Brinkmann coordinates
Encyclopedia
Brinkmann coordinates are a particular coordinate system
for a spacetime
belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor
can be written as
where
, the coordinate vector
field dual to the covector field
, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars
. Physically speaking, it serves as the wave vector
defining the direction of propagation
for the pp-wave.
The coordinate vector field
can be spacelike, null, or timelike at a given event in the spacetime
, depending upon the sign of
at that event. The coordinate vector fields 
are both spacelike vector fields. Each surface 
can be thought of as a wavefront
.
In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended range
of the coordinate variables
. Here we should take
to allow for the possibility that the pp-wave develops a null curvature singularity.
Coordinate system
In geometry, a coordinate system is a system which uses one or more numbers, or coordinates, to uniquely determine the position of a point or other geometric element. The order of the coordinates is significant and they are sometimes identified by their position in an ordered tuple and sometimes by...
for a spacetime
Spacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
belonging to the family of pp-wave metrics. In terms of these coordinates, the metric tensor
Metric tensor
In the mathematical field of differential geometry, a metric tensor is a type of function defined on a manifold which takes as input a pair of tangent vectors v and w and produces a real number g in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean...
can be written as
where
, the coordinate vectorCoordinate vector
In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn....
field dual to the covector field
, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalarsOptical scalars
In general relativity, optical scalars are a set of scalars that describe various properties of null geodesic congruences. The three optical scalars used in general relativity are expansion, shear and twist and were first defined and used by Sachs...
. Physically speaking, it serves as the wave vector
Wave vector
In physics, a wave vector is a vector which helps describe a wave. Like any vector, it has a magnitude and direction, both of which are important: Its magnitude is either the wavenumber or angular wavenumber of the wave , and its direction is ordinarily the direction of wave propagation In...
defining the direction of propagation
Wave propagation
Wave propagation is any of the ways in which waves travel.With respect to the direction of the oscillation relative to the propagation direction, we can distinguish between longitudinal wave and transverse waves....
for the pp-wave.
The coordinate vector field
can be spacelike, null, or timelike at a given event in the spacetimeSpacetime
In physics, spacetime is any mathematical model that combines space and time into a single continuum. Spacetime is usually interpreted with space as being three-dimensional and time playing the role of a fourth dimension that is of a different sort from the spatial dimensions...
, depending upon the sign of
at that event. The coordinate vector fields are both spacelike vector fields. Each surface can be thought of as a wavefrontWavefront
In physics, a wavefront is the locus of points having the same phase. Since infrared, optical, x-ray and gamma-ray frequencies are so high, the temporal component of electromagnetic waves is usually ignored at these wavelengths, and it is only the phase of the spatial oscillation that is described...
.
In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended range
Range
-In mathematics:* Range , the set of all output values produced by a function* Interval , also called a range, a set of real numbers that includes all numbers between any two numbers in the set...
of the coordinate variables
Variable (mathematics)
In mathematics, a variable is a value that may change within the scope of a given problem or set of operations. In contrast, a constant is a value that remains unchanged, though often unknown or undetermined. The concepts of constants and variables are fundamental to many areas of mathematics and...
. Here we should taketo allow for the possibility that the pp-wave develops a null curvature singularity.