Nature's Laws

D (Dirichlet) - Branes

B. E. Baaquie and Marakani Srikant, Department of Physics, National University of Singapore D2-brane

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.

Black Holes

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.

Point of
						 no return

For a body of mass M, the radius of the horizon RS is given by 2GM/c2. For Msun, Rs is ~3km.

Geometry around a black hole

Hawking Radiation

Hawking radiation

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 TH given by (hc)/(8 2kRs).

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 SBH = 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 mc2, we have

Energy conservation :

dE = c2dM = THdS (first law)

Entropy always increases :

dS >= 0 (second law) where S = SBH+Sradiation.

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

Strings and Black Holes

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

pstring ~ eMstr, lstr ~ 10-35m.

pBH ~ eM2BH, Rs ~ 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

Mstr = sqrt (N)/lstr.


MBH/Mstr = Rsc2/2G * lstr/sqrt (N)

GR is valid only if curvature of space is much less than 1/lstr, i.e. the geometry of the space is smooth and Rs is much greater than lstr. Note that the length of the string is sqrt (G)/gstr where gstr is the string coupling constant.

For gstr -> 0, lstr can become very large and we examine the case when lstr~RS~1km. For this case, curvature of space ~ 1/lstr and black holes need to be described by string theory.

		 and black holes

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 T5. Consider spacetime to be a simple product of a Minkowski five-dimensional spacetime M5 with T5, i.e. we take the underlying manifold of superstring theory to be
M10 -> M5 x T5

The black hole is composed of a 5-brane wrapped Q5 number of times on T5 together with a 1-brane wrapped Q1 times. The 1-brane is moving with momentum N. It was shown that

SBH = 2sqrt (Q1Q5 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.

Hawking radiation
					 with strings

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

NUS Core Curriculum Nature's Laws Physics String Theory