compat=1.1.0

# Linear dynamics and classical tests of the gravitational quantum field
theory

Yuan-Kun<EMAIL_ADDRESS>International Centre for
Theoretical Physics Asia-Pacific (ICTP-AP), University of Chinese Academy of
Sciences (UCAS), Beijing 100190, China  Taiji Laboratory for Gravitational
Wave Universe (Beijing/Hangzhou), UCAS, Beijing 100049, China  Da
<EMAIL_ADDRESS>National Astronomical Observatories, Chinese Academy
of Sciences, Beijing 100101, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  International Centre for Theoretical Physics Asia-Pacific
(ICTP-AP), University of Chinese Academy of Sciences (UCAS), Beijing 100190,
China  Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou),
UCAS, Beijing 100049, China  Yong-Liang<EMAIL_ADDRESS>Nanjing
University, Suzhou, 215163, China  School of Fundamental Physics and
Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou
310024, China  Taiji Laboratory for Gravitational Wave Universe
(Beijing/Hangzhou), UCAS, Beijing 100049, China  International Centre for
Theoretical Physics Asia-Pacific (ICTP-AP), University of Chinese Academy of
Sciences (UCAS), Beijing 100190, China  Yong<EMAIL_ADDRESS>School of
Astronomy and Space Sciences, UCAS, Beijing 100049, China  School of
Fundamental Physics and Mathematical Sciences, Hangzhou Institute for Advanced
Study, UCAS, Hangzhou 310024, China  International Centre for Theoretical
Physics Asia-Pacific (ICTP-AP), University of Chinese Academy of Sciences
(UCAS), Beijing 100190, China  Taiji Laboratory for Gravitational Wave
Universe (Beijing/Hangzhou), UCAS, Beijing 100049, China  Yue-Liang
<EMAIL_ADDRESS>International Centre for Theoretical Physics Asia-Pacific
(ICTP-AP), University of Chinese Academy of Sciences (UCAS), Beijing 100190,
China  Taiji Laboratory for Gravitational Wave Universe (Beijing/Hangzhou),
UCAS, Beijing 100049, China  School of Fundamental Physics and Mathematical
Sciences, Hangzhou Institute for Advanced Study, UCAS, Hangzhou 310024, China
CAS key laboratory of theoretical Physics, Institute of Theoretical Physics,
Chinese Academy of Sciences, Beijing 100190, China  Yu-Feng
<EMAIL_ADDRESS>CAS key laboratory of theoretical Physics, Institute
of Theoretical Physics, Chinese Academy of Sciences, Beijing 100190, China
School of Physics, UCAS, Beijing 100049, China  School of Fundamental Physics
and Mathematical Sciences, Hangzhou Institute for Advanced Study, UCAS,
Hangzhou 310024, China  International Centre for Theoretical Physics Asia-
Pacific (ICTP-AP), University of Chinese Academy of Sciences (UCAS), Beijing
100190, China

###### Abstract

We explore the new physics phenomena of gravidynamics governed by the
inhomogeneous spin gauge symmetry based on the gravitational quantum field
theory. Such a gravidynamics enables us to derive the generalized Einstein
equation and an equation beyond it. To simplify the analyses, we linearize the
dynamic equations of gravitational interaction by keeping terms up to the
leading order in the dual gravigauge field. We then apply the linearized
dynamic equations into two particular gravitational phenomena. Firstly, we
consider the linearized equations in the absence of source fields, which is
shown to have five physical propagating polarizations as gravitational waves,
i.e., two tensor modes, two vector modes, and one scalar, instead of two
tensor polarizations in the general relativity. Secondly, we examine the
Newtonian limit in which the gravitational fields and the matter source
distribution are weak and static. By deriving the associated Poisson equation,
we obtain the exact relation of the fundamental interaction coupling in the
gravidynamics with the experimentally measured Newtonian constant. We also
make use of non-relativistic objects and relativistic photons to probe the
Newtonian field configurations. In particular, the experiments from the
gravitational deflection of light rays and the Shapiro time delay can place
stringent constraints on the linearized gravidynamics in the gravitational
quantum field theory.

