# Where does the potential energy go?

1. Aug 29, 2010

### jmd_dk

The Earth is gravitationally bounded to the sun. If the sun were to suddenly disappear (say, it got hit by an anti-sun, and all the mass annihilated completely), the Earth wouldn't be bounded anymore, and it would fly off into space. Would the speed of the Earth now be more than the tangential velocity it had when it was in orbit around the sun? Or where would the (previous) potential energy go?

A similar thought experiment could be...
Take a spring, and attach a mass to each end. Now stretch the string, by moving the masses apart. The system has gained some potential energy. If the spring were to suddenly disappear, the masses would have no way to reclaim the potential energy they had when the spring was there. Where does the (previous) potential energy go? Is the radiation from the spring (if again, it disappeared through annihilation) more energetic? In this example, I can easily imagine the to masses to be at rest after the spring is gone; hence the potential energy does not convert to kinetic energy here.

2. Aug 29, 2010

### uart

Well considering that the annihilation will release close to 10^48 Joules of energy, then sure the Earth will have more energy after the annihilation. It will be in very small pieces of course, but in total it will gain an energy equivalent to about 100,000 times that of it's initial orbital KE!

3. Aug 29, 2010

### jmd_dk

That's totally irrelevant to the argument here. It's a thought experiment, not a practical test. The radiation doesn't have to hit the earth. Of course it will if we tried, but lets assume that the effects of the suns disappearance doesn't reach Earth.

4. Aug 29, 2010

### phyzguy

Mass-energy is conserved, so it can't be created or destroyed, only moved around. So there is no way for the sun's mass to suddenly disappear. In your thought experiment of an anti-sun coming in to annihilate the sun, remember that anti-matter has positive mass, not negative mass. If an antisun hit the sun, it would have to come from somewhere, and at the instant it hit the sun, there would now be two solar masses at the sun's location, not zero. After that, of course, all hell would break loose in an enormous explosion as uart pointed out. But the Earth (until it was destroyed) would continue to respond to the combined gravitational pull of the sun and anti-sun.

5. Aug 29, 2010

### jmd_dk

Once again, that doesn't answer the question. It's a thought experiment. It's the idea that matters, not if this can actually be carried out in practice. The radiation from the annihilation doesn't hit every point in space (it's made up of quantized bits, not a continues substance). Imagine that the Earth happens to be very lucky, and don't get hit at all. Or think of the other example i gave, with the spring, if that makes you happier.
Don't tell me why this experiment can't be done; tell me where the potential energy would go, if we actually did. Thank you!

6. Aug 29, 2010

### Integral

Staff Emeritus
When the statement of your thought experiment posits a nonphysical condition how can you expect a physically meaningful answer?

7. Aug 29, 2010

### phyzguy

You're basically asking what happens to the energy in an imaginary universe where energy is not conserved. Since we don't live in such a universe, you're free to imagine anything you want about what happens to the energy. I'll imagine that it turns into strawberry Jello. Your turn.

8. Aug 29, 2010

### IsometricPion

The only violation of energy conservation that necessarily takes place in a model where the Sun ceases to exist is that due to the kinetic and potential energy of the Sun. Suppose one models the existence failure with a heaviside function: {$$^{m, t<t_{o}}_{0, t \geq t_{o}}$$} for the Sun's mass; clearly any energy terms containing the Sun's mass as a multiplicative factor will be zero after $$t = t_{o}$$. In classical mechanics (excluding forces that depend on variables other than position, such as electromagnetism), this is the end of the story. The potential is gone and there are no unexpected effects. There cannot be any such effects since for all $$t$$ after $$t_{o}$$ the problem is equivalent to one in which the Sun never existed and the initial conditions are given by the state of the system at the instant the Sun suffered an existence failure.

If the system is modeled with general relativistic mechanics, the situation is slightly different. Since the effects of any changes in the system under consideration cannot propagate faster than light, the Earth would not "feel" the effects of the existence failure until about 8 minutes after it occurred. If one only considers the gravitational effects, the only other interesting occurrence is that, analogous to similar situations with the electromagnetic force that produce electromagnetic waves, the existence failure produces a gravitational wave. Since gravitational waves interact relatively weakly with matter (and I don't think it would be an extraordinarily powerful wave anyway), any effects on the path of the Earth should be negligible. So, the main long-term difference between this result and the classical one is that the tangent line is taken slightly farther along in Earth's orbital path (the gravitational wave(s) could alter the velocity vector slightly due to interactions with the mass-energy that makes up the Earth in this model, but this would be an extremely small (and model-dependent) effect).

In summery, in (some simple) classical models the potential just goes away (it can be thought of as part of the violation of energy conservation), in general relativistic models (which, as far as I know, do not have a well-defined potential energy scalar associated with gravitational field, in general) the change in gravity produces a gravitational wave (which would carry away a small amount of what might previously have been thought of as potential energy).

In the case of converting (neglecting any effects due to the physics of the conversion) a spring in a bound system into electromagnetic energy (EME), the total EME released would be greater than that released by a similar non-deformed spring. For a mass under the influence of gravity, conservation of energy dictates that the total EME escaping at infinity would be lower than that of a converted isolated mass (this can be thought of as being due to the gravitational red-shifting of the EME). It is critical to realize that the radiation you mentioned has a certain amount of energy (otherwise the total energy of the system would not be conserved), which it carries away from the two masses.

Note: This analysis does not affect the validity of the other posters' comments. They are entirely correct in their criticisms indicating these are (literally absolutely) impossible thought experiments/imaginings.