Gravity and Strings

6.3 The positive-energy theorem 183
These two terms are manifestly positive. The second one vanishes if we use spinors
satisfying the Witten condition
γ i ei µ ∇µɛ = 0. (6.61)
Thus, we have proven that, if the dominant energy condition is satisfied and we use
spinors satisfying the Witten condition, I (ɛ) is non-negative.
2. We rewrite I (ɛ) as follows:
I (ɛ) = 1
d 2
2 µνɛ µνρσ ¯ɛγ5γσ ∇ρɛ, (6.62)
∂
and expand the integrand around the vacuum ¯gµν (Minkowski spacetime) to which the
solution asymptotically tends. We also impose on the chosen spinors that they admit the
expansion
where r →∞at spatial infinity and
1
ɛ = ɛ0 + O ,
r
(6.63)
¯∇µɛ0 = 0. (6.64)
A spinor satisfying this condition in N = 1 SUGRA is a Killing spinor of the solution ¯gµν.
Since ∇µ = ¯∇µ − 1
4 ωµ ab γab and the integral is taken at spatial infinity,
I (ɛ) =− 1
8
d 2 µνɛ µνρσ ¯ɛ0γ5γσ ∇ρɛ0 = 1
4
d 2 ρµν,γ
µν −3g αβωρ αβ
k0 γ
= 1
4 E(k0), (6.65)
where we have used Eq. (6.34) and the fact that
k a 0 = i ¯ɛ0γ a ɛ0
(6.66)
is, trivially, a Killing vector of the vacuum ¯gµν. When it is timelike, k0 is the generator of
translations in time and E(k0) is just the mass.
This proves that M ≥ 0 and M = 0for Minkowski spacetime.
The above relation between Killing spinors and Killing vectors is quite generic and, with
minor adaptations, is true in most SUGRAs. Just as the Killing vectors of a metric constitute
a Lie algebra that generates its isometry group, the Killing spinors and Killing vectors
of a solution of a SUGRA theory, which may involve other fields apart from the metric,
constitute a superalgebra that generates a supergroup that leaves the solution invariant. The
simplest case is Minkowski spacetime, whose invariance supergroup is the super-Poincaré
one.