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I'm trying to understand where the energy for a quasar comes from and specifically if it comes from converting mass to energy. I know that as matter falls into a black hole gravitational energy is converted to heat and other kinds of energy but that is not converting mass to energy. I keep seeing statements like "The process of converting mass to energy from falling onto a black hole has an efficiency that is over ten times as large as the efficiency of nuclear fusion." That sounds like there is some other process by which mass is converted to energy, some mechanism other than nuclear fusion. But what is that process?

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If something falls in the gravitational field of a black hole it gains kinetic energy. The total mass-energy of the system is unchanged.

If the falling object thermalises in an accretion disk, close to the black hole, and emits radiation that escapes, then the total mass-energy of the black hole plus falling object is diminished.

A way of characterising this is to calculate how much energy escapes as a fraction of the rest mass energy of the falling object. In a similar way, you can characterise the "efficiency" of nuclear fusion in terms of the escaping energy (as photons, neutrinos etc) as a fraction of the rest mass energy of the initial reactants.

Very roughly, a falling object might thermalise at the innermost stable circular orbit of the black hole at the center of a quasar, at 3 times the Schwarzschild radius, $r_s$. Its kinetic energy is split between emitted radiation and the kinetic energy of its orbit; so in a Newtonian approximation, the emitted radiation energy would be $$E \sim \frac{GM_{\rm BH} m}{6r_s},$$ where $m$ is the object rest mass and $r_s = 2GM_{\rm BH}/c^2$.

Dividing by $mc^2$, the efficiency is therefore as high as $\epsilon \sim \frac{1}{12}$

This compares with 0.7 per cent for nuclear fusion via the pp chain.

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As I understand it the core of your objection is: conversion of gravitational potential energy to heat is not conversion of mass to energy.

I won't address that directly, but there is some background that I think is relevant:

In terms of general relativity there is the following property: when you have a bound system (such as two black holes orbiting each other) then all of the energy of that system contributes to the gravitational mass of that system. That includes the kinetic energy of the orbiting motion, and the binding energy.

Let's say that double black hole system has an orbital period of days, and that there are stars orbiting that double black hole with orbital periods in the order of years. The gravitational potential for those orbiting stars arises from the total energy of the double black hole. When the black holes eventually merge there is a very large emission of gravitational waves. The resulting single black hole does not have the same gravitational potential as the previous double black hole system because of the amount of energy that was radiated away. Assuming the orbiting stars survived that event: their orbits will be different from the moment the different potential reaches them, as it is a different potential.

The current simulations for the astrophysics of black hole mergers are sufficiently developed to allow assessment of how much difference in gravitational potential there will be before and after the event.

That is how astrophysicists arrive at statements such as: "In first black hole merger that was detected by LIGO the amount of gravitational energy emitted was the equivalent of three solar masses."

More generally:
Quasar physics is so extreme that the physics is fully in the realm of relativistic physics, and the mass-energy equation takes center stage.

I think that is the background of why astrophysicists feel it's justified to make a comparison between mass conversion in nuclear physics, and energy conversion in quasar physics.

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