Gravity and Strings

4.6 Teleparallelism 147
Lagrangian, and, therefore, gives the vacuum Einstein equations. 20 The Lagrangian turns
out to be invariant under not just global but also local Lorentz transformations and the only
degrees of freedom left are (we know it) those of the graviton. For general values of the
parameters, the analysis is more complicated and it is convenient to start by studying the
linear limit. To this end, we split the Vielbeins into their vacuum (Minkowski) values plus
perturbations. Working in Cartesian coordinates for simplicity, we write
For the inverse Vielbeins, we have
e a µ = δ a µ + A a µ. (4.169)
ea µ = δa µ − δb µ δa ν A b ν + O(A 2 ). (4.170)
To this order we can unambiguously trade curved and flat indices and the above formula
can be rewritten
ea µ = δa µ − A µ a + O(A 2 ), A µ a ≡ δb µ δa ν A b ν. (4.171)
The metric perturbation that we have called hµν in previous chapters is given by the symmetric
part of A at lowest order:
gµν = ηµν + hµν + O(A 2 ), hµν ≡ 2A(µν), bµν ≡ 2A[µν], Aµν ≡ δaµ A a µ.
(4.172)
With these definitions is straightforward to obtain, up to total derivatives, the linear limit of
action for the Lagrangian density Eq. (4.164):
ST[h, b] = d d
1
x
16 (2c1 + c2)∂µhνρ∂ µ h νρ − 1
16 (2c1 + c2 − c3)∂µhνρ∂ ν h µρ
− 1
8 c3∂µh∂νh νµ + 1
16 c3(∂h) 2 − 1
16 [4c1 + 2(c2 + c3)]∂µhνρ∂ ρ b νµ
+ 1
16 ∂µbνρ∂ µ b νρ − 1
16 (2c1 − 3c2 − c3)∂µbνρ∂ ρ b νµ
. (4.173)
The first four terms are familiar to us: up to coefficients, they are the same terms as those
that appear in the Fierz–Pauli Lagrangian Eq. (3.84). The last two terms are also well
known: up to coefficients, they are exactly those that appear in the Lagrangian of the Kalb–
Ramond 2-form field, which we still have not seen. The terms in the third line represent a
coupling (already at the linear level) between these two fields.
Now, it is clear that it is not possible to recover solutions of the vacuum Fierz–Pauli
theory if the coupling terms have a non-zero coefficient: a non-vanishing h field is a source
for a non-vanishing b field and vice-versa. Thus, the only theories which we expect to be
phenomenologically viable are those in the family
2c1 + c2 + c3 = 0. (4.174)
20 In fact, this theory is sometimes referred to as the teleparallel equivalent of GR.