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Reprinted from T HE P HYSICAL RE V I E W , Vol. 110, No. 4, 965-967, May 15, 1958Printed inU. s. A.

Past-Future Asymmetry of the Gravitational Field of a Point Particle

SWIIItrirJs ofTcckdogy, B&oh,Nm Jusq, and Ah York Unhusily, Nm Yak, New York(Received January9,1958)

The analytic c x t e n s i o n of the Schwarzschild exterior solution is given ina closed form valid throughoutempty space-time and possessing no irregularities except that at the origin. The gravitational field of aspherical point particle is then seen not to be invariant under time reversal for any admissible choice of timecoordinate. TheSchwarzschild surfacer=2m is not a singularity but acts as a perfect unidirectional mem-brane: causal influences can cross it but only in one direction. The apparent violation of the principle ofsufficient reason seemssimilar to that which is associated with instabilities in other nonlinear phenomena.

I. INTRODUCTION (c) Invariance under the connected three-parameter

T0 define a gravitational universe we must give an groupanalytic manifold and an analytic quadratic formgc, of correct signature. 2o+&=zo

For the manifold nt associated with a universe con- R,: 1 & + 2= Qd- I , I%= 1, detr= 1. (1.3)taining one point particle, one takes all of 4-space(+c) less the line z*=O. (Greek indices=0, 1, 2, 3;Latin= 1, 2, 3.) We might subject the gravitational (d) Asymptotic to the Lorentz metricfieid g,,,(z) to the following requirements: g,,(r)---)$w, +x+ = l (1 4).

(a) The free space equation of Einstein:(e) Not extendable to the line &=O (true singu-

R,,(r)=0. (1.1) larity). Thisexcludes the trivial case g,,(r)~,,.(b) Invariance under the onc-parameter group (f) Invariant under the discrete group generated by

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