Didn't realize that energy could be converted into gravitational waves. It does make sense I suppose, but I always thought gravity and gravitational waves were just a side-effect of the way mass interacted with space-time.
(Not sure where I first heard this explanation, but I can definitely say that I didn't originate it!)
[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?)
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.
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[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.
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?
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
If I drop my bowling ball, it has lower potential energy, but can't some of that be compensated for by the Earth-ball system radiating a gravitational wave (rather than all being converted into kinetic energy)?
Similarly, don't orbiting bodies emit gravitational waves? I forget the exact dynamics, but the way that they drag just a little bit as they orbit creates waves that radiate power out of the system -- I think the Sun and Jupiter radiate something like 55 watts of gravitational waves.
If you take two things that are each tens of times larger than the Sun and set them spinning an appreciable amount of the speed of light very nearby -- to the point they eventually collide -- it's not surprising they emit quite a bit more than our solar system.
Think of it this way:
Two large objects start orbiting slowly far apart. But because they're large and the system is big, they have a lot of orbital momentum.
They pull each other close, but as they get closer, that orbital momentum has nowhere to dissipate, so they orbit faster and faster!...
...Except that because they're so massive, they slightly tug space around with them as they orbit. And because they're orbiting so fast, space doesn't have a good way to smooth itself out. So some of the momentum from the heavy things spinning far apart gets stored in wrinkles in space as they come together.
Which then radiates away from there to us, ...and slightly stretches one direction of reality as it passes by. Which we detect by measuring how long two perpendicular lines are very accurately a few different places on a nearly spherical object.
...So you can model it as just the way that mass interacts with spacetime if you remember to include things like the distribution of your mass as a source of potential energy that can itself be used to generate gravitational waves. However, this does not make the whole situation any less weird to explain.
Reports just tend to convert the amount of energy released through mechanisms like that in terms of mass, because it's the only thing vaguely comprehensible. (eg, 3 suns)
Nothing wrong with your comment, but I just want to emphasize that for objects smaller than stars, the amount of energy carried by gravitational waves ranges from tiny to unimaginably tiny. The Earth-Moon system emits about 7 microwatts of gravitational waves. A bowling ball in low earth orbit would radiate about 5*10^-41 W, which is roughly equivalent to one photon of visible light every quadrillion years.
(In reality the mass contribution due to the gravitational potential energy is included in the "mass", this is just illustrative.)
Since the negative energy ultimately wins, there must be less and less positive energy holding the bodies apart. The positive energy has to go somewhere, and that somewhere is gravitational radiation.
One could equivalently be very Machian and rotate with the system and consider that the gravitational attraction of "the whirling distant stars" is working to keep the black holes from colliding. The distant matter loses because the binary black holes move positive energy (holding the BHs apart) out towards the distant stars (which also lets the BHs move a bit further away from the distant stars).
s ---+ B1 --++ B2 ---+ s -> s ---++ B1 -- B2 ++---s
Even if we consider inspiralling stars (no horizons to be found), we're lifting relatively little mass-energy out of them, rather than dealing with the energy that keeps them from colliding. The mass-energy of the system (which you measure in Joules or solar masses) when considered in the cosmological frame is what has the excess ~ 12% that gets radiated away.
Finally, in Special Relativity the angular momentum and energy-momentum of localized systems is frame-dependent; the addition of gravitational fields in a linear approximation of GR does not change that, and gravitational waves are a result in linearized gravity. It certainly is useful to discuss dimensionful quantities like the ones you listed, but it's important to remember that different observers are free to disagree violently about the numbers. Thinking about how and why they might do so is often enlightening.
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[1] Although the convention is mostly there because in Newtonian gravity, one takes T + U = const. for a body felling gravity, and since by virtue of having been described earlier T (kinetic energy) is positive so U (gravitational potential) must be negative, with the result that F = - \nabla U. We break the conservation of T + U in General Relativity quite happily, but retain the notion of the gravitational interaction as negative in many useful frames, so in the footnoted paragraph we have a generalized T and U available by a suitable choice of frame of reference in the weak limit.
I'll try an explanation, hopefully it's useful to you or another reader.
General Relativity gives us $\nabla_\mu T^{\mu\nu} = 0$, which reads that the covariant derivative of the contravariant stress-energy tensor (the matter tensor) generally vanishes. When remembering the heart of General Relativity: G_{\mu\nu} = T_{\mu nu}, that is, the gravitational interaction arises from moving matter, we observer that changes in the gravitational field match changes in the arrangements and/or motions of matter and vice-versa.
