Physical quantities and spatial parameters in the complex octonion curved space

Abstract

The paper focuses on finding out several physical quantities to exert an influence on the spatial parameters of complex-octonion curved space, including the metric coefficient, connection coefficient, and curvature tensor. In the flat space described with the complex octonions, the radius vector is combined with the integrating function of field potential to become a composite radius vector. And the latter can be considered as the radius vector in a flat composite-space (a function space). Further it is able to deduce some formulae between the physical quantity and spatial parameter, in the complex-octonion curved composite-space. Under the condition of weak field approximation, these formulae infer a few results accordant with the general theory of relativity. The study reveals that it is capable of ascertaining which physical quantities are able to result in the warping of space, in terms of the curved composite-space described with the complex octonions. Moreover, the method may be expanded into some curved function spaces, seeking out more possible physical quantities to impact the bending degree of curved spaces.

Appendices

Appendix 1: Connection coefficient

In the curved space described with the complex octonions, there is a relation between the metric coefficient and connection coefficient. The definition of metric coefficient must meet the demand of this relation, deducing the connection coefficient from the metric coefficient.

Multiplying the component ${{\varvec{e}}}_\lambda ^*$ from the left of the definition,

$$\begin{aligned} \partial ^2 \mathbb {H} / \partial u^\beta \partial u^{\gamma } = \varGamma _{\beta \gamma }^\alpha {{\varvec{e}}}_\alpha , \end{aligned}$$

(43)

yields,

$$\begin{aligned} ( \partial \mathbb {H}^*/ \partial u^\lambda ) \circ ( \partial ^2 \mathbb {H} / \partial u^\beta \partial u^{\gamma } ) = g_{\overline{\lambda } \alpha } \varGamma _{\beta \gamma }^\alpha , \end{aligned}$$

(44)

while multiplying the component ${{\varvec{e}}}_\lambda$ from the right of the conjugate of Eq. (43) produces,

$$\begin{aligned} ( \partial ^2 \mathbb {H}^*/ \partial u^\beta \partial u^{\gamma } ) \circ ( \partial \mathbb {H} / \partial u^\lambda ) = \overline{\varGamma _{\beta \gamma }^\alpha } g_{\overline{\alpha } \lambda }, \end{aligned}$$

(45)

where $\partial ^2 \mathbb {H}^*/ \partial u^\beta \partial u^{\gamma } = \overline{\varGamma _{\beta \gamma }^\alpha } {{\varvec{e}}}_\alpha ^*$. $\varGamma _{\beta \gamma }^\alpha$ and $\overline{\varGamma _{\beta \gamma }^\alpha }$ are coefficients.

From the last two equations, the partial derivative of the metric tensor, $g_{\overline{\lambda } \gamma } = {{\varvec{e}}}_\lambda ^*\circ {{\varvec{e}}}_\gamma$, with respect to the coordinate value $u^\beta$ generates,

$$\begin{aligned} g_{\overline{\lambda } \alpha } \varGamma _{\beta \gamma }^\alpha + \overline{\varGamma _{\beta \lambda }^\alpha } g_{\overline{\alpha } \gamma } = \partial g_{\overline{\lambda } \gamma } / \partial u^\beta , \end{aligned}$$

(46)

similarly there are,

$$\begin{aligned} g_{\overline{\beta } \alpha } \varGamma _{\gamma \lambda }^\alpha + \overline{\varGamma _{\gamma \beta }^\alpha } g_{\overline{\alpha } \lambda }= & {} \partial g_{\overline{\beta } \lambda } / \partial u^{\gamma }, \end{aligned}$$

(47)

$$\begin{aligned} g_{\overline{\gamma } \alpha } \varGamma _{\lambda \beta }^\alpha + \overline{\varGamma _{\lambda \gamma }^\alpha } g_{\overline{\alpha } \beta }= & {} \partial g_{\overline{\gamma } \beta } / \partial u^\lambda . \end{aligned}$$

(48)

From the last three equations, there are,

$$\begin{aligned} \varGamma _{ \overline{\lambda } , \beta \gamma } = (1/2) ( \partial g_{ \overline{\gamma } \lambda } / \partial u^\beta + \partial g_{ \overline{\lambda } \beta } / \partial u^{\gamma } - \partial g_{ \overline{\gamma } \beta } / \partial u^\lambda ), \end{aligned}$$

