# Potential energy contributes more to gravity

• bobsmith76

#### bobsmith76

can someone explain why the following is true:

a tightly wound watch will contribute slightly more to gravity in virtue of its potential energy than one that has run down

Didn't you just answer this yourself? Because of the potential energy stored in the spring, which slightly increases the effective mass.

I know I can always tell when my watch battery needs replacing, because it feels so much lighter. but i thought potential energy does not weigh anything.

Internal potential energy certainly does. When you compress a spring, you're storing energy in the atomic lattice. On a microscopic scale, you're putting energy in the electric fields between neighboring atoms. This is real, and has an effect on the mass. Energy stored in the nucleus is what makes fission/fusion bombs possible. And it's potential energy, caused by nuclear forces. All of these make a small contribution to the mass of the object, both the inertial mass and the gravitational mass.

thanks i really appreciate your help

Einstein's field equations say that curvature is the result of the "stress-energy tensor" which is more than just the mass (as in Newtonian gravitation). Therefore, energy, as well as less obvious things like pressure and shear stress contribute to the curvature of space-time and therefore to the "gravitational force".

Not for an isolated body. As an example, the Earth itself contains a great deal of internal pressure and stress, yet the external gravitational field it produces (Schwarzschild) displays no effect of stress or pressure, and is a result entirely of its mass. Not just to a good approximation: entirely.

Not for an isolated body. As an example, the Earth itself contains a great deal of internal pressure and stress, yet the external gravitational field it produces (Schwarzschild) displays no effect of stress or pressure, and is a result entirely of its mass. Not just to a good approximation: entirely.

Yes, by spherical symmetry and isolation the field outside Earth is described by one mass parameter (barring tiny effect of rotation). However, that mass is affected by many things. Compared to sum of rest mass of molecules in earth, heat, gravitational binding energy, etc. all contribute to that mass parameter. Of course, you are well aware of this.

Not for an isolated body. As an example, the Earth itself contains a great deal of internal pressure and stress, yet the external gravitational field it produces (Schwarzschild) displays no effect of stress or pressure, and is a result entirely of its mass. Not just to a good approximation: entirely.

I think the angular momentum will also contribute, but O.K. I will buy that argument. This is because you are looking at the external vacuum gravitational field though, where the stress energy tensor is 0. What about the internal gravitational field? I'm sure that that would have to include the full treatment? Or can you still do what you would do in Newtonian mechanics and say that the gravitational field inside the Earth at radius r is due to only the spherical mass that is enclosed by a sphere with radius r where r is the distance to the field point, and the mass outside r must necessarily produce 0 net gravitational effects?

In general, you can't neglect the pressure contributions to gravity. In the case of the Earth the pressure terms probably won't be significant. However, as we were discussing in a recent thread, a contained photon gas will have a much different interior gravitational field than a similar sphere of presureless dust containing the same total energy.

can someone explain why the following is true:

a tightly wound watch will contribute slightly more to gravity in virtue of its potential energy than one that has run down
It seems weird if not wrong to say that potential energy contributes to mass. It's energy stored as a stress that contributes to the mass.

If we can approximate watch as a closed system then mass of the watch will stay the same after running down. In this case stress energy is converted to heat energy and it still contributes to mass.

So in order to reduce mass of the watch as it is running down we have to let it interact with environment where environment acts as heat sink.

Of course the internal gravitational field will be different.

It seems weird if not wrong to say that potential energy contributes to mass. It's energy stored as a stress that contributes to the mass.

If we can approximate watch as a closed system then mass of the watch will stay the same after running down. In this case stress energy is converted to heat energy and it still contributes to mass.

So in order to reduce mass of the watch as it is running down we have to let it interact with environment where environment acts as heat sink.

Yes, I agree with this. I would call it compression energy and it would need to able to radiate away for the unwound system to have lower mass.