Is Energy Conserved in General Relativity?

[DW]

Unless, that is, you consider "pseudo-tensor" and "pseudotensor" to be two different terms with two different meanings.

I can't resist anymore, you coersed me into telling everyone else this joke.
Q: What is the definition of a pseudointellectual?
A: One who pretends to know what pseudo means.
Look it up.

 

[Janus]

TO pmb_phy & DW:

You know what? I've just about had it up to here with the both of you. If you can't leave your personal differences at the door and stick to the subject, maybe neither of you should be here.

Consider this a warning.

 

[pmb_phy]

TO pmb_phy & DW:

You know what? I've just about had it up to here with the both of you. If you can't leave your personal differences at the door and stick to the subject, maybe neither of you should be here.

Consider this a warning.

You're kidding me. dw flames me that you're warning me?

Sorry. I can't think of a reason to want to post at a place where a moderator warns people who are being flamed.

Good luck with dw. You'll certainly need it.

 

[enigma]

You're kidding me. dw flames me that you're warning me?

That's funny. He's saying the exact same thing about you.

Both of you have got your fangs out and your talons sharpened.

Both of you read back what you posted, including the earlier post before you edited it, pmb_phy.

Can you honestly say that you'd talk to someone in the same room as you the same way you've been talking to each other on this thread?

The little flamewar which you're both participating in started with a little jab and then quickly escalated, as it appears to me. That being said, I'll echo Janus and say:

Knock it off, the both of you.

 

[turin]

This was regarding the statement about the coordinate independence of tensors:

This is one of the most important points to understand in relativity.

I have heard this (concerning tensors), and I think I agree, but I don't think I quite have the idea behind 4-acceleration (as a tensor). If it is a tensor (invariant, so vanishing is vanishing in any coordinate system), then I guess I'm just having a little trouble understanding how acceleration can be a tensor in the first place. I can sit on the patio and watch something fall out of the sky. Then I would say, "that thing is accelerating." Then, I could be sitting right on top of the thing and fall with it and say, "this thing is not accelerating."

I guess this has something to do with the proper time dilating more as the speed increases. I can kind of get the idea mathematically, by my intuitive picture seems to be lacking.

 

[DW]

This was regarding the statement about the coordinate independence of tensors:I have heard this (concerning tensors), and I think I agree, but I don't think I quite have the idea behind 4-acceleration (as a tensor). If it is a tensor (invariant, so vanishing is vanishing in any coordinate system), then I guess I'm just having a little trouble understanding how acceleration can be a tensor in the first place. I can sit on the patio and watch something fall out of the sky. Then I would say, "that thing is accelerating." Then, I could be sitting right on top of the thing and fall with it and say, "this thing is not accelerating."

I guess this has something to do with the proper time dilating more as the speed increases. I can kind of get the idea mathematically, by my intuitive picture seems to be lacking.

The "length" of a tensor is frame invariant. So is a tensor equation. The tensor itself is frame covariant. If a tensor is not zero according to one frame it is not zero according to all and this goes for the acceleration four-vector as well. The length of the velocity four-vector is c which is also invariant. This means that in a relativistic interperetation of the motion anything always moves at the speed c in four dimensional spacetime. It is just the direction in four-dimensional spacetime that can be deflected. Naturally an object travels along a geodesic. When a real force is exerted it is expressed as a four-vector force on the object. This never changes the length of the velocity four-vector or the four dimensional speed which is c. What it does is change the deflect the motion or rotate it in spacetime from the geodesic. The four-vector acceleration is a description of this change in motion. When your motion is changed from that of a geodesic all observers will agree on that, even you.
If you must think of the forced and falling object as changing speed then consider its motion with respect to a local free fall frame observer. The length of the acceleration four-vector according to any frame is equal to the magnitude of the coordinate acceleration for a local free fall frame according to which the object is instantaneously at rest. If you are in free fall along with a reference object and then experience a real force, the relative velocity between the two of you will become nonzero and you will feel the force that pushed you so you will be able to determine that you were accelerated just as anyone else will say.

 

[turin]

When your motion is changed from that of a geodesic all observers will agree on that, even you.

