Relativity: Understanding Potential | Explainer

[preet0283]

can anyone explain 2 me the fact that we don't have the concept of potentials in relativity...

 

[robphy]

What do you mean specifically?
We can describe fields theories in relativity, e.g., the 4-potential for the electromagnetic field.
Here are some other "potentials": the Lanczos potential http://en.wikipedia.org/wiki/Lanczos_tensor , Noether potential http://scholar.google.com/scholar?q="noether+potential+" ...

 

[pmb_phy]

can anyone explain 2 me the fact that we don't have the concept of potentials in relativity...

That's wrong. Where did you get such an idea? If you be more precise as to the specifics I can provide examples in both 3+1 language and in geometric language (4-potential in EM, metric tensor as 4-potential etc.)

Pete

 

[pervect]

I would guess that the original poster probably used imprecise language (he said "does not have a potential" when he probably meant "does not have a potential energy").

This would not be a big deal if nearly everyone had not apparently misunderstood the question as a result of a fairly minor omission.

For a source of why GR does not have a concept of gravitational potential energy, consider the following post from Steve Carlip (a recognized expert in the field).

Message-ID: <6csu39$cdb$1@mark.ucdavis.edu>#1/1

Steve Carlip Feb 23 1998, 12:00 am show options
Newsgroups: sci.physics.relativity
From: car...@dirac.ucdavis.edu (Steve Carlip) - Find messages by this author
Date: 1998/02/23
Subject: Re: Gravitational Potential in GR
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Edward Schaefer (schae...@plansys.com) wrote:

: If Phi = -M * (1 - sqrt(g00)) does not apply *throughtout* the
: gravitational field, then what does?

Nothing does. The concept of gravitational potential energy is
a Newtonian concept; it has no equivalent in general relativity.
The best you can do is to define an approximate gravitational
potential energy in regions in which Newtonian gravity is a
good approximation.

The whole concept of gravitational energy in general relativity is
quite tricky. The principle of equivalence implies that there can
be no covariant gravitational stress-energy tensor---one can always
choose coordinates in which the geodesics are arbitrarily close to
straight lines in a small region, which implies that the gravitational
energy in that region is arbitrarily close to zero; but a tensor that
vanishes in any coordinate system vanishes in every coordinate system.
(For a fleshed-out version of this argument, see section 20.4 of
Misner, Thorne, and Wheeler.)

The best one can hope for is a "quasilocal gravitational energy,"
energy defined in a finite region. A nice proposal for such an
energy is given by Brown and York, Phys. Rev. D47 (1993) 1407. It
can be described in terms of the metric, but not in any particularly
somple way. It *certainly* can't be given in terms of a particular
component of the metric---that's a highly non-covariant procedure.

Steve Carlip
car...@dirac.ucdavis.edu

Having been around this barn before, though, I don't at the moment really feel like trying to explain the details of energy in GR to someone who is probably not terribly familiar with the theory while Pete contradicts me.

But I will at least point out that GR does not have an exact equivalent of the Newtonian concept of Gravitational Potential Energy, even if I don't explain the details.

I'll also agree with the remarks that GR does have several different sorts of potentials, even if it doesn't have anything that's exactly analogous to "gravitational potential energy".

It's always possible that I've misunderstood the original question in any event.

 

[pmb_phy]

I would guess that the original poster probably used imprecise language (he said "does not have a potential" when he probably meant "does not have a potential energy").

Why would you make that guess? There's a big difference between the two.

For a source of why GR does not have a concept of gravitational potential energy, consider the following post from Steve Carlip (a recognized expert in the field).

Potential energy is often spoken of within the context of GR. In strong fields the energy of a particle in such a field cannot be broken down into a sum. The energies are bound up together in a non-linear fashion. For this reason some people say that potential energy does not belong in GR. I don't buy it.

Having been around this barn before, though, I don't at the moment really feel like trying to explain the details of energy in GR to someone who is probably not terribly familiar with the theory while Pete contradicts me.

From hereon in I will only state once my position and will not defend it but merely respond to questions. Therefore you have no need for such a position. I'm too tired and in far too much pain to sit here and debate things anymore. Plus my new philosophy of life does not allow for it. However I see no reason to assume the questioner is speaking of GR when the question can apply to SR as well. The concept of "potential" (as opposed to "potential energy" has a very well defined meaning in GR though. The concept is extended beyond its normal meaning in Newtonian physics.

Note: Given my above comment I wish it to be made very clear that my comment on potential energy in GR are my own and are not held by anyone else that I know of. For that reason, and this damn pain, I won't debate it at all.

Pete

 

[pervect]

Well, I really don't want to be in the position of censoring your remarks.

On the other hand, I *still* don't feel like going into all the details of energy, potential energy, and GR. (AT least not right at this very moment).

Probably the most best thing I can do at this point is to provide my "usual link" to a general overview of energy & GR for anyone who is interested, though it may not answer the original question fully. It's really a a rather tricky issue.

That, and take my allergy medication :-).