(49)

where $[ ( g_{\overline{\lambda } \alpha } \varGamma _{\beta \gamma }^\alpha )^*]^T = \overline{\varGamma _{\gamma \beta }^\alpha } g_{\overline{\alpha } \lambda }$, and $[ ( \overline{\varGamma _{\gamma \beta }^\alpha } g_{\overline{\alpha } \lambda } )^*]^T = g_{\overline{\lambda } \alpha } \varGamma _{\beta \gamma }^\alpha$.

On the other hand, multiplying the component ${{\varvec{e}}}_\lambda ^*$ from the left of the definition,

$$\begin{aligned} \partial ^2 \mathbb {H} / \partial \overline{u^\beta } \partial u^{\gamma } = \varGamma _{\overline{\beta } \gamma }^\alpha {{\varvec{e}}}_\alpha , \end{aligned}$$

(50)

deduces,

$$\begin{aligned} ( \partial \mathbb {H}^*/ \partial u^\lambda ) \circ ( \partial ^2 \mathbb {H} / \partial \overline{u^\beta } \partial u^{\gamma } ) = g_{\overline{\lambda } \alpha } \varGamma _{\overline{\beta } \gamma }^\alpha , \end{aligned}$$

(51)

while multiplying the component ${{\varvec{e}}}_\lambda$ from the right of the conjugate of Eq. (50) produces,

$$\begin{aligned} ( \partial ^2 \mathbb {H}^*/ \partial \overline{u^\beta } \partial u^{\gamma } ) \circ ( \partial \mathbb {H} / \partial u^\lambda ) = \overline{\varGamma _{\overline{\beta } \gamma }^\alpha } g_{\overline{\alpha } \lambda }, \end{aligned}$$

(52)

where $\partial ^2 \mathbb {H}^*/ \partial \overline{u^\beta } \partial u^{\gamma } = \overline{\varGamma _{\overline{\beta } \gamma }^\alpha } {{\varvec{e}}}_\alpha ^*$. $\varGamma _{\overline{\beta } \gamma }^\alpha$ and $\overline{\varGamma _{\overline{\beta } \gamma }^\alpha }$ are coefficients.

By means of the last two equations, the partial derivative of the metric tensor, $g_{\overline{\lambda } \gamma } = {{\varvec{e}}}_\lambda ^*\circ {{\varvec{e}}}_\gamma$, with respect to the coordinate value $\overline{u^\beta }$ generates,

$$\begin{aligned} g_{\overline{\lambda } \alpha } \varGamma _{\overline{\beta } \gamma }^\alpha + \overline{\varGamma _{\overline{\beta } \lambda }^\alpha } g_{\overline{\alpha } \gamma } = \partial g_{\overline{\lambda } \gamma } / \partial \overline{u^\beta }, \end{aligned}$$

(53)

similarly there are,

$$\begin{aligned} g_{\overline{\beta } \alpha } \varGamma _{\overline{\gamma } \lambda }^\alpha + \overline{\varGamma _{\overline{\gamma } \beta }^\alpha } g_{\overline{\alpha } \lambda }= & {} \partial g_{\overline{\beta } \lambda } / \partial \overline{u^{\gamma }}, \end{aligned}$$

(54)

$$\begin{aligned} g_{\overline{\gamma } \alpha } \varGamma _{\overline{\lambda } \beta }^\alpha + \overline{\varGamma _{\overline{\lambda } \gamma }^\alpha } g_{\overline{\alpha } \beta }= & {} \partial g_{\overline{\gamma } \beta } / \partial \overline{u^\lambda }. \end{aligned}$$

(55)

From the last three equations, there exist,

$$\begin{aligned} \varGamma _{ \overline{\lambda } , \overline{\beta } \gamma } = (1/2) ( \partial g_{ \overline{\gamma } \lambda } / \partial \overline{u^\beta } + \partial g_{ \overline{\lambda } \beta } / \partial \overline{u^{\gamma }} - \partial g_{ \overline{\gamma } \beta } / \partial \overline{u^\lambda } ), \end{aligned}$$

