## Dark Matter and Dark Energy: A Property of Gravity

The main objective of this post is to show  that

dark matter and dark energy phenomena are just a property of gravity

## 1. Symmetry Dictates Law of Gravity

Gravity is one of the four fundamental interactions/forces of Nature, and is certainly the first interaction/force that people studied over centuries, dating back to Aristotle (4th century BC), to Galileo (late 16th century and early 17th century), to Johannes Kepler (mid 17th century), to Isaac Newton (late 17th century), and to Albert Einstein (1915).

Based solely on two fundamental principles: the principle of general relativity and the principle of interaction dynamics (PID), we have derived, in the previous post, the following gravitational field equations; see also Articles [1, 2]:

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

supplemented by the energy-momentum conservation:

$\displaystyle \nabla^\mu\left[ \frac{8\pi G}{c^4}T_{\mu\nu} + \nabla_{\mu}\Phi_{\nu} \right] =0. \ \ \ \ \ (2)$

Here ${\Phi_\nu}$ is a vector field defined on the 4D space-time manifold ${M}$, and needs to be solved together with the Riemannian metric ${g_{\mu\nu}}$, representing the gravitational potential. Also ${\nabla^\mu}$ is the gradient operator on ${M}$, ${R_{\mu \nu}}$ and ${R}$ are the Ricci and scalar curvatures, and ${T_{\mu\nu}}$ is the energy-momentum of the baryonic matter in the universe.

We emphasize that the field equations (1) are derived solely based on the two first principles. The principle of general relativity, together with the simplicity of laws of Nature, uniquely determines the Lagrangian action for gravity as the Einstein-Hilbert functional:

$\displaystyle L_{EH}(\{g_{\mu\nu}\}) = \int_M \left(R+ \frac{8\pi G}{c^4} S\right)\sqrt{-g}dx. \ \ \ \ \ (3)$

Then as we have demonstrated in the previous post, the presence of dark matter and dark energy implies that the variation of the Einstein-Hilbert functional must be taken under energy-momentum conservation constraint, leading precisely to the gravitational field equations (1). This is is the exact form of PID when applied to gravity, and is the original motivation of PID.

## 2. Duality

The field equations (1) establish a natural duality between the gravitational field ${g_{\mu\nu}}$ and its dual vector field ${\Phi_\mu}$. There are two aspects of this duality. The first is the duality of the two fields: ${g_{\mu\nu}}$, representing massless spin-2 graviton, and ${\Phi_\mu}$, representing massless spin-1 dual vector graviton:

$\displaystyle \text{spin-2 graviton} \ \ \ \ \longleftrightarrow \ \ \ \ \text{spin-1 dual vector graviton}\ \ \ \ \ (4)$

The second aspect of the duality is the duality of the gravitational force. Namely, gravitational interaction generates both attractive and repulsive forces:

$\displaystyle \text{Gravitation force} \ \ \ = \ \ \ \text{attraction due to }\ g_{\mu\nu} + \text{ repelling due to }\ \Phi_{\mu}. \ \ \ \ \ (5)$

We know that the Schwarzschild solution of classical Einstein equations gives rise to the classical Newton’s gravitational force formula. Using the field equations (1), however, we can deduce a revised gravitational force formula. In fact, consider a central matter field with total mass ${M}$ and with spherical symmetry. We can derive an approximate gravitational force formula:

$\displaystyle F=mMG\left[-\frac{1}{r^2} -\frac{k_0}{r} + k_1 r \right], \qquad k_0=4 \times 10^{-18} km^{-1}, \qquad k_1=10^{-57} km^{-3}.$

Here the first term represents the Newton gravitation, the attracting second term stands for dark matter and the repelling third term is the dark energy.

In summary, we have shown that it is the duality between the attracting gravitational field ${\{g_{\mu\nu}\}}$ and the repulsive dual vector field ${\{\Phi_\mu\}}$, together with the nonlinear interaction of these two fields through the field equations (1) and (2), that give rise to gravity, and in particular the gravitational effect of dark energy and dark matter.

