Does Komar Mass Include the Energy of Gravitational Field?

[wLw]

Does Komar mass include the energy of gravitational filed or just only the energy of matter. For instants, we calculate the Korma mass of Schwarzschild space-time and we get the result that is M, which is the mass of central star. Does that mean Komar mass only contain the energy of matter not contain the energy of gravitational filed?

 

[PeterDonis]

Hi @wLw and welcome to PF!

Does Komar mass include the energy of gravitational filed or just only the energy of matter.

It depends on what you mean by "energy of the gravitational field". When you evaluate the Komar mass as a volume integral, it contains a correction for the "redshift factor" due to the metric, which can be viewed as a correction for "energy of the gravitational field". But this "energy" is actually negative--it makes the Komar mass smaller than it would be if you evaluated the integral without the correction for the redshift factor.

we calculate the Korma mass of Schwarzschild space-time and we get the result that is M, which is the mass of central star

Yes, but this mass is smaller than the sum of the masses of the individual constituents of the star, as they would be if the constituents were all very widely separated instead of bound into a single star. This difference is due to the "redshift factor" correction described above. The usual interpretation of this difference is gravitational binding energy: the energy it would take to separate the star into its individual constituent masses, all very widely separated from each other.

 

[wLw]

but if you calculate the Komar mass of RN space-time, it is also M. while i think the whole mass(energy) should include Q(electrical fields contributes ). By the way, do you mean the M in Schw. space-time is not the central star's mass, but the whole space-time mass, which will be bigger than whole space-time mass(energy) due to negative gravitational energy?

 

[PeterDonis]

if you calculate the Komar mass of RN space-time, it is also M. while i think the whole mass(energy) should include Q(electrical fields contributes ).

The mass $M$ of RN spacetime does include a contribution from the energy stored in the static electric field. When you do the Komar mass volume integral the stress-energy tensor will include that energy.

do you mean the M in Schw. space-time is not the central star's mass, but the whole space-time mass

What's the difference?

which will be bigger than whole space-time mass(energy) due to negative gravitational energy?

I already explained how the "negative gravitational energy" contributes to the Komar mass.

I strongly suggest that you look at the actual math. You marked this thread as "A" level so you should be familiar with it. Doing the actual integrals will give a much better understanding of how things work.

 

[wLw]

in schwa. space-time, we can see in its metric : M is described as the mass of central body. but if you calculate the komar mass of schwa. space-time , it is also M. how can they be the same?

 

[PeterDonis]

in schwa. space-time, we can see in its metric : M is described as the mass of central body. but if you calculate the komar mass of schwa. space-time , it is also M. how can they be the same?

Do the integral and see.

 

[pervect]

To anticipate the answer, the Komar mass is the same as the Schwarzschild parameter M. I believe the OP is confused about why this is the case. Unfortunately, I can't quite understand why the OP thinks the two mases should be different.

So the short version is that if one thinks the two masses "should be" different, because it turns out they are the same, one is confused about something. But it's not clear what that "something" might be without more details about why one thought they "should be" different.

 

[wLw]

Do the integral and see.

yeah，i have calculated that, and the result is M. but i am confused that why the komar mass is equal to central body(matter) mass. by definition, komar mass is the total energy(mass) of whole space-time. they should not be equal.

 

[PeterDonis]

yeah，i have calculated that, and the result is M. but i am confused that why the komar mass is equal to central body(matter) mass

Well, you say you've done the integral that gives you that answer. Is there something about the integral that you don't understand?

by definition, komar mass is the total energy(mass) of whole space-time. they should not be equal.

Why not? As I asked before, what is the difference between "the mass of the body" and "the total mass of the whole spacetime"? (Hint: there actually is no difference. But you appear to think there is, so I am asking you to explain why you think that.)

 

[wLw]

Well, you say you've done the integral that gives you that answer. Is there something about the integral that you don't understand?
Why not? As I asked before, what is the difference between "the mass of the body" and "the total mass of the whole spacetime"? (Hint: there actually is no difference. But you appear to think there is, so I am asking you to explain why you think that.)

the energy of whole space-time should include two parts: the energy of matter(here is the central body M); the energy of field (only gravitational filed in this case), so i think the whole space- time energy is not equal to matter energy M. is I right?