Yes, of course! Thank you, DW. It is starting to become a bit more organized. A geodesic is a geodesic, regardless of how one describes it.

I still think that this is drastically different than the pre-relativity (Newton's) picture.

 

[Phobos]

pmb & DW - - We've been through this before. I'm sorry to see that the cease fire has ended. Further flaming will be locked/deleted.

 

[blue_sky]

Energy concept

May I propose a simple problem that could help in understanding the energy concept?
Let consider an astronaut out of his spaceship and connected to it by a wire. He as a tool that can put the wire under tension and let him move towards the spaceship.
The tool can be, for exemplum, a spring loaded so with that tool the astronaut has a definite amount of energy (E*) available.
The astronaut use that energy E* to accelerate towards the spaceship.
If E* is known what will be the relative velocity between the astronaut and the spaceship in Newtonian physics?

 

[pervect]

I'd like to comment on the tidal tensor business. There is a very useful 3x3 tensor in GR that's sometimes called the "Electric part of the Weyl Tensor" (E[v] in GRTensorII software package). Other times a closely related tensor has been called the "electrogravitic tensor"

http://www.lns.cornell.edu/spr/2002-03/msg0039921.html

You basically compute

"Electric part of Weyl"

$$C_{abcd} u^{b}u^{d}$$

"Electrogravitic"
$$R_{abcd} u^{b}u^{d}$$

Where u is the 4-velocity of an observer.

If the matter density is zero where the measurement is made, both of these will be equal.

If you eliminate the index for time, you get a 3x3 tensor that describes the tidal force near the object. If you feed the 3x3 tensor a direction, the result is an acceleration vector - or a force/unit mass. The diagonal terms of this tensor represent "stretching" forces, the off diagonal terms represent torques.

 

[pervect]

Steve Carlip, on sci.physics.research has asserted that energy is not well defined. Paraphrasing him, if you want to have a concept of energy, you need to include gravitational potential energy, right? But you can always switch to a freely falling coordinate system in which gravitational potential is zero. And a tensor, if zero in any coordinate system is zero in every coordinate system. So since this is true at every point of spacetime, either energy is identically zero everywhere, or else it is not well defined, because only tensors are well defined (covariant) in GR.

Going back to the original question/point

I would say that the energy of an isolated system in asymptotically flat space-time is is well defined in GR, but the location of the energy is not.

There's a whole chapter in MTW's "Gravitation"

"Why the energy of the gravitatioanl field cannot be localized" on p 466.

I'd like to add, though, that though I'm fairly confident that the energy and momentum of an isolated but moving system are well defined in GR, I'm currently having a heck of the time with the details for the case in which the system is not stationary (i.e. the gravitational field of a moving black hole).

 

[Haelfix]

"I'm currently having a heck of the time with the details for the case in which the system is not stationary (i.e. the gravitational field of a moving black hole)."

Good luck finding a sensible definition of energy for a nonstationary system. You don't have timelike isometries, and you will run into problems over and over again far away from the linearized theory where the time dependence of the real metric becomes important.

Its really quite simple if you think about it. A metric has ten independant components, and any transformation allows you to introduce 4 extra degrees of freedom.. Only in very particular types of metrics, will it allow you to make your components time independant.

You can sometimes define things in such a way as to give you a conserved quantity that can be interpreted as energy, but its never a general solution, and is case by case for metrics.

 

[pervect]

"I'm currently having a heck of the time with the details for the case in which the system is not stationary (i.e. the gravitational field of a moving black hole)."

Good luck finding a sensible definition of energy for a nonstationary system. You don't have timelike isometries, and you will run into problems over and over again far away from the linearized theory where the time dependence of the real metric becomes important.

Its really quite simple if you think about it. A metric has ten independant components, and any transformation allows you to introduce 4 extra degrees of freedom.. Only in very particular types of metrics, will it allow you to make your components time independant.

You can sometimes define things in such a way as to give you a conserved quantity that can be interpreted as energy, but its never a general solution, and is case by case for metrics.

Well, as near as I can make out, there are supposed to be not just one, but two notions of the energy of a system, that apply in general to asymptotically flat space-times. This is all rather confusingly discussed in one of my textbooks, Wald's "General Relativity" around p 291.

