Compact Non-Linear Gravito-Magnetic Fields as Dark Matter

2. Preliminaries

In Ref. [23], Ellis provides Maxwell’s equations in the 1+3 streamlined form as follows:

$$D^a{E}_a=-2{\omega }^a{H}_a+\varrho ,$$

(1)

$$D^a{H}_a=2{\omega }^a{E}_a,$$

(2)

$${\dot{E}}_{〈a〉}-\nabla \times H_a={\displaystyle \frac{2}{3}}\mathrm{ϴ}{E}_a+{\sigma }_{ab}{E}^b-{\left[\omega ,E\right]}_a+{\left[\dot{u,H}\right]}_a-j_a,$$

(3)

$${\dot{H}}_{〈a〉}+\nabla \times E_a={\displaystyle \frac{2}{3}}\mathrm{ϴ}{H}_a+{\sigma }_{ab}{H}^b-{\left[\omega ,H\right]}_a-{\left[\dot{u,E}\right]}_a,$$

(4)

where $\varrho ={-J}_a{u}^a$ is the electric charge density, $u^a$ the four velocity, $\mathrm{ϴ}=D^a{u}_a$, the shear ${\sigma }_{ab}=D_{{‹}_a{u}_b›}$, ${\omega }_a=-{\displaystyle \frac{1}{2}}\nabla \times u_a$ is the vorticity and ${\dot{u}}_a={\dot{u}}_{〈a〉}$ is the four acceleration.

The above operators obey the following covariant identities:

$$\dot{\left(D_{a}f\right)=D_a\dot{f}-{\displaystyle \frac{1}{3}}}\mathrm{ϴ}{D}_{a}f+{\dot{u}}_a\dot{f}-{{\sigma }_a}^b{D}_{b}f-{\left[\omega ,Df\right]}_a+u_a{\dot{u}}^b{D}_{b}f,$$

(5)

$$\nabla \times D_{a}f=2\dot{f}{\omega }_a,$$

(6)

$$D^a{\left[V,W\right]}_a=W^a\nabla \times V_a-V^a\nabla \times W_a,$$

(7)

And

$$D^a{\left[A,B\right]}_a=B^{ab}\nabla \times A_{ab}-A^{ab}\nabla \times B_{ab},$$

(8)

The Maxwellean equivalent of GR is based on the correspondence between the Weyl tensor $C_{abcd}$and the electromagnetic tensor $F_{ab}$ for any $u_a$. Thus

$$E_{ab}=C_{abcd}{u}^c{u}^d=E_{〈ab〉},H_{ab}={*C}_{abcd}{u}^c{u}^d=H_{〈ab〉}$$

(9)

These gravito-electric/magnetic spatial tensors are in principle physically measurable in the frames of comoving observers, and together they are equivalent to the space-time Weyl tensor.

The gravito-electromagnetic version of the electromagnetic tensor is

$${C_{ab}}^{cd}=4\left\{u_{[a}{u}^{[c}+{h_{[a}}^{[c}\right\}{E_{b]}}^{d]}+2{\epsilon }_{abe}{u}^{[c}{H}^{d]e}+2u_{[a}{H}_{b]e}{\epsilon }^{cde}.$$

(10)

In the 1 + 3 covariant approach to GR, the source of gravity is a fluid in which the energy density , pressure and the gravito-electromagnetic tensors are the fundamental quantities and not the metric. These quantities are governed by the Bianchi identities and the Ricci identities for $u^a$. Einstein’s equations incorporated through the algebraic definition of the Ricci tensor $R_{ab}$ in terms of the energy-momentum tensor $T_{ab}$. The Bianchi identities are

$${\nabla }^a{C}_{abcd}={\nabla }_{[a}\left(-R_{b]c}+{\displaystyle \frac{1}{6}}{Rg}_{b]c}\right).$$

(11)