This is a fancy way of saying that there is a global conservation of stress-energy-momentum, even when you work in coordinates or gauges in which there is no global conservation of energy.
For simplicity, here and below I am setting some constants to 1, only considering classical behaviour, completely ignoring strain-squash modes, and otherwise being only very mildly post-Newtonian.
In a system like a rotating barbell with a very thin, almost massless stick, like in a Cavendish apparatus ( http://en.wikipedia.org/wiki/File:Cavendish_Torsion_Balance_... ) the masses on the end of each bar will influence the motions of masses near them, and vice-versa, because their masses attract one another. We are free to choose whatever coordinates we want to describe this sort of setup, including one in which the two grey M balls move in response to the proximity of the two red m balls, with the point where the bar is suspended taken as an origin of some useful coordinates (e.g. spherical polar ones). In particular, as the ms pass close to the Ms, the rotation of the bar is dragged on, and one is free to model that as angular momentum leaving the rotating barbell under a gravitational interaction.
We can treat two massive objects (stars, for example) in a binary orbit as a barbell-like system; the bar itself simply has no mass and is in free-fall rather than suspended on a wire. However, just like in the cavendish experiment, concentrations in the distant mass on either end of the bar will cause the bar to stretch and the system's rotation to slow, while sparsenesses in the distant mass on either end of the bar will cause the bar to contract (the "bar" itself is just the gravitational interaction between the binary masses).
With a clever change of point of view, we may ignore the influence of nonuniformities in the distant masses and just talk about the momentum leaving the rotating system in response to defined (or discovered) changes in the length of the "bar". That momentum is gravitational radiation, and its amplitude and frequency depend on the masses and orbital periods of the ends of the shrinking "invisible bar" of their mutual gravitational interaction. Near the very end of the inspiral, the bar shrinks rapidly, and the amount of angular momentum that is shed is consequently enormous.
Spacetime enters into the picture here via a linerization of the General Relativity picture of inspiralling massive objects: we start by picking out a flat spacetime (\eta [1]) and a field of perturbations on that spacetime (h) and then get the real metric tensor g via g = \eta + h (+ h^2 + h^3 ...). We drop the higher order perturbations on the spacetime (h^2 + h^3 + ...) and have a linear theory that is approximately General Relativity and is very very very close practically everywhere outside of event horizons and the extremely early universe. Gravitational waves are simply excitations in h that obey the massless wave equation.
Spacetime is pretty much meaningless without a measure of lengths and durations between bits of matter, or equivalently, spacetime is the measure (i.e., geometry) of distances in space and time between bits of matter.
Since gravitational waves are waves in an addend of the metric of spacetime, they are equally waves of spacetime.
We can consider a region of spacetime in which we find the "barbell" of merging compact massive objects (black holes, neutron stars) and discuss how, since the momentum of what's inside it decreases as the "bar" shortens, that momentum must flow outside the region to elsewhere. Matter elsewhere in turn picks up some of that momentum, just as the grey M balls pick up some momentum inwards as the red m balls pass near them.
Gravitational waves are seen when one makes some deliberate choices of coordinates and a linearizable approximation of the "real" metric. On that basis it is perfectly reasonable to treat them as physical, even though you could use a very different description the gravitational interactions between the inspiralling bodies and all the other masses in the universe, in a Machian sort of way. The descriptions are to all practical purposes equivalent, and this can be demonstrated mathematically.
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[1] g_\mu\nu, \eta_mu\nu, and h_\mu\nu all have covariant indices, but I'll just stop writing them now because they are distracting. g is a component of G in the G = T mentioned near the top of this comment, where G is the gravitational field and T is all the stress-energy (the non-gravitational fields, or just "matter").
Correct me if I'm wrong, but another way to explain this would be to say that the mass->energy conversion reduces the general amount of mass in an area, which would then reduce the pull of other objects towards it, which is the same as changing the shape of space nearby. Since this particular energy conversion was so large, and doesn't happen all at once or at a consistent rate, it's detectable like a ripple in a pond.
No, probably it isn't. However there are other readers. :-)
I'm sorry that my answer to the rest of your comment is going to seem very complicated and is certainly going to be very new to you. However, you did literally write "correct me if I'm wrong", and you are partly wrong. :-)
> mass->energy conversion
Special and General Relativity are very different when it comes to this sort of thinking, and unfortunately intuitions gained from experience with the former are usually counterproductive when actual gravitation is involved.
