The Klein-Gordon equation in anti-de Sitter spacetime - Seminario ...

274 K. Yagdjian and A. Galstianwhere M is the curved mass, M ∈ C, and x ∈ R n , t ∈ R. Results of [16] (by meansof the time inversion transformation t →−t) provide us with the fundamental solutionE =E(x,t;x 0 ,t 0 ),E tt − e 2t ∆E + M 2 E = δ(x−x 0 ,t− t 0 ),with support in the “forward light cone” D + (x 0 ,t 0 ), x 0 ∈R n , t 0 ∈R, and for the fundamentalsolution with support in the “backward light cone” D − (x 0 ,t 0 ), x 0 ∈R n , t 0 ∈R,defined as follows{}(3) D ± (x 0 ,t 0 ) := (x,t)∈R n+1 ;|x−x 0 |≤±(e t 0− e t ) .In fact, any intersection of D − (x 0 ,t 0 ) with the hyperplane t = const < t 0 determines theso-called dependence domain for the point(x 0 ,t 0 ), while the intersection of D + (x 0 ,t 0 )with the hyperplane t = const > t 0 is the so-called domain of influence of the point(x 0 ,t 0 ). The equation (2) is non-invariant with respect to time inversion. Moreover, thedependence domain is wider than any given ball if time const > t 0 is sufficiently large,while the domain of influence is permanently, for all time const < t 0 , in the ball of theradius e t 0. In fact, the representation formulas obtained in [16] for the solution of theCauchy problem in the de Sitter spacetime cannot be applied to the solutions of theCauchy problem for the equation in the anti-de Sitter spacetime. The present paper isaimed to fill up that gap.Define for t 0 ∈R in the domain D + (x 0 ,t 0 )∪D − (x 0 ,t 0 ) the function(4)E(x,t;x 0 ,t 0 ) = (4e t 0+t ) iM( (e t + e t 0) 2 −(x−x 0 ) 2) − 1 2 −iM× F( 12 + iM, 1 2 + 0− e t ) 2 −(x−x 0 ) 2 )iM;1;(et (e t 0 + e t ) 2 −(x−x 0 ) 2 ,where F ( a,b;c;ζ ) is the hypergeometric function (See, e.g., [2].). In (4) we use thenotation x 2 =|x| 2 for x∈R n . Let E(x,t;0,t 0 ) be function (4), and setE ± (x,t;0,t 0 ) :={E(x,t;0,t 0 ) in D ± (0,t 0 ),0 elsewhere.Since the function E = E(x,t;0,t 0 ) is smooth in D ± (0,t 0 ) and is locally integrable, itfollows thatE + (x,t;0, t 0 ) andE − (x,t;0, t 0 ) are distributions whose supports are inD + (0,t 0 ) and D − (0,t 0 ), respectively. In order to make the present paper self-containedwe make the transformation t →−t in Theorem 0.1 [16] and introduce the next result.THEOREM 1 ([16]). Suppose that M ∈C. The distributionsE + (x,t;0,t 0 ) andE − (x,t;0,t 0 ) are the fundamental solutions for the operatorS = ∂ 2 t − e2t ∂ 2 x + M2 relativeto the point(0,t 0 ), that isSE ± (x,t;0,t 0 )=δ(x,t− t 0 ), or∂ 2∂t 2E ±(x,t;0,t 0 )−e 2t ∂ 2∂x 2E ±(x,t;0,t 0 )+M 2 E ± (x,t;0,t 0 )=δ(x,t− t 0 ).