One is called the Bondi energy, the concept originated with Bondi before the reformulation in terms of asymptotic flatness, and is associated with null infinity. The other is the ADM energy, and is associated with spatial infinity.

This is in addition to some special-case things one can do, which are supposed to give one a Poincare subgroup of the more general infinite-dimensional groups that arise at the appropriate infinity(ies). These special case things do require some conditions on the metric, though.

The Bondi energy actually isn't strictly speaking an invariant - because it's at null infinity the system can lose energy due to gravitational radiation. I'm describing the known loss of energy at null infinity due to gravitational radiation as "energy being conserved". The ADM energy, being at spatial infinity, doesn't have this feature (or problem, if you prefer), so it's a true invariant. The Bondi and ADM energies can even be compared, with sensible results, they turn out to be the same if there is no gravitational radiation.

Anyway, that's supposed to be the theory. So far I haven't actually managed to calculate anything.

 

[pmb_phy]

Steve Carlip, on sci.physics.research has asserted that energy is not well defined. Paraphrasing him, if you want to have a concept of energy, you need to include gravitational potential energy, right? But you can always switch to a freely falling coordinate system in which gravitational potential is zero. And a tensor, if zero in any coordinate system is zero in every coordinate system. So since this is true at every point of spacetime, either energy is identically zero everywhere, or else it is not well defined, because only tensors are well defined (covariant) in GR.

That is not a reasonable statement. Transforming to a freefall frame and thus transforming to a frame in which the gravitational field vanishes does not mean that the potential energy of a particle vanishes. For example: If you're in a uniform gravitational field and you transform to a freefall frame then the gravitational potentials, g_jk become constants. Thus the metric does not vanish. The existence of a gravitational field does not depend on the vanishing of the metric tensor but on the spacetime variation of the gravitational potentials (when the spatial coordinates are Cartesian). Of course this was all explained/described by Einstein in his 1916 GR paper.

Pete

 

[pervect]

That is not a reasonable statement. Transforming to a freefall frame and thus transforming to a frame in which the gravitational field vanishes does not mean that the potential energy of a particle vanishes. For example: If you're in a uniform gravitational field and you transform to a freefall frame then the gravitational potentials, g_jk become constants. Thus the metric does not vanish. The existence of a gravitational field does not depend on the vanishing of the metric tensor but on the spacetime variation of the gravitational potentials (when the spatial coordinates are Cartesian). Of course this was all explained/described by Einstein in his 1916 GR paper.
Pete

I believe Self Adjoint was the one quoting Steve Carlip, not I. This may seem a minor point, but it does mean that I haven't even seen Mr Carlip's full argument personally.

Regardless of whether the argument was spelled out in enough detail to be believed on its own, I do agree with the conculsion, which is that there is no way to describe potential energy in GR with a tensor quantity. I thought you had agreed with this too, now I'm rather unclear as to your position on the matter.

 

[pmb_phy]

I believe Self Adjoint was the one quoting Steve Carlip, not I.

Sorry. I was simply addressing his comments. My mistake if I got things a bit mixed up.

Regardless of whether the argument was spelled out in enough detail to be believed on its own, I do agree with the conculsion, which is that there is no way to describe potential energy in GR with a tensor quantity. I thought you had agreed with this too, now I'm rather unclear as to your position on the matter.

There is a subtle thing that many people miss here. Notice exactly the comment I was addressing. It was, exactly, this

But you can always switch to a freely falling coordinate system in which gravitational potential is zero.

There is a difference between the gravitational potential and, referring to a particle in a gravitational field, the gravitational potential energy of a particle. There is also gravitational self energy which is the energy related to the gravitational field itself.

The gravitational potential is related to the gravitational force. This means that the gravitational force, in general relativity, is a combination of the derivatives of the gravitational potentials, i.e. g_uv (aka components of the metric tensor) and the velocity of the particle. See Eq. (8a) in

http://www.geocities.com/physics_world/gr/grav_force.htm

The Christoffel symbols (capital gammas) are functions of the gravitational potentials, g_uv. The gravitational potentials are well defined quantities in GR.

