Beyond the Equivalence Principle: Gravitational Magnetic Monopoles

Abstract

We review the hypothesis on the existence of gravitational magnetic monopoles (H-poles for short) defined by analogy with Dirac’s hypothesis on magnetic monopoles in electrodynamics. These hypothetic dual particles violate the equivalence principle and are accelerated by a gravitational field. We propose an expression for the gravitational force exerted upon an H-pole. According to GR, ordinary matter (which we call E-poles) follows geodesics in a background metric $g_{\mu\nu}$. The dual H-poles follows geodesics in an effective metric $\hat{g}_{\mu\nu}$.

Appendix

MATHEMATICAL COMPENDIUM

The Riemann curvature tensor can be decomposed into its irreducible parts by the relation

$$R_{\alpha\beta\mu\nu}=W_{\alpha\beta\mu\nu}+M_{\alpha\beta\mu\nu}-\frac{1}{6}Rg_{\alpha\beta\mu\nu},$$

where $W_{\alpha\beta\mu\nu}$ is the Weyl conformal tensor,

$$2M_{\alpha\beta\mu\nu}=R_{\alpha\mu}g_{\beta\nu}+R_{\beta\nu}g_{\alpha\mu}-R_{\alpha\nu}g_{\beta\mu}-R_{\beta\mu}g_{\alpha\nu},$$

and $g_{\alpha\beta\mu\nu}=g_{\alpha\mu}g_{\beta\nu}-g_{\alpha\nu}g_{\beta\mu}$. The duality operation for an arbitrary antisymmetric tensor $F_{\mu\nu}$ is defined by

$$F^{*}_{\mu\nu}\equiv\frac{1}{2}\eta_{\mu\nu\alpha\beta}F^{\alpha\beta},$$

with

$$\eta_{\alpha\beta\mu\nu}=\sqrt{-g}\varepsilon_{\alpha\beta\mu\nu},$$

$g$ being the determinant of $g_{\mu\nu}$, and $\varepsilon_{\alpha\beta\mu\nu}$ is the Levi-Civita totally antisymmetric quantity. We define the electric vector $E^{\mu}$ and magnetic vector $H^{\mu}$ by setting

$$E_{\alpha}=F_{\alpha\mu}v^{\mu},$$

$$H_{\alpha}=F^{\ast}_{\alpha\mu}v^{\mu}.$$

The Weyl tensor has ten independent components and can also be separated by an arbitrary observer endowed with the four velocity $v^{\mu}$ into its electric ($E_{\alpha\beta}$) and magnetic ($H_{\alpha\beta}$) tensor parts, that is,

$$E_{\alpha\beta}=W_{\alpha\mu\beta\nu}v^{\mu}v^{\nu},$$

$$H_{\alpha\beta}=W^{\ast}_{\alpha\mu\beta\nu}v^{\mu}v^{\nu}.$$

Thus the electric and magnetic tensors are symmetric, traceless and orthogonal to the observer’s velocity:

$$E_{\mu\nu}=E_{\nu\mu},\quad E_{\mu\nu}v^{\mu}=0,\quad{\textrm{and}}\quad E_{\mu\nu}g^{\mu\nu}=0,$$

$$H_{\mu\nu}=H_{\nu\mu},\quad H_{\mu\nu}v^{\mu}=0,\quad{\textrm{and}}\quad H_{\mu\nu}g^{\mu\nu}=0.$$

The kinematic parameters

$$\text{Shear:}\quad\sigma_{\mu\nu}=\frac{1}{2}h^{\alpha}{}_{(\mu}h^{\beta}{}_{\nu)}-\frac{1}{3}\theta h_{\mu\nu};$$

$$\text{Vorticity:}\quad\omega_{\mu\nu}=\frac{1}{2}h^{\alpha}{}_{[\mu}h^{\beta}{}_{\nu]};$$

$$\text{Expansion factor:}\quad\theta=v^{\alpha}{}_{;\alpha};$$

$$\text{Projection:}\quad h_{\mu\nu}=g_{\mu\nu}-v_{\mu}v_{\nu}.$$