# A A problem in Komar mass?

#### 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

Mentor
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

Mentor
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

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#### PeterDonis

Mentor
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

Mentor
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

Mentor
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.

#### wLw

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.
so the gravitational radiation refers to the energy of outside gravitational filed (generated by body), and the gravitational (no filed ,because it is binding energy)energy in komar mass is binding energy，but both of them are nonlocalizable?,is i right.?

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#### PeterDonis

Mentor
so the gravitational radiation refers to the energy of outside gravitational filed (generated by body), and the gravitational (no filed ,because it is binding energy)energy in komar mass is binding energy，but both of them are nonlocalizable?,is i right.?
You are trying to use terminology that is not very suitable. In particular, you are continuing to cling to the concept of "energy of the gravitational field", which is not a good concept. It doesn't help with understanding; it hinders it.

Gravitational waves carry energy because they can do work. For example, if they pass through an object they will heat it up (by a very small amount, but the effect is there). But the waves' ability to do work is not localizable.

Gravitational binding energy of a bound object is there because, in order to take a system consisting of a lot of small, widely separated pieces of matter, and make them into a single bound object like a planet, you need to extract energy from the system (the usual way this happens is for the system to emit electromagnetic radiation that escapes to infinity); or, conversely, if you want to take a single bound object like a planet and make it into a lot of small, widely separated pieces of matter, you need to add energy to (do work on) the system. But, again, this property of the system is not localizable; the transition I just described, in either direction, is a global operation.

I have just described the actual physics of gravitational waves and gravitational binding energy. Using the term "energy of the gravitational field" adds nothing at all to the physics, nor does it help to understand the physics I have described. The best thing you can do is to just forget about the concept altogether and focus on the actual physics.

#### wLw

ok thank you a lot, i am clearer than before

#### PeterDonis

Mentor
wLw

"A problem in Komar mass?"

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