## Principle of Interaction Dynamics

### 1. Dark Energy and Dark Matter Motivation

The original motivation for the theory of unified fields and elementary particles we developed recently is the great mystery of dark matter and dark energy. The understanding of both phenomena necessitates an examination of the law of gravity in the fundamental level based only on first physical principles.

Since Albert Einstein discovered the general theory of relativity in 1915, his two fundamental first principles, the principle of equivalence and the principle of general relativity have gained strong and decisive observational supports. As the law of gravity, the Einstein gravitational field equations are inevitably needed to be modified to account for dark energy and dark matter.

The principle of equivalence says that the space-time manifold with gravity is a 4-dimensional (4D) Riemannian manifold ${M}$ with the Riemannian metric being regarded as the gravitational potentials.

The principle of general relativity provides a fundamental symmetry of Nature, and requires that the physical laws of Nature are covariant under general coordinate transformations. One can demonstrate that this symmetry principle, together with the simplicity of laws of Nature, uniquely determines the Lagrangian action for gravity, which is given by the Einstein-Hilbert functional:

$\displaystyle L_{EH}(\{g_{\mu\nu}\}) = \int_M \left(R+ \frac{8\pi G}{c^4} S\right)\sqrt{-g}dx. \ \ \ \ \ (1)$

Here ${R}$ stands for the scalar curvature of space-time manifold ${M}$, and ${S}$ is the energy-momentum density of matter field in the universe.

The Einstein gravitational field equations are the Euler-Lagrangian equations of the Einstein-Hilbert functional ${L_{EH}}$:

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

where ${T_{\mu\nu}}$ is the usual energy-momentum tensor of visible matter.

By the Bianchi identity, the left-hand side of (2) is divergence-free:

$\displaystyle \text{ div}(R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R)=0, \ \ \ \ \ (3)$

which implies that the usual energy-momentum tensor satisfies

$\displaystyle \text{ div}\ T_{\mu\nu}=0. \ \ \ \ \ (4)$

However, due to the presence of dark matter and dark energy, the energy-momentum tensor of visible matter ${T_{\mu\nu}}$ is no longer conserved, i.e. (4) is not true. Hence we have

$\displaystyle \text{ div}\ T_{\mu\nu}\neq 0,$

which is a contradiction to (2) and (3).

On the other hand, by the Orthogonal Decomposition Theorem (see articles [1, 3] in Books and Articles), ${T_{\mu\nu}}$ can be orthogonally decomposed into

$\displaystyle T_{\mu\nu}=\tilde{T}_{\mu\nu}-\frac{c^4}{8\pi G}\nabla_{\mu}\Phi_{\nu}, \ \ \ \ \ (5)$

and ${\tilde{T}_{\mu\nu}}$ is divergence-free:

$\displaystyle \text{ div}\ \tilde{T}_{\mu\nu}=0. \ \ \ \ \ (6)$

Hence, by (3) and (6) the gravitational field equations (2) should be in the form

$\displaystyle R_{\mu\nu}-\frac{1}{2}g_{\mu\nu}R=-\frac{8\pi G}{c^4}\tilde{T}_{\mu\nu}. \ \ \ \ \ (7)$

By (5) we have

$\displaystyle \tilde{T}_{\mu\nu}=T_{\mu\nu}+\frac{c^4}{8\pi G}\nabla_{\mu}\Phi_{\nu},$

which stands for all energy and momentum including the visible and the invisible matter and energy, and by (6) it is conserved. Thus, the equations (7) are rewritten as

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

with the energy-momentum conservation law given by

$\displaystyle \nabla^\mu\left[ \frac{8\pi G}{c^4}T_{\mu\nu} + \nabla_{\mu}\Phi_{\nu} \right] =0. \ \ \ \ \ (9)$

By our mathematical theory of variation with div-free constraints, we obtain that (8) are just the variation equations of ${L_{EH}}$ with the div-free constraint:

$\displaystyle (\delta L_{EH}(g_{\mu\nu}), X)= \frac{d}{d\lambda}\Big|_{\lambda = 0} L_{EH}(g_{\mu\nu}+ \lambda X_{\mu\nu}) = 0\quad \forall X=\{X_{\mu\nu}\} \text{ with } \nabla^\mu X_{\mu\nu}=0. \ \ \ \ \ (10)$

The term ${\nabla_\mu\Phi_\nu}$ does not correspond to any Lagrangian action density, and is the direct consequence of energy-momentum conservation constraint of the variation element ${X}$ in (10).

We have shown that it is the duality between the attracting gravitational field ${\{ g_{\mu\nu}\}}$ and the repulsive dual field ${\{\Phi_\mu\}}$ in (8), and their nonlinear interaction that give rise to gravity, and in particular the gravitational effect of dark energy and dark matter.

