# Straight cosmic string energy-momentum tensor and the cosmic strings EoS

Consider a simple infinite straight "cosmic" string of negligible thickness, in flat spacetime. The string energy-momentum tensor has the following components (in the string proper frame, and using cartesian coordinates system with the $$z$$ axis oriented along the string) : $$\begin{equation}\tag{1} T^{ab} = \left(\begin{array}{cccc} \rho & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & 0\\ 0 & 0 & 0 & -\, \tau \end{array}\right), \end{equation}$$ where $$\rho$$ is the string's energy density (which includes some Dirac deltas) and $$\tau > 0$$ is the string tension (I'm using the $$\eta = (1, -1, -1, -1)$$ convention). In general $$\tau \ne \rho$$.

Now, consider a large collection of random strings covering all of space. On average, the "fluid" of strings is described by the following tensor: $$\begin{equation}\tag{2} \langle \, T^{ab} \rangle = \left(\begin{array}{cccc} \rho & 0 & 0 & 0\\ 0 & p & 0 & 0\\ 0 & 0 & p & 0\\ 0 & 0 & 0 & p \end{array}\right). \end{equation}$$ So the trace of (1) and (2) give $$\tag{3} T = \rho + \tau = \rho - 3 p.$$ In cosmology it is frequently stated that the equation of state $$p = -\, \frac{1}{3} \: \rho$$ describes a fluid of strings (and $$p = -\, \frac{2}{3} \: \rho$$ is associated to a fluid of "cosmic walls"). Substituting this EoS into (3) gives $$\tau = \rho$$, which is just a special case.

So how can we justify that $$p = -\, \frac{1}{3} \: \rho$$ describes a fluid of strings? How could we justify that $$\tau = \rho$$ for a string? What if $$\tau \ne \rho$$?

How could we justify that $$τ=ρ$$ for a string?

Enhanced symmetry, which amounts to a simpler and more natural description. Note, that the tensor $$T_{ab}=\rho \mathop{\mathrm{diag}}(1,0,0,-1)$$ is invariant under Lorentz boosts along the $$z$$ direction, the string energy momentum tensor is simply proportional to the metric induced on the string with a constant coefficient. Such string does not have a preferred frame and its description could be made invariant under worldsheet coordinates reparametrization. So the dynamic of such string could be derived from Nambu–Goto action.

Plus, we have a “microscopic” mechanism for the appearance of such cosmic strings from phase transition in theories with spontaneous symmetry breaking. The prototypical example of such theory is an Abelian Higgs model with Lagrangian: $$\mathcal{L}=-\frac14 F_{μν} F^{μν}-|D_{μ} \Phi|^2 -V(|\Phi|),$$ where the potential has a typical “mexican hat” shape with its minimum achieved for nonzero vev of $$\Phi$$. A necessary condition for the existence of stable string solution, the nontrivial first homotopy group of vacuum manifold (in this case it is a circle $$\Phi=\eta$$) is fulfilled here and indeed this model has a string with a finite energy per unit length.

What if $$τ≠ρ$$?

Then the string has a preferred frame. This could mean that there is a whole worldsheet field theory “living” on the string, with its own evolution equations that cannot be derived just from energy–momentum conservation. One has to specify such theory and possibly couple it to the background fields other than the metric.

So how can we justify that $$p=−\frac13 ρ$$

The context for such justifications is cosmology. If there are excitation modes of strings that have equation of state corresponding to e.g. massive or massless particles the energy contained in them would be diluted with the expansion of the Universe and we would be left with only $$τ=ρ$$ contributions. Note, that such modes could still leave observable consequences for structure formation, etc.

For more, see the review:

and a more recent but less detailed review:

• Copeland, E. J., & Kibble, T. W. B. (2010). Cosmic strings and superstrings. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 466(2115), 623-657. doi:10.1098/rspa.2009.0591.