The space of physical events will be described by
a real, smooth manifold of dimension
coordinatised by local coordinates
, and provided with
smooth vector fields
with components
and linear forms
, (
) in the local coordinate system, as well as further
geometrical objects such as tensors, spinors, connections13
. At each point,
linearly independent vectors (linear forms) form a linear space,
the tangent space (cotangent space) of
. We will assume that the manifold
is space- and time-orientable. On it, two
independent fundamental structural
objects will now be introduced.
The first is a prescription for the definition
of the distance between two infinitesimally close points on
, eventually corresponding to temporal and spatial
distances in the external world. For
, we need
positivity, symmetry in the two points, and the validity of the
triangle equation. We know that
must be homogeneous of
degree one in the coordinate differentials
connecting the points. This condition is not very
restrictive; it still includes Finsler
geometry [280, 126
, 223] to be
briefly touched, below.
In the following, is linked to a
non-degenerate bilinear form
, called the
first fundamental form; the corresponding quadratic form
defines a tensor field, the metrical
tensor, with
components
such that
From this we note that an antisymmetric part of the metrical tensor does not influence distances and norms but angles.
With the metric tensor having full rank, its
inverse is defined through15
We are used to being a symmetric tensor field, i.e., with
and with
only
components; in this case the metric is
called Riemannian if its eigenvalues
are positive (negative) definite and Lorentzian if its signature is
16
. In the following this
need not hold, so that the decomposition obtains17:
For an asymmetric metric, the inverse
is determined by the relations and turns out to be [356The manifold is called space-time if and the metric
is symmetric and Lorentzian, i.e.,
symmetric and with signature
.
Nevertheless, sloppy contemporaneaous usage of the term
“space-time” includes arbitrary dimension, and sometimes is applied
even to metrics with arbitrary signature.
In a manifold with Lorentzian metric, a non-trivial real conformal structure always exists; from the equation
results an equivalence class of metricsA special case of a space with a Lorentzian metric is Minkowski space, whose metrical components, in Cartesian coordinates, are given by
A geometrical characterization of Minkowski space as an uncurved, flat space is given below. LetThe metric tensor may also be defined
indirectly through
vector fields forming an orthonormal
-leg (-bein)
with
A new physical
aspect will come in if the are considered to be the
basic geometric variables satisfying field equations, not the
metric. Such tetrad-theories (for the
case
) are described well by the concept of fibre bundle. The fibre at each point of the
manifold contains, in the case of an orthonormal
-bein (tetrad), all
-beins (tetrads) related to
each other by transformations of the group
, or the Lorentz group, and so on.
In Finsler geometry,
the line element depends not only on the coordinates of a point on the manifold, but also on the
infinitesimal elements of direction between neighbouring points
:
The second structure to be introduced is a
linear connection
with
components
; it is a geometrical object
but not a tensor field and its
components change inhomogeneously under local coordinate
transformations21
. The connection is a
device introduced for establishing a comparison of vectors in
different points of the manifold. By its help, a tensorial
derivative
, called covariant
derivative is constructed. For each vector field and each
tangent vector it provides another unique vector field. On the
components of vector fields
and linear forms
it is defined by
We have adopted the notational convention used by
Schouten [300, 310
, 389]. Eisenhart and others [121
, 233] change
the order of indices of the components of the connection:
A manifold provided with only a linear connection L is called affine space. From the point of view of group theory, the affine group (linear inhomogeneous coordinate transformations) plays a special role: With regard to it the connection transforms as a tensor (cf. Section 2.1.5).
For a vector density (cf. Section 2.1.5), the covariant
derivative of contains one more term:
A smooth vector field is said to be parallely transported along a parametrised
curve
with tangent vector
if for its
components
holds along the curve. A
curve is called an autoparallel if its
tangent vector is parallely transported along it at each point23
:
A transformation mapping autoparallels to autoparallels is given by:
The equivalence class of autoparallels defined by Equation (18The particular set of connections
withIn Part II of this article, we shall find
the set of transformations playing a role in versions of Einstein’s unified field theory.
From the connection further
connections may be constructed by adding an arbitrary tensor field
to its symmetrised part24
:
The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
For a vector density, the identity is given by with the homothetic curvatureThe curvature tensor (22) satisfies two
algebraic identities:
From both affine curvature tensors we may form
two different tensorial traces each. In the first case , and
is called homothetic curvature, while
is the first of the two affine generalisations from
and
of the Ricci
tensor in Riemannian geometry. We get26
In order to shorten the presentation of affine geometry, we refrain from listing the corresponding set of equations for the other affine curvature tensor (cf., however, [356]).
For a symmetric
affine connection, the preceding results reduce considerably due to
. From Equations (29
, 30
, 32
) we obtain the
identities:
In affine geometry, the simplest way to define a
fundamental tensor is to set , or
. It may be desirable to derive the
metric from a Lagrangian; then the simplest scalar density that
could be used as such is given by
28
.
