## Law of gravity and Gravitational Radiation

Tian Ma & Shouhong Wang, Radiations and Potentials of Four Fundamental Interactions, Hal-preprint, hal-01616874, October 10, 2017

Talk at 2017 Midwest Relativity Meeting. October 13, 2017, Ann Arbor, University of Michigan

The above paper addresses radiation and field particles of the four fundamental interactions, demonstrating that each individual interaction possesses two basic attributes:

• field particles and their radiation, and
• the interaction potential and force formulas.

In this blog, we examine the law of gravity and the gravitational field particle and radiation; see also the talk given in 2017 Midwest Relativity Conference.

### 1. Gravitational field equations

The Einstein theory of general relativity is the most profound scientific theory in the recorded human history. The Einstein theory is built on two first principles: the principle of equivalence (PE) and the principle of general relativity (PGR). In essence, PE amounts to saying that space time is a four-dimensional Riemannian manifold ${(\mathcal M, \{g_{\mu\nu}\})}$ with the metric being the gravitational potential.

The PGR is a symmetry principle, and says that the law of gravity is the same (covariant) under all coordinate systems. In other words, the Lagrangian action of gravity, called the Einstein-Hilbert action, is invariant under all coordinate transformations.

The Einstein–Hilbert functional (1) is uniquely dictated by this profound and simple looking symmetry principle, together with simplicity of laws of Nature, and is given as follows:

$\displaystyle L_{\text{EH}}=\int_{M}\bigg[R+\frac{8\pi G}{c^4}S\bigg]\sqrt{-g}\mbox{d}x. \ \ \ \ \ (1)$

Indeed, in Riemannian geometry, the invariant quantities satisfying the principle of general relativity and containing the second order derivative terms of ${\{g_{\mu\nu}\}}$ is just the scalar curvature ${R}$, which is unique.

The presence of the dark matter and dark energy phenomena requires the inevitable need for modifying the Einstein general theory of relativity. Such modification needs to preserve the following basic physical requirements:

• conservation of energy-momentum,
• inclusion of dark matter and dark energy effect, and
• preservation of Einstein’s two principles: PE and PGR.

We have shown that, under these basic requirements, the unique route for altering the Einstein general theory of relativity is through the principle of interaction dynamics (PID), which takes variation of the Einstein-Hilbert action subject of energy-momentum conservation constraint. This leads to the the following new gravitational field equations; see [Tian Ma & Shouhong Wang, Gravitational field equations and theory of dark matter and dark energy, Discrete and Continuous Dynamical Systems, Ser. A, 34:2 (2014), pp. 335-366; see also arXiv:1206.5078]:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu}-\nabla_\mu\Phi_\nu. \ \ \ \ \ (2)$

Also, we have shown that PID is the direct consequence of the presence of dark energy and dark matter, is the requirement of the presence of the Higgs field for the weak interaction, and is the consequence of the quark confinement phenomena for the strong interaction; see [Tian Ma & Shouhong Wang, Mathematical Principles of Theoretical Physics, Science Press, 524pp., 2015].

### 2. Gravitational field particle

The gravitational field particle is described by the dual field ${\{\Phi_\mu\}}$ in (2), and the governing radiation equations are

$\displaystyle \nabla^\mu\nabla_\mu\Phi_\nu=-\frac{8\pi G}{c^4}\nabla^\mu T_{\mu\nu}, \ \ \ \ \ (3)$

where ${T_{\mu\nu}}$ stands for the energy-momentum tensor of the visible matter, and ${\{\Phi_\mu\}}$ is a massless, spin-1 and electric neutral boson.

In fact, the gravitational field particle ${\{\Phi_\mu\}}$ represents the dark matter that we have been searching for, and the energy that ${\{\Phi_\mu\}}$ carries is the dark energy. Equation (3) is the field equations for dark matter and dark energy. The gravitational effect of the field particle ${\{\Phi_\mu\}}$ is manifested through the mutual coupling and interaction with the gravitational potential ${\{g_{\mu\nu}\}}$, through the field equations (2), leading to both attractive and repulsive behavior of gravity, which is exactly the dark energy and dark matter phenomena.

### 3. Gravitational force formula

Gravitational force formula By the gravitational field equations (2), we derive the following approximate gravitational force formula:

$\displaystyle F(r)=mMG\bigg[-\frac{1}{r^2}-\frac{k_0}{r}+k_1r\bigg],\quad k_0=4\times 10^{-18}\mbox{km}^{-1},\quad k_1=10^{-57}\mbox{km}^{-3}.$

Tian Ma & Shouhong Wang, October 13, 2017