Where does the energy of gravity come from?

[PeterDonis]

Even if we hypothesize a conservative force with a finite range, the region outside the force would be an equipotential region. The potential would be defined, even outside the finite range of the force.

This is true as a general statement about forces and potentials. But in relativity, gravity is not a force, and formulating a valid concept of "potential energy" for gravity must be done without making use of the concept of force. (It is also worth noting that this can only be done in GR for a particular class of spacetimes, the stationary spacetimes.) Doing this correctly in GR does show that, in spacetimes where there is a valid concept of "gravitational potential energy" at all, it is well defined everywhere in the spacetime (or more precisely, everywhere in the stationary region--which might not be the entire spacetime in a case like a black hole, but that is a whole separate issue that is off topic for this thread). But the logic is different from that expressed in the above quote.

 

[Dale]

formulating a valid concept of "potential energy" for gravity must be done without making use of the concept of force. (It is also worth noting that this can only be done in GR for a particular class of spacetimes, the stationary spacetimes.)

And in that class of spacetimes the concept of a gravitational potential is perfectly legitimate and is reasonable fodder for discussion here. And in those spacetimes the gradient of a potential is still a force and fictitious forces also bear the name of force. I understand your opinion, but my logic is also sound. Particularly as I actually stated the matter.

You are just using a more restrictive sense of the word “force” than I am here. I have no qualms about speaking of gravitational forces, using a broader meaning of the term. Nor do I have qualms about stating that gravity is not a force in GR, using a more narrow meaning of the term. Words have more than one meaning.

 

[PeterDonis]

You are just using a more restrictive sense of the word “force” than I am here.

Fair enough. However:

the gradient of a potential is still a force

This still isn't quite true. Take the simplest example, Schwarzschild spacetime. The potential energy (per unit mass--I'm going to do everything per unit mass here to avoid having to carry around a factor of $m$ in all the formulas that does nothing) is

$$
U = \sqrt{1 - 2M / r} - 1
$$

Note that of course this is negative for any finite value of $r$ (and goes to $0$ at $\infty$).

The force (per unit mass) on an object at $r$ (given by, for example, the proper acceleration necessary to "hover" at rest) is (without worrying about the sign, since that just reflects whether we want the actual force on an object at rest, or the fictitious force in that object's frame on an object freely falling radially)

$$
F = \frac{M}{r^2 \sqrt{1 - 2M / r}}
$$

This is $dU / dr$, but that is not the same as the gradient, because $r$ is not the same as radial distance. The gradient will be the derivative with respect to radial distance, which gives

$$
\frac{dU}{d \ell} = \frac{M}{r^2}
$$

i.e., the "redshifted" acceleration "due to gravity". (A similar discrepancy arises when we compute the work required to lift an object "hovering" at finite $r$ out to infinity.)

 

[member 728827]

Even in English it would probably not be the best reference to give. See below for better ones. It only took me a few minutes of web searching to find the references below; that is something you could (and should) have done.

I didn't look into any particular video to talk about the Equivalence Principles, but just thought (guessed, supposed....) that the concept could fit into the naive question "where does the energy of gravity come from...." question in a Relativity Forum, mentioning a video that the IA of Google put in my playlist, mixed with some cat videos, bird training and other stuff.

Is not the best reference in your opinion, but is an enough clear video for me from the former director of the Theoretical Physics Division of the CERN in his endeavor of outreaching some physic concepts to a broad range of people.

And IMHO, the part that reads "...you could (and should) have done..." is not appropriate, and you "should not" use those type of wording in your answers.

 

[phinds]

Is not the best reference in your opinion, but is an enough clear video for me from the former director of the Theoretical Physics Division of the CERN in his endeavor of outreaching some physic concepts to a broad range of people.

Even with only 119 posts, you should be aware by now that pop-science presentations (designed, as you say, for a broad range of people) are never acceptable references here on PF, regardless of the speaker/writer.

 

[Dale]

pop-science presentations (designed, as you say, for a broad range of people) are never acceptable references here on PF

Well, they can be acceptable references for an OP. But we should use sources that are consistent with the professional scientific literature in our responses.

 

[PeroK]

Is not the best reference in your opinion, but is an enough clear video for me from the former director of the Theoretical Physics Division of the CERN in his endeavor of outreaching some physic concepts to a broad range of people.

This misses the point that PF is not here to present popular-science. Nor to discuss the content as presented in popular science.

