preserve the gauge invariance principle in QFT, the regularized tensor-type
and scalar-type ILIs should satisfy the following necessary and sufficient
conditions:

$\displaystyle I_{2\mu\nu}^{R}=\frac{1}{2}g_{\mu\nu}\ I_{2}^{R},\quad
I_{2\mu\nu\rho\sigma}^{R}=\frac{1}{8}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\rho\nu})\
I_{2}^{R},$ $\displaystyle I_{0\mu\nu}^{R}=\frac{1}{4}g_{\mu\nu}\
I_{0}^{R},\quad
I_{0\mu\nu\rho\sigma}^{R}=\frac{1}{24}(g_{\mu\nu}g_{\rho\sigma}+g_{\mu\rho}g_{\nu\sigma}+g_{\mu\sigma}g_{\rho\nu})\
I_{0}^{R},$ (126)

which is referred to as the gauge invariance consistency conditionsLORE1 ;
LORE2 ; LORE3 .

To achieve such consistency conditions, a simple regularization prescription
in LORE method has consistently be realized as follows: firstly rotating the
momentum to the Euclidean space via a Wick rotation and then replacing the
loop integrating variable $k^{2}$ and loop integrating measure $\int{d^{4}k}$
of the ILIs through the corresponding regularized ones $[k^{2}]_{l}$ and
$\int[d^{4}k]_{l}$, i.e.:

$\displaystyle k^{2}$ $\displaystyle\rightarrow$
$\displaystyle[k^{2}]_{l}\equiv k^{2}+M^{2}_{l}\ ,$ (127)
$\displaystyle\int{d^{4}k}\ {\cal F}(k^{2})$ $\displaystyle\rightarrow$
$\displaystyle\int[d^{4}k]_{l}\ {\cal F}(k^{2}+M^{2}_{l})$ (128)
$\displaystyle\equiv$
$\displaystyle\int{d^{4}k}\lim_{N,M_{i}^{2}}\sum_{l=0}^{N}c_{l}^{N}\ {\cal
F}(k^{2}+M^{2}_{l})$ $\displaystyle=$
$\displaystyle\lim_{N,M_{i}^{2}}\sum_{l=0}^{N}c_{l}^{N}\ \int{d^{4}k}\ {\cal
F}(k^{2}+M^{2}_{l}).$

Where $M_{l}^{2}$ ($l=0,1,\ \cdots$) are regarded as regulator masses and
${\cal F}(k^{2})$ is considered to be any integration function. The notation
$\lim_{N,M_{i}^{2}}$ denotes the limiting case
$\lim_{N\to\infty,M_{i}^{2}\rightarrow\infty}$ ($i=1,2,\cdots,N$).

The key point in LORE is to choose the appropriate coefficients $c_{l}^{N}$,
so that any loop divergence with the power counting dimension larger than or
equal to the space-time dimension will vanish. From that, we come to the
general conditions for the coefficients $c_{l}^{N}$ as follows:

$\displaystyle\int d^{4}k\
\lim_{N,M_{i}^{2}}\sum_{l=0}^{N}c_{l}^{N}(k^{2}+M_{l}^{2})^{n}=0,\quad(n=0,1,\cdots),$
(129)

which brings on the following relations for the regulator masses:

$\displaystyle\sum_{l=0}^{N}c_{l}^{N}(M_{l}^{2})^{n}=0,\quad(n=0,1,\cdots),$
(130)

where we shall choose $M_{0}^{2}\equiv\mu_{s}^{2}\to 0$ and $c_{0}^{N}=1$ as
the initial conditions in order to recover the original integrals in the
limits $M_{i}^{2}\to\infty$ ($i=1,2,\cdots,N$ ) and $N\to\infty$.

To completely fix the coefficients $c_{l}^{N}$ and make them independent of
the regulator masses, it is simple to take the following string-mode
regulators:

$M_{l}^{2}=\mu_{s}^{2}+lM_{R}^{2},\qquad l=0,1,2,\cdots,$ (131)

which enables us to determine uniquely the coefficients $c_{l}^{N}$ from the
general relations of regulator masses given in Eq.(130) and obtain the
following explicit solution:

$c_{l}^{N}=(-1)^{l}\frac{N!}{(N-l)!l!}$ (132)

which is known as the sign-associated Combinations. The factor
$(-1)^{l}c_{l}^{N}$ may be regarded as the number of combinations of $N$
regulators taken $l$ at a time.

To be more explicit and concrete, when applying such a regularization
prescription in the LORE method with the simple solution of the regulators to
the quadratically and logarithmically divergent ILIs, the divergent
integrations become well-defined and can safely be performed to obtain the
regularized divergent ILIs $I_{2}^{R}$ and $I^{R}_{2\mu\nu}$ as well as
$I_{0}^{R}$ and $I^{R}_{0\mu\nu}$LORE1 ; LORE2 ; LORE3 . We just represent
them as follows:

$\displaystyle I_{2}^{R}$ $\displaystyle=$
$\displaystyle\frac{-i}{16\pi^{2}}\\{M_{c}^{2}-\mu_{M}^{2}-\mu_{M}^{2}[\ln\frac{M_{c}^{2}}{\mu_{M}^{2}}-\gamma_{E}+\varepsilon(\frac{\mu_{M}^{2}}{M_{c}^{2}})+\varepsilon^{\prime}(\frac{\mu_{M}^{2}}{M_{c}^{2}})-1]\\},$
$\displaystyle I_{0}^{R}$ $\displaystyle=$
$\displaystyle\frac{i}{16\pi^{2}}[\ln\frac{M_{c}^{2}}{\mu_{M}^{2}}-\gamma_{E}+\varepsilon(\frac{\mu_{M}^{2}}{M_{c}^{2}})],$
$\displaystyle I^{R}_{2\mu\nu}$ $\displaystyle=$
$\displaystyle\frac{1}{2}g_{\mu\nu}\ I^{R}_{2};\qquad
I^{R}_{0\mu\nu}=\frac{1}{4}g_{\mu\nu}\ I^{R}_{0},$ (133)

with the definitions,

$\displaystyle\mu_{M}^{2}=\mu_{s}^{2}+M^{2},$ $\displaystyle
M_{c}^{2}\equiv\lim_{N,M_{R}\to\infty}\left(\frac{M_{R}^{2}}{\ln N}\right),$
$\displaystyle\varepsilon(x)=\sum_{n=1}^{\infty}\frac{(-)^{n-1}}{n\
n!}x^{n}=\int_{0}^{x}d\sigma\frac{1-e^{-\sigma}}{\sigma}.$ (134)

Where the primary regulator mass $M_{R}$ is taken to be infinity so as to
recover the original integrals and the regulator number $N$ is set to be
infinity so as to make the regularized theory independent of the
regularization prescription. $\gamma_{E}$ is the Euler constant
($\gamma_{E}=0.577215\cdots$) and $\varepsilon(x)$ is a special function with
$x=\mu_{M}^{2}/M_{c}^{2}$ and
$\varepsilon^{\prime}(x)\equiv\partial_{x}\varepsilon(x)$. It is noted that
$M_{c}$ and $\mu_{s}$ are regarded as intrinsic mass scales. In general, the
mass scale $M_{c}$ can always be made to be finite as long as the ratio
$M_{R}^{2}/\ln N$ is kept as a finite quantity when the primary regulator mass
$M_{R}$ and regulator number $N$ are approaching to infinity.

It is seen that the LORE method truly brings on an infinity-free and symmetry-
preserving regularization method with the presence of two intrinsic mass
scales and the satisifaction of symmetry invariance consistency conditions.
Specifically, $M_{c}$ provides an ultraviolet (UV) ‘cutoff’, and $\mu_{s}$
sets an infrared (IR) ‘cutoff’ for massless case $M^{2}=0$. It is clear that
the LORE method does maintain the original divergent structure of integrals
when taking $M_{c}\to\infty$ and $\mu_{s}\to 0$. In comparison to the
dimensional regularization (DR) scheme, although the DR scheme can also result
in the same symmetry invariance consistency conditions to preserve gauge
invariance, while its regularized quadratic divergent ILIs are actually
suppressed to be a logarithmic divergence multiplying by the mass scale ${\cal
M}^{2}$ or vanishes ($I^{R}_{2}=0$) for the massless case ${\cal M}^{2}=0$.
This is because the dimensional regularization scheme modifies the original
theory by making an analytical extension for the space-time dimensions of
original theory, which makes the DR scheme to be limited in directly applying
to theories which require to keep well-defined spacetime dimensions and
maintain quadratic divergent structure, such as chiral type theories and
supersymmetric theories as well as theories with dynamically spontaneous
symmetry breaking.

Furthermore, the LORE method becomes simple at one loop level and gets very
applicable at high loop level for understanding the overlapping divergence
structure of Feynman diagrams. Nevertheless, LORE method is not motivated for
working out a much simpler regularization scheme but for figuring out an
infinity-free and symmetry-preserving regularization method without modifying
the original theory. It has been demonstrated in HW1 ; HLW  that the
derivation of ILIs from Feynman integrals by adopting UVDP parametrization
brings about the so-called Bjorken-Drell’s circuit analogy between Feynman
diagrams and electric circuitsBD , which enables us to pick up one-to-one
correspondence between the divergences of UVDP parameters and the subdiagrams
of Feynman diagrams.

Therefore, the LORE method does overcome the shortages and limitations
appearing in some widely-used regularization schemes. In fact, the LORE method
has been applied to make numerous interesting applications and carry out
various consistent calculationsCui:2008uv ; Cui:2008bk ; Ma:2005md ; Ma:2006yc
; cui:2011 ; DW ; HW ; Tang:2008ah ; Tang:2010cr ; Tang2011 , such as: how it
preserves non-Abelian gauge symmetry Cui:2008uv  and supersymmetry Cui:2008bk
, how it results in consistent analyses on the chiral anomalyMa:2005md  and
radiatively induced Lorentz and CPT-violating Chern-Simons term in the
extended QEDMa:2006yc  as well as QED trace anomalycui:2011 , how it enables
us to derive the gap equation for the dynamically generated spontaneous chiral
symmetry breaking at low energy QCD and obtain reliably the dynamic quark
masses and mass spectra of light scalar and pseudoscalar mesons in the chiral
effective field theoryDW , and meanwhile understand the chiral symmetry
restoration in chiral thermodynamic modelHW . Especially, the LORE method has
been utilized to perform the consistent computations on the quantum
gravitational contribution to gauge theories and obtain asymptotic free power-
law runningTang:2008ah ; Tang:2010cr ; Tang2011 , and also to clarify the
important issues raised inHGG  for the Higgs decay process
$H\to\gamma\gamma$HTW . It is intriguing to observe that the LORE method has
also been adopted to achieve a dynamical spontaneous symmetry breaking
mechanism for the electroweak symmetry in the SM and arrive at a sliding
electroweak symmetry breaking scale to be around $\mu_{EW}\sim 760$ GeVBCW .
In general, the LORE method was proved to be applicable to theories in high
dimensional spacetimeChapling1 ; Chapling2 .

With the above analyses, we should turn to make discussions on the finiteness
and renormalizability of GQFT. The crucial point in LORE is the appearance of
two intrinsic energy scales $M_{c}$ and $\mu_{s}$ which play the role as
ultraviolet (UV) cut-off and infrared (IR) cut-off of loop integrals to avoid
infinities in QFT without spoiling symmetries of original theory. These two
energy scales should get physically meaningful as the characteristic energy
scale $M_{c}$ and sliding energy scale $\mu_{s}$ indicated from the so-called
folk’s theoremSW1 ; SW2  addressed by Weinberg. Such a folk’s theorem states
that: any quantum theory that at sufficiently low energy and large distances
looks Lorentz invariant and satisfies the cluster decomposition principle will
also at sufficiently low energy look like a quantum field theory. It implies
that there must exist a characteristic energy scale (CES) $M_{c}$ which makes
the statement of sufficiently low energy to be reliable. It also means that
such a CES $M_{c}$ can be either a fundamental mass scale for a basic theory
or a dynamically generated energy scale for an effective theory. In the GQFT,
the CES $M_{c}$ is considered to be a fundamental mass scale, which
distinguishes to the QFT in Minkowski spacetime. The reason is that the
presence of gravigauge field in GQFT will bring about the scaling gauge
invariance of the action, which always allows us to set up a fundamental mass
scale $M_{\kappa}$ by imposing a suitable scaling gauge fixing condition. Such
a fundamental mass scale $M_{\kappa}$ is naturally identified to the CES
$M_{c}$ for characterizing the ultraviolet behavior of the theory, i.e.:

$\displaystyle M_{\kappa}\simeq M_{c},$ (135)

which makes any theory in the framework of GQFT to be infinity-free. Namely,
there exists in principle no divergences in GQFT and any quantum contribution
from high dimensionful interactions in GQFT should be finite and physically
meaningful.

In addition to the above consideration, let us follow along the key concept of
biframe spacetime in GQFT to understand the finiteness and renormalizability
of GQFT. In general, the biframe spacetime forms the fiber bundle structure
with the base spacetime taken to be a globally flat Minkowski spacetime and
the fiber bringing on a locally flat gravigauge spacetime. As the base
spacetime acts as an inertial reference frame for describing the motion of
basic fields, the energy momentum of basic fields in such a base spacetime can
approach to be infinite large and small as it is considered to be a continue
and differential spacetime to ensure the laws of energy momentum conservation,
which becomes the main reason why the path integral in such a globally flat
Minkowski spacetime gets divergences as the virtual particles can in principle
have very short wavelength/high frequency fluctuations or infinite large
energy momentum $k_{\mu}\to\infty$. In contrast, the fiber as a locally flat
gravigauge spacetime emerges as a non-commutative geometry characterized by
the field strength of gravigauge field and reflects the gravitational
interaction, which functions as an interaction frame of basic fields.
Therefore, the actual energy momentum of all basic fields in such a locally
flat gravigauge spacetime is always associated to the dual gravigauge field
and presented explicitly as follows:

$\displaystyle\mathcal{K}_{a}\equiv\hat{\chi}_{a}^{\;\;\mu}(x)k_{\mu}\sim\int[dq]e^{-iq_{\mu}x^{\mu}}\hat{k}_{a}(q),$
$\displaystyle\hat{k}_{a}\equiv\hat{\chi}_{a}^{\;\;\mu}(q)k_{\mu},$ (136)

which provides a physically meaningful energy momentum in GQFT. For
convenience, we may refer to $\hat{k}_{a}$ (or $\mathcal{K}_{a}$) as
gravigauging enengy momentum. Unlike the globally flat Minkowski spacetime,
the locally flat gravigauge spacetime as a matter field spacetime with
noncommutative geometry is in principle no longer to appear as an infinite
continue spacetime, so the gravigauging energy momentum $\hat{k}_{a}$ (or
$\mathcal{K}_{a}$) should not approach to be an infinite large/small quantity
even when the ordinary energy momentum $k_{\mu}$ in globally flat Minkowski
spacetime goes to be infinite, i.e.:

$\displaystyle\hat{k}_{a}\neq\infty,\qquad k_{\mu}\to\infty,$ (137)

which may be comprehended as follows: once at a short distance with infinite
large energy momentum $k_{\mu}\to\infty$ in the base spacetime of coordinates,
the gravigauge field $\chi_{\mu}^{\;\;a}$ will become infinite large
$\chi_{\mu}^{\;\;a}\to\infty$ and its dual/inverse gravigauge field
$\hat{\chi}_{a}^{\;\;\mu}$ gets infinitely small $\hat{\chi}_{a}^{\;\;\mu}\to
0$ , whereas the gravigauging energy momentum $\hat{k}_{a}$ (or
$\mathcal{K}_{a}$) as a combination quantity
$\hat{k}_{a}\equiv\hat{\chi}_{a}^{\;\;\mu}(q)k_{\mu}$ (or
$\hat{\chi}_{a}^{\;\;\mu}(x)k_{\mu}$) remains to keep a physically meaningful
finite energy momentum. Therefore, it appears in principle no infinite
gravigauging energy momentum in GQFT. On the other hand, there should exist a
potential horizon $L_{H}$ for the locally flat gravigauge spacetime as a
matter field spacetime, which can provide a natural infrared cutoff scale,
i.e.:

$\displaystyle\mu_{s}\to 1/L_{H},$ (138)

which allows us to remove the infrared divergence in GQFT.

From the above analyses and discussions, we are in the position to conclude
that any divergence should in principle disappear in GQFT, which provides a
resonable answer to the criticisms raised by Dirac and Feynman about the
treatment of infinities and divergences via the usual renormalization schemes
in QFT. Actually, the globally flat Minkowski spacetime as an inertial frame
of free motion fields is unobservable, only the locally flat gravigauge
spacetime as a gravitational frame of dynamic interaction fields becomes
observable, which constitutes our observed universe. Moreover, by applying for
the concept of renormalization group developed in refs.KGW ; GML , we should
be able to study physical phenomena at any interesting energy scale by
integrating out physical effects at higher energy scale, and define a finite
renormalized theory at any interesting renormalization scale. In fact, the
existence of both the characteristic energy scale (CES) $M_{c}\simeq
M_{\kappa}$ and sliding energy scale(SES) $\mu_{s}\sim 1/L_{H}$ in GQFT allows
us to choose the renormalization scale at any scale of interest between them.
In this sense, GQFT becomes more natural to provide a profound theoretical
framework for describing all basic forces.

##  VII Gravitational relativistic quantum theory on Dirac spinor field

The above various formalisms of the action enable us to derive the equations
of motion for all basic fields based on the least action principle. These
equations can be applied to investigate the dynamics of basic fields in the
gravitational relativistic quantum theory. Let us first study the equation of
motion for the chirality-based Dirac spinor $\Psi_{-}(x)$ in the presence of
gravitational interaction and spin gauge interaction as well as
electromagnetic interaction, which generalizes the usual relativistic quantum
mechanics to gravitational relativistic quantum theory.

###  VII.1  Generalized Dirac equation in gravitational relativistic quantum
theory

Following along the least action principle, we are able to arrive at the
following two types of the equations of motion for the chirality-based Dirac
spinor field:

$\displaystyle\hat{\mathsf{H}}^{\mu\nu}\Sigma_{-a}\mathbf{A}_{\mu}^{\;\;a}i\left({\mathcal{D}}_{\nu}-{\mathbf{V}}_{\nu}\right)\Psi_{-}(x)-\frac{m}{M_{\kappa}}\Sigma_{-}^{6}\Psi_{-}(x)=0,$
$\displaystyle\hat{\chi}^{\mu\nu}\Sigma_{-a}\chi_{\mu}^{\;\;a}i\left({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu}\right)\varPsi_{-}(x)-\frac{m}{M_{\kappa}}\phi(x)\Sigma_{-}^{6}\varPsi_{-}(x)=0,$
(139)

which correspond to the scaling gauge invariant dimensionless fields and local
scaling charged fields. Where we have introduced the following vector-like
field:

$\displaystyle{\mathbf{V}}_{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\mathfrak{A}\,\hat{\mathbf{A}}_{b}^{\;\;\nu}{\mathcal{D}}_{\nu}(\hat{\mathfrak{A}}\mathbf{A}_{\mu}^{\;\;b})=-\frac{1}{2}\mathbf{A}_{\mu
a}\hat{\mathbf{A}}_{b}^{\;\;\nu}\mathbf{A}_{\nu}^{ab},$
$\displaystyle{\mathsf{V}}_{\mu}$ $\displaystyle\equiv$
$\displaystyle\frac{1}{2}\chi\,\hat{\chi}_{b}^{\;\;\nu}{\mathcal{D}}_{\nu}(\hat{\chi}\chi_{\mu}^{\;\;b})=-\frac{1}{2}\chi_{\mu
a}\hat{\chi}_{b}^{\;\;\nu}\mathsf{A}_{\nu}^{ab},$ (140)

which is given by the gravigauge field and spin covariant gauge field. Such a
vector-like field preserves the scaling gauge invariance for the equation of
motion, which can be verified explicitly from the second equation in
Eq.(VII.1) when all fields are represented as local scaling charged fields.
For convenience, we may refer to such a vector-like field as a graviscaling
induced gauge field, which is distinguished from the ordinary gauge field of
Lie group due to the associated imaginary factor in the covariant derivative.

For simplicity of discussions, let us consider the equation of motion of
chirality-based Dirac spinor by taking the scaling gauge fixing condition to
be in Einstein basis, i.e.:

$\displaystyle\hat{\chi}^{\mu\nu}\Sigma_{-a}\chi_{\mu}^{\;\;a}i\left({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu}\right)\varPsi_{-}(x)-m\Sigma_{-}^{6}\varPsi_{-}(x)=0.$
(141)

In terms of the vector-like spinor representation of Dirac spinor field in
four-dimensional Hilbert space, the above equation can be simplified into the
following form:

$\displaystyle\gamma^{a}\hat{\chi}_{a}^{\;\;\mu}i\left({\mathcal{D}}_{\mu}-{\mathsf{V}}_{\mu}\right)\psi-m\psi=0,$
(142)

with the covariant derivative defined as follows:

$\displaystyle i({\mathcal{D}}_{\mu}-{\mathsf{V}}_{\mu})\equiv
i\partial_{\mu}+\frac{1}{2}g_{G}i\chi_{\mu
a}\hat{\chi}_{b}^{\;\;\nu}\mathsf{A}_{\nu}^{ab}+(\mathsf{\Omega}_{\mu}^{bc}+g_{G}\mathsf{A}_{\mu}^{bc})\frac{1}{2}\Sigma_{bc}+g_{E}A_{\mu}.$
(143)

Let us now investigate the quadratic form of the equation of motion for the
Dirac spinor in the presence of gravitational interaction and spin gauge
interaction as well as electromagnetic interaction. Its explicit form is found
to be:

$\displaystyle\hat{\chi}^{\mu\nu}(\nabla_{\mu}-{\mathsf{V}}_{\mu})({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})\psi+m^{2}\psi$
$\displaystyle\;\;=\Sigma^{cd}\hat{\chi}_{c}^{\;\;\mu}\hat{\chi}_{d}^{\;\;\nu}[{\mathcal{F}}_{\mu\nu}^{ab}\frac{1}{2}\Sigma_{ab}+g_{E}F_{\mu\nu}-g_{G}{\mathbf{F}}_{\mu\nu}^{a}\hat{\chi}_{a}^{\;\;\rho}i({\mathcal{D}}_{\rho}-{\mathsf{V}}_{\rho})-i{\mathsf{V}}_{\mu\nu}\,]\psi,$
(144)

with the following definitions:

$\displaystyle\nabla_{\mu}({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})\equiv{\mathcal{D}}_{\mu}({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})-\varGamma_{\mu\nu}^{\rho}({\mathcal{D}}_{\rho}-{\mathsf{V}}_{\rho}),$
$\displaystyle\varGamma_{\mu\nu}^{\rho}\equiv\hat{\chi}_{a}^{\;\;\rho}{\mathcal{D}}_{\mu}\chi_{\nu}^{\;\;a}=\hat{\chi}_{a}^{\;\;\rho}(\partial_{\mu}\chi_{\nu}^{\;\;a}+{\mathcal{A}}_{\mu\,b}^{a}\chi_{\nu}^{\;\;b}\,)\equiv\Gamma_{\mu\nu}^{\rho}+g_{G}\mathsf{A}_{\mu}^{ab}\chi_{\nu
b}\hat{\chi}_{a}^{\;\;\rho},$
$\displaystyle{\mathbf{F}}_{\mu\nu}^{a}\equiv{\mathcal{D}}_{\mu}\chi_{\nu}^{\;\;a}-{\mathcal{D}}_{\nu}\chi_{\mu}^{\;\;a}\equiv\mathsf{A}_{\mu}^{ab}\chi_{\nu
b}-\mathsf{A}_{\nu}^{ab}\chi_{\mu b},$
$\displaystyle{\mathsf{V}}_{\mu\nu}\equiv\partial_{\mu}{\mathsf{V}}_{\nu}-\partial_{\nu}{\mathsf{V}}_{\mu}.$
(145)

Where ${\mathcal{F}}_{\mu\nu}^{ab}$ and $F_{\mu\nu}$ are spin gauge field
strength and electromagnetic field strength, respectively.
$\Gamma_{\mu\nu}^{\rho}$ denotes the affine connection and
${\mathsf{V}}_{\mu\nu}$ represents the graviscaling induced gauge field
strength.

