Schwarzschild-de Sitter Metric and Inertial Beltrami Coordinates

Under consideration of coordinate conditions, we get the Schwarzschild-Beltrami-de Sitter (S-BdS) metric solution of the Einstein field equations with a cosmological constant $Λ$. A brief review to the de Sitter invariant special relativity (dS-SR), and de Sitter general relativity (dS-GR, or GR with a $Λ$) is presented. The Beltrami metric $B_{μν}$ provides inertial reference frame for the dS-spacetime. By examining the Schwarzschild-de Sitter (S-dS) metric $g_{μν}^{(M)}$ existed in literatures since 1918, we find that the existed S-dS metric $g_{μν}^{(M)}$ describes some mixing effects of gravity and inertial-force, instead of a pure gravity effect arisen from "solar mass" $M$ in dS-GR. In this paper, we solve the vacuum Einstein equation of dS-GR, with the requirement of gravity-free metric $g_{μν}^{(M)}|_{M\rightarrow 0}=B_{μν}$. In this way we find S-BdS solution of dS-GR, written in inertial Beltrami coordinates. This is a new form of S-dS metric. Its physical meaning and possible applications are discussed.

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