Anyway, the "usual link" is
http://math.ucr.edu/home/baez/physics/Relativity/GR/energy_gr.html.

 

[nemosum]

I didn't have the time to look through that whole link, so I was wondering if anyone could give a quick definition (not too technical please) of Gavitational Potential Energy.

 

[Garth]

I didn't have the time to look through that whole link, so I was wondering if anyone could give a quick definition (not too technical please) of Gavitational Potential Energy.

Classically if you lift up a body, say a brick, you are doing work against the gravitational force, this work is then 'stored' as Gravitational Potential Energy and released when the body is dropped.

Energy is conserved so GPE (and other forms of PE) + KE = Constant.

In GR the problem is that there is no gravitational force as such. Gravitation is explained as the effect of space-time curvature. Free-falling bodies travel along straight lines (geodesics) through curved space-time. Like two ants crawling across the dip around the stalk of an apple, the geodesics of a dropped brick and the Earth converge because the Earth's mass (and to a much lesser extent the brick's) has curved the space-time through which they both 'fall'.

The force of 'weight' is a non-inertial force perturbing the brick from its free-falling geodesic. Release it and the brick suffers no weight at all, it is weightless.

The problem arises when you try to work out where the energy used to lift the brick goes to.

Lift a brick and put it on a shelf. The rest energy of the brick has not changed, so where has the GPE, the work expended lifting it, gone to?

The standard answer is "into the field", the presence of the brick higher up in the Earth's gravitational field has altered it slightly, the components of the GR Riemannian tensor describing that field has changed.

However if the brick now falls off the shelf it is momentarily still stationary but accelerating downwards. In that free-falling state the space-time locally around becomes flat. This is called the Einstein Equivalence Principle, "the physics of a small enough volume around a free-falling test particle becomes indistinguishable from that of SR", its coordinate system locally is Lorentzian.

So what has happened to the GPE now?

This example demonstrates that GR is an improper energy theorem in which energy is not to be expected to be locally conserved. To define energy in any sensible way you have to go out from a gravitating mass to an asymptotically null infinity where the space-time becomes that of SR again.

The problem is the curvature of space-time, energy is a frame dependent non-tensorial, concept that loses its definition in the presence of space-time curvature.

I hope this helps.

Garth

 

[pmb_phy]

Well, I really don't want to be in the position of censoring your remarks.

That wasn't what I meant. I meant that I can no longer get into such debates due to health reasons and a new philosophy of life. You have an opinion. I see no reason for me to want to change it by debating or to make some sort of effort to prove you wrong when nobody has asked me to. Its the new me as one might say. If I don't start acting in this way it has the potential to actually kill me. It sounds bizzare, I know. But it is due to a recent brush with Mr. Grip Reaper.

re - "On the other hand, I *still* don't feel like going into all the details of energy, potential energy, and GR."

Me too.

However the above comments apply to GR and not to SR. SR is a different story.

Pete

 

[Juan R.]

For a source of why GR does not have a concept of gravitational potential energy, consider the following post from Steve Carlip (a recognized expert in the field).

Carlip is trivializing point. When spacetime is asymptotically flat or fields are associated to certain background metric (e.g. string theory) one recovers concept of gravitational energy.

Some relativists specially those working in quantum gravity believe that an concept of energy still hold And in fact the question is far from being solved. Many people is searching a correct tensorial expression for energy t_munu.

Now i do not remember but i think that Penrose argues that potential energy is defined but is non-local.

 

[pervect]

Carlip is trivializing point. When spacetime is asymptotically flat or fields are associated to certain background metric (e.g. string theory) one recovers concept of gravitational energy.

Some relativists specially those working in quantum gravity believe that an concept of energy still hold And in fact the question is far from being solved. Many people is searching a correct tensorial expression for energy t_munu.
Now i do not remember but i think that Penrose argues that potential energy is defined but is non-local.

Locality is one of the key issues. When space-time is asymptotically flat, one does have a definite notion of energy (well, actually at least *two* definite notions of energy - ADM mass and Bondi mass, and the associated 4-vectors which define energy and momentum), however, the energy can't be localized. This means that it cannot be unambiguously "split up" into a kinetic energy part and a potential energy part. There is a "total energy", obtained in the ADM case by integrating some force-like pseudotensors dereived from the metric over an enclosing surface in a manner similar to Gauss's law, but there is no unambiguous way to split the total energy up into parts or to assign it a location. Thus potential energy doesn't have a definite meaning in a flat space-time, though total energy does.

The other issue is that non-asymptotically flat space-times, such as the FRW cosmologies, do not have a well defined notion of energy at all.

There is an important case where something very similar to potential energy does exist in relativity, however. This is the case of a static metric, and the associated Komar mass. In this case, the energy can be localized, and one will frequently see remarks about the "gravitational effective potential". MTW has numerous such references, an online example of this sort of usage would be

http://www.fourmilab.ch/gravitation/orbits/

However, that's the only case where the concept of potential energy has any meaning in GR. An asymptotically flat metric has a notion of energy, but no way to split it up into parts, and the most general metric doesn't have a notion of energy at all.