In
physics
Physics is the natural science that studies matter, its fundamental constituents, its motion and behavior through space and time, and the related entities of energy and force. "Physical science is that department of knowledge which r ...
, particularly
special relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory regarding the relationship between space and time. In Albert Einstein's original treatment, the theory is based on two postulates:
# The laws o ...
, light-cone coordinates, introduced by
Paul Dirac
Paul Adrien Maurice Dirac (; 8 August 1902 – 20 October 1984) was an English theoretical physicist who is regarded as one of the most significant physicists of the 20th century. He was the Lucasian Professor of Mathematics at the Univer ...
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 number
In algebra, a split complex number (or hyperbolic number, also perplex number, double number) has two real number components and , and is written z=x+yj, where j^2=1. The ''conjugate'' of is z^*=x-yj. Since j^2=1, the product of a number wi ...
s which is acted upon by elements of the
unit hyperbola
In geometry, the unit hyperbola is the set of points (''x'',''y'') in the Cartesian plane that satisfy the implicit equation x^2 - y^2 = 1 . In the study of indefinite orthogonal groups, the unit hyperbola forms the basis for an ''alternative radi ...
to effect Lorentz boosts. This number plane has axes corresponding to time and space. An alternative
basis
Basis may refer to:
Finance and accounting
*Adjusted basis, the net cost of an asset after adjusting for various tax-related items
*Basis point, 0.01%, often used in the context of interest rates
*Basis trading, a trading strategy consisting of ...
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
null vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which .
In the theory of real number, real bilinear forms, definite quadrat ...
s 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
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of ...
)
:
,
with
we have
:
with
,
and
.
Both
and
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.
A boost in the
plane shows up as the
squeeze mapping
In linear algebra, a squeeze mapping, also called a squeeze transformation, is a type of linear map that preserves Euclidean area of regions in the Cartesian plane, but is ''not'' a rotation or shear mapping.
For a fixed positive real number , th ...
,
,
. 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
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
.
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
In theoretical physics, light cone gauge is an approach to remove the ambiguities arising from a gauge symmetry. While the term refers to several situations, a null component of a field ''A'' is set to zero (or a simple function of other variables ...
of string theory.
Light-cone coordinates in 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
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
Noether's theorem or Noether's first theorem states that every differentiable symmetry of the action of a physical system with conservative forces has a corresponding conservation law. The theorem was proven by mathematician Emmy Noether in ...
. 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.
[L. Susskind and J. Lindesay, Black Holes, Information and the String Theory Revolution, World Scientific (2004), , p. 163.]
Free particle motion in light-cone coordinates
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
*
Newman–Penrose formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Sp ...
References
{{Reflist
Theory of relativity