##  I Introduction

Among the fundamental interactions, the gravitation is the weakest and the
most mysterious. The standard theory of gravitation at present is Einstein’s
General Relativity (GR) [1, 2], in which the gravity is described by the
Riemannian geometry of curved spacetime. Since its birth in 1915, the GR has
withstood all the astrophysical and cosmological tests, which include four
classical ones: (i) the gravitational redshift of photons [3, 4, 5]; (ii) the
deflection of light [6, 7, 8, 9]; (iii) the perihelion advance of Mercury
[10]; and (iv) the time delay of light [11, 12] (for a recent review on the
classical tests of GR, see e.g. Ref. [13] and references therein). The
discovery of the binary pulsar B1913+16 by Hulse and Taylor in 1974 [14] has
verified the GR by providing the first evidence of the existence of
gravitational radiation [15, 16]. In 2015, the first direct observation by
advanced LIGO observatories of the gravitational wave (GW) signal emitted from
the merger of binary black holes [17, 18] has provided a further support to
the GR.

On the other hand, the electromagnetic, weak and strong interactions are all
described by quantum field theories (QFTs) based on the gauge principle [19].
Thus, it is tempting to write down the gravity in terms of the gauge language,
and numerous efforts were explored in early studies (for review articles see
e.g. Refs. [20, 21, 22]. Whereas most gravity gauge theories were formulated
based on Riemannian geometry on curved space-time or Poincaré group in
coordinate space-time rather than in Hilbert space of fields. Recently, a
gauge theory of gravity has been constructed in the framework of gravitational
quantum field theory (GQFT) [23] and unified with the standard model to build
the hyperunified field theory(see Ref. [24] for the latest review of the GQFT
and the hyperunified field theory as well as the references therein). It has
been shown that the gravitational force and the spin gauge force are governed
by the inhomogeneous spin gauge symmetry ${\rm WS}(1,3)={\rm
SP}(1,3)\rtimes{\rm W}^{1,3}$, in which the spin gauge field corresponding to
the spin gauge symmetry ${\rm SP}(1,3)$ is introduced to characterize the spin
gauge interaction and the gravigauge field, while the translation-like ${\cal
W}_{e}$-spin gauge symmetry ${\rm W}^{1,3}$ in Hilbert space of spinor fields
emerges to describe the gravitational force. A biframe spacetime appears to
reveal the nature of spacetime in which the globally flat Minkowski spacetime
plays the role of the base spacetime while the locally flat gravigauge
spacetime acts as a fiber. As a result, the gravitational dynamics is
described in terms of the gravigauge and spin gauge fields.
Phenomenologically, there have already been many studies in the framework of
GQFT [24], including discussions on inflation [25], dark matter, and particle
physics.

Note that the gravitational dynamics in the GQFT is highly nonlinear [23, 24],
which makes its application rather difficult. However, when the gravitational
effects are weak, the gravitational dynamics becomes easy and tractable, so
that we can expand the equations of motion in terms of perturbations at the
linear level. In the present work, we would like to explore the gravitational
physics in this linearized GQFT. We start by deriving the linearized
gravitational field equations, which are then applied to two special
situations. Firstly, we shall consider the free gravitational equations
without any matter sources, aiming to investigate how many and what physical
polarizations are contained in the propagating GWs. Next, we turn to the
Newtonian limit, in which the gravitational field and matter sources are weak
and static. By solving the linearized equations, we can obtain the well-known
Poisson equation which governs the associated Newtonian potential. We will
also make use of both non-relativistic and relativistic test bodies to probe
this Newtonian configuration. In particular, when the probing particles are
photons, we can constrain the GQFT with existing experiments, such as the
deflection of a light ray and the Shapiro time delay.

The paper is organized as follows. In Sec. II, we shall derive the linearized
gravitational equations of gravigauge fields in the GQFT. We shall apply in
Sec. III the obtained equations to the free field case, and examine the
physical degrees contained in the propagating GWs. In Sec. IV, we turn to the
Newtonian limit, which is tested by the non-relativistic test bodies and
relativistic photons. Finally, we conclude and comment in Sec. V.

