Gravity and Strings

6.3 The positive-energy theorem 181
with a matter energy–momentum tensor satisfying the dominant energy condition
Tmatter µν kµnν ≥ 0, ∀ nµ, kµ non-spacelike, (6.46)
is always non-negative, vanishing only for flat spacetime. This result was first obtained
by Schoen and Yau in [830]. A new proof based on spinor techniques inspired by SUGRA
was afterwards presented by Witten in [958] and subsequently by Nester in [719] and Israel
and Nester in [596]. Previously, the positivity of mass in SUGRA and GR (as the bosonic
part of N = 1 SUGRA) had been established in [310, 480]. Here we are going to use this
Witten–Nester–Israel (WNI) technique because it can be generalized to more complicated
cases and because it has a strong relation to supergravity that we will also use later on in
Chapter 13.
The positive-energy theorem is a very important result associated with the cosmiccensorship
conjecture: inthe gravitational collapse of a star, the gravitational binding
energy, which is negative, grows in absolute value. If the process continued indefinitely,
the total energy of the collapsing star would become negative. However, according to the
positive-mass theorem, this cannot happen and we expect a black-hole horizon to appear
before the mass becomes negative.
The WNI technique starts with the construction of the Nester 2-form E µν .Inthis case
(pure d = 4 gravity; the extension to higher dimensions is straightforward) it is simply
E µν (ɛ) =+ i
2 ¯ɛγ µνρ ∇ρɛ + c.c., (6.47)
where ɛ is a commuting Dirac spinor. The Nester form is manifestly real. Then, we define
the integral I ,
I (ɛ) =
⋆
E(ɛ), (6.48)
∂
where is a three-dimensional spacelike hypersurface (for instance, a constant time slice)
whose boundary ∂ is a 2-sphere at infinity S2 ∞ . Observe that the Nester form can be
rewritten in the form
E µν (ɛ) =+i¯ɛγ µνρ
∇ρɛ +∇ρ − i
2 ¯ɛγ µνρ
ɛ , (6.49)
and only the first term contributes to I .
The proof has two parts.
1. Prove that, for suitably chosen spinors ɛ and Tmatter µν satisfying the dominant energy
condition, I (ɛ) ≥ 0.
2. Relate I (ɛ) to conserved charges.
1. We use Stokes’ theorem
⋆
I (ɛ) = E(ɛ) =
∂
d ⋆ E(ɛ) =− 1
2
d 3
ν −i∇µ ¯ɛγ µνρ ∇ρɛ − i ¯ɛγ µνρ ∇µ∇ρɛ ,
(6.50)