#### PeterDonis

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- What is the gravitational mass of a ball of photons?

This is based on "Concept Question 10.4" in Andrew Hamilton's

Suppose we have a spherically symmetric ball of stress-energy surrounded by vacuum. More precisely, the ball contains a perfect fluid with an equation of state given by ##p = w \rho##, where ##w## is a constant that can vary according to the type of stress-energy. We will consider three commonly used values for ##w## taken from cosmology: ##w = \frac{1}{3}##, which is usually called "radiation" (the original question in Hamilton's book used "photons" to refer to this, hence the title of this thread); ##w = 0##, which is usually called "matter"; and ##w = -1##, which is usually called "dark energy" (Hamilton calls it "vacuum").

Since the spacetime is spherically symmetric, the geometry of the vacuum region is known; by Birkhoff's Theorem, it must be the Schwarzschild geometry. This geometry has one free parameter, ##M##, which is the externally measured mass of whatever is inside the non-vacuum region. This obviously gives rise to the question: how does ##M## relate to the stress-energy tensor inside the non-vacuum region? In particular, how does it relate to the equation of state parameter ##w## (since that is the key thing that characterizes the stress-energy inside the non-vacuum region)?

We can sharpen this question by considering two possible cases:

(1) The ball of stress-energy has no restriction at its boundary: it is free to expand or contract, as long as it remains spherically symmetric. For this case, does the mass ##M## depend on ##w##? If so, how?

(2) The ball of stress-energy is contained by a stationary wall with a fixed surface area ##A##. The wall itself must contain stress-energy, so there will be some externally measured mass ##M_{\text{wall}}## associated with it. The question for this case is, for a given value of ##w##, is the "net" mass ##M - M_{\text{wall}}##, which can be interpreted as the externally measured mass of the stress-energy inside the wall (i.e., the same stress-energy as in #1 above) the same as the mass ##M## in #1 above? If not, how does it change?

Now, this question occurs in a part of Hamilton's book which is discussing FRW spacetime, which seems to suggest (at least to me) that the intended answers to the above are, for case #1, that ##M##

However, I can think of two simple heuristic arguments that seem to lead to the opposite conclusion:

For case #1, we can extend the obvious Schwarzschild coordinates in the external vacuum region inside the region containing the stress-energy, and then consider a spacelike slice of constant time ##t## in these coordinates. In such a slice, from the known properties of the Einstein Field Equation for the case of spherical symmetry, we should have the following:

$$

M = \int_0^R 4 \pi r^2 \rho (r) dr

$$

where ##\rho (r)## is the energy density and ##R## is the areal radius of the surface of the ball. But this integral only contains ##\rho##; it does not contain ##p##. So the equation of state does not affect this integral at all. So ##M## should be independent of ##w##.

For case #2, we can do the same integral as above, but now just split up into two pieces:

$$

M = M_{\text{net}} + M_{\text{wall}} = \int_0^R 4 \pi r^2 \rho (r) dr + \int_R^{R_{\text{wall}}} 4 \pi r^2 \rho (r) dr

$$

where ##R## is the areal radius of the inner surface of the wall and ##R_{\text{wall}}## is the areal radius of its outer surface. The first term in this integral is the same as above, so we should have ##M_{\text{net}}## in the above be the same as ##M## for case #1 (with the obvious proviso that we have to compare them for the same value of ##R##, i.e., at an instant where the areal radius in case #1 is the same as that of the inner surface of the wall in case #2).

So what do others think? Is the answer the one that I think Hamilton seems to expect? Or are my heuristic arguments correct?

*General Relativity, Black Holes, and Cosmology*. I have modified the question somewhat in order to focus on what seem to me to be the key issues.Suppose we have a spherically symmetric ball of stress-energy surrounded by vacuum. More precisely, the ball contains a perfect fluid with an equation of state given by ##p = w \rho##, where ##w## is a constant that can vary according to the type of stress-energy. We will consider three commonly used values for ##w## taken from cosmology: ##w = \frac{1}{3}##, which is usually called "radiation" (the original question in Hamilton's book used "photons" to refer to this, hence the title of this thread); ##w = 0##, which is usually called "matter"; and ##w = -1##, which is usually called "dark energy" (Hamilton calls it "vacuum").

Since the spacetime is spherically symmetric, the geometry of the vacuum region is known; by Birkhoff's Theorem, it must be the Schwarzschild geometry. This geometry has one free parameter, ##M##, which is the externally measured mass of whatever is inside the non-vacuum region. This obviously gives rise to the question: how does ##M## relate to the stress-energy tensor inside the non-vacuum region? In particular, how does it relate to the equation of state parameter ##w## (since that is the key thing that characterizes the stress-energy inside the non-vacuum region)?

We can sharpen this question by considering two possible cases:

(1) The ball of stress-energy has no restriction at its boundary: it is free to expand or contract, as long as it remains spherically symmetric. For this case, does the mass ##M## depend on ##w##? If so, how?

(2) The ball of stress-energy is contained by a stationary wall with a fixed surface area ##A##. The wall itself must contain stress-energy, so there will be some externally measured mass ##M_{\text{wall}}## associated with it. The question for this case is, for a given value of ##w##, is the "net" mass ##M - M_{\text{wall}}##, which can be interpreted as the externally measured mass of the stress-energy inside the wall (i.e., the same stress-energy as in #1 above) the same as the mass ##M## in #1 above? If not, how does it change?

Now, this question occurs in a part of Hamilton's book which is discussing FRW spacetime, which seems to suggest (at least to me) that the intended answers to the above are, for case #1, that ##M##

*should*vary with ##w## (since the dynamics of an FRW spacetime certainly does), and for case #2, that the presence of the wall*should*make a difference (since the wall prevents the stress-energy inside from exhibiting the same dynamics as it would in the free case).However, I can think of two simple heuristic arguments that seem to lead to the opposite conclusion:

For case #1, we can extend the obvious Schwarzschild coordinates in the external vacuum region inside the region containing the stress-energy, and then consider a spacelike slice of constant time ##t## in these coordinates. In such a slice, from the known properties of the Einstein Field Equation for the case of spherical symmetry, we should have the following:

$$

M = \int_0^R 4 \pi r^2 \rho (r) dr

$$

where ##\rho (r)## is the energy density and ##R## is the areal radius of the surface of the ball. But this integral only contains ##\rho##; it does not contain ##p##. So the equation of state does not affect this integral at all. So ##M## should be independent of ##w##.

For case #2, we can do the same integral as above, but now just split up into two pieces:

$$

M = M_{\text{net}} + M_{\text{wall}} = \int_0^R 4 \pi r^2 \rho (r) dr + \int_R^{R_{\text{wall}}} 4 \pi r^2 \rho (r) dr

$$

where ##R## is the areal radius of the inner surface of the wall and ##R_{\text{wall}}## is the areal radius of its outer surface. The first term in this integral is the same as above, so we should have ##M_{\text{net}}## in the above be the same as ##M## for case #1 (with the obvious proviso that we have to compare them for the same value of ##R##, i.e., at an instant where the areal radius in case #1 is the same as that of the inner surface of the wall in case #2).

So what do others think? Is the answer the one that I think Hamilton seems to expect? Or are my heuristic arguments correct?

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