# Mass - Energy Equivalence

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## Main Question or Discussion Point

Suppose you have two particles of mass 'm'.
Their combined rest energy will be: E_rest=2mc^2

Its said that when they interact (gravitationaly), they're total energy will decrease due to the negative gravitational potential energy. The rest of the energy is stored in the field (they say).
E_total=E_rest - |V| , where: V= - Gmm/r
And (they say) we assign: E_total = m' c^2
And with this equation we argue that the new mass of the system is smaller than the sum of the constituents: m' < 2m.

But i noticed this: If i place the two masses so close to each other that |V| becomes greater than E_rest, then E_total < 0 !! And if we assign again: E_total=m'c^2, then we get the beautiful result: m'<0..

What's that? Antigravity?

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bcrowell
Staff Emeritus
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I don't think you even need two particles in order to get this paradox. You can get it with one. In a classical field theory like GR, it's not generally valid to talk about potential energy in terms of an equation relating to the distance between particles. In general what you want to talk about in a classical field theory is the energy in the field itself. The electric field has energy density $\rho=(1/8\pi k)|E|^2$, in SI units, where k is the Coulomb constant. For a Newtonian gravitational field, $\rho=(-1/8\pi G)|g|^2$, where G is the gravitational constant and g is the gravitational field. The minus sign is because the gravitational force between two positive masses is attractive (versus the electrical case, where it would be repulsive). So for any point mass, the total gravitational energy stored in the mass's field is negative infinity. (If you're allergic to point masses, you can still get a negative total mass-energy by taking any material object and compressing it enough.)

The resolution of the paradox is that you can't calculate an energy density for the gravitational field in general relativity. The equivalence principle guarantees that the gravitational field can be made to vanish at any given point by a choice of an appropriate set of coordinates. Therefore there can be no generally valid, useful energy density calculated from the gravitational field. There are approximate energy densities that you can write down in the limit of weak fields (the Newtonian expression being one of them), but they're only weak field approximations. As an example of the use of these approximations, you can calculate the energy carried by a gravitational wave.

Another way of looking at this kind of thing is in terms of energy conditions. The point of the energy conditions (e.g., the null energy condition) is to quantify what it would mean for something to have negative mass, as you describe. The energy conditions can probably be violated, but not as simply as in the kind of Newtonian example you have in mind.

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The resolution of the paradox is that you can't calculate an energy density for the gravitational field in general relativity. The equivalence principle guarantees that the gravitational field can be made to vanish at any given point by a choice of an appropriate set of coordinates. Therefore there can be no generally valid, useful energy density calculated from the gravitational field. There are approximate energy densities that you can write down in the limit of weak fields (the Newtonian expression being one of them), but they're only weak field approximations. As an example of the use of these approximations, you can calculate the energy carried by a gravitational wave.

Another way of looking at this kind of thing is in terms of energy conditions. The point of the energy conditions (e.g., the null energy condition) is to quantify what it would mean for something to have negative mass, as you describe. The energy conditions can probably be violated, but not as simply as in the kind of Newtonian example you have in mind.
First of all thanks for your response! I'd like to ask you some questions:

1) Firstly, you say that we cannot calculate energy densities in GR while later you say that calculating of the energy densities of weak fields can be done. So, can it be or not be done? Im confused.. If it can be done, why its only valid for weak fields and not for strong fields as well?

2) I didnt quite understand the last paragraph. What 'energy conditions' are being violated here?

Thanks

bcrowell
Staff Emeritus
Gold Member
First of all thanks for your response! I'd like to ask you some questions:

1) Firstly, you say that we cannot calculate energy densities in GR while later you say that calculating of the energy densities of weak fields can be done. So, can it be or not be done? Im confused.. If it can be done, why its only valid for weak fields and not for strong fields as well?
It's valid in Newtonian gravity. It can't be generally valid in GR, because of the equivalence principle. Since Newtonian gravity is a weak-field approximation of GR, it makes sense that it's valid only in the weak-field approximation of GR. I think pretty much any GR textbook should have a discussion of this, typically in the material on weak fields and gravitational waves.

2) I didnt quite understand the last paragraph. What 'energy conditions' are being violated here?
http://en.wikipedia.org/wiki/Energy_conditions

bcrowell
Staff Emeritus