content_image/374049.jpg content_image/374050.jpg content_image/374055.jpg
The exact Schwarzschild metric, which describes a black hole or the exterior of any spherically symmetric body, is given in the same coordinates as Eq.(9) by 24 

<img src='content_image/374049.jpg'>

This expression is correct only in matter-free space outside the body. At large distances from the body it is approximately 

<img src='content_image/374050.jpg'>

which agrees with the Newtonian limit Eq.(9). If the body is small enough so that r = GM/ 2 c 2 lies outside of it then the coefficient of the time term in Eq.(11) vanishes, which means that the proper time ds is zero, and a clock at that position would appear to a distant observer to stop. This radius defines the surface of the black hole. Light cannot escape from the surface since it undergoes a red shift to zero frequency. A black hole of typical stellar mass is about a km in radius. 3 

A minor caveat concerning Eq.(11) is in order. The coordinate r used in Eq.(11) is called the isotropic radial coordinate and is not the same as that usually used for the Schwarzschild metric, the Schwarschild radial coordinate. We will not use the Schwarzschild radial coordinate here (or even define it) but only note that it is asymptotically equal to the isotropic coordinate r in Eq.(11) but differs from it at the black hole surface by a factor of 4; that is the black hole surface is at radius 2 GM/c 2 in the Schwarzschild coordinate; 2 GM/c 2 is widely known as the Schwarzschild radius. 24 

It is generally believed that if enough mass M is squeezed into a roughly spherical volume of size about r ≈ GM/c 2 then it must collapse to form a black hole, regardless of internal pressure or other opposing forces; however if the mass is needle or pancake shaped the question of collapse is not yet clearly settled. 25,26 

In summary the lesson to take away from the above paragraphs is that spacetime dis- tortion is a measure of the gravitational field. Specifically, since the line element gives the physical distance between nearby spacetime points or events, such distances are given roughly by Eq.(10). This corresponds to a fractional distortion given by 

<img src='content_image/374055.jpg'>