 

[PeterDonis]

the energy of whole space-time should include two parts: the energy of matter(here is the central body M); the energy of field (only gravitational filed in this case), so i think the whole space- time energy is not equal to matter energy M. is I right?

No, you are not right, because the mass of the central body also includes both parts: "the energy of matter" and "the energy of field". The mass of the central body includes a (negative) contribution from gravitational binding energy, which is "the energy of field". I have already explained how this works in previous posts, and it is all there in the Komar mass integral.

 

[wLw]

No, you are not right, because the mass of the central body also includes both parts: "the energy of matter" and "the energy of field". The mass of the central body includes a (negative) contribution from gravitational binding energy, which is "the energy of field". I have already explained how this works in previous posts, and it is all there in the Komar mass integral.

1.But i calculated the Komar mass of R -N space-time and the result is still M, which is the mass of central body, but it is known that central body also has electric charges , so the whole energy of R-N space-time should include something with parameter Q, which is the energy of electronic filed. what is wrong on this .?
2.By the way,according to your idea, the mass in GR is very different from Newton's . for instance, we can see a 100kg matter in Newton but in GR it makes no sense because if i say 100kg matter in GR, which must contain the gravitational filed contributes so the actual mass of that matter (100kg in Newton) is bigger than 100kg due to negative energy of gravitational filed. I can not get my head around it.
3. it is known that the komar mass integral are independent with choice of spherical surface that contains all matter. However, it will cause a problem :if i choose a small surface (contains all matter) and an infinite surface which contains the whole space, but i calculated and the results are both M, so there is a question: now that komar mass is the whole energy within surface, and there is gravitational filed between two surface. where is the energy of that?

 

[PeterDonis]

calculated the Komar mass of R -N space-time and the result is still M, which is the mass of central body, but it is known that central body also has electric charges , so the whole energy of R-N space-time should include something with parameter Q, which is the energy of electronic filed. what is wrong on this .?

I'm sorry, but just continuing to repeat the same questions when I've already responded to them isn't going to get anywhere. As I've repeatedly asked you: look at the actual integrals. Feel free to post them here and ask questions about them if there's something about them you don't understand. But there's no point in waving your hands with intuitive guesses when you say you've done the actual math. The physics is in the actual math.

according to your idea, the mass in GR is very different from Newton's

Yes, that's true.

we can see a 100kg matter in Newton but in GR it makes no sense

Huh? An object having a mass of 100 kg makes no sense in GR? Why not?

if i say 100kg matter in GR, which must contain the gravitational filed contributes so the actual mass of that matter (100kg in Newton) is bigger than 100kg due to negative energy of gravitational filed

No, the mass of the 100 kg object is 100 kg. But, if you took that 100 kg and dissasembled it into its constituent atoms, and separated all the atoms so that they were widely separated from each other, you would have to add energy to the system, so the system consisting of all those atoms widely separated from each other would have a mass greater than 100 kg. The difference is the gravitational binding energy.

You can think of this, if you insist, as the 100 kg object consisting of a "mass of matter" greater than 100 kg, and a negative "mass of gravitational field" that brings the total mass of the object back down to 100 kg. But it seems like that way of thinking of it is confusing you, so my advice would be to stop thinking of it that way. There is no requirement to think of it that way in GR; it's just a heuristic that helps some people but not others.

now that komar mass is the whole energy within surface, and there is gravitational filed between two surface. where is the energy of that?

There is no "energy of the gravitational field" between the two surfaces. What you are calling the "energy of the gravitational field" is not a thing that is contained in vacuum regions of space. It's just an abstract number that you can obtain if you insist on thinking about the mass of a bound object a certain way. But as I said above, if that way of thinking about it is confusing you, the thing to do is to stop thinking of it that way. Nothing requires you to think of it that way in GR.

 

[pervect]

the energy of whole space-time should include two parts: the energy of matter(here is the central body M); the energy of field (only gravitational filed in this case), so i think the whole space- time energy is not equal to matter energy M. is I right?

That much is mostly right, modulo some sticky questions about how well defined "the energy of the gravitational field" is.

Sometimes mostly right is more dangerous than wrong, by the way. However, I won't go into much more detail on this point at the moment - perhaps a little bit at the end. There's something else to resolve first.

