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MATHEMATICS
R. I. PIMENOV
APPLICATION OF SEMI-RIEMANNIAN GEOMETRY TO A UNIFIED FIELD THEORY
(Presented by Academician V. I. Smirnov on 9 III 1964)
1°. In \((^1)\) we considered semi-Euclidean spaces. Semi-Riemannian spaces \({}^{l_1\ldots l_r}V_n^{m_1\ldots m_{r-1}}\) are spaces that are infinitesimally semi-Euclidean; they carry a degenerate Riemannian metric (cf. \((^{2-4})\)). Here we shall study a special case of semi-Riemannian spaces \({}^{l}V_n^m\). For this purpose we construct a generalization of tensor calculus. We define invariants of the spaces \({}^{l}V_n^m\), as well as curvature flagtensors and ersatz-curvatures. The direct interpretation of \({}^{3}V_5^4\) gives 10 gravitational potentials \(g_{ik}\) and 4 electromagnetic ones \(g_{5i}/\sqrt{g_{55}}\). The invariants are the interval, velocity, and electromagnetic action. From the field equations, which directly generalize Einstein’s equations of gravitation, one obtains Einstein’s equations and Maxwell’s equations. This solves that part of the unified-field problem which is formulated as follows: to construct, in an invariant way, a geometry that would describe gravitation and electromagnetism \((^5)\).
2°. We shall call a manifold \(\mathfrak M_n^m\) a flag manifold if the following restriction is imposed on its group of admissible transformations*:
\[
x^\prime{}^i{}_{,\mu}=\partial_\mu x^i=D_\mu^i=0
\]
\[
(1\le i,j,k,l\le m<\mu,\nu,\pi\le n;\quad 1\le \alpha,\beta\le n),
\tag{1}
\]
i.e., the admissible transformations in \(\mathfrak M_n^m\) have the form
\[ x^{i'}=f^i(x^1,\ldots,x^m);\qquad x^{\mu'}=f^\mu(x^1,\ldots,x^n). \tag{2} \]
A flagtensor \(g_{\alpha\beta}\) is an object which, under (2), transforms according to the law
\[ g_{i'j'}=g_{ij}D_{i'}^{i}D_{j'}^{j},\qquad g_{\alpha'\mu'}=g_{\alpha\beta}D_{\alpha'}^{\alpha}D_{\mu'}^{\beta}. \tag{3} \]
An object of flag-affine connection \(\Gamma_{\beta\gamma}^{\alpha}\) is an object transforming according to the usual law \((^6)\) and satisfying the equalities
\[ \Gamma_{\alpha\mu}^{i}=0 \qquad (1\le i\le m<\mu\le n;\quad 1\le \alpha\le n). \tag{4} \]
The covariant derivative of a flagtensor is the flagtensor
\[ g_{ij;\alpha}=g_{ij,\alpha}-\Gamma_{i\alpha}^{l}g_{lj}-\Gamma_{j\alpha}^{l}g_{il},\qquad g_{\beta\mu;\alpha}=g_{\beta\mu,\alpha}-\Gamma_{\beta\alpha}^{\gamma}g_{\gamma\mu}-\Gamma_{\mu\alpha}^{\gamma}g_{\beta\gamma}. \tag{5} \]
A semi-Riemannian space \({}^{l}V_n^m\) is a \(\mathfrak M_n^m\) in which a torsion-free flag-affine connection and a symmetric
* We denote by the symbol \({}_{,\alpha}\) ordinary differentiation with respect to \(x^\alpha\), and by the symbol \({}_{;\alpha}\) covariant differentiation.
