Transport equation for Nambu-Goto strings

We consider a covariant approach to coarse graining a network of interacting Nambu-Goto strings. A transport equation is constructed for a spatially flat Friedmann universe. In Minkowski space and with no spatial dependence this model agrees with a previous model. Thus it likewise converges to an equilibrium with a factorizability property. We present an argument that this property does not depend on a “string chaos” assumption on the correlations between strings. And in contrast to the earlier model, this transport equation agrees with conservation equations for a fluid of strings derived from a different perspective.

Figure 1

The worldsheet of a piecewise linear string. The horizontal dotted lines represent time slices at which each segment has coordinate length $\mathrm{\Delta }\sigma$. The values of $A$ and $B$ are constant within each diamond-shaped worldsheet patch. The values of $A$ move to the right along the string (as represented by the thick line), and the values of $B$ move to the left (as represented by the thick dotted line). As we trace the history of a segment $(A,B)$, we encounter other arbitrary values of $A$ and $B$ denoted by primes.