# A A problem in Komar mass?

#### PeterDonis

Mentor
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

Mentor
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

Mentor
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

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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"A problem in Komar mass?"

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