Here $R={R_a}^a$ and $R_{ab}=\kappa \left(T_{ab}-{\displaystyle \frac{1}{2}}{T_c}^c{g}_{ab}\right)$ where κ is the Einstein constant. The gravitational equivalents of Maxwell’s equations are therefore

$$D^a{E}_{ab}=-3{\omega }^a{H}_{ab}+{\displaystyle \frac{1}{3}}{D}_a\rho +{\left[\sigma ,H\right]}_a,$$

(12)

$$D^a{H}_{ab}=3{\omega }^a{E}_{ab}-{\displaystyle \frac{1}{3}}{\left(\rho +p\right)\omega }_a\rho -{\left[\sigma ,E\right]}_a,$$

(13)

$${\dot{E}}_{〈ab〉}-\nabla \times H_{ab}=\mathrm{ϴ}{E}_{ab}+{\sigma }_{c⟨a}{E_{b⟩}}^c-{\omega }^c{\epsilon }_{cd\left(a\right.}{E_{\left.b\right)}}^d+2{\dot{u}}^c{\omega }^c{\epsilon }_{cd\left(a\right.}{H_{\left.b\right)}}^d-{\displaystyle \frac{1}{2}}\left(\rho +p\right){\sigma }_{ab},$$

(14)

$${\dot{H}}_{〈ab〉}-\nabla \times E_{ab}=-\mathrm{ϴ}{H}_{ab}+{\sigma }_{c⟨a}{H_{b⟩}}^c-{\omega }^c{\epsilon }_{cd\left(a\right.}{H_{\left.b\right)}}^d-2{\dot{u}}^c{\omega }^c{\epsilon }_{cd\left(a\right.}{E_{\left.b\right)}}^d.$$

(15)

These are the full nonlinear gravito-electromagnetic equations in covariant form. In both electromagnetism and gravito-electromagnetism vorticity produces source terms however, gravity has additional sources from a tensor coupling of the shear to the field. Furthermore, in electromagnetism, there is no magnetic charge sources, but the gravito- magnetic field $H_{ab}$ has the source $\left(\rho +p\right){\omega }_a$. Since $\rho +p$ is the relativistic inertial mass-energy density, $\left(\rho +p\right){\omega }_a$ is the ’angular momentum density’, which we identify as a gravito- magnetic ’charge’ density.

3. Modification of General Relativity

The electromagnetic analogy suggests a further interesting interpretation of the vorticity. In flat spacetime, relative to inertial observers, the electric and magnetic vectors may be written as

$$\overrightarrow{E}=\overrightarrow{\nabla }V-{\partial }_t\overrightarrow{\alpha },\overrightarrow{H}=\overrightarrow{\nabla }\times \overrightarrow{\alpha },$$

(16)

where, $V$ is the electric scalar potential and $\overrightarrow{\alpha }$ is the gravitomagnetic vector potential. We express the gravitomagnetic vector potential as $\overrightarrow{\alpha }=\overrightarrow{v}={\mu \left(t\right)\overrightarrow{H}}_0\times \overrightarrow{r}$. Here, the magnitude of ${\overrightarrow{H}}_0$ is the Hubble constant, $\overrightarrow{r}$the radius vector, $\overrightarrow{v}$ the tangential velocity and $\mu \left(t\right)$ the time dependent relative gravito-magnetic permeability. The curl of the tangential velocity for a given $r$ results in an angular velocity about the z direction of

$${\omega =\mu \left(t\right)H}_0.$$

(17)

