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Let's do a thought experiment:

Assume that I have a machine that can convert energy into matter with 100% efficiency i.e. it can create any amount of matter in a small space so long as I pay it's energy cost according to $E=mc^2$.

Now, before I use the machine, I have a bunch of masses at rest scattered around a point, say on average 10,000 km away, and those masses are as large as they can be, yet small enough that their potential energy relative to the point is effectively zero.

Now, I use my machine to create a black hole with the mass of the sun at that point. The scattered masses now suddenly have a nontrivial amount of potential energy due to the gravitational field, and this energy will now start being converted to kinetic energy as the masses feel the acceleration due to gravity.

This begs the questions:

  1. Does my machine have to pay the potential energy cost added to the scattered masses? If not, where does it come from?
  2. If my machine has to pay the cost, how does it know that it needs to pay for the potential energy of the masses?
  3. And given causality requirements, how is the potential energy "transmitted" to the scattered masses? One would think it bound up somehow in gravitational waves, but does our current understanding of physics support energy transfer like this?
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    $\begingroup$ This isn't really a physics question. Since you create the mass by magic, don't you also create any potential energy by magic as well? If you had a physics mechanism that produced the mass (moved it in from far away or something), you could come up with a physics explanation of energy. $\endgroup$
    – mmesser314
    Commented Dec 5, 2023 at 2:17
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    $\begingroup$ @mmesser314 It's only slightly magic. You could just as easily assume the mass comes out of a wormhole. The sun would never magically disappear either, but we still do thought experiments about what would happen to the Earth's orbit if it did and use physics to explain that it would take 8 minutes for the Earth to begin traveling tangent to its orbit. The mass isn't created by magic, I already explained you have to pay for it with E=mc^2. Just ignoring the rest of the thought experiment because you don't understand how the machine works is a cheap cop out. $\endgroup$
    – stix
    Commented Dec 5, 2023 at 2:18
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    $\begingroup$ Here's another way to see that this isn't really a fair question. If the energy comes from nowhere, then the stress-energy tensor $T$ does not obey local energy conservation, and so $\nabla_\mu T^{\mu\nu} \neq 0$. However, the Einstein tensor must obey the equation $\nabla_\mu G^{\mu\nu} = 0$. Therefore, the Einstein equations which govern GR ($G^{\mu \nu} = k T^{\mu\nu}$) cannot be valid. If you're going to invalidate these equations, you'll need to explain what you're replacing them with before you can expect any theoretical answer. $\endgroup$
    – J. Murray
    Commented Dec 5, 2023 at 3:18
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    $\begingroup$ Voting to reopen. I have removed the unfortunate reference to "magic". The questioner has clarified that they are not talking about creating energy, just converting it. Converting energy to matter is certainly within the bounds of mainstream physics $\endgroup$
    – gandalf61
    Commented Dec 5, 2023 at 16:21
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    $\begingroup$ I still think this questions should be reopened. The questioner has an incorrect understanding about how gravitational fields work but that does not make the question non-mainstream. We don't (and shouldn't) close every question that contains a misunderstanding. $\endgroup$
    – gandalf61
    Commented Dec 6, 2023 at 15:05

2 Answers 2

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There are many things wrong with this question. Unfortunately, the only way to answer it is to correct the problems:

Assume that I have a slightly magic machine that can create any amount of matter in a small space so long as I pay it's energy cost according to E=mc2.

Since this machine already has energy, it also has mass already. The production of matter doesn’t change the amount of energy or the amount of mass. The only way for the mass/energy to change is for mass/energy to move from outside to inside the machine. Producing matter will not do it.

those masses are as large as they can be, yet small enough that their potential energy relative to the point is effectively zero

The only way for them to have a negligible potential is for them to be a negligible total mass.

Now, I use my machine to create a black hole with the mass of the sun at that point.

Then your machine already has the mass of the sun. It just collapsed it to within its Schwarzschild.

The scattered masses now suddenly have a nontrivial amount of potential energy due to the gravitational field,

Changing your solar mass machine into a solar mass black hole doesn’t change the PE of the scattered masses. If they were negligible before then they are still negligible after. It is the same.

this energy will now start being converted to kinetic energy as the masses feel the acceleration due to gravity

Gravitational PE is negative. The negative PE isn’t converted to KE. Instead they get more negative PE as they gain KE.

Does my machine have to pay the potential energy cost added to the scattered masses?

There was no change in PE, and it doesn’t really make sense to speak of paying a negative PE cost.

If my machine has to pay the cost, how does it know that it needs to pay for the potential energy of the masses?

There is no cost to pay so there is nothing that it needs to know.

And given causality requirements, how is the potential energy "transmitted" to the scattered masses?

Nothing is transmitted in this scenario. There is no effect, so there is also no cause.

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I think the OP could basically be describing a Kugelblitz$^†$ formed by light rays coming from nearly all directions, focused on the point in question, which is a physical scenario.

My attempt at answering the question: the important thing to remember is that potential energy is negative, in a sense. Objects drifting through empty space have a potential energy set at zero, which is a good convention as we will see.

In your scenario, your drifting objects pre-Kugelblitz have basically zero potential energy with respect to the origin. If a bunch of light rays then converge and create a black hole at the origin (or the BH simply "appears" there - it makes no difference to us), the objects potential energy is still zero - we are assuming none of them are very close to the BH. No energy has been imparted to the drifting asteroids (let's call them). But now, there is a new location in space they could occupy, with a very large negative potential energy. As an asteroid falls towards the origin, it gains after some time, say $1000$ in kinetic energy. At that time it has also lost $1000$ in potential energy -- it went from a point in the gravity well with $PE = 0$ to $PE = -1000$.

Its total energy, before:

$$E_1 = K_1 + P_1 = (0)+(0)$$

and after:

$$E_2 = K_2 + P_2 = (1000)+(-1000)$$

is unchanged.

So there is no "infusion" of energy necessary to the system (of asteroids),* and the system's energy before and after is the same. It simply has a new available state with much lower potential energy and much higher kinetic energy, but the same total energy.

Some additional notes:

  • There is no need for the central object to be a black hole. Any massive object will do.
  • We created the caveat that all the asteroids started off far away from the central object (or black hole). If an asteroid were present at say 2 Schwarzschild radii from the origin when the light rays converged, where the local potential energy is say $-1,000,000$ (in some units), the transfer of potential to kinetic energy would be the same idea, except it would happen in a transient and non-obvious way, as a result of the incoming energy (and the asteroid will almost definitely not survive intact) - possibly as the light rays pass before they reach the origin. This is kind of like if a nuke were detonated in the ocean and caused a tidal wave to spread out, the far away boats would be disturbed by the wave eventually, while the boat that was 100 ft away from the blast would be disturbed - differently.

$^†$ Convergence of light rays, basically the Death Star but from nearly all directions, with sufficient energy in a small enough space to form an event horizon.

$^*$ System means the combined system of all the asteroids. Of course the central BH will have its own mass-energy, but this is irrelevant to the asteroids' motion.

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