global lorentzian geometry ii - univie.ac.at

GLOBAL LORENTZIAN GEOMETRY II

JAMES D.E. GRANT

Contents

Synopsis 2

Part 1. Witten’s proof of the Positive Energy/Mass Theorem 31. Orthonormal frames and connections 32. Spinors 62.1. Spin representations 112.2. The Lichnerowicz theorem 123. The positive energy/mass theorem 143.1. Modifications required to prove the Positive Energy Theorem [Non-examinable] 25

Part 2. Appendices [Non-examinable] 27Appendix A. Justification for definition of E and P 27References 29Index 30

2 JAMES D.E. GRANT

Lecture 1Synopsis

The first part of the course gives an introduction to the Positive Mass/Energy theorem inGeneral Relativity. In physical terms, the Positive Energy Theorem states that if a Lorentzianmanifold obeys the Einstein field equations with “sensible” matter content (i.e. matter thatsatisfies the Dominant Energy Condition) then the energy of the system is necessarily non-negative.The original proof of this result is due to Schoen and Yau (see [25, 23, 27, 26]), but we follow alater argument due to Witten [29]. The original Positive Energy Theorem is a result concerningfour-dimensional Lorentzian manifolds. We will, however, prove a simpler result (the RiemannianPositive Mass Theorem) that holds on Riemannian spin manifolds of any dimension (and non-spinmanifolds of dimension n ≤ 7). We then indicate the modifications required to prove the PositiveEnergy Theorem. Our approach is based on the papers of Parker and Taubes [19], Bartnik [6], andthe Appendix of Lee and Parker’s paper on the Yamabe problem [17]. A useful summary of theapproaches to the Positive Energy Theorem around 1983 is the article by Choquet-Bruhat [10].

In the second part of the course, we outline Schoen’s use of the Positive Mass Theorem to solvethe Yamabe problem in dimensions n ≤ 5. In the Yamabe problem, we are given a closed (i.e.compact without boundary), connected Riemannian manifold (M,g) and we ask whether we canfind a conformally rescaled metric g = f2g, where f is a smooth, positive function on M , withthe property that the scalar curvature of the metric g is constant. In dimension 2, all metrics arelocally conformally flat, and the existence of such a metric is then essentially a corollary of theUniformisation theorem. In particular, if M = S2 then there exists a metric with scalar curvature2 (hence the sectional curvature equals 1), if M = T 2 then there exists a flat metric with scalarcurvature 0, and on a higher genus Riemann surface, there exists a metric with scalar curvature −2that we may construct as the quotient under a discrete group of isometries of the upper half-planewith its standard hyperbolic metric. In higher dimensions, Yamabe considered the problem offinding a metric of constant scalar curvature within a conformal equivalence class as a first steptowards proving the Poincare conjecture. Although he claimed to have solved the problem in hisoriginal paper [30], there is an error in his paper (as pointed out by Trudinger [28]) and the generalproof took some time to emerge. The proof in dimensions n ≥ 6 in the case where the Yamabeinvariant of (M,g), λg, is strictly less than that of the standard sphere, λ(Sn), was completed byAubin [4]. The proof in dimensions n = 3, 4, 5 and the remaining cases with λg = λ(Sn) for n ≥ 6are due to Schoen [22]. His solution involves the use of the Positive Mass theorem. For generalinformation on the Yamabe problem, I would recommend the article by Lee and Parker [17] and,once you are more familiar with the material, perhaps Chapters 2 and 5 of Aubin’s book [5].

If there is time, the third part of the course will present some recent work of Akutagawa, Ishidaand LeBrun [1] (see also [2]) between the Yamabe invariant (or sigma constant) of a manifold,which arises in the proof of the Yamabe conjecture, and the λ-invariant defined by Perelman inhis proof of the Poincare conjecture [20].

GLOBAL LORENTZIAN GEOMETRY II 3

Lecture 2Part 1. Witten’s proof of the Positive Energy/Mass Theorem

Witten’s proof of the Positive Mass/Energy Theorem uses spinors in a fundamental way. Tounderstand spinor fields on manifolds, we first need to understand orthonormal coframes and Clif-ford algebras. To do this properly would require setting up a large amount of technical machinerythat we do not have the time for (e.g. principal G-bundles (in particular, bundles of orthonormal(co)frames), connections on principal bundles, associated vector bundles, ...). As such, I will takea much more concrete, calculation-based approach which, unfortunately, is not the most aestheti-cally appealing approach from a mathematical point of view. For those of you that want to knowabout the abstract machinery, I will write an appendix that will give you the details.

1. Orthonormal frames and connections

Let (M,g) be a semi-Riemannian manifold of dimension n, with g a semi-Riemannian metricof signature (p, q) on M (with p + q = n). An orthonormal coframe on an open set U ⊆ M is a(continuous) basis ε1, . . . , εn for the cotangent bundle T ∗M (restricted to the set U) with theproperty that the metric g takes the form

g = ε1 ⊗ ε1 + · · ·+ εp ⊗ εp − εp+1 ⊗ εp+1 − · · · − εn ⊗ εn. (1.1)

It is often useful to define an n× n matrix η, with components ηij = diag[1, . . . , 1,−1, . . . ,−1] interms of which the above can be written as

g =n∑

i,j=1

ηijεi ⊗ εj . (1.2)

Remark 1.1. Given a semi-Riemannian manifold (M,g), it will generally not be possible to definesuch a coframe globally on M . Such a global coframe will exists if and only if the cotangent bundleof M is trivial.

There is no unique way to construct an orthonormal basis. Given a point p ∈ M and anorthonormal basis εi for T ∗pM , consider another basis of T ∗pM , εi =

∑nj=1 Λi

jεj , where Λ is a

non-singular n× n real matrix. Then, in order for εi to be orthonormal, we require

g =∑i,j

ηij εi ⊗ εj =

∑i,j,k,l

ηijΛikΛj

lεk ⊗ εl =

∑i,j

ηijεi ⊗ εj .

Comparing coefficients of εi ⊗ εj , we require that∑i,j

ηijΛikΛj

l = ηkl,

or, in matrix notation, thatΛtηΛ = η. (1.3)

The set of such matrices is the orthogonal group of η. We denote this group by

Op,q = Λ ∈ GLn(R) : ΛtηΛ = η.As such, there is a natural action of O(η) on the space of orthonormal coframes at each pointp ∈ M . Taking the determinant of (1.3), we deduce that (det Λ)2 = 1, so det Λ = ±1. We thendefine the special orthogonal group

SOp,q = Λ ∈ O(η) : det Λ = 1.This group acts on the space of orthonormal coframes, and preserves the orientation of the coframe(i.e. ε1 ∧ · · · ∧ εn = ε1 ∧ · · · ∧ εn).

