In physics, particularly special relativity, light-cone coordinates, introduced by Paul Dirac[1] 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.
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.
In a light-cone coordinate system, two of the coordinates are null vectors and all the other coordinates are spatial.
Assume we are working with a (d,1) Lorentzian signature.
Instead of the standard coordinate system (using Einstein notation) with
can act as "time" coordinates.
[2]: 21 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
The parabolic transformations show up as
Another set of parabolic transformations show up as
Light cone coordinates can also be generalized to curved spacetime in general relativity.
Sometimes calculations simplify using light cone coordinates.
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.
A closed string is a generalization of a particle.
The spatial coordinate of a point on the string is conveniently described by a parameter
Time is appropriately described by a parameter
Associating each point on the string in a D-dimensional spacetime with coordinates
, these coordinates play the role of fields in a
dimensional field theory.
and treat this degree of freedom as the time variable.
is not an independent degree of freedom anymore.
can be identified as the corresponding Noether charge.
is the Noether charge, we obtain: This result agrees with a result cited in the literature.
[3] For a free particle of mass
the action is In light-cone coordinates
as time variable: The canonical momenta are The Hamiltonian is (
): and the nonrelativistic Hamilton equations imply: One can now extend this to a free string.