardless of which direction a ray of light is traveling it'll always find itself back inside the event horizon.

(Not sure where I first heard this explanation, but I can definitely say that I didn't originate it!)

[nerfhammer]

Man, that made me realize that black holes really are literally "holes".

[lomnakkus]

Yup. A "hole" in space in some sense, though I must admit I haven't thought of it such extremely concrete terms[1]. I imagine that realization went into the naming :).

[1] I mean, we usually imagine we can retrieve things when they drop into a hole, but that's not really a thing in this case. (Excepting Hawking radiation and the absurd amount of computation/time you would need to reconstruct the original state information from that. If you thought the time-of-evaporation for a Black Hole was long, you've got another thing coming when it comes to reconstructing the state from that evaporated radiation! Not that this has has any practical bearing, but it's fun to think about, right?)

[raattgift]

Spherically symmetrical masses (even ones that are rotating and even ones that aren't exactly spherical as a result) do not shed gravitational radiation. If you put two such masses on a parallel course through space, although they'll eventually collide, they won't release gravitational radiation.

Barbell-like configurations of matter emit gravitational radiation when they spin about an axis perpendicular to a line through both bodies' centres of mass. A binary star system is barbell-like, with two heavy masses at opposite ends of an so-infinitesimally-thin-it's-not-really-there bar. This "barbell" rotates around the barycentre, which for all practical purposes lies along a line through the two stars' centres-of-mass. Even more massive binary objects can have a rotating-barbell type of arrangement.

More detail here if you want it: https://www.wikiwand.com/en/Gravitational_wave#/Sources

Rotation of each black hole is relevant in black hole mergers in the late stages of the inspiral, and there have been several numerical relativity studies of BHs with extreme rotation rates (each BH rotating around its own axis, and possessing only axisymmetry rather than approximately spherical symmetry). The results are interesting, but don't really have an impact on the shedding of gravitational radiation; that comes down to the barbell-like configuration.

In order for there to be frame-dragging one must first fix a frame of reference. If you fix Cartesian coordinates with the origin at the middle of a bucket of water, and then you spin that bucket around on a rope, you're dragging those coordinates around with the rope. If the coordinates serve as the basis of a frame of reference then presto, you have frame-dragging. But if you have a suitably heavy bucket of water and spin it hard enough, you'll notice the barycentre shifts outside of your body along the rope towards the bucket. Congratulations, you're now shedding gravitational radiation!

That gravitational radiation is there even if you only ever calculate in coordinates where the origin is on the floor of the corner of the room in which you and the bucket are spinning; that ("laboratory") frame of reference is not being dragged by you.[1] In the lab frame and in the bucket frame, your precession as you struggle with your footing are pretty different: the rope remains the same length so you are at a constant position in bucket-basis coordinates, but you are moving around relative to the lab coordinates.

You can even fix (0,0,0,t) on some part of your own body; you are dragging those coordinates around as you spin the bucket, and the bucket eventually settles at some fixed point in your coordinates. The walls of the lab move around in those coordinates though, and might move inwards and outwards from you as you spin around.

The Lense-Thirring effect is similar to how the walls move around in your coordinate basis as you spin around. It arises when one uses an exact solution of the Einstein Field Equations (typically the Kerr metric) and a set of coordinates suitable for a static observer. A static observer is one who sees [a] no change to the gravitational or matter fields over time and [b] can slice spacetime into space+time in a particular way. Static observers are not realistic observers in a universe full of moving matter. The Kerr metric is unsuitable for the real Earth, since it's lumpy inside. These approximation choices are mathematically convenient, but come with the side effect of needing to import fictitious forces to explain orbital precessions.

The extra mathematics of Lense-Thirring precession however are much much more convenient than dealing with a more realistic metric and set of coordinates for Earth. Moreover, it is perfectly reasonable to do physics in preferred frames of reference (Kerr + Boyer-Lindquist, in this case) as long as you admit to yourself that that is what you are doing. There exists a Bogoliubov transformation from this preferred frame of reference to any other reasonable frame of reference, so you're not "stuck" with Lense-Thirring.

