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A spacecraft falling from infinity directly towards a celestial body accelerates until it impacts the surface at the surface escape velocity. For any celestial body, the escape velocity is proportional to the square root of 1/radius from center of mass. So dense bodies have higher surface escape velocity than less dense bodies of the same mass

This means the potential gravitational energy of an object at a given orbital radius varies with the radius of the celestial body. The surface escape velocity of all celestial bodies, independent of their mass, approaches infinity as their radius decreases towards zero. With conventional celestial bodies, this “approach to infinity” is limited by a finite radius.

The singularity of a black hole has no radius. When an object reaches the event horizon, it still has significant gravitational potential energy. How is this energy accounted for when the falling object crosses the event horizon? The object’s mass and angular momentum is added to the mass of the black hole, but what happens to the “unused” gravitational potential?

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    $\begingroup$ This sounds like a physics question, not a space exploration question. $\endgroup$
    – Wyck
    Feb 16 at 17:36
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    $\begingroup$ @Wyck ... Its a problem for my Newtonian brain to understand their Relativistic answers. I was hoping someone here could provide an answer i could understand. $\endgroup$
    – Woody
    Feb 16 at 18:01
  • $\begingroup$ Unfortunately the best description we have of gravitational potential energy at scales where relativity comes into play (like black holes) is the metric tensor which is a generalization of the gravitational potential of Newtonian gravitation. (In general relativity, gravitational potential is replaced by the metric tensor). I fear that Newtonian brains aren't mathematically ready for the answer, and you'll be perpetually unsatisfied thinking about it with the Newtonian formula. Brush up on your Riemannian geometry? $\endgroup$
    – Wyck
    Feb 16 at 18:24
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    $\begingroup$ @Wyck ... I read in economist.com/science-and-technology/2023/02/15/… that compression of infalling mass can lead to vacuum energy which neatly accounts for Dark Energy. I was hoping (Newtonian) gravity potential would equal the vacuum energy, tying all the loose ends together, $\endgroup$
    – Woody
    Feb 16 at 19:24

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The object's mass and energy together add to the mass of the black hole. Mass is a form of energy. There's even a concept of a kugelblitz formed by nothing but photons. While photons are massless, focusing enough light on a central point theoretically will result in a black hole from which light cannot escape.

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    $\begingroup$ This is where I get confused... the object's rest mass or its mass as it approaches the speed of light at the event horizon? The kinetic energy of the object, or does it include the "Newtonian" gravitational potential energy of falling all the way to the singularity? $\endgroup$
    – Woody
    Feb 16 at 23:50

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