It is intriguing to notice that although the gauge invariant action is
hermitian, while the equation of motion of Dirac spinor field gets an emergent
dissipative term characterized by the graviscaling induced gauge field
${\mathsf{V}}_{\mu}$ and field strength ${\mathsf{V}}_{\mu\nu}$ due to the
presence of gravitational interaction and spin gauge interaction via the
gravigauge field $\chi_{\mu}^{\;a}$ and spin covariant gauge field
$\mathsf{A}_{\mu}^{ab}$.

To display explicitly the effects of gravitational interaction and spin gauge
interaction on the dynamics of Dirac fermion, we represent the above equation
of motion into the following form:

$\displaystyle\hat{\chi}^{\mu\nu}({\mathcal{D}}_{\mu}-{\mathsf{V}}_{\mu})({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})\psi+\left(\partial_{\mu}\hat{\chi}^{\mu\nu}+2\hat{\chi}^{\mu\nu}\partial_{\mu}\ln\chi+g_{G}\mathsf{A}_{\mu}^{ab}\hat{\chi}_{a}^{\;\mu}\hat{\chi}_{b}^{\;\nu}\right)({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})\psi$
$\displaystyle+(m^{2}+\frac{1}{16}R+\frac{1}{16}g_{G}\mathsf{F}_{\mu\nu}^{ab}\hat{\chi}_{b}^{\;\mu}\hat{\chi}_{a}^{\;\nu}-\frac{1}{8}g_{G}\epsilon_{abcd}\mathsf{F}_{\mu\nu}^{ab}\hat{\chi}^{c\mu}\hat{\chi}^{d\nu}i\gamma_{5})\psi$
$\displaystyle\;=\varSigma^{cb}[\hat{\chi}_{c}^{\;\;\mu}\hat{\chi}_{b}^{\;\;\nu}(g_{E}F_{\mu\nu}-i{\mathsf{V}}_{\mu\nu})+\hat{\chi}_{c}^{\;\mu}\left(ig_{G}\mathsf{F}_{\mu\nu
b}^{a}-2g_{G}\mathsf{A}_{\mu
b}^{a}i({\mathcal{D}}_{\nu}-{\mathsf{V}}_{\nu})\right)\hat{\chi}_{a}^{\;\nu}\,]\psi,$
(146)

where we have used the algebra relation of $\gamma$-matrices:

$\displaystyle\Sigma^{cd}\Sigma^{ab}=\frac{1}{2}i(\Sigma^{cb}\eta^{ad}-\Sigma^{db}\eta^{ac}-\Sigma^{ca}\eta^{db}+\Sigma^{da}\eta^{cb})+\frac{1}{16}(\eta^{ca}\eta^{db}-\eta^{da}\eta^{cb})+\frac{1}{4}\epsilon^{abcd}i\gamma_{5},$
(147)

and the properties of Riemann and Ricci curvature tensors:

$\displaystyle
R_{\mu\rho\nu\sigma}+R_{\rho\nu\mu\sigma}+R_{\nu\mu\rho\sigma}=0,\quad
R_{\mu\nu}=R_{\nu\mu}.$ (148)

###  VII.2  Gravigauge Dirac equation in locally flat gravigauge spacetime

From the gauge and scaling invariant action presented in Eqs.(80)-(85) and
(88), it is noticed that the Dirac fermion actually couples to the spin gauge
field
${\mathcal{A}}_{c}^{ab}\equiv\hat{\chi}_{c}^{\;\mu}{\mathcal{A}}_{\mu}^{ab}$
defined in locally flat gravigauge spacetime. Therefore, it is particularly
interesting to derive the equation of motion of Dirac spinor field in locally
flat gravigauge spacetime via a hidden coordinate formalism. Taking the
scaling gauge fixing condition to be in Einstein basis, we can rewrite the
equation of motion for the Dirac spinor into the following simple form:

$\displaystyle\gamma^{c}i\left({\mathcal{D}}_{c}-{\mathsf{V}}_{c}\right)\psi-m\psi=0,$
(149)

which appears formally to be analogous to the Dirac equation in globally flat
Minkowski spacetime. Whereas they are essentially distinguished due to the
presence of spin gauge field ${\mathcal{A}}_{c}^{ab}$ and the emergent non-
commutation relation of covariant derivative operator in locally flat
gravigauge spacetime shown in Eq.(V.2), which can be seen from the following
definitions and relations:

$\displaystyle i{\mathcal{D}}_{c}\equiv
i\eth_{c}+g_{G}{\mathcal{A}}_{c}^{ab}\frac{1}{2}\Sigma_{ab}+g_{E}A_{a},$
$\displaystyle[\eth_{c}\,,\eth_{d}]=\mathsf{\Omega}_{[cd]}^{a}\mspace{1.0mu}\eth_{a},\quad\eth_{c}\equiv\hat{\chi}_{c}^{\;\mu}\partial_{\mu},$
$\displaystyle{\mathsf{V}}_{c}\equiv\frac{1}{2}g_{G}\mathsf{A}_{bc}^{b}\equiv\frac{1}{2}g_{G}\hat{\chi}_{b}^{\;\;\nu}\mathsf{A}_{\nu
c}^{b}.$ (150)

To be more explicit, let us check the covariant quadratic form for the
equation of motion in locally flat gravigauge spacetime:

$\displaystyle(\nabla_{c}-{\mathsf{V}}_{c})({\mathcal{D}}^{c}-{\mathsf{V}}^{c})\psi+m^{2}\psi$
$\displaystyle=\Sigma^{cd}[{\mathcal{F}}_{cd}^{\,ab}\frac{1}{2}\Sigma_{ab}+g_{E}F_{cd}-g_{G}{\mathbf{F}}_{cd}^{a}i({\mathcal{D}}_{a}-{\mathsf{V}}_{a})-i{\mathsf{V}}_{cd}\,]\psi,$
(151)

with the definitions:

$\displaystyle\nabla_{c}({\mathcal{D}}_{d}-{\mathsf{V}}_{d})\equiv{\mathcal{D}}_{c}({\mathcal{D}}_{d}-{\mathsf{V}}_{d})-{\mathcal{A}}_{cd}^{a}({\mathcal{D}}_{a}-{\mathsf{V}}_{a}),$
$\displaystyle{\mathbf{F}}_{cd}^{a}\equiv{\mathcal{A}}_{[cd]}^{a}-\mathsf{\Omega}_{[cd]}^{a}\equiv\mathsf{A}_{[cd]}^{a}\equiv\mathsf{A}_{cd}^{a}-\mathsf{A}_{dc}^{a},$
$\displaystyle{\mathsf{V}}_{cd}\equiv\eth_{c}{\mathsf{V}}_{d}-\eth_{d}{\mathsf{V}}_{c}-\mathsf{\Omega}_{[cd]}^{a}{\mathsf{V}}_{a}.$
(152)

Where all the gauge fields and field strengths are defined in locally flat
gravigauge spacetime as shown in Eqs.(V.2) and (V.2). It can be further
demonstrated that Eq.(VII.2) can be rewritten into the following explicit form
in locally flat gravigauge spacetime:

$\displaystyle[({\mathcal{D}}_{c}-{\mathsf{V}}_{c})({\mathcal{D}}^{c}-{\mathsf{V}}^{c})+(\partial_{\mu}\hat{\chi}_{c}^{\;\mu}+\eth_{c}\ln\chi+\mathsf{A}_{a}^{ac})({\mathcal{D}}^{c}-{\mathsf{V}}^{c})]\psi$
$\displaystyle+(m^{2}+\frac{1}{16}R+\frac{1}{16}g_{G}\mathsf{F}_{ab}^{\,ba}-\frac{1}{8}g_{G}\epsilon^{cdab}\mathsf{F}_{cdab}i\gamma_{5})\psi$
$\displaystyle=\varSigma^{cb}[g_{E}F_{cb}-2g_{G}\mathsf{A}_{c\,b}^{a}i({\mathcal{D}}_{a}-{\mathsf{V}}_{a})+ig_{G}\mathsf{F}_{ca\mspace{1.0mu}b}^{a}-i{\mathsf{V}}_{cb})\,]\psi.$
(153)

It is interesting to notice that the Ricci curvature scalar $R\equiv
R_{cb}\eta^{cb}\equiv R_{\mu\sigma}\hat{\chi}^{\mu\sigma}$ and the spin
covariant gauge field strength scalar
$\mathsf{F}_{cd}^{ab}\eta_{b}^{\;c}\eta_{a}^{\;d}\equiv\mathsf{F}_{\mu\nu}^{ab}\hat{\chi}_{b}^{\;\mu}\hat{\chi}_{a}^{\;\nu}$
emerge as mass-like terms in the quadratic form of equation of motion for the
Dirac fermion, which indicates that a non-zero Ricci curvature scalar $R\neq
0$ appears to generate an effective mass for the Dirac fermion when it
propagates in locally flat gravigauge spacetime. Meanwhile, the totally
antisymmetric tensor $\epsilon^{cdab}\mathsf{F}_{cdab}$ arising from the spin
covariant gauge field strength behaves as a pseudoscalar.

In conclusion, the equation of motion of Dirac spinor field in the presence of
gravitational interaction and spin gauge interaction characterized by the spin
gravigauge field (or gravigauge field) and spin gauge field generalizes the
relativistic quantum mechanics described by Dirac equation in globally flat
Minkowski spacetime to obtain a gravitational relativistic quantum theory in
locally flat gravigauge spacetime with the emergence of non-commutative
geometry.

##  VIII Gravidynamics with gauge-type and geometric gravitational equations
as extension to Einstein theory of general relativity

To study the dynamics of gauge fields, we are going to take the gravigauge
field $\mathbf{A}_{\mu}^{\;\;a}(x)\equiv\phi(x)\chi_{\mu}^{\;\;a}(x)$ and spin
gauge field ${\mathcal{A}}_{\mu}^{ab}(x)$ as independent degrees of freedom
based on the inhomogeneous spin gauge symmetry WS(1,3). In particular, we will
discuss the potential new effects of gravidynamics beyond the Einstein theory
of general relativity.

###  VIII.1  Gravidynamics with gauge-type gravitational equations of
gravigauge field

To describe the gravidynamics, let us derive the equation of motion for the
${\mathcal{W}}_{e}$-spin invariant-gauge field as gravigauge field. The gauge
covariant and scaling invariant equations of motion in correspondence to the
scaling gauge invariant gravigauge field $\mathbf{A}_{\mu}^{\;\;a}$ and
scaling charged gravigauge field $\chi_{\mu}^{\;\;a}$ are found to be:

$\displaystyle\partial_{\nu}\tilde{\mathsf{F}}^{\mu\nu}_{a}=\boldsymbol{{\mathcal{J}}}_{a}^{\;\;\mu},$
$\displaystyle\tilde{d}_{\nu}\tilde{\mathrm{F}}^{\mu\nu}_{a}={\mathcal{J}}_{a}^{\;\;\mu},$
(154)

where the field strengths $\tilde{\mathsf{F}}^{\mu\nu}_{a}$ and
$\tilde{\mathrm{F}}^{\mu\nu}_{a}$ are defined as follows:

$\displaystyle\tilde{\mathsf{F}}^{\mu\nu}_{a}\equiv
M_{\kappa}^{-1}\mathfrak{A}\mspace{1.0mu}\tilde{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}\mathsf{F}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},\quad\mathsf{F}_{\mu\nu}^{a}\equiv\partial_{\mu}\mathbf{A}_{\nu}^{\;a}-\partial_{\nu}\mathbf{A}_{\mu}^{\;a},$
$\displaystyle\tilde{\mathrm{F}}^{\mu\nu}_{a}\equiv\chi\mspace{1.0mu}M_{\kappa}^{-2}\phi^{2}\tilde{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}\mathrm{F}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},\quad\mathrm{F}_{\mu\nu}^{a}\equiv
d_{\mu}\chi_{\nu}^{\;a}-d_{\nu}\chi_{\mu}^{\;a},$
$\displaystyle\tilde{d}_{\mu}\equiv\partial_{\mu}-S_{\mu},\quad
d_{\mu}=\partial_{\mu}+S_{\mu},\quad S_{\mu}\equiv\partial_{\mu}\ln\phi,$
(155)

and the currents $\boldsymbol{{\mathcal{J}}}_{a}^{\;\;\mu}$ and
${\mathcal{J}}_{a}^{\;\;\mu}$ are given explicitly by:

$\displaystyle\boldsymbol{{\mathcal{J}}}_{a}^{\;\;\mu}$ $\displaystyle\equiv$
$\displaystyle 16\pi
G_{N}\mspace{1.0mu}\hat{{\mathbf{J}}}_{a}^{\;\;\mu}+\tilde{{\mathbf{J}}}_{a}^{\;\;\mu},\quad\hat{{\mathbf{J}}}_{a}^{\;\;\mu}\equiv{\mathbf{J}}_{a}^{\;\;\mu}+\widetilde{{\mathbf{J}}}_{a}^{\;\;\mu},$
$\displaystyle\tilde{{\mathbf{J}}}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\hat{\mathbf{A}}_{a}^{\;\rho}\mathsf{F}_{\rho\nu}^{c}\tilde{\mathsf{F}}^{\mu\nu}_{c}-\frac{1}{4}\hat{\mathbf{A}}_{a}^{\;\mu}\mspace{1.0mu}\mathsf{F}_{\rho\nu}^{c}\tilde{\mathsf{F}}^{\rho\nu}_{c},$
$\displaystyle{\mathbf{J}}_{a}^{\;\;\mu}$ $\displaystyle=$ $\displaystyle
M_{\kappa}\mathfrak{A}\mspace{1.0mu}\\{(\hat{\mathbf{A}}_{a}^{\;\rho}\hat{\mathbf{A}}_{c}^{\;\mu}-\hat{\mathbf{A}}_{a}^{\;\mu}\hat{\mathbf{A}}_{c}^{\;\rho})(\bar{\Psi}_{-}\varSigma_{-}^{c}i{\mathcal{D}}_{\rho}\Psi_{-}+H.c.)$
$\displaystyle+$
$\displaystyle\hat{\mathbf{A}}_{a}^{\;\mu}\frac{m}{M_{\kappa}}\bar{\Psi}_{-}\Gamma^{6}\Psi_{-}-g_{E}^{-2}(\hat{\mathsf{H}}^{\mu\mu^{\prime}}\hat{\mathbf{A}}_{a}^{\;\rho}-\frac{1}{4}\hat{\mathsf{H}}^{\rho\mu^{\prime}}\hat{\mathbf{A}}_{a}^{\;\mu})\hat{\mathsf{H}}^{\nu\nu^{\prime}}\mspace{1.0mu}F_{\rho\nu}F_{\mu^{\prime}\nu^{\prime}}\\},$
$\displaystyle\widetilde{{\mathbf{J}}}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle
M_{\kappa}\mathfrak{A}\mspace{1.0mu}\\{-g_{G}^{-2}(\hat{\mathsf{H}}^{\mu\mu^{\prime}}\hat{\mathbf{A}}_{a}^{\;\rho}-\frac{1}{4}\hat{\mathsf{H}}^{\rho\mu^{\prime}}\hat{\mathbf{A}}_{a}^{\;\mu})\hat{\mathsf{H}}^{\nu\nu^{\prime}}\mspace{1.0mu}{\mathcal{F}}_{\rho\nu}^{bc}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}bc}$
(156) $\displaystyle+$
$\displaystyle\frac{m_{G}^{2}}{M_{\kappa}^{2}}(\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\mathbf{A}}_{a}^{\;\;\rho}-\frac{1}{4}\bar{\mathsf{H}}^{[\rho\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\mathbf{A}}_{a}^{\;\;\mu}){\mathcal{F}}_{\rho\nu}^{b}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}\\}$
$\displaystyle-$
$\displaystyle\frac{m_{G}^{2}}{M_{\kappa}}{\mathcal{D}}_{\nu}(\mathfrak{A}\,\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}),$

and

$\displaystyle{\mathcal{J}}_{a}^{\;\;\mu}$ $\displaystyle\equiv$
$\displaystyle 16\pi
G_{N}\mspace{1.0mu}\hat{{\mathsf{J}}}_{a}^{\;\;\mu}+\tilde{{\mathsf{J}}}_{a}^{\;\;\mu},\quad\hat{{\mathsf{J}}}_{a}^{\;\;\mu}\equiv{\mathsf{J}}_{a}^{\;\;\mu}+\widetilde{{\mathsf{J}}}_{a}^{\;\;\mu},$
$\displaystyle\tilde{{\mathsf{J}}}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\hat{\chi}_{a}^{\;\rho}\mathrm{F}_{\rho\nu}^{c}\tilde{\mathrm{F}}^{\mu\nu}_{c}-\frac{1}{4}\hat{\chi}_{a}^{\;\mu}\mspace{1.0mu}\mathrm{F}_{\rho\nu}^{c}\tilde{\mathrm{F}}^{\rho\nu}_{c},$
$\displaystyle{\mathsf{J}}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\mspace{1.0mu}\\{(\hat{\chi}_{a}^{\;\;\rho}\hat{\chi}_{c}^{\;\;\mu}-\hat{\chi}_{a}^{\;\;\mu}\hat{\chi}_{c}^{\;\;\rho})(\bar{\varPsi}_{-}\varSigma_{-}^{c}i{\mathcal{D}}_{\rho}\varPsi_{-}+H.c.)$
$\displaystyle+$
$\displaystyle\hat{\chi}_{a}^{\;\;\mu}\frac{m}{M_{\kappa}}\phi\bar{\varPsi}_{-}\Gamma^{6}\varPsi_{-}-(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\hat{\chi}^{\rho\mu^{\prime}}\hat{\chi}_{a}^{\;\;\mu})\hat{\chi}^{\nu\nu^{\prime}}\mspace{1.0mu}F_{\rho\nu}F_{\mu^{\prime}\nu^{\prime}}\\},$
$\displaystyle\widetilde{{\mathsf{J}}}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\mspace{1.0mu}\\{-(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}_{a}^{\;\rho}-\frac{1}{4}\hat{\chi}^{\rho\mu^{\prime}}\hat{\chi}_{a}^{\;\mu})\hat{\chi}^{\nu\nu^{\prime}}\mspace{1.0mu}{\mathcal{F}}_{\rho\nu}^{bc}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}bc}$
(157) $\displaystyle+$
$\displaystyle\frac{m_{G}^{2}}{M_{\kappa}^{2}}\phi^{2}(\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\bar{\chi}^{[\rho\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\mu})({\mathbf{F}}_{\rho\nu}^{b}+S_{\rho\nu}^{b})({\mathbf{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}+S_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}})\\}$
$\displaystyle-$
$\displaystyle\frac{m_{G}^{2}}{M_{\kappa}}{\mathcal{D}}_{\nu}\left(\chi\,\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}\phi({\mathbf{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}+S_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}})\right).$

In the above equations, we have introduced the following definitions:

$\displaystyle{\mathcal{D}}_{\nu}(\mathfrak{A}\,\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}})=\partial_{\nu}(\mathfrak{A}\,\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}})-\mathfrak{A}\,g_{G}{\mathcal{A}}_{\nu
a}^{b}\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},$
$\displaystyle\bar{\mathsf{H}}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\hat{\mathbf{A}}_{c}^{\;\,\mu}\hat{\mathbf{A}}_{d}^{\;\,\nu}\hat{\mathbf{A}}_{c^{\prime}}^{\;\,\mu^{\prime}}\hat{\mathbf{A}}_{d^{\prime}}^{\;\,\nu^{\prime}}\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}},\quad\tilde{\mathsf{H}}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\hat{\mathbf{A}}_{c}^{\;\,\mu}\hat{\mathbf{A}}_{d}^{\;\,\nu}\hat{\mathbf{A}}_{c^{\prime}}^{\;\,\mu^{\prime}}\hat{\mathbf{A}}_{d^{\prime}}^{\;\,\nu^{\prime}}\tilde{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}},$
$\displaystyle\bar{\chi}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\hat{\chi}_{c}^{\;\,\mu}\hat{\chi}_{d}^{\;\,\nu}\hat{\chi}_{c^{\prime}}^{\;\,\mu^{\prime}}\hat{\chi}_{d^{\prime}}^{\;\,\nu^{\prime}}\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}},\quad\tilde{\chi}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\hat{\chi}_{c}^{\;\,\mu}\hat{\chi}_{d}^{\;\,\nu}\hat{\chi}_{c^{\prime}}^{\;\,\mu^{\prime}}\hat{\chi}_{d^{\prime}}^{\;\,\nu^{\prime}}\tilde{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}},$
$\displaystyle\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}}\equiv\frac{1}{2}(\bar{\eta}^{cdc^{\prime}d^{\prime}}_{aa^{\prime}}-\bar{\eta}^{dcc^{\prime}d^{\prime}}_{aa^{\prime}}),\quad\tilde{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}}\equiv\frac{1}{2}(\tilde{\eta}^{cdc^{\prime}d^{\prime}}_{aa^{\prime}}-\tilde{\eta}^{dcc^{\prime}d^{\prime}}_{aa^{\prime}}).$
(158)

