Light-cone coordinates

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. The former can be denoted $x^+$ and $x^-$ and the latter $x\_\backslash perp$.

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

Instead of the standard coordinate system (using Einstein notation)

:$ds^2=-dt^2+\backslash delta\_\{ij\}dx^i\; dx^j$,

with $i,j=1,\backslash dots,d$ we have

:$ds^2=-2dx^+dx^-\; +\; \backslash delta\_\{ij\}dx^i\; dx^j$

with $i,j=1,\backslash dots,d-1$, $x^+=\backslash frac\{t+x\}\{\backslash sqrt\{2\}\}$ and $x^-=\backslash frac\{t-x\}\{\backslash sqrt\{2\}\}$.

Both $x^+$ and $x^-$ can act as "time" coordinates.{{Cite book|last=Zwiebach|first=Barton|url=https://www.worldcat.org/oclc/560236176|title=A first course in string theory|date=2004|publisher=Cambridge University Press|isbn=978-0-511-21115-7|location=New York|oclc=560236176|author-link=Barton Zwiebach}}{{Rp|page=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 $(t,x)$ plane shows up as the squeeze mapping $x^+\; \backslash to\; e^\{+\backslash beta\}x^+$, $x^-\; \backslash to\; e^\{-\backslash beta\}x^-$, $x^i\; \backslash to\; x^i$. A rotation in the $(i,j)$-plane only affects $x\_\backslash perp$.

The parabolic transformations show up as $x^+\; \backslash to\; x^+$, $x^-\; \backslash to\; x^-\; +\; \backslash delta\_\{ij\}\backslash alpha^i\; x^j\; +\; \backslash frac\{\backslash alpha^2\}\{2\}\; x^+$, $x^i\; \backslash to\; x^i\; +\; \backslash alpha^i\; x^+$. Another set of parabolic transformations show up as $x^+\; \backslash to\; x^+\; +\; \backslash delta\_\{ij\}\backslash alpha^i\; x^j\; +\; \backslash frac\{\backslash alpha^2\}\{2\}\; x^-$, $x^-\; \backslash to\; x^-$ and $x^i\; \backslash to\; x^i\; +\; \backslash alpha^i\; x^-$.

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.

A closed string is a generalization of a particle. The spatial coordinate of a point on the string is conveniently described by a parameter $\backslash sigma$ which runs from $0$ to $2\backslash pi$. Time is appropriately described by a parameter $\backslash sigma\_0$. Associating each point on the string in a D-dimensional spacetime with coordinates $x\_0,\; x$ and transverse coordinates $x\_i,\; i=2,...,D$, these coordinates play the role of fields in a $1+1$ dimensional field theory. Clearly, for such a theory more is required. It is convenient to employ instead of $x\_0=\backslash sigma\_0$ and $x$, light-cone coordinates $x\_\{\backslash pm\}$ given by

:$x\_\{\backslash pm\}=\backslash frac\{1\}\{\backslash sqrt\; 2\}(x\_0\; \backslash pm\; x)$

so that the metric $ds^2$ is given by

:$ds^2\; =\; 2dx\_+dx\_-\; -(dx\_i)^2$

(summation over $i$ understood).

There is some gauge freedom. First, we can set $x\_+=\backslash sigma\_0$ and treat this degree of freedom as the time variable. A reparameterization invariance under $\backslash sigma\; \backslash rightarrow\; \backslash sigma\; +\; \backslash delta\backslash sigma$ can be imposed with a constraint $\{\backslash mathcal\; L\}\_0\; =0$ which we obtain from the metric, i.e.

:$\{\backslash mathcal\; L\}\_0\; =\; \backslash frac\{dx\_-\}\{d\backslash sigma\}\; -\; \backslash frac\{dx\_i\}\{d\backslash sigma\}\backslash frac\{dx\_i\}\{d\backslash sigma\_0\}=0.$

Thus $x\_-$ is not an independent degree of freedom anymore. Now $\{\backslash mathcal\; L\}\_0$ can be identified as the corresponding Noether charge. Consider $\{\backslash mathcal\; L\}\_0(x\_-,x\_i)$. Then with the use of the Euler-Lagrange equations for $x\_i$ and $x\_-$ one obtains

:$\backslash delta\{\backslash mathcal\; L\}\_0\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash sigma\}\backslash bigg(\backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\_0\}\{\backslash partial(\backslash partial\; x\_i/\backslash partial\; \backslash sigma)\}\backslash delta\; x\_i\; +\; \backslash delta\; x\_-\backslash bigg).$

Equating this to

:$\backslash delta\{\backslash mathcal\; L\}\_0\; =\; \backslash frac\{\backslash partial\}\{\backslash partial\; \backslash sigma\}(Q\backslash delta\backslash sigma),$

where $Q$ is the Noether charge, we obtain:

:$Q=\backslash frac\{\backslash partial\{\backslash mathcal\; L\}\_0\}\{\backslash partial(\backslash partial\; x\_i/\backslash partial\backslash sigma)\}\backslash frac\{\backslash delta\; x\_i\}\{\backslash delta\; \backslash sigma\}$

+ \frac{\delta x_-}{\delta \sigma}

= -\frac{dx_i}{d\sigma_0}\frac{\delta x_i}{\delta \sigma}

+ \frac{\delta x_-}{\delta \sigma} = {\mathcal L}_0.

This result agrees with a result cited in the literature.L. Susskind and J. Lindesay, Black Holes, Information and the String Theory Revolution, World Scientific (2004), {{ISBN|978-981-256-083-4}}, p. 163.

For a free particle of mass $m$ the action is

:$S=\backslash int\{\backslash mathcal\; L\}d\backslash sigma,\; \backslash ;\backslash ;\backslash ;\; \{\backslash mathcal\; L\}\; =\; -\backslash frac\{1\}\{2\}\backslash bigg[\backslash frac\{dx^\{\backslash mu\}\}\{d\backslash sigma\}\backslash frac\{dx\_\{\backslash mu\}\}\{d\backslash sigma\}\; +m^2\backslash bigg].$

In light-cone coordinates $\{\backslash mathcal\; L\}$ becomes with $\backslash sigma\; =\; x\_+$ as time variable:

:$\{\backslash mathcal\; L\}\; =\; -\backslash frac\{dx\_-\}\{d\backslash sigma\}\; +\; \backslash frac\{1\}\{2\}\backslash bigg(\backslash frac\{dx\_i\}\{d\backslash sigma\}\backslash bigg)^2\; -\; \backslash frac\{m^2\}\{2\}.$

The canonical momenta are

:$p\_-=\backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial(dx\_-/d\backslash sigma)\}=\; -1,\; \backslash ;\backslash ;\backslash ;\; p\_i\; =\backslash frac\{\backslash partial\; \{\backslash mathcal\; L\}\}\{\backslash partial(\; dx\_i/d\backslash sigma)\}=\backslash frac\{dx\_i\}\{d\backslash sigma\}.$

The Hamiltonian is ($\backslash hbar\; =\; c\; =\; 1$):

:$\{\backslash mathcal\; H\}\; =\; \backslash dot\{x\}\_-p\_-\; +\; \backslash dot\{x\}\_ip\_i\; -\; \{\backslash mathcal\; L\}\; =\; \backslash frac\{1\}\{2\}p\_i^2\; +\; \backslash frac\{1\}\{2\}m^2,$

and the nonrelativistic Hamilton equations imply:

:$x\_i(\backslash sigma)\; =\; p\_i\backslash sigma\; +\; \{\backslash it\; const.\}.$

One can now extend this to a free string.

• Newman–Penrose formalism

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Category:Theory of relativity