To study the Lie algebra of SOp,q, we let Λ = exp sA, where A ∈ R(n), and take d/ds of thecondition (1.3) at t = 0, giving

0 =d

ds

(exp(sAt)η exp(sA)

)∣∣∣∣t=0

= Atη + ηA.

4 JAMES D.E. GRANT

In particular, if we consider p = n, q = 0, then we deduce that

son = A ∈ R(n) : At = −Ais the Lie algebra of skew-symmetric (n×n) matrices. As such, we deduce that dim son = 1

2n(n−1).

Example 1.2. We previously considered the constant curvature, two-dimensional metrics. GivenK ∈ R, we defined the functions

snK(r) :=

1√K

sin(√

Kr)

K > 0,

r K = 0,1√|K|

sinh(√

|K|r)

K < 0(1.4)

where r ∈ [0, π/√K] for K > 0, and r ∈ [0,∞) for K ≤ 0. We then define the metric

gK := dr2 + snK(r)2dθ2, (1.5)

where θ ∈ [0, 2π). In this case, one possible orthonormal coframe would be given by the basis

ε1 := dr, ε2 := snK(r)dθ,

in terms of whichg = ε1 ⊗ ε1 + ε2 ⊗ ε2.

Note, however, that we could equally well choose a coframe

ε1α := cosα ε1 + sinα ε2, ε2

α := cosα ε2 − sinα ε1,

for any function α : M → R, and we would still have

g = ε1α ⊗ ε1

α + ε2α ⊗ ε2

α.

Given an orthonormal coframe εi, we may construct the spin connection, which is the ana-logue of the Levi-Civita connection. This is a collection of 1-forms Γi

j , i, j = 1, . . . , n, that obeyCartan’s first structure equation

dεi = −n∑

j=1

Γij ∧ εj , (1.6)

(which is the analogue of the vanishing torsion condition for the Levi-Civita connection) and thesymmetry condition that

n∑k=1

(ηikΓk

j + ηjkΓki

)= 0. (1.7)

This is the analogue of the metric property of the Levi-Civita connection1. In the same way asthe Levi-Civita connection is uniquely determined by the conditions that it be torsion-free andmetric, it turns out that equations (1.6) and (1.7) uniquely determine the 1-forms Γi

j .Given the connection Γ, we define the curvature as the collection of 2-forms given by Cartan’s

second structure equation

Rij = dΓi

j +n∑

k=1

Γik ∧ Γk

j ,

and has the symmetry property thatn∑

k=1

(ηikRk

j + ηjkRki

)= 0.

The components of the curvature tensor are then defined by the relation

Rij =

12

∑k,l

Rklijε

k ∧ εl,

1The point is that we want the covariant derivative of g to vanish, and we want to organise the connection so that∇εi = 0. This implies that we require∇ηij = 0, which then implies that 0 = ∇ηij = dηij−

Pk Γk

iηkj−P

k Γkjηik.

Since the ηij are constant, this implies equation (1.7).


and obey the symmetry conditionRkl

ij = −Rlk

ij .

The components of the Ricci tensor with respect to the orthonormal frame are then defined bythe sum

Ricjl =∑

i

Rilij .

Since the Ricci-tensor itself is tensorial, we deduce that

Ric =∑i,j

Ricijεi ⊗ εj .

Finally, the scalar curvature is given by

s =∑i,j

ηijRicij .

Example 1.3. We return to our example above, with η = diag[1, 1] and

ε1 := dr, ε2 := snK(r) dθ.

Equation (1.7) implies that we have

Γ11 = Γ2

2 = 0, Γ12 + Γ2

1 = 0,

so the only non-trivial information on the connection is contained in Γ12, say. Equation (1.6) with

i = 1 implies thatdε1 = d(dr) = 0 = −Γ1

2 ∧ ε2.

Therefore Γ12 = λε2, for some function λ. Equation (1.6) with i = 2 gives

dε2 = d(snK(r)dθ) = sk′k(r)dr ∧ dθ =sk′k(r)skk(r)

ε1 ∧ ε2 = −Γ21 ∧ ε1 = Γ1

2 ∧ ε1 = λε2 ∧ ε1.

Hence

λ = −sk′k(r)

skk(r).

and

Γ12 = −sk

′k(r)

skk(r)ε2 = −sk′k(r)dθ.

Finally, the second Cartan structure equation implies that the only non-trivial curvature two-formis

R12 = dΓ1

2 = d(−sk′k(r)dθ) = −sk′′k (r)dr ∧ dθ = −sk′′k (r)

skk(r)ε1 ∧ ε2.

Therefore, we have

R1212 = −sk

′′k (r)

skk(r)= K,

where the final relationship follows from the form of the functions snK(r). The non-vanishingcomponents of the Ricci tensor are then

Ric11 = Ric22 = R1212 = K.

Therefore the Ricci tensor is of the form

Ric = Kε1 ⊗ ε1 +Kε2 ⊗ ε2 = Kg.

and the scalar curvature iss = Ric11 +Ric22 = 2K.

Lecture 3

6 JAMES D.E. GRANT

2. Spinors

Our treatment in this section largely follows [21, Chapter 3]. For much more details on Cliffordalgebras, see, e.g., [16].

Let V be a finite-dimensional vector space over R, with inner product q(·, ·). A Clifford algebrafor V is an algebra with unit, A, equipped with a map ϕ : V → A such that ϕ(v)2 = −q(v,v)1,for all v ∈ V , that is universal among algebras equipped with such maps.

Remark 2.1. Cl(V,g) may be equivalently defined as the algebra generated by the vector spaceV ⊂ Cl(V,g) and the identity element 1 subject to the relation

v · v = −g(v,v)1, ∀v ∈ V.

By polarisation, it follows that

v ·w + v ·w = −2g(v,w)1, ∀v,w ∈ V.

Remark 2.2. For any given (V,q), such an algebra exists, and is unique up to isomorphism. Wedenote it by Cl(V,q).