Taking the scaling gauge fixing condition to be in Einstein basis, we arrive
at the following gauge-type gravitational equation:

$\displaystyle\partial_{\nu}\tilde{F}^{\mu\nu}_{a}=J_{a}^{\;\mu},$ (159)

where the field strength $\tilde{F}^{\mu\nu}_{a}$ and current $J_{a}^{\;\mu}$
are defined as follows:

$\displaystyle\tilde{F}^{\mu\nu}_{a}$ $\displaystyle\equiv$
$\displaystyle\chi\mspace{1.0mu}\tilde{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}F_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},$
(160)

and

$\displaystyle J_{a}^{\;\mu}$ $\displaystyle\equiv$ $\displaystyle 16\pi
G_{N}\hat{J}_{a}^{\;\;\mu}+\tilde{J}_{a}^{\;\;\mu},\quad\hat{J}_{a}^{\;\;\mu}\equiv\mathrm{J}_{a}^{\;\;\mu}+\widetilde{J}_{a}^{\;\;\mu},$
$\displaystyle\tilde{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\hat{\chi}_{a}^{\;\rho}F_{\rho\nu}^{c}\tilde{F}^{\mu\nu}_{c}-\frac{1}{4}\hat{\chi}_{a}^{\;\mu}\mspace{1.0mu}F_{\rho\nu}^{c}\tilde{F}^{\rho\nu}_{c},$
$\displaystyle\mathrm{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\mspace{1.0mu}\\{(\hat{\chi}_{a}^{\;\;\rho}\hat{\chi}_{c}^{\;\;\mu}-\hat{\chi}_{a}^{\;\;\mu}\hat{\chi}_{c}^{\;\;\rho})\frac{1}{2}(\bar{\psi}\gamma^{c}i{\mathcal{D}}_{\rho}\psi+H.c.)$
$\displaystyle+$
$\displaystyle\hat{\chi}_{a}^{\;\;\mu}m\bar{\psi}\psi-(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\hat{\chi}^{\rho\mu^{\prime}}\hat{\chi}_{a}^{\;\;\mu})\hat{\chi}^{\nu\nu^{\prime}}\mspace{1.0mu}F_{\rho\nu}F_{\mu^{\prime}\nu^{\prime}}\\},$
$\displaystyle\widetilde{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\mspace{1.0mu}\\{-(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}_{a}^{\;\rho}-\frac{1}{4}\hat{\chi}^{\rho\mu^{\prime}}\hat{\chi}_{a}^{\;\mu})\hat{\chi}^{\nu\nu^{\prime}}\mspace{1.0mu}{\mathcal{F}}_{\rho\nu}^{bc}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}bc}$
(161) $\displaystyle+$ $\displaystyle
m_{G}^{2}(\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\bar{\chi}^{[\rho\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\mu}){\mathbf{F}}_{\rho\nu}^{b}{\mathbf{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}\\}$
$\displaystyle-$ $\displaystyle
m_{G}^{2}{\mathcal{D}}_{\nu}(\chi\,\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}),$

with the conserved current,

$\displaystyle\partial_{\mu}J_{a}^{\;\mu}=0.$ (162)

Such equations of motion for gravigauge field are obtained to describe the
gravidynamics and referred to as gauge-type gravitational equations.

###  VIII.2  Gauge-type gravidynamics in locally flat gravigauge spacetime

From the gauge and scaling invariant action built in hidden coordinate
formalisms in Eqs.(80)-(85) and (88), it is seen that all interactions emerge
in locally flat gravigauge spacetime spanned by the gravigauge bases. It
should be appropriate to derive the gauge-type gravitational equation in
locally flat gravigauge spacetime. When taking the scaling gauge fixing
condition to be in Einstein basis, we arrive at the following equation of
motion:

$\displaystyle\tilde{D}_{d}\tilde{F}^{cd}_{b}=J_{b}^{\;\;c},$ (163)

with the definitions:

$\displaystyle\tilde{F}^{cd}_{a}\equiv\tilde{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}}F_{c^{\prime}d^{\prime}}^{a^{\prime}},\quad
F_{cd}^{a}\equiv\hat{\chi}_{c}^{\;\;\mu}\hat{\chi}_{d}^{\;\;\nu}F_{\mu\nu}^{a},$
$\displaystyle\tilde{D}_{d}\tilde{F}^{cd}_{b}\equiv\eth_{d}\tilde{F}^{cd}_{b}+\mathsf{\Omega}_{da}^{c}\tilde{F}^{ad}_{b}+\mathsf{\Omega}_{da}^{d}\tilde{F}^{ca}_{b},$
(164)

for the field strength and covariant derivative, and

$\displaystyle J_{b}^{\;\;c}$ $\displaystyle\equiv$ $\displaystyle 16\pi
G_{N}\hat{J}_{b}^{\;\;c}+\tilde{J}_{b}^{\;c},\quad\hat{J}_{b}^{\;c}\equiv\mathrm{J}_{b}^{\;c}+\widetilde{J}_{b}^{\;c},$
$\displaystyle\tilde{J}_{b}^{\;c}$ $\displaystyle=$ $\displaystyle
F_{bd}^{a}\tilde{F}^{cd}_{a}-\frac{1}{4}\eta_{b}^{\;c}\mspace{1.0mu}F_{cd}^{a}\tilde{F}^{cd}_{a},$
$\displaystyle\mathrm{J}_{b}^{\;c}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\bar{\psi}\gamma_{c}i{\mathcal{D}}_{b}\psi+H.c.)-\eta_{b}^{\;c}[\frac{1}{2}(\bar{\psi}\gamma^{a}i{\mathcal{D}}_{a}\psi+H.c.)-m\bar{\psi}\psi]$
$\displaystyle-$
$\displaystyle(\eta_{b}^{\;a}\eta^{cc^{\prime}}-\frac{1}{4}\eta_{b}^{\;c}\eta^{ac^{\prime}})\eta^{dd^{\prime}}F_{ad}F_{c^{\prime}d^{\prime}},$
$\displaystyle\widetilde{J}_{b}^{\;c}$ $\displaystyle=$
$\displaystyle-(\eta_{b}^{\;a}\eta^{cc^{\prime}}-\frac{1}{4}\eta_{b}^{\;c}\eta^{ac^{\prime}})\eta^{dd^{\prime}}F_{ad}^{a^{\prime}b^{\prime}}F_{c^{\prime}d^{\prime}a^{\prime}b^{\prime}}$
(165) $\displaystyle+$ $\displaystyle
m_{G}^{2}(\eta_{b}^{\;c^{\prime\prime}}\eta_{b^{\prime}}^{\;c}\bar{\eta}^{[b^{\prime}d]c^{\prime}d^{\prime}}_{aa^{\prime}}-\frac{1}{4}\eta_{b}^{\;c}\bar{\eta}^{[c^{\prime\prime}d]c^{\prime}d^{\prime}}_{aa^{\prime}}){\mathbf{F}}_{c^{\prime\prime}d}^{a}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}}$
$\displaystyle-$ $\displaystyle
m_{G}^{2}\bar{{\mathcal{D}}}_{d}(\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{ba^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}}),$

for the currents. The gauge covariant derivative in the last term is defined
as follows:

$\displaystyle\bar{{\mathcal{D}}}_{d}(\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{ba^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}})$
$\displaystyle\equiv$
$\displaystyle\eth_{d}(\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{ba^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}})+(\mathsf{\Omega}_{da}^{c}\bar{\eta}^{[ad]c^{\prime}d^{\prime}}_{ba^{\prime}}+\mathsf{\Omega}_{da}^{d}\bar{\eta}^{[ca]c^{\prime}d^{\prime}}_{ba^{\prime}}-g_{G}{\mathcal{A}}_{db}^{a}\bar{\eta}^{[cd]c^{\prime}d^{\prime}}_{aa^{\prime}}){\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}}.$
(166)

Such equations of motion characterize the gauge-type gravidynamics in locally
flat gravigauge spacetime.

###  VIII.3  Geometric gravidynamics and extension to Einstein theory of
general relativity

To obtain geometric gravitational equations, let us take into account the
gauge-gravity and gravity-geometry correspondences shown in Eqs.(94)-(95) and
(98). When ignoring the total derivative term, the least action principle
leads to the following relations:

$\displaystyle
2\mathbf{R}_{\rho\sigma}\hat{\mathbf{A}}_{a}^{\;\rho}\hat{\mathsf{H}}^{\mu\sigma}-\hat{\mathbf{A}}_{a}^{\;\mu}\mathbf{R}=-16\pi
G_{N}\mspace{1.0mu}\hat{{\mathbf{J}}}_{a}^{\;\;\mu},$ $\displaystyle
2\phi^{2}\mathsf{R}_{\rho\sigma}\hat{\chi}_{a}^{\;\rho}\hat{\chi}^{\mu\sigma}-\hat{\chi}_{a}^{\;\mu}\phi^{2}\mathsf{R}=-16\pi
G_{N}M_{\kappa}^{2}\mspace{1.0mu}\hat{{\mathsf{J}}}_{a}^{\;\;\mu},$ (167)

with

$\displaystyle\mathsf{R}_{\rho\sigma}\equiv
R_{\rho\sigma}-6\partial_{\rho}\ln\phi\partial_{\sigma}\ln\phi,$
$\displaystyle\mathsf{R}\equiv\hat{\chi}^{\rho\sigma}\mathsf{R}_{\rho\sigma}=R-6\hat{\chi}^{\rho\sigma}\partial_{\rho}\ln\phi\partial_{\sigma}\ln\phi.$
(168)

In the Einstein basis, we can rewrite the above relation into the following
simple form:

$\displaystyle R_{a}^{\;\mu}-\frac{1}{2}\hat{\chi}_{a}^{\;\mu}R=-8\pi
G_{N}\mspace{1.0mu}\hat{J}_{a}^{\;\;\mu},$ (169)

with

$\displaystyle
R_{a}^{\;\mu}\equiv\hat{\chi}_{a}^{\;\mu^{\prime}}\hat{\chi}_{a^{\prime}}^{\;\nu^{\prime}}\hat{\chi}_{b^{\prime}}^{\;\mu}R_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}b^{\prime}}.$
(170)

When projecting the tensor and current in biframe spacetime into the base
spacetime of coordinates by applying for the gravigauge field
$\chi_{\mu}^{\;a}$ ($\hat{\chi}_{a}^{\;\mu}$), we arrive at the following two
geometric gravitational equations:

$\displaystyle R_{\mu\nu}-\frac{1}{2}\chi_{\mu\nu}R+8\pi
G_{N}\mspace{1.0mu}\widetilde{T}_{\mu\nu}=-8\pi
G_{N}\mspace{1.0mu}T_{\mu\nu},$ (171)

and

$\displaystyle\bar{\nabla}_{\rho}\bar{{\mathbf{F}}}_{\;\,[\mu\nu]}^{\rho}+\widetilde{T}_{[\mu\nu]}=-m_{G}^{-2}T_{[\mu\nu]}.$
(172)

The above two equations correspond to the symmetric tensor and antisymmetric
tensor in the base spacetime of coordinates.

The geometric gravitational equation presented in Eq.(171) brings on the
extension to Einstein theory of general relativity in the presence of spin
gauge field, which is characterized by the symmetric tensor
$\widetilde{T}_{\mu\nu}$. The explicit forms of the tensors are given as
follows:

$\displaystyle
T_{\mu\nu}+\widetilde{T}_{\mu\nu}=T_{\nu\mu}+\widetilde{T}_{\nu\mu}\equiv\frac{1}{2}\hat{\chi}(\chi_{\mu\rho}\hat{J}_{a}^{\;\rho}\chi_{\nu}^{\;a}+\chi_{\nu\rho}\hat{J}_{a}^{\;\rho}\chi_{\mu}^{\;a}),$
$\displaystyle
T_{\mu\nu}=\frac{1}{4}[\chi_{\mu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\nu}\psi+\chi_{\nu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\mu}\psi+H.c.]$
$\displaystyle\qquad-\chi_{\mu\nu}[\frac{1}{2}(\bar{\psi}\gamma^{c}\hat{\chi}_{c}^{\;\rho}i{\mathcal{D}}_{\rho}\psi+H.c.)-m\bar{\psi}\psi]$
$\displaystyle\qquad-(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}\mspace{1.0mu}F_{\rho\sigma}F_{\rho^{\prime}\sigma^{\prime}},$
$\displaystyle\widetilde{T}_{\mu\nu}=-(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}\mspace{1.0mu}{\mathcal{F}}_{\rho\sigma}^{ab}{\mathcal{F}}_{\rho^{\prime}\sigma^{\prime}ab}+m_{G}^{2}\bar{\nabla}_{\rho}\bar{{\mathbf{F}}}_{\;\,(\mu\nu)}^{\rho}$
$\displaystyle\qquad+m_{G}^{2}[\frac{1}{2}(\chi_{\mu\lambda}\eta_{\nu}^{\;\rho}+\chi_{\nu\lambda}\eta_{\mu}^{\;\rho})\bar{\chi}^{[\lambda\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}-\frac{1}{4}\hat{\chi}_{\mu\nu}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}]{\mathbf{F}}_{\rho\sigma}^{a}{\mathbf{F}}_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}},$
(173)

with the definitions:

$\displaystyle\bar{\nabla}_{\rho}\bar{{\mathbf{F}}}_{\;\,(\mu\nu)}^{\rho}\equiv\partial_{\rho}\bar{{\mathbf{F}}}_{(\mu\nu)}^{\rho}-\Gamma_{\rho\lambda}^{\rho}\bar{{\mathbf{F}}}_{(\mu\nu)}^{\lambda}-\frac{1}{2}(\Gamma_{\rho\mu}^{\lambda}\bar{{\mathbf{F}}}_{\lambda\nu}^{\rho}+\Gamma_{\rho\nu}^{\lambda}\bar{{\mathbf{F}}}_{\lambda\mu}^{\rho}+g_{G}{\mathcal{A}}_{\rho\nu}^{\lambda}\bar{{\mathbf{F}}}_{\mu\lambda}^{\rho}+g_{G}{\mathcal{A}}_{\rho\mu}^{\lambda}\bar{{\mathbf{F}}}_{\nu\lambda}^{\rho}),$
$\displaystyle\bar{{\mathbf{F}}}_{\mu\nu}^{\rho}\equiv\chi_{\mu\sigma}\chi_{\nu}^{\;\;a}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}}\equiv\bar{{\mathbf{F}}}_{(\mu\nu)}^{\rho}+\bar{{\mathbf{F}}}_{[\mu\nu]}^{\rho},$
$\displaystyle\bar{{\mathbf{F}}}_{(\mu\nu)}^{\rho}\equiv\frac{1}{2}(\bar{{\mathbf{F}}}_{\mu\nu}^{\rho}+\bar{{\mathbf{F}}}_{\nu\mu}^{\rho})=\frac{1}{2}(\chi_{\mu\sigma}\chi_{\nu}^{\;\;a}+\chi_{\nu\sigma}\chi_{\mu}^{\;\;a})\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}}.$
(174)

Where $T_{\mu\nu}$ is the usual four energy momentum tensor arising from the
Dirac fermion and electromagnetic gauge field in the presence of spin gauge
field and gravigauge field, and $\widetilde{T}_{\mu\nu}$ is purely attributed
to the inhomogeneous spin gauge field governed by the inhomogeneous spin gauge
symmetry.

The geometric gravitational equation given in Eq.(172) is obtained from the
antisymmetric tensors, which indicates that the gauge invariant field strength
$\tilde{\mathsf{F}}_{\mu\nu}^{\rho}$ is governed by the relevant antisymmetric
tensors $T_{[\mu\nu]}$ and $\widetilde{T}_{[\mu\nu]}$. Their explicit forms
are defined as follows:

$\displaystyle\bar{\nabla}_{\rho}\bar{{\mathbf{F}}}_{\;\,[\mu\nu]}^{\rho}\equiv\partial_{\rho}\bar{{\mathbf{F}}}_{[\mu\nu]}^{\rho}-\Gamma_{\rho\lambda}^{\rho}\bar{{\mathbf{F}}}_{[\mu\nu]}^{\lambda}-\frac{1}{2}(\Gamma_{\rho\mu}^{\lambda}\bar{{\mathbf{F}}}_{\lambda\nu}^{\rho}-\Gamma_{\rho\nu}^{\lambda}\bar{{\mathbf{F}}}_{\lambda\mu}^{\rho}+g_{G}{\mathcal{A}}_{\rho\nu}^{\lambda}\bar{{\mathbf{F}}}_{\mu\lambda}^{\rho}-g_{G}{\mathcal{A}}_{\rho\mu}^{\lambda}\bar{{\mathbf{F}}}_{\nu\lambda}^{\rho}),$
$\displaystyle\bar{{\mathbf{F}}}_{[\mu\nu]}^{\rho}\equiv\frac{1}{2}(\bar{{\mathbf{F}}}_{\mu\nu}^{\rho}-\bar{{\mathbf{F}}}_{\nu\mu}^{\rho})=\frac{1}{2}(\chi_{\mu\sigma}\chi_{\nu}^{\;\;a}-\chi_{\nu\sigma}\chi_{\mu}^{\;\;a})\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}},$
$\displaystyle\widetilde{T}_{[\mu\nu]}\equiv\frac{1}{2}(\chi_{\mu\lambda}\eta_{\nu}^{\;\rho}-\chi_{\nu\lambda}\eta_{\mu}^{\;\rho})\bar{\chi}^{[\lambda\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{\rho\sigma}^{a}{\mathbf{F}}_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}},$
$\displaystyle
T_{[\mu\nu]}\equiv\frac{1}{4}\left(\chi_{\mu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\nu}\psi-\chi_{\nu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\mu}\psi+H.c\right),$
(175)

which reflect the basic feature of gravigauge field $\chi_{\mu}^{\;a}$ as the
fundamental gravitational field instead of the symmetric gravimetric field
$\chi_{\mu\nu}$.

###  VIII.4  Geometric gravidynamics in locally flat gravigauge spacetime

Let us now project all tensors and currents into the ones in locally flat
gravigauge spacetime by adopting the gravigauge field $\chi_{\mu}^{\;a}$
($\hat{\chi}_{a}^{\;\mu}$) as Goldstone-like boson. The following
gravitational equations are obtained:

$\displaystyle R_{cb}-\frac{1}{2}\eta_{cb}R+8\pi
G_{N}\mspace{1.0mu}\widetilde{T}_{cb}=-8\pi G_{N}\mspace{1.0mu}T_{cb},$ (176)

and

$\displaystyle\bar{{\mathcal{D}}}_{d}\bar{\mathsf{F}}_{[cb]}^{d}+\widetilde{T}_{[cb]}=-m_{G}^{-2}T_{[cb]},$
(177)

where the tensors are given by the field strength of spin gravigauge field
$\mathsf{\Omega}_{c}^{ab}$:

$\displaystyle R_{cb}=\eta^{da}R_{cdab},\quad R=\eta^{cb}R_{cb},$
$\displaystyle
R_{cdab}=\eth_{c}\mathsf{\Omega}_{dab}-\eth_{d}\mathsf{\Omega}_{cab}+\mathsf{\Omega}_{cae}\mathsf{\Omega}_{db}^{e}-\mathsf{\Omega}_{dae}\mathsf{\Omega}_{cb}^{e}-\mathsf{\Omega}_{[cd]}^{e}\mathsf{\Omega}_{eab}.$
(178)

The symmetric tensors $T_{cb}$ and $\widetilde{T}_{cb}$ are provided by the
following explicit forms:

$\displaystyle T_{cb}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[\bar{\psi}\gamma_{c}i{\mathcal{D}}_{b}\psi+\bar{\psi}\gamma_{b}i{\mathcal{D}}_{c}\psi+H.c.]$
$\displaystyle-$
$\displaystyle\eta_{cb}[\frac{1}{2}(\bar{\psi}\gamma^{a}i{\mathcal{D}}_{a}\psi+H.c.)-m\bar{\psi}\psi]$
$\displaystyle-$
$\displaystyle(\eta_{c}^{\;a}\eta_{b}^{\;c^{\prime}}-\frac{1}{4}\eta_{cb}\eta^{ac^{\prime}})\eta^{dd^{\prime}}F_{ad}F_{c^{\prime}d^{\prime}},$
$\displaystyle\widetilde{T}_{cb}$ $\displaystyle=$
$\displaystyle-(\eta_{c}^{\;c^{\prime\prime}}\eta_{b}^{\;c^{\prime}}-\frac{1}{4}\eta_{cb}\eta^{c^{\prime\prime}c^{\prime}})\eta^{dd^{\prime}}{\mathcal{F}}_{c^{\prime\prime}d}^{ab^{\prime}}{\mathcal{F}}_{c^{\prime}d^{\prime}ab^{\prime}}+m_{G}^{2}\bar{{\mathcal{D}}}_{d}\bar{{\mathbf{F}}}_{\;(cb)}^{d}$
(179) $\displaystyle+$ $\displaystyle
m_{G}^{2}[\frac{1}{2}(\eta_{cb^{\prime}}\eta_{b}^{\;c^{\prime\prime}}+\eta_{bb^{\prime}}\eta_{c}^{\;c^{\prime\prime}})\bar{\eta}^{[b^{\prime}d]c^{\prime}d^{\prime}}_{aa^{\prime}}-\frac{1}{4}\eta_{cb}\bar{\eta}^{[c^{\prime\prime}d]c^{\prime}d^{\prime}}_{aa^{\prime}}]{\mathbf{F}}_{c^{\prime\prime}d}^{a}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}},$

with the definitions:

