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Black holes in string theory
We can think of the metric (10.2) or (10.7) as being that which arises
following the spherically symmetric collapse of a star with mass M ~ 1.52m0' It is instructive to consider a series of light Hashes emitted near the centre
of the collapsing star, which is assumed to be made of transparent matter. In
the early stages, the density of the star is low, the wavefront of the light will be
approximately spherical and its area proportional to the square of the time elapsed
since the emission of the flash. However, the gravitational attraction of the stellar
matter through which the light is passing deHects neighbouring rays towards each
other, reducing the rate at which they are diverging from each other. In other
words, the gravitational effect of the matter is to focus the light and to reduce the
area of the wavefront from what it otherwise would have been. In the early stages,
the wavefront continues to increase in area, crossing the surface of the collapsing
star and eventually reaching infinity. As the collapse continues, the matter density
increases and so does the focusing effect, until a critical wavefront emerges from
the surface of the star with zero divergence. Outside of the star, this wavefront
will remain constant and will be the surface r = 2M discussed earlier whose
spacetime evolution is the event horizon. It is the boundary of the spacetime
region from which it is not possible to escape to infinity. It is generated by null
geodesics which have no future endpoint but which do have past endpoints (at the
emission of the flash.) The divergence of these null geodesic generators is positive
during the collapse phase and zero in the final time-independent state. The area
of a two-dimensional section of the event horizon increases monotonically from
zero to the final value (10.8). Subsequent Hashes will be focused so much by the
stronger gravitational focusing that their rays begin to converge and the area of
the wavefront decreases.
Now consider what happens when a thin spherical shell of matter of mass 8M
collapses from infinity at some later time and hits the singularity at r = O. During
the collapse, the metric is spherically symmetric but, of course, time-dependent.
Afterwards, it will have the form (10.2) or (10.7) but with M replaced by M +8M.
Since 8M is necessarily positive, the area of the two-dimensional section of the
event horizon must increase:
8AH = 321rM8M > O.
(10.12)
These results illustrate general results for black holes that are true even
without spherical symmetry. The focusing or converging effect that follows
from the fact that the gravitational mass is always positive can be described
quantitatively using the positive-definiteness of the energy density. Consider a set
of null geodesics, and let'I' = dx'" /dv be a null tangent vector to these geodesics,
where v is an affine parameter for the geodesic. At each point, we can define two
unit spacelike vectors a'" and b'" that are orthogonal to each other and to ' I' . It is
convenient to define the complex vectors
m' " = 1
_'"
1
- ../2(al' + ib"') and m == ../2(a'" - ib"').
(10.13)
Black holes in string theory
We can think of the metric (10.2) or (10.7) as being that which arises
following the spherically symmetric collapse of a star with mass M ~ 1.52m0' It is instructive to consider a series of light Hashes emitted near the centre
of the collapsing star, which is assumed to be made of transparent matter. In
the early stages, the density of the star is low, the wavefront of the light will be
approximately spherical and its area proportional to the square of the time elapsed
since the emission of the flash. However, the gravitational attraction of the stellar
matter through which the light is passing deHects neighbouring rays towards each
other, reducing the rate at which they are diverging from each other. In other
words, the gravitational effect of the matter is to focus the light and to reduce the
area of the wavefront from what it otherwise would have been. In the early stages,
the wavefront continues to increase in area, crossing the surface of the collapsing
star and eventually reaching infinity. As the collapse continues, the matter density
increases and so does the focusing effect, until a critical wavefront emerges from
the surface of the star with zero divergence. Outside of the star, this wavefront
will remain constant and will be the surface r = 2M discussed earlier whose
spacetime evolution is the event horizon. It is the boundary of the spacetime
region from which it is not possible to escape to infinity. It is generated by null
geodesics which have no future endpoint but which do have past endpoints (at the
emission of the flash.) The divergence of these null geodesic generators is positive
during the collapse phase and zero in the final time-independent state. The area
of a two-dimensional section of the event horizon increases monotonically from
zero to the final value (10.8). Subsequent Hashes will be focused so much by the
stronger gravitational focusing that their rays begin to converge and the area of
the wavefront decreases.
Now consider what happens when a thin spherical shell of matter of mass 8M
collapses from infinity at some later time and hits the singularity at r = O. During
the collapse, the metric is spherically symmetric but, of course, time-dependent.
Afterwards, it will have the form (10.2) or (10.7) but with M replaced by M +8M.
Since 8M is necessarily positive, the area of the two-dimensional section of the
event horizon must increase:
8AH = 321rM8M > O.
(10.12)
These results illustrate general results for black holes that are true even
without spherical symmetry. The focusing or converging effect that follows
from the fact that the gravitational mass is always positive can be described
quantitatively using the positive-definiteness of the energy density. Consider a set
of null geodesics, and let'I' = dx'" /dv be a null tangent vector to these geodesics,
where v is an affine parameter for the geodesic. At each point, we can define two
unit spacelike vectors a'" and b'" that are orthogonal to each other and to ' I' . It is
convenient to define the complex vectors
m' " = 1
_'"
1
- ../2(al' + ib"') and m == ../2(a'" - ib"').
(10.13)