flag-tensor \(g_{\alpha\beta}\), satisfying the requirements:
\[ \det g_{ij}\ne 0,\qquad \det g_{\mu\nu}\ne 0\quad (1\leq i,j\leq m<\mu,\nu\leq n), \tag{6} \]
\[ g_{ij,\mu}=0, \tag{7} \]
\[ g_{ij;k}=0,\qquad g_{\mu\nu;\pi}=0; \tag{8} \]
here \(l\) is an index of \(g_{ik}\). It is easy to see that (6)—(8) are invariant with respect to (2). There exists a unique flag-tensor \(g^{\alpha\beta}\), which is symmetric and is determined by the formulas
\[ g_{ik}g^{kj}=\delta_i^j,\qquad g_{i\alpha}g^{\alpha\mu}=\delta_i^\mu,\qquad g_{\mu\pi}g^{\pi\nu}=\delta_\mu^\nu . \tag{9} \]
\(3^\circ\). We say that \(\operatorname{ord}\xi^\alpha=1\) if there exists \(\xi^i\ne0\). Otherwise \(\operatorname{ord}\xi^\alpha=2\). For a vector of the first order the expression \(g_{ij}\xi^i\xi^j=|\xi|^2\), and for a vector of the second order \(g_{\mu\nu}\xi^\mu\xi^\nu=|\xi|^2\), is an invariant (the square of the length of the vector). For a pair \(\xi\) and \(\eta\) of vectors of the same order, the expression \(g_{ij}\xi^i\eta^j\) or, respectively, \(g_{\mu\nu}\xi^\mu\eta^\nu\) is an invariant, called their scalar product \(\xi\cdot\eta\). For vectors of different order the expressions
\[ g_{ij}\xi^i\eta^j=\xi\cdot\eta\qquad (\operatorname{ord}\xi<\operatorname{ord}\eta), \tag{10} \]
\[ g_{\mu\alpha}\xi^\mu\eta^\alpha=\xi\cdot\eta\qquad (\operatorname{ord}\xi>\operatorname{ord}\eta) \tag{11} \]
are invariants, called their scalar products. The scalar product is not commutative. Vectors \(\xi^\alpha,\eta^\alpha\) are called 1-collinear if \(\xi^i=\lambda\cdot\eta^i\), but \(\xi^\alpha\ne\lambda\cdot\eta^\alpha\). The angle of two same-order non-1-collinear vectors is the complex number \(\varphi\) such that
\[ \cos\varphi=\frac{\xi\cdot\eta}{|\xi|\cdot|\eta|}. \tag{12} \]
By the angle of same-order 1-collinear vectors we mean \(\varphi\)
\[ \varphi=\frac{|\xi-\lambda\eta|}{|\xi|} = \frac{ \sqrt{ g_{ij}g_{\mu\nu} \left|\begin{matrix} \xi^i\xi^\mu\\ \eta^i\eta^\mu \end{matrix}\right| \cdot \left|\begin{matrix} \xi^j\xi^\nu\\ \eta^j\eta^\nu \end{matrix}\right| } }{ |\xi|\cdot|\eta| }. \tag{13} \]
Formula (12) would give in this case \(\cos\varphi\equiv1\). By the angle \(\psi\) of different-order vectors, supplementary to the straight angle, we mean
\[ \psi=\frac{\xi\cdot\eta}{|\xi|\cdot|\eta|}. \tag{14} \]
Thus defined, \(\varphi\) and \(\psi\) are additive functions of pairs of vectors. From (7) it follows that \(|\xi|,|\eta|\) and \(\varphi\) form a complete system of invariants of the pair of vectors \(\xi\) and \(\eta\). Under parallel transport \(d|\xi|=0\), \(d\varphi=0\), but \(d\psi\ne0\).
\(4^\circ\). We construct, as usual, the flag-tensor of curvature \(R_{\beta\gamma\delta}^{\alpha}\). In order that in \(V_n^{l,m}\) one could choose a system of coordinates in which \(\Gamma_{\beta\gamma}^{\alpha}=0\), it is necessary and sufficient that \(R_{\beta\gamma\delta}^{\alpha}=0\). But not one of the components \(R_{\alpha\beta}\) is a function of \(g_{i\mu}\). In order that in a domain one could choose a system of coordinates in which \(g_{\alpha\beta}=\mathrm{const}\), it is necessary and sufficient that \(R_{\beta\gamma\delta}^{\alpha}+Q_{\beta\gamma\delta}^{\alpha}=0\), where
\[ Q_{ijkl}=0,\qquad Q_{\alpha\beta\gamma\mu}=g_{\mu[\alpha;\beta]\gamma},\qquad Q_{i\mu jk}=g_{\mu[i;k]l}. \tag{15} \]
In order that at a point it be possible to choose a locally Euclidean coordinate system \((g_{\alpha\beta,\gamma}=0)\), it is necessary and sufficient that in it \(g_{(\mu;\nu)}=0\). In order that the angle between a first-order vector and a geodesic of the second order not change under parallel transport of it along the latter, it is necessary and sufficient that \(g_{i(\mu;\nu)}=0\). A space \({}^{l}V_n^m\) in which \(g_{i(\mu;\nu)}=0\) we call skew.
