STATIC CYLINDRICALLY SYMMETRIC EXACT SOLUTION OF FIELD EQUATION 038

In this section the static cylindrically symmetric gravitomagnetic field and the the evolution of wavefuction of photon in gravitomagnetic charge are considered. The form of linear element describing the static cylindrically symmetric gravitomagnetic field is assumed to be \EQN{0}{1}{}{0,}{\RD{\CELL{ds^{2}=dx^{0}-dx^{2}-dy^{2}-dz^{2}+2g_{0x}(y)dx^{0}dx+2g_{0y}(x)dx^{0}dy,}}{2}{20}{eq20}{}}where we assume that the gravitomagnetic potentials $g_{0x}$ and $g_{0y}$ are functions of $y$ and $x$, respectively. Thus we obtain all the nonvanishing Christoffel symbols as follows:\EQN{1}{1}{}{0,}{\RD{\CELL{\Gamma _{0,xy} &=&\Gamma _{0,yx}=\frac{1}{2}(\frac{\partial g_{0x}}{\partial y}+\frac{\partial g_{0y}}{\partial x}),\quad }}{0}{}{}{}\RD{\CELL{\Gamma _{x,0y} &=&\Gamma _{x,y0}=-\Gamma _{y,0x}=-\Gamma _{y,x0}=\frac{1}{2}(\frac{\partial g_{0x}}{\partial y}-\frac{\partial g_{0y}}{\partial x}).\bigskip }}{2}{21}{}{}} \ Since the field equation of gravitomagnetic matter is the antisymmetric equation, we might as well take into account a simple case of the following equation\EQN{0}{1}{}{0,}{\RD{\CELL{\epsilon ^{0\alpha \beta \gamma }R_{\quad \alpha \beta \gamma }^{0}=\rho _{M}}}{2}{22}{eq22}{}}with $\rho _{M}$ being the parameter associared with the coupling constants and gravitomagnetic charge. It is therefore apparent that Eq. (\QTSN{ref}{eq22}) agrees with Eq. (\QTSN{ref}{eq13}). Hence, the solution of the former equation also satisfies the latter. For the reason of the completely antisymmetric property of the Levi-Civita tensor, the contravariant indices $\alpha ,\beta ,\gamma $ should be taken to be $x,y,z$, respectively, namely,\EQN{0}{1}{}{0,}{\RD{\CELL{\epsilon ^{0\alpha \beta \gamma }R_{\quad \alpha \beta \gamma }^{0}=\epsilon ^{0xyz}(R_{\quad xyz}^{0}+R_{\quad zxy}^{0}+R_{\quad yzx}^{0}).}}{2}{23}{eq23}{}}There exist the products of two Christoffel symbols, i.e., $g^{\sigma \tau }(\Gamma _{\tau ,\alpha \gamma }\Gamma _{\lambda ,\sigma \beta }-\Gamma _{\tau ,\alpha \beta }\Gamma _{\lambda ,\sigma \gamma })$ in the definition of the Riemann curvature, $R_{\lambda \alpha \beta \gamma }$. Apparently, the products of two Christoffel symbols (the nonlinear terms of field equation) contain the total indices, $x,y,z$ of three-dimensional space cordinate and therefore vanish, in the light of the fact that the Christoffel symbol with index $z$ is vanishing in terms of Eq. (\QTSN{ref}{eq21}). In view of the above discussion, one can concluds that Eq. (\QTSN{ref}{eq22}) can be exactly reduced to a linear equation.