$\displaystyle\bar{{\mathcal{D}}}_{d}\bar{{\mathbf{F}}}_{\;(cb)}^{d}\equiv\eth_{d}\bar{{\mathbf{F}}}_{(cb)}^{d}+\mathsf{\Omega}_{da}^{d}\bar{{\mathbf{F}}}_{(cb)}^{a}-\frac{1}{2}(\mathsf{\Omega}_{dc}^{a}\bar{{\mathbf{F}}}_{ab}^{d}+\mathsf{\Omega}_{db}^{a}\bar{{\mathbf{F}}}_{ac}^{d}+g_{G}{\mathcal{A}}_{db}^{a}\bar{{\mathbf{F}}}_{ca}^{d}+g_{G}{\mathcal{A}}_{dc}^{a}\bar{{\mathbf{F}}}_{ba}^{d}),$
$\displaystyle\bar{{\mathbf{F}}}_{cb}^{d}\equiv\eta_{cb^{\prime}}\eta_{b}^{\;a}\bar{\eta}^{[db^{\prime}]c^{\prime}d^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}}\equiv\bar{{\mathbf{F}}}_{(cb)}^{d}+\bar{{\mathbf{F}}}_{[cb]}^{d},$
$\displaystyle\bar{{\mathbf{F}}}_{(cb)}^{d}\equiv\frac{1}{2}(\bar{{\mathbf{F}}}_{cb}^{d}+\bar{{\mathbf{F}}}_{bc}^{d})=\frac{1}{2}(\eta_{cb^{\prime}}\eta_{b}^{\;a}+\eta_{bb^{\prime}}\bar{\eta}_{c}^{\;a})\bar{\eta}^{[db^{\prime}]c^{\prime}d^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}},$
(180)

The antisymmetric tensors are defined as follows:

$\displaystyle\bar{{\mathcal{D}}}_{d}\bar{{\mathbf{F}}}_{\;[cb]}^{d}\equiv\eth_{d}\bar{{\mathbf{F}}}_{[cb]}^{d}+\mathsf{\Omega}_{da}^{d}\bar{{\mathbf{F}}}_{[cb]}^{a}-\frac{1}{2}(\mathsf{\Omega}_{dc}^{a}\bar{{\mathbf{F}}}_{ab}^{d}-\mathsf{\Omega}_{db}^{a}\bar{{\mathbf{F}}}_{ac}^{d}+g_{G}{\mathcal{A}}_{db}^{a}\bar{{\mathbf{F}}}_{ca}^{d}-g_{G}{\mathcal{A}}_{dc}^{a}\bar{{\mathbf{F}}}_{ba}^{d}),$
$\displaystyle\bar{{\mathbf{F}}}_{[cb]}^{d}\equiv\frac{1}{2}(\bar{{\mathbf{F}}}_{cb}^{d}-\bar{{\mathbf{F}}}_{bc}^{d})=\frac{1}{2}(\eta_{cb^{\prime}}\eta_{b}^{\;a}-\eta_{bb^{\prime}}\eta_{c}^{\;a})\bar{\eta}^{[db^{\prime}]c^{\prime}d^{\prime}}_{aa^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}},$
(181)

and

$\displaystyle\widetilde{T}_{[cb]}$ $\displaystyle=$
$\displaystyle\frac{1}{2}(\eta_{cb^{\prime}}\eta_{b}^{\;c^{\prime\prime}}-\eta_{bb^{\prime}}\eta_{c}^{\;c^{\prime\prime}})\bar{\eta}^{[b^{\prime}d]c^{\prime}d^{\prime}}_{aa^{\prime}}\mathsf{F}_{c^{\prime\prime}d}^{a}\mathsf{F}_{c^{\prime}d^{\prime}}^{a^{\prime}},$
$\displaystyle T_{[cb]}$ $\displaystyle=$
$\displaystyle\frac{1}{4}[\bar{\psi}\gamma_{c}i{\mathcal{D}}_{b}\psi-\bar{\psi}\gamma_{b}i{\mathcal{D}}_{c}\psi+H.c.].$
(182)

The gravitational equations in Eqs.(176) and (177) provide the equivalent
formalisms to characterize the gravidynamics in locally flat gravigauge
spacetime spanned by the gravigauge bases.

###  VIII.5 New effects of gravidynamics beyond Einstein theory of general
relativity

The gravidynamics described by the geometric gravitational equation presented
in Eq.(171) or Eq.(176) brings about the extension to Einstein theory of
general relativity, which is explicitly shown from the additional energy
momentum tensor $\widetilde{T}_{\mu\nu}$ in coordinate spacetime or
$\widetilde{T}_{bc}$ in gravigauge spacetime. Such an additional energy
momentum tensor arises solely from the inhomogeneous spin gauge field which is
governed by the inhomogeneous spin gauge symmetry.

To explore potential new effects caused from such a gravidynamics beyond the
Einstein theory of general relativity, let us turn to investigate the property
of spin gauge field. It is interesting to notice from the action shown in
Eq.(88) that the term given by the spin gauge invariant gravigauge field
strength ${\mathbf{F}}_{cd}^{a}$ can be rewritten into the following form:

$\displaystyle\frac{1}{4}\bar{\eta}^{cdc^{\prime}d^{\prime}}_{aa^{\prime}}m_{G}^{2}{\mathbf{F}}_{cd}^{a}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}}$
$\displaystyle=$
$\displaystyle\frac{3}{4}(\alpha_{G}+\frac{3}{2}\alpha_{W})m_{G}^{2}g_{G}^{2}({\mathcal{A}}_{[cab]}-\mathsf{\Omega}_{[cab]}/g_{G})({\mathcal{A}}^{[cab]}-\mathsf{\Omega}^{[cab]}/g_{G})$
(183) $\displaystyle+$
$\displaystyle\frac{1}{4}(\alpha_{G}-\frac{1}{2}\alpha_{W})m_{G}^{2}g_{G}^{2}({\mathcal{A}}_{cab}-\mathsf{\Omega}_{cab}/g_{G})({\mathcal{A}}^{cab}-\mathsf{\Omega}^{cab}/g_{G})$
$\displaystyle\equiv$
$\displaystyle\frac{1}{2}m_{T}^{2}({\mathcal{A}}_{[cab]}-\mathsf{\Omega}_{[cab]}/g_{G})({\mathcal{A}}^{[cab]}-\mathsf{\Omega}^{[cab]}/g_{G})$
$\displaystyle+$
$\displaystyle\frac{1}{2}m_{S}^{2}({\mathcal{A}}_{(cab]}-\mathsf{\Omega}_{(cab]}/g_{G})({\mathcal{A}}^{(cab]}-\mathsf{\Omega}^{(cab]}/g_{G}),$

with

$\displaystyle m_{T}^{2}\equiv 2(\alpha_{G}+\alpha_{W})m_{G}^{2}g_{G}^{2},$
$\displaystyle
m_{S}^{2}\equiv\frac{1}{2}(\alpha_{G}-\frac{1}{2}\alpha_{W})m_{G}^{2}g_{G}^{2}.$
(184)

Where we have introduced the following definitions:

$\displaystyle{\mathcal{A}}_{cab}\equiv{\mathcal{A}}_{[cab]}+{\mathcal{A}}_{(cab]},\quad\mathsf{\Omega}_{cab}\equiv\mathsf{\Omega}_{[cab]}+\mathsf{\Omega}_{(cab]},$
(185)

with

$\displaystyle{\mathcal{A}}_{[cab]}\equiv\frac{1}{3}({\mathcal{A}}_{cab}+{\mathcal{A}}_{abc}+{\mathcal{A}}_{bac}),$
$\displaystyle{\mathcal{A}}_{(cab]}\equiv\frac{1}{3}(2{\mathcal{A}}_{cab}-{\mathcal{A}}_{abc}-{\mathcal{A}}_{bac}).$
(186)

A similar definition holds for $\mathsf{\Omega}_{[cab]}$ and
$\mathsf{\Omega}_{(cab]}$. Where ${\mathcal{A}}_{[cab]}$ defines a totally
antisymmetric spin gauge field which couples directly to spinor field, and
${\mathcal{A}}_{(cab]}$ represents a mixing symmetric spin gauge field.

It indicates that the action of spin gauge invariant gravigauge field strength
${\mathbf{F}}_{cd}^{a}$ brings a gauge invariant mass term for the spin gauge
field. In general, the totally antisymmetric spin gauge field
${\mathcal{A}}_{[cab]}$ as a torsion like field and the mixing symmetric spin
gauge field ${\mathcal{A}}_{(cab]}$ have different masses. Only when the
coupling constants satisfy a special relation, they arrive at the same mass,
i.e.:

$\displaystyle\alpha_{W}=-\frac{2}{3}\alpha_{G},$ $\displaystyle
m_{T}^{2}=m_{S}^{2}=\frac{2}{3}\alpha_{G}m_{G}^{2}g_{G}^{2}\equiv m_{A}^{2},$
(187)

which leads to the mass $m_{A}$ for whole spin gauge field
${\mathcal{A}}_{\mu}^{ab}$.

As the spin gauge field couples to all spinor fields, i,e., leptons and quarks
in the standard model, so that it can in general decay into leptons and quarks
when its mass becomes much larger than the masses of those particles. On the
other hand, if the spin gauge field is not heavy enough, it should be produced
from the colliders of the electron and hadron. As there is no any signal
indicated from the experiments on such a spin gauge field, the large hadron
collider LHC as the highest energy collider at present provides the most
stringent constraint on the mass of spin gauge field up to several TeV for the
spin gauge coupling constant being compatible to the electroweak coupling
constant. In fact, various tests on the general relativity at short distance
arrive at the same order of magnitude though its consistency has well been
established in the macroscopic world.

Let us first consider the gravidynamics at the distance $l_{G}$ which is
larger than the one determined by the mass scale of spin gauge field,
$l_{G}>l_{A}=1/m_{A}$. In such a case, the spin gauge field goes away due to
its decay into other light particles, so the gravidynamics is characterized by
the following simplified equation:

$\displaystyle
R_{\mu\nu}-\frac{1}{2}\chi_{\mu\nu}R+\gamma_{G}\widetilde{R}_{\mu\nu}=-8\pi
G_{N}\mspace{1.0mu}T_{\mu\nu},$ (188)

with the definitions:

$\displaystyle\gamma_{G}\equiv 8\pi G_{N}m_{G}^{2},$
$\displaystyle\widetilde{R}_{\mu\nu}=\bar{\nabla}_{\rho}\bar{F}_{\;\,(\mu\nu)}^{\rho}+[\frac{1}{2}(\chi_{\mu\lambda}\eta_{\nu}^{\;\rho}+\chi_{\nu\lambda}\eta_{\mu}^{\;\rho})\bar{\chi}^{[\lambda\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}-\frac{1}{4}\hat{\chi}_{\mu\nu}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}]F_{\rho\sigma}^{a}F_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}},$
$\displaystyle\bar{\nabla}_{\rho}\bar{F}_{\;\,(\mu\nu)}^{\rho}\equiv\partial_{\rho}\bar{F}_{(\mu\nu)}^{\rho}-\Gamma_{\rho\lambda}^{\rho}\bar{F}_{(\mu\nu)}^{\lambda}-\frac{1}{2}(\Gamma_{\rho\mu}^{\lambda}\bar{F}_{\lambda\nu}^{\rho}+\Gamma_{\rho\nu}^{\lambda}\bar{F}_{\lambda\mu}^{\rho}),$
$\displaystyle\bar{F}_{\mu\nu}^{\rho}\equiv\chi_{\mu\sigma}\chi_{\nu}^{\;\;a}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}F_{\rho^{\prime}\sigma^{\prime}}^{a^{\prime}},\quad
F_{\rho\sigma}^{a}=\partial_{\rho}\chi_{\sigma}^{\;\;a}-\partial_{\sigma}\chi_{\rho}^{\;\;a},$
(189)

and

$\displaystyle
T_{\mu\nu}=\frac{1}{4}[\chi_{\mu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\nu}\psi+\chi_{\nu}^{\;a}\bar{\psi}\gamma_{a}i{\mathcal{D}}_{\mu}\psi+H.c.]$
$\displaystyle\qquad-\chi_{\mu\nu}[\frac{1}{2}(\bar{\psi}\gamma^{c}\hat{\chi}_{c}^{\;\rho}i{\mathcal{D}}_{\rho}\psi+H.c.)-m\bar{\psi}\psi]$
$\displaystyle\qquad-(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}\mspace{1.0mu}F_{\rho\sigma}F_{\rho^{\prime}\sigma^{\prime}},$
$\displaystyle i{\mathcal{D}}_{\mu}=i\partial_{\mu}+g_{E}A_{\mu},$ (190)

where $T_{\mu\nu}$ is the usual energy momentum tensor arising from the Dirac
fermion and electromagnetic gauge field.

It is seen from the above equation that the new effect beyond Einstein theory
of general relativity will arise from the tensor $\widetilde{R}_{\mu\nu}$
which is characterized by the gravigauge field $\chi_{\mu}^{\;\;a}$. As the
spin gauge symmetry always allows us to make the gravigauge field to be
symmetric $\chi_{\mu a}=\chi_{a\mu}$ by taking the physical unitary gauge, so
that its degrees of freedom are the same as those of gravimetric field
$\chi_{\mu\nu}$ with the relation $\chi_{\mu\nu}=\chi_{\mu
a}\eta^{ab}\chi_{b\nu}\equiv(\chi)^{2}_{\mu\nu}$. As the tensor
$\widetilde{R}_{\mu\nu}$ involves a quadratic derivative over the gravigauge
field, it brings on a similar effect as that of Ricci tensor $R_{\mu\nu}$. To
be more explicit, let us write down its gauge-type gravitational equation
based on Eqs.(159)-(VIII.1) as follows:

$\displaystyle\partial_{\nu}\tilde{F}^{\mu\nu}_{a}=\tilde{J}_{a}^{\;\;\mu}+\gamma_{G}\check{J}_{a}^{\;\;\mu}+16\pi
G_{N}\mathrm{J}_{a}^{\;\;\mu}$ (191)

with

$\displaystyle\tilde{F}^{\mu\nu}_{a}$ $\displaystyle\equiv$
$\displaystyle\chi\mspace{1.0mu}\tilde{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}F_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},$
$\displaystyle\tilde{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\hat{\chi}_{a}^{\;\rho}F_{\rho\nu}^{c}\tilde{F}^{\mu\nu}_{c}-\frac{1}{4}\hat{\chi}_{a}^{\;\mu}\mspace{1.0mu}F_{\rho\nu}^{c}\tilde{F}^{\rho\nu}_{c},$
$\displaystyle\check{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\,(\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\bar{\chi}^{[\rho\nu]\mu^{\prime}\nu^{\prime}}_{ba^{\prime}}\hat{\chi}_{a}^{\;\;\mu})F_{\rho\nu}^{b}F_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}-\partial_{\nu}(\chi\,\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}F_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}}),$
$\displaystyle\mathrm{J}_{a}^{\;\;\mu}$ $\displaystyle=$
$\displaystyle\chi\mspace{1.0mu}\\{(\hat{\chi}_{a}^{\;\;\rho}\hat{\chi}_{c}^{\;\;\mu}-\hat{\chi}_{a}^{\;\;\mu}\hat{\chi}_{c}^{\;\;\rho})\frac{1}{2}(\bar{\psi}\gamma^{c}i{\mathcal{D}}_{\rho}\psi+H.c.)$
(192) $\displaystyle+$
$\displaystyle\hat{\chi}_{a}^{\;\;\mu}m\bar{\psi}\psi-(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}_{a}^{\;\;\rho}-\frac{1}{4}\hat{\chi}^{\rho\mu^{\prime}}\hat{\chi}_{a}^{\;\;\mu})\hat{\chi}^{\nu\nu^{\prime}}\mspace{1.0mu}F_{\rho\nu}F_{\mu^{\prime}\nu^{\prime}}\\},$

where $\check{J}_{a}^{\;\;\mu}$ reflects the new effect of gravidynamics
beyond Einstein theory of general relativity. It is seen that the usual
gravitational current $\tilde{J}_{a}^{\;\;\mu}$ and the extra current
$\check{J}_{a}^{\;\;\mu}$ are all described by the gravigauge field strength
$F_{\mu\nu}^{a}$.

It is manifest that the magnitude of new effect caused from the gravidynamics
based on the inhomogeneous spin gauge symmetry at a distance larger than
$1/m_{G}$ is characterized by the mass scale $m_{G}$ via the mass ratio
$\gamma_{G}$:

$\displaystyle\gamma_{G}=8\pi G_{N}m_{G}^{2}=\frac{8\pi
m_{G}^{2}}{M_{P}^{2}}\equiv\frac{m_{G}^{2}}{\bar{M}_{P}^{2}},$ (193)

with $M_{P}$ the Planck mass. Such a ratio indicates that when the mass of
spin gauge field is much smaller than the Planck mass, i.e.,
$m_{G}^{2}\ll\bar{M}_{P}^{2}$ and $\gamma_{G}\ll 1$, the resulting new effect
only brings a tiny modification to the general relativity. It is expected that
the future experiments could provide a reliable constraint on the mass of spin
gauge field.

Let us now discuss an alternative case that the considering energy scale
$\mu_{G}$ is higher than the mass scale of spin gauge field, i.e.,
$\mu_{G}>m_{A}$. For such a short distance case, it is useful to take into
account the gravitational origin of spin gauge symmetry, which brings the spin
gauge field decomposed into two parts as shown in Eq.(58), i.e.,
${\mathcal{A}}_{\mu}^{ab}\equiv\mathsf{\Omega}_{\mu}^{ab}+\mathsf{A}_{\mu}^{ab}$.
By applying for such a decomposition of spin gauge field with the replacement
$\mathsf{A}_{\mu}^{ab}\to g_{G}\mathsf{A}_{\mu}^{ab}$, we can rewrite the
gravidynamic equation given in Eq.(171) into the following form:

$\displaystyle R_{\mu\nu}-\frac{1}{2}\chi_{\mu\nu}R+8\pi
G_{N}\widetilde{G}_{\mu\nu}=-8\pi
G_{N}\mspace{1.0mu}(T_{\mu\nu}+\check{T}_{\mu\nu}),$ (194)

with the definitions:

$\displaystyle\widetilde{G}_{\mu\nu}=(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}\mspace{1.0mu}R_{\rho\sigma\lambda^{\prime}}^{\lambda}R_{\rho^{\prime}\sigma^{\prime}\lambda}^{\lambda^{\prime}}=R_{\mu\rho}^{\;\sigma\lambda}R_{\;\nu\sigma\lambda}^{\rho}-\frac{1}{4}\chi_{\mu\nu}R_{\rho\sigma}^{\;\rho^{\prime}\sigma^{\prime}}R_{\;\rho^{\prime}\sigma^{\prime}}^{\sigma\rho},$
$\displaystyle\check{T}_{\mu\nu}=-(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}(2g_{G}^{-1}\mathsf{F}_{\rho\sigma}^{ab}R_{\rho^{\prime}\sigma^{\prime}ab}+\mathsf{F}_{\rho\sigma}^{ab}\mathsf{F}_{\rho^{\prime}\sigma^{\prime}ab})+m_{G}^{2}g_{G}\bar{\nabla}_{\rho}\bar{\mathsf{A}}_{\;\,(\mu\nu)}^{\rho}$
$\displaystyle\qquad+m_{G}^{2}g_{G}^{2}[\frac{1}{2}(\chi_{\mu\lambda}\eta_{\nu}^{\;\rho}+\chi_{\nu\lambda}\eta_{\mu}^{\;\rho})\bar{\chi}^{[\lambda\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}-\frac{1}{4}\hat{\chi}_{\mu\nu}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}]\mathsf{A}_{[\rho\sigma]}^{a}\mathsf{A}_{[\rho^{\prime}\sigma^{\prime}]}^{a^{\prime}},$
(195)

and

$\displaystyle\bar{\nabla}_{\rho}\bar{\mathsf{A}}_{\;\,(\mu\nu)}^{\rho}\equiv\partial_{\rho}\bar{\mathsf{A}}_{(\mu\nu)}^{\rho}-\Gamma_{\rho\lambda}^{\rho}\bar{\mathsf{A}}_{(\mu\nu)}^{\lambda}-\Gamma_{\rho\mu}^{\lambda}\bar{\mathsf{A}}_{\lambda\nu}^{\rho}-\Gamma_{\rho\nu}^{\lambda}\bar{\mathsf{A}}_{\lambda\mu}^{\rho}-\frac{1}{2}g_{G}(\mathsf{A}_{\rho\nu}^{\lambda}\bar{\mathsf{A}}_{\mu\lambda}^{\rho}+\mathsf{A}_{\rho\mu}^{\lambda}\bar{\mathsf{A}}_{\nu\lambda}^{\rho}),$
$\displaystyle\bar{\mathsf{A}}_{\mu\nu}^{\rho}\equiv\chi_{\mu\sigma}\chi_{\nu}^{\;\;a}\bar{\chi}^{[\rho\sigma]\rho^{\prime}\sigma^{\prime}}_{aa^{\prime}}\mathsf{A}_{[\rho^{\prime}\sigma^{\prime}]}^{a^{\prime}},\quad\bar{\mathsf{A}}_{(\mu\nu)}^{\rho}\equiv\frac{1}{2}(\bar{\mathsf{A}}_{\mu\nu}^{\rho}+\bar{\mathsf{A}}_{\nu\mu}^{\rho}),\quad\mathsf{A}_{[\rho\sigma]}^{a}\equiv\mathsf{A}_{\rho
b}^{a}\chi_{\sigma}^{\;b}-\mathsf{A}_{\sigma b}^{a}\chi_{\rho}^{\;b},$ (196)

with $\mathsf{F}_{\rho\sigma}^{ab}$ and $R_{\rho\sigma}^{ab}$ defined in
Eq.(V.1).