### 2. Principle of Interaction Dynamics

Motivated mainly by presence of dark energy and dark matter as discussed above and by the spontaneous symmetry breaking mechanism to be discussed in the next section, we postulated in [3] a basic principle, PID, for all fundamental interactions/forces of Nature. PID takes the variation of the action of interaction(s) under energy-momentum conservation constraint, and can be stated as follows:

#### Principle of Interaction Dynamics (PID)

1) For all physical interactions there are Lagrangian actions

$\displaystyle L(g,A,\psi )=\int_M\mathcal{L}(g_{\mu\nu},A,\psi )\sqrt{-g}dx, \ \ \ \ \ (11)$

which satisfy the invariance of general relativity, Lorentz invariance, gauge invariance and the gauge representation invariance.

2) The states ${(g,A,\psi )}$ are the extremum points of ${L}$ with the ${\text{ div}_A}$-free constraint, which satisfy

$\displaystyle \frac{\delta}{\delta g_{\mu\nu}}L(g,A,\psi ) = (\nabla_{\mu}+\alpha_bA^b_{\mu})\Phi_{\nu},$

$\displaystyle \frac{\delta}{\delta A^a_{\mu}}L(g,A,\psi) = (\nabla_{\mu}+\beta^a_bA^b_{\mu})\varphi^a,$

$\displaystyle \frac{\delta}{\delta\psi}L(g,A,\psi ) = 0.$

Here the terms on the right-hand side are due to PID, ${\Phi_\nu}$ stands for the dual vector graviton, and ${\varphi^a}$ stand for dual scalar particles for other interactions.

The mathematical foundations of this principle is based on the div${_A}$ free constraint variation and the related orthogonal decomposition theorems obtained in Articles [1, 3] listed in Books and Articles.

### 3. PID-Induced Higgs Mechanism

Higgs mechanism plays a crucial role in the electroweak theory and the standard model of particle physics. As another important motivation for postulating PID, we examine the key idea of the PID-induced Higgs mechanism. In comparison with the classical Higgs mechanism, the PID-induced mechanism is based on first principles, and is more natural and much simpler.

For simplicity, we demonstrate the essence of this new mechanism using ${U(1)}$ gauge theory, and we refer interested readers to Article [4] in Books and Articles for the ${SU(2)}$ gauge theory of weak interactions, and to later our future posts on spontaneous symmetry breaking.

The Yang-Mills action density for the ${U(1)}$ gauge field ${A_\mu}$ is in the form

$\displaystyle \mathcal{L}_{YM}=-\frac{1}{4}g^{\mu\alpha}g^{\nu\beta}(\partial_{\mu}A_{\nu}-\partial_{\nu}A_{\mu})(\partial_{\alpha}A_{\beta}-\partial_{\beta}A_{\alpha})+ \bar{\psi}(i\gamma^{\mu}D_{\mu}-m)\psi , \ \ \ \ \ (12)$

where ${\psi}$ is the Dirac spinor, ${g^{\mu\nu}}$ is the Minkowski metric, and

$\displaystyle D_{\mu}\psi =(\partial_{\mu}+igA_{\mu})\psi . \ \ \ \ \ (13)$

It is clear that the action (12) is invariant under the following ${U(1)}$ gauge transformation

$\displaystyle \psi\rightarrow e^{i\theta}\psi ,\ \ \ \ A_{\mu}\rightarrow A_{\mu}-\frac{1}{g}\partial_{\mu}\theta . \ \ \ \ \ (14)$

Based on PID, the variation equations of the Yang-Mills action (12) with the ${\text{ div}_A}$-free constraint are

$\displaystyle \partial^{\nu}(\partial_{\nu}A_{\mu}-\partial_{\mu}A_{\nu})-gJ_{\mu}= \left[\partial_{\mu}-\frac{1}{4}\left(\frac{m_Hc}{\hbar}\right)^2x_{\mu}+\alpha A_{\mu}\right]\phi, \ \ \ \ \ (15)$

$\displaystyle \left(i\gamma^{\mu}D_{\mu}-\frac{m_fc}{\hbar}\right)\psi =0, \ \ \ \ \ (16)$

where ${\phi}$ is a scalar Higgs field, ${-\frac{1}{4}\left(\frac{m_H c}{\hbar}\right)^2x_{\mu}}$ is the mass potential of ${\phi}$, and is also regarded as the interacting range of ${\phi}$.

Let ${\phi_0=\rho}$ be a nonzero ground state of ${\phi}$, then for the translation

$\displaystyle \phi =\tilde{\phi}+\rho ,\ \ \ \ A_{\mu}=\tilde{A}_{\mu},\ \ \ \ \psi =\tilde{\psi},$

then (15) becomes

$\displaystyle \partial^{\nu}(\partial_{\nu}\tilde{A}_{\mu}-\partial_{\mu}\tilde{A}_{\nu})-\left(\frac{m_0c}{\hbar}\right)^2\tilde{A}_{\mu}-g\tilde{J}_{\mu}=\left[\partial_{\mu} -\frac{1}{4}\left(\frac{m_Hc}{\hbar}\right)^2x_{\mu}+\alpha \tilde{A}_{\mu}\right]\tilde{\phi}, \ \ \ \ \ (17)$

where ${\left({m_0c}/{\hbar}\right)^2=\alpha\rho}$. Thus the mass

$\displaystyle m_0=\frac{\hbar}{c}\sqrt{\alpha \rho}$

is generated in (17) as the Yang-Mills action takes the ${\text{ div}_A}$-free constraint variation. Moreover, when we take divergence on both sides of (17), and by