The Schwarzschild mass does not claim to be "the energy of the matter, not including field energy". You haven't said explicitly that you think it does, but the only way your remarks make sense to me is if I assume that is what you were trying to say, even though you haven't said it explicitly. The other alternative I see would be if you believed that the Komar mass represented "the energy of the matter, divorced from the gravitational field". But I don't think you are claiming that. Perhaps you are claiming something other than one of these two possibilities, but if so I really don't follow.

At a guess, you think that the Schwarzschild mass represent the "energy of the matter", because you multiply 4/3 pi r^3 by rho.

Unfortunately, 4/3 pi r^3 does not represent the volume element.

To define the volume element in the sense we are talking about, the 3-volume, or spatial volume, one needs to split space-time into space + time. Rather than belabor this point, we will assume that we use the Schwarzschild time coordinate "t" to do this split. Exploring what happens if we don't do this will raise other, important concerns that I alluded to earlier. But for now, let us assume that we are defining "volume" using the space-time split in which the Schwarzschild t coordinate represent time.

Then the 3-volume element is the square root of the determinant of the spatial metric, which in this case is just the space-time metric with the time component removed. More formally, we might talk about using a projection operator to project the 4-d line element into a 3-d line element.

So in spherical coordinates in flat space-time, if we assume the line element is
$$dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$$

the volume element is $\sqrt{|g|} dr \, d\theta \, d\phi = r^2 \sin \theta \, dr \, d\theta \, d\phi$

However, in Schwarzschild coordinate,s the line element is
$$\frac{dr^2}{{1-2M/r}} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$$

(with G=c=1).

which gives a different volume element from the case of spherical coordinates in flat space-time due to a multiplicative factor if $1/\sqrt{1-2M/r}$. The net result is that the integral of the energy density over the correct volume element is different (larger) than the Schwarzschild mass M. So the Schwarzschild mass M is not equal to "the energy of matter", it is lower.

There is a section in the textbook MTW that goes through this calculation in more detail, that I could perhaps dig up if prodded.

 

[wLw]

No, the mass of the 100 kg object is 100 kg. But, if you took that 100 kg and dissasembled it into its constituent atoms, and separated all the atoms so that they were widely separated from each other, you would have to add energy to the system, so the system consisting of all those atoms widely separated from each other would have a mass greater than 100 kg. The difference is the gravitational binding energy.

You can think of this, if you insist, as the 100 kg object consisting of a "mass of matter" greater than 100 kg, and a negative "mass of gravitational field" that brings the total mass of the object back down to 100 kg. But it seems like that way of thinking of it is confusing you, so my advice would be to stop thinking of it that way. There is no requirement to think of it that way in GR; it's just a heuristic that helps some people but not others.

How about the gravitational field out of body, that energy should be accounted when you calculate energy of system.

 

[wLw]

I'm sorry, but just continuing to repeat the same questions when I've already responded to them isn't going to get anywhere. As I've repeatedly asked you: look at the actual integrals. Feel free to post them here and ask questions about them if there's something about them you don't understand. But there's no point in waving your hands with intuitive guesses when you say you've done the actual math. The physics is in the actual math.
Yes, that's true.
Huh? An object having a mass of 100 kg makes no sense in GR? Why not?
No, the mass of the 100 kg object is 100 kg. But, if you took that 100 kg and dissasembled it into its constituent atoms, and separated all the atoms so that they were widely separated from each other, you would have to add energy to the system, so the system consisting of all those atoms widely separated from each other would have a mass greater than 100 kg. The difference is the gravitational binding energy.

You can think of this, if you insist, as the 100 kg object consisting of a "mass of matter" greater than 100 kg, and a negative "mass of gravitational field" that brings the total mass of the object back down to 100 kg. But it seems like that way of thinking of it is confusing you, so my advice would be to stop thinking of it that way. There is no requirement to think of it that way in GR; it's just a heuristic that helps some people but not others.
There is no "energy of the gravitational field" between the two surfaces. What you are calling the "energy of the gravitational field" is not a thing that is contained in vacuum regions of space. It's just an abstract number that you can obtain if you insist on thinking about the mass of a bound object a certain way. But as I said above, if that way of thinking about it is confusing you, the thing to do is to stop thinking of it that way. Nothing requires you to think of it that way in GR.

so, do you mean that region has no energy?, but there is gravitational filed there, i think gravitational should carry energy. what is wrong??

 

[wLw]

That much is mostly right, modulo some sticky questions about how well defined "the energy of the gravitational field" is.