Our goal is to provide a community for people (whether students, professional scientists, or hobbyists) to learn and discuss science as it is currently generally understood and practiced by the professional scientific community.

Reference: https://www.physicsforums.com/insights/about-physics-forums/

 

[PeterDonis]

IMHO, the part that reads "...you could (and should) have done..." is not appropriate, and you "should not" use those type of wording in your answers.

I think you need to consider that you should be respectful of other poster's time. You should not depend on other posters to go and find valid references for you. You should be doing it yourself.

 

[PeterDonis]

I didn't look into any particular video to talk about the Equivalence Principles, but just thought (guessed, supposed....) that the concept could fit into the naive question "where does the energy of gravity come from...." question in a Relativity Forum

What does the Equivalence Principle have to do with energy?

 

[vanhees71]

This is true as a general statement about forces and potentials. But in relativity, gravity is not a force, and formulating a valid concept of "potential energy" for gravity must be done without making use of the concept of force. (It is also worth noting that this can only be done in GR for a particular class of spacetimes, the stationary spacetimes.) Doing this correctly in GR does show that, in spacetimes where there is a valid concept of "gravitational potential energy" at all, it is well defined everywhere in the spacetime (or more precisely, everywhere in the stationary region--which might not be the entire spacetime in a case like a black hole, but that is a whole separate issue that is off topic for this thread). But the logic is different from that expressed in the above quote.

Do you have a reference (a review article or textbook?) about the status of this complicated issue. The question of the energy of the gravitational field. It lead to the famous work by Emmy Noether on symmetries and conservation laws, and this work was more or less complete in the sense that it dealt with both, "true symmetries" (global symmetries), which imply conservation laws, i.e., each one-parameter Lie group is generated by a quantity that is conserved (Noether's 1st theorem) as well as local gauge symmetries (in the case of GR it's GL(4), i.e., the general covariance), which leads to the existence of a set of arbitrary functions, which are irrelevant for the physics.

In electrodynamics the four-potential is only determined up to a gradient of a scalar field, but this doesn't matter, because the physics is indeed completely described by the four-potential modulo gauge transformations, and observables must be gauge-invariant. In GR everything is defined by $g_{\mu \nu}$ modulo arbitrary coordinate transformations (general covariance).

Concerning "gravitational field energy" the best I know is the pseudotensor a la Landau and Lifshitz, but I gueess there might be some progress in the matter since then.

Besices this, I think the gravitational interaction still is an interaction described by a local field in GR as all the other interactions. I

In GR gravitation is a true interaction and not only "inertial forces". Forces a la Newton, which is an action-at-a-distance concept, don't exist already in SR in the literal sense. My argument that gravitation in GR is a "true interaction" simply is that you can only transform away the gravitational interaction locally and approximately over spacetime regions, where the gravitational field can be considered as homogeneous. That's always approximate, because for a true gravitational field the curvature tensor is non-vanishing, and that's valid in any reference frame/choice of coordinates, i.e., you cannot get rid of the "tidal forces" also in local free-falling reference frames.

 

[PeterDonis]

I think the gravitational interaction still is an interaction described by a local field in GR

In classical GR, the only "local field" you can use is the metric itself or some tensor derived from it (such as the Riemann tensor or the Einstein tensor). But the metric doesn't describe an "interaction"--it describes the spacetime geometry. That is why it is commonly stated that gravity is not a force in classical GR.

If you view classical GR as the classical limit of the quantum field theory of a massless spin-2 field, then yes, you can view that field as a local field describing a "gravitational interaction". But you will still not be able to construct any local invariant that corresponds to the "energy" of this field. The various pseudotensors that have been used for this purpose in the literature are not local invariants; they are coordinate dependent. That is why there has never been general acceptance of any of them as a proper description of "the energy of the gravitational field" or as describing a classical "gravitational interaction".

There are "energy" concepts that are well defined in particular classes of spacetimes: the ADM energy and Bondi energy in asymptotically flat spacetimes, the Komar energy and the energy at infinity of a test object in stationary spacetimes. (Note that all but the last of these are integral quantities so they are not "local".) In the stationary case, the energy at infinity can be used to define "gravitational potential energy" as I discussed in what you quoted from me. I discussed some further issues involved in post #38 of this thread.

