The Sehwarzschild Space-Time in the Background of the Flat Robertson-Walker Space-Time

Abstract

The Schwarzschild space-time is well known in describing the gravitational field of an object in an otherwise empty universe. The Schwarzschild space-time was derived by Karl Schwarzschild ( 1916) considering the merger of the Schwarzschild space-time with the Lorentz metric as the boundary (!)_ However, the Loremtz metric cannot be used in investigations of non empty large scale space-times, the whole universe being one such case. Thus, the cosmologists use the Robertson-Walker space-times, in describing the universe (2. -'i. As a result it becomes necessary to investigate the gravitational field of an object in the background of the Robertson-Walker space-time, We have studied the merger of the isotropic Schwarzschild space-time with the flat Robertson-Walker space-time. In this scenario, the flat Robertson-Walker space-time was considered for simplicity. The expressions for the radial coordinates r11 and rJl at the merger of the flat Robertson-Walker space-time and the isotropic Schwarzschild space-time were derived in terms of the scale factor R(t) and a constant R* and found to be given by An analytic expression for the time coordinate ( t) of the Schwarzschild space-time was obtained in the case of the de-Sitter universe, l = 2T0 In[- 1 !R' _l where To is the reciprocal of the Hubble constant (2'. 2~ I( - Jf?(t) J Schwarzschild Flat Robertson-Walker space-time space-time Figure: The radial coordinates and the time coordinates of the Schwarzschild space-time and the t1at Robertson Walker space-time at the merger The derived expressions for the radial coordinates '~, and rJI imply that an object in the universe begins to communicate with the "outside world" after a particular time, before which r11 and rfl are negative. At this particular time, R(t) approaches the constant R* and r,, , rfl tend to infinity. It could be said that the object comes into existence as far as the rest of the universe is concerned at this particular instant. The values of r11 and rf.i decrease with increase of time. When the time coordinate of the Schwarzschild space-time tends to infinity, rfl achieves the value (;) , the value of the Schwarzschild radius in isotropic coordinates.