Our theoretical understanding of the underlying structure has progressed from a point particles (local quantum fields) to strings. Why stop at the string? Shouldn't nature have quantas of energy which are structures that are two-dimensional (membrane), three-dimensional (3-brane) ... all the way to nine-dimensional?

The answer, found only in 1996, is that YES, nature indeed
has higher dimensional structures, generically called
Dirichlet p-branes, with p = 0,1,2,...,9. The reason they
are called Dirichlet is because they are structures on
which **open** strings can start and end (Dirichlet
boundary conditions).

Since the open string extends in a dimension orthogonal to the D-brane, if one confines oneself to the D-brane, the beginning and end of the open string appear as point like charges (like electrons, quarks and so on).

Hence, if we consider the 3-dimensional world that we see around us as a 3-brane embedded in a 10 dimensional spacetime, then point charges like the electron and quark are the end points of open strings.

D-branes can also emit and absorb not only open strings, but also closed strings as well as D-branes.

General Relativity predicts that a star of mass greater than three solar mass will undergo gravitational collapse and form a black hole. A black hole has all of its matter squeezed into a single point, called a singularity, and which is enclosed by a horizon. Particles crossing the horizon can never escape, and will eventually hit the singularity. The analogy of the horizon is the point of no return for a waterfall. |
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For a body of mass M, the radius of the horizon
R |
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A black hole in the presence of a quantum field emits radiation. The reason is the following. Due to quantum fluctuations, if a particle/antiparticle pair is created near the horizon, the particle may cross the horizon (and hence disappear forever), and due to momentum conservation, 'kick' the antiparticle away from the horizon. To an observer outside the black hole, it will appear as if the black hole is emitting radiation.

Amazingly enough, Hawking showed that the radiation from
a black hole is identical to the radiation from a
*black body* at **temperature** T_{H}
given by (hc)/(8 ^{2}kR_{s}).

As a black hole radiates, it becomes smaller (as it loses mass) and hence its temperature increases. Hence, the final result of Hawking radiation is a spectacular explosion when the black hole disappears.

The **entropy** of the black hole is given by the
famous area law S_{BH} = A/2Gh, where A is the
area of the horizon.

The black holes also satisfy the First and Second Laws
of Thermodynamics. Since Energy of a black hole is
mc^{2}, we have

Entropy always increases :dE = c

^{2}dM = T_{H}dS (first law)

dS >= 0 (second law) where S = S

_{BH}+S_{radiation}.

As the black hole radiates, its entropy goes to zero, but the total entropy always increases.

Why should strings have any connection with black holes?
In particular, the Schwarzchild radius is typically
R_{s} ~ 10km where the length of the string
l_{str} ~ 10^{-35}m. The density of
states measures the number of microstates which
correspond to say the total energy having a given
value. We have

p

_{string}~ e^{Mstr}, l_{str}~ 10^{-35}m.p

_{BH}~ e^{M2BH}, R_{s}~ 10km.

So how do we justify applying string theory to study the physics of a black hole?

For a excited state of a string specified by integer N, we have

HenceM

_{str}= sqrt (N)/l_{str}.

M

_{BH}/M_{str}= R_{s}c^{2}/2G * l_{str}/sqrt (N)

GR is valid only if curvature of space is much less than
1/l_{str}, i.e. the geometry of the space is
smooth and R_{s} is much greater than
l_{str}. Note that the length of the string is
sqrt (G)/g_{str} where g_{str} is the
string coupling constant.

For g_{str} -> 0, l_{str} can become
very large and we examine the case when
l_{str}~R_{S}~1km. For this case,
curvature of space ~ 1/l_{str} and black holes
need to be described by string theory.

It was shown in 1996 that a 5-dimensional super
symmetric black hole is a bound state of
10-dimensional string theory in which the extra 5
dimensions are curled into a five-dimensional torus
T^{5}. Consider spacetime to be a simple
product of a Minkowski five-dimensional spacetime
M_{5} with T^{5}, i.e. we take the
underlying manifold of superstring theory to be

M_{10} -> M_{5} x T^{5}

The black hole is composed of a 5-brane wrapped
Q_{5} number of times on T^{5}
together with a 1-brane wrapped Q_{1}
times. The 1-brane is moving with momentum N. It was
shown that

S_{BH} = 2sqrt
(Q_{1}Q_{5} N)

~ Area of black hole's horizon

This is an exact match with Hawking's result.

This result has been extended to four-dimensional black holes and the black body spectrum has also been derived.

It has also been shown that Hawking radiation consists of the emission and absorption of closed strings from the bound state of D-branes which comprise a black hole. |

Last updated: 06 March, 2000