By Erwin Schrödinger
Traditionally fascinating. an entire replica.
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Extra resources for Expanding Universes
Sample text
On null geoilesics A geodesic may be defined as giving a stationary value to the line integral of the interval, for fixed limits, dfds:0, ils2:gord,rod,rr. r:s, ilsz ' \lmJ d,s ds - "' if (I) (z) after performing the aariation with an a,rbitrary pd,ra,rneter one with s. dr,gr- d: constant. (3) X'rom the second equation (1) the constant is 1. ) ft follows that the first member of (3) and therefore the second member of the second equation (l) can vanish nowhere on our curYe. X'rom general theory the solutions of (2) exist for any initial values of q, d,rrf ils,and (3) is always a first integral.
Q (:const'ant')' a+z uAv o: ,_"n l"-\ So we have to take* \ . \A+zl. (40) (41) The planes (40), which trace the co-ordinate lines 7: constant on the hyperboloid (H) pass throughthe spatial origin O. They are the bundle through the asymptote n:0, A+z:0. Since * By a slightly more involved procedure g ca,n be determined directly and turns out to be an arbitrary function of the tuo argurttonts rl@ az) and u,*A,-2,. But the second is constant on (I{). I, 6] LEMATTBE,S FBAME the planes pa,ss through O, bhe co-ordinate lines are geodesics and, of course, time-like, as c&n be seen directly from (40); for the cosine of the angle between the z-axis and the normal to this plane is l"l , I JO+,A-P (42) for any not infinibe a.
Again (28) needs in general a matter tensor to support it,, but we are not concerned with it here. As in (18) we use the'quadratic' variational principle, which for brevity we write ,ilfl'd,r:0, (2s) or, a little more elaborately, ,l [ - ezoo Rz (#)' + R, ffi1d, : o (30) The a is not a variable, but its derivative with respect to t is an 'angular velocity'. As before we have the first integral GJ':, (31) The variations with respect to y,0, Q would here be a little more involved, but need not be carried out, because we see that, using again a parametet r'as in (21), we are faced with the minimum problem of the three-dimensional static hypersphere.