(56)

where $[ ( g_{\overline{\lambda } \alpha } \varGamma _{\overline{\beta } \gamma }^\alpha )^*]^T = \overline{\varGamma _{\overline{\gamma } \beta }^\alpha } g_{\overline{\alpha } \lambda }$, and $[ ( \overline{\varGamma _{\overline{\gamma } \beta }^\alpha } g_{\overline{\alpha } \lambda } )^*]^T = g_{\overline{\lambda } \alpha } \varGamma _{\overline{\beta } \gamma }^\alpha$.

Appendix 2: Curvature tensor

In the curved space described with the complex octonions, for a tensor $Y^{\gamma }$ with contravariant rank 1, the covariant derivative is,

$$\begin{aligned} \nabla _\beta Y^{\gamma } = \partial Y^{\gamma } / \partial u^\beta + \varGamma _{\lambda \beta }^{\gamma } Y^\lambda , ~~~~~ \nabla _{\overline{\alpha }} Y^{\gamma } = \partial Y^{\gamma } / \partial \overline{u^\alpha } + \varGamma _{\lambda \overline{\alpha }}^{\gamma } Y^\lambda , \end{aligned}$$

(57)

meanwhile, for a mixed tensor, $Z_\nu ^{\gamma }$, with contravariant rank 1 and covariant rank 1, the covariant derivative can be written as,

$$\begin{aligned} \nabla _\beta Z_\nu ^{\gamma }= & {} \partial Z_\nu ^{\gamma } / \partial u^\beta - \varGamma _{\beta \nu }^\lambda Z_\lambda ^{\gamma } + \varGamma _{\beta \lambda }^{\gamma } Z_\nu ^\lambda , \end{aligned}$$

(58)

$$\begin{aligned} \nabla _{\overline{\alpha }} Z_\nu ^{\gamma }= & {} \partial Z_\nu ^{\gamma } / \partial \overline{u^\alpha } - \varGamma _{\overline{\alpha } \nu }^\lambda Z_\lambda ^{\gamma } + \varGamma _{\overline{\alpha } \lambda }^{\gamma } Z_\nu ^\lambda , \end{aligned}$$

(59)

where $Y^{\gamma }$ and $Z_\nu ^{\gamma }$ both are scalar.

Apparently, the above equations deduce,

$$\begin{aligned} \nabla _{\overline{\alpha }} ( \nabla _\beta Y^{\gamma } ) = \partial ( \nabla _\beta Y^{\gamma } ) / \partial \overline{u^\alpha } - \varGamma _{\beta \overline{\alpha } }^\lambda ( \nabla _\lambda Y^{\gamma } ) + \varGamma _{\lambda \overline{\alpha } }^{\gamma } ( \nabla _\beta Y^\lambda ), \end{aligned}$$

(60)

similarly there is,

$$\begin{aligned} \nabla _\beta ( \nabla _{\overline{\alpha }} Y^{\gamma } ) = \partial ( \nabla _{\overline{\alpha }} Y^{\gamma } ) / \partial u^\beta - \varGamma _{\overline{\alpha } \beta }^\lambda ( \nabla _\lambda Y^{\gamma } ) + \varGamma _{\lambda \beta }^{\gamma } ( \nabla _{\overline{\alpha }} Y^\lambda ). \end{aligned}$$

(61)

As a result, making a subtraction in last two equations yields,

$$\begin{aligned}&\nabla _{\overline{\alpha }} ( \nabla _\beta Y^{\gamma } ) - \nabla _\beta ( \nabla _{\overline{\alpha }} Y^{\gamma } ) \nonumber \\&\quad = \left\{ \partial \varGamma _{\nu \beta }^{\gamma } / \partial \overline{u^\alpha } + \varGamma _{\lambda \overline{\alpha }}^{\gamma } \varGamma _{\nu \beta }^\lambda - \partial \varGamma _{\nu \overline{\alpha }}^{\gamma } /\partial u^\beta - \varGamma _{\lambda \beta }^{\gamma } \varGamma _{\nu \overline{\alpha }}^\lambda \right\} Y^\nu + T_{\beta \overline{\alpha }}^\lambda (\nabla _\lambda Y^{\gamma }), \end{aligned}$$

(62)

where $\partial ( \partial Y^{\gamma } / \partial \overline{u^\alpha } ) / \partial u^\beta - \partial ( \partial Y^{\gamma } / \partial u^\beta ) / \partial \overline{u^\alpha } = 0$. The torsion tensor is, $T_{\beta \overline{\alpha }}^\lambda = \varGamma _{\overline{\alpha } \beta }^\lambda - \varGamma _{\beta \overline{\alpha }}^\lambda$.