Example 2.3. Consider V = Rn with the standard inner product

g(x,y) := x1y1 + · · ·+ xnyn,

where x = (x1, . . . , xn) ∈ Rn, etc. In this case, we denote the corresponding Clifford algebra byCln (or Cln,0).

n = 1 In this case, V = R with g(x, y) = xy for x, y ∈ R. We may then choose V to be spannedby the vector x = (1). Cl1 is then the algebra generated by 1 and v (defined to be the image of xunder the embedding given above) subject to the relation

v2 := v · v = −g(x,x) = −1.

We therefore require the algebra generated by 1,v with v2 = −1. As such, Cl1 is isomorphic tothe algebra of complex numbers C.

n = 2 In this case, V = R2, with g(x,y) = x1y1 + x2y2 for x,y ∈ R2. We may then choose V

to be spanned by the vector x =(

10

), y =

(01

). Cl2 is then the algebra generated by 1,u,v

subject to the relationsu2 = v2 = −1, u · v + v · u = 0.

Defining w := u · v, we deduce that Cl2 is the space generated by 1,u,v,w subject to therelations

u2 = v2 = −1, u · v = w = −v · u.From these relations it follows that Cl2 is isomorphic to the algebra of quaternions H.

More generally, the Clifford algebras, Cln, with n = 0, . . . , 7 are given in Table 1 where, forK = R/C/H, the notation K(n) denotes the algebra of (n × n) matrices with entries in K. In

n 1 2 3 4 5 6 7 8Cln C H H⊕H H(2) C(4) R(8) R(8)⊕ R(8) R(16)

Table 1. Clifford algebras up to dimension 8.

dimensions n > 8, the Clifford algebras may be deduced from the isomorphism

Cln+8∼= Cln ⊗ R(16).


Representations: Let S be a complex vector space equipped with an R-linear map c : V →EndC(S) with the property that c(v)2 = −q(v,v). S is then a Clifford module and we have theClifford multiplication map V × S → S : (v, s) 7→ c(v)s, which we will also denote by v · s.

Bundles: If (M,g) is a Riemannian manifold2 then the fibres of TM are inner product spaces, sowe form the Clifford bundle Cl(TM,g), the fibres of which are the Clifford algebras Cl(TpM,gp),for p ∈M . Similarly, we define the bundle of Clifford modules S, with fibre Sp defined as above.

We want to define a connection on the bundle S. Following [21], we have the following:

Definition 2.4. Let S be a bundle of Clifford modules over Riemannian manifold (M,g). S isa Clifford bundle if it is equipped with a Hermitian metric (·, ·) and a compatible connection ∇such that

i) the action of v ∈ TpM on Sp is skew-adjoint with respect to (·, ·):

(v · s1, s2) + (s1,v · s2) = 0, ∀v ∈ TpM, ∀s1, s2 ∈ Sp.

ii) The connection on S is compatible with the Levi-Civita connection

∇X (v · s) = (∇Xv) · s+ v · (∇Xs) , ∀X,v ∈ X(M), ∀s ∈ C∞(S).

We then define the Dirac operator , D, on a Clifford bundle S to be the first order partialdifferential operator C∞(S) → C∞(S) defined as the composition

C∞(S) ∇−→ C∞(T ∗M ⊗ S)g−→ C∞(TM ⊗ S) ·−→ C∞(S).

Here the first map is covariant differentiation, the second is contraction with the metric g, andthe third is Clifford multiplication. In terms of a local orthonormal frame ei for TM , this maybe written in the form

Ds =n∑

i=1

ei · ∇eis

for s ∈ C∞(S).

Example 2.5. Let S = Λ∗(M), the exterior algebra on M . We define Clifford multiplication byv ∈ X(M) by

v · s = g(v, ·) ∧ s+ v s,

for all s ∈ Ω∗(M) ≡ C∞(S). In this case, the Dirac operator takes the form D = d + d∗, whered∗ is the formal adjoint of d with respect to the inner product on S induced by the Riemannianmetric.

For many purposes, we will need to know the square of the Dirac operator D2 : C∞(S) →C∞(S). It is most straightforward to calculate this at a point p ∈ M using an orthonormalcoframe ei defined on a neighbourhood of p that obeys the condition that ∇eiej

∣∣p

= 0. Note that

2We will mention later the modifications for the semi-Riemannian case.

8 JAMES D.E. GRANT

this condition implies that [ei, ej ]∣∣p

= 0. At p, we then have

D2s =∑

i

ei · ∇ei

∑j

ej · ∇ejs

=∑i,j

ei · ej ·(∇ei∇ejs

)(using ∇eiej

∣∣p

= 0)

=∑i,j

12

(ei · ej + ej · ei + ei · ej − ej · ei) ·(∇ei∇ejs

)=∑i,j

12−2g(ei, ej)∇ei∇ejs+

∑i,j

12ei · ej ·

[∇ei ,∇ej

]s

= −∑

i

∇ei∇eis+∑i,j

12ei · ej ·

[∇ei ,∇ej

]−∇[ei,ej ]

s (using [ei, ej ]

∣∣p

= 0)

= −∑

i

∇ei∇eis+∑i,j

12ei · ej ·K(ei, ej)s,

where K ∈ Ω2(End(S)) is the curvature of the connection on S defined by

K(X,Y) =[∇ei ,∇ej −∇[ei,ej ]

]s, ∀X,Y ∈ X(M), ∀s ∈ C∞(S)

We therefore have

D2s = −∑

i

∇ei∇eis+∑i,j

12ei · ej ·K(ei, ej)s (2.1)

Lecture 4

Remark 2.6. The bundle T ∗M ⊗ S has a natural (pointwise) inner product, defined using theHermitian metric on S and the inner product on T ∗M induced by the metric g. Given ϕ1, ϕ2 ∈C∞(T ∗M ⊗ S), this takes the form

(ϕ1, ϕ2)T∗M⊗S :=∑i,j

gij (ϕ1i, ϕ2j) ≡∑

i

(ϕ1(ei), ϕ2(ei)) ,

where the first expression is in local coordinates, the second is with respect to a local orthonormalframe, and (·, ·) is the inner product on S. We will also denote the inner product (·, ·)T∗M⊗S by(·, ·), when no confusion can arise. Integrating these inner products over M , we define the L2

inner products

〈s1, s2〉 =∫

M

(s1, s2) dvolg, 〈ϕ1, ϕ2〉 =∫

M

(ϕ1, ϕ2) dvolg,

for s1, s2 ∈ C∞(S), ϕ1, ϕ2 ∈ C∞(T ∗M ⊗ S).