 

[PeterDonis]

My argument that gravitation in GR is a "true interaction" simply is that you can only transform away the gravitational interaction locally and approximately over spacetime regions, where the gravitational field can be considered as homogeneous. That's always approximate, because for a true gravitational field the curvature tensor is non-vanishing

This is not an argument for gravitation being a "true interaction". (Misner, Thorne, & Wheeler, in particular, would vigorously object to any such claim.) It is just an often necessary clarification to what aspect of "gravity" is truly invariant, i.e., independent of any choice of coordinates. That aspect is spacetime curvature, i.e., tidal gravity.

 

[Dale]

In classical GR, the only "local field" you can use is the metric itself or some tensor derived from it (such as the Riemann tensor or the Einstein tensor)

I think you mean, "the only local tensor field" or something similar. You certainly can use many non-covariant fields if you wish.

 

[vanhees71]

This is not an argument for gravitation being a "true interaction". (Misner, Thorne, & Wheeler, in particular, would vigorously object to any such claim.) It is just an often necessary clarification to what aspect of "gravity" is truly invariant, i.e., independent of any choice of coordinates. That aspect is spacetime curvature, i.e., tidal gravity.

My point is that you cannot transform away a true "gravitational field", i.e., a curved spacetime doesn't become flat by any coordinate transformation. In this sense gravity is not merely an "inertial/fictitious force". In MTW you find the expression "tidal gravitational forces" a zillion of times!

 

[PeterDonis]

I think you mean, "the only local tensor field" or something similar.\

Yes.

You certainly can use many non-covariant fields if you wish.

But doing that is open to serious objections, which are discussed in detail in, for example, Misner, Thorne & Wheeler.

 

[PeterDonis]

My point is that you cannot transform away a true "gravitational field", i.e., a curved spacetime doesn't become flat by any coordinate transformation.

Of course not.

In this sense gravity is not merely an "inertial/fictitious force".

The "gravity" that is sometimes called a "fictitious force" is not spacetime curvature. Spacetime curvature is tidal gravity.

In MTW you find the expression "tidal gravitational forces" a zillion of times!

Yes, but none of them are referring to the "force of gravity" that is sometimes called a "fictitious force". They are referring to the tidal stretching and squeezing of objects, for example as the singularity of a black hole is approached. When you look closely at such cases, the actual forces involved are the internal forces between the parts of the object (which are electromagnetic) that try to resist the tidal effects, and ultimately fail.Objects moving solely under the influence of tidal gravity, with no non-gravitational interactions present, feel no force and are in free fall.

 

[vanhees71]

Ok, that's again a semantic discussion. The tidal gravity is what I meant when I said that there is a "true interaction".

 

[PeterDonis]

The tidal gravity is what I meant when I said that there is a "true interaction".

The issue is whether "true interaction" is an appropriate term for that. I have given reasons why it isn't, and referred to a classic textbook that makes that same case in great detail.

 

[vanhees71]

Where in this textbook?

 

[PeterDonis]

Where in this textbook?

Pretty much everywhere. MTW is well known as a classic exposition of the "gravity is not a force" geometric viewpoint, as well as the viewpoint that all physics is contained in invariant or covariant objects (which does not include the various "gravitational energy" pseudotensors). The fact that they often happen to use the expression "tidal forces" does not contradict this, for reasons I have already explained in post #51.

 

[vanhees71]

But in #51 you admitted that there is "tidal stretching and squeezing of objects". So there is an interaction. It's of course not a force. There are no forces in relativistic physics, only local field descriptions, but that's also semantics...

 

[PeterDonis]

in #51 you admitted that there is "tidal stretching and squeezing of objects".

I stated that, yes. But such objects, in the absence of non-gravitational interactions, feel no force due to tidal gravity and are in free fall. Just like any other objects that are moving solely under the influence of the spacetime geometry, with no non-gravitational interactions present.

So there is an interaction

If an object subjected to tidal gravity feels a force (or an interaction, if you prefer), the interaction it feels is non-gravitational--internal non-gravitational interactions between the parts of the object that, as I said, are trying to resist the effects of tidal gravity. The stretching or squeezing is what is left over after those non-gravitational interactions have done as much as they can. The tidal gravity itself is not felt at all. Which, once again, is a key reason why gravity is not considered to be a force or an interaction in classical GR.

 

[vanhees71]

Of course, a free point particle moves along a geodesic, and you can say that "free fall" is the analogue to force-free motion in Newtonian and special relativistic physics, as long as you only look locally. Already a cloud of "dust" may collapse under its own gravity, as you've just nicely described in your recent Insights article. I'd call this an interaction, but I can live with just saying "gravity".

 

[Demystifier]

In all current physical theories, energy comes from initial conditions. Where do energy conditions come from, that's a question that current theories don't answer.