The gravitational potential energy of a particle, at least to me, is the energy of the particle by virtue of its position in a gravitational field. The energy of a particle as a function of position is also well defined in GR. Just because the energy is not a linear sum of rest, kinetic and potential energy, it doesn't mean that they are not well defined or meaningless.

I believe that what Carlip was referring to was the fact that if you tell me the position and velocity of a particle in a strong gravitational field that I will not be able to give you a specific value for something and meaningfully call it "potential energy". However if the field is weak I can do this and do it in general relativity.

Pete

 

[selfAdjoint]

Pete, why don't you post on s.p.s in that Carlip thread and let's get the issues cleared up. With all respect, I trust his claims more than yours, but I think it's probably a semantic difference. But it should be resolved, and I don't see how I can improve the conversation as a middleman.

 

[pmb_phy]

Pete, why don't you post on s.p.s in that Carlip thread and let's get the issues cleared up.

What is "s.p.s"? If its a newsgroup then I don't post in newsgroups anymore. Carlip and I have discussed this before anyway.

With all respect, I trust his claims more than yours, but I think it's probably a semantic difference. But it should be resolved, and I don't see how I can improve the conversation as a middleman.

I don't wish to post anywhere outside this forum again. If someone wants to discuss this within a forum here then I'll be glad to chime in and clarify/backup anything I say/post.

Pete

 

[marcus]

What is "s.p.s"? If its a newsgroup then I don't post in newsgroups anymore. Carlip and I have discussed this before anyway.

Pete I think it was SPR (sci.physics.research) that was meant and that by mistake SPS got typed----that would stand for sci.physics.strings
but that would not make sense in this context.

SelfAdjoint originally gave a link to something on SPR by Carlip.
June 6, 2004
Re: Tired Light
http://groups.google.com/groups?hl=en&lr=&ie=UTF-8&selm=c9tajo$d6h$2@woodrow.ucdavis.edu

 

[marcus]

Pete, I have an idea for you

Go to this thread here at PhysicsForums
https://www.physicsforums.com/showthread.php?t=26625

look at post #15

here is the direct link to it:
https://www.physicsforums.com/showthread.php?p=227804&posted=1#post227804

PF has "eaten" your old Alma Mater newsgroup sci.physics.research

The look and feel is nicer in the PF version

Now you can enjoy corresponding with Steve Carlip in the comfort of PF accomodations! Have fun!

 

[pervect]

The gravitational potential is related to the gravitational force. This means that the gravitational force, in general relativity, is a combination of the derivatives of the gravitational potentials, i.e. g_uv (aka components of the metric tensor) and the velocity of the particle. See Eq. (8a) in

http://www.geocities.com/physics_world/gr/grav_force.htm

The Christoffel symbols (capital gammas) are functions of the gravitational potentials, g_uv. The gravitational potentials are well defined quantities in GR.

Now that I know that you are calling the metric coefficients g_uv "gravitational potentials", some of your remarks make more sense. But I should note that this usage is not standard in any of my textbooks, and I would imagine it would confuse most readers as much as it confused me. Since we have a perfectly good name for the metric coefficients already ("metric coefficients"), and since this usage causes *no* confusion, I would like to suggest that we continue to call metric coefficients metric coefficients.

The gravitational potential energy of a particle, at least to me, is the energy of the particle by virtue of its position in a gravitational field. The energy of a particle as a function of position is also well defined in GR. Just because the energy is not a linear sum of rest, kinetic and potential energy, it doesn't mean that they are not well defined or meaningless.

Since the energy of a particle will in general depend on the path it takes, I don't see how to define "the energy of the particle by virtue of its position".

I believe that what Carlip was referring to was the fact that if you tell me the position and velocity of a particle in a strong gravitational field that I will not be able to give you a specific value for something and meaningfully call it "potential energy". However if the field is weak I can do this and do it in general relativity.

Oh, weak fields. Sure, one can define potential energy for weak fields, the first thought that comes to mind is to use PPN. But I thought we were talking about GR, not the weak-field version therof.

 

[pmb_phy]

Going back to the original question/point

I would say that the energy of an isolated system in asymptotically flat space-time is is well defined in GR, but the location of the energy is not.