Given the operator ∇ : C∞(S) → C∞(T ∗M ⊗ S), we may define its formal adjoint ∇∗ :C∞(T ∗M ⊗ S) → C∞(S). Recall that we introduced an orthonormal frame ei above with theproperty that ∇eiej |p = 0. In the next result, we also use the dual orthonormal coframe, denotedεi which has the property that ∇eiε

j∣∣p

= 0

Lemma 2.7. The operator ∇∗ : C∞(T ∗M ⊗ S) → C∞(S) is given in terms of the orthonormalcoframe ei by the formula

∇∗(∑

i

εi ⊗ si

)= −

∑i

∇isi, (2.2)

where si ∈ C∞(S). (From now on we will use the notation ∇i to denote ∇ei .)

Proof. To show that ∇∗ is the formal adjoint of ∇ with respect to the inner products 〈·, ·〉, weneed to show that

〈ϕ,∇s〉 − 〈∇∗ϕ, s〉 = boundary term,


for all s ∈ C∞(S) and ϕ ∈ C∞(T ∗M ⊗ S). (If we want to be more formal then, if M is, forexample, an open set with compact closure, we show that the left-hand-side of the above is zerowhen s, ϕ have compact support contained in M .) To show that the assertion of the Lemma iscorrect, we need to calculate

(s,∇∗ϕ)− (∇s, ϕ)

for s ∈ C∞(S), ϕ ∈ C∞(T ∗M ⊗ S) with ∇∗ defined as in (2.2), and show that this gives aboundary term when integrated over M . Working at point p ∈ M with a basis ei and dual basisεi as above, we then, without loss of generality, let ϕ =

∑i εi⊗ si, where si ∈ C∞(S). We then

calculate

(s,∇∗ϕ)− (∇s, ϕ) =

(s,∇∗

(∑i

εi ⊗ si

))−

(∇s,

∑i

εi ⊗ si

)

=

(s,−

∑i

∇isi

)−∑

i

(∇is, si)

= −∑

i

∇i (s, si)

= −div v,

where v :=∑

i (s, si) ei. Stokes’ theorem then implies that

〈s,∇∗ϕ〉 = 〈∇s, ϕ〉 −∫

∂M

v · dS.

Therefore ∇∗ is the formal adjoint of ∇ with respect to the L2 inner product, as required.

Remark 2.8. By similar methods, one can show that the Dirac operator is formally self-adjointwith respect to the L2 inner product on S i.e. if s1, s2 ∈ C∞(S), at least one of which has compactsupport, then

〈s1, Ds2〉 = 〈Ds1, s2〉.

To summarise: at this stage, equation (2.1) may be rewritten in the form

D2s = ∇∗∇s+∑i,j

12ei · ej ·K(ei, ej)s (2.3)

We now aim to simplify the second term in this equation.

Proposition 2.9.

D2s = ∇∗∇s+R

4s+ FSs, (2.4)

where R denotes the scalar curvature of the Riemannian metric g and

FS :=12

∑i,j

c(ei)c(ej)F(ei, ej) ∈ C∞(End(S)),

with F ∈ Ω2(End(S)) commuting with Clifford multiplication.

We prove this Proposition in a sequence of smaller steps.

Lemma 2.10. As endomorphisms of S,

[K(X,Y), c(Z)] = c(R(X,Y)Z),

for all X, Y, Z ∈ X(M).

10 JAMES D.E. GRANT

Proof. Since the connection is compatible with Clifford multiplication, we have([∇X,∇Y]−∇[X,Y]

)(Z · s) = ∇X ((∇YZ) · s+ Z · ∇Ys)−∇Y ((∇XZ) · s+ Z · ∇Xs)

−((∇[X,Y]Z

)· s+ Z · ∇[X,Y]s

)= (∇X∇YZ) · s+ (∇YZ) · ∇Xs+ (∇XZ) · ∇Ys+ Z · ∇X∇Ys

− (∇Y∇XZ) · s− (∇XZ) · ∇Ys− (∇YZ) · ∇Xs+ Z · ∇Y∇Xs

−(∇[X,Y]Z

)· s− Z · ∇[X,Y]s

=((

[∇X,∇Y]−∇[X,Y]

)Z)· s+ Z ·

(([∇X,∇Y]−∇[X,Y]

)s)

= R(X,Y)Z · s+ Z ·K(X,Y)s,

for all X, Y, Z ∈ X(M) and all s ∈ C∞(S). Since the left-hand-side of this equation equalsK(X,Y) (Z · s), we have the required result.

Given the Riemann tensor R we define R ∈ Ω2(End(S)) by

R(X,Y) :=14

∑k,l

c(ek)c(el)g(R(X,Y)ek, el).

Lemma 2.11. As endomorphisms of S,

[R(X,Y), c(Z)] = c(R(X,Y)Z),

for all X, Y, Z ∈ X(M).

Proof. This is simply a calculation:

[R(X,Y), c(Z)] =14

∑k,l

(ek · el · Z− Z · ek · el)g(R(X,Y)ek, el)

=14

∑k,l

(ek · (−Z · el − 2g(Z, el))− Z · ek · el)g(R(X,Y)ek, el)

=14

∑k,l

(− (−Z · ek − 2g(Z, ek)) · el − 2g(Z, el)ek − Z · ek · el)g(R(X,Y)ek, el)

=14

∑k,l

(2g(Z, ek)el − 2g(Z, el)ek)g(R(X,Y)ek, el)

=14

∑k,l

g(Z, ek)g(R(X,Y)ek, el)el

=∑

l

g(R(X,Y)Z, el)el

= R(X,Y)Z,

as required.

Corollary 2.12. As endomorphisms of S

[R(X,Y), c(Z)] = [K(X,Y), c(Z)] .

HenceK(X,Y) = R(X,Y) + F(X,Y), (2.5)

where F ∈ Ω2(End(S)) commutes with the action of the Clifford algebra.

Proof of Proposition 2.9. From (2.5), we have

12

∑i,j

c(ei)c(ej)K(ei, ej) =12

∑i,j

c(ei)c(ej)R(ei, ej) +12

∑i,j

c(ei)c(ej)F(ei, ej).


The second term is the object denoted FS in our Proposition. The first term, we can simplify.Note that

12

∑i,j

c(ei)c(ej)R(ei, ej) =18

∑i,j,k,l

c(ei)c(ej)g(R(ei, ej)ek, el)c(ek)c(el)

=18

∑i,j,k,l

c(eiejekel)g(R(ei, ej)ek, el).