As a final result in this section, we give the
curvature tensor calculated from the connection (cf. Equation (20
)), expressed by the
curvature tensor of
and by the tensor
:
From the symmetric part of the first fundamental
form , a connection may be constructed, often
called after Levi-Civita
[203
],
With the help of the symmetric affine connection,
we may define the tensor of non-metricity by29
Then the following identity holds:
where the contorsion tensorThe inner product of two tangent vectors is not conserved under
parallel transport of the vectors along
if the
non-metricity tensor does not vanish:
A connection for which the non-metricity tensor vanishes, i.e.,
holds, is called metric-compatible31. J. M. Thomas introduced a combination of the terms appearing inEinstein later used as a constraint on the metrical tensor
a condition that cannot easily be interpreted geometrically [97Connections that are not metric-compatible have been used in
unified field theory right from the beginning. Thus, in Weyl’s theory [397, 395
] we have
We may also abbreviate the last term in the
identity (42) by introducing
Riemann-Cartan geometry is the
subcase of a metric-affine geometry in which the metric-compatible connection contains
torsion, i.e., an antisymmetric part
; torsion is a tensor field to be linked to physical
observables. A linear connection whose antisymmetric part
has the form
Riemannian geometry
is the further subcase with vanishing
torsion of a metric-affine geometry with metric-compatible
connection. In this case, the connection is derived from the
metric: , where
is the usual Christoffel symbol (40
). The covariant
derivative of
with respect to the Levi-Civita connection
is abbreviated by
. The Riemann
curvature tensor is denoted by
An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:
This is an invariant characterisation irrespective of whether the Minkowski metricIn Riemanian geometry, the so-called geodesic equation,
determines the shortest and the straightest curve between two infinitesimally close points. However, in metric affine and in mixed geometry geodesic and autoparallel curves will have to be distinguished.A conformal transformation of the metric,
with a smooth functionEven before Weyl, the question had been asked
(and answered) as to what extent the conformal and the projective
structures were determining the geometry: According to Kretschmann
(and then to Weyl) they fix the metric up to a
constant factor ([195]; see also [401
], Appendix 1; for a
modern approach, cf. [67]).
The geometry needed for the pre- and non-relativistic approaches to unified field theory will have to be dealt with separately. There, the metric tensor of space is Euclidean and not of full rank; time is described by help of a linear form (Newton-Cartan geometry, cf. [65, 66]). In the following we shall deal only with relativistic unified field theories.
If we define , with
, then
transforms like a tangent
vector under point transformations of the
, and as a covariant vector under homogeneous
transformations of the
. The
may be used to relate covariant vectors
and
by
Thus,
the metric tensor in the space of homogeneous coordinates
and the metric tensor
of
are related by
with
The inverse relationship is given by
with
The covariant derivative for tensor fields in the
space of homogeneous coordinates is defined as before
(cf. Section 2.1.2):
In this section, we briefly present Cartan’s one-form formalism in
order to make understandable part of the literature. Cartan introduces one-forms (
) by
The reciprocal basis in tangent space
is given by
. Thus,
. The metric is then given by
. The covariant derivative of a tangent
vector with bein-components
is defined via Cartan’s first structure
equations,
By further external derivation35
on we arrive at the second structure relation of Cartan,
A relative tensor
of type
and of weight
at a point
on the manifold
transforms like
In connection with conformal transformations , the concept of the gauge-weight of a tensor is introduced. A
tensor
is said to be of gauge weight
if it transforms by
Equation (56
) as
Objects that transform as in Equation (67) but with respect to a
subgroup, e.g., the linear group,
affine group
, orthonormal group
, or the Lorentz group
, are tensors in a
restricted sense; sometimes they are named affine or Cartesian tensors. All the subgroups
mentioned are Lie-groups, i.e., continuous groups with a finite number of parameters. In general
relativity, both the “group” of general coordinate transformations
and the Lorentz group are present. The concept of tensors used in
Special Relativity is restricted to a representation
of the Lorentz group; however, as soon as the theory is to be given
a coordinate-independent (“generally covariant”) form, then the
full tensor concept comes into play.
Then, by a transformation from
,
Now, contravariant
2-spinors (
) are the
elements of a complex linear space, spinor
space, on which the matrices
are acting39
. The spinor is called
elementary if it transforms under a
Lorentz-transformation as
Higher-order spinors with dotted and undotted
indices transform correspondingly.
For the raising and lowering of indices now a real, antisymmetric
-matrix
with components
is needed, such that
Next to a spinor, bispinors of the form , etc. are the simplest quantities (spinors of
2nd order). A vector
can be represented by a bispinor
,
In order to write down spinorial field equations, we need a spinorial derivative,
withDirac- or 4-spinors with 4 components ,
, may be constructed from
2-spinors as a direct sum of contravariant undotted and covariant
dotted spinors
and
: For
, we enter
and
; for
, we enter
and
In connection with Dirac spinors,
instead of the Pauli-matrices the Dirac
-matrices (
-matrices) appear; they
satisfy
The generally-covariant formulation of spinor
equations necessitates the use of -beins
, whose internal “rotation” group, operating on the
“hatted” indices, is the Lorentz group. The group of coordinate
transformations acts on the Latin indices. In Cartan’s one-form formalism
(cf. Section 2.1.4), the covariant
derivative of a 4-spinor is defined by
Equation (89) is a special case of
the general formula for the covariant derivative of a tensorial form
, i.e., a vector in some
vector space
, whose components are differential
forms,
A Riemannian space is called (locally) stationary if it admits a timelike Killing
vector; it is called (locally) static
if this Killing vector is hypersurface orthogonal. Thus if, in a
special coordinate system, we take then from
Equation (91
) we conclude that
stationarity reduces to the condition
. If we take
to be the tangent vector field to the congruence of
curves
, i.e., if
, then a necessary and sufficient
condition for hypersurface-orthogonality is
A generalisation of Killing vectors are conformal Killing vectors for which with an arbitrary smooth function
holds. In purely affine spaces, another type of
symmetry may be defined:
; they are
called affine motions [425].