You mentioned the chapter in MTW on this point. Note that they say that its the gravitational energy that is not localizable, not the energy (i.e. other forms such as mass-energy, EM energy etc.).

That's a nice section in MTW by the way.

Pete

 

[pmb_phy]

Now you can enjoy corresponding with Steve Carlip in the comfort of PF accomodations! Have fun!

Thanks but no thanks. I have no desire to post to any of those newsgroups for my own personal reasons. Thanks anyway.

Pete

 

[pmb_phy]

Now that I know that you are calling the metric coefficients g_uv "gravitational potentials", some of your remarks make more sense. But I should note that this usage is not standard in any of my textbooks, and I would imagine it would confuse most readers as much as it confused me.

You're kidding me! Almost all of my GR texts call them that as did Einstein. For example: The following texts refer to g_uv as "gravitational potentials" -

Gravitation and Spacetime, Ohanian & Ruffini, WW Norton n& Co., (1994)

Introducing Einstein’s Relativity, D’Inverno, Oxford Univ. Press, (1992)

Basic Relativity, Mould, Springer Verlag, (1994)

I also recall seeing it in MTW and in Wald but I can't locate it at the moment

Since we have a perfectly good name for the metric coefficients already ("metric coefficients"), and since this usage causes *no* confusion, I would like to suggest that we continue to call metric coefficients metric coefficients.

I've never heard them called that. They are called the components of the metric. But you're free to call them what you like. But when you start to discuss gravitational potentials in GR then you're talking about g_uv whether you want to call them that or not.

Since the energy of a particle will in general depend on the path it takes, I don't see how to define "the energy of the particle by virtue of its position".

The energy of a particle is a function of velocity, position and rest mass. The functionality of position is what I mean by potential energy. I did not say that you can separate these energies into individual pieces. Let me quote Ohanian, page 157

$$P_0 = \simeq \frac{m}{\sqrt{1-v^2}} - \frac{1}{2}\frac{m}{(1-v^2)^{3/2}} h_{\mu\nu}\frac{dx^{\mu}}{dt}\frac{dx^{\nu}}{dt}+mh_{0\alpha}u^{\alpha}$$

The first term on the right side is of the form of the usual rest-mass and kinetic energy; the other terms represent gravitational interaction (potential energy).

Oh, weak fields. Sure, one can define potential energy for weak fields, the first thought that comes to mind is to use PPN. But I thought we were talking about GR, not the weak-field version therof.

You understand that by weak I do not mean Newtonian, right? Why do you think that the weak field approximation is not part of general relativity. I don't see any need to make such a distinction myself. I understand that you think that they are different but I don't see them that way.

In any case there are cases where, even in strong gravitational fields, the gravitational force is given in terms of the gravitational potential as

$$\bold G = -m\nabla \Phi$$

where m = inertial mass (aka relativistic mass). See derivation (and meaning of phi = gravitational potential) at
http://www.geocities.com/physics_world/gr/grav_force.htm

Its my turn to ask you something - A particle in a gravitational field has energy P₀ where P is the 4-momentum of the particle. Do you think that a particle in a gravitational field has rest energy? If so then do you think that the rest energy is part of the energy P₀? Its a mixture of these energies which one cannot separate into nifty pieces. Notice that Carlip said that potential energy is not well defined. He did not say it does not exist or that it is totally meaningless.

By the way, I posted a web page quoting that part of D'Inverno on this metric = potentials in case you don't have that text. See
http://www.geocities.com/physics_world/gr_potential.htm

As you can see, the term "potential" is used here in a similar way to its used in EM where the magnetic field is the curl (which also involves derivatives) of something called the magnetic vector potential. The force on a charged particle can therefore be written in terms of the derivatives of potentials, i.e. the Coulomb potential and the magnetic vector potential. That is the reason for calling the components of the metric "potentials".

Pete

 

[Haelfix]

Pervect.. Ok.

The ADM paradigm is a little bit tricky, but it does reproduce somewhat sensible results.

