In physics, particularly special relativity, light-cone coordinates, introduced by Paul Dirac and also known as Dirac coordinates, are a special coordinate system where two coordinate axes combine both space and time, while all the others are spatial.

Motivation

A spacetime plane may be associated with the plane of split-complex numbers which is acted upon by elements of the unit hyperbola to effect Lorentz boosts. This number plane has axes corresponding to time and space. An alternative basis is the diagonal basis which corresponds to light-cone coordinates.

Light-cone coordinates in special relativity

In a light-cone coordinate system, two of the coordinates are with respect to null vectors.

Assume we are working with a (d,1) Lorentzian signature.

Instead of the standard coordinate system (using Einstein notation)

1. ,

we have

.

can act as "time" coordinates.

One nice thing about light cone coordinates is that the causal structure is partially included into the coordinate system itself.

plane shows up as the squeeze mapping.

.

Light cone coordinates can also be generalized to curved spacetime in general relativity. Sometimes calculations simplify using light cone coordinates. See Newman–Penrose formalism. Light cone coordinates are sometimes used to describe relativistic collisions, especially if the relative velocity is very close to the speed of light. They are also used in the light cone gauge of string theory.

Light-cone coordinates in string theory

given by

is given by

which we obtain from the metric, i.e.

can be identified as the corresponding Noether charge one obtains

Equating this to

is the Noether charge, we obtain:

This result agrees with a result cited in the literature.

Free particle motion in light-cone coordinates

For a free particle the action is

as time variable:

The canonical momenta are

):

and the nonrelativistic Hamilton equations imply:

One can now extend this to a free string.

See also

• Newman–Penrose formalism

References