Does Potential Energy Have Mass, Even When Negative and Non-Unique?

[ralqs]

By the equivalence of mass and energy, does potential energy have mass? Surely it does, right? But if the potential energy is negative, is it possible that the mass of a system could be negative? And if the potential energy function is not uniquely defined (because of an arbitrary constant), wouldn't the mass likewise be non-unique?

 

[Dale]

But if the potential energy is negative, is it possible that the mass of a system could be negative?

The mass of a system bound by some potential energy will be less than the sum of the masses of the unbound constituents. This is known as the mass deficit or the binding energy:
http://en.wikipedia.org/wiki/Binding_energy

 

[ardie]

the actual value of the potential energy is not negative. by definition zero of potential is defined at infinity, which is why potential energy at some point will be some integral from this point to infinity, which is why it takes negative values. the energy of the particle isn't negative however, the value of energy at any point is absolute. the convention just helps to calculate the energy you need to make the particle escape the potential well.
now back on to the more interesting issue of wether there is mass in that energy. it turns out mass and energy are the same thing. however let's make the crucial distinction here that intertial mass is quite different from rest mass. in general objects in a gravitational field moving at some speed should experience the same gravitational force than objects at rest. otherwise one would find a falling rock would experience a higher and higher gravitational force as it falls faster and faster, which isn't the case. a better description of this is in Einstein's publications.

 

[ralqs]

The mass of a system bound by some potential energy will be less than the sum of the masses of the unbound constituents. This is known as the mass deficit or the binding energy:
http://en.wikipedia.org/wiki/Binding_energy

But is it possible that the total energy (KE + PE) is less than zero? How would this physically make sense?

the actual value of the potential energy is not negative. by definition zero of potential is defined at infinity, which is why potential energy at some point will be some integral from this point to infinity, which is why it takes negative values. the energy of the particle isn't negative however, the value of energy at any point is absolute. the convention just helps to calculate the energy you need to make the particle escape the potential well.

What's your justification? As I've always understood, the potential energy doesn't mean anything in itself, only it's change has any significance.

now back on to the more interesting issue of wether there is mass in that energy. it turns out mass and energy are the same thing. however let's make the crucial distinction here that intertial mass is quite different from rest mass. in general objects in a gravitational field moving at some speed should experience the same gravitational force than objects at rest. otherwise one would find a falling rock would experience a higher and higher gravitational force as it falls faster and faster, which isn't the case. a better description of this is in Einstein's publications.

So, what does this mean? Are you saying that PE changes the rest mass?

 

[Dale]

But is it possible that the total energy (KE + PE) is less than zero?

Not in GR, that would violate the weak energy condition. In fact the weak energy condition is stronger than that, since the weak energy condition requires that the energy density is greater than or equal to 0 everywhere.

Of course, in Newtonian gravity the PE is arbitrary so it could certainly be sufficiently negative that the total energy is less than 0. But even in such a case the total energy would be conserved.

 

[ralqs]

Not in GR, that would violate the weak energy condition. In fact the weak energy condition is stronger than that, since the weak energy condition enforces that the energy density is greater than or equal to 0 everywhere.

Of course, in Newtonian gravity the PE is arbitrary so it could certainly be sufficiently negative that the total energy is less than 0.

I'm getting the impression that PE is very different in relativity than in classical mechanics...at least, I guess that PE becomes uniquely defined. Is there a text on relativistic dynamics that covers this in more depth? Or will I need to understand classical field theories to get it?

 

[Dale]

I'm getting the impression that PE is very different in relativity than in classical mechanics.

Well, it is not just PE, but also KE and mass. In GR mass and energy are tricky to define globally, and even in SR there are multiple distinct concepts and they differ from their non-relativistic counterparts.

 

[ardie]

i really cannot try to explain it a lot simpler than that. the rest mass is unique to particles and its description is the fruit of the standard model of particle physics. the potential and kinetic energies associated with particles are both positive valued objects, however the PE is written as a negative value to make calculations easier and no other reason. in geenral the KE and PE do not effect the rest mass of the object, but its inertial mass. so in answering the question, does adding kinetic energy to a particle increase its mass, one can respond: well its rest mass is the same and cannot be effected by the standard model. however its inertial mass appears to increase as it will become harder and harder to accelerate an object with an ever increasing kinetic energy. same is true for the PE. so it helps not to label the KE and PE as the mass of the object, because in most cases mass refers to rest mass in textbooks.

 

[PAllen]

Generally agreeing with Ardie, I would like to add a couple of examples that might clarify a few points. First, where Ardie mentions inertial mass, the implicit assumption (as per GR) is that this will be the same as active (how much gravitational influence is produced) and passive gravitational mass (how a body responds to gravity from another source).

Talking about the inertial mass of a single rapidly moving particle is tricky because it is frame dependent, and any curvature invariant (rather than coordinate dependent components) will be the same as the frame in which it is at rest.

So, I think slightly sharper examples:

1) Consider two spherical bodies made of the same particles (thus same total rest mass), same temperature, but one is larger, more diffuse, than the other (thus more potential energy; the smaller one will have given up potential energy - radiated it away to be at the same temperature). Then the larger one will have more total mass, in any of the senses mentioned above.

2) Consider two spherical bodies made of the same particles. The bodies are the same size. However, one is hotter than the other (thus its particles are moving faster and have more kinetic energy). Then the hotter one will have more mass in all the described senses. Note that here, there is no frame which can make the extra kinetic energy disappear - the temperature difference is primarily frame invariant, thus contributes to curvature scalars.

 

[Dale]

the PE is written as a negative value to make calculations easier and no other reason

I think this is incorrect. The PE is actually negative in the case of an attractive force. I.e. the mass of a system of particles bound by an attractive force is LESS than the mass of the unbound constituents. This mass deficit is exactly equal to the binding energy (aka change in PE). I don't know how else you can explain this besides the PE being negative.

http://en.wikipedia.org/wiki/Binding_energy

What is not negative is the energy density of the fields and matter.

 

[ralqs]

DaleSpam, how does the total energy conspire to be positive in GR? For example, wouldn't the total energy of objects in orbit around the Sun have to be negative?

 

[Dale]

DaleSpam, how does the total energy conspire to be positive in GR? For example, wouldn't the total energy of objects in orbit around the Sun have to be negative?

No, the binding energy is less (much, much, less actually) than the rest energy. Remember E=mc^2. So even a small amount of mass has a lot of energy. Even in a black hole the weak energy condition is not violated.