2.1 Geometry

It is very easy to get lost in the many constructive possibilities underlying the geometry of unified field theories. We briefly describe the mathematical objects occurring in an order that goes from the less structured to the more structured cases. In the following, only local differential geometry is taken into account12.

The space of physical events will be described by a real, smooth manifold MD of dimension D coordinatised by local coordinates xi, and provided with smooth vector fields X, Y,... with components i i X ,Y ,... and linear forms ω, ν,..., (ωi,νi) in the local coordinate system, as well as further geometrical objects such as tensors, spinors, connections13. At each point, D linearly independent vectors (linear forms) form a linear space, the tangent space (cotangent space) of M D. We will assume that the manifold M D is space- and time-orientable. On it, two independent fundamental structural objects will now be introduced.

2.1.1 Metrical structure

The first is a prescription for the definition of the distance ds between two infinitesimally close points on MD, eventually corresponding to temporal and spatial distances in the external world. For ds, we need positivity, symmetry in the two points, and the validity of the triangle equation. We know that ds must be homogeneous of degree one in the coordinate differentials i dx connecting the points. This condition is not very restrictive; it still includes Finsler geometry [281126Jump To The Next Citation Point224] to be briefly touched, below.

In the following, ds is linked to a non-degenerate bilinear form g (X, Y ), called the first fundamental form; the corresponding quadratic form defines a tensor field, the metrical tensor, with 2 D components gij such that

∘ --------- ds = gijdxidxj, (1 )
where the neighbouring points are labeled by xi and xi + dxi, respectively14. Besides the norm of a vector ∘ -----i-j- |X | := gijX X, the “angle” between directions X, Y can be defined by help of the metric:
gijXiY j cos(⁄ (X, Y )) :=-------. |X ||Y |

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 gik is defined through15

gmigmj = δji (2 )

We are used to g being a symmetric tensor field, i.e., with gik = g(ik) and with only D (D + 1)∕2 components; in this case the metric is called Riemannian if its eigenvalues are positive (negative) definite and Lorentzian if its signature is ± (D − 2)16. In the following this need not hold, so that the decomposition obtains17:

gik = γ(ik) + ϕ[ik]. (3 )
An asymmetric metric was considered in one of the first attempts at unifying gravitation and electromagnetism after the advent of general relativity.

For an asymmetric metric, the inverse

gik = h(ik) + f[ik] = hik + fik (4 )
is determined by the relations
γ γik = δk, ϕ ϕik = δk, h hik = δk, f fik = δk, (5 ) ij j ij j ij j ij j
and turns out to be [356Jump To The Next Citation Point]
γ ϕ h(ik) = --γik +--ϕimϕkn γmn, (6 ) g g (ik) ϕ- ik γ-im kn f = g ϕ + gγ γ ϕmn, (7 )
where g, ϕ, and γ are the determinants of the corresponding tensors gik, ϕik, and γik. We also note that
γ- kl mn g = γ + ϕ + 2γ γ ϕkm ϕln, (8 )
where g := det gik, ϕ := detϕik, γ := detγik. The results (6View Equation,7View Equation,8View Equation) were obtained already by Reichenbächer ([273], pp. 223–224)18 and also by Schrödinger [320]. Eddington also calculated Equation (8View Equation); in his expression the term ∼ ϕik⋆ϕik is missing (cf. [59Jump To The Next Citation Point], p. 233).

The manifold is called space-time if D = 4 and the metric is symmetric and Lorentzian, i.e., symmetric and with signature sigg = ±2. 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

g(X, X ) = 0 (9 )
results an equivalence class of metrics {λg} with λ 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 [67Jump To The Next Citation Point]. For an asymmetric metric, this structure can exist as well; it then is determined by the symmetric part γik = γ (ik) 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

0 0 1 1 2 2 3 3 ηik = δi δi − δi δi − δi δi − δi δi . (10 )
A geometrical characterization of Minkowski space as an uncurved, flat space is given below. Let ℒX be the Lie derivative with respect to the tangent vector X19; then ℒXp ηik = 0 holds for the Lorentz group of generators Xp.