$\displaystyle \partial^{\mu}\partial^{\nu}(\partial_{\nu}\tilde{A}_{\mu}-\partial_{\mu}\tilde{A}_{\nu})=0,\ \ \ \ \partial^{\mu}\tilde{J}_{\mu}=0,$

we derive the following field equation for the Higgs field ${\tilde{\phi}}$:

$\displaystyle \partial^{\mu}\partial_{\mu}\tilde{\phi}-\left(\frac{m_Hc}{\hbar}\right)^2\tilde{\phi}=-\alpha\partial^{\mu}(\tilde{A}_{\mu}\tilde{\phi})+ \frac{1}{4}\left(\frac{m_Hc}{\hbar}\right)^2x_{\mu}\partial^{\mu}\tilde{\phi}. \ \ \ \ \ (18)$

This equation (18) is the Higgs field equation with mass ${m_H}$ for the Higgs bosonic particle ${\tilde{\phi}}$.

### 4. Other Physical Supports of PID

Other physical supports for PID include the Ginzburg-Landau model of superconductivity and the non well-posedness of both the classical Einstein field equations with matter field and the classical ${SU(N)}$ non-abelian gauge field equations with nonzero Dirac spinors. Here we demonstrate the classical Einstein gravitational field equations (2) form an over-determined system and may lead to non well-posedness.

It is known that the metric of central gravitational field takes the form

$\displaystyle ds^2=c^2g_{00}dt^2+g_{11}dr^2+r^2(d\theta^2+\sin^2\theta d\varphi^2), \ \ \ \ \ (19)$

where ${(ct,r,\theta ,\varphi )}$ is the spheric coordinates, and the metric ${g_{\mu\nu}}$ can be expressed as

$\displaystyle \begin{array}{ll} g_{00}=-e^u&(u=u(r)),\\ g_{11}=e^v&(v=v(r)),\\ g_{22}=r^2,\\ g_{33}=r^2\sin^2\theta ,\\ g_{\mu\nu}=0,&\text{ for}\ \mu\neq\nu . \end{array} \ \ \ \ \ (20)$

The presence of dark matter or the influence of microwave background radiation implies that the energy-momentum tensors can be approximatively written as

$\displaystyle T_{\mu\nu}=\left(\begin{matrix} -g_{00}\rho&0\\ 0&0\end{matrix}\right), \ \ \ \ \ (21)$

where ${\rho}$ is the energy density, a constant.

For the metric (20), the nonzero Ricci tensors are

$\displaystyle \begin{array}{lcl} \displaystyle R_{00} &= &-e^{\mu -\nu}\left[\frac{u^{\prime\prime}}{2}+\frac{u^{\prime}}{r}+\frac{u^{\prime}}{4}(u^{\prime}-v^{\prime})\right], \\ \displaystyle R_{11} &= &\frac{u^{\prime\prime}}{2}-\frac{v^{\prime}}{r}+\frac{u^{\prime}}{4}(u^{\prime}-v^{\prime}), \\ \displaystyle R_{22} & = &e^{-v}\left[1-e^v+\frac{r}{2}(u^{\prime}-v^{\prime})\right],\\ \displaystyle R_{33} &= &\sin^2\theta R_{22}. \end{array} \ \ \ \ \ (22)$

On the other hand, the Einstein equations (2) can be equivalently written as

$\displaystyle R_{\mu\nu}=-\dfrac{8\pi G}{c^4}(T_{\mu\nu}-\dfrac{1}{2}g_{\mu\nu}T), \ \ \ \ \ (23)$

and by (21), ${T}$ is as

$\displaystyle T=g^{\mu\nu}T_{\mu\nu}=g^{00}T_{00}=-\rho .$

Thus, the Einstein field equations for the spherically symmetric gravitation fields (23) are in the form

$\displaystyle \begin{array}{lcl} \displaystyle R_{00} &= &\frac{4\pi G}{c^4}g_{00}\rho , \\ \displaystyle R_{11} & = & -\frac{4\pi G}{c^4}g_{11}\rho ,\\ \displaystyle R_{22} & = & -\frac{4\pi G}{c^4}g_{22}\rho ,\\ \displaystyle D^{\mu}T_{\mu\nu} & = &0. \end{array} \ \ \ \ \ (24)$

Now, we show that the equations (24) have no solutions. In fact, we derive from ${D^{\mu}T_{\mu\nu}=0}$ that

$\displaystyle \Gamma^0_{10}T_{00}=\frac{1}{2}u^{\prime}\rho =0.$

Hence ${u^{\prime}=0}$. Then by (22) we have

$\displaystyle R_{00}=0,$

which is a contradiction to the first equation of (24). Therefore the equations (24) have no solutions.

However, if we consider this example by using the PID-induced gravitational field equations (8) and (9), then the problem has a solution; see Article [1] in Books and Articles. In summary, the simple example presented here offers another strong physical support for PID.