String Theory Demystified

52 String Theory Demystifi ed
The Energy-Momentum Tensor
Let’s quickly review a few things before getting started. Recall that the intrinsic
distance on the worldsheet can be determined using the induced metric h αβ . This is
given by
α β
ds h dσ dσ
(3.1)
2 = αβ
0 1
where σ = τ, σ = σ are the coordinates which parameterize points on the
worldsheet. A set of functions X µ ( σ, τ)
describe the shape of the worldsheet and
the motion of the string with respect to the background space-time, where
µ = 01 , , ..., D −1for
a D-dimensional space-time. To fi nd the dynamics of the
string, we can minimize the Polyakov action [Eq. (2.27)]:
S
P
T 2 αβ µ ν
=− ∫ d σ −det( h) h ∂αX ∂βX
η
(3.2)
µν
2
Minimizing SP (by minimizing the area of the worldsheet) gives us the equations
of motion for the X µ ( σ, τ),
and hence the dynamics of the string. In the quiz at the
end of Chap. 2 in Prob. 4, you were invited to show that the Polyakov and Nambu-
Goto actions were equivalent by considering the energy-momentum or stress-energy
tensor Tαβ which is given by
T
αβ
2 1 δS
=−
T −h
δh
P
αβ
(3.3)
In this book we’ll go mostly by the name energy-momentum tensor. In a nutshell,
the energy-momentum tensor describes the density and fl ux of energy and
momentum in space-time. You should be familiar with the basics of what T αβ is
from some exposure to or study of quantum fi eld theory, so we’re just going to go
with that and describe how it works in string theory. When working out the solution
to Prob. 4 in Chap. 2, you should have found that
( µν )
µ ν 1 ρσ µ ν
Tαβ =∂αX ∂βX ηµν − hαβh ∂ρX ∂σX
η
2
(3.4)
The fi rst property that we will establish for the energy-momentum tensor is that it
has zero trace. We can calculate the trace using the induced metric:
α αβ
Tr( T ) = T =
h T
αβ
α
αβ