The metric tensor g may also be defined indirectly through D vector fields forming an orthonormal D-leg (-bein) k hˆı with

glm = hlˆjhmˆkηˆjˆk, (11 )
where the hatted indices (“bein-indices”) count the number of legs spanning the tangent space at each point (ˆj = 1,2,...,D ) and are moved with the Minkowski metric20. From the geometrical point of view, this can always be done (cf. theories with distant parallelism). By introducing 1-forms 𝜃ˆk := hˆkdxl l, Equation (11View Equation) may be brought into the form ds2 = 𝜃ˆı𝜃 ˆkη ˆıˆk.

A new physical aspect will come in if the k hˆı are considered to be the basic geometric variables satisfying field equations, not the metric. Such tetrad-theories (for the case D = 4) are described well by the concept of fibre bundle. The fibre at each point of the manifold contains, in the case of an orthonormal D-bein (tetrad), all D-beins (tetrads) related to each other by transformations of the group O (D ), or the Lorentz group, and so on.

In Finsler geometry, the line element depends not only on the coordinates i x of a point on the manifold, but also on the infinitesimal elements of direction between neighbouring points dxi:

ds2 = gij(xn, dxm)dxidxj. (12 )
Again, gij is required to be homogeneous of rank 1.

2.1.2 Affine structure

The second structure to be introduced is a linear connection L with D3 components L k ij; 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 X and linear forms ω it is defined by

+ ∂Xi + ∂ω ∇k Xi = ---k + L kij Xj, ∇k ωi = ---ik − Lkijωj. (13 ) ∂x ∂x
The expressions + ∇k Xi and ∂Xik ∂x are abbreviated by Xi ∥k and Xi,k, respectively, while for a scalar f covariant and partial derivative coincide: -∂f ∇if = ∂xi ≡ ∂if ≡ f,i.

We have adopted the notational convention used by Schouten [300Jump To The Next Citation Point310Jump To The Next Citation Point389]. Eisenhart and others [121Jump To The Next Citation Point234] change the order of indices of the components of the connection:

− ∂Xi − ∂ ωi ∇k Xi = ---k + LjkiXj, ∇k ωi = ---k − Likjωj. (14 ) ∂x ∂x
As long as the connection is symmetric, this does not make any difference as + i − i i j ∇k X − ∇k X = 2L [kj]X . For both kinds of derivatives we have:
+ l ∂(vlwl)- − l ∂-(vlwl)- ∇k (v wl) = ∂xk , ∇k (v wl) = ∂xk (15 )
Both derivatives are used in versions of unified field theory by Einstein and others22.

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 ˆX contains one more term:

i i + ˆXi = ∂Xˆ-+ L iX ˆj − L rXˆi, − Xˆi = ∂X-- + L i ˆXj − L r ˆXi. (16 ) ∇k ∂xk kj kr ∇k ∂xk jk rk

A smooth vector field Y is said to be parallely transported along a parametrised curve λ (u ) with tangent vector X if for its components Yi∥kXk (u) = 0 holds along the curve. A curve is called an autoparallel if its tangent vector is parallely transported along it at each point23:

i k i X ∥kX (u) = σ(u)X . (17 )
By a particular choice of the curve’s parameter, σ = 0 may be imposed.

A transformation mapping autoparallels to autoparallels is given by:

j Lik → Likj + δj(iωk). (18 )
The equivalence class of autoparallels defined by Equation (18View Equation) defines a projective structure on MD [404403].

The particular set of connections

(p)Lijk := Lijk −--2---δk(iLj) (19 ) D + 1
with L := L m j im is mapped into itself by the transformation (18View Equation[348Jump To The Next Citation Point].

In Part II of this article, we shall find the set of transformations j j j ∂ω- Lik → Lik + δ i∂xk playing a role in versions of Einstein’s unified field theory.

From the connection L ijk further connections may be constructed by adding an arbitrary tensor field T to its symmetrised part24:

¯Lijk = L(ij)k + Tijk = Γ ijk + Tijk. (20 )
By special choice of T we can regain all connections used in work on unified field theories. We will encounter examples in later sections. The antisymmetric part of the connection, i.e.,
k k k Sij = L [ij] = T[ij] (21 )
is called torsion; it is a tensor field. The trace of the torsion tensor Si := S l il is called torsion vector; it connects to the two traces of the affine connection l Li := Lil; &tidle; l Lj := Llj, as 1 &tidle; Si = 2(Li − Li).

