## Friday, November 04, 2011 ... / / ### Equations of state

When we discuss things such as the content of the Universe and the different "substances" it is composed of (e.g. in this question at a forum), an important concept needed to quantify these issues is the so-called "equation of state".

In rudimentary thermodynamics, an equation of state is any equation relating state variables such as pressure, volume, temperature, energy, and so on. In cosmology and fundamental physics, the term is most often used for equations describing the relationship between the energy density $$\rho$$ and pressure $$p$$:
$p = p(\rho)$ That's because these two variables are entries in the stress-energy tensor that directly enters Einstein's equations, those that determine the evolution of the spacetime geometry. The simplest relationship you may have is a proportionality law:
$p = w\cdot \rho$ where $$w$$ is a numerical constant. Why is it dimensionless? It's because dimensionally speaking, pressure is force per unit area which is the same thing as energy per unit distance and unit area which is the same as energy per unit volume i.e. energy density. Now, the number $$w$$ can't ever jump out of the interval $$[-1,+1]$$. Why?

The reason is that the speed of sound $$v$$ can't exceed the speed of light in the vacuum, $$c$$. The speed of sound may be calculated from:
$\frac{ v^2}{c^2} = \left( \frac{\partial p}{\partial \rho} \right)^2$ If you substitute a linear dependence of the pressure on the energy density, both sides are simply equal to $$w^2$$ which therefore can't be greater than $$+1$$. Similar constraints are equivalent to some of the "energy conditions" but I don't want to discuss numerous energy conditions that have been analyzed in some older TRF blog entries.

Important values of $$w$$

Instead, let us look at important values of $$w$$ in the allowed interval. The lowest allowed value is
$w=-1$ Is there a material that may give you this highly negative pressure? Well, it's not really a "material": it's the cosmological constant, the dominant model for the "dark energy" we know (or believe) to constitute 73% of the energy density in the Universe. Note that if you set $$p=-\rho$$, the 4-dimensional stress-energy tensor will be
$T_{\mu\nu} = {\rm diag} (+1,-1,-1,-1)\rho$ which means that it will be proportional to the metric tensor. Such a tensor doesn't pick any preferred reference frame; it locally preserves the Lorentz symmetry. Globally, when you require Einstein's equations to hold, you will find many solutions to these equations. Among them, the most importat ones will be the "maximally symmetric" universes that include, aside from the flat Minkowski space for $$\rho=0$$, de Sitter space for a positive $$\rho$$ and anti de Sitter space for a negative $$\rho$$ as well.

The existence of the cosmological constant may also be deduced from a Lagrangian; the relevant term in the action is proportional simply to
$\int {\rm d}^4x \,\sqrt{g}$ so the Lagrangian density is constant, assuming that you include the correct "proper spacetime volume" integration measure.

The negative-energy-density space, the anti de Sitter space, may be viewed as a Lorentzian-signature hyperboloid of a sort. It contains globally timelike Killing vectors which is why you may define "globally static" frames. That's necessary for supersymmetry – because supercharges' anticommutators include time-like translations. And indeed, anti de Sitter spaces are among the spaces that are most fully understood within string theory (and its approximations) because of the supersymmetry that may remain unbroken.

More realistically, we may also consider de Sitter space with a positive value of $$\rho$$. Our Universe is almost certainly an example. Supersymmetry has to be broken in such a space; well, in the real world, it's broken much more intensely than by the minimum breaking required by the de Sitter curvature. De Sitter space may also be visualized as a "static space" but only if you cut the "exterior space" beyond the cosmic horizon. Also, de Sitter space may be imagined as a space whose spatial volume is exponentially increasing – a picture that is essential in cosmology (both during inflation which has an approximate de Sitter space, as well as during the recent and future expansion of our Universe that is dominated by a much smaller positive value of the vacuum energy than the huge value during inflation).

The supernova observations indicate that most of the energy density in the Universe indeed has $$w$$ close to $$-1$$ which supports the idea that the cosmological constant (or something very similar to it) is the right detailed name of the vague beast known as "dark energy". Are there other negative values of $$w$$ that you should know? You bet.

Cosmic strings and domain walls

In string theory (whether the real one or its partially inconsistent imitations and approximations), one may find one-dimensional and two-dimensional objects, the cosmic strings and the domain walls. The cosmic strings may be nothing else than the fundamental strings that have grown to astronomical dimensions (usually because of the extreme conditions of the early Universe) but they may also be other kinds of string-like objects that string theory admits that just happen to be large. The same thing applies to the two-dimensional domain walls (more generally, domain walls are objects of co-dimension one) which may be some membranes or membrane-shaped entities found in string theory.

What is the equation of state of a material composed of cosmic strings? Well, that's not hard to calculate. Their histories in the spacetime look like two-dimensional world sheets. The energy density within these world sheet is given by the stress-energy tensor. It's proportional to a two-dimensional delta-function, so that the energy density is only nonzero at the locus of the world sheet. However, it must be proportional to
$T_{\mu\nu} = {\rm diag} (+1,-1,0,0) \rho$ because in the two-dimensional "spacetime" (world sheet) generated by the first two dimensions, the (world sheet) Lorentz symmetry must continue to be unbroken so the corresponding stress-energy tensor has to be proportional to the metric tensor again. It's just like the case of the cosmological constant but now it only applies to two spacetime dimensions – time and the dimension along the string.

