Levi-Civita simplifies Einstein. The Ricci rotation coefficients and unified field theories

Abstract

This paper concerns late 1920 s attempts to construct unitary theories of gravity and electromagnetism. A first attempt using a non-standard connection—with torsion and zero-curvature—was carried out by Albert Einstein in a number of publications that appeared between 1928 and 1931. In 1929, Tullio Levi-Civita discussed Einstein’s geometric structure and deduced a new system of differential equations in a Riemannian manifold endowed with what is nowadays known as Levi-Civita connection. He attained an important result: Maxwell’s electromagnetic equations and the gravitational equations were obtained exactly, while Einstein had deduced them only as a first order approximation. A main feature of Levi-Civita’s theory is the essential use of the Ricci’s rotation coefficients, introduced by Gregorio Ricci Curbastro many years before. We trace the history of Ricci’s coefficients that are still used today, and highlight their geometric and mechanical meaning.

Appendices

Appendix A The vector-valued 1-form $-\Delta ^{\ \nu }_{\mu \sigma } A^{\mu }\textrm{d}x^\sigma$ is closed

$$\begin{aligned}{} & {} \Theta ^{(\nu )}_{\sigma }\textrm{d}x^\sigma :=- \Delta _{\mu \sigma }^{\nu } A^{\mu } \textrm{d}x^{\sigma } \qquad \text {where}\qquad \Delta _{\mu \sigma }^{\nu } = h_{a}^{\nu } \frac{\partial h_{\mu a}}{\partial x^{\sigma }} \\{} & {} \begin{array}{rll} \Theta ^{(\nu )}_{\sigma ,\beta }&{}=(-\Delta ^{\ \nu }_{\mu \sigma } A^{\mu })_{,\beta }= \left( -h^\nu _a \, h_{\mu a,\sigma }\, A^\mu \right) _{,\beta }&{}\ \\ \\ \ {} &{}= -h^\nu _{a,\beta } \, h_{\mu a,\sigma }\, A^\mu -h^\nu _a \, h_{\mu a,\sigma \beta }\, A^\mu -h^\nu _a \, h_{\mu a,\sigma }\, A^\mu _{,\beta } &{}\ \\ \end{array} \end{aligned}$$

we continue by elaborating just the third term in the above rhs:

$$\begin{aligned}{} & {} -h^\nu _a \, h_{\mu a,\sigma }\, A^\mu _{,\beta } = -h^\nu _a \, h_{\mu a,\sigma }\, (-\Delta ^\mu _{\rho \beta }A^\rho ) = -h^\nu _a \, h_{\mu a,\sigma }\, (-h^\mu _b \, h_{\rho b, \beta }A^\rho ) \\{} & {} \quad = h^\nu _a \, h_{\mu a,\sigma }\, h^\mu _b \, h_{\rho b, \beta }A^\rho = - h^\nu _{a,\sigma } \, h_{\mu a}\, h^\mu _b \, h_{\rho b, \beta }A^\rho = - h^\nu _{a,\sigma } \, h_{\rho a, \beta }A^\rho \\ {}{} & {} \quad = - h^\nu _{a,\sigma } \, h_{\mu a, \beta }A^\mu \end{aligned}$$

Finally,

$$\begin{aligned} \Theta ^{(\nu )}_{\sigma ,\beta }=-h^\nu _{a,\beta } \, h_{\mu a,\sigma }\, A^\mu -h^\nu _a \, h_{\mu a,\sigma \beta }\, A^\mu - h^\nu _{a,\sigma } \, h_{\mu a, \beta }A^\mu = \Theta ^{(\nu )}_{\beta , \sigma } \qquad \square \ \end{aligned}$$

Hence:

$$\begin{aligned} \oint \textrm{d}A^\nu =\oint \Theta ^{(\nu )}_{\beta }\textrm{d}x^\beta =0, \end{aligned}$$

the connection $\Delta ^{\ \nu }_{\mu \sigma }$ is curvature-free.

Appendix B Levi-Civita proposal and comparison with the modern expression of Maxwell’s equations

The final step⁶⁷ performed by Levi-Civita is the definition of the skew-symmetrical object $\xi _{ik}$ (formula (12) in Levi-Civita 1929c):

$$\begin{aligned} \xi _{ik}:= \sum _{\ell =0}^{n-1} \frac{\textrm{d}\gamma _{ik\ell }}{\textrm{d}s_\ell }\qquad \qquad \text {where}\ \ \frac{\textrm{d}}{\textrm{d}s_\ell }{\Phi \dots }= {\Phi \dots }_{;\mu }\,\lambda _{\ell |}^{\ \ \mu } \end{aligned}$$

(B1)

if one transforms it into

$$\begin{aligned} \xi _{\mu \nu }:=\xi ^{ik}\lambda _{i|\mu }\lambda _{k|\nu } \end{aligned}$$

(B2)

one discovers that

$$\begin{aligned} \xi ^{\mu \nu }_{\ \;\mu \nu }=0. \end{aligned}$$

(B3)

Then realizing what is now called Hodge transformation of $\xi _{\mu \nu ;\sigma }$,

$$\begin{aligned} p_{\mu }:=\epsilon ^{\mu \nu \rho \sigma }\,\xi _{\nu \rho ;\sigma } \end{aligned}$$

(B4)

one gains that

$$\begin{aligned} p^{\mu }_{\;\mu }=0 \end{aligned}$$

(B5)

The Faraday tensor (2-form) is recognized⁶⁸:

$$\begin{aligned} F_{\mu \nu }=v\, \xi _{\mu \nu },\quad v: \text {constant} \end{aligned}$$

(B6)

Formulae (B3) and (B5) represent Maxwell’s equations in vacuum. Today they are written respectively, using the exterior differential d, as

$$\begin{aligned} \textrm{d}F=0,\qquad \textrm{d}^*F=0. \end{aligned}$$

(B7)