The problem is essentially trying to reproduce covariant hamiltonians for a suitable definition of asymptotic flatness called k-asymptotic flatness (believe it or not, there are several definitions of this, the weakest being the usual flat hypersurface at infinity). As this will lead to a method of constructing an asymptotic noether current, both for energy-density and angular momentum, etc.

AFAIR, the weakest form does indeed reproduce a good quantity for energy-density for a large subclass of metrics k > 1/2. It had some problems, I think, b/c it wasn't in agreement with a few other methods, but again AFAIR it ended up being ok.

Now, the non tensorial nature of this quantity had people perplexed for awhile, so they felt the whole thing was contrived and useless for more general metrics.

Others disagreed, as it was important in several fields including quantum gravity... Particuraly to find sensible definitions of observables in GR (the socalled quasi-locality theorems that Witten and others worked on). So believe it or not, ADM works in some situations even with nonstationary metrics with no asymptotic flatness assumptions.

 

[DW]

Steve Carlip is 100% right in this argument that the Newtonian potential concept you are arguing over breaks down and is not general relativistic at all. Also relativistic mass as I have already proven is not the same thing as inertial mass. In fact it has no place in modern relativity.

 

[pervect]

I also recall seeing it in MTW and in Wald but I can't locate it at the moment

If you can find it in MTW or Wald, I'll conceed the point. It's certainly not in the index of either one of them (of course MTW's index **** ###). I really don't see the point of giving the metric coefficients Yet Another Name, though. It's easy enough for me to remember that when you say "relativistic mass" I think "energy", and when you say "gravitational force" I think "Christofel symbol", but I'm starting to build up quite a dictionary here :-(.

You understand that by weak I do not mean Newtonian, right? Why do you think that the weak field approximation is not part of general relativity.

The issue we are talking about, energy, is trivial in PPN, but much less so in the general theory. Unforutnately PPN theory doesn't illuminate much of the general problem.

Its my turn to ask you something - A particle in a gravitational field has energy P₀ where P is the 4-momentum of the particle. Do you think that a particle in a gravitational field has rest energy?

My view is that when we talk about the energy-momentum 4 vector of a particle, we are talking about the energy relative to some specific observer and some specific coordinate system. We need to define the coordinate system and the observer to measure the energy in GR, just as we need to define the "frame" we are talking about when we measure the energy of a body in Newtonian physics.

I also believe that given an energy-momentum 4-vector, we have an invariant mass^2 that's equal to $$g_{uv} E^u E^v$$ -- of course we do to know the metric coefficients as well as the energy-momentum 4-vector to compute the mass

I'll leave to you to do the translation into your own terminology, I hope my meaning is clear.

The other point I want to make is that issues do arise when attempting to parallel transport the energy-momentum 4-vectors in order to do some sort of intergal to get "total energy". So having a definition of the energy-momentum 4 vector at a point isn't all that's needed to be able to define the energy of a system. For many applications, if the region is small enough, the parallel transport issue can be ignored, but this isn't always true.

 

[pmb_phy]

If you can find it in MTW or Wald, I'll conceed the point.

Okay, but that seems to be a bit of a narrow view. The GR literature uses the same terminology. I'm just letting you know what's out there. But I found where I saw it in MTW. Its on page 434 where they are quoting Hilbert. I've e-mailed someone to find it in Wald. I'll let you know the result.

To see where Einstein used it in the most notable place, its in his Nobel lecture. See - http://www.nobel.se/physics/laureates/1921/einstein-lecture.pdf

It's certainly not in the index of either one of them (of course MTW's index **** ###).

Indexes don't always list all terms used in a text

I really don't see the point of giving the metric coefficients Yet Another Name, though.

John Stachel wrote a paper which touched on this point in How Einstein Discovered General Relativity: A Historical Tale with some Contemporary Morals. Stachel is a famous GRist and GR historian. He points out

The main difficulty at this stage was to grasp the dual nature of the metric tensor: it is both the mathematical object which represents the space-time structure (chronogeometry) and the set of 'potentials,' from which represents the gravitational field (Christoffel symbols) and the tidal 'forces' (Riemann tensor) may be derived.

I'm doing something bad right now, i.e. I'm sitting and typing against docs orders so I can't get into all the details of that paper and the context. If you want more then I'll get to it at a later date.

