Binding Mass Distribution in Gravitating Bodies

[zonde]

For bound system to form it must radiate away binding energy. By mass energy equivalence mass of the bound system is reduced proportionally to amount of binding energy.
The question is from which mass this "binding mass" is subtracted. Common answer that I have seen is that this binding mass is subtracted from system as a whole and it does not make sense to ask from which parts of the system this binding mass is taken away.

Now as I see this answer leads to hidden assumption if we consider internal state of gravitating system. Say Oppenheimer-Volkoff limit is derived assuming homogeneous mass distribution within gravitating body. But that assumes that this negative binding mass is distributed homogeneously as well. But then particles deep inside the body are deeper in gravitational well than particles on the surface. And it does not seem reasonable at all that binding mass is distributed homogeneously as you have to use more energy to take to infinity particle from the middle of the body than for particle from the surface of the body.

So it seems to me that the question from which parts of the system this binding mass is taken away is very sensible. What do you think?

 

[PeterDonis]

By mass energy equivalence mass of the bound system is reduced proportionally to amount of binding energy.

More precisely, the externally measured mass of the system is reduced.

The question is from which mass this "binding mass" is subtracted

It isn't. The externally measured mass is not reduced because some mass got removed. It is reduced because the spacetime geometry inside the body changes as it gets more tightly bound, and that change reduces the externally measured mass of the body, without changing the locally measured mass of any of its parts. (We're assuming that the binding energy is carried away by radiation, not by ejecting matter.)

 

[Mister T]

So it seems to me that the question from which parts of the system this binding mass is taken away is very sensible. What do you think?

It's not a sensible question. The question is based on the misconception that the mass of a composite body equals the sum of the masses of the constituents. This issue addresses the core of the meaning of the Einstein mass-energy equivalence.

 

[PeterDonis]

Say Oppenheimer-Volkoff limit is derived assuming homogeneous mass distribution within gravitating body

It isn't. The maximum mass limit for a gravitating body is not derived assuming a homogeneous mass distribution.

that assumes that this negative binding mass is distributed homogeneously as well

The "negative binding mass" is not a local property; it's a global property. There is no way to localize where any particular amount of "negative binding mass" is taken away. This is a consequence of the fact that "energy stored in the gravitational field" is not well defined in GR and does not have any representation as a tensor. (There are various possible pseudo-tensors that some physicists interpret as "energy stored in the gravitational field", but as the term
"pseudo-tensor" indicates, these aren't actual tensors and don't correspond to any localizable quantity.)

 

[PAllen]

There is way you can play the game 'where is binding energy released from?' as opposed to where is it located. Given a collapsing cloud, for the collapse to proceed, radiation is released, decreasing kinetic energy of constituents. Each radiative interaction can be be given a location. But this really doesn't equate to where the 'missing mass' of binding energy is.

 

[SiennaTheGr8]

...
(There are various possible pseudo-tensors that some physicists interpret as "energy stored in the gravitational field", but as the term "pseudo-tensor" indicates, these aren't actual tensors and don't correspond to any localizable quantity.)

When you say "pseudo-tensor" here, do you mean it in the technical sense of a tensor that changes sign under certain coordinate transformations? Or do you mean it in a loose sense (i.e., not actually a tensor at all)?

 

[PAllen]

When you say "pseudo-tensor" here, do you mean it in the technical sense of a tensor that changes sign under certain coordinate transformations? Or do you mean it in a loose sense (i.e., not actually a tensor at all)?

They are not tenors at all. A good presentation of the approach is in Landau and Lifschitz “Classical Theory of Fields”.

 

[PeterDonis]

When you say "pseudo-tensor" here, do you mean it in the technical sense of a tensor that changes sign under certain coordinate transformations? Or do you mean it in a loose sense (i.e., not actually a tensor at all)?

The pseudo-tensors I refer to are not tensors at all.

 

[Heikki Tuuri]

The Einstein-Hilbert action of a static system is essentially an integral over a slice of spacetime of the mass-energy L_M in the system plus the Ricci curvature R:

1 / (2 kappa) * R + L_M

If the spacetime outside the system is asymptotically Minkowski, we can take the slice according to the global Minkowski time coordinate, and extend the slice in some natural way to the system.

Time flows slower in a low gravitational potential, and the integral over L_M is reduced, because in the slice we integrate over a shorter time interval (t_0, t_1) in such an area. But R is positive there and makes the integral bigger. The system settles in the state where the integral is at the minimum.

From where do we take the binding energy of the system? If we interpret the integral as some kind of energy, then each atom in a low potential will contribute less because time flows slower there (equivalently, there is a redshift if that atom sends a photon to distant space). Let us denote by E the energy freed. We can break down E to each atom in a natural way.

The integral over R we can interpret as the energy of spacetime deformation. Some of the energy freed from atoms goes to the deformation. The deformation is a collective process. We cannot say which atom is responsible for which part of R. Also, we cannot say where the energy of R is exactly located. Let us denote by D the energy of spacetime deformation.

The binding energy is E - D. We cannot locate D, and D is a collective phenomenon. We cannot say "from which part" did each joule of the binding energy come from.

An analogous system is a slippery rubber sheet on which we put coins as weights. The coins do not overlap. The coins tend to slide together, so that they can hang lower. From where do we take the "binding energy" in this system? E is the potential energy lost by coins. There is a natural way to locate E.