2.1.3 Different types of geometry

2.1.3.1 Affine geometry
Various subcases of affine spaces will occur, dependent on whether the connection is asymmetric or symmetric, i.e., with k k Lij = Γ ij. In physical applications, a metric always seems to be needed; hence in affine geometry it must be derived solely by help of the connection or, rather, by tensorial objects constructed from it. This is in stark contrast to Riemannian geometry where, vice versa, the connection is derived from the metric. Such tensorial objects are the two affine curvature tensors defined by25

+ i i i i m i m K jkl = ∂kLlj − ∂lL kj + Lkm Llj − L lm Lkj , (22 ) − i i i i m i m K jkl = ∂kLjl − ∂lL jk + Lmk Ljl − L ml Ljk , (23 )
respectively. In a geometry with symmetric affine connection both tensors coincide because of
1 + − -(K ijkl− K ijkl) = ∂[kS]lji+ 2Sj[k mSl]mi+ Lm i[kSl]j m − L j[mk Sl]mi. (24 ) 2
In particular, in Riemannian geometry, both affine curvature tensors reduce to the one and only Riemann curvature tensor.

The curvature tensors arise because the covariant derivative is not commutative and obeys the Ricci identity:

+ + 1 + + ∇[j∇k ] Ai =--K irjkAr − Sjkr∇r Ai (25 ) 2 −∇ −∇ Ai = 1-K− i Ar + S r −∇ Ai (26 ) [j k] 2 rjk jk r
For a vector density, the identity is given by
+ + 1 + + 1 ∇[j∇k ] ˆAi =--K irjkAˆr − Sjkr∇r Aˆi + -VjkAˆi, (27 ) 2 2
with the homothetic curvature Vjk to be defined below in Equation (31View Equation).

The curvature tensor (22View Equation) satisfies two algebraic identities:

+ i K j[kl] = 0, (28 ) + i i i m K {jkl} = 2∇ {jS kl} + 4S m{j Skl} , (29 )
where the curly bracket denotes cyclic permutation:
i i i i K{jkl} := K jkl + K ljk + K klj.
These identities can be found in Schouten’s book of 1924 ([300Jump To The Next Citation Point], p. 88, 91) as well as the additional single integrability condition, called Bianchi identity:
+ i i r K j{kl∥m} = 2K r{klSm }j . (30 )
A corresponding condition obtains for the curvature tensor − K from Equation (23View Equation).

From both affine curvature tensors we may form two different tensorial traces each. In the first case Vkl := Kiikl = V [kl], and Kjk := Kijki.Vkl is called homothetic curvature, while Kjk is the first of the two affine generalisations from + K and − K of the Ricci tensor in Riemannian geometry. We get26

Vkl = ∂kLl − ∂lLk, (31 )
and the following identities hold:
m m Vkl + 2K [kl] = 4∇ [kSl] + 8Skl Sm + 2∇mS kl , (32 ) − − − m m V kl +2 K [kl] = − 4 ∇[k Sl] + 8Skl Sm + 2∇mS kl , (33 )
where S := S l k kl. While V kl is antisymmetric, K jk has both tensorial symmetric and antisymmetric parts:
m m m r K [kl] = − ∂[kL&tidle;l] + ∇mS kl + LmS kl + 2L[l|r S m|k], (34 ) K = ∂ &tidle;L − ∂ L m − L&tidle; L m + L nL m . (35 ) (kl) (k l) m (kl) m (kl) (k|m | l)n
We use the notation A (i|k|l) in order to exclude the index k from the symmetrisation bracket27. 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 m Skl = 0. From Equations (29View Equation,30View Equation,32View Equation) we obtain the identities:

Ki{jkl} = 0, (36 ) i K j{kl∥m } = 0, (37 ) Vkl + 2K [kl] = 0, (38 )
i.e., only one independent trace tensor of the affine curvature tensor exists. For the antisymmetric part of the Ricci tensor K = − ∂ &tidle;L [kl] [k l] 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 g := αK ij (ij), or g := α − ij K (ij). 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 det (Kij)28.