If i, j, k are distinct, then the expression c(eiejekel) is skew-symmetric in i, j, k. Since we aresumming, this means we would get a cyclic sum over the components R(ei, ej)ek, which theBianchi identity tells us is zero3. As such, the only contributions to the final sum are those withi = k 6= j and j = k 6= i. We therefore want to calculate the quantity

12

∑i,j

c(ei)c(ej)R(ei, ej) =18

∑l

∑i=k 6=j

+∑

i=k 6=j

c(eiejekel)g(R(ei, ej)ek, el)

=14

∑l

∑i,j

c(eiejeiel)g(R(ei, ej)ei, el)

=14

∑l

∑i,j

c(−eieiejel)g(R(ei, ej)ei, el)

=14

∑l

∑i,j

c(ejel)g(R(ei, ej)ei, el) (since c(ei)2 = −1)

= −14

∑j,l

c(ejel)Ric(ej , el)

= −14

∑j,l

−g(ej , el)Ric(ej , el)

=14R,

where R denotes the Ricci scalar.

Therefore,

D2s = ∇∗∇s+R

4s+ FSs,

as required. Lecture 5

2.1. Spin representations. We have thus far worked with general Clifford modules and Cliffordbundles. However, we now restrict ourselves to a particular class of modules and bundles thatarise directly from the representation theory of Clifford algebras.

Claim: Let M be a spin manifold4 of dimension n. If n = 2m is even, or n = 2m + 1 is odd,then there exists a complex vector bundle over M , ∆, over M of rank 2m with the followingproperty: Let S be a Clifford bundle over M . Then there exists a vector bundle V , equipped witha Hermitian metric and connection, with the property that S ∼= ∆⊗V (as Clifford bundles). ∆ isreferred to as the spin bundle. The curvature of the natural connection on S is then of the formK∆ ⊗ 1 + 1⊗KV . In particular, the first term is the R of (2.5) and the second is the F term.

If you wish to see details of the proof of this result, see e.g. [16] or [21]. From now on, wewill assume that we are working on the bundle ∆. In this case, we may explicitly write down(in local form) the compatible covariant derivative on ∆ in terms of local bases, etc. If εi is

3Recall R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0.4I will not formally define what this term means. It is basically the requirement that the bundles mentioned

later in this paragraph are globally well-defined. If M is not a spin manifold, then the bundles ∆ and V will notbe globally defined, but the tensor product ∆⊗ V will be globally well-defined.

12 JAMES D.E. GRANT

an orthonormal coframe for T ∗M and ei the dual orthonormal frame, then (in terms of a localtrivialisation of ∆) we may write

∇ψ = dψ +12

n∑i,j=1

Γij

(−1

4[c(ei), c(ej)]

)ψ,

where ψ ∈ C∞(∆) and Γij are the components of the spin-connection that we introduced inSection 15. The curvature of this connection then coincides with R ∈ Ω2(End(∆)) introducedabove, and explicitly has FS = 0.

Remark 2.13. Letting Σij := − 14 [c(ei), c(ej)] ∈ C∞(End(∆)), it is straightforward to show that

the Σ’s define a representation of the orthogonal group SOn in the sense that

[Σij ,Σkl] = −δikΣjl + δjkΣil + δilΣjk − δjlΣik,

for i, j, k, 1 = 1, . . . , n where δ is the Kronecker symbol

δij =

1 i = j,

0 i 6= j.

2.2. The Lichnerowicz theorem. 6

In the case where the bundle S is the spin-bundle, ∆, then FS = 0 and we have the Lichnerowiczformula [18] for the square of the Dirac operator7

D2ψ = ∇∗∇ψ +s

4ψ. (2.6)

for any section ψ ∈ C∞(∆).

Proposition 2.14. Let (M,g) be a closed, Riemannian spin manifold with non-negative scalarcurvature that is somewhere positive. Then there are no non-trivial sections ψ ∈ C∞(∆) withDψ = 0 on M .

Proof. Let g be a metric of positive scalar curvature, and let ψ lie in the kernel of D (i.e. Dψ = 0).Then D2ψ = D(Dψ) = 0. Taking the product of D2ψ with ψ and integrating over M , we deducethat

0 =∫

M

〈ψ,D2ψ〉 =∫

M

⟨ψ,∇∗∇ψ +

s

4ψ⟩

=∫

M

(|∇ψ|2 +

s

4|ψ|2

).

Since the right-hand-side is a sum of non-negative terms, it follows that both terms must vanish.In particular, ∇ψ = 0 (so ψ is necessarily parallel) and s

4 |ψ|2 = 0. Therefore, as long as there

exists p ∈M at which s(p) > 0, then we deduce that ψ = 0.

Remark 2.15. The Atiyah-Singer index theorem implies that, on a spin manifold, the index of theDirac operator indD = n+ − n− (where n± are non-negative integers) is given by the A-genusA(M). For example, in four-dimensions this implies that

n+ − n− = −τ8, (2.7)

where τ := b2+− b2− is an integer. For S4, τ = 0, for CP2, τ = −1, and for a K3 surface, τ = 16.(Note that in the case of CP2, the right-hand-side of (2.7) is not an integer. This is a consequenceof the fact that CP2 is not a spin manifold.)

Theorem 2.16. Let M be a closed spin manifold. If M admits a Riemannian metric of positivescalar curvature, then A(M) = 0.

Proof. If M admits a metric of positive scalar curvature then, by the previous theorem, n+ =n− = 0. Therefore A(M) = 0.

5We have lowered one of the indices but, since we are in Riemannian signature, this doesn’t make any difference.6This material actually appeared in Lecture 3, but it is more logically consistent to put it here.7I have now returned to my usual notation where s denotes the scalar curvature of the metric g.


Corollary 2.17. Let M be a closed spin manifold with A(M) 6= 0. Then M admits no Riemann-ian metric of positive scalar curvature.

Remark 2.18. Note that the conditions that M be closed, a spin manifold, and that A(M) 6= 0are all topological statements8. As such, in the above result, topological restrictions have directimplications for the geometrical properties of the manifold.

If one considers manifolds that do not admit a spin structure, then the above results tell usnothing. In particular, it is known that if M is a closed, oriented, simply-connected manifoldwith dimM ≥ 5, then M admits a metric of positive scalar curvature (see, e.g., [8], where a moregeneral result is proved).

On the other hand, there is no obstruction to negative scalar curvature metrics, in the sensethat any closed manifold M with dimM ≥ 3 admits a metric of negative scalar curvature9 [11, 15].

