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 [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 that14. Besides the norm of a vector , the “angle” between directions , can be defined by help of the metric:
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 , pp. 223–224)18 and also by Schrödinger . Eddington also calculated Equation (8); in his expression the term is missing (cf. , p. 233).
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 equationcausal structure is provided by the equivalence class of metrics . 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 by19; then holds for the Lorentz group of generators .
The metric tensor may also be defined indirectly through vector fields forming an orthonormal -leg (-bein) with20. From the geometrical point of view, this can always be done (cf. theories with distant parallelism). By introducing 1-forms , Equation (11) may be brought into the form .
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, 234] change the order of indices of the components of the connection:symmetric, this does not make any difference as For both kinds of derivatives we have: 22.
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:projective structure on [404, 403].
The particular set of connections.
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 part24:antisymmetric part of the connection, i.e., torsion; it is a tensor field. The trace of the torsion tensor is called torsion vector; it connects to the two traces of the affine connection ; , as
The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:
The 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 get2627. 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, ).
For a symmetric affine connection, the preceding results reduce considerably due to . From Equations (29,30,32) we obtain the identities: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 :, p. 141).
From the symmetric part of the first fundamental form , a connection may be constructed, often called after Levi-Civita ,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 by29
Then the following identity holds:contorsion tensor , a linear combination of torsion , is defined by30
The 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.,metric-compatible31. J. M. Thomas introduced a combination of the terms appearing in and to define a covariant derivative for the metric (, p. 188), . We will have to deal with Equation (47) in Section 6.1 and, more intensively, in Part II of this review.
Connections that are not metric-compatible have been used in unified field theory right from the beginning. Thus, in Weyl’s theory [397, 395] we havesemi-metrical [300, 310]. According to Equation (44), in Weyl’s theory the inner product multiplies by a scalar factor under parallel transport:
We may also abbreviate the last term in the identity (42) by introducingtorsionless affine space is given by
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 formsemi-symmetric .
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:
In Riemanian geometry, the so-called geodesic equation,geodesic and autoparallel curves will have to be distinguished.
A conformal transformation of the metric,conformally flat. In , for , the vanishing of the Weyl curvature tensor , p. 404, , 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 (; see also , Appendix 1; for a modern approach, cf. ).
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,34. The covariant derivative of a tangent vector with bein-components then is
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 liketensor 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 by
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 ,38. Moreover, which, according to Equation (70), expresses the invariance of the space-time distance to the origin:
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 ascontravariant dotted spinors are those transforming with the complex-conjugate matrix : inverse matrices,
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,Weyl 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[53, 54]:
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,40.
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 .
© Max Planck Society and the author(s)