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,
connections^{13}.
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 [281, 126, 224] 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

where the neighbouring points are labeled by and , respectivelyFrom 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
through^{15}

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
obtains^{17}:

For an asymmetric metric, the inverse

is determined by the relations and turns out to be [356] where , , and are the determinants of the corresponding tensors , , and . We also note that where , . The results (6,7,8) were obtained already by Reichenbächer ([273], pp. 223–224)The 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 metrics with being an arbitrary smooth function. In view of the physical interpretation of the light cone as the locus of light signals, a causal structure is provided by the equivalence class of metrics [67]. For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part of the metric alone taken to be Lorentzian.A 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. Let be the Lie derivative with respect to the tangent vector XThe metric tensor may also be defined indirectly through vector fields forming an orthonormal -leg (-bein) with

where the hatted indices (“bein-indices”) count the number of legs spanning the tangent space at each point and are moved with the Minkowski metricA 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 :

Again, is required to be homogeneous of rank 1.

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
transformations^{21}.
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, 234] change the order of indices of the components of the connection:

As long as the connection is symmetric, this does not make any difference as For both kinds of derivatives we have: Both derivatives are used in versions of unified field theory by Einstein and othersA 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
point^{23}:

A transformation mapping autoparallels to autoparallels is given by:

The equivalence class of autoparallels defined by Equation (18) defines a projective structure on [404, 403].The particular set of connections

with is mapped into itself by the transformation (18) [348].In 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 part^{24}:

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 curvature to be defined below in Equation (31).The curvature tensor (22) satisfies two algebraic identities:

where the curly bracket denotes cyclic permutation: 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
get^{26}

For a symmetric affine connection, the preceding results reduce considerably due to . From Equations (29,30,32) we obtain the identities:

i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor holds. This equation will be important for the physical interpretation of affine geometry. 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 :

where is the covariant derivative formed with the connection (cf. also [310], p. 141).

From the symmetric part of the first fundamental form , a connection may be constructed, often called after Levi-Civita [204],

and from it the Riemannian curvature tensor defined as in Equation (22) with (cf. Section 2.1.3); is called the Christoffel symbol. Thus, in metric-affine and in mixed geometry, two different connections arise in a natural way. In the remaining part of this section we will deal with a symmetric fundamental form only, and denote it by . With the help of the symmetric affine connection, we may define the tensor of non-metricity byThen the following identity holds:

where the contorsion tensor , a linear combination of torsion , is defined byThe 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-compatibleConnections 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

In case of such a relationship, the geometry is called semi-metrical [300, 310]. According to Equation (44), in Weyl’s theory the inner product multiplies by a scalar factor under parallel transport: This means that the light cone is preserved by parallel transport.We may also abbreviate the last term in the identity (42) by introducing

Then, from Equation (39), the curvature tensor of a torsionless affine space is given by where is the covariant derivative formed with the Christoffel symbol.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

is called semi-symmetric [300].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 metric is given in Cartesian coordinates as in Equation (10), or in an arbitrary coordinate system. We also have where is the Lie-derivative (see below under “symmetries”), and stands for the generators of the Lorentz group.In 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 function changes the components of the non-metricity tensor, as well as the Levi-Civita connection, with As a consequence, the Riemann curvature tensor is also changed; if, however, can be reached by a conformal transformation, then the corresponding space-time is called conformally flat. In , for , the vanishing of the Weyl curvature tensor is a necessary and sufficient condition for to be conformally flat ([397], p. 404, [300], p. 170).Even 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 ([196]; 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):

The covariant derivative of the quantity interconnecting both spaces is given by

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,

where is the connection-1-form, and is the torsion-2-form, . We have The link to the components of the affine connection is given by By further external derivation^{35}
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

An example is given by the totally antisymmetric object with , or depending on whether is an even or odd permutation of (0123), or whether two indices are alike. for ; in this case, the relative tensor is called tensor density. We can form a tensor from by introducing , where is a Lorentz-metric. Note that The dual to a 2-form (skew-symmetric tensor) then is defined byIn 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 ,

where is the Hermitian conjugate matrix Now, contravariant 2-spinors () are the elements of a complex linear space, spinor space, on which the matrices
are acting^{39}.
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 ,

where () is a quantity linking the tangent space of space-time and spinor space. If numerates the matrices and , designate rows and columns, then we can chose to be the unit matrix while for the other three indices are taken to be the Pauli matrices. Often the quantity is introduced. The reciprocal matrix is defined by whereasIn order to write down spinorial field equations, we need a spinorial derivative,

with The simplest spinorial equation is the Weyl equation: The next simplest spinor equation for two spinors would be where is a mass. Equation (85) is the 2-spinor version of Dirac’s equation.Dirac- 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 Dirac equation is in 4-spinor formalism [53, 54]: with the 4-component Dirac spinor . In the first version of Dirac’s equation, - and -matrices were used, related to the ’s by where the matrices and are given by , .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

where .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,

where is a particular representation of the corresponding Lie algebra in with basis vectors For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used

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].

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