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Traditionally the cosmological equation of state of cold matter (so-called dust) is simply: $$p = 0.$$ But, in Newtonian terms, each particle is gravitationally attracting every other particle.

Therefore could one say that the dust actually has a negative pressure $p$?

In support of this view I would say that Einstein's Field Equations say that spacetime curvature is equivalent to stress-energy.

This negative pressure might then act as a source of cosmological acceleration which would to some degree counteract the standard deceleration caused by the dust's positive mass density.

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  • $\begingroup$ I think this question is actualy a very important one. I was thinking about this "negative pressure" thing since a few years ago, and I still don't have any convincing answer yet. It's actually very debated even today, since the Einstein's field equation is non-linear and may imply some very subtile "back reaction" effects. $\endgroup$ – Cham Jun 10 '17 at 17:11
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No, there is no contribution to the pressure from the gravitational attraction between the particles.

To see this you need to appreciate that the pressure is an ensemble property, and look at the stress-energy tensor for a single point particle. This is:

$$ T^{\alpha\beta}({\bf x},t) = \gamma m v^\alpha v^\beta \delta\left( x - x_p(t) \right) $$

where $v$ is the velocity $(1, \frac{d{\bf x}}{dt})$ not the four velocity. The $\delta$ function just makes $T^{\alpha\beta}$ zero everywhere except at the particle position, so let's assume we are at the particle position and drop it. Then if you look at the diagonal elements that we normally consider to be pressure we get entries like:

$$ T^{11} = \gamma m (v^{1})^2 $$

which is basically just the relativistic kinetic energy of the particle. If you consider an ensemble of particles with random velocities (e.g. thermal velocities) then the kinetic energy is simply related to the pressure, and that's why the diagonal terms are effectively a pressure.

In a dust we assume the particles have negligable velocities, so the kinetic energy of the dust grains is zero and hence so is the pressure. If you have a collapsing dust cloud then it's certainly true that the dust grains will starts falling inwards and will therefore acquire a velocity, but the grain velocities aren't random because all the grains fall in the same direction, so this doesn't constitute a pressure.

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  • $\begingroup$ The start premise is wrong, or just an over simplification, or an hypothsis at best if you prefer. So no wonders you arrive at the "no pressure" result since it's already implied from the start. The gas of particles with "internal" interactions don't have this free particles energy-momentum tensor. Think of a VanderWalls gas for example, which have short range interactions. Its energy-momentum tensor is certainly not like this simple form. So I think that the OP question is a sensible one, and it's still debated even today from a "gravity back reaction" view point. $\endgroup$ – Cham Jun 10 '17 at 17:08
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Therefore could one say that the dust actually has a negative pressure p?

In mechanics of solids, negative pressure (positive tension) means the internal forces resist expansion of the body due to external forces.

If you have dust (rarified set of particles) in a syringe acting on each other with non-negligible gravitational forces, the gravitational forces will also act on the piston and pull it inside. If this force is greater than the opposite force of the impacts of the moving particles on the piston, the piston would be pulled inside and that would mean negative pressure as in the above case.

Such gravitating gas system does not settle into uniform density state and normal thermodynamics does not apply to it. So even if we introduce negative pressure, it is not easy to see how to use it in thermodynamic sense.

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The following isn't exactly an "answer", but an opinion or an interesting hypothesis to be debated (at least from my own view point !).

It feels very "natural" to me that the cosmological fluid of "dust" matter should be under tension (i.e. should produce some negative pressure), a bit like some kind of Van der Walls gas, or a kind of polymer fluid. At very long range, a given particle doesn't produces any noticeable global attraction on the whole. But at close range, it certainly creates some attraction on its surrounding (i.e. on a few other close particles). So the gas particles has some close range interaction, that is not described by the standard "dust" equation of state ($p = 0$).

In the standard FLRW cosmology, it is already assumed from the start that spacetime has exact local isotropy and homogeneity, while matter doesn't respect that symetry principle in reality (the symetry is only "statistical", on a very large scale). It always striked me that cosmological mega-structures (galactic superclusters, "matter bridges", cosmic filaments, ...) are looking like some stretched material that is under tension. So it's natural to ask if the "dust" gas of galaxies is really well described by the equation of state $p = 0$. I now seriously think this is actually a very bad idealisation, and it's not a surprise to me that we now see some weird cosmological effects like "repulsion", "dark matter", etc. We may be applying and interpreting general relativity in a very wrong way !

It is well known that the Einstein field equation is highly non-linear and may show some subtle "back reaction" effects from the sub-levels on the higher levels. The importance of "back reactions" in general relativity is very much debated today, and there is still no clear agreements on it yet. There's a LOT of litterature on that subject, while is it not well known enough in the scientific community (AFAIK).

I personally highly suspect that general relativity is trying to tell us something very important about gravity : it is hierarchical (the Einstein' equation is not scale invariant). Gravity may act differently at different scales.

So when we transpose the $p = 0$ dust gas (that comes from the small human scale) to the much larger scale of a fluid of galaxies, we may actually do a very huge mistake !

It's possible that, because of the exact symetry requirement, the standard RWFL cosmology is simply neglecting the galactic interactions in the "dust" gas (i.e. the small scale short range interactions between the particles).

I think that we cannot transpose (from the human scale again) the usual dust equation of state to the cosmological fluid. The rules are not the same at the large scale.

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