\(5^\circ\). Consider the skew \({}^{3}V_5^4\). Locally this space satisfies the system of postulates from (1): \((\Pi_0^*,\ \Pi_1^*,\ I_2^*,\ I_3^*,\ II_4^*)\). Put \(A_i=g_{5i}\cdot(g_{55})^{-1/2}\), \(F_{ik}=2A_{[i;k]}\). Then, without additional assumptions, in any coordinate system we obtain
\[ A_{i'}=A_iD^i_{i'}+\partial_{i'}f(x^1,x^2,x^3,x^4,x^5) \qquad (1\leq i\leq 4), \tag{16} \]
\[ g_{ik,5}=0,\qquad F_{ik,5}=0. \tag{17} \]
Here \(f\) is an arbitrary function; (16) coincides with the gauge (gradient) transformations of the covector-potential, while (17) are the cylindricity conditions. The invariants \(|dx^\alpha|\) and \(\varphi\) are interpreted trivially. The invariant \(\psi\) is equal to
\[ \psi=A_i\frac{dx^i}{ds}+ \frac{df(x^1,\ldots,x^5)}{ds}. \]
This makes it possible in (8) to identify \(e^{\int_M^N \psi\,ds}\) with the action of an electromagnetic field; the constant \(e^{\int_M^N df}\), by which our expression differs from formula (8), § 16, is immaterial under variation. The coordinate \(x^5\) has dimension \(\dfrac{h}{e}\), i.e. volt-sec.
We choose the matter–electricity flag tensor \(T_{\alpha\beta}\) in the form:
\[ (T_{\alpha\beta})= \begin{pmatrix} T_{ik} & a\cdot j_k\\ a\cdot j_i & 0 \end{pmatrix} \qquad (1\leq i,k\leq 4;\ 1\leq \alpha,\beta\leq 5), \tag{18} \]
where \(a\) is a constant, \(j_k=g_{kl}j^l\), \(j^k\) is the current vector in \({}^{3}V_4\), and \(T_{ik}\) is the usual matter tensor.
We write the field equations in the form
\[ g^{hl}(R_{lihk}+Q_{lihk}) =\varkappa\left(T_{ik}-\frac12 g_{ik}T^j_j\right), \tag{19} \]
\[ g^{hl}(R_{l5h\alpha}+Q_{l5h\alpha}) =\varkappa\left(T_{5\alpha}-\frac12 g_{5\alpha}T^5_5\right). \tag{20} \]
They are invariant. Since \(T^5_5=0=T_{55}\), they give the Einstein equations and the second pair of Maxwell equations (8):
\[ R_{ik}=\varkappa\left(T_{ik}-\frac12 g_{ik}T\right) \qquad (1\leq i,k\leq 4), \tag{21} \]
\[ F^p_{k;\,p}=\lambda j_k, \tag{22} \]
while the first pair of Maxwell equations \(F_{[ik;\,l]}=0\) is obtained directly from the symmetries of \(g_{\alpha\beta}\) and \(\Gamma^\alpha_{\beta\gamma}\). No superfluous physically uninterpretable functions arise. If in (18) the 0 is replaced by an arbitrary scalar, then the possibility opens of introducing a new small parameter into the theory.
Received
4 III 1964
CITED LITERATURE
- R. I. Pimenov, DAN, 155, No. 1 (1964).
- I. V. Panasyuk, Tr. II nauchn. konf. matem. kafedr ped. inst. Povolzhya, issue 1, 1963, p. 178.
- Ch. Yan Kei Ch., Bull. Polish Acad. Sci., ser. 3, 2, 305, No. 7 (1954).
- E. Yu. Yasinskaya, DAN, 137, No. 6, 1327 (1961).
- V. Bargmann, Relativity, Theoretical Physics in 20-th Century, N. Y., 1960.
- P. K. Rashevskii, Riemannian Geometry and Tensor Analysis, 1953.
- G. Weyl, Classical Groups, Their Invariants and Representations, 1947.
- L. D. Landau, E. M. Lifshitz, The Theory of Fields, 1962.