But there is the question of "Why would I want to know that?" wherein I'd respond "To understand what you're reading when you encounter it." For example. See - http://arcturus.mit.edu/8.962/notes/gr6.pdf

This was written by Edmund Bertschinger, MIT cosmologist (GR prof). What do you think he means when he says

Although the gravitational potentials represent physical metric perturbations,...

What are these "gravitational potentials" that he refers to?

It's easy enough for me to remember that when you say "relativistic mass" I think "energy", and when you say "gravitational force" I think "Christofel symbol", but I'm starting to build up quite a dictionary here :-(.

That's a good thing.

If you have MTW then see what they say about the stress-energy pseudo-tensor of the gravitational field. It's in MTW page 465 Eq. (20.18).

My view is that when we talk about the energy-momentum 4 vector of a particle, we are talking about the energy relative to some specific observer and some specific coordinate system.

That is inaccurate. The 4-vector itself has a geometric signifigance which has nothing to do with coordinates. When you wish to choose a coordinate system then you've chosen a basis and a particular value of the energy. Once you do that, my question remains - Is the rest energy of the particle part of that energy?

The other point I want to make is that issues do arise when attempting to parallel transport the energy-momentum 4-vectors in order to do some sort of intergal to get "total energy".

I have no idea what you're talking about. What is it that you're adding? Total energy of what? I was referring to the energy of a single particle in a G-field.

So having a definition of the energy-momentum 4 vector at a point isn't all that's needed to be able to define the energy of a system.

Who was speaking of the energy of a system? I was speaking of the energy of a particle in a g-field. Those are two different topics.

Pete

 

[pervect]

When you wish to choose a coordinate system then you've chosen a basis and a particular value of the energy. Once you do that, my question remains - Is the rest energy of the particle part of that energy?

As far as "rest energy" goes, do you mean something other than the invariant associated with the 4-vector? We both agree that the energy-momentum 4-vector exists, and that it has an associated invariant. It seems to me that that should be sufficient.

Who was speaking of the energy of a system? I was speaking of the energy of a particle in a g-field. Those are two different topics.

Most of my remarks in this thread have been addressed to the issue of energy conservation in GR. This requires that one consider the notion of the nergy of a system as well as the energy of a particle.

 

[pmb_phy]

As far as "rest energy" goes, do you mean something other than the invariant associated with the 4-vector?

I don't mean something other. I'm referring to rest energy, i.e. the product of a particle's proper mass with c².

We both agree that the energy-momentum 4-vector exists, and that it has an associated invariant. It seems to me that that should be sufficient.

For what? The topic here is energy in GR, correct? There are different subtopics on that topic. One subtopic is P₀ = energy of a particle in a g-field. Another is the mass-energy that creates the g-field, i.e, T⁰⁰, and then there is the self energy of the g-field which is represented be a pseudo-tensor (which I'm not all that familiar with).

Most of my remarks in this thread have been addressed to the issue of energy conservation in GR. This requires that one consider the notion of the nergy of a system as well as the energy of a particle.

So do you think that energy is not conserved?

By the way, I just looked at the index to Wald. It does not contain the term "invariant." However I don't think that Wald considered it to be useless in GR.

Pete

 

[pmb_phy]

Pete, I have an idea for you

Go to this thread here at PhysicsForums
https://www.physicsforums.com/showthread.php?t=26625

look at post #15

here is the direct link to it:
https://www.physicsforums.com/showthread.php?p=227804&posted=1#post227804

PF has "eaten" your old Alma Mater newsgroup sci.physics.research

The look and feel is nicer in the PF version

Now you can enjoy corresponding with Steve Carlip in the comfort of PF accomodations! Have fun!

Okay. I've changed my mind. Since its a moderated forum I'll take a look. But Carlip and I have discussed something similar before. It turned into a discussion of semantics. But he obviously knows GR better than most of us so I recommend that people pay close attention to what he says.

Pete

 

[selfAdjoint]

Good show, Pete. We'll be paying attention for sure.

 

[pmb_phy]

Good show, Pete. We'll be paying attention for sure.