##  II Gravitational Equations of Gravigauge Field at the Linear Level

In this section, we linearize the gravitational equations of gravigauge fields
in the GQFT [23]. Our derivation begins with the gauge-type formulation of the
gravidynamics given in Eq. (159) of Ref. [23]

$\displaystyle\partial_{\nu}\tilde{F}_{a}^{\,\mu\nu}=J_{a}^{\,\mu}\,,$ (1)

where the definition of $\tilde{F}_{a}^{\,\mu\nu}$ and various contributions
to the source $J_{a}^{\,\mu}$ are defined in Ref. [23]. Note that the basic
ingredient in the gravidynamics is the gravigauge field $\chi_{\,\mu}^{a}$,
which can always be written as follows:

$\displaystyle\chi_{\,\mu}^{a}\equiv\eta^{a}_{\mu}+\frac{1}{2}h^{a}_{\,\mu}\,,$
(2)

where $\eta^{a}_{\mu}$ is regarded as the background field while
$h^{a}_{\,\mu}$ is the redefinition of the field variable $\chi^{a}_{\,\mu}$.
If we further require that $h^{a}_{\mu}$ is a weak perturbation, we can expand
the dual gravigauge field as
$\hat{\chi}^{\,\mu}_{a}=(\eta^{a}_{\mu}+\frac{1}{2}h^{a}_{\,\mu})^{-1}$ and
the determinant $\chi$ in terms of $h^{a}_{\,\mu}$. As a result, the field
strength $F^{a}_{\,\mu\nu}$ can be reduced into

$\displaystyle F^{a}_{\,\mu\nu}$ $\displaystyle=$
$\displaystyle\partial_{\mu}\chi^{a}_{\,\nu}-\partial_{\nu}\chi^{a}_{\,\mu}=(\partial_{\mu}h^{a}_{\,\nu}-\partial_{\nu}h^{a}_{\,\mu})/2\,,$
(3)

and the leading-order gravitational field equation in the GQFT is given by

$\displaystyle\left[\square
h_{a}^{\,\rho}-\partial^{\rho}\partial_{\nu}h_{a}^{\,\nu}-\partial_{\nu}\partial_{a}h^{\nu\rho}+\partial_{a}\partial^{\rho}h+\delta^{\rho}_{a}(\partial_{\nu}\partial_{\sigma}h^{\nu\sigma}-\square
h)\right]+\gamma_{W}(\square
h_{a}^{\,\rho}-\partial^{\rho}\partial_{\nu}h_{a}^{\,\nu})=-16\pi{G}_{\kappa}{\rm
J}_{a}^{\,\rho}\,,$ (4)

where the $\gamma_{W}$-independent terms on the left-hand side (LHS) are
derived from the LHS of Eq. (1) while the $\gamma_{W}$-dependent part from the
term $m_{G}^{2}{\cal
D}_{\nu}\left(\chi\bar{\chi}_{aa^{\prime}}^{[\mu\nu]\mu^{\prime}\nu^{\prime}}\mathbf{F}^{a^{\prime}}_{\,\mu^{\prime}\nu^{\prime}}\right)$
contained in $J_{a}^{\,\mu}$. Here
$\gamma_{W}\equiv\gamma_{G}(\alpha_{G}-\alpha_{W}/2)$ with $\gamma_{G}$ and
$\alpha_{G(W)}$ defined in Ref. [23]. The current ${\rm J}_{a}^{\,\rho}$ on
the right-hand side is composed of ordinary matter fields which sources the
gravitational perturbation $h^{a}_{\,\mu}$. In Eq. (4) we have fixed the gauge
conditions for the local ${\rm SP}(1,3)$ symmetry so that $h_{a}^{\,\mu}$ is a
symmetric tensor with $h_{a}^{\,\mu}=h^{\mu}_{\,a}$. We also have
$h\equiv\eta^{\mu}_{a}h^{a}_{\,\mu}$ and
$h_{\mu\nu}\equiv\eta^{a}_{\nu}\eta_{\mu\rho}h^{\,\rho}_{a}$. However, the
indices $a$ and $\mu$ in Eq. (4) do not possess any symmetry property, so that
we can decompose this equation into the symmetric and anti-symmetric parts as
follows