8Strictly speaking, bA(M) is sensitive to the differentiable structure on M , rather than just the topologicalstructure.

9In fact, such an M admits a metric with constant negative scalar curvature.

14 JAMES D.E. GRANT

3. The positive energy/mass theorem

In [29], Witten used the properties of a particular type of Dirac operator to give a proof ofthe positive energy/mass theorem. We will give a proof of a simpler result, sometimes called theRiemannian positive mass theorem, based on the papers of Parker-Taubes [19] and the appendix ofLee-Parker [17]. We will then, briefly, mention the modifications that are needed to treat the fullversion of the positive energy/mass theorem. Details of the full proof are given in an Appendix.For additional background material and information on Witten’s proof and the Schoen-Yau proofof the positive energy and mass theorems, see, e.g., the article by Choquet-Bruhat [10]. Bray’sarticle [9] is an interesting introduction to the ideas involved in the positive mass theorem.

Let (M,g) be a four-dimensional Lorentzian manifold. We assume that we are given a symmetrictensor field T ∈ T 0

2 (M), interpreted physically as the energy-momentum tensor of any matter thatlives in our space-time. It is assumed that the Ricci curvature, Ric, and the scalar curvature, s,of the metric g are related to T by the Einstein equations

Ric− s

2g = 8πT.

(For those that know of such things, we will use units in which Newton’s constant, G, and thespeed of light, c, are equal to one.). It is assumed that the matter in our manifold has “non-negative local mass density”. Mathematically speaking, this condition is imposed by asking thatthe energy momentum tensor, T, satisfy the dominant energy condition [14]:

Definition 3.1. The energy momentum tensor, T ∈ T 02 (M) satisfies the dominant energy con-

dition if, for every time-like vector field V ∈ X(M), we have T(V,V) ≥ 0 and the vector fieldcorresponding to the one-form T(V, ·) is causal.

Remark 3.2. This condition has the physical interpretation that the energy density measured byany local observer is non-negative, and the energy flow-vector is causal (i.e. non-space-like). Ina local orthonormal frame ea3a=0, the dominant energy condition is equivalent to the conditionthat the components Tab := T(ea, eb) obey the condition

T00 ≥ |Tab|, for a, b = 0, 1, 2, 3,

and that

T 200 ≥

3∑i=1

(T0i)2.

This condition is satisfied for most sensible forms of matter (e.g. scalar, Dirac and Maxwell fields).

Definition 3.3. Let Σ ⊂ M be a complete, oriented, 3-dimensional, space-like hypersurface. Σis asymptotically flat if the following conditions are satisfied. There exists a compact set K ⊂ Σwith the property that Σ \K is the disjoint union of a finite number of subsets Σ1, . . . ,ΣN of Σ –the ends of Σ – each of which is diffeomorphic to the complement of a contractible, compact setin R3. Under this diffeomorphism the metric on Σl ⊂M should be of the form

gij = δij + aij ,

in the standard coordinates, xi3i=1, on R3, with

aij = O(r−1), ∂iajk = O(r−2), ∂i∂jakl = O(r−3),

where r := |x| ≡√∑3

i=1 (xi)2. Furthermore, the components of the second fundamental form10

of Σl ⊂M , kij , should satisfy

kij = O(r−2), ∂ikjk = O(r−3).

Remark 3.4. We will often, without comment, identify the ends Σl with the corresponding set inR3.

Lecture 6

10Denote the unit (time-like) normal vector field to Σ by n ∈ Γ(Σ, TM). Given vector fields tangent to Σ, werecall that the second fundamental form of Σ, k, is defined by the formula k(v,w) := g(∇vn,w).


Example 3.5. If we, for the moment, drop the assumption of completeness, then the basic exampleof an asymptotically flat space-like hypersurface is a surface t = constant in the Schwarzschildsolution

g = −(

1− 2mr

)dt2 +

(1− 2m

r

)−1

dr2 + r2(dθ2 + sin2 θdφ2

),

where (t, r, θ, φ) ∈ R× (2m,∞)× [0, π]× [0, 2π). Letting Σ be the space-like hyper-surface t = t0,then the induced metric on Σ is

h =(

1− 2mr

)−1

dr2 + r2(dθ2 + sin2 θdφ2

).

If we rewrite this in terms of Cartesian coordinates (x, y, z) = (r sin θ cosφ, r sin θ sinφ, r cos θ),then the metric takes the form

h = |dx|2 +

[(1− 2m

r

)−1

− 1

]· (d|x|)2

= |dx|2 +2m

|x| − 2m(d|x|)2 = |dx|2 +

2m|x| − 2m

(1|x|

x · dx)2

= |dx|2 +2m

(|x| − 2m)|x|2∑i,j

xixjdxi ⊗ dxj .

Therefore the metric takes the required form with

aij =2m

(|x| − 2m)|x|2xixj .

Then aij = O(|x|−1) as |x| → ∞, etc, so the metric is asymptotically flat in the above sense.

If one adopts a Hamiltonian point of view to the Einstein field equations (i.e. one writes them asthe Euler-Lagrange equations of an action functional, and then performs a Legendre transformationto go to a Hamiltonian version) then a by-product of such an approach is a definition of the totalenergy and the total momentum of an asymptotically flat manifold [3]. These quantities aredefined in each asymptotic end Σl as limits over spheres Sr,l of radius r in Σl ⊂ R3:

El :=1

16πlim

r→∞

∫Sr,l

3∑i,j=1

(∂jgij − ∂igjj) dΣi, (3.1a)

(pl)k :=1

16πlim

r→∞

∫Sr,l

23∑

i=1

kik − δik

3∑j=1

kjj

dΣi, (3.1b)

where dΣi denotes the area element of the sphere S(r) in R3 of radius r := |x| (so dΣi =xi

r r2dvolS2). The justification of the definition of these quantities would require a lot of physics,

so we will content ourselves with calculating a relevant example to show that the above givesensible answers11.

11See Appendix A if you want a brief justification of these expressions.

16 JAMES D.E. GRANT

Example 3.6. In the case of the Schwarzschild solution, with Σ := t = t0, then k = 0, sop = 0. In terms of the above form of the metric, we find that∑

j

(∂jgij − ∂igjj) =∑

j

(∂jaij − ∂iajj)

=∑

j

(∂j

(2m

(|x| − 2m)|x|2xixj

)− ∂i

(2m

(|x| − 2m)|x|2xjxj

))=

2m(|x| − 2m)|x|2

∑j

(∂j (xixj)− ∂i (xjxj))

=2m

(|x| − 2m)|x|2∑

j

(δijxj + xiδjj − (2δijxj))

=2m

(|x| − 2m)|x|2(xi + 3xi − 2xi)

= 4mxi

(|x| − 2m)|x|2.