 

[haushofer]

In classical GR, the only "local field" you can use is the metric itself or some tensor derived from it (such as the Riemann tensor or the Einstein tensor). But the metric doesn't describe an "interaction"--it describes the spacetime geometry. That is why it is commonly stated that gravity is not a force in classical GR.

In classical GR. But classical GR is equivalent to classical Fierz-Pauli theory, i.e. an interacting spin-2 theory on flat spacetime. So in that sense the metric field does introduce an interaction.

 

[PeterDonis]

But classical GR is equivalent to classical Fierz-Pauli theory, i.e. an interacting spin-2 theory on flat spacetime.

As long as the global topology of spacetime is $\mathbb{R}^4$. Otherwise the assumption of a flat background spacetime won't work.

 

[member 728827]

What does the Equivalence Principle have to do with energy?

I don't know, is just this guy that at least at the beginnings thought than has something to do with it.

 

[PeterDonis]

I don't know, is just this guy that at least at the beginnings thought than has something to do with it.

This is an early paper on what would eventually become General Relativity. Many of the things in it were later found not to work. If you are studying the history of how Einstein discovered GR, these papers are very interesting and useful. But they can be worse than useless if you try to use them to understand the physics. You should be looking at the latest modern treatments, not historical attempts that have long since been superseded.

 

[Wes Tausend]

"Where does the energy of gravity come from?"
If Einstein is at all correct, and I believe he is, the energy of gravity simply comes from e=mC², as do all other energies, including all accelerations, the Big Bang, Dark Energy etc. --wes

 

[PeterDonis]

If Einstein is at all correct, and I believe he is, the energy of gravity simply comes from e=mC²

No, that is not correct.

as do all other energies, including all accelerations, the Big Bang, Dark Energy etc. --wes

That is not correct either.

 

[Wes Tausend]

In my 1960's physics class, our book stated something to the effect that, in our universe, "The total energy equaled the the total mass times the speed of light squared and that energy and the mass were constant & interchangeable throughout the universe." I tend to believe this, following along the lines of https://en.wikipedia.org/wiki/Zero-energy_universe. The whole thing seems simple enough to me.

Peter stated that my post was not correct with no reason given. I am interested what his theory is, or that of another theory is, that accounts for his apparent claim that Einstein's theory is not correct in the above respect.

In dlgoff's post, the wiki link given includes the phrase , "In general relativity gravitational energy is extremely complex, and there is no single agreed upon definition of the concept." This line is offered at the beginning of the General Relativity section (https://en.wikipedia.org/wiki/Gravitational_energy#General_relativity). In that, the given link is not so helpful other than it is admittedly not well understood. -wes

 

[jbriggs444]

In my 1960's physics class, our book stated something to the effect

That is not an acceptable reference in these forums.

 

[PeterDonis]

In my 1960's physics class

We've learned a lot about this subject since the 1960s.

our book

What book?

I tend to believe this, following along the lines of https://en.wikipedia.org/wiki/Zero-energy_universe.

Wikipedia is not a valid reference. But even leaving that aside, that article does not say what you were saying in the post that I said was wrong.

In @dlgoff's post

I'll respond to that separately.

 

[haushofer]

As long as the global topology of spacetime is $\mathbb{R}^4$. Otherwise the assumption of a flat background spacetime won't work.

That's a good point, but does the interpretation of gravity as an interaction then depend on the global topology of spacetime?

E.g., how is that done in string theory? There you quantize a string on a flat background, and after a lot of calculations (conformal invariance etc.) you show that there are gravitons as vibrational modes obeying the Einstein equations (plus corrections). Would you call gravity an interaction in string theory, or only within certain topologies? To me that sounds a bit overcomplicated.

To be honest, I've never understood the problem here, but that could be my deficit understanding. You start out in Fierz Pauli theory on, say, Minkowski spacetime, use gauge invariance to include non-linear terms, and in the end you end up with the full Einstein equations, which show that you could have started from every background you wanted as long as it is a solution to the obtained field equations. I.e., to me that shows how one can obtain a backghround independent theory by starting from a background dependent theory, and I've used this argument sometimes when string theory critics complained that string theory is not background independent.

 

[vanhees71]

Gravitation can be described as gauging the Poincare symmetry of SRT. In this sense it's an interaction as all the other (known) ones. Of course, you can also reinterpret it in the sense of geometrodynamics. I'd not make a religion of either point of view.