It is seen that the tensor $\widetilde{G}_{\mu\nu}$ is solely described by the
Riemann tensor $R_{\mu\nu\sigma}^{\rho}$, while the tensor
$\check{T}_{\mu\nu}$ is proportionally given by the spin covariant gauge field
$\mathsf{A}_{\mu}^{ab}$. Actually, the tensor $\widetilde{G}_{\mu\nu}$
provides the radiative energy momentum of gravitational field with the
following traceless property:

$\displaystyle\hat{\chi}^{\mu\nu}\widetilde{G}_{\mu\nu}=\hat{\chi}^{\mu\nu}(\eta_{\mu}^{\;\rho}\eta_{\nu}^{\;\rho^{\prime}}-\frac{1}{4}\chi_{\mu\nu}\hat{\chi}^{\rho\rho^{\prime}})\hat{\chi}^{\sigma\sigma^{\prime}}\mspace{1.0mu}R_{\rho\sigma\lambda^{\prime}}^{\lambda}R_{\rho^{\prime}\sigma^{\prime}\lambda}^{\lambda^{\prime}}=0.$
(197)

As the tensor term $\widetilde{G}_{\mu\nu}$ concerns the quadratic form of
Riemann tensor, it involves high derivatives over the gravimetric field.
Therefore, when the energy scale $\mu_{G}$ is considered to be much below the
Planck scale but above the mass scale $m_{A}$ of spin gauge field, the effect
of gravidynamics is small, i.e.:

$\displaystyle 8\pi G_{N}\widetilde{G}_{\mu\nu}\ll R_{\mu\nu},\quad
m_{A}<\mu_{G}\ll M_{P},$ (198)

Nevertheless, once the energy scale $\mu_{G}$ is compatible to the Planck mass
scale, such an effect becomes significant:

$\displaystyle 8\pi G_{N}\widetilde{G}_{\mu\nu}\sim R_{\mu\nu},\quad
m_{A}<\mu_{G}\sim M_{P}.$ (199)

In conclusion, at a very short distance close to or even smaller than the
Planck length $l_{P}=1/M_{P}$, the new effect caused from the gravidynamics
based on the inhomogeneous spin gauge symmetry will get significantly
important, it may even change the statement on the singularity of spacetime
based on the Einstein theory of general relativity. It is interesting to make
a quantitive study and general proof, while it is beyond the scope of present
paper and will be investigated elsewhere. Actually, our starting point in this
paper is mainly based on the consideration of effective theory. When
approaching to the Planck scale, we should begin with exploring a more
fundamental theory which is expected to involve more contents of new physics.

##  IX Spinodynamics via equations of motion of spin gauge field in biframe
spacetime

Following along the least action principle and taking the spin gauge field and
gravigauge field as independent degrees of freedom, we are able to derive the
following equations of motion to describe the spinodynamics:

$\displaystyle{\mathcal{D}}_{\nu}\boldsymbol{\hat{{\mathcal{F}}}}_{ab}^{\mu\nu}-\frac{m_{G}^{2}}{M_{\kappa}^{2}}\bar{{\mathcal{F}}}_{a}^{\mu\nu}\mathbf{A}_{\nu
b}=\hat{{\mathbf{J}}}_{ab}^{\mu},$
$\displaystyle{\mathcal{D}}_{\nu}\hat{{\mathcal{F}}}_{ab}^{\mu\nu}-m_{G}^{2}g_{G}\bar{{\mathbf{F}}}_{a}^{\mu\nu}\chi_{\nu
b}=\hat{{\mathsf{J}}}_{ab}^{\mu},$ (200)

where the second equation is resulted by making the scaling gauge fixing
condition to be in Einstein basis. The field strength and gauge covariant
derivative are defined as follows:

$\displaystyle\boldsymbol{\hat{{\mathcal{F}}}}_{ab}^{\mu\nu}\equiv\mathfrak{A}\hat{\mathsf{H}}^{\mu\mu^{\prime}}\hat{\mathsf{H}}^{\nu\nu^{\prime}}g_{G}^{-2}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}ab},$
$\displaystyle{\mathcal{D}}_{\nu}\boldsymbol{\hat{{\mathcal{F}}}}_{ab}^{\mu\nu}\equiv\partial_{\nu}\boldsymbol{\hat{{\mathcal{F}}}}_{ab}^{\mu\nu}-{\mathcal{A}}_{\nu
a}^{c}\boldsymbol{\hat{{\mathcal{F}}}}_{cb}^{\mu\nu},$
$\displaystyle\bar{{\mathcal{F}}}_{a}^{\mu\nu}\equiv\mathfrak{A}\bar{\mathsf{H}}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},$
(201)

and

$\displaystyle\hat{{\mathcal{F}}}_{ab}^{\mu\nu}\equiv\chi\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}^{\nu\nu^{\prime}}{\mathcal{F}}_{\mu^{\prime}\nu^{\prime}ab},$
$\displaystyle{\mathcal{D}}_{\nu}\hat{{\mathcal{F}}}_{ab}^{\mu\nu}\equiv\partial_{\nu}\hat{{\mathcal{F}}}_{ab}^{\mu\nu}-g_{G}{\mathcal{A}}_{\nu
a}^{c}\hat{{\mathcal{F}}}_{cb}^{\mu\nu},$
$\displaystyle\bar{{\mathbf{F}}}_{a}^{\mu\nu}\equiv\chi\bar{\chi}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}{\mathbf{F}}_{\mu^{\prime}\nu^{\prime}}^{a^{\prime}},$
(202)

with

$\displaystyle\bar{\mathsf{H}}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\frac{1}{2}(\bar{\mathsf{H}}_{aa^{\prime}}^{\mu\nu\mu^{\prime}\nu^{\prime}}-\bar{\mathsf{H}}_{aa^{\prime}}^{\nu\mu\mu^{\prime}\nu^{\prime}}),$
$\displaystyle\bar{\chi}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\frac{1}{2}(\bar{\chi}_{aa^{\prime}}^{\mu\nu\mu^{\prime}\nu^{\prime}}-\bar{\chi}_{aa^{\prime}}^{\nu\mu\mu^{\prime}\nu^{\prime}}).$
(203)

Where $\bar{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}$ and
$\bar{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}_{aa^{\prime}}$ are defined in
Eqs.(VI.3) and (VI.3), respectively. The field strengths
${\mathcal{F}}_{\mu\nu}^{ab}$, ${\mathcal{F}}_{\mu\nu}^{a}$ and
${\mathbf{F}}_{\mu\nu}^{a}$ are defined correspondingly in Eqs.(III.1),
(III.1) and (V.2). The corresponding currents are given by the following
explicit forms:

$\displaystyle\hat{{\mathbf{J}}}_{ab}^{\mu}\equiv\mathfrak{A}\hat{\mathbf{A}}^{c\mu}\bar{\Psi}_{-}\Sigma_{cab}\Psi_{-},$
$\displaystyle\hat{{\mathsf{J}}}_{ab}^{\mu}\equiv\chi\hat{\chi}^{c\mu}g_{G}\bar{\psi}\Sigma_{cab}\psi.$
(204)

The above equations of motion characterize the spinodynamics of spin gauge
field.

The inhomogeneous spin gauge symmetry brings the spin gauge field
${\mathcal{A}}_{\mu}^{ab}$ as a vector field in Minkowski spacetime to couple
with Dirac fermion via a vector gauge field
$\boldsymbol{{\mathcal{A}}}_{c}^{ab}\equiv\hat{\mathbf{A}}_{c}^{\;\mu}{\mathcal{A}}_{\mu}^{ab}$
or
${\mathcal{A}}_{c}^{ab}\equiv\hat{\chi}_{c}^{\;\mu}{\mathcal{A}}_{\mu}^{ab}$
due to the presence of gravigauge field, which is regarded as a vector spin
gauge field defined in locally flat gravigauge spacetime, which motivates us
to derive the equation of motion of spin gauge field in locally flat
gravigauge spacetime. The equation of motion for such a spin gauge field can
directly be derived from the actions presented in Eqs.(85) and (88) based on
the least action principle. In the Einstein basis, the equation of motion is
obtained as follows:

$\displaystyle{\mathcal{D}}_{d}{\mathcal{F}}_{ab}^{\;cd}-m_{G}^{2}g_{G}\bar{{\mathbf{F}}}_{a}^{cd}\eta_{db}={\mathsf{J}}_{ab}^{\;c},$
(205)

with the definitions:

$\displaystyle{\mathcal{F}}_{ab}^{cd}\equiv\eta^{cc^{\prime}}\eta^{dd^{\prime}}{\mathcal{F}}_{c^{\prime}d^{\prime}ab},\quad\bar{{\mathbf{F}}}_{a}^{cd}\equiv\bar{\eta}_{aa^{\prime}}^{[cd]c^{\prime}d^{\prime}}{\mathbf{F}}_{c^{\prime}d^{\prime}}^{a^{\prime}},$
$\displaystyle{\mathcal{D}}_{d}{\mathcal{F}}_{ab}^{\;cd}\equiv\eth_{d}{\mathcal{F}}_{ab}^{\;cd}+\mathsf{\Omega}_{dc^{\prime}}^{c}{\mathcal{F}}_{ab}^{\;c^{\prime}d}+\mathsf{\Omega}_{dd^{\prime}}^{d}{\mathcal{F}}_{ab}^{\;cd^{\prime}}-g_{G}{\mathcal{A}}_{da}^{a^{\prime}}{\mathcal{F}}_{a^{\prime}b}^{\;cd},$
(206)

for the field strength and gauge covariant derivative, and

$\displaystyle{\mathsf{J}}_{ab}^{\;c}\equiv
g_{G}\eta^{cd}\bar{\psi}\Sigma_{dab}\psi,$ (207)

for the corresponding current.

By applying the relations shown in Eqs.(183)-(VIII.5), we are able to rewrite
the above equation of motion into the following forms:

$\displaystyle{\mathcal{D}}_{d}{\mathcal{F}}_{\;\;[cab]}^{d}+m_{T}^{2}{\mathcal{A}}_{[cab]}=m_{T}^{2}g_{G}^{-1}\mathsf{\Omega}_{[cab]}-{\mathsf{J}}_{cab},$
$\displaystyle{\mathcal{D}}_{d}{\mathcal{F}}_{\;\;(cab]}^{d}+m_{S}^{2}{\mathcal{A}}_{(cab]}=m_{S}^{2}g_{G}^{-1}\mathsf{\Omega}_{(cab]},$
(208)

with

$\displaystyle{\mathcal{F}}_{\;\;cab}^{d}\equiv\eta^{dd^{\prime}}{\mathcal{F}}_{d^{\prime}cab}\equiv{\mathcal{F}}_{\;\;[cab]}^{d}+{\mathcal{F}}_{\;\;(cab]}^{d},$
$\displaystyle{\mathcal{F}}_{\;\;[cab]}^{d}\equiv\frac{1}{3}({\mathcal{F}}_{\;\;cab}^{d}+{\mathcal{F}}_{\;\;abc}^{d}+{\mathcal{F}}_{\;\;bca}^{d}),$
$\displaystyle{\mathcal{F}}_{\;\;(cab]}^{d}\equiv\frac{1}{3}(2{\mathcal{F}}_{\;\;cab}^{d}-{\mathcal{F}}_{\;\;abc}^{d}-{\mathcal{F}}_{\;\;bca}^{d}),$
$\displaystyle{\mathsf{J}}_{cab}\equiv g_{G}\bar{\psi}\Sigma_{cab}\psi,$ (209)

where the totally antisymmetric spin gauge field ${\mathcal{A}}_{[cab]}$ with
mass $m_{T}$ reflects the so-called torsion like field.

The above equations of motion for the spin gauge field describe the
spinodynamics in locally flat gravigauge spacetime, which can also be derived
from Eq.(IX) by projecting it into locally flat gravigauge spacetime via the
gravigauge field $\chi_{\mu}^{\;a}$.

##  X Electrodynamics in gravitational quantum field theory and generalized
Maxwell equations in biframe spacetime

Based on the maximal joint symmetry for the action of chirality-based Dirac
spinor, namely the inhomogeneous spin gauge symmetry and inhomogeneous Lorentz
symmetry (or Poincaré symmetry), we have provided a detailed analysis and
investigation on the gravitational relativistic quantum theory of Dirac spinor
as well as the gravidynamics and spinodynamics, which brings about the
theoretical framework of gravitational quantum field theory. In this section,
we are going to investigate systematically the electrodynamics in the presence
of gravitational interaction and to derive the generalized Maxwell equations
in Minkowski spacetime of coordinates and also in locally flat gravigauge
spacetime spanned by the gravigauge basis.

###  X.1 Gravigeometry-medium electrodynamics and generalized Maxwell
equations with gravigeometry-medium electromagnetic field

Following along the least action principle, we obtain the following equations
of motion for the electromagnetic gauge field:

$\displaystyle\partial_{\nu}\widehat{{\mathbf{F}}}^{\mu\nu}=-\widehat{{\mathbf{J}}}^{\mu},$
$\displaystyle\partial_{\nu}\widehat{F}^{\mu\nu}=-\widehat{J}^{\mu},$ (210)

with the field strengths and currents defined by

$\displaystyle\widehat{{\mathbf{F}}}^{\mu\nu}\equiv\mathfrak{A}\hat{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}F_{\mu^{\prime}\nu^{\prime}},$
$\displaystyle\widehat{F}^{\mu\nu}\equiv\chi\hat{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}F_{\mu^{\prime}\nu^{\prime}},$
$\displaystyle\hat{\mathsf{H}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\frac{1}{2}(\hat{\mathsf{H}}^{\mu\mu^{\prime}}\hat{\mathsf{H}}^{\nu\nu^{\prime}}-\hat{\mathsf{H}}^{\nu\mu^{\prime}}\hat{\mathsf{H}}^{\mu\nu^{\prime}}),$
$\displaystyle\hat{\chi}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\equiv\frac{1}{2}(\hat{\chi}^{\mu\mu^{\prime}}\hat{\chi}^{\nu\nu^{\prime}}-\hat{\chi}^{\nu\mu^{\prime}}\hat{\chi}^{\mu\nu^{\prime}}),$
(211)

and

$\displaystyle\widehat{{\mathbf{J}}}^{\mu}\equiv\mathfrak{A}\hat{\mathbf{A}}_{a}^{\;\mu}{\mathbf{J}}^{a},\quad{\mathbf{J}}^{a}=-g_{E}\bar{\Psi}_{-}\Gamma^{a}\Psi_{-},$
$\displaystyle\widehat{J}^{\mu}\equiv\chi\hat{\chi}_{a}^{\;\mu}J^{a},\quad
J^{a}=-g_{E}\bar{\psi}\gamma^{a}\psi.$ (212)

The second equation in Eq.(171) holds in the Einstein basis for the gauge
fixing condition of scaling gauge symmetry. Where
$\widehat{{\mathbf{F}}}^{\mu\nu}$ ($\widehat{F}^{\mu\nu}$) and
$\widehat{{\mathbf{J}}}^{\mu}$ ($\widehat{J}^{\mu}$) define the
electromagnetic gauge field strength and the corresponding current density in
the presence of gravimetric field, and ${\mathbf{J}}^{a}$ ($J^{a}$) is
regarded as a free current density in the absence of gravitational
interaction.

Meanwhile, the gauge field strength $F_{\mu\nu}$ satisfies the Bianchi
identity,

$\displaystyle\varepsilon^{\mu\nu\rho\sigma}\partial_{\nu}F_{\rho\sigma}=0.$
(213)

The above equations of motion in the Einstein basis can be rewritten into the
following four sets of equations:

$\displaystyle\nabla\cdot\widehat{\mathbf{E}}=\widehat{J}^{0},$
$\displaystyle\nabla\times\widehat{\mathbf{B}}=\boldsymbol{\widehat{J}}+\frac{\partial\widehat{\mathbf{E}}}{\partial
x^{0}},$ $\displaystyle\nabla\cdot\mathbf{B}=0,$
$\displaystyle\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial
x^{0}},$ (214)

where we have introduced the definitions:

$\displaystyle\widehat{\mathbf{E}}\equiv(\widehat{E}^{1},\widehat{E}^{2},\widehat{E}^{3})=(-\widehat{F}^{01},-\widehat{F}^{02},-\widehat{F}^{03}),$
$\displaystyle\widehat{\mathbf{B}}\equiv(\widehat{B}^{1},\widehat{B}^{2},\widehat{B}^{3})=(-\widehat{F}^{23},-\widehat{F}^{31},-\widehat{F}^{12}),$
$\displaystyle\mathbf{E}\equiv(E^{1},E^{2},E^{3})=(-F^{01},-F^{02},-F^{03}),$
$\displaystyle\mathbf{B}\equiv(B^{1},B^{2},B^{3})=(-F^{23},-F^{31},-F^{12}).$
(215)

The vector fields $\widehat{\mathbf{E}}$ and $\widehat{\mathbf{B}}$ define the
electric field and magnetic field in the presence of gravimetric field which
is introduced to characterize the Riemann geometry of curved spacetime. For
convenience, we may refer to $\widehat{\mathbf{E}}$ and $\widehat{\mathbf{B}}$
as gravigeometry-medium electric field and gravigeometry-medium magnetic
field, respectively. The generalized Maxwell equations presented in Eq.(X.1)
may be mentioned as gravigeometry-medium Maxwell equations, and the
corresponding electrodynamics is referred to as gravigeometry-medium
electrodynamics.

In the ordinary Maxwell equations yielded in globally flat Minkowski
spacetime, the corresponding field strength with
$\mathbf{E}\equiv\\{E^{i}=-F^{0i}\\}$ (i=1,2,3) and
$\mathbf{B}\equiv\\{B^{i}=-\frac{1}{2}\epsilon^{i}_{\;\;jk}F^{jk}\\}$
(i,j,k=1,2,3) defines a free-motion electromagnetic field in the absence of
gravitational interaction. In the gravigeometry-medium Maxwell equations shown
in Eq.(X.1) due to the presence of gravimetric field, the field strength
defined via
$\widehat{\mathbf{E}}\equiv\\{\widehat{E}^{i}=-\widehat{F}^{0i}\\}$ and
$\widehat{\mathbf{B}}\equiv\\{\widehat{B}^{i}=-\frac{1}{2}\epsilon^{i}_{\;\;jk}\widehat{F}^{jk}\\}$
are regarded as gravigeometry-medium electromagnetic field. The relations
between the gravigeometry-medium electromagnetic field and the free motion
electromagnetic field can directly be written down from the definitions in
Eq.(III.1) as follows:

$\displaystyle\widehat{F}^{0i}=\chi(\hat{\chi}^{00}\hat{\chi}^{ij}-\hat{\chi}^{0i}\hat{\chi}^{0j})F_{0j}+\chi\hat{\chi}^{ik}\hat{\chi}^{0j}F_{jk}\equiv-\widehat{E}^{i},$
$\displaystyle\widehat{F}^{ij}=\chi(\hat{\chi}^{i0}\hat{\chi}^{jk}-\hat{\chi}^{j0}\hat{\chi}^{ik})F_{0k}+\chi\hat{\chi}^{ik}\hat{\chi}^{jl}F_{kl}\equiv\epsilon^{ij}_{\;\;\;k}\widehat{B}^{k},$
(216)

which can be expressed into the following simple forms:

$\displaystyle\widehat{E}^{i}=\hat{\varepsilon}^{i}_{\;j}E^{j}+\hat{\alpha}^{i}_{\;j}B^{j},$
$\displaystyle\widehat{B}^{i}=\hat{\mu}^{i}_{\;j}B^{j}-\hat{\alpha}^{\;\,i}_{j}E^{j},$
(217)

with the coefficient functions:

$\displaystyle\hat{\varepsilon}^{i}_{\;j}(x)\equiv\chi(\hat{\chi}^{0}_{\;0}\hat{\chi}^{i}_{\;j}-\hat{\chi}^{i}_{\;0}\hat{\chi}^{0}_{\;j}),$
$\displaystyle\hat{\mu}^{i}_{\;j}(x)\equiv-\frac{1}{2}\chi\epsilon^{i}_{\;\;kl}\hat{\chi}^{kk^{\prime}}\hat{\chi}^{ll^{\prime}}\epsilon_{k^{\prime}l^{\prime}j},$
$\displaystyle\hat{\alpha}^{i}_{\;j}(x)\equiv\chi\hat{\chi}^{ik}\hat{\chi}^{l0}\epsilon_{klj}.$
(218)

Note that all the coefficients are functions of coordinate spacetime and they
are in general the vectors and tensors in spatial dimensions.

It is useful to rewrite the above relations into the following forms in the
vector representation:

$\displaystyle\widehat{\mathbf{E}}\equiv\widehat{\boldsymbol{\varepsilon}}\cdot\mathbf{E}+\widehat{\boldsymbol{\alpha}}\cdot\mathbf{B},$
$\displaystyle\widehat{\mathbf{B}}\equiv\widehat{\boldsymbol{\mu}}\cdot\mathbf{B}-\widehat{\boldsymbol{\alpha}}^{T}\cdot\mathbf{E},$
(219)

with the bi-vector coefficient functions:

$\displaystyle\widehat{\boldsymbol{\varepsilon}}(x)\equiv\\{\hat{\varepsilon}^{i}_{\;j}(x)\\},\quad\widehat{\boldsymbol{\mu}}(x)\equiv\\{\hat{\mu}^{i}_{\;j}(x)\\},$
$\displaystyle\widehat{\boldsymbol{\alpha}}(x)\equiv\\{\hat{\alpha}^{i}_{\;j}(x)\\},\quad\widehat{\boldsymbol{\alpha}}^{T}(x)\equiv\\{\hat{\alpha}_{j}^{\;i}(x)\\}.$
(220)

These coefficient functions are in principle governed by the gravidynamics.

###  X.2  Gravigauge-mediated electrodynamics and generalized Maxwell
equations in locally flat gravigauge spacetime

In the absence of gravitational interaction, the electromagnetic gauge field
$A_{\mu}$ is regarded as a vector field in globally flat Minkowski spacetime.
The inhomogeneous spin gauge symmetry with the introduction of gravigauge
field $\chi_{\mu}^{\;a}$ ($\hat{\chi}_{a}^{\;\mu}$) brings $A_{\mu}$ to couple
with Dirac fermion via a vector gauge field
$A_{a}\equiv\hat{\chi}_{a}^{\;\mu}A_{\mu}$ defined in locally flat gravigauge
spacetime. The least action principle leads the equation of motion of gauge
field $A_{a}$ to have the following form in locally flat gravigauge spacetime:

$\displaystyle D_{b}F^{ab}=-J^{a},$ (221)

with the field strength and covariant derivative defined as follows:

$\displaystyle D_{b}F^{ab}$ $\displaystyle\equiv$
$\displaystyle\eth_{b}F^{ab}+\mathsf{\Omega}_{bc}^{a}F^{cb}+\mathsf{\Omega}_{bc}^{b}F^{ac},$
$\displaystyle F^{ab}$ $\displaystyle\equiv$
$\displaystyle\eta^{ac}\eta^{bd}F_{cd}\equiv\hat{\chi}^{a\mu}\hat{\chi}^{b\nu}F_{\mu\nu},\quad
F_{\mu\nu}=\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu},$ $\displaystyle
F_{cd}$ $\displaystyle\equiv$ $\displaystyle
D_{c}A_{d}-D_{d}A_{c}=\eth_{c}A_{d}-\eth_{d}A_{c}+\mathsf{\Omega}_{cd}^{b}A_{b}-\mathsf{\Omega}_{dc}^{b}A_{b},$
(222)

and the current density is simply given by

$\displaystyle J^{a}=-g_{E}\bar{\psi}\gamma^{a}\psi,$ (223)

which is considered as the current density for freely moving Dirac fermion in
locally flat gravigauge spacetime.

