Implication of Spatial and Temporal Variations of the Fine-Structure Constant

Abstract

Temporal and spatial variations of fine-structure constant $\alpha \equiv e^{2}/\hbar c$ in cosmology have been reported in analysis of combination Keck and VLT data. This paper studies the variations based on consideration of basic spacetime symmetry in physics. Both laboratory α ₀ and distant α _z are deduced from relativistic spectrum equations of atoms (e.g., hydrogen atom) defined in inertial reference systems. When Einstein’s Λ≠0, the metric of local inertial reference systems in SM of cosmology is Beltrami metric instead of Minkowski, and the basic spacetime symmetry has to be de Sitter (dS) group. The corresponding special relativity (SR) is dS-SR. A model based on dS-SR is suggested. Comparing the predictions on α-varying with the data, the parameters are determined. The best-fit dipole mode in α’s spatial varying is reproduced by this dS-SR model. α-varyings in whole sky are also studied. The results are generally in agreement with the estimations of observations. The main conclusion is that the phenomenon of α-varying cosmologically with dipole mode dominating is due to the de Sitter (or anti de Sitter) spacetime symmetry with a Minkowski point in an extended special relativity called de Sitter invariant special relativity (dS-SR) developed by Dirac-Inönü-Wigner-Gürsey-Lee-Lu-Zou-Guo.

Appendices

Appendix A: Beltrami Metric and de Sitter Invariant Special Relativity

In this Appendix we briefly interpret the Beltrami metric and the de Sitter invariant Special Relativity (dS-SR).

• 1. Beltrami metric:

  We derive the expression of Beltrami metric (1.9) in the text. We consider a 4-dimensional pseudo-sphere (or hyperboloid) $\mathcal {S}_{\Lambda }$ embedded in a 5-dimensional Minkowski spacetime with metric η _{A B} =d i a g(1,−1,−1,−1,−1):

  $$\begin{array}{@{}rcl@{}} \mathcal{S}_{\Lambda} :&&\eta_{AB}\xi^{A}\xi^{B}=-R^{2},\\ && ds^{2}=\eta_{AB}d\xi^{A}d\xi^{B}, \end{array} $$

  (A.127)

  where index A, B={0,1,2,3,5}, R ²:=3Λ⁻¹ and Λ is the cosmological constant. $\mathcal {S}_{\Lambda }$ is also called de Sitter pseudo-spherical surface with radii R. Defining

  $$ x^{\mu}:=R\frac{\xi^{\mu}}{\xi^{5}},~~\text{with}~~\xi^{5}\neq 0,~\text{and}~\mu=\{0,1,2,3\}. $$

  (A.128)

  and treating x ^μ are Cartesian-type coordinates of a 4-dimensional spacetime with metric g _{μ ν} (x)≡B _{μ ν} (x), denoting this 4-dimensional spacetime as $\mathcal {B}_{\Lambda }$ (call it Beltrami spacetime), we derive B _{μ ν} (x) by means of the geodesic projection of $\{\mathcal {S}_{\Lambda }\mapsto \mathcal {B}_{\Lambda }\}$ (see Fig. 11).

  Fig. 11

  

  Sketch of the geodesic projection from de Sitter pseudo-spherical surface $\mathcal {S}_{\Lambda }$ to the Beltrami spacetime $\mathcal {B}_{\Lambda }$ via (A.128)

  From the definition (A.127), we have

  $$\begin{array}{@{}rcl@{}} ds^{2}&=&\eta_{AB}d\xi^{A} d\xi^{B}|_{\xi^{A,B}\in \mathcal{S}_{\Lambda}}\\ &=&\eta_{\mu\nu}d\xi^{\mu} d\xi^{\nu}-(d\xi^{5})^{2}\\ &:=&B_{\mu\nu}(x)dx^{\mu} dx^{\nu}. \end{array} $$

  (A.129)