Einstein’s field equations (EFE) need to be modified such that they include the contribution by the magnetic charge. We now proceed with a similar approach as in [24]. As stated earlier, in the 1 + 3 covariant approaches to GR, the source of gravity is a fluid in which the energy density, pressure and the gravito-electromagnetic tensors are the fundamental quantities and not the metric. This allows us to intuitively make the assumption that the gravito-magnetic field is a compact and expanding region of spacetime that can be described as a Ricci soliton of the form $R_{\mu \nu }-{\displaystyle \frac{1}{2}}R{g}_{\mu \nu }=k{g}_{\mu \nu }$ with the Ricci tensor of the compact manifold expressed as a compact energy-momentum tensor field $R_{ab}=\kappa \left(T_{ab}-{\displaystyle \frac{1}{2}}T{g}_{ab}\right)=k{g}_{ab}$. Here, $\kappa$ is the Einstein gravitational constant $k$ is a constant proportional to the cosmological constant. These expressions can only be self-consistent if and only if the traces are $T=0$ and $R=0$. Here, the compact stress momentum tensor field $\left(T_{ab}-{\displaystyle \frac{1}{2}}T{g}_{ab}\right)$ is considered as a barotropic fluid. A system consisting of gravitationally bound baryonic matter can assume this barotropic fluid like morphology when modeled at a sufficiently large scale. For example, the solar system, star and galactic clusters can be considered as fluid at sufficiently large scales relative to the intrinsic scale of the gravitationally bound system. This approach has been successfully applied at cosmic scales to model the universe using the FLRW metric. Here the approach is applied at much smaller clustering of gravitationally bound astrophysical systems. By modelling the system as barotropic, it satisfies the condition of being homogeneous and isotropic in the volume it occupies. Einstein’s field equations (EFE) are then modified such that they include the GEM field modelled as 4-D compact Einstein manifold and thus expressing the EFE as follows:

$$G_{ab}-k{g}_{ab}+\Lambda g_{ab}={\kappa T}_{ab}.$$

(18)

Here, $k={\mu }^2\Lambda =3{\left({\displaystyle \frac{{\mu H}_0}{c}}\right)}^2$ with a trace $R=0$. Here, ${\mu H}_0$ is the angular velocity of a Ricci soliton of reduced radius $r=$ ${\displaystyle \frac{c}{{2\pi \mu H}_0}}$. This implies a rotational velocity on the surface of the soliton of ${v=H}_{0}r={\displaystyle \frac{c}{2\pi \mu }}$. This circular motion on the surface of the Ricci soliton arises from the traceless condition $T=0$. This condition demands marginally stable or zero energy orbits on the surface in which the kinetic energy is equal to the gravitational potential energy.

Equation(18) must therefore satisfy a metric solution of the form

$$-c^2{\tau }^2=-c^2{t}^2+\mu {\left(t\right)}^{2}d{\Sigma }^2$$

(19)

Where $d{\Sigma }^2={\displaystyle \frac{d{r}^2}{1-k{r}^2}}+r^2\left(d{\theta }^2+{sin}^2\theta d{\varphi }^2\right)$ in reduced-circumference polar coordinates.

The analytic solutions to Equation (19) in which the energy momentum tensor is isotropic and homogeneous are the following equations:

$${\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2+{\displaystyle \frac{{\mu }^2\Lambda c^2}{3{\mu }^2}}-{\displaystyle \frac{\Lambda c^2}{3}}={\displaystyle \frac{\kappa c^4}{3}}\rho \to {\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2+{\displaystyle \frac{2\Lambda c^2}{3}}={\displaystyle \frac{\kappa c^4}{3}}\rho ,$$

(20)

$$2{\displaystyle \frac{\ddot{\mu }}{\mu }}+{\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2+{\displaystyle \frac{{\mu }^2\Lambda c^2}{{\mu }^2}}-\Lambda c^2=-\kappa c^{2}p\to 2{\displaystyle \frac{\ddot{\mu }}{\mu }}+{\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2=-\kappa c^{2}p,$$

(21)

$$R=-6\left[{\displaystyle \frac{\ddot{\mu }}{\mu }}+{\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2+\Lambda c^2\right]=0$$

(22)

From Equation (22) we obtain

$${\displaystyle \frac{\ddot{\mu }}{\mu }}=-{\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2-\Lambda c^2.$$

(23)

Substituting Equation (23) into Equation (21) yields

$${\left({\displaystyle \frac{\dot{\mu }}{\mu }}\right)}^2=\kappa c^{2}p-2\Lambda c^2.$$

(24)

Substituting Equation (24) into Equation (20) and taking$p=-\rho c^2$ yields

$$\Lambda =-\kappa c^2\rho$$

(25)

Given a measured value of the cosmological constant of $1.1\times {10}^{-52}{m}^{-2}$ we compute a critical baryonic matter density for the Ricci soliton of ${\rho }_{cb}={\rho }_{DE}=5.6\times {10}^{-27}kg{m}^{-3}$.

