By Valeri P. Frolov, Igor D. Novikov (auth.)

It isn't an exaggeration to claim that some of the most intriguing predictions of Einstein's concept of gravitation is that there may well exist "black holes": putative gadgets whose gravitational fields are so robust that no actual our bodies or indications can separate from in their pull and break out. The facts that black holes do exist, and an research in their houses, could have a value going some distance past astrophysics. certainly, what's concerned isn't just the invention of another whether tremendous awesome, astro­ actual item, yet a try of the correctness of our knowing of the houses of house and time in tremendous robust gravitational fields. Theoretical examine into the homes of black holes, and into the prospective corol­ laries of the speculation that they exist, has been performed with unique vigour because the starting of the 1970's. as well as these particular gains of black holes which are vital for the translation in their attainable astrophysical manifestations, the speculation has printed a few unforeseen features of actual interactions regarding black holes. by way of the center of the 1980's a pretty certain knowing were completed of the homes of the black holes, their attainable astrophysical manifestations, and the specifics of a number of the actual tactics concerned. although a totally trustworthy detection of a black gap had now not but been made at the moment, numerous items between these scrutinized by way of astrophysicists have been regarded as robust applicants to be proven as being black holes.

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For radial motion in the Schwarzschild metric E = mc2 / / (1 where A = 1 — — rg /r, and i dr/(cdt). 7) CHAPTER 2. SPHERICALLY SYMMETRIC BLACK HOLES 20 If the falling body approaches rg , the physical velocity v ds/dr constantly increases: —+ c as r —+ r g • By the clock of the distant observer, the velocity dx/dt tends to zero as r r g , as in the case of the photon. This fact reflects the slowing down of time as r —+ rg . What is the time required for a body falling from a point r = r 1 to reach the gravitational radius rg (by the clock of the distant observer)?

Nevertheless, such a "system" of test photons proves to be convenient. One needs to remember, though, that y (usually called an advanced time) is a lightlike coordinate (neither spatial nor temporal). For a second coordinate, we can choose the familiar coordinate r. 12) is regular on r = rg . Indeed, the coefficient with dy 2 vanishes on rg , but the presence of the term 2 dy dr ensures that the metric (and hence the coordinate system) remains non-degenerate. 4. Coordinate lines of constant y, representing ingoing radial null rays, are plotted on a 45-degree slant, just as they would be in flat spacetime.

9 clearly illustrates the general situation. Recall that the r = 0 line is spacelike so that a reference frame exists in which all events on this line are simultaneous. 9 and 2A0) that first the singularity r 0 existed in vacuum, and then the matter of the spherical cloud began to expand from the singularity. These events are not connected by a timelike interval. 9). Note that: no particle coming from the spatial infinity (or from any region of r > rg ) can penetrate the expanding T+-region. Such regions of spacetime are called white holes [Novikov (1964b), Ne'eman (1965) 1 .

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