In the case there is, $T_{\beta \overline{\alpha }}^\lambda = 0$, the above can be reduced to,

$$\begin{aligned} \nabla _{\overline{\alpha }} ( \nabla _\beta Y^{\gamma } ) - \nabla _\beta ( \nabla _{\overline{\alpha }} Y^{\gamma } ) = R_{\beta \overline{\alpha } \nu }^{~~~~~~\gamma } Y^\nu , \end{aligned}$$

(63)

where the curvature tensor is, $R_{\beta \overline{\alpha } \nu }^{~~~~~~\gamma } = \partial \varGamma _{\nu \beta }^{\gamma } / \partial \overline{u^\alpha } - \partial \varGamma _{\nu \overline{\alpha }}^{\gamma } /\partial u^\beta + \varGamma _{\lambda \overline{\alpha }}^{\gamma } \varGamma _{\nu \beta }^\lambda - \varGamma _{\lambda \beta }^{\gamma } \varGamma _{\nu \overline{\alpha }}^\lambda$.

Appendix 3: Tangent space

In the curved spaces described with the complex octonions, there are some particular situations, in which the tangent spaces are different from the underlying spaces. They include but not limited to the following cases:

(1) The underlying space is the complex octonion space $\mathbb {O}$, while some tangent spaces respectively relate with the complex-octonion radius vector $\mathbb {H}$, composite radius vector $\mathbb {U}$, and angular momentum $\mathbb {L}$ and so forth. Therefore their components of tangent frames are respectively the partial derivatives of the complex-octonion radius vector, composite radius vector, and angular momentum and so on, with respect to the coordinate value $u^\alpha$ of complex-octonion radius vector.

(2) The underlying space is the complex-octonion composite space $\mathbb {O}_U$, while several tangent spaces are respectively relevant to the complex-octonion composite radius vector, angular momentum, and torque $\mathbb {W}$ and so forth. Accordingly their components of tangent frames are respectively the partial derivatives of the complex-octonion composite radius vector, angular momentum, and torque and so on, with respect to the coordinate value $U^\alpha$ of complex-octonion composite radius vector.

(3) The underlying space is the complex-octonion angular momentum space $\mathbb {O}_L$, while a part of tangent spaces are respectively involved with the complex-octonion angular momentum, torque, force $\mathbb {N}$ and so forth. Consequently their components of tangent frames are respectively the partial derivatives of the complex-octonion angular momentum, torque, and force and so on, with respect to the component $L^\alpha$ of complex-octonion angular momentum.

Apparently, in these curved spaces, there are some special cases, that is, the metric coefficient is a dimensionless parameter, and able to be approximately degenerated into, $g_{\overline{\alpha } \beta } \approx 1 + h_{\alpha \beta }$. For example, (a) The underlying space and tangent space both are the complex octonion space $\mathbb {O}$. (b) The underlying space and tangent space both are the complex-octonion composite space $\mathbb {O}_U$. (c) The underlying space and tangent space both are the complex-octonion angular momentum space $\mathbb {O}_L$. (d) The underlying space is the complex octonion space $\mathbb {O}$, while the tangent space is the complex-octonion composite space $\mathbb {O}_U$.

For the majority of the curved spaces described with the complex octonions, it may cause the metric coefficient to be seized of one certain dimension in the physics. As a result, they are incapable of deducing some equations to meet the requirement of the academic thought, that is, the existence of field will dominate the bending of space. The weak approximate method may be not suitable to these curved spaces, and even it may be hard to comprehend a few physical meanings may be in these curved spaces.