In
differential geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multili ...
and
mathematical physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and t ...
, a spin connection is a
connection on a
spinor bundle In differential geometry, given a spin structure on an n-dimensional orientable Riemannian manifold (M, g),\, one defines the spinor bundle to be the complex vector bundle \pi_\colon\to M\, associated to the corresponding principal bundle \pi_\c ...
. It is induced, in a canonical manner, from the
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
. It can also be regarded as the
gauge field
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
generated by local
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
s. In some canonical formulations of general relativity, a spin connection is defined on spatial slices and can also be regarded as the gauge field generated by local
rotations.
The spin connection occurs in two common forms: the ''Levi-Civita spin connection'', when it is derived from the
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
, and the ''affine spin connection'', when it is obtained from the affine connection. The difference between the two of these is that the Levi-Civita connection is by definition the unique
torsion-free connection, whereas the affine connection (and so the affine spin connection) may contain torsion.
Definition
Let
be the local Lorentz
frame fields or
vierbein
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independent ...
(also known as a tetrad), which is a set of orthonormal space time vector fields that diagonalize the metric tensor
:
where
is the spacetime metric and
is the
Minkowski metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (rel ...
. Here, Latin letters denote the local
Lorentz
Lorentz is a name derived from the Roman surname, Laurentius, which means "from Laurentum". It is the German form of Laurence. Notable people with the name include:
Given name
* Lorentz Aspen (born 1978), Norwegian heavy metal pianist and keyboar ...
frame indices; Greek indices denote general coordinate indices. This simply expresses that
, when written in terms of the basis
, is locally flat. The Greek vierbein indices can be raised or lowered by the metric, i.e.
or
. The Latin or "Lorentzian" vierbein indices can be raised or lowered by
or
respectively. For example,
and
The
torsion-free spin connection is given by
:
where
are the
Christoffel symbols
In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distance ...
. This definition should be taken as defining the torsion-free spin connection, since, by convention, the Christoffel symbols are derived from the
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
, which is the unique metric compatible, torsion-free connection on a Riemannian Manifold. In general, there is no restriction: the spin connection may also contain torsion.
Note that
using the gravitational covariant derivative
of the contravariant vector
. The spin connection may be written purely in terms of the vierbein field as
[M.B. Green, J.H. Schwarz, E. Witten, "Superstring theory", Vol. 2.]
:
which by definition is anti-symmetric in its internal indices
.
The spin connection
defines a covariant derivative
on generalized tensors. For example, its action on
is
:
Cartan's structure equations
In the
Cartan formalism
The tetrad formalism is an approach to general relativity that generalizes the choice of basis for the tangent bundle from a coordinate basis to the less restrictive choice of a local basis, i.e. a locally defined set of four linearly independen ...
, the spin connection is used to define both torsion and curvature. These are easiest to read by working with
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
s, as this hides some of the profusion of indexes. The equations presented here are effectively a restatement of those that can be found in the article on the
connection form In mathematics, and specifically differential geometry, a connection form is a manner of organizing the data of a connection using the language of moving frames and differential forms.
Historically, connection forms were introduced by Élie Carta ...
and the
curvature form In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie algebra ...
. The primary difference is that these retain the indexes on the vierbein, instead of completely hiding them. More narrowly, the Cartan formalism is to be interpreted in its historical setting, as a generalization of the idea of an
affine connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values i ...
to a
homogeneous space
In mathematics, particularly in the theories of Lie groups, algebraic groups and topological groups, a homogeneous space for a group ''G'' is a non-empty manifold or topological space ''X'' on which ''G'' acts transitively. The elements of ' ...
; it is not yet as general as the idea of a
principal connection
In mathematics, and especially differential geometry and gauge theory, a connection is a device that defines a notion of parallel transport on the bundle; that is, a way to "connect" or identify fibers over nearby points. A principal ''G''-conne ...
on a
fiber bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a p ...
. It serves as a suitable half-way point between the narrower setting in
Riemannian geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to poin ...
and the fully abstract fiber bundle setting, thus emphasizing the similarity to
gauge theory
In physics, a gauge theory is a type of field theory in which the Lagrangian (and hence the dynamics of the system itself) does not change (is invariant) under local transformations according to certain smooth families of operations (Lie groups) ...
. Note that Cartan's structure equations, as expressed here, have a direct analog: the
Maurer–Cartan equations for
Lie group
In mathematics, a Lie group (pronounced ) is a group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additio ...
s (that is, they are the same equations, but in a different setting and notation).
Writing the vierbeins as differential forms
:
for the orthonormal coordinates on the
cotangent bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may ...