As a final result in this section, we give the curvature tensor calculated from the connection L¯ k = Γ k + T k ij ij ij (cf. Equation (20View Equation)), expressed by the curvature tensor of Γ k ij and by the tensor k T ij:

i ¯ i (Γ ) i m i m i K jkl(L) = K jkl(Γ ) + 2 ∇[kTl]j − 2T [k|j| Tl]m + 2Skl Tmj , (39 )
where (Γ )∇ is the covariant derivative formed with the connection Γ k ij (cf. also [310Jump To The Next Citation Point], p. 141).

2.1.3.2 Mixed geometry
A manifold carrying both structural elements, i.e., metric and connection, is called a metric-affine space. If the first fundamental form is taken to be asymmetric, i.e., to contain an antisymmetric part g[ik] := 12(gij − gji), we speak of a mixed geometry. In principle, both metric-affine space and mixed geometry may always be re-interpreted as Riemannian geometry with additional geometric objects: the 2-form field ϕ (f ) (symplectic form), the torsion S, and the non-metricity Q (cf. Equation 41View Equation). It depends on the physical interpretation, i.e., the assumed relation between mathematical objects and physical observables, which geometry is the most suitable.

From the symmetric part of the first fundamental form hij = g(ij), a connection may be constructed, often called after Levi-Civita [204Jump To The Next Citation Point],

1 {kij} := -γkl(γli,j + γlj,i − hij,), (40 ) 2
and from it the Riemannian curvature tensor defined as in Equation (22View Equation) with L ijk = {kij} (cf. Section 2.1.3); {kij} 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 γij only, and denote it by gij. With the help of the symmetric affine connection, we may define the tensor of non-metricity k Q ij by29
Q k := gkl∇ g . (41 ) ij l ij

Then the following identity holds:

k k k 1 k k k Γij = {ij} + K ij + 2(Q ij + Q ji − Qj i), (42 )
where the contorsion tensor K k ij, a linear combination of torsion S k ij, is defined by30
k k k k k Kij := S ji + S ij − S ij = − K i j. (43 )

The inner product of two tangent vectors i k A ,B is not conserved under parallel transport of the vectors along Xl if the non-metricity tensor does not vanish:

+ Xk ∇k (AnBmgnm ) = QnmlAnBmXl ⁄= 0. (44 )

A connection for which the non-metricity tensor vanishes, i.e.,

+ ∇k gij = 0 (45 )
holds, is called metric-compatible31. J. M. Thomas introduced a combination of the terms appearing in + ∇ and − ∇ to define a covariant derivative for the metric ([346Jump To The Next Citation Point], p. 188),
r r gik∕l := gik,l − grkΓ il − girΓlk , (46 )
and extended it for tensors of arbitrary rank ≥ 3. Einstein later used as a constraint on the metrical tensor
0 = gik∥l:= gik,l − grkΓ r − girΓ r, (47 ) +− il lk
a condition that cannot easily be interpreted geometrically [97Jump To The Next Citation Point]. We will have to deal with Equation (47View Equation) 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 [397Jump To The Next Citation Point395Jump To The Next Citation Point] we have

Qijk = Qk gij. (48 )
In case of such a relationship, the geometry is called semi-metrical [300Jump To The Next Citation Point310Jump To The Next Citation Point]. According to Equation (44View Equation), in Weyl’s theory the inner product multiplies by a scalar factor under parallel transport:
+ Xk ∇k (AnBmgnm ) = (QlXl )AnBmgnm. (49 )
This means that the light cone is preserved by parallel transport.

We may also abbreviate the last term in the identity (42View Equation) by introducing

k k k k X ij := Q ij + Qji − Qj i. (50 )
Then, from Equation (39View Equation), the curvature tensor of a torsionless affine space is given by
Kijkl(¯Γ ) = Kijkl({nrm }) + 2 ({ijk})∇ [kX l]ji− 2X [k|j|mX l]mi, (51 )
where ({i }) jk ∇ 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 L k [ij]; torsion is a tensor field to be linked to physical observables. A linear connection whose antisymmetric part k S ij has the form

S ijk = S [iδkj] (52 )
is called semi-symmetric [300Jump To The Next Citation Point].