Sometimes mostly right is more dangerous than wrong, by the way. However, I won't go into much more detail on this point at the moment - perhaps a little bit at the end. There's something else to resolve first.

The Schwarzschild mass does not claim to be "the energy of the matter, not including field energy". You haven't said explicitly that you think it does, but the only way your remarks make sense to me is if I assume that is what you were trying to say, even though you haven't said it explicitly. The other alternative I see would be if you believed that the Komar mass represented "the energy of the matter, divorced from the gravitational field". But I don't think you are claiming that. Perhaps you are claiming something other than one of these two possibilities, but if so I really don't follow.

At a guess, you think that the Schwarzschild mass represent the "energy of the matter", because you multiply 4/3 pi r^3 by rho.

Unfortunately, 4/3 pi r^3 does not represent the volume element.

To define the volume element in the sense we are talking about, the 3-volume, or spatial volume, one needs to split space-time into space + time. Rather than belabor this point, we will assume that we use the Schwarzschild time coordinate "t" to do this split. Exploring what happens if we don't do this will raise other, important concerns that I alluded to earlier. But for now, let us assume that we are defining "volume" using the space-time split in which the Schwarzschild t coordinate represent time.

Then the 3-volume element is the square root of the determinant of the spatial metric, which in this case is just the space-time metric with the time component removed. More formally, we might talk about using a projection operator to project the 4-d line element into a 3-d line element.

So in spherical coordinates in flat space-time, if we assume the line element is
$$dr^2 + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$$

the volume element is $\sqrt{|g|} dr \, d\theta \, d\phi = r^2 \sin \theta \, dr \, d\theta \, d\phi$

However, in Schwarzschild coordinate,s the line element is
$$\frac{dr^2}{{1-2M/r}} + r^2 d\theta^2 + r^2 \sin^2 \theta d\phi^2$$

(with G=c=1).

which gives a different volume element from the case of spherical coordinates in flat space-time due to a multiplicative factor if $1/\sqrt{1-2M/r}$. The net result is that the integral of the energy density over the correct volume element is different (larger) than the Schwarzschild mass M. So the Schwarzschild mass M is not equal to "the energy of matter", it is lower.

There is a section in the textbook MTW that goes through this calculation in more detail, that I could perhaps dig up if prodded.

thanks for you. so you mean the different volume element of different space-time causes the different mass and the central body ?. by the way, does komar mass includes both matter energy and gravitational filed energy, or just the matter mass?

 

[PeterDonis]

i think gravitational should carry energy

You are wrong. It doesn't, at least not in the sense you are using the term. There is zero stress-energy in any vacuum region.

does komar mass includes both matter energy and gravitational filed energy, or just the matter mass?

I already answered this question way back in post #2.

 

[wLw]

i think i should spend some time to think it over and to make my mind clear, i am now chaotic . I will ask later Thanks for your all comments and your patience.

 

[PeterDonis]

That much is mostly right, modulo some sticky questions about how well defined "the energy of the gravitational field" is.

I don't think the OP is "mostly right". You have left out some key items in discussing the Komar mass integral.

First, the OP talks about Schwarzschild spacetime but also mentions a "central star". So we need to clarify exactly what spacetime we are talking about. (This is one reason why I have repeatedly asked the OP to write down an actual integral instead of waving his hands.) The easier case to deal with (and the one I think the OP is referring to, based on what he has said in various posts in this thread) is the case of a spherically symmetric massive body surrounded by vacuum. This is the case I will discuss in this post (I'll save the more difficult case of maximally extended Schwarzschild spacetime, where there is vacuum everywhere, in a follow-up post).

The relevant form of the Komar mass integral as a volume integral for this case is

$$
M = \int_0^{\infty} \left( \rho + 3 p \right) \sqrt{g_{rr} g_{tt}} 4 \pi r^2 dr
$$

For details on how this is derived, see my Insights article here:

https://www.physicsforums.com/insights/gravity-gravitate-part-2-sequel/
The factor $\rho + 3 p$ is what you get when you evaluate the factor $\left( 2 T_{ab} - g_{ab} T \right) u^a u^b$ for a spherically symmetric, static perfect fluid.

