# Where is gravitational energy stored?

Suppose you separate two masses from each other and expend a certain amount of energy in doing so, where exactly has this energy been transfered to?

I was thinking perhaps the mass of both objects increases much like how separating protons from a nucleus makes them heavier, but then that just leads me to believe then that mass in general is only present in objects because of the fact that objects are separated ~ all objects in the universe together would have no mass.

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pervect
Staff Emeritus
In general, it's impossible to localize gravitational energy in any precise manner. See for instance MTW's textbook, Gravitation. The title of section $20.4 is "Why the energy of the gravitational field cannot be localized". Probably the most convincing argument is the equivalence principle. MTW said: One can always find in any given locale a frame of reference in which all "local gravitational fields" vanish. ... No local gravitational field means no local gravitational energy momentum. Like for E and B fields we can say the energy is stored in the field, but we cant say this for Gravitational fields? Is the energy contained in the tensor T? pervect Staff Emeritus Science Advisor The stress-energy tensor T contains the source terms. In Newtonian theory mass could be considered to be the source of gravity, but in General Relativity the source terms contained in the stress-energy tensor T are energy, momentum, and pressure. However, the stress-energy tensor is not like the E and B fields in classical mechanics, and there is no general way to localize energy in GR as there is in electromagnetism. http://en.wikipedia.org/w/index.php?title=Mass_in_general_relativity&oldid=366036138 The main reason for this is that "gravitational field energy" is not a part of the energy-momentum tensor; instead, what might be identified as the contribution of the gravitational field to a total energy is part of the Einstein tensor on the other side of Einstein's equation (and, as such, a consequence of these equations' non-linearity). There's another way of looking at the problem that might also be helpful. The conservation of energy is, in general, associated with a time translation symmetry by Noether's theorem. In particular, if one has a one-parameter sort of symmetry, there is a corresponding conserved quantity. In flat space-time, or in Newtonian mechanics, space-time always has the required symmetry. In GR, this is not necessarily the case. However, in those cases in which the necessary time-translation symmetry exists (static or stationary space-times, for instance, or asymptotically flat space-times), the time-translation symmetry of the space-time does give rise to a useful notion of globally conserved energy. None of this should be taken to mean that GR doesn't have a local notion of conserved energy. The stress-energy tensor has to obey a law that it's divergence is zero, which can be considered to be a form of local energy conservation. But these local laws volume don't allow one to assign a global number E to some particular system of non-zero volume and state that that is the energy of the system (or contained within the volume). The topic is rather technical - the best reference on the topic is probably Wald, though as I mentioned MTW has some information on it too. There's only so much one can do in a post such as this. On a less technical level, one might try http://www.desy.de/user/projects/Physics/Relativity/GR/energy_gr.html. Last edited: Suppose you separate two masses from each other and expend a certain amount of energy in doing so, where exactly has this energy been transfered to? I was thinking perhaps the mass of both objects increases much like how separating protons from a nucleus makes them heavier, but then that just leads me to believe then that mass in general is only present in objects because of the fact that objects are separated ~ all objects in the universe together would have no mass. Are you speaking of mass or weight? If you separate two objects that are attracted to each other, the energy used is transformed to potential energy. This potential energy is released if the thing that's holding apart the two objects that are attracted to each other is removed. Upon release, potential transforms to kinetic energy and during collision, potential energy is transformed to heat, sound, etc.. That is what I've learned from college physics 10 years ago, unless I'm missing something. Last edited: That is what I've learned from college physics 10 years ago, unless I'm missing something. I'm thinking a bit more technical than that. In general, it's impossible to localize gravitational energy in any precise manner. See for instance MTW's textbook, Gravitation. The title of section$20.4 is "Why the energy of the gravitational field cannot be localized".

Probably the most convincing argument is the equivalence principle.
Thanks, i'll have a look.

The potential energy is mass in this case.

It's easy to define and localize the gravitational field energy in the Newtonian case.
Consider an object of mass M; the magnitude of the gravitational force field at a distance r is:
ForceField(r)=G M/r^2 (Units of force/mass)
The gravitational energy density of this field can be defined as

EnergyDensity(r)=-(1/8pi)*(1/G)*ForceField(r)^2=-(1/4pi)G M^2/r^4.

It's easy to see that this quantity has dimensions of energy/volume, and the minus sign indicates the force is attractive, but why the 8pi factor? And does the definition make sense?
To find out, Integrate EnergyDensity(r) to get total energy;then calculate the work it takes to remove a portion of the mass to infinity.
You'll find that this work value equals the change in total energy.

Last edited:
bcrowell
Staff Emeritus
Gold Member
Upon release, potential transforms to kinetic energy and during collision, potential energy is transformed to heat, sound, etc..

That is what I've learned from college physics 10 years ago, unless I'm missing something.
Yes, but that doesn't work in GR, for the reasons pervect gave in #5.

It's easy to define and localize the gravitational field energy in the Newtonian case.
Yes, but the OP was asking about general relativity, not Newtonian gravity.

"...where exactly has this energy been transfered to?"