$\displaystyle\widetilde{G}_{\mu\nu}\equiv\frac{1}{2}\left[\square
h_{\mu\nu}-2\partial^{\sigma}\partial_{(\mu}h_{\nu)\sigma}+\partial_{\mu}\partial_{\nu}h+\eta_{\mu\nu}(\partial^{\rho}\partial^{\sigma}h_{\rho\sigma}-\square
h)\right]+\frac{\gamma_{W}}{2}[\square
h_{\mu\nu}-\partial^{\sigma}\partial_{(\mu}h_{\nu)\sigma}]=-8\pi
G_{\kappa}T_{(\mu\nu)}\,,$ (5)
$\displaystyle\widetilde{G}_{[\mu\nu]}\equiv-\frac{\gamma_{W}}{2}\partial^{\sigma}\partial_{[\mu}h_{\nu]\sigma}=-8\pi{G}_{\kappa}T_{[\mu\nu]}\,,$
(6)

where $\widetilde{G}_{\mu\nu}$ represents the generalized Einstein tensor and
the source terms are given by

$\displaystyle T_{(\mu\nu)}$ $\displaystyle\equiv$
$\displaystyle(\eta_{\mu\rho}\eta^{a}_{\,\nu}+\eta_{\nu\rho}\eta^{a}_{\,\mu}){\rm
J}_{a}^{\,\rho}/2\,,$ $\displaystyle T_{[\mu\nu]}$ $\displaystyle\equiv$
$\displaystyle(\eta_{\mu\rho}\eta^{a}_{\,\nu}-\eta_{\nu\rho}\eta^{a}_{\,\mu}){\rm
J}_{a}^{\,\rho}/2\,,$ (7)

Eqs. (5) and (6) comprises the complete linearized equations governing
gravitational fields in the GQFT.

##  III Free Field Equations and Physical Degrees of Freedom

As the first application of the general linearized field equations in the
GQFT, this section is devoted to exploring the free equations in the absence
of matter sources $T_{(\mu\nu)}=0$ and $T_{[\mu\nu]}=0$, paying attention to
the question how many GW degrees of freedom (dofs) are propagating in this
theory.

First of all, we follow the standard procedure in Refs. [26, 27, 28] to
decompose $h_{\mu\nu}$ into the following polarization modes:

  * •

Spin-2 tensor modes: $\hat{h}_{ij}$

$\displaystyle\hat{h}_{it}=\hat{h}_{tt}=0\,,\,\hat{h}^{i}_{i}=0\,,\,\partial^{i}\hat{h}_{ij}=0\,,$
(8)

  * •

Spin-1 vector modes: $S_{i}$ and $F_{i}$

$\displaystyle
h_{tt}=0\,,\,h_{it}=S_{i}\,,\,h_{ij}=2\partial_{(i}F_{j)}\,,\,\partial^{i}S_{i}=\partial^{i}F_{i}=0\,,$
(9)

  * •

Spin-0 scalar modes: $\phi$, $B$, $\psi$ and $E$

$\displaystyle
h_{tt}=-2\phi\,,\,h_{it}=-\partial_{i}B,\,h_{ij}=-2\psi\delta_{ij}+2\partial_{i}\partial_{j}E.$
(10)

We can conveniently summarize these fields with the following line element

$\displaystyle ds^{2}$
$\displaystyle=(1+2\phi)dt^{2}-2(S_{i}-\partial_{i}B)dx^{i}dt$ (11)
$\displaystyle-$
$\displaystyle[\hat{h}_{ij}-(1-2\psi)\eta_{ij}+2\partial_{(i}F_{j)}+2\partial_{i}\partial_{j}E]dx^{i}dx^{j}\,,$

with $\eta_{ij}\equiv-\delta_{ij}$. Furthermore, one can prove that the field
equations in Eqs. (5) and (6) are invariant under the scalar-type gauge
transformation $\delta h_{\mu\nu}=\partial_{\mu}\partial_{\nu}\zeta$, which
leads the scalar fields $\phi$, $B$ and $E$ to transform as

$\displaystyle\phi\to\phi+\partial_{t}^{2}\zeta/2\,,\quad B\to
B+\partial_{t}\zeta\,,\quad E\to E-\zeta/2\,.\,\,$ (12)

Therefore, we can define the following two gauge-invariant variables

$\displaystyle\Phi=\phi-\partial_{t}B/2\,\quad A=B+2\partial_{t}E\,.$ (13)

With the above polarization fields, the symmetric equation in Eq. (5) can give
us the following independent equations

  * •

$(t,t)$ component

$\displaystyle
2\partial^{k}\partial_{k}\psi+\gamma_{W}\partial^{k}\partial_{k}\Phi=0\,.$
(14)

