Finslerian Ressiner-Nordstrom spacetime

Abstract

We have obtained Finslerian Ressiner-Nordstrom solution where it is asymptotic to a Finsler spacetime with constant flag curvature while $r\rightarrow\infty$. The covariant derivative of modified Einstein tensor in Finslerian gravitational field equation for this solution is conserved. The symmetry of the special Finslerian Ressiner-Nordstrom spacetime, namely, Finsler spacetime with constant flag curvature, has been investigated. It admits four independent Killing vectors. The Finslerian Ressiner-Nordstrom solution differs from Ressiner-Nordstrom metric only in two dimensional subspace. And our solution requires that its two dimensional subspace have constant flag curvature. We have obtained eigenfunction of Finslerian Laplace operator of "Finslerian sphere", namely, a special subspace with positive constant flag curvature. The eigenfunction is of the form $\bar{Y}_l^m=Y_l^m+\epsilon^2(C_{l+2}^m Y_{l+2}^m+C_{l-2}^m Y_{l-2}^m)$ in powers of Finslerian parameter $\epsilon$, where $C_{l+2}^m$ and $C_{l-2}^m$ are constant. However, the eigenvalue depends on both $l$ and $m$. The eigenvalues corresponded to $Y_1^0$ remain the same with Riemannian Laplace operator and the eigenvalues corresponded to $Y_1^{\pm1}$ are different. This fact just reflect the symmetry of "Finslerian sphere", which admits a $z$-axis rotational symmetry and breaks other symmetry of Riemannian sphere. The eigenfunction of Finslerian Laplace operator implies that monopolar and dipolar terms of multipole expansion of gravitational potential are unchanged and other multipole terms are changed.