# Gravity due to pressure

• I
How do I calculate the gravitational mass of a cylinder of compressed gas, including the effects of pressure? By gravitational mass, I mean what I would measure on an ideal mass balance.
(I know that the pressure is negligibly small in a realistic container, but I want to have a conceptual understanding.)

My understanding is that the time time component of the Ricci curvature is $$R_{00}=\frac{1}{2}\left(\rho_E+P_x+P_y+P_z\right)$$
so pressure should have an analogous contribution to energy on gravity. But I've never seen it applied to any sort of ordinary objects so I'm having a hard time connecting it to reality.

Suppose the gas has a energy of ##E_g##, a pressure of ##P##, and a volume ##V##. The cylinder has an energy ##E_c## and a wall tension of ##-P## due to the confinement of the gas and a surface area ##A##.

If we ignore pressure, then the gravitational mass is just ##(E_g+E_c)/c^2##.

Is the pressure contribution then just ##3PV/c^2##? So the total gravitational mass is ##(E_g+E_c+3PV)/c^2##?

Is the wall tension irrelevant because the wall has 0 volume?

• Paul Colby

PeterDonis
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2020 Award
How do I calculate the gravitational mass of a cylinder of compressed gas, including the effects of pressure?
For a static container of compressed gas, the gravitational effect of pressure in the gas is exactly cancelled by the gravitational effect of tension (which is negative pressure) in the walls of the container. So the gravitational mass is just the ordinary mass (density times volume) of the gas.

I thought that might be the case, but I don't understand how the math works that way, since the volume of the gas is much greater than the volume of the container walls.

I thought that might be the case, but I don't understand how the math works that way, since the volume of the gas is much greater than the volume of the container walls.
I'll admit that I knew nothing about this to begin with, but I think I understand why the math works this way. It shouldn't have anything to do with the volume. As long as the cylinder is in equilibrium (i.e. the gas isn't breaking the walls due to pressure), then the summation of forces at every point will equal zero. As @PeterDonis pointed out: there is an equal and opposite force for every point where pressure is acting. So the equation would be...
ΣF=0
P + (-T) = 0
P = T
Same deal for the normal force cancelling out the force caused by gravity when you have an object resting on the Earth.

PeterDonis
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2020 Award
I don't understand how the math works that way, since the volume of the gas is much greater than the volume of the container walls.
Comeback City's response is basically correct (but see my follow-up post to him). For a more technical answer, look up "Komar Mass"; that is the actual calculation you would do in GR to find the total mass of an isolated static object by "adding up" the contributions from all the stress-energy in it.

PeterDonis
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2020 Award
there is an equal and opposite force for every point where pressure is acting
Yes, but force is not the same as pressure. The force balance turns out to lead to a relationship between the pressure in the gas and the tension in the walls of the container, but that relationship is not equality. For a spherical container, the relationship is

$$\sigma = \frac{p r}{2t}$$

where ##p## is the gas pressure, ##r## is the inner radius of the sphere, and ##t## is its thickness, which is assumed to be much smaller than ##r## (the usual constraint is ##r / t > 10##). Note that as ##t## gets smaller, ##\sigma## gets larger in relation to ##p##, and in the limit of a zero thickness wall, ##t## [Edit: actually ##\sigma##] increases without bound. That is a key part of the resolution to the issue @Khashishi raises.

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• Comeback City
σ=pr/2t
What does σ actually represent in this?
and in the limit of a zero thickness wall, t increases without bound.
Shouldn't σ increase without bound, not t?

PeterDonis
Mentor
2020 Award
What does σ actually represent in this?
The tension in the container wall.

Shouldn't σ increase without bound, not t?
Sorry, yes, ##\sigma## increases without bound as ##t \rightarrow 0##. I have edited my previous post to correct this.