Therefore

E =1

16πlim

|x|→∞

∫S2

3∑i=1

4mxi

(|x| − 2m)|x|2xi

rr2 dvolS2

=1

16π

∫S2

4mdvolS2 =1

16π(4m)(4π) = m.

The physical intuition is that a gravitational system with non-negative matter density shouldhave non-negative total energy. It is not clear, however, that the dominant energy condition andasymptotic flatness imply anything about the integrals (3.1).

Positive Energy Theorem. Under the conditions of asymptotic flatness and the dominant en-ergy condition, El − |pl| ≥ 0 on each end Σl. If El = 0 for some l then Σ has only one end andM is flat along Σ.

Remark 3.7. Viewing P = (El,pl) as a vector in R3,1, the Positive Energy Theorem implies thatP is a future-directed, causal vector.

An important special case of the Positive Energy Theorem occurs if one assumes that the metricon Σ has the asymptotic form

gij =(1 +

ml

2r

)4

δij + pij (3.2)

in the end Σl, with pij = O(r−2), ∂kpij = O(r−3) and ∂l∂kpij = 0(r−4). In this case, the PositiveEnergy Theorem is equivalent to the following result:

Positive Mass Theorem. Under the conditions of asymptotic flatness, the dominant energycondition and (3.2), then ml ≥ 0 for each l, with equality if and only if Σ is flat along M .

What we will prove, for the moment, is a simpler result. If we consider a space-like hypersurfaceΣ the extrinsic curvature of which is zero (i.e. k = 0) then automatically we have pl = 0. Thereforethe positive energy theorem is simply the statement that E ≥ 0, under the condition that Σ beasymptotically flat, and the dominant energy condition hold on M . Such a condition is necessarilysatisfied (via the Gauss-Codazzi relations) if the scalar curvature of the induced metric on Σ isnon-negative.

Note that the asymptotically flat condition on Σ and the non-negativity of the scalar curvatureare conditions that we require of Σ, viewed as an abstract Riemannian manifold (i.e. withoutreference to Σ being a space-like hyper-surface in a Lorentzian manifold M .) As such, we definea Riemannian manifold (M,g) to be asymptotically flat if there exist ends, M1, . . . ,MN , of M , as


in 3.3. We may then define the corresponding energies on the ends Ml by

El :=1

16πlim

r→∞

∫Sr,l

3∑i,j=1

(∂jgij − ∂igjj) dΣi.

We then study the following version of the positive mass theorem:

Riemannian Positive Mass Theorem. Let (M,g) be a complete, Riemannian spin manifoldof dimension n with non-negative scalar curvature that is asymptotically flat12. If El < ∞ forl = 1, . . . , N , then El ≥ 0. Moreover, if El = 0 for some l then M has only one end and is flat.

Remark 3.8. Note that the expressions for El depend only on the asymptotic information aboutthe metric g. This result implies that if we wish to localise the scalar curvature of our manifoldin a small region and have the manifold sufficiently flat asymptotically that the energies El arezero, then this is not possible while maintaining non-negative scalar curvature. Similarly, if wehave an asymptotically flat manifold for which El < 0, then it cannot be possible that the metricg is complete with positive scalar curvature.

Remark 3.9. This result is the version of the positive mass theorem that is required to solve theYamabe problem.

Lecture 7

Definition 3.10. 13 Let R ≥ 1 be large enough so that each end Ml ⊂ R3 contains the exteriorof the ball BR of radius R. For each l and each r ≥ R, set Ml,r := Ml \ Br (considered either asa subset of R3 or of M). Fix a smooth function ρ on M with the following properties:

(i) ρ ≥ 1;(ii) ρ = r in Ml,2R;

(iii) ρ = 1 in M \(⋃

l

Ml,R

).

For 1 ≤ p < ∞, δ ∈ R, we define the weighted Lebesgue spaces, Lpδ(M), to be the completion

of C∞c (M) with respect to the norm

‖u‖p,δ :=(∫

M

|u|p ρ−pδ−ndvolg

)1/p

.

We then define weighted Sobolev spaces, W k,pδ (M), as the completion of C∞c (M) with respect to

the norm

‖u‖k,p,δ :=k∑

j=0

∥∥∂ju∥∥

p,δ−j.

Similarly, given the spinor bundle ∆ we define the weighted Lebesgue and Sobolev spaces ofsections, Lp

δ(∆) and W k,pδ (∆) where, for the pointwise norm, we use where

‖ψ‖p =(∫

Σ

〈ψ,ψ〉p/2

)1/p

is the Lp norm.

Remark 3.11. The weighting, δ, gives a measure of the asymptotic growth properties of elementsof Lp

δ(M). Given a smooth function, u with ‖u‖p,δ <∞, we find that u = o(rδ) as r →∞.

We say that an asymptotically flat manifold M is asymptotically flat of order τ if there existsq > n and τ ≥ n−2

2 such that, for each end Ml, we have

gij − δij ∈W 2,p−τ (Ml).

We then study the following version of the positive mass theorem:

12We will actually need to modify, slightly, what we mean by the term “asymptotically flat”for the case of narbitrary. We will do this in the next lecture.

13See, e.g., [6]

18 JAMES D.E. GRANT

Riemannian Positive Mass Theorem. Let (M,g) be a complete, Riemannian spin manifoldwith non-negative scalar curvature s ∈ L1(M) that is asymptotically flat of order τ for someτ ≥ n−2

2 . If El < ∞ for l = 1, . . . , N , then El ≥ 0. Moreover, if El = 0 for some l then M hasonly one end and is flat.

Remark 3.12. The result can also be proved with τ = n− 2 in the case where dimM ≤ 7, and Mis not assumed to be a spin manifold [24].

Remark 3.13. For τ ≥ n−22 , the mass exists and is unique (i.e. independent of the choices of

asymptotic coordinates, the choice of the function ρ, etc.). For τ > n− 2, the mass is zero.

Remark 3.14. If you prefer not to think in terms of Sobolev spaces, then one has the same resultif the metric g is smooth, and has asymptotic behaviour

gij − δij = O(r−τ ), ∂g = O(r−τ−1), ∂2g = O(r−τ−2),

with τ > n−22 . (This is what Aubin [5], for example, does.) We will actually use this notation,

although, strictly speaking, it should be interpreted as saying that a particular object lies in aparticular weighted Sobolev space.