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: Γ ijk = {kij}, where {k } ij is the usual Christoffel symbol (40View Equation). The covariant derivative of A with respect to the Levi-Civita connection k {ij} ∇ is abbreviated by A;k. The Riemann curvature tensor is denoted by

Ri = ∂ {i} − ∂ {i } + { i} {m} − { i} {m}. (53 ) jkl k lj lkj km lj lm kj

An especially simple case of a Riemanian space is Minkowski space, the curvature of which vanishes:

Rijkl(η) = 0. (54 )
This is an invariant characterisation irrespective of whether the Minkowski metric η is given in Cartesian coordinates as in Equation (10View Equation), or in an arbitrary coordinate system. We also have ℒX Rijkl = 0 where ℒ is the Lie-derivative (see below under “symmetries”), and X stands for the generators of the Lorentz group.

In Riemanian geometry, the so-called geodesic equation,

Xi Xk (u ) = σ(u)Xi, (55 ) ;k
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,

′ gik → gik = λgik, (56 )
with a smooth function λ changes the components of the non-metricity tensor,
Qijk → Q ijk+ gij gkl∂lσ, (57 )
as well as the Levi-Civita connection,
1 {kij} → {kij} + -(σiδkj − σjδki + gijgklσl), (58 ) 2
with σi := λ−1∂iλ. As a consequence, the Riemann curvature tensor Rijkl is also changed; if, however, R ′ijkl = 0 can be reached by a conformal transformation, then the corresponding space-time is called conformally flat. In MD, for D > 3, the vanishing of the Weyl curvature tensor
i i ---2-- i i ------2R--------i C jkl := R jkl + D − 2(δ[kRl]j + gj[lR k]) + (D − 1)(D − 2)δ [lgk]j (59 )
is a necessary and sufficient condition for MD to be conformally flat ([397Jump To The Next Citation Point], p. 404, [300Jump To The Next Citation Point], 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 ([196Jump To The Next Citation Point]; see also [401Jump To The Next Citation Point], 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. [6566]). In the following we shall deal only with relativistic unified field theories.

2.1.3.3 Projective geometry
Projective geometry is a generalisation of Riemannian geometry in the following sense: Instead of tangent spaces with the light cone ηikdxidxk = 0, where η is the Minkowski metric, in each event now a tangent space with a general, non-degenerate surface of second order γ will be introduced. This leads to a tangential cone g dxidxk = 0 ik in the origin (cf. Equation (9View Equation)), and to a hyperplane, the polar plane, formed by the contact points of the tangential cone and the surface γ. In place of the D inhomogeneous coordinates xi of MD, D + 1 homogeneous coordinates X α (α = 0,1,2,...,D) are defined32 such that they transform as homogeneous functions of first degree:

α ∂X-′ν- ′ν X ∂X μ = X . (60 )
The connection to the inhomogeneous coordinates i x is given by homogeneous functions of degree zero, e.g., by i --Xi- x = ϕαX α33. Thus, the X α themselves form the components of a tangent vector. Furthermore, the quadratic form gαβX αX β = 𝜖 = ±1 is adopted with gαβ being a homogeneous function of degree − 2. A tensor field T m1m2 m3... n1n2 n3 ... (cf. Section 2.1.5) depending on the homogeneous coordinates X μ with u contravariant (upper) and l covariant (lower) indices is required to be a homogeneous function of degree r := u − l.