  Since $\xi ^{A,B}\in \mathcal {S}_{\Lambda }$, and from (A.128) and (A.127), it is easy to obtain:

  $$\begin{array}{@{}rcl@{}} &&\xi^{\mu}=\frac{x^{\mu} }{ R}\xi^{5},~~d\xi^{\mu}=\frac{1}{ R}(\xi^{5}dx^{\mu}+x^{\mu} d\xi^{5}),~~ (\xi^{5})^{2}=\frac{R^{2}}{ \sigma(x)},\\ && d\xi^{5}=\eta_{\mu\nu} \frac{\xi^{\mu}}{ \xi^{5}}d\xi^{\nu}=\frac{1}{ R}\eta_{\mu\nu}x^{\mu} d\xi^{\nu} =\frac{\eta_{\mu\nu}x^{\mu} dx^{\nu}}{ \xi^{5}\sigma(x)^{2}}, \end{array} $$

  where

  $$ \sigma(x)= 1-\frac{\eta_{\mu\nu}x^{\mu} x^{\nu} }{ R^{2}}. $$

  (A.130)

  Substituting them into (A.129), we have

  $$ds^{2}=\frac{\eta_{\mu\nu}dx^{\mu} dx^{\nu}}{ \sigma(x)}+\frac{(\eta_{\mu\nu}x^{\mu} dx^{\nu})^{2}}{ R^{2} \sigma(x)^{2}}:= B_{\mu\nu}(x)dx^{\mu} dx^{\nu}. $$

  Then, we obtain the Beltrami metric as follows

  $$ B_{\mu\nu}(x)=\frac{\eta_{\mu\nu}}{\sigma(x)}+\frac{\eta_{\mu\lambda}x^{\lambda} \eta_{\mu\rho}x^{\rho}}{ R^{2} \sigma(x)^{2}}, $$

  (A.131)

  which is just (1.9) in the text.

• 2. Inertial reference coordinates and principle of relativity: The first Newtonian law is the foundation of the relativity. This law claims that the free particle moves with uniform velocity and along straight line. There exist systems of reference in which the first Newtonian motion law holds. Such reference systems are defined to be inertial. And the Newtonian motion law is always called the inertial moving law. If two reference systems move uniformly relative to each other, and if one of them is an inertial system, then clearly the other is also inertial. Experiment, e.g., the observations in the Galileo-boat which moves uniformly, shows that the so-called principle of relativity is valid. According to this principle all the law of nature are identical in all inertial systems of reference.

Theorem 1

The motion of particle with mass m ₀ and described by the following Lagrangian

$$ L_{Newton}=\frac{1}{2}m_{0}\mathbf{v}^{2}=\frac{1}{2}m_{0}\dot{\mathbf{x}}^{2} $$

(A.132)

satisfy the first Newtonian motion law, or the motion is inertial. In (A.132), the Cartesian expression of the velocity is as follows

$$ \mathbf{v}\equiv \dot{\mathbf{x}},~~\text{and}~~\mathbf{x}=x^{1}\mathbf{i}+x^{2}\mathbf{j}+x^{3}\mathbf{k}, $$

(A.133)

where i⋅i=j⋅j=k⋅k=1, and i⋅j=i⋅k=j⋅k=0.

Proof

By means of the Euler-Lagrangian equation

$$ \frac{\partial L }{ \partial x^{i}}=\frac{d}{dt} \frac{\partial L}{\partial \dot{x}^{i}},~~\text{or}~~\frac{\partial L }{ \partial \mathbf{x}}=\frac{d }{ dt} \frac{\partial L }{ \partial \dot{\mathbf{x}}} $$

(A.134)

(where ∂/∂ x≡∇:=(∂/∂ x ¹)i+(∂/∂ x ²)j+(∂/∂ x ³)k and etc) and L=L _{N e w t o n} we obtain

$$ \ddot{x}^{i}=0,~~~~\dot{x}^{i}=v^{i}=constant,~~\text{or}~~\dot{\mathbf{x}}=\mathbf{v}=constant.~~~~QED. $$

(A.135)

□

Theorem 2

The motion of particle in Minkowski spacetime described by

$$ L_{Einstein}=-m_{0}c\frac{ds}{ dt}=-m_{0}c\frac{\sqrt{\eta_{\mu\nu}dx^{\mu} dx^{\nu}}}{ dt}=-m_{0}c^{2}\sqrt{1-\frac{\dot{\mathbf{x}}^{2}}{ c^{2}}} $$

(A.136)

is inertial.