Well, I didn't want to appear closed minded. This is a topic that I'm highly interested in. One problem in addressing it is that energy itself has never been a well-defined quantity so when one says that its not well-defined in GR I'm not really surprised. But I do want to learn more about these problems and the meaning of the stress-energy-momemtum pseudo-tensor of the gravitational field.

Pete

 

[kurious]

If energy is not conserved in GR but it is conserved in QM this would make it difficult to see how QM and GR can be reconciled in one theory.Energy is conserved in every other branch of physics so why should GR be the exception? Can't the energy of microwave background photons for example be absorbed by something else in the universe? This has got to be more likely than a lack of energy conservation.
And if one looks at the entropy of the universe as a whole,are we to believe that
heat energy is lost from the universe? This would imply that the second law of thermodynamics can be violated or that the energy passes into another universe.
There is no proof for either of these.

 

[pervect]

For what? The topic here is energy in GR, correct? There are different subtopics on that topic. One subtopic is P₀ = energy of a particle in a g-field. Another is the mass-energy that creates the g-field, i.e, T⁰⁰, and then there is the self energy of the g-field which is represented be a pseudo-tensor (which I'm not all that familiar with).
So do you think that energy is not conserved?

OK, we've discussed most of the other subtopics, but we haven't discussed how the stress-energy tensor arises from the energy-momentum 4 vector in this thread.

I'm not sure if we need to talk about the stress energy tensor or not. I would describe it informally as the density per unit volume of the energy-momentum 4 vector. I could make this description more formal if there is a point to doing so, but I won't bother unless there is some interest and/or flames about my informal description. Well, I will mention that one can measure volume with both a three-form and with the dual (Hodges dual) of a three form, which is a formal way of saying that any volume element defines a unique perpendicular time-like vector. Hopefully that terse description may be useful in communicating my meaning.

The fact that the self energy of the field is represented by a pseudo-tensor and not a real tensor means that the notion of self-energy of a gravitational field isn't a geometric object. Different observers define it differently. So, when we require a pseudo tensor to measure system energy, "red flags" should be raised, because we are combining geometric objects, and something that's not geometric. Unfortuately, we can't simply ignore or drop the self-energy term and retain a conserved energy.

There's an amusing quote from MTW on this topic, I'll try and find it, after I resolve some technical difficulties with my computer - look for a short separate post, hopefully coming soon, unless I can't find it - did I mention that the index in MTW **** ###?.

In any event, defining energy with the pseudo-tensor approach is, at least in my opinion, a little bit like adding together all the physical energies, plus the well-known "Finnagle Factor" from engineering, where the "Finnagle Factor" takes on whatever value is required to make the answer come out right. It's not surprising that one can come up with a conserved quantity by adding the Finnagle number F to *any* general quantity, conserved or not, but it's not particularly physically significant to be able to do so.

Fortunately, there are a number of more physicaly motivated ways of defnining the total energy of a system in a manner which retains the desired conservation property. One way of doing this is to use the approach of Cartan moments, described in MTW. Another way of doing this, also mentioned in MTW, is to look at the behavior of an object orbiting the system "a long ways away" from the system, and to use agreement with Newtonian theory to define the energy of the system.

A more modern approach ito defining energy in GR s to focus on the behavior of the gravitational field at infinity, the behavior of the field at null infinity gives the Bondi energy, and the behavior of the field at spacelike infinity gives the ADM energy. Look at my previous posts in the thread for a reference to the pages in Wald where these are discussed. Wald discusses the relationship between the Bondi and ADM energy as well, according to Wald they eventually turn out to be closely related, though it takes a considerable amount of work to establish the relationship. I believe that the Bondi energy is also closely related to the first two definitions of energy I mentioned in the previous paragraph, but I'm not absolutely positive on this partciular point (I'm still working on undertanding Wald's discussion more fully, it's pretty technical).

The unifying thread to all of the methods for dealing with the energy problem that I"ve mentioned is the notion of the observer at infinity. Apparently this isn't quite the last word, though, based on the comments of another poster. It's a bit scary to think that there are probably more definitions of energy in GR than I've posted even in this very long post - but probably accurate.