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In a system of binary black holes where one is 31 solar masses and the other is 25 they have event horizon radii of 93 and 75 km, respectively. At the point the

In a system of binary black holes where one is 31 solar masses and the other is 25 they have event horizon radii of 93 and 75 km, respectively. At the point their event horizons start to touch they have a relative distance of 168 km. Each of them have a gravitational potential energy in the other's gravitational field of G m1 m2 / r or 1.22e48 Joules. Both of them together have a total gravitational potential energy of 2.44e48 Joules or 1.36e31 kg (6.8 solar masses). (In reality the mass contribution due to the gravitational potential energy is included in the "mass", this is just illustrative.)

raattgift 9y ago
Sure, but the crucial thing is that offsetting the gravitational potential energy is angular momentum; we can treat the former as "negative energy" with the idea that it takes positive energy to pull apart two bodies attracting each other gravitationally. [1] 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. - -- [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.

raattgift 9y ago

Sure, but the crucial thing is that offsetting the gravitational potential energy is angular momentum; we can treat the former as "negative energy" with the idea that it takes positive energy to pull apart two bodies attracting each other gravitationally. [1] 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. - -- [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.