The proof is the same as above, because both L _Newton and L _Einstein are coordinates x ⁱ -independent. Generally, any x-free and time t-free Lagrangian functions $L(\dot {\mathbf {x}})$ can always reach the result of (A.135). However, when Lagrangian function is time-dependent that rule will become invalid. A useful example is as follows:

$$ L_{\Lambda}(t,\mathbf{x},\dot{\mathbf{x}}) =-m_{0}c^{2} \sqrt{3/{\Lambda}} \sqrt{\frac{3/{\Lambda} (c^{2}- \dot{\mathbf{x}}^{2})-\mathbf{x}^{2}\dot{\mathbf{x}}^{2}+(\mathbf{x}\cdot\dot{\mathbf{x}})^{2}+c^{2}(\mathbf{x}-\dot{\mathbf{x}}t)^{2} }{ c^{2} (3/{\Lambda} +\mathbf{x}^{2}-c^{2}t^{2})^{2}}}, $$

(A.137)

where a constant Λ ≠ 0. The stick-to-itive readers can verify the following identity via straightforward calculations from (A.137):

$$ \frac{\partial L_{\Lambda} }{ \partial \mathbf{x}}=\frac{\partial }{\partial t} \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}+\left( \dot{\mathbf{x}}\cdot\frac{\partial }{ \partial\mathbf{x}}\right) \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}. $$

(A.138)

Noting that the Euler-Lagrange (A.134) reads

$$ \frac{\partial L_{\Lambda} }{ \partial \mathbf{x}}=\frac{d }{ dt} \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}} =\frac{\partial }{ \partial t} \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}+\left( \dot{\mathbf{x}}\cdot\frac{\partial }{ \partial\mathbf{x}}\right) \frac{\partial L_{\Lambda}}{\partial \dot{\mathbf{x}}}+ \left( \ddot{\mathbf{x}}\cdot\frac{\partial }{ \partial\dot{\mathbf{x}}}\right) \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}, $$

(A.139)

and substituting (A.138) to (A.139), we have

$$ \left( \ddot{\mathbf{x}}\cdot\frac{\partial }{ \partial\dot{\mathbf{x}}}\right) \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}=0. $$

(A.140)

Since

$$ \|\frac{\partial }{ \partial\dot{\mathbf{x}}} \frac{\partial L_{\Lambda} }{ \partial \dot{\mathbf{x}}}\|\equiv \det \left( \frac{\partial^{2}L_{\Lambda}}{ \partial x^{i}\partial x^{j}}\right)\neq 0 $$

(A.141)

we have

$$ \ddot{\mathbf{x}}=0,~~~~\dot{\mathbf{x}}=\mathbf{v}=constant, $$

(A.142)

which indicates that the particle motion described by Lagrangian function (A.137) is inertial, and the first Newton motion law holds. Thus, the corresponding inertial reference systems can be built. Noting

$$ \lim_{\Lambda\rightarrow 0}L_{\Lambda}=L_{Einstein}, $$

(A.143)

it is essential and remarkable that a new kind of Special Relativity based on L _Λ (A.137) serving as an extension of the Einstein’s Special Relativity (E-SR) may exist.

