We have obtained a Finslerian Reissner-Nordström solution where it is asymptotic to a Finsler spacetime with a constant flag curvature while $r\to \infty$. The covariant derivative of a modified Einstein tensor in a Finslerian gravitational field equation for this solution is conserved. The symmetry of the special Finslerian Reissner-Nordström spacetime, namely, Finsler spacetime with a constant flag curvature, has been investigated. It admits four independent Killing vectors. The Finslerian Reissner-Nordström solution differs from the Reissner-Nordström metric only in two-dimensional subspace, and our solution requires that its two-dimensional subspace has a constant flag curvature. We have obtained the eigenfunction of the Finslerian Laplacian operator of the “Finslerian sphere,” namely, a special subspace with a positive constant flag curvature. The eigenfunction is of the form ${\overline{Y}}_l^m=Y_l^m+{\epsilon }^2(C_{l+2}^m{Y}_{l+2}^m+C_{l-2}^m{Y}_{l-2}^m)$ in powers of the 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 corresponding to $Y_1^0$ remain the same with the Riemannian Laplacian operator and the eigenvalues corresponding to $Y_1^{\pm 1}$ are different. This fact just reflects the symmetry of the Finslerian sphere, which admits a $z$-axis rotational symmetry and breaks other symmetry of the Riemannian sphere. The eigenfunction of the Finslerian Laplacian operator implies that monopolar and dipolar terms of the multipole expansion of gravitational potential are unchanged and other multipole terms are changed.

• Received 25 May 2018

DOI:https://doi.org/10.1103/PhysRevD.98.084030