The field strength $F_{ab}$ satisfies the following identity:

$\displaystyle\epsilon^{cabd}D_{c}F_{ab}=0,$ (224)

which results from the Bianchi identity of Riemann curvature tensor
$\epsilon^{cabd}R_{cab}^{b^{\prime}}=0$ in locally flat gravigauge spacetime.
It can directly be verified as follows:

$\displaystyle\epsilon^{cabd}D_{c}F_{ab}=2\epsilon^{cabd}D_{c}(D_{a}A_{b})=-\epsilon^{cabd}R_{cab}^{b^{\prime}}A_{b^{\prime}}=0.$
(225)

It can be proved from Eq.(221) that the current density $J^{a}$ is conserved
in a gauge covariant form, i.e.:

$\displaystyle D_{a}J^{a}$ $\displaystyle=$ $\displaystyle-
D_{a}D_{b}F^{ab}=\frac{1}{2}(R_{abc}^{a}F^{cb}+R_{abc}^{b}F^{ac})$ (226)
$\displaystyle=$ $\displaystyle
R_{cb}F^{cb}=\frac{1}{2}(R_{cb}-R_{bc})F^{cb}=0,$

where the last equality is due to the symmetric feature of Ricci curvature
tensor $R_{cb}=R_{bc}$ and the antisymmetric property of field strength
$F^{cb}=-F^{bc}$. It is straightforward to write down the quadratic form for
the equation of motion of gauge field $A_{a}$ shown in Eq.(221) as follows:

$\displaystyle D_{c}D^{c}A_{a}+R_{ac}A^{c}-D_{a}(D_{c}A^{c})=J_{a}.$ (227)

When taking the following covariant gauge fixing condition:

$\displaystyle D_{c}A^{c}=\eth_{c}A^{c}+\mathsf{\Omega}_{cd}^{c}A^{d}=0,$
(228)

and utilizing the equation of motion of gravidynamics presented in Eq.(176),
we arrive at the following equation of motion for the gauge field $A_{a}$ in
locally flat gravigauge spacetime:

$\displaystyle D_{c}D^{c}A_{a}+\frac{1}{2}R\mspace{1.0mu}A_{a}-8\pi
G_{N}(T_{ac}+\widetilde{T}_{ac})A^{c}=J_{a},$ (229)

which indicates that a non-zero Ricci curvature scalar $R\neq 0$ brings on an
effective mass to the electromagnetic gauge field $A_{a}$ when it propagates
in locally flat gravigauge spacetime.

It is useful to rewrite the equations of motion in Eqs.(221) and (224) into
the following forms:

$\displaystyle\eth_{b}F^{ba}-\breve{A}_{b}F^{ba}-\frac{1}{2}F_{bc}^{a}F^{bc}=J^{a},$
$\displaystyle\epsilon^{abcd}(\eth_{b}F_{cd}+F_{bc}^{c^{\prime}}F_{c^{\prime}d})=0,$
(230)

with the definitions:

$\displaystyle
F_{bc}^{a}\equiv-\mathsf{\Omega}_{[bc]}^{a}=\mathsf{\Omega}_{cb}^{a}-\mathsf{\Omega}_{bc}^{a}\equiv\hat{\chi}_{b}^{\;\;\mu}\hat{\chi}_{c}^{\;\;\nu}F_{\mu\nu}^{a}\equiv\hat{F}_{cb}^{\mu}\chi_{\mu}^{\;\;a},$
$\displaystyle\hat{F}_{cb}^{\mu}\equiv\eth_{c}\hat{\chi}_{b}^{\;\;\mu}-\eth_{b}\hat{\chi}_{c}^{\;\;\mu},\quad\eth_{b}\equiv\hat{\chi}_{b}^{\;\;\nu}\partial_{\nu},$
$\displaystyle\breve{A}_{b}\equiv
F_{cb}^{c}\equiv-\mathsf{\Omega}_{cb}^{c}=-\hat{\chi}\partial_{\nu}(\chi\hat{\chi}_{b}^{\;\;\nu}),$
(231)

where $F_{bc}^{a}$ is the gravigauge field strength in locally flat gravigauge
spacetime. The equations of motion in locally flat gravigauge spacetime with
$a,b=(0,\alpha)$ ($\alpha=1,2,3$) can be expressed as follows:

$\displaystyle\eth_{\alpha}\widetilde{E}^{\alpha}-\breve{A}_{\alpha}\widetilde{E}^{\alpha}-\breve{E}_{\alpha}\widetilde{E}^{\alpha}=J^{0}-\breve{B}_{\alpha}\widetilde{B}^{\alpha},$
$\displaystyle\epsilon^{\beta\alpha\gamma}(\eth_{\beta}\widetilde{B}_{\gamma}-\breve{A}_{\beta}\widetilde{B}_{\gamma})+\breve{B}_{\beta}^{\;\;\alpha}\widetilde{B}^{\beta}=J^{\alpha}+\eth_{0}\widetilde{E}^{\alpha}-\breve{A}_{0}\widetilde{E}^{\alpha}+\breve{E}_{\beta}^{\;\;\alpha}\widetilde{E}^{\beta},$
$\displaystyle\eth_{\alpha}\widetilde{B}^{\alpha}+2\breve{A}_{\alpha}\widetilde{B}^{\alpha}+2\breve{E}_{\alpha}\widetilde{B}^{\alpha}=\breve{B}_{\alpha}\widetilde{E}^{\alpha},$
$\displaystyle\eth_{0}\widetilde{B}^{\alpha}+2\breve{A}_{0}\widetilde{B}^{\alpha}-2\breve{E}_{\beta}^{\;\;\alpha}\widetilde{B}^{\beta}=-\epsilon^{\beta\alpha\gamma}(\eth_{\beta}\widetilde{E}_{\gamma}+\breve{E}_{\beta}\widetilde{E}_{\gamma})-\breve{B}^{\;\;\alpha}_{\beta}\widetilde{E}^{\beta},$
(232)

with the field strengths defined to be:

$\displaystyle\widetilde{E}^{\alpha}=-F^{0\alpha},\quad\widetilde{B}^{\alpha}=-\frac{1}{2}\epsilon^{\alpha}_{\;\;\beta\gamma}F^{\beta\gamma},$
$\displaystyle\breve{E}_{\alpha}=-F_{0\alpha}^{0},\quad\breve{B}_{\alpha}=-\frac{1}{2}\epsilon_{\alpha}^{\;\;\gamma\delta}F_{\gamma\delta}^{0},$
$\displaystyle\breve{E}_{\alpha}^{\;\;\beta}=-F_{0\alpha}^{\beta},\quad\breve{B}_{\alpha}^{\;\;\beta}=-\frac{1}{2}\epsilon_{\alpha}^{\;\;\gamma\delta}F_{\gamma\delta}^{\beta}.$
(233)

The field strengths $\widetilde{E}^{\alpha}$ and $\widetilde{B}^{\alpha}$ are
defined as electromagnetic field in locally flat gravigauge spacetime, which
may be called as gravigauge-dressed electric field and gravigauge-dressed
magnetic field, respectively, in order to distinguish with the free-motion
electromagnetic field $\mathbf{E}$ and $\mathbf{B}$ defined in Minkowski
spacetime. $\breve{E}_{\alpha}$ and $\breve{B}_{\alpha}$ as well as
$\breve{E}_{\alpha}^{\;\;\beta}$ and $\breve{B}_{\alpha}^{\;\;\beta}$ are
defined as electromagnetic-like gravigauge fields to characterize the
gravitational interactions. For convenience, $\breve{E}_{\alpha}$ and
$\breve{B}_{\alpha}$ are referred to as electric-like gravigauge field and
magnetic-like gravigauge field, respectively, while
$\breve{E}_{\alpha}^{\;\;\beta}$ and $\breve{B}_{\alpha}^{\;\;\beta}$ are
mentioned as electric-like gravigauge tensor field and magnetic-like
gravigauge tensor field, respectively. $\breve{A}_{0}$ and
$\breve{A}_{\alpha}$ are related to the electromagnetic-like gravigauge field
as follows:

$\displaystyle\breve{A}_{0}=\eta_{\alpha}^{\;\;\beta}\breve{E}_{\beta}^{\;\;\alpha},\quad\breve{A}_{\alpha}=-\breve{E}_{\alpha}-\epsilon_{\alpha\beta}^{\;\;\;\;\,\gamma}\breve{B}_{\gamma}^{\;\;\beta}.$
(234)

The equations of motion given in Eqs.(221) and (224) or Eqs.(X.2)-(X.3)
characterize the electrodynamics in locally flat gravigauge spacetime, which
may be referred to as gravigauge-mediated electrodynamics.

In light of vector representation, the equations of motion for the gravigauge-
dressed electromagnetic fields $\widetilde{\mathbf{E}}$ and
$\widetilde{\mathbf{B}}$ as shown in Eq.(X.2) can be rewritten into the
following forms in locally flat gravigauge spacetime:

$\displaystyle\breve{\nabla}\cdot\widetilde{\mathbf{E}}-\breve{\mathbf{A}}\cdot\widetilde{\mathbf{E}}-\breve{\mathbf{E}}\cdot\widetilde{\mathbf{E}}=J^{0}+\breve{\mathbf{B}}\cdot\widetilde{\mathbf{B}},$
$\displaystyle\breve{\nabla}\times\widetilde{\mathbf{B}}-\breve{\mathbf{A}}\times\widetilde{\mathbf{B}}+\breve{\mathbb{B}}\cdot\widetilde{\mathbf{B}}={\mathbf{J}}+\eth_{0}\widetilde{\mathbf{E}}-\breve{A}_{0}\widetilde{\mathbf{E}}+\breve{\mathbb{E}}\cdot\widetilde{\mathbf{E}},$
$\displaystyle\breve{\nabla}\times\widetilde{\mathbf{E}}+\breve{\mathbf{E}}\times\widetilde{\mathbf{E}}-\breve{\mathbb{B}}\cdot\widetilde{\mathbf{E}}=-\eth_{0}\widetilde{\mathbf{B}}-2\breve{A}_{0}\widetilde{\mathbf{B}}+\breve{\mathbb{E}}\cdot\widetilde{\mathbf{B}},$
$\displaystyle\breve{\nabla}\cdot\widetilde{\mathbf{B}}+2\breve{\mathbf{A}}\cdot\widetilde{\mathbf{B}}+2\breve{\mathbf{E}}\cdot\widetilde{\mathbf{B}}=\breve{\mathbf{B}}\cdot\widetilde{\mathbf{E}},$
(235)

where we have introduced the following vector-like notations:

$\displaystyle\widetilde{\mathbf{E}}\equiv(\widetilde{E}^{1},\widetilde{E}^{2},\widetilde{E}^{3}),\quad\widetilde{\mathbf{B}}\equiv(\widetilde{B}^{1},\widetilde{B}^{2},\widetilde{B}^{3}),$
$\displaystyle\breve{\mathbf{E}}\equiv(\breve{E}_{1},\breve{E}_{2},\breve{E}_{3}),\quad\breve{\mathbf{B}}\equiv(\breve{B}_{1},\breve{B}_{2},\breve{B}_{3}),$
$\displaystyle\breve{\mathbf{E}}^{\alpha}\equiv(\breve{E}_{1}^{\;\;\alpha},\breve{E}_{2}^{\;\;\alpha},\breve{E}_{3}^{\;\;\alpha}),\quad\breve{\mathbf{B}}^{\alpha}\equiv(\breve{B}_{1}^{\;\;\alpha},\breve{B}_{2}^{\;\;\alpha},\breve{B}_{3}^{\;\;\alpha}),$
$\displaystyle\breve{\mathbf{A}}\equiv(\breve{A}_{1},\breve{A}_{2},\breve{A}_{3}),$
(236)

and

$\displaystyle\breve{\mathbb{E}}\equiv\\{\breve{\mathbf{E}}^{\alpha}\\}\equiv\\{\breve{\mathbf{E}}^{1},\breve{\mathbf{E}}^{2},\breve{\mathbf{E}}^{3}\\}$
$\displaystyle\quad\equiv\\{(\breve{E}_{1}^{\;\;1},\breve{E}_{2}^{\;\;1},\breve{E}_{3}^{\;\;1}),(\breve{E}_{1}^{\;\;2},\breve{E}_{2}^{\;\;2},\breve{E}_{3}^{\;\;2}),(\breve{E}_{1}^{\;\;3},\breve{E}_{2}^{\;\;3},\breve{E}_{3}^{\;\;3})\\},$
$\displaystyle\breve{\mathbb{B}}\equiv\\{\breve{\mathbf{B}}^{\alpha}\\}\equiv\\{\breve{\mathbf{B}}^{1},\breve{\mathbf{B}}^{2},\breve{\mathbf{B}}^{3}\\}$
$\displaystyle\quad\equiv\\{(\breve{B}_{1}^{\;\;1},\breve{B}_{2}^{\;\;1},\breve{B}_{3}^{\;\;1}),(\breve{B}_{1}^{\;\;2},\breve{B}_{2}^{\;\;2},\breve{B}_{3}^{\;\;2}),(\breve{B}_{1}^{\;\;3},\breve{B}_{2}^{\;\;3},\breve{B}_{3}^{\;\;3})\\},$
$\displaystyle\breve{\nabla}\equiv(\eth_{1},\eth_{2},\eth_{3})\equiv\\{\hat{\chi}_{\alpha}^{\;\;0}\partial_{0}+\hat{\chi}_{\alpha}^{\;\;i}\partial_{i}\\},\quad\eth_{0}\equiv\hat{\chi}_{0}^{\;\;\mu}\partial_{\mu}\equiv\hat{\chi}_{0}^{\;\;0}\partial_{0}+\hat{\chi}_{0}^{\;\;i}\partial_{i},$
(237)

with $\breve{\mathbb{E}}$ and $\breve{\mathbb{B}}$ denoting electromagnetic-
like gravigauge bi-vector fields. Where $\breve{\nabla}$ and $\eth_{0}$
represent the gravi-coordinate derivatives, which are related to the ordinary
coordinate derivatives as follows:

$\displaystyle\breve{\nabla}\equiv(\eth_{1},\eth_{2},\eth_{3})\equiv\\{\hat{\chi}_{\alpha}^{\;\;0}\partial_{0}+\hat{\chi}_{\alpha}^{\;\;i}\partial_{i}\\}\equiv\breve{\boldsymbol{\kappa}}\frac{\partial}{\partial
x^{0}}+\mathring{\boldsymbol{\kappa}}\cdot\nabla,$
$\displaystyle\eth_{0}\equiv\hat{\chi}_{0}^{\;\;\mu}\partial_{\mu}=\hat{\chi}_{0}^{\;\;0}\partial_{0}+\hat{\chi}_{0}^{\;\;i}\partial_{i}\equiv\breve{\kappa}\frac{\partial}{\partial
x^{0}}+\hat{\boldsymbol{\kappa}}\cdot\nabla,$ (238)

with the definitions:

$\displaystyle\breve{\boldsymbol{\kappa}}(x)\equiv\\{\hat{\chi}_{\alpha}^{\;\;0}\\}\equiv(\hat{\chi}_{1}^{\;\;0},\hat{\chi}_{2}^{\;\;0},\hat{\chi}_{3}^{\;\;0}),\quad\hat{\boldsymbol{\kappa}}_{\alpha}(x)\equiv\\{\hat{\chi}_{\alpha}^{\;\;i}\\}\equiv(\hat{\chi}_{\alpha}^{\;\;1},\hat{\chi}_{\alpha}^{\;\;2},\hat{\chi}_{\alpha}^{\;\;3}),$
$\displaystyle\mathring{\boldsymbol{\kappa}}(x)\equiv\\{\hat{\boldsymbol{\kappa}}_{1},\hat{\boldsymbol{\kappa}}_{2},\hat{\boldsymbol{\kappa}}_{3}\\}\equiv\\{(\hat{\chi}_{1}^{\;\;1},\hat{\chi}_{1}^{\;\;2},\hat{\chi}_{1}^{\;\;3}),(\hat{\chi}_{2}^{\;\;1},\hat{\chi}_{2}^{\;\;2},\hat{\chi}_{2}^{\;\;3}),(\hat{\chi}_{3}^{\;\;1},\hat{\chi}_{3}^{\;\;2},\hat{\chi}_{3}^{\;\;3})\\},$
$\displaystyle\breve{\kappa}(x)\equiv\hat{\chi}_{0}^{\;\;0},\quad\hat{\boldsymbol{\kappa}}(x)\equiv\\{\hat{\chi}_{0}^{\;\;i}\\}\equiv(\hat{\chi}_{0}^{\;\;1},\hat{\chi}_{0}^{\;\;2},\hat{\chi}_{0}^{\;\;3}).$
(239)

In terms of the above definitions and notations for the gravi-coordinate
derivatives, the gravigauge Maxwell equations in Eq.(X.2) can be expressed as
follows:

$\displaystyle\breve{\boldsymbol{\kappa}}\cdot\frac{\partial\widetilde{\mathbf{E}}}{\partial
x^{0}}+(\mathring{\boldsymbol{\kappa}}\cdot\nabla)\cdot\widetilde{\mathbf{E}}-\breve{\mathbf{A}}\cdot\widetilde{\mathbf{E}}-\breve{\mathbf{E}}\cdot\widetilde{\mathbf{E}}=J^{0}+\breve{\mathbf{B}}\cdot\widetilde{\mathbf{B}},$
$\displaystyle\breve{\boldsymbol{\kappa}}\times\frac{\partial\widetilde{\mathbf{B}}}{\partial
x^{0}}+(\mathring{\boldsymbol{\kappa}}\cdot\nabla)\times\widetilde{\mathbf{B}}-\breve{\mathbf{A}}\times\widetilde{\mathbf{B}}+\breve{\mathbb{B}}\cdot\widetilde{\mathbf{B}}={\mathbf{J}}+\breve{\kappa}\frac{\partial\widetilde{\mathbf{E}}}{\partial
x^{0}}+(\hat{\boldsymbol{\kappa}}\cdot\nabla)\widetilde{\mathbf{E}}-\breve{A}_{0}\widetilde{\mathbf{E}}+\breve{\mathbb{E}}\cdot\widetilde{\mathbf{E}},$
$\displaystyle\breve{\boldsymbol{\kappa}}\times\frac{\partial\widetilde{\mathbf{E}}}{\partial
x^{0}}+(\mathring{\boldsymbol{\kappa}}\cdot\nabla)\times\widetilde{\mathbf{E}}+\breve{\mathbf{E}}\times\widetilde{\mathbf{E}}-\breve{\mathbb{B}}\cdot\widetilde{\mathbf{E}}=-\breve{\kappa}\frac{\partial\widetilde{\mathbf{B}}}{\partial
x^{0}}-(\hat{\boldsymbol{\kappa}}\cdot\nabla)\widetilde{\mathbf{B}}-\breve{A}_{0}\widetilde{\mathbf{E}}-2\breve{A}_{0}\widetilde{\mathbf{B}}+\breve{\mathbb{E}}\cdot\widetilde{\mathbf{B}},$
$\displaystyle\breve{\boldsymbol{\kappa}}\cdot\frac{\partial\widetilde{\mathbf{B}}}{\partial
x^{0}}+(\mathring{\boldsymbol{\kappa}}\cdot\nabla)\cdot\widetilde{\mathbf{B}}+2\breve{\mathbf{A}}\cdot\widetilde{\mathbf{B}}+2\breve{\mathbf{E}}\cdot\widetilde{\mathbf{B}}=\breve{\mathbf{B}}\cdot\widetilde{\mathbf{E}}.$
(240)

It is noticed that the equations in Eq.(X.2) and Eq.(X.2) (or Eq.(X.2)) bring
on the generalized Maxwell equations in locally flat gravigauge spacetime. To
distinguish with the gravigeometry-medium Maxwell equations presented in
Eq.(X.3) and Eq.(X.1)-(X.1) in coordinate spacetime, they may be referred to
as gravigauge-mediated Maxwell equations which are derived to describe the
gravigauge-mediated electrodynamics in locally flat gravigauge spacetime.

The gravigauge-mediated Maxwell equations show explicitly how the gravigauge-
dressed electromagnetic fields $\widetilde{\mathbf{E}}$ and
$\widetilde{\mathbf{B}}$ interact directly with the electromagnetic-like
gravigauge fields $\breve{\mathbf{E}}$ and $\breve{\mathbf{B}}$ as well as
electromagnetic-like gravigauge vector fields $\breve{\mathbb{E}}$ and
$\breve{\mathbb{B}}$ in locally flat gravigauge spacetime, which reveals how
the gravitational effect emerges to bring the gravigauge-mediated
electrodynamics in locally flat gravigauge spacetime.

###  X.3 Gravimetric-gauge-mediated electrodynamics and generalized Maxwell
equations in dynamic Riemannian spacetime

The equations of motion for the gauge field $A_{\mu}$ can also be expressed as
the following form in dynamic Riemannian spacetime:

$\displaystyle\nabla^{\nu}F_{\mu\nu}=-J_{\mu},$
$\displaystyle\varepsilon^{\mu\nu\rho\sigma}\partial_{\rho}F_{\mu\nu}=0,$
(241)

with the covariant derivative defined as follows:

$\displaystyle\nabla^{\nu}F_{\mu\nu}\equiv\hat{\chi}^{\nu\rho}\nabla_{\rho}F_{\mu\nu}=\hat{\chi}^{\nu\rho}(\partial_{\rho}F_{\mu\nu}-\Gamma_{\rho\mu}^{\lambda}F_{\lambda\nu}-\Gamma_{\rho\nu}^{\lambda}F_{\mu\lambda}).$
(242)

The second equation in Eq.(X.3) is the Bianchi identity which can also be
rewritten in terms of the covariant derivative:

$\displaystyle\varepsilon^{\mu\nu\rho\sigma}\partial_{\rho}F_{\mu\nu}\equiv\varepsilon^{\mu\nu\rho\sigma}\nabla_{\rho}F_{\mu\nu}\equiv\varepsilon^{\mu\nu\rho\sigma}(\partial_{\rho}F_{\mu\nu}-\Gamma_{\rho\mu}^{\lambda}F_{\lambda\nu}-\Gamma_{\rho\nu}^{\lambda}F_{\mu\lambda})=0,$
(243)

where we have used the symmetric property of the affine connection
(Christoffel symbol) $\Gamma_{\rho\mu}^{\lambda}=\Gamma_{\mu\rho}^{\lambda}$.

