Gravity and Strings

84 A perturbative introduction to general relativity
This is an equation in the constant coefficients a, b, c, d,....Tosolve it, we first observe
that all the terms with the structure h∂∂∂h on the l.h.s. must vanish because they do not
occur on the r.h.s. Then, we also impose the vanishing of all the terms with the structure
∂h(∂∂h) β on the l.h.s. for the same reason. Finally, we identify the terms with structures
∂ β h(∂∂h) and ∂h β (∂∂h) on both sides of the above equation. The result can be expressed
in terms of two parameters x and y, which are left undetermined:
a =− 1
, b = 2 − y, c =−1, d = 1 − y, e = y, 2 f =−1,
g =−1− x, i = 1, j =−1 + x, k = 1
, l =−1 − x,
4 2
m = 1
2
, n = 1
2
, p =−1,
q = y, r = x.
4
(3.198)
On substituting these into the general expression for L (1) µσ and collecting all the terms
proportional to the two parameters x and y,weobtain
L (1) µσ = L (1)
GR µσ + total derivatives,
L (1)
GR µσ =−1 2∂µh νρ ∂σ hνρ − ∂ ν h ρ µ∂νhρσ + ∂ ν h ρ (µ|∂ρhν|σ)
+ 2∂ ν h ρ (µ∂σ)hνρ − ∂(µhσ) ν ∂νh − ∂ ν hµσ ∂ρh ρ ν
− ∂νh ν (µ∂σ)h + ∂νhµσ ∂ ν h + 1
2 ∂µh∂σ h + ηµσ LFP,
(3.199)
an unambiguous, unique, answer (up to total derivatives), which leads to a Lagrangian
L (1) that is invariant to first order in χ under the gauge transformations Eq. (3.181). The
equations of motion are fully determined and the gravitational energy–momentum tensor
is the piece of these equations of motion that is proportional to χ,given by Eq. (3.196), or,
more explicitly, by
t (0) µσ 1
GR = 2∂µ hλδ∂ σ h λδ + ∂λhδ µ ∂ λ h δσ − ∂λhδ µ ∂ δ h λσ + ∂λh µσ ∂δh δλ
− 2∂ (µ h σ) δ∂λh λδ − 1
2∂λh µσ ∂ λ h + ∂ (µ h σ)λ ∂λh
+ η µσ − 3
4∂αhβγ∂ α h βγ + 1
2∂αhβγ∂ β h αγ + ∂λh λα ∂δh δ α
− ∂λh λα ∂αh + 1
4∂λh∂ λ h
+ h αβ
∂α∂βh µσ − 2∂α∂ (µ h σ) β + ∂ µ ∂ σ hαβ + 2η (σ α ˆD µ) β(h)
− 1
2 ηµ αη σ β ˆD ρ ρ(h) − 1
2 ηαβD µσ (h) − η µσ ˆDαβ(h)
. (3.200)
This is clearly the energy–momentum tensor we were looking for. It is related to the
Rosenfeld energy–momentum tensor Eq. (3.190) by
t (0) µσ (0)
GR − (t can µσ + D ρµ (h)hρ σ + ∂ρ ρµσ ) ≡ ∂ρ ρµσ
GR−Ros ,
ρµσ σ [ρ µ]ν λδ
GR−Ros = ∂ν η η h hλδ + 2η ν[ρ h µ] λh λσ − 2η σ [ρ h µ]λ hλ ν
(3.201)
+ η σ [ρ h µ]ν h + η ν[µ h ρ]σ h − 1
2ην[µ η ρ]σ h 2 .
Summarizing: the Noether procedure allows us to find corrections to the free Fierz–Pauli
theory order by order in the parameter χ, making it self-consistent to that given order.