• 3. de Sitter invariant Special Relativity (dS-SR): Following the Landau-Lifshitz formulation of Lagrangian [46] (see (A.136)), we examine the motion of free particle in the spacetime with Beltrami metric (A.131). From (2.18) in text

  $$ L_{dS}=-m_{0}c \frac{ds}{dt} =-m_{0}c\frac{\sqrt{B_{\mu\nu}(x)dx^{\mu} dx^{\nu}}}{ dt}=-m_{0}c{\sqrt{B_{\mu\nu}(x)\dot{x}^{\mu} \dot{x}^{\nu}}}, $$

  (A.144)

  we derive its expression in Cartesian coordinates. Setting up the time t=x ⁰/c, B _{μ ν} (x) can be rewritten as follows

  $$\begin{array}{@{}rcl@{}} ds^{2}\hskip-0.06in &=&\hskip-0.06in B_{\mu\nu}(x) dx^{\mu} dx^{\nu} =\widetilde{g}_{00}d(ct)^{2}+\widetilde{g}_{ij}\left[(dx^{i}+N^{i}d(ct)) (dx^{j}+N^{j}d(ct))\right]\\ &=& c^{2} (dt)^{2} \left[\widetilde{g}_{00} +\widetilde{g}_{ij}(\frac{1}{ c}\dot{x}^{i}+N^{i}) (\frac{1}{ c}\dot{x}^{j}+ N^{j})\right], \end{array} $$

  (A.145)

  where

  $$\begin{array}{@{}rcl@{}} \widetilde{g}_{00}&=&\frac{R^{2}}{ \sigma(x) (R^{2}-c^{2}t^{2})}, \end{array} $$

  (A.146)
  $$\begin{array}{@{}rcl@{}} \widetilde{g}_{ij}&=&\frac{\eta_{ij}}{ \sigma (x)}+ \frac{1}{ R^{2}\sigma(x)^{2}}\eta_{il}\eta_{jm}x^{l}x^{m}, \end{array} $$

  (A.147)
  $$\begin{array}{@{}rcl@{}} N^{i}&=&\frac{ctx^{i} }{ R^{2}-c^{2}t^{2}}. \end{array} $$

  (A.148)

  Substituting (A.145)–(A.148) into (A.144), we obtain the Lagrangian for free particle in $\mathcal {B}_{\Lambda }$:

  $$ L_{dS}=-m_{0}c^{2} \sqrt{\widetilde{g}_{00} +\widetilde{g}_{ij}(\frac{1}{ c}\dot{x}^{i}+N^{i}) (\frac{1}{ c}\dot{x}^{j}+ N^{j})}. $$

  (A.149)

  By using Cartesian notations (A.133) and expressions of (A.130) (A.146) (A.147) and (A.148), the explicit expression of Lagrangian (A.149) is:

  $$\begin{array}{@{}rcl@{}} L_{dS}&=& -m_{0}c^{2}\left[\frac{R^{4}}{ (R^{2}+\mathbf{x}^{2}-c^{2}t^{2})(R^{2}-c^{2}t^{2})}\right.\\ && +\left.\frac{-R^{2}}{ R^{2}+\mathbf{x}^{2}-c^{2}t^{2}}\right.\left( \frac{\dot{\mathbf{x}}^{2}}{ c^{2}}+\frac{c^{2}t^{2}\mathbf{x}^{2}}{ (R^{2}-c^{2}t^{2})^{2}}+\frac{2t(\mathbf{x}\cdot\dot{\mathbf{x}})}{ R^{2}-c^{2}t^{2}}\right)\\ && +\left.\frac{R^{2}}{ (R^{2}+\mathbf{x}^{2}-c^{2}t^{2})^{2}}\left( \frac{\dot{\mathbf{x}}\cdot\mathbf{x}}{ c}+\frac{ct\mathbf{x}^{2}}{ R^{2}-c^{2}t^{2}}\right)^{2}\right]^{1/2}\\ &=& -m_{0}c^{2} R \sqrt{\frac{R^{2} (c^{2}- \dot{\mathbf{x}}^{2})-\mathbf{x}^{2}\dot{\mathbf{x}}^{2}+(\mathbf{x}\cdot\dot{\mathbf{x}})^{2}+c^{2}(\mathbf{x}-\dot{\mathbf{x}}t)^{2}}{ c^{2} (R^{2}+\mathbf{x}^{2}-c^{2}t^{2})^{2}}}, \end{array} $$

