String Theory Demystified

258 String Theory Demystifi ed
The Holographic Principle and M-Theory
Now let’s make our description more quantitative. In the fi nal chapter of the book
we discuss stringy cosmology. There we will encounter a model of space-time that
has sprung out of string/M-theory that might in fact describe our actual universe.
That same model has a nice application in the topic of this chapter as well. The
model is a fi ve-dimensional AdS space. It can be described as follows.
We start with a fi ve-dimensional AdS space. In a nutshell, this is a four-
dimensional spatial ball and an infi nite time axis. The radius of the ball is 0≤ r < 1.
The radius of curvature is denoted by R, and we lump the remaining spatial
dimensions together into a unit three-sphere denoted by Ω. The metric which
describes the AdS is written as
ds
2
2
R
2 2 2 2 2
= [( 1+ r ) dt −4dr −4r
d ]
2 2
( 1−
r ) Ω
Note that there are different, equivalent ways to write this metric which you might
encounter elsewhere. AdS space has negative curvature and acts like a cavity of size
R with refl ecting walls. Light or objects and refl ect off the boundary and return to the
center (see “The Illusion of Gravity” by Juan Maldacena in Scientifi c American,
November 2005, for a nice popular level description of AdS).
For us, we are interested in superstring theory. The number of space-time
dimensions in superstring theory is D = 10. So the complete space is
AdS ⊗ S
5
where S5 is a unit fi ve-sphere containing the remaining dimensions from string theory.
If we denote the extra fi ve coordinates by y they are incorporated into our metric by
5
2
adding a term Rdy . We can imagine compactifying these dimensions to a very small
5
size so that they can be effectively ignored. So the universe can be effectively treated
as the fi ve-dimensional “bulk” which is the interior of the sphere and the boundary
which is the surface. The surface has three spatial dimensions and time.
In the M-theory picture, the world we know is in essence a “shadow” or hologram
living on the boundary of a larger dimensional universe. The physics is divided as
follows:
• The boundary conformal theory lives on the surface of the sphere at x = 1.
These are the particles and interactions of the standard model, plus any
supersymmetric extension of it.
• Gravity is everywhere.