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> Didn't realize that energy could be converted into gravitational waves. I'll try an explanation, hopefully it's useful to you or another reader. General Rel

> Didn't realize that energy could be converted into gravitational waves. 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_Diagram.svg 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. - -- [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").

owyn 9y ago
I applaud the level of detail, and I don't understand any of it, so I'm not sure if it counts as a useful explanation to someone with a very general question about the energy->gravitation relationship expressed in the parent comment. 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.
- raattgift 9y ago
  > I'm not sure if it counts as a useful explanation to someone with a very general question 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.

owyn 9y ago

I applaud the level of detail, and I don't understand any of it, so I'm not sure if it counts as a useful explanation to someone with a very general question about the energy->gravitation relationship expressed in the parent comment. 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.

raattgift 9y ago
> I'm not sure if it counts as a useful explanation to someone with a very general question 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.