'Energy' has not been transferred. Energy is a theoretical, mechanical type, idea of 'cause'. Energy is not a cause. We do not know what cause is. Energy is force multiplied by distance. Force is the cause. Force is not defined or explained. Energy is a sum total calculated after an event has occurred. When two masses are separated by applying forces to them, they have what is called potential energy. Potential energy is not a property in itself. Potential energy is the recognition that a force exists, and, if that force is permitted to be applied over a distance then, the calculation of force multiplied times that distance can be made. That result of that calculation is energy. It is a number. Before that event occurs there is no energy, there is a force waiting to be applied.

James

Potential is just a scalar field, there is no such things as force.
At least this is what quantum field theory says.

Force is cause. There must be cause. I am not speaking about a specific mechanical notion of a specific cause such as electric charge. I am simply saying that there is not even a thimble full of energy anywhere for physicists to experiment upon. And, that effects do exist. Effects result from cause. Cause must exist whether it is named force or some other name. In any case, energy remains the sum total of cause multiplied by distance.

James

The cause is the scalar potential. The field is the potential gradient. The force is the effect of the field.

We only know about intial conditions and final conditions. Equations express this relationship for us. Potential possibilities are contained in initial conditions. Whatever it is that these initial conditions change into, it is for reasons that we cannot know. We do not know what cause is. Initial conditions change into final conditions. A=B. We observe patterns of behavior of various kinds where initial conditions change into final conditions. We never know the cause.

Theory is the practice of inventing or proposing causes. Theory replaces ignorance with educated guesses. Force is never an effect. Force is always the cause. Gradients are parts of patterns. Patterns are observed in effects only. Gradients will lead to change, but, for reasons that we cannot empirically know. Whatever it is that they can do, it is because of the cause that first gave rise to them. Gradients are intermediary steps in implementing the effects of the fundamental cause that formed the pattern that has a gradient.

'Potential' means only that a cause of some type exists but has not yet been activated. We know about cause because of patterns in effects. We do not observe the cause. So, we identify potential as initial conditions that will change when the unobserved cause is freed to act.

James

Potential energy is mass times potential.
See? No force needed.

Mass is resistance to force. Force is always involved in any change. I use the word force to represent cause. We do not know what cause is; however, we know it exists because effects exist. If there are effects, then there is cause. In the case of potential anything, no force is necessary in the equations because no effects occur. When change is observed, force is always involved.

James

According to general relativity, there is no force, planet just move in a straight line in curved spacetime.

Yes, I know. But, relativity theory is theory. Theory involves bringing educated guesses to the fore about why effects occur. We do not know why effects occur. We do have theory that takes the place of actually knowing why effects occur. The theory is not the cause of effects. The theory is chosen to fit the patterns that occur in the effects for reasons that we cannot know. Theory is the practice of making educated guesses about what cause is. No matter how closely the theory fits the patterns in effects, it does not prove the theory. It only proves that the theorist very closely fit their theory to match empirical evidence. If they are lucky, the patterns that they trusted in will continue to occur. When this is the case, theory will appear to predict effects. All the while we do not know what cause is. We can know our theories. It is a matter of taste whether or not to believe in them.

James

I thought experimental and observational evidence, rather than taste, is the true test of a theory?

quantum123,

You make very good points. I haven't posted much here. The guidelines look pretty firm. I am straying from the thread subject. I don't think that starting my own thread about evaluating theory would be permitted. One point that might be acceptable here is with regard to your statement: "According to general relativity, there is no force, planet just move in a straight line in curved spacetime."

You are correct of course. However, I look at what occurs and I see objects accelerating. For me, this means there is a force involved. In a generic sense, the force of gravity appears to be as much a force as any other kind. Theory can portray it as being different or absent. However, what empirical evidence is there to show that gravity is not a force? I thought this point would fit with the thread subject, but, It is starting to feel that what I am saying might be not be permitted. I guess I will wait and see what happens with this message.

James

quantum123,

Allright, no problem with posting yet. Lets back up a little bit then. I say theory, including relativity theory, is a matter of taste. The reason for saying this is that theory is the practice of imagining what cause may be. We do not know what cause is. So, theoretical physics makes educated guesses about causes. Everything we learn, mechanically speaking, is by discerning patterns in changes of velocity. When we see patterns that appear to be so different that we cannot imagine how they might be the result of a single cause, we introduce two causes. This is why I say that theory is a matter of taste. Why introduce two or more causes into the analysis of a universe that must be the result of a single cause? This practice of introducing disunity into theory makes it impossible to achieve a unified theory without introducing additional theoretical properties that do not have empirical support, purely for the purpose of forcibly rejoining that which we have theoretically made disjointed.

I will wait to see if this post is deemed to be improper before continuing.

James

James

Because your Newtonian viewpoint cannot explain what we know about space time mass and energy in the 20th century.

Because your Newtonian viewpoint cannot explain what we know about space time mass and energy in the 20th century.
Theoretical viewpoints are not relevant to what I stated. Anyway I see you have some notion about what my view is and find it lacking. Since the subject of this thread is not what do I think about space, time, mass, and energy, all of which I have no problem explaining, I will desist. Thank you for the conversation.

James