The current density in Eq.(X.3) is defined by,

$\displaystyle
J_{\mu}\equiv\chi_{\mu}^{\;\;a}J_{a}=-g_{E}\chi_{\mu}^{\;\;a}\bar{\psi}\gamma_{a}\psi.$
(244)

which is conserved via the covariant derivative:

$\displaystyle\nabla^{\mu}J_{\mu}$ $\displaystyle=$
$\displaystyle-\nabla^{\mu}\nabla^{\nu}F_{\mu\nu}=\frac{1}{2}(R^{\mu\nu\rho}_{\mu}F_{\rho\nu}+R^{\mu\nu\rho}_{\nu}F_{\mu\rho})=R^{\mu\rho}F_{\mu\rho}=0,$
(245)

where the last equality is due to the symmetric feature of Ricci curvature
tensor $R^{\mu\rho}=R^{\rho\mu}$ and the antisymmetric property of
electromagnetic field strength $F_{\mu\rho}=-F_{\rho\mu}$.

In can be checked that the equation of motion of gauge field $A_{\mu}$ gets
the following form:

$\displaystyle\nabla^{\nu}\nabla_{\nu}A_{\mu}+R_{\mu\nu}A^{\nu}-\nabla_{\mu}(\nabla^{\nu}A_{\nu})=J_{\mu}.$
(246)

By imposing the covariant gauge fixing condition $\nabla^{\nu}A_{\nu}=0$, we
come to the following equation of motion for the gauge field $A_{\mu}$:

$\displaystyle\nabla^{\nu}\nabla_{\nu}A_{\mu}+R_{\mu\nu}A^{\nu}=J_{\mu}.$
(247)

When applying for the geometric gravitational equation shown in Eq.(171), we
have,

$\displaystyle(\nabla^{\nu}\nabla_{\nu}+\frac{1}{2}R)A_{\mu}-8\pi
G_{N}(T_{\mu\nu}+\widetilde{T}_{\mu\nu})A^{\nu}=J_{\mu},$ (248)

which indicates that a non-zero Ricci curvature scalar $R\neq 0$ leads to an
effective mass-like term for the electromagnetic gauge field $A_{\mu}$ when it
is propagating in a curved Riemannian spacetime.

To reflect the effect of geometric gravidynamics on the electrodynamics, it is
useful to express the equations of motion in Eq.(X.3) into the following form:

$\displaystyle(\hat{\partial}^{\nu}+\check{A}^{\nu})F_{\nu\mu}+\frac{1}{2}\hat{\chi}^{\rho\nu}\hat{\chi}^{\sigma\nu^{\prime}}F_{\rho\sigma}^{\lambda}\eta_{\lambda\mu}F_{\nu\nu^{\prime}}=J_{\mu},$
(249)

where we have introduced the definitions:

$\displaystyle
F_{\rho\sigma}^{\lambda}\equiv\partial_{\rho}\chi_{\sigma}^{\;\lambda}-\partial_{\sigma}\chi_{\rho}^{\;\lambda},\quad\chi_{\rho}^{\;\lambda}\equiv\chi_{\rho\mu}\eta^{\mu\lambda},$
$\displaystyle\check{A}^{\nu}\equiv\frac{1}{2}\hat{\chi}^{\nu\rho}\hat{\chi}^{\mu\sigma}F_{\rho\sigma}^{\lambda}\eta_{\mu\lambda}\equiv\hat{\partial}^{\nu}\ln\chi+\partial_{\rho}\hat{\chi}^{\rho\nu},$
$\displaystyle\hat{\partial}^{\nu}\equiv\hat{\chi}^{\nu\rho}\partial_{\rho}.$
(250)

Such a defined $F_{\rho\sigma}^{\lambda}$ is considered to be a gauge field
strength when viewing the gravimetric field $\chi_{\rho}^{\;\lambda}$ to be a
gauge-type field (which is actually a combined field of gauvigauge field
$\chi_{\rho}^{\;\lambda}\equiv\chi_{\rho}^{\;a}\chi_{\;a}^{\lambda}$). For
convenience of mention, $\chi_{\rho}^{\;\lambda}$ is referred to as
gravimetric-gauge field and $F_{\rho\sigma}^{\lambda}$ is called as
electromagnetic-like gravimetric-gauge field strength.

The equations of electromagnetic field in dynamic Riemannian spacetime are
obtained as follows:

$\displaystyle(\hat{\partial}^{i}+\check{A}^{i})E_{i}+\hat{\chi}^{\rho
i}\hat{\chi}^{\sigma 0}F_{\rho\sigma}^{0}E_{i}+\frac{1}{2}\hat{\chi}^{\rho
j}\hat{\chi}^{\sigma
k}F_{\rho\sigma}^{0}\epsilon_{jk}^{\;\;\;\;i}B_{i}=J_{0},$
$\displaystyle(\hat{\partial}^{j}+\check{A}^{j})\epsilon_{jk}^{\;\;\;\;\;i}B_{i}+\frac{1}{2}\hat{\chi}^{\rho
k}\hat{\chi}^{\sigma l}F_{\rho\sigma
i}\epsilon_{kl}^{\;\;\;\;j}B_{j}=J_{i}+(\hat{\partial}^{0}+\check{A}^{0})E_{i}+\hat{\chi}^{\rho
0}\hat{\chi}^{\sigma j}F_{\rho\sigma i}E_{j},$ (251)

which can simply be expressed as follows:

$\displaystyle(\hat{\partial}^{i}+\check{A}^{i})E_{i}+\check{{\mathcal{E}}}^{i}E_{i}+\check{\mathcal{B}}^{i}B_{i}=J_{0},$
$\displaystyle(\hat{\partial}^{j}+\check{A}^{j})\epsilon_{jk}^{\;\;\;\;\;i}B_{i}+B_{j}\check{\mathcal{B}}^{j}_{\;\;i}=J_{i}+(\hat{\partial}^{0}+\check{A}^{0})E_{i}-E_{j}\check{{\mathcal{E}}}^{j}_{\;\;i}.$
(252)

Where we have introduced the definitions:

$\displaystyle\check{{\mathcal{E}}}^{i}\equiv\varepsilon^{ij}\check{E}_{j}+\alpha^{ij}\check{B}_{j},$
$\displaystyle\check{\mathcal{B}}^{i}\equiv\mu^{ij}\check{B}_{j}-\alpha^{ji}\check{E}_{j},$
$\displaystyle\check{{\mathcal{E}}}^{j}_{\;\;i}\equiv\varepsilon^{jk}\check{E}_{k\,i}+\alpha^{jk}\check{B}_{k\,i},$
$\displaystyle\check{\mathcal{B}}^{j}_{\;\;i}\equiv\mu^{jk}\check{B}_{k\,i}+\alpha^{kj}\check{E}_{k\,i},$
(253)

with the coefficient functions and field strengths defined as follows:

$\displaystyle\varepsilon^{ij}(x)\equiv\hat{\chi}^{ij}\hat{\chi}^{00}-\hat{\chi}^{i0}\hat{\chi}^{j0},$
$\displaystyle\mu^{ij}(x)\equiv\frac{1}{2}\epsilon^{i}_{\;\;kl}\hat{\chi}^{kk^{\prime}}\hat{\chi}^{ll^{\prime}}\epsilon_{k^{\prime}l^{\prime}}^{\;\;\;\;\;j},$
$\displaystyle\alpha^{ij}(x)\equiv\hat{\chi}^{ik}\hat{\chi}^{0l}\epsilon^{j}_{\;\;kl},$
(254)

and

$\displaystyle\check{E}_{i}=-F_{0i}^{0},\quad\check{B}_{i}=-\frac{1}{2}\epsilon_{i}^{\;\;jl}F_{jl}^{0},$
$\displaystyle\check{E}_{k}^{\;\;i}=-F_{0k}^{i},\quad\check{B}_{k}^{\;\;i}=-\frac{1}{2}\epsilon_{k}^{\;\;jl}F_{jl}^{i}.$
(255)

It is noticed that $\check{E}_{i}$ and $\check{B}_{i}$ as well as
$\check{E}_{k}^{\;\;i}$ and $\check{B}_{k}^{\;\;i}$ are introduced as
electromagnetic-like gravimetric-gauge fields to characterize the
gravitational interactions in dynamic Riemannian spacetime. In analogous to
electromagnetic field, $\check{E}_{i}$ and $\check{B}_{i}$ are referred to as
electric-like gravimetric-gauge field and magnetic-like gravimetric-gauge
field, respectively, while $\check{E}_{k}^{\;\;i}$ and $\check{B}_{k}^{\;\;i}$
are called as electric-like gravimetric-gauge tensor field and magnetic-like
gravimetric-gauge tensor field, respectively.

In terms of vector representation, the equations presented in Eq.(X.3) and
Eq.(X.3) can be rewritten into the following forms:

$\displaystyle\hat{\nabla}\cdot\mathbf{E}+\check{\mathbf{A}}\cdot\mathbf{E}+\check{\boldsymbol{{\mathcal{E}}}}\cdot\mathbf{E}+\check{\boldsymbol{\mathcal{B}}}\cdot\mathbf{B}=J_{0},$
$\displaystyle\hat{\nabla}\times\mathbf{B}+\check{\mathbf{A}}\times\mathbf{B}+\widehat{\boldsymbol{\mathcal{B}}}\cdot\mathbf{B}={\mathbf{J}}+(\hat{\partial}^{0}+\check{A}^{0})\mathbf{E}-\widehat{\boldsymbol{{\mathcal{E}}}}\cdot\mathbf{E},$
$\displaystyle\nabla\cdot\mathbf{B}=0,$
$\displaystyle\nabla\times\mathbf{E}=-\frac{\partial\mathbf{B}}{\partial
x^{0}},$ (256)

which bring on the generalized Maxwell equations in dynamic Riemannian
spacetime. Where we have used the vector notations
$\check{\boldsymbol{{\mathcal{E}}}}\equiv\\{\check{{\mathcal{E}}}_{i}\\}$ ,
$\check{\boldsymbol{\mathcal{B}}}\equiv\\{\check{\mathcal{B}}_{i}\\}$,
$\check{\mathbf{A}}\equiv\\{\check{\mathbf{A}}_{i}\\}$,
$\hat{\nabla}\equiv\\{\hat{\partial}_{i}\\}$ and bi-vector notations
$\widehat{\boldsymbol{{\mathcal{E}}}}\equiv\\{\check{{\mathcal{E}}}_{j}^{\;\;i}\\}$,
$\widehat{\boldsymbol{\mathcal{B}}}\equiv\\{\check{\mathcal{B}}_{j}^{\;\;i}\\}$.

To distinguish with the gravigauge-mediated Maxwell equations in locally flat
gravigauge spacetime and gravigeometry-medium Maxwell equations in coordinate
spacetime, we may refer to the above generalized Maxwell equations as
gravimetric-gauge-mediated Maxwell equations, which are provided to describe
the gravimetric-gauge-mediated electrodynamics in dynamic Riemannian
spacetime.

##  XI General covariance of dynamic equations and electrodynamics in any
motional and spinning reference frames

It is shown that the hermitian action constructed in locally flat gravigauge
spacetime based on the gauge invariance principle becomes independent of the
choice of coordinates. Namely, such an action holds in any coordinate system
and gets invariant under the general coordinate transformation. In other word,
such a gauge invariant action possesses automatically a hidden general linear
group symmetry GL(1,3,R) which is known to lay the foundation of Einstein’s
general theory of relativity. Such a consequence indicates that the gauge
invariance principle brings naturally on the emergence of general linear group
symmetry GL(1,3,R) as a local symmetry in coordinate spacetime.

###  XI.1 General covariance of dynamic equations

To be concrete, the general coordinate transformation is defined as an
arbitrary reparametrization of coordinate systems, i.e.:

$\displaystyle x^{\mu}\to x^{{}^{\prime}\mu}\equiv x^{{}^{\prime}\mu}(x),$
(257)

which provides a local transformation in curved Riemannian spacetime and leads
the displacement and derivative of coordinate systems to satisfy the following
transformation laws:

$\displaystyle dx^{\mu}\to
dx^{{}^{\prime}\mu}=T^{\mu}_{\;\;\;\nu}\,dx^{\nu},\quad
T^{\mu}_{\;\;\;\nu}\equiv\frac{\partial x^{{}^{\prime}\mu}}{\partial
x^{\nu}},$ $\displaystyle\partial_{\mu}\equiv\frac{\partial}{\partial
x^{\mu}}\to\partial^{\prime}_{\mu}\equiv\frac{\partial}{\partial
x^{{}^{\prime}\mu}}=T_{\mu}^{\;\;\nu}\,\partial_{\nu},\quad
T_{\mu}^{\;\;\nu}\equiv\frac{\partial x^{\nu}}{\partial x^{{}^{\prime}\mu}},$
(258)

which preserves the inner product to be invariant under the general coordinate
transformation,

$\displaystyle\langle\partial^{\prime}_{\mu},dx^{{}^{\prime}\nu}\rangle=T_{\mu}^{\;\,\rho}T^{\nu}_{\;\;\;\sigma}\langle\partial_{\rho},dx^{\sigma}\rangle=T_{\mu}^{\;\,\rho}T^{\nu}_{\;\;\;\sigma}\eta_{\rho}^{\;\;\sigma}=\eta_{\mu}^{\;\;\nu}\equiv\langle\partial_{\mu},dx^{\nu}\rangle.$
(259)

From the above transformation law, $\partial_{\mu}$ is usually regarded as
covariant vector and $dx^{\mu}$ as contravariant vector in coordinate
spacetime. In general, any covariant vector field ${\mathsf{V}}_{\mu}(x)$ with
a lower index and contravariant vector field $\hat{{\mathsf{V}}}^{\mu}(x)$
with an upper index transform as follows:

$\displaystyle{\mathsf{V}}^{\prime}_{\mu}(x^{\prime})=T_{\mu}^{\;\;\nu}\mspace{1.0mu}{\mathsf{V}}_{\nu}(x),\quad\hat{{\mathsf{V}}}^{{}^{\prime}\mu}(x^{\prime})=T^{\mu}_{\;\;\;\nu}\mspace{1.0mu}\hat{{\mathsf{V}}}^{\nu}(x),$
(260)

which keeps the scalar product between the covariant vector field and
contravariant vector field to be invariant:

$\displaystyle{\mathsf{V}}^{\prime}_{\mu}(x^{\prime})\hat{{\mathsf{V}}}^{{}^{\prime}\mu}(x^{\prime})=T_{\mu}^{\;\;\nu}T^{\mu}_{\;\;\;\nu^{\prime}}{\mathsf{V}}_{\nu}(x)\hat{{\mathsf{V}}}^{\nu^{\prime}}(x)=\eta_{\;\nu^{\prime}}^{\nu}{\mathsf{V}}_{\nu}(x)\hat{{\mathsf{V}}}^{\nu^{\prime}}(x)\equiv{\mathsf{V}}_{\mu}(x)\hat{{\mathsf{V}}}^{\mu}(x).$
(261)

The gravigauge field $\mathbf{A}_{\mu}^{\;a}(x)$($\chi_{\mu}^{\;a}(x)$) and
its dual field $\hat{\mathbf{A}}_{a}^{\;\mu}(x)$($\hat{\chi}_{a}^{\;\mu}(x)$)
are considered as bi-covariant vector fields in biframe spacetime. Where
$\mathbf{A}_{\mu}^{\;a}(x)$($\chi_{\mu}^{\;a}(x)$) is regarded as a covariant
vector field in coordinate spacetime and a contravariant vector field in
locally flat gravigauge spacetime, while
$\hat{\mathbf{A}}_{a}^{\;\mu}(x)$($\hat{\chi}_{a}^{\;\mu}(x)$) is viewed as a
contravariant vector field in coordinate spacetime and a covariant vector
field in locally flat gravigauge spacetime. Namely, they transform as follows:

$\displaystyle\mathbf{A}_{\mu}^{{}^{\prime}\;a}(x^{\prime})=T_{\mu}^{\;\;\nu}\mspace{1.0mu}\mathbf{A}_{\nu}^{\;a}(x),\quad\chi_{\mu}^{{}^{\prime}\;a}(x^{\prime})=T_{\mu}^{\;\;\nu}\mspace{1.0mu}\chi_{\nu}^{\;a}(x),$
$\displaystyle\hat{\mathbf{A}}_{a}^{{}^{\prime}\;\mu}(x^{\prime})=T^{\mu}_{\;\;\;\nu}\mspace{1.0mu}\hat{\mathbf{A}}_{a}^{\;\nu}(x),\quad\hat{\chi}_{a}^{{}^{\prime}\;\mu}(x^{\prime})=T^{\mu}_{\;\;\;\nu}\mspace{1.0mu}\hat{\chi}_{a}^{\;\nu}(x),$
(262)

under the general coordinate transformation of general linear group symmetry
GL(1,3; R), and

$\displaystyle\mathbf{A}_{\mu}^{{}^{\prime}\;a}(x)=\Lambda^{a}_{\;\;b}(x)\mspace{1.0mu}\mathbf{A}_{\mu}^{\;b}(x),\quad\chi_{\mu}^{{}^{\prime}\;a}(x)=\Lambda^{a}_{\;\;b}(x)\mspace{1.0mu}\chi_{\mu}^{\;b}(x)$
$\displaystyle\hat{\mathbf{A}}_{a}^{{}^{\prime}\;\mu}(x)=\Lambda_{a}^{\;\;b}(x)\mspace{1.0mu}\hat{\mathbf{A}}_{b}^{\;\mu}(x),\quad\hat{\chi}_{a}^{{}^{\prime}\;\mu}(x)=\Lambda_{a}^{\;\;b}(x)\mspace{1.0mu}\hat{\chi}_{b}^{\;\mu}(x),$
(263)

under the gauge transformation of spin gauge group symmetry SP(1,3). So the
gravicoordinate displacement $\delta\varkappa^{a}$ ($\delta\chi^{a}$) and
derivative $\hat{\eth}_{a}$ ($\eth_{a}$) defined in locally flat gravigauge
spacetime become invariant under the general coordinate transformation.
Explicitly, we have,

$\displaystyle\delta\varkappa^{a}\equiv\mathbf{A}_{\mu}^{\;a}(x)dx^{\mu}=\mathbf{A}_{\mu}^{{}^{\prime}\;a}(x^{\prime})dx^{{}^{\prime}\mu},\quad\delta\chi^{a}\equiv\chi_{\mu}^{\;a}(x)dx^{\mu}=\chi_{\mu}^{{}^{\prime}\;a}(x^{\prime})dx^{{}^{\prime}\mu},$
$\displaystyle\hat{\eth}_{a}\equiv\hat{\mathbf{A}}_{a}^{\;\mu}(x)\partial_{\mu}=\hat{\mathbf{A}}_{a}^{{}^{\prime}\;\mu}(x^{\prime})\partial^{\prime}_{\mu},\quad\eth_{a}\equiv\hat{\chi}_{a}^{\;\mu}(x)\partial_{\mu}=\hat{\chi}_{a}^{{}^{\prime}\;\mu}(x^{\prime})\partial^{\prime}_{\mu},$
(264)

which indicates that the gravicoordinate displacement $\delta\varkappa^{a}$
($\delta\chi^{a}$) and derivative $\hat{\eth}_{a}$ ($\eth_{a}$) introduced in
Eqs.(44) and (IV) are the scalar product between the covariant vector and
contravariant vector in curved Riemannian spacetime of coordinates. Therefore,
the gravicoordinate displacement $\delta\varkappa^{a}$ ($\delta\chi^{a}$) and
derivative $\hat{\eth}_{a}$ ($\eth_{a}$) naturally possess the general linear
group symmetry GL(1,3,R). It can be checked that the inner product of
$\delta\varkappa^{a}$ ($\delta\chi^{a}$) and derivative $\hat{\eth}_{a}$
($\eth_{a}$) becomes invariant under the spin gauge transformation,

$\displaystyle\langle\hat{\eth}^{\prime}_{a},\delta\varkappa^{{}^{\prime}b}\rangle=\Lambda_{a}^{\;\,c}\Lambda^{b}_{\;\;\;d}\mspace{1.0mu}\langle\hat{\eth}_{c},\delta\varkappa^{d}\rangle=\Lambda_{a}^{\;\,c}\Lambda^{b}_{\;\;\;d}\mspace{1.0mu}\eta_{c}^{\;\;d}=\eta_{a}^{\;\;b}\mspace{1.0mu}\equiv\langle\hat{\eth}_{a},\delta\varkappa^{b}\rangle.$
(265)

On the other hand, the gravimetric field $\mathsf{H}_{\mu\nu}(x)$
($\chi_{\mu\nu}(x)$) and dual gravimetric field $\hat{\mathsf{H}}^{\mu\nu}(x)$
($\hat{\chi}^{\mu\nu}(x)$) defined in Eqs.(III.2) and (IV) as the scalar
products of bi-covariant vector field
$\mathbf{A}_{\mu}^{\;a}(x)$($\chi_{\mu}^{\;a}(x)$) and
$\hat{\mathbf{A}}_{a}^{\;\mu}(x)$($\hat{\chi}_{a}^{\;\mu}(x)$) in locally flat
gravigauge spacetime correspond to the covariant tensor and contravariant
tensor fields in curved Riemannian spacetime of coordinates. They obey the
following transformation laws:

