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I've recently been discussing the time symmetry or lack thereof for GR. One take that I see frequently is that GR is time symmetric, and therefore conserves energy, on local scales (e.g. those where universal expansion is negligible).

This lead me to consider the simple case of throwing a ball in the air. The ball loses kinetic energy as it travels upwards. We say that energy conservation is satisfied by converting this kinetic energy into GPE. But is energy actually stored this way?

If I grant a ball kinetic energy from my arm, this increases the mass, increasing the gravitational force felt by the ball. But since the ball now has more mass, this also changes the GPE (at the starting moment). And the change in the gravity due to the extra mass also changes the GPE. But since GPE should be equivalent to mass as well under $e=mc^2$, this causes a recursive self-interaction. I attempted to calculate these self-interactions and it seems like under most circumstances they converge, but in more "fun" cases like a 1kg mass some distance away from Sagittarius A*, it seems like you can get infinities for GPE. For things below the event horizon that feels weird but not necessarily broken, but it should definitely not occur for objects 100 light years above.

For conservation of energy to hold even in the simplest situations like throwing a ball, it seems like GPE should obey the normal rules of energy, like causing curvature and being equivalent to mass. But considering what happens in these cases seems to lead to some strange outcomes which I feel are big red flags. Is GPE actually a real energy? Or would it be more accurate to say that it's more of a mathematical convenience?

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    $\begingroup$ Unlike kinetic energy, potential energy is a property of a system, not an object. The gravitational potential energy of a thrown ball belongs to the ball-planet system, not to the ball, and we would not include it in any calculations of the ball's mass-energy. $\endgroup$
    – g s
    Jun 25, 2021 at 2:06
  • $\begingroup$ But that energy must still exist to satisfy local conservation, and therefore still lead to curvature and possible self-interaction, right? $\endgroup$
    – DeadMG
    Jun 26, 2021 at 18:22
  • $\begingroup$ @gs Also curious how this is defined when the ball/planet system becomes a ball/black hole system where the ball is 100 light years away from the black hole.. seems like the energy can't just disappear into the total system without taking 100 years to do so. But the ball can start losing KE basically immediately. $\endgroup$
    – DeadMG
    Jun 26, 2021 at 18:33
  • $\begingroup$ Not sure I follow the question, but since neither the black hole not the ball spontaneously popped into existence out of nothing, but have been made of things that have been part of the universe since the beginning of time, time delay shouldn't be an issue. $\endgroup$
    – g s
    Jun 26, 2021 at 18:50

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We say that energy conservation is satisfied by converting this kinetic energy into GPE. But is energy actually stored this way?

Yes, but there is a caveat. The gravitational potential energy is frame variant, but that shouldn't be terribly surprising since energy itself is frame variant anyway.

If I grant a ball kinetic energy from my arm, this increases the mass

This is not correct. The famous $E=mc^2$ is only valid at rest. For the ball with kinetic energy you have to use the more general $m^2 c^2 = E^2/c^2 - p^2$ which obviously reduces to the famous equation for $p=0$. Then you throw a ball $E$ increases as well as $p$, and if you work out the math $m$ remains constant (unless you change the internal state, e.g. by compressing or heating the ball).

this causes a recursive self-interaction

Gravity is non-linear, but not because of this.

For conservation of energy to hold even in the simplest situations like throwing a ball, it seems like GPE should obey the normal rules of energy, like causing curvature and being equivalent to mass.

One frequent problem is that people sometimes think that energy is the source of gravity, but it is not. The source of gravity is the stress energy tensor. The stress energy tensor includes energy, but it also includes momentum, pressure, and shear stress. And the relationship to curvature is complicated and non-linear. Throwing the ball increases both its energy and also its momentum. So the curvature is affected, but not in a straightforward manner.

When you consider the whole stress energy tensor then you get a self-consistent situation that behaves according to the same physical law in all frame, but winds up difficult to describe in any frame verbally. In frames where there is GPE it works according to the same laws as in frames where there is no GPE.

Is GPE actually a real energy? Or would it be more accurate to say that it's more of a mathematical convenience?

I personally am not sure that these two are mutually exclusive.

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