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This question might be a little misguided, given that I have never studied GR or quantum field theory, but...

As I understand it, dark energy is caused by negative pressure in the gravity field. Gravity relates to pressure, as it results from the stress-energy tensor? Sometimes this pressure can be negative, giving rise to dark energy that pushes the universe apart.

What about here on the surface of a planet, where there is a positive atmospheric pressure? Is this the same kind of pressure that is talked about in the context of the gravity field? And does being under pressure cause Earth's gravity to be increased as a result?

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Yes, (positive) pressure increases gravity.

Consider an ideal gas and ask what you mean by the pressure. The pressure of a gas is the result of moving gas molecules colliding with the walls of their contained and bouncing off. The force the molecules exert on the walls of the container is simply the rate of change of their momentum as they bounce off those walls, and the rate of change of momentum is related to how fast the gas molecules are travelling - faster gas molecules = more pressure.

But faster gas molecules also means a greater kinetic energy. So if we increase the pressure of a gas (while keeping the density and volume fixed) we are increasing the total energy of the gas. And if we increase the total energy we are increasing the total mass (obtained by dividing the energy by $c^2$). And if we are increasing the total mass we are increasing the gravity that gas creates.

Admittedly this is a grossly simplified version of how pressure creates spacetime curvature, but I think it serves to give you the general idea. More precisely we should say that Einstein's equation relates the curvature to the stress-energy tensor, and the pressure makes a contribution to the diagonal entries in this tensor.

The obvious next question is what negative pressure means, and there isn't a simple picture of this like gas molecules producing a positive pressure. I explore this question a bit in my answer to 'Negative pressure' counteracting gravity?.

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As I understand it, dark matter is caused by negative pressure in the gravity field.

Sorry, that doesn't sound right. Dark matter is a term associated with a gravitational anomaly wherein there's more gravity than can be accounted for by the visible ordinary matter.

Gravity relates to pressure, as it results from the stress-energy tensor?

Yes, see the stress-energy-momentum tensor which "describes the density and flux of energy and momentum in spacetime":

enter image description herePublic domain image by Maschen, based on an image by Bamse see Wikipedia

Note the energy-pressure diagonal mentioned by John Rennie? That pressure isn't just pressure in air. The stress-energy-momentum tensor doesn't describe pressure in air, it's in spacetime, and since it's static it's effectively in space. A gravitational field is in essence "a pressure gradient in space" which is modelled as curved spacetime. Think of something like this image, but pushing out instead of pulling in.

Sometimes this pressure can be negative, giving rise to dark energy that pushes the universe apart.

That's what people tend to say, but remember that negative pressure is tension. Think of the balloon analogy for the expanding universe. The ballooon is in vacuum, and the pressure of the air inside is balanced by the tension in the skin. For the balloon to expand you either need to increase the pressure or reduce the tension.

enter image description here Image courtesy of the one-minute astronomer.

The tension reduces for a bubble-gum balloon, whereupon the skin gets thinner, so the tension reduces further so the balloon expands further, and so on. Since dark energy is associated with negative pressure which is tension, IMHO reducing tension fits with what we see better than increasing pressure.

What about here on the surface of a planet, where there is a positive atmospheric pressure? Is this the same kind of pressure that is talked about in the context of the gravity field?

Yes. You need to add energy to a spring to compress it. It's the same for a volume of gas, like John Rennie was saying. Or a block of elastic. Rather surprisingly, general relativity treats space as a kind of gin-clear ghostly elastic continuum. Hence the shear stress term in the stress-energy-momentum tensor. So it's the same for space too.

And does being under pressure cause Earth's gravity to be increased as a result?

Not the air pressure. Like Coilkid said, the Earth's atmospheric pressure is there because of the gravity anyway.

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