The energy D is the rubber deformation energy. In rubber, we CAN locate where the deformation energy is stored. We can cut the sheet in small pieces and measure the deformation energy of each piece. The deformation energy is spread under the coins and also in their neighborhood.

I am currently studying this discrepancy between the rubber model and general relativity. What are its implications?

 

[zonde]

More precisely, the externally measured mass of the system is reduced.

Right

It isn't. The externally measured mass is not reduced because some mass got removed. It is reduced because the spacetime geometry inside the body changes as it gets more tightly bound, and that change reduces the externally measured mass of the body, without changing the locally measured mass of any of its parts. (We're assuming that the binding energy is carried away by radiation, not by ejecting matter.)

Well, there can be more than one "because" that are consistent with each other. Obviously there are several parts, one is radiation escaping the body (which can be observed externally), other one is reduction of externally observable mass and then there is the model of inside of the body and related changes in configuration. The first two parts provide a motivation to look for third part.

As I understand your take on explanation is that changes in space time geometry is all that is needed to explain reduced external mass. This explanation can not be tested (at least practically) by observation, however it can be tested by checking it's consistency. I will keep it in mind and will try to see if some inconsistency does not creep up.

It isn't. The maximum mass limit for a gravitating body is not derived assuming a homogeneous mass distribution.

You are right. My error.

The "negative binding mass" is not a local property; it's a global property. There is no way to localize where any particular amount of "negative binding mass" is taken away. This is a consequence of the fact that "energy stored in the gravitational field" is not well defined in GR and does not have any representation as a tensor. (There are various possible pseudo-tensors that some physicists interpret as "energy stored in the gravitational field", but as the term "pseudo-tensor" indicates, these aren't actual tensors and don't correspond to any localizable quantity.)

The claim that "negative binding mass" can not be localized follows from your explanation. But I'm quite skeptical about this.
Let's say some nuclear reaction takes place within gravitating body and it releases energy in the form of radiation. Let's say this radiation escapes to "infinity". In this case "negative binding mass" can be clearly localized. On the other hand the changes in space-time geometry as radiation travels from place of nuclear reaction to infinity should be no different than the changes in space-time geometry as radiation travels from different source.

And then there is redshift of radiation as it escapes the gravity well. But again there is no difference if radiation comes from nuclear reaction or it is thermal radiation that come from kinetic energy of massive particles.

 

[zonde]

There is way you can play the game 'where is binding energy released from?' as opposed to where is it located. Given a collapsing cloud, for the collapse to proceed, radiation is released, decreasing kinetic energy of constituents. Each radiative interaction can be be given a location. But this really doesn't equate to where the 'missing mass' of binding energy is.

Well, I would say that usually one would take trivial explanation if there is one and look for more complicated one when the trivial explanation fails.
As I see in this case trivial assumption is that "missing mass" comes from location of radiative interaction. Do you say it breaks down? Or are you simply careful about such trivial assumption?

 

[PAllen]

Well, I would say that usually one would take trivial explanation if there is one and look for more complicated one when the trivial explanation fails.
As I see in this case trivial assumption is that "missing mass" comes from location of radiative interaction. Do you say it breaks down? Or are you simply careful about such trivial assumption?

I meant my comment to be a little bit tongue in cheek. If you think about, it would have the location of binding energy depend on random history, and most of it would be located outside the final boundary of the collapsed body. However, it is true that for an isolated body in asymptotically flat spacetime, all of final binding energy of body formed from collapse from an infinitely diffuse state will correspond to emitted radiation (but you must include gravitational as well as EM, and for the latter, there is no unambiguous emission location in the first place). This follows from the formulation of Bondi mass, which is a well accepted, invariant approach - but as a result, does not answer the location question.

There are two respected research programs that try provide some meaningful answer to location of gravitational energy in GR. Neither is generally considered to be fully successful, but some find them useful.

One is ongoing work on stress energy pseudotensors. On this score, there is one highly respected researcher who has pursued the idea that to avoid the ambiguities and odd features of the pseudotensor in arbitrary coordinates, you should accept that this nontensor object should be interpreted only in special coordinates, specifically, harmonic coordinates. Google Nakanishi + pseudotensor to find links for this program.

The other approach is to define tensorial objects that have most but not all of the properties expected for a localized energy momentum (it is proven that you cannot formulate an object that has all expected properties). This is the quasilocal energy momentum program. One way into this, expressing also my bias as to the most successful part of this effort, is to google Bartnik mass.

 

[PeterDonis]

The Einstein-Hilbert action of a static system

Or any system. The E-H action applies in any spacetime, not just a static spacetime.

 

[PeterDonis]

Obviously there are several parts, one is radiation escaping the body (which can be observed externally), other one is reduction of externally observable mass

These two have to be the same.

changes in space time geometry is all that is needed to explain reduced external mass

Changes in spacetime geometry due to the system emitting radiation and becoming more tightly bound. The spacetime geometry can't just change out of nowhere. There has to be a process that carries away energy. We are using radiation in this example as that process because radiation does not carry away any rest mass.

Let's say some nuclear reaction takes place within gravitating body and it releases energy in the form of radiation. Let's say this radiation escapes to "infinity". In this case "negative binding mass" can be clearly localized.

No, it can't, because while the radiation carries stress-energy, the change in spacetime geometry does not. There is no invariant quantity that corresponds to "the change in negative binding energy due to the change in spacetime geometry". And you can't localize something that doesn't correspond to any invariant quantity.