, the affine spin connection one-form is
:
The
torsion
Torsion may refer to:
Science
* Torsion (mechanics), the twisting of an object due to an applied torque
* Torsion of spacetime, the field used in Einstein–Cartan theory and
** Alternatives to general relativity
* Torsion angle, in chemistry
Bi ...
2-form is given by
:
while the
curvature 2-form is
:
These two equations, taken together are called Cartan's structure equations.
[Tohru Eguchi, Peter B. Gilkey and Andrew J. Hanson,]
Gravitation, Gauge Theories and Differential Geometry
, ''Physics Reports'' 66 (1980) pp 213-393.
Consistency requires that the
Bianchi identities In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case.
Definition
Let ''G'' be a Lie group with Lie alge ...
be obeyed. The first Bianchi identity is obtained by taking the exterior derivative of the torsion:
:
while the second by differentiating the curvature:
:
The covariant derivative for a generic
differential form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, ...
of degree ''p'' is defined by
:
Bianchi's second identity then becomes
:
The difference between a connection with torsion, and the unique torsionless connection is given by the
contorsion tensor The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subje ...
. Connections with torsion are commonly found in theories of
teleparallelism
Teleparallelism (also called teleparallel gravity), was an attempt by Albert Einstein to base a unified theory of electromagnetism and gravity on the mathematical structure of distant parallelism, also referred to as absolute or teleparallelism. In ...
,
Einstein–Cartan theory
In theoretical physics, the Einstein–Cartan theory, also known as the Einstein–Cartan–Sciama–Kibble theory, is a classical theory of gravitation similar to general relativity. The theory was first proposed by Élie Cartan in 1922. Einstein ...
,
gauge theory gravity
Gauge theory gravity (GTG) is a theory of gravitation cast in the mathematical language of geometric algebra. To those familiar with general relativity, it is highly reminiscent of the tetrad formalism although there are significant conceptual d ...
and
supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
.
Derivation
Metricity
It is easy to deduce by raising and lowering indices as needed that the
frame fields defined by
will also satisfy
and
. We expect that
will also annihilate the Minkowski metric
,
:
This implies that the connection is anti-symmetric in its internal indices,
This is also deduced by taking the gravitational covariant derivative
which implies that
thus ultimately,
. This is sometimes called the metricity condition;
it is analogous to the more commonly stated metricity condition that
Note that this condition holds only for the Levi-Civita spin connection, and not for the affine spin connection in general.
By substituting the formula for the Christoffel symbols
written in terms of the
, the spin connection can be written entirely in terms of the
,
:
where antisymmetrization of indices has an implicit factor of 1/2.
By the metric compatibility
This formula can be derived another way. To directly solve the compatibility condition for the spin connection
, one can use the same trick that was used to solve
for the Christoffel symbols
. First contract the compatibility condition to give
:
.
Then, do a cyclic permutation of the free indices
and
, and add and subtract the three resulting equations:
:
where we have used the definition
. The solution for the spin connection is
:
.
From this we obtain the same formula as before.
Applications
The spin connection arises in the
Dirac equation
In particle physics, the Dirac equation is a relativistic wave equation derived by British physicist Paul Dirac in 1928. In its free form, or including electromagnetic interactions, it describes all spin- massive particles, called "Dirac par ...
when expressed in the language of
curved spacetime
Curved space often refers to a spatial geometry which is not "flat", where a flat space is described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry though some simple cases can be described in other ways. Cu ...
, see
Dirac equation in curved spacetime
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold.
Mathematical formulation
Spacetime
In full ...
. Specifically there are problems coupling gravity to
spinor
In geometry and physics, spinors are elements of a complex vector space that can be associated with Euclidean space. Like geometric vectors and more general tensors, spinors transform linearly when the Euclidean space is subjected to a sligh ...
fields: there are no finite-dimensional spinor representations of the
general covariance group. However, there are of course spinorial representations of the
Lorentz group
In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (non-gravitational) physical phenomena. The Lorentz group is named for the Dutch physicis ...