## 3. Law of Gravity, Dark Matter and Dark Energy

Here we summarize the law of gravity as we understand now.

Law of gravity:

• (Principle of Equivalence) The space-time is a 4D Riemannian manifold ${M}$, with the metric ${\{g_{\mu\nu}\}}$ being the gravitational potential.
• The gravitational potential ${\{g_{\mu\nu}\}}$ and its dual vector field ${\Phi_\mu}$ satisfy the law of gravity, the gravitational field equations (1) and (2), which are completely determined by the principle of general relativity and PID.
• Gravitational effect is achieved through that the space-time curvature dictated by the gravitational potential ${\{g_{\mu\nu}\}}$.
• Gravity can display both attractive and repulsive effect, caused by the duality between the attracting gravitational field ${\{g_{\mu\nu}\}}$ and the repulsive dual vector field ${\{\Phi_\mu\}}$, together with their nonlinear interactions governed by the field equations (1) and (2).

Also, we have demonstrated in this and in the previous post that PID– the energy-momentum conservation constraint variation– is simply the direct and unique consequence of the presence of dark energy and dark matter. In return, we are able to show the following:

Dark energy and dark matter phenomena are simply a property of gravity.

We end this post with a few remarks.

First, over the centuries, mankind has been searching for ultimate laws of Nature, including, in particular, the law of gravity. Albert Einstein was the first to believe that the laws of Nature are dictated by symmetry and a few fundamental principles. This is also what we believe in and what we intend to achieve in a series of recent papers. Again, we reiterate that this new law of gravity is derived using solely fundamental principles.

The missing information when Einstein discovered his theory of general relativity was the dark matter and dark energy phenomena. The dark matter phenomenon was initially discovered by Fritz Zwicky in early 1930s, Kent Ford and Vera Cooper Rubin in 1970s, and the dark energy phenomenon was discovered in 1990s by Saul Perlmutter, Brian P. Schmidt and Adam G. Riess.

The great mystery of dark matter and dark energy has been the main source of  inspiration for numerous attempts to alter the Einstein gravitational field equations. Most of these attempts, if not all, focus on altering the Einstein-Hilbert action with fine tunings, and therefore are phenomenological. The key observation for our study is that the presence of dark matter and dark energy induces variation of the Einstein-Hilbert action under energy-momentum conservation constrains, leading to the postulation of PID for all four fundamental interactions. The new term ${\nabla_\mu \Phi}$ in the new field equations (1) is the natural and unique consequence of PID, thanks for the theory on divergence-free constrain variations and orthogonal decomposition theorems that we developed recently.

Second, it is then clear that the term ${\nabla_\mu\Phi_\nu}$ does not correspond to any Lagrangian action density, and is the direct consequence of PID. In fact, as pointed out by one of the referees of Article [1], if one intends to derive this new term by adding into the Einstein-Hilbert action density something like:

$\displaystyle g^{\mu\nu} \nabla_\mu\Phi_\nu,$

then two problems arise. First, field equation would contain not only ${\nabla_\mu \Phi_\nu}$, but also additional terms:

$\displaystyle g^{\mu\nu} \delta(\nabla_\mu \Phi_\nu),$

as the covariant derivative ${\nabla_\mu}$ is metric dependent. The second problem is that Stokes formula would imply that the added density is nil:

$\displaystyle \int_M g^{\mu\nu} \nabla_\mu\Phi_\nu =0.$

Third, if we take the cosmic microwave background radiation into consideration, the field equations are in a more general form with the vector field ${\Phi_\nu}$:

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}T_{\mu\nu}-(\nabla_{\mu}+\frac{e}{\hbar c}A_{\mu})\Phi_{\nu}, \ \ \ \ \ (6)$

where the term ${\frac{e}{\hbar c}A_{\mu}\Phi_{\nu}}$ represents the coupling between the gravitation and the microwave background radiation.

Tian Ma and Shouhong Wang