Note that there are two correction factors in the integral, $\sqrt{g_{rr}}$ and $\sqrt{g_{tt}}$. The former is the one you discussed, the "volume element" correction. But you did not mention a critical point: you need $g_{rr}$ in the matter region, whose geometry is not the Schwarzschild vacuum geometry, so the value of $g_{rr}$ in this region will not be the one you gave. In the vacuum region, where $g_{rr}$ has the Schwarzschild vacuum value you gave, $\rho$ and $p$ are zero (vacuum, no stress-energy), so this region makes zero contribution to the volume integral.

The second correction factor is the "redshift factor" correction that I have mentioned in previous posts. The net effect of the combined corrections is to give a factor that reduces the total value of the integral just enough to compensate for the addition of the $3p$ factor in the integrand. In other words, the actual value of the integral is identical to the value of the following integral:

$$
M = \int_0^{\infty} 4 \pi \rho r^2 dr
$$

which, as a matter of fact, can be derived directly from the $tt$ component of the Einstein Field Equation, without ever having to worry about either pressure (note that only $\rho$ appears in the above integral) or any correction factors for the volume element or the redshift. Some sources refer to this phenomenon as pressure just balancing gravitational binding energy in hydrostatic equilibrium.

the Schwarzschild mass M is not equal to "the energy of matter", it is lower.

To state this more precisely, the mass $M$ obtained from the above integrals is smaller than the following:

$$
\bar{M} = \int_0^{\infty} 4 \pi \rho r^2 \sqrt{g_{rr}} dr
$$

which is, intuitively, what you get by trying to integrate the density of the matter over the actual proper volume of the matter. However, this integral does not represent anything physical, because the reason why the proper volume of the matter is larger than the Euclidean value is that the matter is in a bound system in static equilibrium, and there is no way to put the matter in a bound system and have it be in static equilibrium and have only the $\sqrt{g_{rr}}$ correction factor. You have to also include the $\sqrt{g_{tt}}$ correction factor for redshift, and you have to include pressure, because the object has to have pressure to support itself against its own gravity, and pressure is part of the stress-energy tensor so it is a source of gravity and you can't just leave it out.

 

[PeterDonis]

This follow-up is to discuss the more difficult case I referred to in my last post, of the Komar mass of maximally extended Schwarzschild spacetime. As I think has been discussed in previous PF threads, there is no well-defined way to evaluate this as a volume integral, because there is no well-defined "spatial volume" with the required properties: the geometry of a spacelike slice orthogonal to the timelike Killing vector field (which is what you need to evaluate the Komar mass as a volume integral) is not $R^3$, it's $S^2 \times R$.

It is possible to evaluate the Komar mass as a surface integral, and doing that for maximally extended Schwarzschild spacetime gives the Schwarzschild mass $M$. However, this way of doing the integral does not have a physical interpretation as "integrating over the mass of the matter", so it isn't relevant to the discussion in this thread.

 

[wLw]

I don't think the OP is "mostly right". You have left out some key items in discussing the Komar mass integral.

what does OP mean?

 

[PeterDonis]

what does OP mean?

Original Poster; the person whose post started the thread (or sometimes that first post itself). In this case, you.

 

[wLw]

To anticipate the answer, the Komar mass is the same as the Schwarzschild parameter M. I believe the OP is confused about why this is the case. Unfortunately, I can't quite understand why the OP thinks the two mases should be different.

So the short version is that if one thinks the two masses "should be" different, because it turns out they are the same, one is confused about something. But it's not clear what that "something" might be without more details about why one thought they "should be" different.

Because by definition, komar mass is the whole energy of a system in GR, which includes the energy of matter and the energy of gravitational field . so there are different, but the calculating shows the same result :M, so that is what i am confused

 

[PeterDonis]

the calculating shows the same result :M

Calculating the Komar mass gives $M$. But you have not shown any calculation of "the energy of matter" which gives a different result. In fact you have shown no calculations at all, despite my repeated requests. Can you show a calculation of "the energy of matter" which gives a different result from $M$?

 

[wLw]

Calculating the Komar mass gives $M$. But you have not shown any calculation of "the energy of matter" which gives a different result. In fact you have shown no calculations at all, despite my repeated requests. Can you show a calculation of "the energy of matter" which gives a different result from $M$?

show what? my confusion is why they are equal. because komar mas is the total energy of a system. and $$E_{\mathrm{total}} = E_{\mathrm{matter}}+E_{\mathrm{field}}$$ And by definition,komar mass are defined as :$$M_{\mathrm{K}} =-\frac{1}{8 \pi} \int_{S} \varepsilon_{a b c d} \nabla^{c} \xi^{d}$$, which is total energy but M is central body's mass,which is matter,not field . they should be different. but as you said, "As I asked before, what is the difference between "the mass of the body" and "the total mass of the whole spacetime"? (Hint: there actually is no difference.) ", i think you discuss a different thing from mine.