If we define i γμi:= ∂∂xXμ-, with γμiX μ = 0, then γμi transforms like a tangent vector under point transformations of the xi, and as a covariant vector under homogeneous transformations of the X α. The γ i μ may be used to relate covariant vectors a i and A μ by A = γ ia . μ μ i Thus, the metric tensor in the space of homogeneous coordinates gαβ and the metric tensor gik of MD are related by β gik = γiαγk gαβ with μ γ iμ γk = δik. The inverse relationship is given by gαβ = γαiγβkgik + 𝜖X αX β with X α = gαβX β. The covariant derivative for tensor fields in the space of homogeneous coordinates is defined as before (cf. Section 2.1.2):

β ∇ A β(X ) = ∂A--(X-)+ Γ β(X )A ν(X ). (61 ) α ∂X α αν
The covariant derivative of the quantity γ μ k interconnecting both spaces is given by
μ ∂γkμ μ σ m l μ ∇ ργk = ∂x-ρ-+ Γ ρσ γk − {kl}γρ γm . (62 )

2.1.4 Cartan’s method

In this section, we briefly present Cartan’s one-form formalism in order to make understandable part of the literature. Cartan introduces one-forms 𝜃ˆa (ˆa = 1, ...,4) by 𝜃ˆa := h ˆaldxl. The reciprocal basis in tangent space is given by e = hl-∂- ˆj ˆj∂xl. Thus, 𝜃ˆa(e) = δˆa ˆj ˆj. The metric is then given by η 𝜃ˆı ⊗ 𝜃ˆk ˆıˆk. The covariant derivative of a tangent vector with bein-components ˆ X k is defined via Cartan’s first structure equations,

i ˆı ˆı ˆı ˆl Θ := D 𝜃 = d𝜃 + ω ˆl ∧ 𝜃 , (63 )
where ωˆıˆ k is the connection-1-form, and Θˆı is the torsion-2-form, Θ ˆı = − Sˆ ˆı 𝜃ˆl ∧ 𝜃ˆm lmˆ. We have ω ˆıˆk = − ωˆkˆı. The link to the components k L[ij] of the affine connection is given by ω ˆıˆk = h ˆılhmˆk L ˆrm l𝜃 ˆr34. The covariant derivative of a tangent vector with bein-components X ˆk then is
DX ˆk := dX ˆk + ωˆk X ˆl. (64 ) ˆl

By further external derivation35 on Θ we arrive at the second structure relation of Cartan,

ˆk ˆk ˆl D Θ = Ω ˆl ∧ 𝜃 . (65 )
In Equation (65View Equation) the curvature-2-form Ω ˆkˆ= 1R ˆkˆ 𝜃ˆm ∧ 𝜃ˆn l 2 lˆm ˆn appears, which is given by
ˆk ˆk kˆ ˆk Ω ˆl = d ω ˆl + ω ˆl ∧ ω ˆl. (66 )
Ω ˆk ˆk is the homothetic curvature.

2.1.5 Tensors, spinors, symmetries

2.1.5.1 Tensors
Up to here, no definitions of a tensor and a tensor field were given: A tensor Tp(MD ) of type (r,s) at a point p on the manifold MD is a multi-linear function on the Cartesian product of r cotangent- and s tangent spaces in p. A tensor field is the assignment of a tensor to each point of MD. Usually, this definition is stated as a linear, homogeneous transformation law for the tensor components in local coordinates:

′ ′′ ∂xn1 ∂xn2 ∂xn3 ∂xk′1 ∂xk ′2 ∂xk′3 T kl′1k2l′kl3′......= Tmn1mn2 mn3......---′----′----′---------------- ⋅⋅⋅ (67 ) 1 2 3 1 2 3 ∂xl1 ∂xl2 ∂xl3 ∂xm1 ∂xm2 ∂xm3
where xk′ = xk ′(xi) with smooth functions on the r.h.s. are taken from the set (“group”) of coordinate transformations (diffeomorphisms). Strictly speaking, tensors are representations of the abstract group at a point on the manifold36.

