A First Course in String Theory - Chapter 03 - ism & Gravitation in Various Dimen.pdf

Chapter 3
ism and
Gravitation in Various
Dimensions
String theory is promising because it includes Maxwell electrodynamics and
its nonlinear cousins, as well as gravitation. We review the relativistic for-
mulation of electrodynamics and show how it facilitates the definition of elec-
trodynamics in other dimensions. We give a brief description of Einstein’s
gravity, and use the Newtonian limit to discuss the relation between Planck’s
length and the gravitational constant in various dimensions. We study the
effect pactification on the gravitational constant, and explain how large
extra dimensions could escape detection.
Classical Electrodynamics
Unlike Newtonian mechanics, classical electrodynamics is a relativistic the-
ory. In fact, Einstein was led by electrodynamic considerations to formulate
the special theory of relativity. ism has a particularly elegant
formulation where the relativistic character of the theory is manifest. This
relativistic formulation allows a natural extension of ic theory
to higher dimensions. Before we discuss the relativistic formulation, how-
ever, we will review the equations of Maxwell. These equations describe the
dynamics of electric and ic fields.
Although most undergraduate and graduate courses in ism
53
54 CHAPTER 3. ISM AND GRAVITATION
nowadays use the international system of units (SI units), the Gaussian sys-
tem of units is far more appropriate for discussions involving relativity. In
Gaussian units, Maxwell’s equations take the following form:
1 ∂B

∇×E
= −, ()
c ∂t
∇·B
=0, ()
∇·E
=4πρ, ()
4π 1 ∂E

∇×B
=
j + . ()
c c ∂t
The above equations imply that in Gaussian units E
and B
are measured
with the same units. The first two equations are the source-free Maxwell
equations. The second two involve sources: the charge density ρ, with units
of charge per unit volume, and the current density
j, with units of current per
unit area. The Lorentz force law, wh