4 3 6 D O C . 4 4 3 G E N E R A L R E L A T I V I T Y A N D M O T I O N . Thereby we retain only terms that can contribute something finite to the integral over an infinitely small sphere. Due to the above choice of coordinates, this must be substituted by , or, more precisely, by . We now multiply this by and integrate the result over the sphere’s surface. The first term yields something finite only for the second for the third for . The result of this integration is , where we sum over α running from 1 to 3. If this calculation is performed for all the terms of , one obtains . If one now integrates this expression over x4, between two limits where van- ishes, then the integral vanishes completely.1) For that reason, as in the case of the stationary field, the integral over the inner sheath M reduces to the integral over . From this it follows, exactly as above, that the motion of the singular point is characterized by a geodesic of the “outer” field . 1) Regarding the calculation it should be noted that the last term of line 2 and line 3 within the brackets of (15) vanish because they make no contribution of the character of at all to the inte- grand. gμν === gαβ ==== τν β ξ,τμ τμ α ξ,τμ -- - ∂γτα ∂xμ ---------- ∂γμα ∂xτ ---------- - ∂γτμ· ∂xα ---------¸ -–+ ¹ ¨ ξτ,μ r ----- α τ, μ α = = μ α, τ α = = τ μ = 8πm 3 ---------- -ξ,α α 4πm( ξ,α α ξ,4 4 ) B α r Sd ³Bα----- 16πmξ,4 4 = ξα r 2– αξτσ γμν
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