  (A.150)

  where x ²=(x⋅x). Noting R ²=3/Λ (see, e.g., Eq. (15) in Ref. [30]), and comparing L _{d S} with $L_{\Lambda }(t,\mathbf {x},\dot {\mathbf {x}})$ of (A.137), we find

  $$ L_{dS}=L_{\Lambda}(t,\mathbf{x},\dot{\mathbf{x}})=-m_{0}c{\sqrt{B_{\mu\nu}(x)\dot{x}^{\mu} \dot{x}^{\nu}}}, $$

  (A.151)

  which is the Lagrangian for free particle mechanics of dS-SR. Since (A.143), when |R|→∞, the dS-SR goes back to E-SR.

• 4. de Sitter transformation to preserve Beltrami metric B _{μ ν} : In [26] (see Eqs. (41)–(43) in [26]), we have shown that under Lu-Zou-Guo (LZG) transformation (see also (B.161) below) preserves Beltrami metric B _{μ ν} . When space rotations were neglected temporarily for simplify, the transformation both due to a Lorentz-like boost and a space-transition in the x ¹ direction with parameters $\beta =\dot {x}^{1}/c$ and a ¹ respectively and due to a time transition with parameter a ⁰ can be explicitly written as follows:

  $$\begin{array}{@{}rcl@{}} \begin{array}{rcl} t\rightarrow \tilde{t}&=& \frac{\sqrt{\sigma(a)}}{c \sigma(a,x)} \gamma \left[ct-\beta x^{1}-a^{0}+ \beta a^{1} +\frac{a^{0}-\beta a^{1}}{R^{2}}\frac{a^{0} ct-a^{1} x^{1}-(a^{0})^{2} +(a^{1})^{2} } { \sigma(a)+\sqrt{\sigma(a)}} \right] \\ x^{1}\rightarrow \tilde{x}^{1}&=& \frac{\sqrt{\sigma(a)}}{ \sigma(a,x)}\gamma \left[ x^{1}-\beta ct +\beta a^{0} -a^{1} +\frac{a^{1}- \beta a^{0}}{R^{2}} \frac{a^{0} ct-a^{1} x^{1}-(a^{0})^{2} +(a^{1})^{2}}{ \sigma(a)+\sqrt{\sigma(a)}}\right]\\ x^{2}\rightarrow \tilde{x}^{2}&=&\frac{\sqrt{\sigma(a)}}{\sigma(a,x)}x^{2} \\ x^{3}\rightarrow \tilde{x}^{3}&=&\frac{\sqrt{\sigma(a)}}{\sigma(a,x)}x^{3} \end{array} \end{array} $$

  (A.152)

  where $\gamma =1/\sqrt {1-\beta ^{2}}$. It is easy to check when R→∞ the above transformation goes back to Poincaré transformation (or inhomogeneous Lorentz group I S O(1,3) transformation) in E-SR.

• 5. Conserved Noether charges of S O(4,1) of dS-SR: The external spacetime symmetry of dS-SR is S O(4,1). According to Neother theorem, the corresponding 10-Noether charges are energy E, momentums p ⁱ, boost charges K ⁱ and angular-momentums L ⁱ. All have been derived in [26]. The results are as follows

  $$\begin{array}{@{}rcl@{}} \begin{array}{rcl} &&{\text{ Noether charges for Lorentz boost:}}\, ~~ K^{i}=m_{0} {\Gamma} c (x^{i}- t \dot{x}^{i}) \\ &&\text{Charges for space-transitions (momenta):}~~~ p^{i}=m_{0} {\Gamma} \dot{x}^{i}, \\ &&\text{Charge for time-transition(energy):}~~~ E= m_{0} c^{2} {\Gamma} \\ &&\text{Charges for rotations in space (angular momenta):}~~~ L^{i} = \epsilon^{i}_{jk}x^{j}p^{k}, \end{array} \end{array} $$

