By Harrison D.M.

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**Additional resources for About Mass-Energy Equivalence**

**Example text**

45) 0. 41) in the form and put m = k, i = j to get which is equivalent to Eq. 45). 1 Write down the equations of a null geodesic in the space· time given by the line-element ds 2 = dt 2 - 2ex1 dt dx2 - (dxl)2 + ! l (x2)2 ez"l] 2 constant, where ~ dxz - [(x2)2exl ~ + 2e-"'1) dt ~ is an affine parameter. ') ds = 0 ds ds gik - - where s is the proper length parameter, gives the usual geodesics. 3 Two metrics g 111 ;k and g 121 ;k on a given space-time give the same geodesic curves. Show that their respective Christoffel symbols r il)i kl and r IZ)i kl satisfy a relation of the form where Vk are the components of a vector.

35) This process of symmetrization can be extended to more than two indices. Thus for any tensor of rank n, we define sik s

1 at P 2, say. 8) If we consider a family of cunes, all starting from P1 and ending at P 2 we get different values of s(P1 , P 2 1 ~). We shall look for that curve for which s(P1 , P 2 1')is stationary. 9) This i'l the classh,;al Euler-Lagrange problem and the solution h 43 SPACE-TIME CURVATURE given by the Euler-Lagrange equation. Writing Bq. 8) in the form s 1 =f 0 F(xi, xi) dA. , the Euler-Lagrange equation is d (oF)_ oF_ 0 dA. 11) = (gik x 1 x~<)i we have oF --: ox' = -1 gik X. k • -oF. 12) x" x' = 0.

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