  * •

four-dimensional trace

$\displaystyle
3\square\psi=\partial^{i}\partial_{i}\left(\psi+\Phi-\partial_{t}A/2\right)\,.$
(15)

  * •

$(t,i)$ component

$\displaystyle\left[\partial^{k}\partial_{k}(S_{i}-\partial_{t}F_{i})+4\partial_{t}\partial_{i}\psi\right]+(\gamma_{W}/2)\left[\square
S_{i}-\square\partial_{i}A\right.$ (16) $\displaystyle+$
$\displaystyle\left.\partial^{k}\partial_{k}(S_{i}-\partial_{t}F_{i})+2\partial_{i}\partial_{t}(\Phi-\psi+\partial_{t}A/2)\right]=0\,.$

  * •

$(i,j)$ component

$\displaystyle(1+\gamma_{W})\square\hat{h}_{ij}+\gamma_{W}\square\partial_{(i}F_{j)}-{(\gamma_{W}+2)}\partial_{t}\partial_{(i}[S-\partial_{t}F]_{j)}$
$\displaystyle+{2\eta_{ij}(\gamma_{W}+1)}\square\psi$
$\displaystyle+2\partial_{i}\partial_{j}\left[{(1-\gamma_{W})}\psi-\Phi+{(\gamma_{W}+1)}\partial_{t}A/2\right]=0\,,$
(17)

It is clear that only the gauge-invariant fields appear in the final
equations.

Our next task is to solve these field equations in the GQFT. We begin with the
scalar sector. Eq. (14) implies the following constraint for $\psi$

$\displaystyle\psi=-\gamma_{W}\Phi/2\,,$ (18)

which can be easily seen by transforming Eq. (14) into the Fourier space with
a nonzero wave number. Moreover, we can obtain the additional independent
relations

$\displaystyle\frac{\gamma_{W}}{2}\square A$ $\displaystyle=$
$\displaystyle\partial_{t}\left[(4-\gamma_{W})\psi+\gamma_{W}\Phi+\frac{\gamma_{W}}{2}\partial_{t}A\right]\,,$
(19) $\displaystyle\square\psi$ $\displaystyle=$
$\displaystyle\partial_{j}\partial^{j}\left[\frac{\gamma_{W}-1}{\gamma_{W}+1}\psi+\Phi-\frac{1}{2}\partial_{t}A\right]\,,$
(20)

by acting one and two spatial derivatives on Eqs. (16) and (• ‣ III),
respectively. By solving Eqs. (18), (15), (19) and (20), we can yield the
equation of motion for $\Phi$

$\displaystyle\square\Phi=0\,,$ (21)

and the constraint for $A$:

$\displaystyle\partial_{t}A=-(\gamma_{W}-2)\Phi\,.$ (22)

We now turn to the vector sector containing $F_{i}$ and $S_{i}$. By taking
into account Eqs. (21), (18), and (22), all terms related to scalars are
cancelled out in Eqs. (16) and (• ‣ III), which results in the following
reduced equations

$\displaystyle\square
S_{i}+\frac{(\gamma_{W}+2)}{\gamma_{W}}\partial_{k}\partial^{k}(S_{i}-\partial_{t}F_{i})=0\,,$
(23) $\displaystyle\square
F_{j}-\frac{(\gamma_{W}+2)}{\gamma_{W}}\partial_{t}(S_{j}-\partial_{t}F_{j})=0\,,$
(24)

where the second equality follows by taking a divergence $\partial_{i}$ on Eq.
(• ‣ III). In order to fully shed light on the vector dynamics, we also need
to consider the anti-symmetric gravitational equations in Eq. (6), which can
be written in terms of component fields as

  * •

$(t,i)$ component

$\displaystyle\square
S_{i}-\partial^{j}\partial_{j}(S_{i}-\partial_{t}F_{i})=0\,.$ (25)

  * •

$(i,j)$ component

$\displaystyle\square\partial_{[i}F_{j]}+\partial_{t}\partial_{[i}(S-\partial_{t}F)_{j]}=0\,,$
(26)

where scalars are cancelled out totally due to their dynamics. By solving Eqs.
(23)-(26), one can show for $\gamma_{W}\neq-1$

$\displaystyle\square F_{i}=0\,,$ (27)

with the constraint

$\displaystyle S_{i}=\partial_{t}F_{i}\,.$ (28)

Finally, by considering the relations in the scalar and vector sectors, Eq. (•
‣ III) gives us

$\displaystyle\square\hat{h}_{ij}=0\,,$ (29)

which is the wave equation for the two massless tensor dofs like in the GR.

