4 2 D O C . 1 7 G R A V I T A T I O N A N D E L E C T R I C I T Y . (10b) By cyclically exchanging the indices μ, ν, and α twice, two further equations come about. The Γ can be calculated from these three equations similarly to the symmet- ric case. One obtains . (14) Equation (4) reduces to the first and third terms. If one then inserts the expression for from (14), one obtains . (15) Before we continue to consider (15), we develop equation (7). First, from (13) it follows that in the approximation we are interested in now we have . (16) Taking this into account, (7) becomes . (17) Now we insert the expressions given in (13) into (15) and obtain, taking (17) into account, (18) (19) Equations (18), which we know can be simplified by an appropriate choice of co- ordinates, are the same as those for the case when an electromagnetic field is ab- sent. Neither the electromagnetic equations (17) nor (19) contain the quantities which refer to the gravitational field. Therefore, to first approximation, both fields are independent of each other—in accord with experience. Equations (17) and (19) are almost entirely equivalent to the Maxwell equations for empty space. (17) is the first set of Maxwell’s equations. The expressions , ∂xα ∂gμν Γμα ν Γαν μ + + + 0 = α –Γμν -- - ∂xμ ∂gαν ∂xν ∂gμα ∂xα ∂gνμ· –+ ¹ ¨ ¸ = [p. 418] Γμν α 2 2gνμ ∂xα 2gαμ ∂xν∂xα 2gαν ∂xμ∂xα 2gαα ∂xμ∂xν + + 0 = gμν δμν γμν φμν = ∂xν ∂φμν 0= 2 2γμν ∂xα 2γμα ∂xν∂xα 2γνα ∂xμ∂xα 2γαα ∂xμ∂xν + + 0 = 2φμν ∂xα2 0. = γμν 2 ∂xα ∂φμν ∂xμ ∂φνα ∂xν ∂φαμ + +
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