What we can do is draw a boundary around a region of spacetime and calculate the stress-energy within that region. That quantity will include contributions from matter, and from the matter's movement, and from quantities like pressure, stress and shear. The quantity is expressed in a sort of a 4x4 matrix called the Stress-Energy Tensor, and at every point in our region there will be a tensorial value.
In General Relativity, stress-energy is conserved; this is more general than the conservation of energy, and in many physically reasonable curved spacetimes the conservation is violated enormously on the largest scales, but stress-energy is conserved. In the case of our selected region of spacetime, it lets us say things like all the stress-energy that went into it in the past can exit it in the future.
That stress-energy includes the quantity that keeps the binary black holes from quickly falling onto each other. Typically you would want to work in a frame of reference in which that quantity is identified with angular momentum, but that's not obligatory in General Relativity.
We can instead do something like put ourselves above the barycentre of the binary black hole, rotating with the system, so that (wilfully blinding ourselves to distant stars and so forth) we do not see angular momentum, we only see something preventing the two masses from falling straight onto one another. This is making a choice of frame of reference in which we can work with fictitious forces to our advantage.
Einstein had an analogy here: consider two identical elastic spheroids alone in space, not too close to one another. One is exactly spherical, one is an oblate spheroid; they are arranged on a line that extends along the semimajor axis of the oblate spheroid through the centre of the exact sphere. An small observer at the midway point on this line between the objects can adjust its rotation so that either -- but not both -- objects appear to be rotating according to the observer. How does the observer decide if either object is the one the observer should be still with respect to?
Our observer can do perfectly reasonable physics if either or neither is rotating with respect to her. The choice often introduces fictitious (specifically, d'Alembert) forces into the picture; an example being centripetal or centrifugal forces.
The solution in modern physics is to be agnostic about whether either has "absolute rotation", and rely upon modern physics written in generally covariant form, and use tensors; or to bite the bullet and not be afraid of d'Alembert forces.
So we have just taken the latter path.
d'Alembert forces are encoded in the stress-energy tensor (an expert screaming "ARGH!" here should consider f^\lambda = -\Gamma^lambda_\mu\nu u^\mu P^\nu; whether you want to call them forces of affine connection, inertial forces, or whatever, they can be locally transformed away into a LFF frame).
So we have a fictitious-force quantity that's separating the binary black holes, and it's part of the local stress-energy, and we can export stress-energy from a region. We just need a mechanism.
Here we introduce another "ARGH!"-able thing: a stress-energy-momentum pseudotensor. We choose it carefully so that it can disappear in a local free falling frame, and we do so not so much for mathematical convenience but for conceptual utility.
For this comment, let's just call it a tool that lets us treat some of the left hand side of the Einstein Field Equations as the right hand side: we treat some of the "gravitational field" as if it were a component of the stress-energy tensor. Our goal is to transform our fictitious force separating the binary black holes into something that we can radiate out of the region of spacetime we've drawn around them. That something is identifiable with gravitational radiation.
So, the vanishing of the "force" separating the binary black holes as the black holes fall towards each other is tied to the emission of gravitational radiation. Moreover, as the binary black holes get ever closer, the amount of the force needed to keep them from just falling onto each other is ever higher, so the amount of gravitational radiation that is emitted at each tiny shortening of the distance between them goes up. When they are very very close together, the amount of gravitational radiation goes up to an extreme.
It is perfectly reasonable to say, "hang on, don't you have to keep adjusting the rotation rate of your frame of reference so that the black holes stay on, say, the X axis at all times?" Sure, but we're allowed to do that; in fact, it's almost encouraged in General Relativity.
> changing the shape of space
This you have right enough. Shifting contents of the stress-energy tensor outside our chosen region of spacetime must change the curvature of the spacetime in that region because in General Relativity spacetime curvature and matter must balance.
It's not quite that easy, though. Distances and lengths (which describe the geometry of spacetime) are encoded in the metric tensor. The arrangement of the stress-energy in our region determines the metric. Our region with the binary black hole sources some metric that we do not know exactly, but there are several approximate metrics that would describe the region reasonably well. Alternatively, we can take a well-known exact metric and perturb it based on the stress-energy in the region. We can use the post-Newtonian expansion in v/c. Or we can integrate the field equations numerically. Any of these approaches is reasonable, but one has to think about the two approaches a little differently, in order to avoid being misled about what is physical and what is not.