  (A.153)

  where the Lorentz factor of dS-SR is:

  $$ {\Gamma} =\frac{1}{ \sqrt{1-\frac{\dot{\mathbf{x}}^{2}}{ c^{2}}+\frac{(\mathbf{x}\cdot \dot{\mathbf{x}})^{2} -\mathbf{x}^{2}\dot{\mathbf{x}}^{2}}{ c^{2}R^{2}}+\frac{(\mathbf{x}-\dot{\mathbf{x}}t)^{2}}{ R^{2}}}}. $$

  (A.154)

  It can be checked that $\dot {E}=\dot {p^{i}}=\dot {K^{i}}=\dot {L^{i}}=0$ under the equation of motion $\ddot {x}^{i}=0$ (or $\ddot {\mathbf {x}}=0$) [26].

Appendix B: Modified Beltrami Metric and de Sitter Invariant Special Relativity

We provide a brief introduction to Modified Beltrami metric (M-Beltrami metric) and the corresponding dS-SR.

• 1. M-Beltrami metric: (1.10) and (1.11) are the definition of M-Beltrami metric ${B}_{\mu \nu }^{(M)}(x)$. Being different from B _{μ ν} (x), the coordinate components of Minkowski point for ${B}_{\mu \nu }^{(M)}(x)$ is M ^μ instead of the origin of spacetime system x ^μ. Introducing notation

  $$ y^{\mu}\equiv x^{\mu}-M^{\mu} $$

  (B.155)

  then

  $$ {B}_{\mu\nu}^{(M)}(x)=B_{\mu\nu}(y). $$

  (B.156)

  The Landau-Lifshitz action is

  $$ S= -mc\int \sqrt{{B}_{\mu\nu}^{(M)}(x)dx^{\mu} dx^{\nu}}= -mc\int \sqrt{B_{\mu\nu}(y)dy^{\mu} dy^{\nu}}, $$

  (B.157)

  where d M ^μ=0 were used duo to constancy of M ^μ. The Lagrangian L _{M−d S} reads

  $$ L_{M-dS}=-mc\sqrt{{B}_{\mu\nu}^{(M)}(x)\dot{x}^{\mu}\dot{x}^{\nu}}. $$

  (B.158)

  Then, from δ S=0 and (A.142), we have

  $$ \frac{d^{2}{\mathbf y}}{ d(y^{0})^{2}}=0, $$

  (B.159)

  where y=x−M, y ⁰=c t _y =c t−M ⁰ (see (B.155)). Equation (B.159) becomes

  $$ \frac{d^{2}{\mathbf x}}{ dt^{2}}\equiv \ddot{\mathbf x}=0, $$

  (B.160)

  which means that the free particle moves with uniform velocity and along straight line in the dS-SR based M-Beltrami metric. Consequently, the first Newtonian law holds for L _{M−d S} (B.158) and inertial coordinate frames are well defined.

• 2. Spacetime symmetries of M-Beltrami metric and the motion integrals. In [26] (see Eqs. (41)–(43) in [26]), we have shown that under Lu-Zou-Guo (LZG) transformation

  $$\begin{array}{@{}rcl@{}} y^{\mu} \longrightarrow{LZG} \tilde{y}^{\mu} &=& \pm \sigma(a)^{1/2} \sigma(a,x)^{-1} (y^{\nu}-a^{\nu})D_{\nu}^{\mu}, \\ D_{\nu}^{\mu} &=& L_{\nu}^{\mu}+R^{-2} \eta_{\nu \rho}a^{\rho} a^{\lambda} (\sigma (a) +\sigma^{1/2}(a))^{-1} L_{\lambda}^{\mu} ,\\ L : &=& (L_{\nu}^{\mu})\in SO(1,3), \\ \sigma(y)&=& 1-\frac{1 }{ R^{2}}{\eta_{\mu \nu}y^{\mu} y^{\nu}}, \\ \sigma(a,y)&=& 1-\frac{1 }{ R^{2}}{\eta_{\mu \nu}a^{\mu} y^{\nu}}, \end{array} $$