$\displaystyle\mathsf{H}^{\prime}_{\mu\nu}(x^{\prime})=\frac{\partial
x^{\sigma}}{\partial x^{{}^{\prime}\mu}}\frac{\partial x^{\sigma}}{\partial
x^{{}^{\prime}\nu}}\mspace{1.0mu}\mathsf{H}_{\rho\sigma}(x)=T_{\mu}^{\;\;\rho}T_{\nu}^{\;\;\sigma}\,\mathsf{H}_{\rho\sigma}(x),\quad\chi^{\prime}_{\mu\nu}(x^{\prime})=T_{\mu}^{\;\;\rho}T_{\nu}^{\;\;\sigma}\,\chi_{\rho\sigma}(x),$
$\displaystyle\hat{\mathsf{H}}^{{}^{\prime}\mu\nu}(x^{\prime})=\frac{\partial
x^{{}^{\prime}\mu}}{\partial x^{\rho}}\frac{\partial
x^{{}^{\prime}\nu}}{\partial
x^{\sigma}}\mspace{1.0mu}\hat{\mathsf{H}}^{\rho\sigma}(x)=T^{\mu}_{\;\;\rho}T^{\nu}_{\;\;\sigma}\,\hat{\mathsf{H}}^{\rho\sigma}(x),\quad\hat{\chi}^{{}^{\prime}\mu\nu}(x^{\prime})=T^{\mu}_{\;\;\rho}T^{\nu}_{\;\;\sigma}\,\hat{\chi}^{\rho\sigma}(x).$
(266)

So the Christoffel symbols given in Eq.(LABEL:HSGMF) transform as follows:

$\displaystyle\mathbf{\Gamma}_{\mu\nu}^{{}^{\prime}\rho}(x^{\prime})$
$\displaystyle=$ $\displaystyle(\frac{\partial x^{\mu^{\prime}}}{\partial
x^{{}^{\prime}\mu}}\frac{\partial x^{\nu^{\prime}}}{\partial
x^{{}^{\prime}\nu}})\left(\,\frac{\partial x^{{}^{\prime}\rho}}{\partial
x^{\rho^{\prime}}}\mathbf{\Gamma}_{\mu^{\prime}\nu^{\prime}}^{\rho^{\prime}}(x)-\frac{\partial^{2}x^{{}^{\prime}\rho}}{\partial
x^{\mu^{\prime}}\partial x^{\nu^{\prime}}}\,\right)$ (267)
$\displaystyle\equiv$ $\displaystyle
T_{\mu}^{\;\;\mu^{\prime}}T_{\nu}^{\;\;\nu^{\prime}}T^{\rho}_{\;\;\;\rho^{\prime}}\left(\,\mathbf{\Gamma}_{\mu^{\prime}\nu^{\prime}}^{\rho^{\prime}}(x)-T^{\rho^{\prime}}_{\;\;\,\sigma}\partial_{\mu^{\prime}}T^{\sigma}_{\;\;\,\nu^{\prime}}\right),$

which provides the local transformation property of Christoffel symbols with
appearance of inhomogeneous term under general linear group symmetry
GL(1,3,R), which is analogous to the gauge-type transformation of spin gauge
group symmetry SP(1,3) in locally flat gravigauge spacetime. Such a property
of transformations keeps the covariant derivative on the covariant vector
field ${\mathsf{V}}_{\mu}(x)$ and contravariant vector field
$\hat{{\mathsf{V}}}^{\mu}(x)$ to be general covariance under the general
coordinate transformations, i.e.:

$\displaystyle\nabla^{\prime}_{\mu}{\mathsf{V}}^{\prime}_{\nu}(x^{\prime})=T_{\mu}^{\;\;\rho}T_{\nu}^{\;\;\sigma}\nabla_{\rho}{\mathsf{V}}_{\sigma}(x),\quad\nabla_{\mu}{\mathsf{V}}_{\nu}(x)=\partial_{\mu}{\mathsf{V}}_{\nu}(x)-\mathbf{\Gamma}_{\mu\nu}^{\rho}(x){\mathsf{V}}_{\rho}(x),$
$\displaystyle\nabla^{\prime}_{\mu}\hat{{\mathsf{V}}}^{{}^{\prime}\nu}(x^{\prime})=T_{\mu}^{\;\;\rho}T^{\nu}_{\;\;\sigma}\nabla_{\rho}\hat{{\mathsf{V}}}^{\sigma}(x),\quad\nabla_{\mu}\hat{{\mathsf{V}}}^{\nu}(x)=\partial_{\mu}\hat{{\mathsf{V}}}^{\nu}(x)+\mathbf{\Gamma}_{\mu\rho}^{\nu}(x)\hat{{\mathsf{V}}}^{\rho}(x).$
(268)

With the above transformation laws, it can be verified explicitly that all
actions constructed in the previous sections are invariant under the general
coordinate transformation via an arbitrary reparametrization of coordinates
shown in Eqs.(257)-(259), which displays the emergent general linear group
symmetry GL(1,3,R) of the actions. As a consequence, we come to the statement
that the gauge invariance principle naturally leads the action formulated in
locally flat gravigauge spacetime to possess a maximal joint symmetry:

$\displaystyle G_{S}=\mbox{GL(}1,3,\mbox{R)}\Join\mbox{WS(}1,3),$

which implies that the Poincaré group symmetry PO(1,3) in globally flat
Minkowski spacetime will automatically be extended to the local general linear
group symmetry GL(1,3,R), which brings on the appearance of Riemann geometry
in curved Riemannian spacetime.

Such a joint symmetry of the action indicates that the dynamic equations of
motion for the basic fields derived based on the least action principle all
become covariance in biframe spacetime. Namely, the gravidynamics and
spinodynamics as well as electrodynamics discussed in the previous sections
should obey the general coordinate covariance in curved Riemannian spacetime
and spin gauge covariance in locally flat gravigauge spacetime.

###  XI.2 Gravigeometry-medium electrodynamics in any motional reference frame

Let us examine explicitly the general coordinate covariance and spin gauge
covariance for the electrodynamics. The general coordinate covariance
indicates that the dynamic equations of electromagnetic gauge field presented
in Eqs. (171)-(X.1) hold in any reference frame in the Einstein basis for the
scaling gauge symmetry, i.e.:

$\displaystyle\partial^{\prime}_{\nu}\widehat{F}^{{}^{\prime}\mu\nu}(x^{\prime})=-\widehat{J}^{{}^{\prime}\mu}(x^{\prime}),$
(269)

with

$\displaystyle\widehat{F}^{{}^{\prime}\mu\nu}(x^{\prime})\equiv\chi^{\prime}(x^{\prime})\tilde{\chi}^{{}^{\prime}[\mu\nu]\rho\sigma}(x^{\prime})F^{\prime}_{\rho\sigma}(x^{\prime}),$
$\displaystyle
F^{\prime}_{\rho\sigma}(x^{\prime})\equiv\partial^{\prime}_{\rho}A^{\prime}_{\sigma}(x^{\prime})-\partial^{\prime}_{\sigma}A^{\prime}_{\rho}(x^{\prime}),$
$\displaystyle\tilde{\chi}^{{}^{\prime}[\mu\nu]\rho\sigma}(x^{\prime})\equiv\frac{1}{2}[\hat{\chi}^{{}^{\prime}\mu\rho}(x^{\prime})\hat{\chi}^{{}^{\prime}\nu\sigma}(x^{\prime})-\hat{\chi}^{{}^{\prime}\nu\rho}(x^{\prime})\hat{\chi}^{{}^{\prime}\mu\sigma}(x^{\prime})],$
$\displaystyle\widehat{J}^{{}^{\prime}\mu}(x^{\prime})=\chi^{\prime}(x^{\prime})\hat{\chi}_{a}^{{}^{\prime}\;\mu}(x^{\prime})J^{a}(x^{\prime}),\quad
J^{a}(x^{\prime})\equiv-
g_{E}\bar{\psi}(x^{\prime})\gamma^{a}\psi(x^{\prime}),$ (270)

where the relevant fields and vectors in arbitrary new reference frame obey
the following transformation properties:

$\displaystyle\partial_{\mu}\equiv\frac{\partial}{\partial
x^{\mu}}\to\partial^{\prime}_{\mu}\equiv\frac{\partial}{\partial
x^{{}^{\prime}\mu}}=T_{\mu}^{\;\;\nu}\partial_{\nu},$ $\displaystyle
A_{\mu}(x)\to A^{\prime}_{\mu}(x^{\prime})=T_{\mu}^{\;\;\nu}A_{\nu}(x),$
$\displaystyle F_{\mu\nu}(x)\to
F^{\prime}_{\mu\nu}(x^{\prime})=T_{\mu}^{\;\;\rho}T_{\nu}^{\;\;\sigma}F_{\rho\sigma}(x),$
$\displaystyle\hat{\chi}_{a}^{\;\;\mu}(x)\to\hat{\chi}_{a}^{{}^{\prime}\;\mu}(x^{\prime})=T^{\mu}_{\;\;\nu}\mspace{1.0mu}\hat{\chi}_{a}^{\;\;\nu}(x),$
$\displaystyle\hat{\chi}^{\mu\nu}(x)\equiv\eta^{ab}\hat{\chi}_{a}^{\;\;\mu}(x)\hat{\chi}_{b}^{\;\;\nu}(x)\to\hat{\chi}^{{}^{\prime}\mu\nu}(x^{\prime})=T^{\mu}_{\;\;\rho}T^{\nu}_{\;\;\sigma}\hat{\chi}^{\rho\sigma}(x).$
(271)

It is noticed that the above dynamic equations possess a hidden gauge
invariance.

To demonstrate the general coordinate covariance, it is useful to rewrite the
equation of motion in terms of the covariant derivative. Explicitly, we have
the following identity:

$\displaystyle\partial_{\nu}\widehat{F}^{\mu\nu}(x)\equiv\chi(x)\nabla_{\nu}\hat{F}^{\mu\nu}(x)\equiv\chi(x)[\partial_{\nu}\hat{F}^{\mu\nu}(x)+\Gamma_{\nu\rho}^{\mu}\hat{F}^{\rho\nu}(x)+\Gamma_{\nu\rho}^{\nu}\hat{F}^{\mu\rho}(x)],$
(272)

where we have used the following definition and property:

$\displaystyle\hat{F}^{\mu\nu}(x)\equiv\hat{\chi}^{\mu\rho}(x)\hat{\chi}^{\nu\sigma}(x)F_{\rho\sigma}(x)\equiv\tilde{\chi}^{[\mu\nu]\rho\sigma}(x)F_{\rho\sigma}(x)=-\hat{F}^{\nu\mu}(x),$
$\displaystyle\Gamma_{\nu\rho}^{\mu}\equiv\Gamma_{\rho\nu}^{\mu},\quad\Gamma_{\nu\rho}^{\nu}\equiv\partial_{\rho}\ln\chi(x),$
$\displaystyle\Gamma_{\nu\rho}^{\mu}\widehat{F}^{\rho\nu}(x)\equiv\frac{1}{2}(\Gamma_{\nu\rho}^{\mu}-\Gamma_{\rho\nu}^{\mu})\widehat{F}^{\rho\nu}(x)=0,$
$\displaystyle\chi(x)\Gamma_{\nu\rho}^{\nu}\widehat{F}^{\mu\rho}(x)\equiv(\partial_{\rho}\chi(x))\widehat{F}^{\mu\rho}(x).$
(273)

So the equations of motion in terms of the explicit covariant form can be
expressed as follows:

$\displaystyle\nabla_{\nu}\hat{F}^{\mu\nu}(x)=-\hat{J}^{\mu}(x),$ (274)

with the current density,

$\displaystyle\hat{J}^{\mu}(x)\equiv\hat{\chi}_{a}^{\;\mu}(x)J^{a}(x)=-g_{E}\hat{\chi}_{a}^{\;\mu}(x)\bar{\psi}(x)\gamma^{a}\psi(x).$
(275)

Therefore, the gravigeometry-medium Maxwell equations hold in any motional
reference frame and can be expressed into the following forms:

$\displaystyle\nabla^{\prime}\cdot\widehat{\mathbf{E}}^{\prime}=\widehat{J}^{{}^{\prime}0},$
$\displaystyle\nabla^{\prime}\times\widehat{\mathbf{B}}^{\prime}=\widehat{{\mathbf{J}}}^{\prime}+\frac{\partial\widehat{\mathbf{E}}^{\prime}}{\partial
x^{{}^{\prime}0}},$ $\displaystyle\nabla^{\prime}\cdot\mathbf{B}^{\prime}=0,$
$\displaystyle\nabla^{\prime}\times\mathbf{E}^{\prime}=-\frac{\partial\mathbf{B}^{\prime}}{\partial
x^{{}^{\prime}0}},$ (276)

with

$\displaystyle\widehat{\mathbf{E}}^{\prime}(x^{\prime})\equiv\widehat{\boldsymbol{\varepsilon}}^{\prime}\cdot\mathbf{E}^{\prime}+\widehat{\boldsymbol{\alpha}}^{\prime}\cdot\mathbf{B}^{\prime},\quad\mbox{or}\quad\widehat{E}^{{}^{\prime}i}(x^{\prime})=\hat{\varepsilon}^{{}^{\prime}i}_{\;j}E^{{}^{\prime}j}+\hat{\alpha}^{{}^{\prime}i}_{\;\;j}B^{{}^{\prime}j},$
$\displaystyle\widehat{\mathbf{B}}^{\prime}(x^{\prime})\equiv\widehat{\boldsymbol{\mu}}^{\prime}\cdot\mathbf{B}^{\prime}-\widehat{\boldsymbol{\alpha}}^{{}^{\prime}T}\cdot\mathbf{E}^{\prime},\quad\mbox{or}\quad\widehat{B}^{{}^{\prime}i}(x^{\prime})=\hat{\mu}^{{}^{\prime}i}_{\;j}B^{{}^{\prime}j}-\hat{\alpha}^{{}^{\prime}\;i}_{j}E^{{}^{\prime}j},$
$\displaystyle\hat{J}^{{}^{\prime}0}(x^{\prime})\equiv\breve{\kappa}^{\prime}J^{{}^{\prime}0}+\breve{\boldsymbol{\kappa}}^{\prime}\cdot{\mathbf{J}}^{\prime},\quad\hat{{\mathbf{J}}}^{\prime}(x^{\prime})\equiv\mathring{\boldsymbol{\kappa}}\cdot{\mathbf{J}}^{\prime}+\hat{\boldsymbol{\kappa}}^{\prime}J^{{}^{\prime}0},$
(277)

and

$\displaystyle\widehat{\boldsymbol{\varepsilon}}^{\prime}(x^{\prime})\equiv\\{\hat{\varepsilon}^{{}^{\prime}i}_{\;j}(x^{\prime})\\},\quad\mbox{or}\quad\hat{\varepsilon}^{{}^{\prime}i}_{\;j}(x^{\prime})\equiv\chi^{\prime}(x^{\prime})(\hat{\chi}^{{}^{\prime}0}_{\;0}(x^{\prime})\hat{\chi}^{{}^{\prime}i}_{\;j}(x^{\prime})-\hat{\chi}^{{}^{\prime}i}_{\;0}(x^{\prime})\hat{\chi}^{{}^{\prime}0}_{\;j}(x^{\prime})),$
$\displaystyle\widehat{\boldsymbol{\mu}}^{\prime}(x^{\prime})\equiv\\{\hat{\mu}^{{}^{\prime}i}_{\;j}(x^{\prime})\\},\quad\mbox{or}\quad\hat{\mu}^{{}^{\prime}i}_{\;j}(x^{\prime})\equiv-\frac{1}{2}\chi^{\prime}(x^{\prime})\epsilon^{i}_{\;\;kl}(x^{\prime})\hat{\chi}^{kk^{\prime}}(x^{\prime})\hat{\chi}^{{}^{\prime}ll^{\prime}}(x^{\prime})\epsilon_{k^{\prime}l^{\prime}j},$
$\displaystyle\widehat{\boldsymbol{\alpha}}^{\prime}(x^{\prime})\equiv\\{\hat{\alpha}^{{}^{\prime}i}_{\;j}(x^{\prime})\\},\quad\mbox{or}\quad\hat{\alpha}^{{}^{\prime}i}_{\;j}(x^{\prime})\equiv\chi^{\prime}(x^{\prime})\hat{\chi}^{{}^{\prime}ik}(x^{\prime})\hat{\chi}^{{}^{\prime}l0}(x^{\prime})\epsilon_{klj},$
$\displaystyle\breve{\kappa}^{\prime}(x^{\prime})\equiv\hat{\chi}_{0}^{{}^{\prime}\;0}(x^{\prime}),\quad\breve{\boldsymbol{\kappa}}^{\prime}(x^{\prime})\equiv\\{\hat{\chi}_{\alpha}^{{}^{\prime}\;0}(x^{\prime})\\},$
$\displaystyle\mathring{\boldsymbol{\kappa}}^{\prime}(x^{\prime})\equiv\\{\hat{\chi}_{\alpha}^{{}^{\prime}\;i}(x^{\prime})\\},\quad\hat{\boldsymbol{\kappa}}^{\prime}(x^{\prime})\equiv\\{\hat{\chi}_{0}^{{}^{\prime}\;i}(x^{\prime})\\},$
(278)

where the coefficient functions are in general governed by the gravidynamics
in the new reference frame.

###  XI.3 Gravigauge-mediated electrodynamics in any spinning reference frame

We now turn to discuss the electrodynamics in locally flat gravigauge
spacetime. The dynamic equation presented in Eq.(221) is spin gauge
covariance, which holds in any spinning reference frame in locally flat
garvigauge spacetime, i.e.:

$\displaystyle D^{\prime}_{b}F^{{}^{\prime}ab}(x)=-J^{{}^{\prime}a}(x),$ (279)

with

$\displaystyle
D^{\prime}_{b}F^{{}^{\prime}ab}(x)\equiv\eth^{\prime}_{b}F^{{}^{\prime}ab}(x)+\mathsf{\Omega}_{bc}^{{}^{\prime}a}(x)F^{{}^{\prime}cb}(x)+\mathsf{\Omega}_{bc}^{{}^{\prime}b}(x)F^{{}^{\prime}ac}(x),$
$\displaystyle
F^{{}^{\prime}ab}(x)\equiv\eta^{ac}\eta^{bd}F^{\prime}_{cd}(x)\equiv\hat{\chi}^{{}^{\prime}a\mu}(x)\hat{\chi}^{{}^{\prime}b\nu}(x)F_{\mu\nu}(x),$
$\displaystyle F^{\prime}_{cd}(x)\equiv
D^{\prime}_{c}A^{\prime}_{d}(x)-D^{\prime}_{d}A^{\prime}_{c}(x)=\eth^{\prime}_{c}A^{\prime}_{d}(x)-\eth^{\prime}_{d}A^{\prime}_{c}(x)+\mathsf{\Omega}_{cd}^{{}^{\prime}b}(x)A^{\prime}_{b}(x)-\mathsf{\Omega}_{dc}^{{}^{\prime}b}(x)A^{\prime}_{b}(x),$
$\displaystyle
J^{{}^{\prime}a}(x)=-g_{E}\bar{\psi}^{\prime}(x)\gamma^{a}\psi^{\prime}(x),$
(280)

where all fields in arbitrary spinning reference frame obey the following
transformation properties under the gauge transformation of spin gauge
symmetry SP(1,3):

$\displaystyle F^{ab}(x)\to
F^{{}^{\prime}ab}(x)=\Lambda^{a}_{\;\;c}(x)\Lambda^{b}_{\;\;d}(x)F^{cd}(x),\quad
A_{c}(x)\to A^{\prime}_{c}(x)=\Lambda_{c}^{\;\;d}(x)A_{d}(x),$ $\displaystyle
J^{a}(x)\to
J^{{}^{\prime}a}(x)=\Lambda^{a}_{\;\;c}(x)J^{c}(x),\quad\psi(x)\to\psi^{\prime}(x)=S(\Lambda)\psi(x),\quad
S(\Lambda)=e^{i\varpi_{ab}(x)\Sigma^{ab}},$
$\displaystyle\mathsf{\Omega}_{cb}^{a}(x)\to\mathsf{\Omega}_{cb}^{{}^{\prime}a}(x)=\Lambda_{c}^{\;\;c^{\prime}}(x)[\Lambda^{a}_{\;\;a^{\prime}}(x)\Lambda_{b}^{\;\;b^{\prime}}(x)\mathsf{\Omega}_{c^{\prime}b^{\prime}}^{a^{\prime}}(x)+\frac{1}{2}\Lambda^{a}_{\;\;d}(x)\eth_{c^{\prime}}\Lambda_{b}^{\;\;d}(x)-\Lambda_{b}^{\;\;d}(x)\eth_{c^{\prime}}\Lambda^{a}_{\;\;d}(x)],$
$\displaystyle\eth_{c}\equiv\hat{\chi}_{c}^{\;\;\mu}(x)\partial_{\mu}\to\eth^{\prime}_{c}=\Lambda_{c}^{\;\;d}(x)\eth_{d},\quad\hat{\chi}_{c}^{\;\;\mu}(x)\to\hat{\chi}_{c}^{{}^{\prime}\;\,\mu}(x)=\Lambda_{c}^{\;\;d}(x)\hat{\chi}_{d}^{\;\;\mu}(x).$
(281)

So the gravigauge-mediated Maxwell equations in Eq.(X.2) are valid in any
spinning reference frame, which can be expressed as follows:

$\displaystyle\breve{\nabla}^{\prime}\cdot\widetilde{\mathbf{E}}^{\prime}-\breve{\mathbf{A}}^{\prime}\cdot\widetilde{\mathbf{E}}^{\prime}-\breve{\mathbf{E}}^{\prime}\cdot\widetilde{\mathbf{E}}^{\prime}=J^{{}^{\prime}0}+\breve{\mathbf{B}}^{\prime}\cdot\widetilde{\mathbf{B}}^{\prime},$
(282)
$\displaystyle\breve{\nabla}^{\prime}\times\widetilde{\mathbf{B}}^{\prime}-\breve{\mathbf{A}}^{\prime}\times\widetilde{\mathbf{B}}^{\prime}+\breve{\mathbb{B}}^{\prime}\cdot\widetilde{\mathbf{B}}^{\prime}={\mathbf{J}}^{\prime}+\eth^{\prime}_{0}\widetilde{\mathbf{E}}^{\prime}-\breve{A}^{\prime}_{0}\widetilde{\mathbf{E}}^{\prime}+\breve{\mathbb{E}}^{\prime}\cdot\widetilde{\mathbf{E}}^{\prime},$