Optical equations for null strings

TL;DR

This paper derives an optical equation for null strings, analogous to Sachs' equations for null geodesics, using a complex scalar function to describe string expansion, rotation, and their behavior in various spacetime backgrounds.

Contribution

It introduces a novel optical equation for null strings expressed through a complex scalar, enabling visualization of string dynamics in different gravitational backgrounds.

Findings

Null strings' behavior can be represented by complex Z-plane diagrams.

In asymptotically flat spacetimes, Z approaches zero at null infinity.

Gravitational waves and matter flows create ripples on the strings.

Abstract

An optical equation for null strings is derived. The equation is similar to Sachs' optical equations for null geodesic congruences. The string optical equation is given in terms of a single complex scalar function $Z$, which is a combination of spin coefficients at the string trajectory. Real and imaginary parts of $Z$ determine expansion and rotation of strings. Trajectories of strings can be represented by diagrams in a complex $Z$-plane. Such diagrams allow one to draw some universal features of null strings in different backgrounds. For example, in asymptotically flat space-times $Z$ vanishes as future null infinity is approached, that is, gradually shapes of strings are "freezing out". Outgoing gravitational radiation and flows of matter leave ripples on the strings. These effects are encoded in subleading terms of $Z$. String diagrams are demonstrated for rotating and expanding strings in a flat space-time and in cosmological models.