My doubt lies in the fact that gravitational waves are produced even before the merger so I kindly ask an expert to make me a clear picture of the source of energy of grav. waves before the merger and the energy after the merger that we are told is gained by the vanished masses of the merging objects.
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2 Answers
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Waving your arms around causes gravitational waves, even though they are undetectably weak. Where does the energy for that come from? Your muscles, which get the energy from your food and the oxygen in the air. The plants that were grown to make food require sunlight to grow, which ultimately came from a mass defect during a nuclear fusion reaction inside the sun. Where does the energy of gravitational waves produced during a BH merger come from? It comes from the loss of total mass of the two merging black holes. All radiation comes, ultimately, from a loss of rest mass of a system. The only thing that differs is the mechanism by which that mass is converted to the energy of the radiation.
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Krešimir Bradvica wrote: "My doubt lies in the fact that gravitational waves are produced even before the merger."
The gravitational radiation is not due to the loss of the black hole's rest mass (that can't escape a black hole by definition since the irreducible mass of a Schwarzschild black hole is its rest mass), but due to the loss of kinetic energy and angular momentum, which happens during the inspiral.
If the black holes approach each other the total of the negative potential energy increases, while the positive kinetic energy doesn't cancel that exactly as it would if there was no loss due to gravitational radiation.
If the test particle's mass was neglible compared to the black hole its total energy would be a conserved quantity and the radiation would be neglible as well, but if we leave the test particle regime and have both masses pull on each other the field is no longer static and the total energy is not conserved anymore since some kinetic energy gets converted into waves.
The total energy of a test particle in the field of a Schwarzschild black hole is $\rm E=mc^2 \sqrt{1-r_s/r}\div\sqrt{1-v^2/c^2}$, so if you could get it to $\rm r \to r_s$ and extract its kinetic energy so that $\rm v \to 0$ you get $\rm E \to 0$, so such a particle would not make the black hole heavier since $\rm M+0/c^2=M$.
In the case of a merger not all, but some velocity gets lost since the black holes are pulled not towards each others intantaneous but somewhat delayed positions so that the kinetic energy can no longer keep up with the potential energy.