A relative tensor Tp(MD ) of type (r,s) and of weight ω at a point p on the manifold MD transforms like

[ ( s) ]ω n1 n2 n3 k′1 k′2 k′3 T k′1′ k′2′, k′′3..= det ∂x-- T m1 m2 m3...∂x-′-∂x-′-∂x-′-∂x---∂x----∂x-- ⋅⋅⋅ (68 ) l1 l2 l3.. ∂x ′r n1 n2 n3...∂xl1 ∂xl2 ∂xl3 ∂xm1 ∂xm2 ∂xm3
An example is given by the totally antisymmetric object 𝜖ijkl with 𝜖ijkl = ±1, or 𝜖ijkl = 0 depending on whether (ijkl) is an even or odd permutation of (0123), or whether two indices are alike. ω = − 1 for 𝜖 ijkl; in this case, the relative tensor is called tensor density. We can form a tensor from 𝜖ijkl by introducing √ --- ηijkl := − g 𝜖ijkl, where gik is a Lorentz-metric. Note that ηijkl := √-1−g-𝜖ijkl. The dual to a 2-form (skew-symmetric tensor) then is defined by ∗F ij = 1ηijklFkl. 2

In connection with conformal transformations g → λg, the concept of the gauge-weight of a tensor is introduced. A tensor T ...... is said to be of gauge weight q if it transforms by Equation (56View Equation) as

′... q ... T ...= λ T .... (69 )

Objects that transform as in Equation (67View Equation) but with respect to a subgroup, e.g., the linear group, affine group G(D ), orthonormal group O(D ), or the Lorentz group ℒ, 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.

2.1.5.2 Spinors
Spinors are representations of the Lorentz group only; as such they are related strictly to the tangent space of the space-time manifold. To see how spinor representations can be obtained, we must use the 2–1 homomorphism of the group SL (2,C ) and the proper orthochronous Lorentz group, a subgroup of the full Lorentz group37. Let A ∈ SL (2,C ); then A is a complex (2-by-2)-matrix with det A = 1. By picking the special Hermitian matrix

∑ S = x01 + σpxp, (70 ) p
where 1 is the (2 by 2)-unit matrix and σp are the Pauli matrices satisfying
σiσk + σkσi = 2δik. (71 )

Then, by a transformation A from SL (2, C),

S′ = ASA+, (72 )
where A+ is the Hermitian conjugate matrix38. Moreover, detS = detS ′ which, according to Equation (70View Equation), expresses the invariance of the space-time distance to the origin:
′ ′ ′ ′ (x 0 )2 − (x 1 )2 − (x 2 )2 − (x 3)2 = (x 0)2 − (x 1)2 − (x 2)2 − (x 3)2. (73 )
The link between the representation of a Lorentz transformation Lik in space-time and the unimodular matrix A mapping spin space (cf. below) is given by
1 L(A )ik = -tr(σiA σkA+ ). (74 ) 2
Thus, the map is two to one: +A and − A give the same Lik.

Now, contravariant 2-spinors ξA (A = 1,2) are the elements of a complex linear space, spinor space, on which the matrices A are acting39. The spinor is called elementary if it transforms under a Lorentz-transformation as

A ′ A C ξ = ±A C ξ . (75 )
Likewise, contravariant dotted spinors ζA˙ are those transforming with the complex-conjugate matrix A¯:
ζB˙′ = ± ¯A ˙B˙ ζC˙. (76 ) D
Covariant and covariant dotted 2-spinors correspondingly transform with the inverse matrices,
ξB′ = ± (A −1)C ξC, (77 ) B
and
˙ ξB˙′ = ± (¯A −1)C˙B ξ ˙C. (78 )
The space of 2-spinors can be used as a representation space for the (proper, orthochronous) Lorentz group, with the 2-spinors being the elements of the most simple representation D (1∕2,0).

Higher-order spinors with dotted and undotted indices SA...˙B...˙ C...D... transform correspondingly. For the raising and lowering of indices now a real, antisymmetric (2 × 2)-matrix 𝜖 with components AB A 2 A 1 𝜖 = δ1 δB − δ2 δB = 𝜖AB is needed, such that

A AB B ξ = 𝜖 ξB, ξA = ξ 𝜖BA. (79 )

Next to a spinor, bispinors of the form AB AB˙ ζ ,ξ, etc. are the simplest quantities (spinors of 2nd order). A vector Xk can be represented by a bispinor ˙ XA B,

