On the Spectral Shift of Light Rays Emitted from a Pulsating Object

Abstract

As a sequel to previous papers on a bouncing or pulsating relativistic fluid sphere, we discuss the spectral shift z of light rays which are emitted from the surface of the pulsating sphere and to be received by a distant observer at rest in the exterior Schwarzschild region. The relation between z and the observer's proper time 7: is numerically analyzed for various models corresponding to different sets of four parameters, (M, e, 51> 52), which specify the total mass, the degree of non-homogeneity of the density distribution, the minimum and maximum sizes of the pulsating object, respectively. The relation between the period-density and the adiabatic exponent is also examined in connection with Eddington's pulsation theory of variable stars. In previous papers,l),2 l ,*) one of the authors (I-LN.) has studied a class of models for the gravitational collapse with pressure gradient, but without flow.' The first modePl is an extended version of Buchdahl's3) static fluid energy sphere resembling the Emden poly trope of index;) to the non-static case and it is a bouncing model free from any singularity \vhose spatial curvature is zero, i.e. I~ O. Of the. remain ing two models,2) one is a pulsating model whose spatial metric is of a positive constant curvature (k = 1) and the other a bouncing model such as k = -l. In [IJ, we have also discussed the spectral shift z of light rays emitted from the boundary surface of the collapsing object 'with k c:=: 0 and to be received by a distant observer \vhich is at rest in the exterior Sc1nvarzschild region. The formula for z is an extended version of the one 4 ) for the Oppenheimer­ Snyder modeP) and it permits us in principle to derive the expression for z as a function of the observer's proper time r.**) From the standpoint of studying the z versus r relation, however, the pul­ sating model (k = 1) is more interesting than the bouncing model (k = 0), because we may regard the former as a simple model for a pulsating star with finite amplitude whose gravitational field strength is so strong that the Newtonian treatment is no longer valid. In this connection, it would not be useless to