Introduction to String Theory and D–Branes - School of Natural ...

with ˜ Φ = Φ−Φ0, Looking at the part involving the Ricci scalar, we see that we have the form of the standard
Einstein–Hilbert action (i.e., we have removed the factor involving the dilaton Φ), with Newton’s constant
set by
κ ≡ κ0e Φ0 = (8πGN) 1/2 . (102)
The standard terminology to note here is that the action (96) written in terms of the original fields is
called the “string frame action”, while the action (101) is referred to as the “Einstein frame action”. It
is in the latter frame that one gives meaning to measuring quantities like gravitational mass–energy. It is
important to note the means, equation (99), to transform from the fields of one to another, depending upon
dimension.
2.14 A Quick Look at Geometry
Now that we are firmly in curved spacetime, it is probably a good idea to gather some concepts, language
and tools which will be useful to us in many places later on.
2.14.1 Working with the Local Tangent Frames
We can introduce “vielbeins” which locally diagonalize the metric 5 :
gµν(x) = ηabe a µ(x)e b ν(x).
The vielbeins form a basis for the tangent space at the point x, and orthonormality gives
e a µ (x)eµb (x) = η ab .
These are interesting objects, connecting curved and tangent space, and transforming appropriately under
the natural groups of each. It is a covariant vector under general coordinate transformations x → x ′ :
and a contravariant vector under local Lorentz:
e a µ → e ′a µ = ∂xν
∂x ′µ ea ν ,
e a µ (x) → e′a µ (x) = Λab(x)e b µ (x) ,
where Λ a b(x)Λ c d(x)ηac = ηbd defines Λ as being in the Lorentz group SO(1, D−1).
So we have the expected freedom to define our vielbein up to a local Lorentz transformation in the
tangent frame. In fact the condition Λ is a Lorentz transformation guarantees that the metric is invariant
under local Lorentz:
gµν = ηabe ′a
µe ′b
ν . (103)
Notice that we can naturally define a family of inverse vielbiens as well, by raising and lowering indices
in the obvious way, e µ a = ηabg µν e b ν. (We use the same symbol for the vielbien, but the index structure will
make it clear what we mean.) Clearly,
g µν = η ab e µ ae ν b , e µ
b ea µ = δ a b . (104)
In fact, the vielbien may be thought of as simply the matrix of coefficients of the transformation (that always
exists, by the Equivalence Principle) which finds a locally inertial frame ξ a (x) from the general coordinates
x µ at the point x = xo:
e a µ(x) = ∂ξa (x)
∂x µ
5 “Vielbein” means “many legs”, adapted from the German. In D = 4 it is called a “vierbein”. We shall offend the purists
henceforth and not capitalise nouns taken from the German language into physics, such as “ansatz”, “bremsstrahlung” and
“gedankenexperiment”.
28
x=xo
.