In summary, for a generic value of $\gamma_{W}\neq 0$ or $-1$, we have found
five massless propagating GW dofs in the GQFT: two tensor modes
$\hat{h}_{ij}$, two vector modes $F_{i}$, and one scalar $\Phi$, which can
determine the dynamics of other component fields defined in Eq. (11) via
various relations.

##  IV Newtonian Limits

Let us now apply the linearized gravitational equations in the GQFT to explore
the gravitational fields in the Newtonian limit. As is well known, the
Newtonian limit [29, 30] is the situation in which the gravitational field is
static and weak. Thus, in the following, all of the time derivatives in the
linearized gravitational field equations can be ignored. Also, the matter is
composed of dust and the configuration is static, so that the only nonzero
component of the energy momentum tensor is only $T_{(00)}=\rho(\textbf{x})$,
where the mass density $\rho$ is assumed to be a function of spatial
coordinates x. The antisymmetric energy-momentum tensor $T_{[\mu\nu]}$
vanishes by assumption. Thus, the modified Einstein equations are given by

$\displaystyle\widetilde{G}_{00}=\partial_{i}\partial^{i}(2\psi+\beta\Phi)=-8\pi{G}_{\kappa}\rho({\bf
x})\,,$
$\displaystyle\widetilde{G}_{0i}=-\partial_{k}\partial^{k}\left[(1+\gamma_{W})S_{i}-\gamma_{W}\partial_{i}A/2\right]/2=0\,,$
(30)
$\displaystyle\widetilde{G}_{ij}=-(1/2)\\{(1+\gamma_{W})\partial_{k}\partial^{k}\hat{h}_{ij}+\gamma_{W}\partial_{k}\partial_{k}\partial_{(i}F_{j)}$
$\displaystyle-2\partial_{i}\partial_{j}[(\gamma_{W}-1)\psi+\Phi]-2\eta_{ij}\partial_{k}\partial^{k}[(1-\gamma_{W})\psi-\Phi]\\}=0\,.$

By solving these equations, the Newtonian potential $\Phi$ is shown to obey
the conventional Poisson equation

$\displaystyle\triangle\Phi\equiv-\partial_{k}\partial^{k}\Phi=4\pi
G_{N}\rho({\bf x})\,,$ (31)

with the scalar $\psi$ determined by

$\displaystyle\psi=\Phi/(1-\gamma_{W})\,,$ (32)

while other fields $\hat{h}_{ij}$, $S_{i}$, $F_{i}$ and $A$ all vanish
identically, where we have defined the measured Newtonian constant in terms of
the fundamental coupling ${G}_{\kappa}$ [29, 30] as follows

$\displaystyle
G_{N}\equiv\frac{1-\gamma_{W}}{(1-\gamma_{W}/2)(1+\gamma_{W})}{G}_{\kappa}\,.$
(33)

One can probe this Newtonian field configuration with either non-relativistic
test bodies or relativistic particles like photons. At low-energy limit of the
GQFT, all these objects propagate along the geodesics described by:

$\displaystyle\frac{d^{2}x^{\rho}}{d\tau^{2}}+\Gamma^{\rho}_{\,\mu\nu}\frac{dx^{\mu}}{d\tau}\frac{dx^{\nu}}{d\tau}=0\,,$
(34)

where $\Gamma^{\rho}_{\,\mu\nu}$ is the usual Christoffel symbol defined with
respect to the effective metric in Eq. (11).

If the probing object is massive and moves very slowly, i.e., $dx^{i}/d\tau\ll
dt/d\tau$, the geodesic equation in Eq. (34) dictates the object to follow the
path

$\displaystyle\frac{d^{2}x^{i}}{dt^{2}}=\partial^{i}\Phi(\textbf{x})=-\partial_{i}\Phi(\textbf{x})\,.$
(35)

which is nothing but the acceleration of a massive body moving in the static
Newtonian potential. Furthermore, Eq. (35) also implies that, regardless of
the value of its mass, the test body would always experience the same
acceleration and propagate with the same trajectory in this weak gravitational
field configuration, which has been well tested by the experiments examining
the Weak Equivalence Principle (WEP). Note that the best constraints on the
WEP are provided by the E$\ddot{\rm o}$t-Wash group [31, 32] and MICROSCOPE
[33].