AB˙ AB˙ k X = σk X , (80 )
where σAkB˙ (k = 0,...,3) is a quantity linking the tangent space of space-time and spinor space. If k numerates the matrices and A, B˙ designate rows and columns, then we can chose σA B˙ 0 to be the unit matrix while for the other three indices AB˙ σj are taken to be the Pauli matrices. Often the quantity sAk˙B = √1-σAkB˙ 2 is introduced. The reciprocal matrix sk ˙ AB is defined by
A ˙B k k sj sAB˙= δj, (81 )
whereas
AB˙ j C ˙D AC B˙˙D sj s = 𝜖 𝜖 . (82 )

In order to write down spinorial field equations, we need a spinorial derivative,

k ∂AB˙= sA ˙B∂k (83 )
with ∂ ˙∂A ˙B = ∂k∂k. AB The simplest spinorial equation is the Weyl equation:
∂ ψA = 0, B˙ = 1,2. (84 ) A ˙B
The next simplest spinor equation for two spinors ˙ χB, ψA would be
C˙ 2π C ˙B 2π ˙B ∂AC˙χ = − √----m ψA; ∂ ψC = √----m χ , (85 ) 2 h 2 h
where m is a mass. Equation (85View Equation) is the 2-spinor version of Dirac’s equation.

Dirac- or 4-spinors with 4 components ψk, k = 1,...,4, may be constructed from 2-spinors as a direct sum of contravariant undotted and covariant dotted spinors ψ and ϕ: For k = 1,2, we enter 1 ψ and 2 ψ; for k = 3,4, we enter ϕ ˙1 and ϕ2˙. In connection with Dirac spinors, instead of the Pauli-matrices the Dirac γ-matrices (4 × 4-matrices) appear; they satisfy

i k k i ik γ γ + γ γ = 2η 1. (86 )
The Dirac equation is in 4-spinor formalism [5354]:
( ∂ ) iγl---l + κ χ = 0, (87 ) ∂x
with the 4-component Dirac spinor χ. In the first version of Dirac’s equation, α- and β-matrices were used, related to the γ’s by
γ0 = β, γm = β αm, m = 1,2,3, (88 )
where the matrices β and αm are given by ( ) 0 − σi ( σi 0 ), ( ) 0 − 1 ( 1 0 ).

The generally-covariant formulation of spinor equations necessitates the use of n-beins hk ˆı, 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

1- ˆıˆk D ψ = dψ + 4 ωˆıˆkσ ψ, (89 )
where ˆıˆk 1 ˆı ˆk σ := 2[γ γ ].

Equation (89View Equation) is a special case of the general formula for the covariant derivative of a tensorial form ψ, i.e., a vector in some vector space V, whose components are differential forms,

D ψ = dψ + ωˆıˆkρ (e )ψ, (90 ) ˆıˆk α
where ρ(eα) is a particular representation of the corresponding Lie algebra in V with basis vectors eα. For the example of the Dirac spinor, the adjoint representation of the Lorentz group must be used40.

2.1.5.3 Symmetries
In Section 2.1.1 we briefly met the Lie derivative of a vector field ℒX with respect to the tangent vector X defined by k k i k i k (LX Y) := ([X, Y ]) = X ∂iY − Y ∂iX. With its help we may formulate the concept of isometries of a manifold, i.e., special mappings, also called “motions”, locally generated by vector fields X satisfying

ℒX gik := ∂k (gij)Xk + glj∂iXl + gil∂jXl = 0. (91 )
The generators X solving Equation (91View Equation) given some metric, form a Lie group Gr, the group of motions of MD. If a group Gr is prescribed, e.g., the group of spatial rotations O (3), then from Equation (91View Equation) the functional form of the metric tensor having O (3) as a symmetry group follows.

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 Xi = δi0 then from Equation (91View Equation) we conclude that stationarity reduces to the condition ∂0gik = 0. If we take X to be the tangent vector field to the congruence of curves xi = xi(u), i.e., if k dxk X = du, then a necessary and sufficient condition for hypersurface-orthogonality is ijkl 𝜖 XjX [k,l] = 0.

A generalisation of Killing vectors are conformal Killing vectors for which ℒX gik = Φgik with an arbitrary smooth function Φ holds. In purely affine spaces, another type of symmetry may be defined: ℒ Γ l= 0 X ik; they are called affine motions [425].


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