However, strings may have arbitrary directions in space. For all of them, the trace of the spatial part of the stress energy tensor is the same. But you may average over the directions, anyway. The outcome is an averaged value of the stress-energy tensor:
$T_{\mu\nu} = {\rm diag} (+1,-1/3,-1/3,-1/3) \rho$ Well, we just divided $$-1$$ into three pieces, to make the spatial part rotationally symmetric while preserving the trace. Without much ado, we see that cosmic strings have $$w=-1/3$$, already too close to zero to be consistent with the "dark energy" seen in the Universe. The domain walls have a Lorentz-invariant stress-energy tensor in the 2+1-dimensional world volumes,
$T_{\mu\nu} = {\rm diag} (+1,-1,-1,0) \rho,$ which after averaging over all the directions of the membrane gives us
$T_{\mu\nu} = {\rm diag} (+1,-2/3,-2/3,-2/3) \rho$ i.e. $$w=-2/3$$. It's more negative than for cosmic strings but still too close to zero.

Dust

The dust, i.e. matter particles that are almost at rest and not moving much, have $$p=0$$ which means $$w=0$$ as well. Pressure is force exerted by the material on the walls but if the particles inside the box are not moving at all, they don't act on the walls of the box and the pressure vanishes. I have actually used this logic above – for cosmic strings and domain walls – because this realization was needed to justify why the "last components" of the stress-energy tensor were set to zero. It's also because the objects are separated from the walls in the transverse dimensions and they're moving, so they can't exert a pressure in those directions.

All regular "slow and cold" materials belong to the category of "dust", whether they're gold or something much less valuable.

When particles get faster, the pressure keeps on increasing. An important point is when the particles move by the speed of light. In that case, we have $$w=+1/3$$. Note that the sign is opposite than for the cosmic strings. Why is it exactly $$+1/3$$? Well, consider a photon confined in a box of volume $$L^3$$. The photon moves by the velocity $$v_x$$ in the $$x$$-direction so it takes $$L/v_x$$ for it to get from the left wall to the right one or vice versa. Every time it hits the wall, it gets reflected: the momentum changes from $$p_x$$ to $$-p_x$$ or vice versa. Regardless of the sign, the photon delivers the outward momentum of $$2p_x$$ to the left or right walls. Recall it takes $$L/v_x$$ of time so the momentum per unit time, counting only the left and right walls, is the ratio $$2p_x / (L/v_x) = 2p_x\cdot v_x/L$$. Add the same "outward force" contributions from the other two pairs of the faces of the box to get a force equal to $$F=2p\cdot v / L$$. When you divide this force by the area, you get
$p_{\rm pressure} = F/A = \frac{2p\cdot v}{6L^3} = \frac{\rho}{3}$ because $$p\cdot v = c|p| = E$$ is the energy of the photon and, when divided by $$L^3$$, you get the energy density. This proves $$p=\rho/3$$ for radiation. The same derivation holds for any particles that move (nearly) by the speed of light, including neutrinos or gravitons or whatever you like (even though gravitons won't really get reflected from any wall you may construct). It follows that $$w=+1/3$$ for radiation.

Dense black hole gas

For the sake of completeness, I want to mention a favorite "material" of Tom Banks, a somewhat unrealistic "dense black hole gas", which allows you to reach $$p=+\rho$$, the opposite extreme value of $$p$$ than the vacuum energy density. The relationship may be at least formally derived if you imagine a region with lots of black holes treated as "densely packed balls" and if you use the Schwarzschild formulae for the radius-mass relationship etc. Don't expect this material to occur in your lab: it's mostly a purely theoretical visual way to imagine how the opposite extreme has to look like.

Cosmological evolution

For different values of $$w$$, you get very different cosmological evolutions. The negative pressure offered by the cosmological constant, cosmic strings, or domain walls will be able to accelerate the expansion of the Universe – although the cosmological constant is the only one that quantitatively agrees with the observed acceleration of the expansion. Now, you may re-read the article about energy non-conservation in cosmology.

I was explaining that the total mass-energy carried by the dust remains constant when the Universe is expanding. However, the radiation sees its wavelength to increase proportionally to $$a$$, the linear dimensions of the expanding Universe, which is why the energy of each photon (or another particle) goes down like $$1/a$$: recall that the energy is proportional to the frequency or inversely proportional to the wavelength. The volume goes like $$a^3$$ so the energy density, $$E/V$$, goes like $$1/a^4$$ for radiation.

The energy stored in the cosmological constant has a constant energy density; that's why the cosmological constant has the word "constant" in it. So the energy density goes like $$1/a^0$$ and the total energy goes up as $$a^3$$. During inflation, this is the reason why the total mass of the Universe expands much like the volume, a reason why Alan Guth says that while people believe that there's no free lunch, inflation is the ultimate free lunch. For other equations of state, the energy density goes like
$\rho \sim a^{-3(w+1)}.$ Of course, one may generalize those comments to more complicated, nonlinear equations of state but it's important for a physicist to cover the landscape of possibilities by friendly, well-understood situations so that no place ends up being "completely unfamiliar".

Summary

To summarize, for the purpose of cosmology, the dominant property describing the type of material is $$w=p/\rho$$ which is $$0$$ for ordinary matter composed of particles and similarly $$-d/3$$ for large objects with $$d$$ spatial dimensions (the cosmological constant may be uniformly counted as a bulk-filling three-brane). Positive values of $$w$$ are possible if the objects have a high velocity; $$w=+1/3$$ is valid for radiation (at the speed of light). Higher values of $$w$$ are extreme, unrealistic, and at $$w=+1$$, they may be linked to a dense gas of black holes. Each nonzero value of $$w$$ quantifies how much the total energy will fail to be conserved during the cosmological evolution.    