 

[wLw]

Calculating the Komar mass gives $M$. But you have not shown any calculation of "the energy of matter" which gives a different result. In fact you have shown no calculations at all, despite my repeated requests. Can you show a calculation of "the energy of matter" which gives a different result from $M$?

i am clearer than before now. the M in Schw. space has already includes the energy of matter and the bind gravity energy and komar mass is the whole energy so they are same. but as you said "You are wrong. It doesn't, at least not in the sense you are using the term. There is zero stress-energy in any vacuum region. "why that part of gravitational field has no energy. it is known that that the energy of no -matter electromegemtic field has a energy density:$$\frac{E^2+B^2}{8\pi}$$ , so that gravitational field also has energy, despite there is difference between electro. and gravitational field due to no energy density in GR

 

[PeterDonis]

my confusion is why they are equal.

Why what are equal? So far we have only discussed one actual quantity, the Komar mass. Are you confused about why it is equal to itself?

$$
E_{\mathrm{total}} = E_{\mathrm{matter}}+E_{\mathrm{field}}
$$

Where are you getting this equation from? Do you have a reference to a textbook or peer-reviewed paper that gives this equation? (Hint: You don't because there isn't one. As I have already said, there is no such thing as "energy of the gravitational field".)

i think you discuss a different thing from mine.

I think you do not have a valid concept to discuss other than the one I am discussing.

electromegemtic field has a energy density

Yes.

so that gravitational field also has energy

No. The gravitational field is not the same as the EM field in this respect. The EM field has a local stress-energy; that's what the expression you wrote down represents. The gravitational field does not have a local stress-energy; that's why the stress-energy is zero in the vacuum region around a gravitating mass.

 

[PeterDonis]

komar mass are defined as

You gave the surface integral definition, not the volume integral definition. As I have already said, the surface integral definition is not relevant to this discussion, because it does not capture the intuitive idea of adding up the masses of all the individual pieces of matter that make up the body; the volume integral definition is the one that does that.

 

[wLw]

Calculating the Komar mass gives $M$. But you have not shown any calculation of "the energy of matter" which gives a different result. In fact you have shown no calculations at all, despite my repeated requests. Can you show a calculation of "the energy of matter" which gives a different result from $M$?

i am clearer than before now. the M in Schw. space has already includes the energy of matter and the bind gravity energy and komar mass is the whole energy so they are same. but i have one more confusion as you said

Where are you getting this equation from? Do you have a reference to a textbook or peer-reviewed paper that gives this equation? (Hint: You don't because there isn't one. As I have already said, there is no such thing as "energy of the gravitational field".)

here, in section 4

 

[PeterDonis]

here, in section 4

This is a paper on Newtonian gravity, not relativity.

 

[wLw]

but if gravitational filed has no energy how can you explain the gravitational radiation

 

[PeterDonis]

if gravitational filed has no energy how can you explain the gravitational radiation

Gravitational radiation is not the same as the "gravitational field" you are talking about, nor is the kind of energy that gravitational radiation carries the same as the kind of "energy" that appears in the Komar mass integral.

 

[wLw]

Gravitational radiation is not the same as the "gravitational field" you are talking about, nor is the kind of energy that gravitational radiation carries the same as the kind of "energy" that appears in the Komar mass integral.

in Newton case we have$$\nabla^2\phi=4\pi\rho$$. does the $\rho$ include binding energy or just only matter mass

 

[PeterDonis]

does the ρ\rho include binding energy or just only matter mass

In Newtonian physics $\rho$ only includes "matter mass". But that is also true in General Relativity. Look at the Komar mass integral again: the "binding energy" factor is not $\rho$ (or more generally $\rho + 3p$, which is the "source" factor that comes from the matter). If you have to pick a particular factor in the integral that represents "binding energy", it would be $\sqrt{g_{tt}}$, the redshift factor. But really "binding energy" is not localized at all; it's a global property of the system--it's the fact that the mass $M$ of the bound system is less than the total mass of all the constituents would be if they were all widely separated from each other.