On the other hand, the gravitational field profile in the Newtonian limit can
be detected by photons, which can lead to the gravitational light deflection
[6, 7, 8, 9], the Shapiro time delay [11, 12] and the gravitational redshift
[3, 4, 5] of photon frequencies. Note that these phenomena provide three of
the most important tests of the GR in the history (see e.g. Refs. [13, 30] for
recent reviews and references therein). In what follows, we shall compute the
relevant observables in the GQFT and use the current data to constrain the
parameter $\gamma_{W}$.

Since most observations for gravitational light deflections and time delays
have been performed in the Solar system, we shall work in the following
effective metric:

$\displaystyle ds^{2}=(1+2\Phi)dt^{2}-(1-2\psi)\delta_{ij}dx^{i}dx^{j}\,,$
(36)

where all other gravitational component fields vanish in the Newtonian limit
according to previous discussions. Here we have chosen the gauge with $E=0$,
so that $\phi=\Phi$, where $\Phi({\mathbf{x}})$ is the gauge-invariant
Newtonian potential in the Solar system represented by

$\displaystyle\Phi=-G_{N}M_{\odot}/r\,,$ (37)

with $M_{\odot}$ denoting the solar mass and $r$ as the radial distance from
the Sun.

Figure 1: An illustration of the light ray deflection with the deflection
angle $\delta\theta$ and the Shapiro time delay $\delta t_{\rm Shapiro}$
caused by the Newtonian field of a massive object, such as the Sun.

Following the methods presented in Refs. [30, 34], we can derive, as
illustrated in Fig. 1, the expressions for the deflection angle $\delta\theta$
and the Shapiro time delay $\delta t_{\rm Shapiro}$ as follows

$\displaystyle\delta\theta$ $\displaystyle\approx$
$\displaystyle\left(2+\gamma_{W}\right)\left({G_{N}M_{\odot}}/{b}\right)\left(1+\cos\theta_{0}\right)\,,$
(38) $\displaystyle\delta t_{\rm Shapiro}$ $\displaystyle\approx$
$\displaystyle
2(2+\gamma_{W})G_{N}M_{\odot}\ln\left({4r_{e}r_{s}}/{b^{2}}\right)\,,$ (39)

where $\theta_{0}$ is the elongation angle of the emitter relative to the Sun,
$b$ is the closest approach from the Solar barycentric point to the line
connecting the source and the Earth, and $r_{e(s)}$ is the radial distance of
the Earth (source) from the Sun. Comparing with the well-known formulas in the
parametrized post-Newtonian (PPN) formalism [13, 30], it is seen that
$\gamma_{W}$ is closely related to the PPN parameter $\gamma$ by
$\gamma_{W}\approx\gamma-1$.

Recent tremendous developments in the very-long-baseline radio interferometry
(VLBI) and the radar time-delay experiments enable us to put strong
constraints on the GQFT. In particular, the VLBI observations of the
deflection angles of lights from quasars and radio galaxies have yielded
$\gamma_{W}=(-0.8\pm 1.2)\times 10^{-4}$ at the $1\sigma$ CL [8, 9], while the
most precise measurement of the Shapiro time delay is provided by the Cassini
spacecraft [12], which has given the best limit to date on $\gamma_{W}=(2.1\pm
2.3)\times 10^{-5}$.

Finally, we would like to discuss the gravitational redshift of photons [3, 4,
5] in the GQFT. Based on the arguments given in Ref. [29], we can derive the
ratio of the photon frequencies $\omega({\bf x})$ at different locations ${\bf
x}_{1}$ and ${\bf{x}_{2}}$:

$\displaystyle\frac{\omega({\bf x}_{2})}{\omega({\bf
x}_{1})}=\left(\frac{1+2\Phi({\bf x}_{1})}{1+2\Phi({\bf
x}_{2})}\right)^{1/2}\approx 1+\Phi({\bf x}_{1})-\Phi({\bf x}_{2})\,,$ (40)

where we have expanded the expression for $\Phi({\bf x})\ll 1$. Note that the
photon frequency modification only depends on the Newtonian potential
$\Phi({\textbf{x}})$, without any reference to $\psi$, so that the
gravitational redshift effect in the GQFT should be the same as in the GR.