You are however stuck with the physical result that something orbiting such that 24.something orbits "should" give it the same view below (and above!) but doesn't. We've shown this around various artificial satellites put into polar orbits around various bodies in the solar system.

The underlying source of this precession is a combination of gravitational and special-relativistic time dilation. These are not "forces", though.

- --

[1] It is being dragged around by the Earth's movements though; if you treat it as an exact inertial frame and do extremely sensitive experiments you'll see deviations in motions of things that you may be tempted to describe using e.g. coriolis forces. If you use exquisitely precise atomic clocks and insist on Cartesian coordinates, you will eventually discover that you are not in an inertial frame after all, and will have to take the local curvature of spacetime into account in the basis of your frame of reference, or shrug and use fictitious forces as corrections to Newtonian or special relativistic geometry. The magnitudes of the fictitious forces will be small.

[greeneggs]

Thank you for this explanation. Do you know why 3 solar masses were converted into gravitational energy? (31+25-3=53) What kinds of black holes collisions convert more or less mass?

Is there a rule of thumb one can give to predict this, for 31 and 25 solar mass black holes? Is there a limit that we expect (e.g., 2 <= mass lost <= 5), or could it go up to 31+25-56=0?

[raattgift]

We can extract something like a rule of thumb from the equation for the power going into gravitational waves by a pair of isolated, not-very-close masses in circular orbits around their barycentre:

P = -32/5 G^4/c^5 \frac{ (m1 m2)^2 (m1 + m2) }{r^5}.

The key adjustable here is the orbital distance r. As it goes down, power goes up, but it's still extremely even for large masses until the two masses are practically in contact. You can plug in kilograms for masses m1 and m2 and use metres for r5. We are already doing a linearized approximation here, so we can bite the bullet and circularize the orbit of Earth-Moon at the average orbital distance and find a value that's a few microwatts; doing the same for Earth-Sun gives us a couple of hectowatts. Putting a pair of tens-of-solar masses at r ~ 1000 km gives us a much much larger power.

We can take the integral of power P over time and arrive at the radiated energy. We can in turn consider an r that shrinks over time. We can also flip this on its head and consider that it is equally reasonable to say that r shrinks in response to energy loss or energy loss increases in response to r shrinking. More on that below.

The calculation grows much more complicated for realistic orbits especially when the orbital distance is small, but you only asked for a rule of thumb, not a detailed explanation of, say, the Bondi-Sachs mass loss equation. :-)

> Do you know why [large mass] were converted into gravitational energy?

Sure. Very roughly, angular momentum depends on frame of reference. If you choose to work in a frame of reference in which a black hole binary has significant angular momentum in the mutual orbit (e.g. in the cosmological frame), then you have to get rid of a chunk of that for two reasons.

Firstly, angular momentum needs to drop in order for bodies of constant mass to drop into a closer mutual orbit. Although there are other interactions at work when the distances are large for a binary BH in a galactic core, when the binary BHs are near the end of their inspiral, the orbital distance shrinks very quickly. The only reasonable mechanism to shed that angular momentum that quickly is gravitational radiation.

Secondly, the "no hair" theorem means that there is only a three-component angular momentum J an at any given time coordinate for a black hole once it has stabilized. But as each of the merging BHs has its own nonzero angular momentum there are excess multipole moments that count as hair on the asymmetric configuration at the end of the inspiral, and so balding has to happen. This causes excess momentum either to be swallowed into the merged black hole (which will have some final angular momentum) or radiated to infinity. Swallowing is limited by the final mass of the merged black hole. Superextremal black holes are forbidden, for example, and in practice merged BHs are unlikely to rotate near he maximum possible speed for their mass. Consequently the rest must disappear as gravitational radiation, since it can't escape in any other way.

> could [the whole configuration be radiated away]

A real answer would be quite deep since mass in General Relativity is not so straightforward. However, the individual black hole masses m1 and m2 will change very very little during the inspiral and merger; and the end black hole will be very close to m1+m2. What gets radiated away is maybe best understood as some of the quantity that works against the BHs simply falling straight down onto one another at the speed of light, which is something no observer sees. I discussed this a bit more at https://news.ycombinator.com/item?id=15353970