In the mathematical description of
general relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of grav ...
, the Boyer–Lindquist coordinates
are a generalization of the
coordinates used for the
metric of a
Schwarzschild black hole that can be used to express the metric of a
Kerr black hole.
The
Hamiltonian
Hamiltonian may refer to:
* Hamiltonian mechanics, a function that represents the total energy of a system
* Hamiltonian (quantum mechanics), an operator corresponding to the total energy of that system
** Dyall Hamiltonian, a modified Hamiltonian ...
for particle motion in Kerr spacetime is separable in Boyer–Lindquist coordinates. Using
Hamilton–Jacobi theory one can derive a fourth constant of the motion known as
Carter's constant.
The 1967 paper introducing Boyer–Lindquist coordinates
was a posthumous publication for Robert H. Boyer, who was killed in the 1966
University of Texas tower shooting.
Line element
The
line element for a black hole with a total
mass equivalent , angular momentum
, and charge
in Boyer–Lindquist coordinates and
geometrized units (
) is
:
or equivalently
:
where
:
called the ''discriminant'',
:
and
:
called the ''Kerr parameter''.
Note that in geometrized units
,
, and
all have units of length. This line element describes the
Kerr–Newman metric. Here,
is to be interpreted as the
mass
Mass is an Intrinsic and extrinsic properties, intrinsic property of a physical body, body. It was traditionally believed to be related to the physical quantity, quantity of matter in a body, until the discovery of the atom and particle physi ...
of the black hole, as seen by an observer at infinity,
is interpreted as the
angular momentum
Angular momentum (sometimes called moment of momentum or rotational momentum) is the rotational analog of Momentum, linear momentum. It is an important physical quantity because it is a Conservation law, conserved quantity – the total ang ...
, and
the
electric charge
Electric charge (symbol ''q'', sometimes ''Q'') is a physical property of matter that causes it to experience a force when placed in an electromagnetic field. Electric charge can be ''positive'' or ''negative''. Like charges repel each other and ...
. These are all meant to be constant parameters, held fixed. The name of the discriminant arises because it appears as the discriminant of the quadratic equation bounding the time-like motion of particles orbiting the black hole, ''i.e.'' defining the ergosphere.
The coordinate transformation from Boyer–Lindquist coordinates
,
,
to Cartesian coordinates
,
,
is given (for
) by:
[Matt Visser, arXiv:0706.0622v3, eqs. 60-62]
Vierbein
The
vierbein one-forms can be read off directly from the line element:
:
:
:
:
where
which is
outside the outer horizon,
between the inner and outer horizons, and
inside the inner horizon. The line element is given by
:
In the region outside the outer horizon,
:
where
is the flat-space
Minkowski metric
In physics, Minkowski space (or Minkowski spacetime) () is the main mathematical description of spacetime in the absence of general_relativity, gravitation. It combines inertial space and time manifolds into a four-dimensional model.
The model ...
.
Spin connection
The
torsion-free spin connection
In differential geometry and mathematical physics, a spin connection is a connection (vector bundle), connection on a spinor bundle. It is induced, in a canonical manner, from the affine connection. It can also be regarded as the gauge field gene ...
is defined by
:
The
contorsion tensor gives the difference between a connection with torsion, and a corresponding connection without torsion. By convention, Riemann manifolds are always specified with torsion-free geometries; torsion is often used to specify equivalent, flat geometries.
The spin connection is useful because it provides an intermediate way-point for computing the
curvature two-form:
:
It is also the most suitable form for describing the coupling to
spinor fields, and opens the door to the
twistor formalism.
All six components of the spin connection are non-vanishing. These are:
[Pietro Giuseppe Frè, "Gravity, a Geometrical Course, Volume 2: Black Holes, Cosmology and Introduction to Supergravity", (2013) Springer-Verlag]
: