Light-cone coordinates

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.

Motivation

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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

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In a light-cone coordinate system, two of the coordinates are null vectors and all the other coordinates are spatial. The former can be denoted   and   and the latter  .

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

Instead of the standard coordinate system (using Einstein notation)

 ,

with   we have

 

with  ,   and  .

Both   and   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.

A boost in the   plane shows up as the squeeze mapping  ,  ,  . A rotation in the  -plane only affects  .

The parabolic transformations show up as  ,  ,  . Another set of parabolic transformations show up as  ,   and  .

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

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A closed string is a generalization of a particle. The spatial coordinate of a point on the string is conveniently described by a parameter   which runs from   to  . Time is appropriately described by a parameter  . Associating each point on the string in a D-dimensional spacetime with coordinates   and transverse coordinates  , these coordinates play the role of fields in a   dimensional field theory. Clearly, for such a theory more is required. It is convenient to employ instead of   and  , light-cone coordinates   given by

 

so that the metric   is given by

 

(summation over   understood). There is some gauge freedom. First, we can set   and treat this degree of freedom as the time variable. A reparameterization invariance under   can be imposed with a constraint   which we obtain from the metric, i.e.

 

Thus   is not an independent degree of freedom anymore. Now   can be identified as the corresponding Noether charge. Consider  . Then with the use of the Euler-Lagrange equations for   and   one obtains

 

Equating this to

 

where   is the Noether charge, we obtain:

 

This result agrees with a result cited in the literature.[3]

Free particle motion in light-cone coordinates

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For a free particle of mass   the action is

 

In light-cone coordinates   becomes with   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.

See also

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References

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  1. ^ Dirac, P. A. M. (1 July 1949). "Forms of Relativistic Dynamics". Reviews of Modern Physics. 21 (392): 392–399. Bibcode:1949RvMP...21..392D. doi:10.1103/RevModPhys.21.392.
  2. ^ Zwiebach, Barton (2004). A first course in string theory. New York: Cambridge University Press. ISBN 978-0-511-21115-7. OCLC 560236176.
  3. ^ L. Susskind and J. Lindesay, Black Holes, Information and the String Theory Revolution, World Scientific (2004), ISBN 978-981-256-083-4, p. 163.