  (B.161)

  the Beltrami metric transformation reads:

  $$ B_{\mu\nu}(y) \longrightarrow{LZG} \widetilde{B}_{\mu\nu}(\widetilde{y})=\frac{\partial y^{\lambda} }{ \partial \widetilde{y}^{\mu}}\frac{\partial y^{\rho} }{ \partial \widetilde{y}^{\nu}}B_{\lambda\rho}(y)=B_{\mu\nu}(\widetilde{y}). $$

  (B.162)

  Equation (B.162) leads to the invariance of action of (B.157):

  $$ S\longrightarrow{LZG} \widetilde{S}=S. $$

  (B.163)

  The corresponding Noether chargers or conserved motion integrals are as follows:

  $$\begin{array}{@{}rcl@{}} \begin{array}{rcl} &&{\text{ Noether charges for Lorentz boost:}}\, ~~ K^{i}=m {\Gamma} c (y^{i}- t_{y} \frac{dy^{i}}{ dt_{y}}) \\ &&\text{Charges for space-transitions (momenta):}~~~ p^{i}=m {\Gamma} \frac{dy^{i}}{ dt_{y}}, \\ &&\text{Charge for time-transition (energy): }~~~ E= m c^{2} {\Gamma} \\ &&\text{Charges for rotations in space (angular momenta):}~~~ L^{i} = \epsilon^{i}_{jk}y^{j}p^{k}, \end{array} \end{array} $$

  (B.164)

  where the Lorentz factor of dS-SR is:

  $$ {\Gamma} =\frac{1}{\sqrt{1-\frac{1}{c^{2}}\left( \frac{d\mathbf{y}}{ dt_{y}}\right)^{2}+\frac{1}{c^{2}R^{2}}\left[\left( \mathbf{y}\cdot \frac{d\mathbf{y}}{ dt_{y}}\right)^{2} -\mathbf{y}^{2}\left( \frac{d\mathbf{y}}{ dt_{y}}\right)^{2}\right]+\frac{1}{ R^{2}}\left( \mathbf{y}- t_{y} \frac{d\mathbf{y}}{ dt_{y}}\right)^{2}}}. $$

  (B.165)

  Using (B.155), we have the expressions in x frame:

  $$\begin{array}{@{}rcl@{}} &&K^{i}=m {\Gamma} c [x^{i}-M^{i}- (t-M^{0}/c)\dot{x}^{i}], \end{array} $$

  (B.166)
  $$\begin{array}{@{}rcl@{}} && p^{i}=m {\Gamma} \dot{x}^{i}, \end{array} $$

  (B.167)
  $$\begin{array}{@{}rcl@{}} && E= m c^{2} {\Gamma}, \end{array} $$

  (B.168)
  $$\begin{array}{@{}rcl@{}} && L^{i} = \epsilon^{i}_{jk}(x-M)^{j}p^{k}, \end{array} $$

  (B.169)

  and

  $$\begin{array}{@{}rcl@{}} {\Gamma} =\frac{1}{ \sqrt{1-\frac{\dot{\mathbf{x}}^{2}}{ c^{2}}+\frac{[(\mathbf{x}-\mathbf{M})\cdot \dot{\mathbf{x}}]^{2}-(\mathbf{x}-\mathbf{M})^{2}\dot{\mathbf{x}}^{2}}{ c^{2}R^{2}}+\frac{[\mathbf{x}-\mathbf{M}-\dot{\mathbf{x}}(t-M^{0}/c)]^{2}}{ R^{2}}}}. \end{array} $$

  (B.170)

  It is straightforward to check that $\dot {E}=\dot {p^{i}}=\dot {K^{i}}=\dot {L^{i}}=0$ under the equation of motion $\ddot {x}^{i}=0$ (or $\ddot {\mathbf {x}}=0$) and M ^μ=const..