##  V Conclusion and Discussion

Understanding the nature of gravity and its quantization is one of main goals
in the modern physics. Unlike the Einstein’s GR which was based on the Riemann
geometry, the GQFT [23] has constructed the gravitational interaction based on
the gauge principle that has been well-tested by other three fundamental
interactions. In the present work, we have explored the fundamental physics
and phenomenology in the weak gravity limit of the GQFT, so that the theory
can be examined at the linear level of the perturbation $h_{\mu\nu}$. In order
to realize this aim, we have derived the linearized gravitational field
equations in the GQFT. It is found that, different from the usual
diffeomorphism symmetry in the GR, the gauge symmetry in this theory is
reduced to a scalar-type one parametrized by the infinitesimal gauge parameter
$\zeta(x)$. Moreover, the difference between the GQFT and the GR at the
linearized level can be parametrized by one single parameter $\gamma_{W}$.

After establishing this linearized theory, we then apply this formalism to two
special situations of important physical interest. In the first application,
we examine the free linearized gravitational field equations in the absence of
any matter fields. In particular, we focus on one crucial question: how many
and what physical propagating GW dofs are contained in this theory? As a
result, different from the GR which includes only two massless tensor modes,
there are five physical polarizations: two tensor modes, two vector modes and
one scalar mode, all of which are massless.

In the second application, we turn to the Newtonian limit in which the
gravitational field is weak and the matter source fields are static. By
solving the obtained field equations, we can obtain the conventional Poisson
equation which connects the Newtonian potential with the matter density
distribution. As a byproduct, we have obtained the exact relationship between
the fundamental coupling ${G}_{\kappa}$ defined in the GQFT and the
experimentally measured Newtonian constant $G_{N}$. We then make use of the
non-relativistic objects and photons to probe the yielded gravitational field
configuration. For a slowly-moving object, regardless of the value of its
mass, it would always experience the same acceleration and follow the same
trajectory in the gravitational field, which has been well tested by the
experiments examining the WEP. Finally, we consider the motion of a photon in
this Newtonian background, and investigate three classical tests: (i) the
deflection of light, (ii) the time delay of light, and (iii) the gravitational
redshift,. It turns out that the GQFT gives exactly the same prediction of the
gravitational redshift effect as in the GR, so that this kind of experiments
cannot be used to distinguish these two theories. On the other hand, the light
deflection and the Shapiro time delay do predict differently in the GQFT than
in the GR, due to the dependence of the parameter $\gamma_{W}$. Thus, we can
probe and constrain the GQFT with the associated experiments. In particular,
the radar time-delay experiment carried out at the Cassini spacecraft provided
the most stringent bound on $\gamma_{W}\lesssim{\cal O}(10^{-5})$.

Besides the phenomena investigated in the present work, there are still many
other aspects of the GQFT waiting us to explore. One classical test of the GR
is provided by the perihelion advance of Mercury [10], which requires the
calculation of the gravitational field configuration beyond the linear level
[30]. Furthermore, in the view of the additional propagating GW polarizations,
some important questions are raised: How do the extra GW modes couple to
matter fields? What are the corresponding coupling strengths? Is it possible
to observe these GW dofs by the on-going and forthcoming GW observatories,
such as LIGO-Virgo-KAGRA [35, 36, 37, 38], LISA [39], Taiji [40], and TianQin
[41]? We shall come back to these issues in the near future.

## Acknowledgements

This work was supported in part by the National Key Research and Development
Program of China under Grant No. 2020YFC2201501, No. 2021YFC2203003, No.
2021YFC2202900, No. 2021YFC2201901, and No. 2017YFA0402204, and the National
Natural Science Foundation of China (NSFC) under Grant No. 12005254, No.
12347103, No. 11825506, No. 11821505, and No. 12047503, and also the Strategic
Priority Research Program of the Chinese Academy of Sciences as well as the
Fundamental Research Funds for the Central Universities.

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