The Ricci soliton therefore expands exponentially when $p=-\rho c^2$ with a scale factor $\mu \left(t\right)={\mu }_o{e}^{H_{0}t}$. Here $H_o=\sqrt{\kappa c^{2}p-2\Lambda c^2}=\sqrt{\kappa c^4\rho }$. This expansion occurs when the covariant flatness condition of Equation (19) is satisfied. Under such conditions the centripetal acceleration on the surface of the soliton is given by the expression

$${\displaystyle \frac{v^2}{r}}={\displaystyle \frac{{\left(H_{0}r\right)}^2}{r}}={H_0}^{2}r={\displaystyle \frac{{H_0}^{2}c}{2\pi \mu H_0}}={\displaystyle \frac{H_{0}c}{2\pi \mu }}$$

(26)

In the comoving reference frame $\mu =1$ which implies a scale invariant acceleration

$${\mu }_0={\displaystyle \frac{H_{0}c}{2\pi }}$$

(27)

Under such conditions the radius is computed from Equation (26) as

$$r={\displaystyle \frac{2\pi v^2}{H_{0}c}}$$

(28)

Substituting $r$ in the expression ${\displaystyle \frac{GM\left(r\right)}{r}}=v^2$ we obtain

$$v^4={\displaystyle \frac{GM\left(r\right)H_{0}c}{2\pi }}$$

(29)

This is the baryonic Tully Fisher relation and $a_0={\displaystyle \frac{H_{0}c}{2\pi }}\cong 1.1\times {10}^{-10}m/s^2$ is the empirically observed Milgrom’s acceleration constant [60,61]. The theoretically computed value is in agreement with observations.

Since $r=r_0{e}^{H_{0}t}$ then the velocity evolves with the scale factor as follows:

$$v^4={\displaystyle \frac{GM\left(r\right)H_{0}c}{2\pi }}{e}^{4H_{0}t}$$

(30)

From the above equations, we also obtain the following equations of galactic and galactic cluster evolution

$$r={{\displaystyle \frac{1}{H_0}}e}^{(H_{0}t)}{({GM}_B(r){\displaystyle \frac{H_0}{2\pi }}c)}^{{\displaystyle \frac{1}{4}}}={\displaystyle \frac{v_n}{H_0}}$$

(31)

$$v={\displaystyle \frac{dr}{dt}}=e^{(H_{0}t)}{({GM}_B(r){\displaystyle \frac{H_0}{2\pi }}c)}^{{\displaystyle \frac{1}{4}}}=H_{0}r$$

(32)

$$a={\displaystyle \frac{dv}{dt}}={H_{0}e}^{(H_{0}t)}{({GM}_B{\displaystyle \frac{H_0}{2\pi }}c)}^{{\displaystyle \frac{1}{4}}}$$

(33)

4. Discussion and Concluding Remarks

The gravito-electromagnetism formulation of Roy Maartens and Bruce Bassett draws parallels between the equations governing electromagnetism and gravity. This analogy helps in developing a unified theoretical framework that simplifies the understanding of gravitational phenomena using well-established electromagnetic principles. The concept of super-energy and the super-Poynting vector offers deeper insights into the dynamics of spacetime. In this paper we have taken insights from the Roy Maartens and Bruce Bassett formulation to model dark matter as a compact gravito-magneto field with the same properties as a Ricci soliton which was first discussed in Ref. [24]. This model of dark matter offers a unique perspective compared to traditional dark matter models. It provides a classical, geometric approach to dark matter, contrasting with the particle-based explanations of traditional models. This could lead to new insights and potentially broaden our understanding of dark matter and dark energy.