. This fact is utilized by employing tetrad fields describing a flat tangent space at every point of spacetime. The
Dirac matrices
In mathematical physics, the gamma matrices, \left\ , also called the Dirac matrices, are a set of conventional matrices with specific anticommutation relations that ensure they generate a matrix representation of the Clifford algebra Cl1,3(\ma ...
are contracted onto vierbiens,
:
We wish to construct a generally covariant Dirac equation. Under a flat tangent space
Lorentz transformation
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that moves at a constant velocity relative to the former. The respective inverse transformation i ...
the spinor transforms as
:
We have introduced local Lorentz transformations on flat tangent space generated by the
's, such that
is a function of space-time. This means that the partial derivative of a spinor is no longer a genuine tensor. As usual, one introduces a connection field
that allows us to gauge the Lorentz group. The covariant derivative defined with the spin connection is,
:
and is a genuine tensor and Dirac's equation is rewritten as
:
The generally covariant fermion action couples fermions to gravity when added to the first order
tetradic Palatini action
The Einstein–Hilbert action for general relativity was first formulated purely in terms of the space-time metric. To take the metric and affine connection as independent variables in the action principle was first considered by Palatini. It is ...
,
:
where
and
is the curvature of the spin connection.
The tetradic Palatini formulation of general relativity which is a first order formulation of the
Einstein–Hilbert action
The Einstein–Hilbert action (also referred to as Hilbert action) in general relativity is the action that yields the Einstein field equations through the stationary-action principle. With the metric signature, the gravitational part of the ac ...
where the tetrad and the spin connection are the basic independent variables. In the 3+1 version of Palatini formulation, the information about the spatial metric,
, is encoded in the triad
(three-dimensional, spatial version of the tetrad). Here we extend the metric compatibility condition
to
, that is,
and we obtain a formula similar to the one given above but for the spatial spin connection
.
The spatial spin connection appears in the definition of
Ashtekar–Barbero variables which allows 3+1 general relativity to be rewritten as a special type of
Yang–Mills gauge theory. One defines
. The Ashtekar–Barbero connection variable is then defined as
where
and
is the extrinsic
curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.
For curves, the canonic ...
and
is the
Immirzi parameter. With
as the configuration variable, the conjugate momentum is the densitized triad
. With 3+1 general relativity rewritten as a special type of
Yang–Mills gauge theory, it allows the importation of non-perturbative techniques used in
Quantum chromodynamics
In theoretical physics, quantum chromodynamics (QCD) is the theory of the strong interaction between quarks mediated by gluons. Quarks are fundamental particles that make up composite hadrons such as the proton, neutron and pion. QCD is a type ...
to canonical quantum general relativity.
See also
*
Ashtekar variables
In the ADM formulation of general relativity, spacetime is split into spatial slices and a time axis. The basic variables are taken to be the induced metric q_ (x) on the spatial slice and the metric's conjugate momentum K^ (x), which is related ...
*
Dirac operator
In mathematics and quantum mechanics, a Dirac operator is a differential operator that is a formal square root, or half-iterate, of a second-order operator such as a Laplacian. The original case which concerned Paul Dirac was to factorise formall ...
*
Cartan connection
In the mathematical field of differential geometry, a Cartan connection is a flexible generalization of the notion of an affine connection. It may also be regarded as a specialization of the general concept of a principal connection, in which the ...
*
Levi-Civita connection
In Riemannian or pseudo Riemannian geometry (in particular the Lorentzian geometry of general relativity), the Levi-Civita connection is the unique affine connection on the tangent bundle of a manifold (i.e. affine connection) that preserves th ...
*
Ricci calculus
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern name for what used to be cal ...
*
Supergravity
In theoretical physics, supergravity (supergravity theory; SUGRA for short) is a modern field theory that combines the principles of supersymmetry and general relativity; this is in contrast to non-gravitational supersymmetric theories such as ...
*
Torsion tensor
In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve. The torsion of a curve, as it appears in the Frenet–Serret formulas, for instance, quantifies the twist of a curve ...
*
Contorsion tensor The contorsion tensor in differential geometry is the difference between a connection with and without torsion in it. It commonly appears in the study of spin connections. Thus, for example, a vielbein together with a spin connection, when subje ...
*
Dirac equation in curved spacetime
In mathematical physics, the Dirac equation in curved spacetime is a generalization of the Dirac equation from flat spacetime (Minkowski space) to curved spacetime, a general Lorentzian manifold.
Mathematical formulation
Spacetime
In full ...
References
{{Reflist
*Hehl, F.W.; von der Heyde, P.; Kerlick, G.D.; Nester, J.M. (1976)
"General relativity with spin and torsion: Foundations and prospects" Rev. Mod. Phys. 48, 393.
*
Kibble, T.W.B. (1961)
"Lorentz invariance and the gravitational field" J. Math. Phys. 2, 212.
*
Poplawski, N.J. (2009), "Spacetime and fields"
arXiv:0911.0334*
Sciama, D.W. (1964)
"The physical structure of general relativity" Rev. Mod. Phys. 36, 463.
Connection (mathematics)
Spinors
Differential geometry