| • Science | • People | • Locations | • Timeline |
where ∇ is the affine connection and u and v are vector fields and [,] is the Lie bracket. (see Lie algebra for the Lie bracket definition)
A Riemannian geometry is a Riemann-Cartan geometry with zero torsion, so it is uniquely determined by a metric tensor.
In (pseudo) Riemannian geometry, we have an n dimensional differential manifold M and a Riemannian metric g which is a linear map mapping two tangent vectors into a real number and a Levi-Civita connectionIn Riemannian geometry, the Levi-Civita connection (named for Tullio Levi-Civita) is the torsion-free connection of the tangent bundle, preserving a given Riemannian metric (or pseudo-Riemannian metric). The Fundamental theorem of Riemannian geometry stat which is a connection over the tangent bundleIn mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. The tangent bundle of manifold M is usually denot which can be associatedIn mathematics, the theory of fiber bundles with a structure group (a topological group) allows an operation of creating an associated bundle in which the typical fiber of a bundle changes from to , which are both topological spaces with a group action of with a connectionIn differential geometry, the connection form describes connection on principal bundles (or vector bundles). It can be considered as an generalization/alternative for Christoffel symbols. Principal bundles For a principal G bundle , for each let denote th over a principal GL(n,R)-bundle although it turns out the holonomyDifferential geometry Riemannian geometry In differential geometry, the holonomy of a given structure (for example a Riemannian metric, or more general G-structure) at a point P on a smooth manifold M is the group of all linear maps transforming the tange is merely SO(p,q). The structure group is the general linear groupAbstract algebra Algebra Linear algebra Lie groups In mathematics, the general linear group of degree n over a field F (such as R or C , written as GL ''n F , is the group of n ''n invertible matrices with entries from F with the group operation that of o GL(n,R).
The Einstein-Cartan is formulated differently. We still work with M, but this time we work with ANOTHER vector bundle T (and also possibly spinor bundles S) with the structure group R4spin(p,q) where p+q=n is the metric signature (i.e. the double cover of the Poincaré group). Here, T is invariant under translations (i.e. T isn't faithful). Although we can have ANOTHER affine bundle (note the affine part) V where translations act freely on V.
Note that this is different from the tetrad approach where the structure group is spin(p,q) AND we have tetrads (an isomorphism between TM and T) and the metric g is the push forward via the tetrad and the connection is associated with a connection over a principal spin(p,q)-bundle.
In the Einstein-Cartan theory, the metric g is a linear map mapping two elements of the fiber of T (not TM) to a real number and the connection is a connection over a principal R4spin(p,q)-bundle. This can be associated with a LINEAR connection over T, but only in a one directional manner. It can also be associated with an AFFINE connection (affine in the TRUE sense of the word) over V in a bidirectional manner.
It turns out the Riemann tensor is the curvature form for (generalized to include boosts) rotations (i.e. the spin(p,q) part) while torsion is the curvature form for translations (R4.
As the master theory of classical physics, general relativity has one known flaw: it cannot adequately describe exchange of intrinsic angular momentum ( spin) and orbital angular momentum . The problem is rooted in the foundations of general relativity. General relativity is based on Riemannian geometry, in which the Ricci curvature tensor
must be symmetric in i and j (that is, Rij = Rji . In general relativity, Rij models local gravitational forces, and its symmetry causes the momentum tensor
to be symmetric, so that general relativity cannot accommodate the general equation of conservation of angular momentum
A geometric interpretation of affine torsion comes from continuum mechanics of solid materials. Affine torsion is the continuum approximation to the density of dislocations that are studied in metallurgy and crystallography. The simplest kinds of dislocations in real crystals are
We can think of a Riemann-Cartan geometry as uniquely determined by the lengths and angles of vectors and the density of dislocations in the affine structure of the space.
General relativity set the affine torsion to zero, because it did not appear necessary to provide a model of gravitation (with a consistent set of equations that led to a well-defined initial value problem).