The proof of the Riemannian Positive Mass Theorem will be broken into several smaller steps.Recall from before that we have the Lichnerowicz formula for the square of the Dirac operator:

D∗Dψ = D2ψ = ∇∗∇ψ +s

4ψ. (3.3)

Lemma 3.15. Let ψ ∈ C∞(∆). Then

|∇ψ|2 +s

4|ψ|2 − |Dψ|2 =

n∑i=1

∇i (〈ψ,∇iψ + ei ·Dψ〉) . (3.4)

Given a subset Σ ⊆M with boundary ∂Σ, the integral form of (3.4) is

∫Σ

(|∇ψ|2 +

s

4|ψ|2 − |Dψ|2

)µ =

n∑i=1

∫∂Σ

[〈ψ,∇iψ + ei ·Dψ〉 ei µ] (3.5a)

=12

n∑i,j=1

∫∂Σ

⟨ψ, [ei·, ej ·] · ∇ej

ψ⟩ei µ, (3.5b)

where [ei·, ej ·] := ei · ej · − ej · ei· is the commutator in the Clifford algebra, and

µ := ε1 ∧ · · · ∧ εn

is the volume form on M .

Proof. All that we need to do is work out the boundary terms that appear when computing theformal adjoints. Choosing our orthonormal frame ein

i=1 as before such that, at point p ∈M , wehave ∇eiej

∣∣p

= 0. Therefore, as before, we have

D2ψ = −n∑

i=1

∇i∇iψ +s

4ψ.


Taking the inner product with ψ, we require

〈ψ,D2ψ〉 =n∑

i,j=1

〈ψ, ei · ∇i (ej · ∇jψ)〉

=n∑

i,j=1

∇i〈ψ, ei · (ej · ∇jψ)〉 − 〈∇iψ, ei · (ej · ∇jψ)

=n∑

i=1

∇i〈ψ, ei ·Dψ〉+ 〈∇iei · ψ, ej · ∇jψ

=n∑

i=1

∇i〈ψ, ei ·Dψ〉+ 〈Dψ,Dψ〉

Similarly,

n∑i=1

〈ψ,∇i∇iψ〉 =n∑

i=1

(∇i〈ψ,∇iψ〉 − 〈∇iψ,∇iψ〉)

= − |∇ψ|2 +n∑

i=1

∇i〈ψ,∇iψ〉.

Hence

|∇ψ|2 +s

4|ψ|2 − |Dψ|2 = |∇ψ|2 +

s

4|ψ|2 − 〈ψ,D2ψ〉+

n∑i=1

∇i〈ψ, ei ·Dψ〉

= |∇ψ|2 +s

4|ψ|2 −

⟨ψ,−

n∑i=1

∇i∇iψ +s

4ψ

⟩+

n∑i=1


= |∇ψ|2 +

⟨ψ,

n∑i=1

∇i∇iψ

⟩+

n∑i=1


=n∑

i=1

∇i 〈ψ,∇iψ〉+n∑

i=1


=n∑

i=1

∇i 〈ψ,∇iψ + ei ·Dψ〉 ,

as required.Integrating this equality over Σ gives

∫Σ

(|∇ψ|2 +

s

4|ψ|2 − |Dψ|2

)µ =

∫Σ

n∑i=1

∇i 〈ψ,∇iψ + ei ·Dψ〉µ

=∫

∂Σ

n∑i=1

〈ψ,∇iψ + ei ·Dψ〉 ei µ

20 JAMES D.E. GRANT

where the second equality has used the Stokes formula. To verify the use of the Stokes formula,calculating at p ∈M then ∇ei

ej |p = 0 implies that dεi∣∣p

= 0. At p we therefore have

d

[n∑

i=1

〈ψ,∇iψ + ei ·Dψ〉 ei µ

]= d

[n∑

i=1

〈ψ,∇iψ + ei ·Dψ〉

]ei µ

=n∑

i,j=1

∇j〈ψ,∇iψ + ei ·Dψ〉εj ∧ (ei µ)

=n∑

i,j=1

∇j〈ψ,∇iψ + ei ·Dψ〉 (δijµ)

=n∑

i=1

∇i〈ψ,∇iψ + ei ·Dψ〉µ,

as required.Lecture 8Finally, equation (3.5b) follows using the following rearrangement

∇eiψ + ei ·Dψ = ∇eiψ +n∑

j=1

ei · ej · ∇ejψ

= ∇eiψ +

12

n∑j=1

(ei · ej ·+ej · ei ·+ [ei·, ej ·] · ∇ej

)ψ

= ∇eiψ −

n∑j=1

g(ei, ej)∇ejψ +

12

n∑j=1

[ei·, ej ·]∇ejψ

=12

n∑j=1

[ei·, ej ·]∇ejψ.

The Weitzenbock formula (3.5) leads to a vanishing theorem discovered by Witten [29]. Namely,under the conditions of the Riemannian Positive Mass Theorem, if ψ is a spinor field on M thatsatisfies Dψ = 0 and that vanishes sufficiently quickly at infinity that the boundary term in (3.5)is zero, then ψ ≡ 0. This leads to the following result.

Proposition 3.16. For 0 < η < n− 1, the Dirac operator

D : W 2,p−η (∆) →W 1,p

−η−1(∆)

is an isomorphism.

Sketch of Proof. Unfortunately, this result needs a fair bit of PDE technology to prove properly.Essentially, we need to show that D and its adjoint

D∗ = D : W 2,p′

η+1−n(∆) →W 1,p′

η−n(∆)

(where p′ is defined by 1p + 1

p′ = 1) have trivial kernel. Let ψ ∈ W 2,p−η (∆) with Dψ = 0. Since D

is an elliptic operator, elliptic regularity implies that ψ is smooth. We then have

∆|ψ|2 = 〈∆ψ,ψ〉+ 2|∇ψ|2 + 〈ψ,∆ψ〉

=⟨s

4ψ,ψ

⟩+ 2|∇ψ|2 +

⟨ψ,s

4ψ⟩

= 2(s

4|ψ|2 + |∇ψ|2

)≥ 0,

using the fact that s ≥ 0. By the maximum principle, any maximum of |ψ|2 will occur on theboundary of Σ. Since |ψ|2 → 0 at infinity (because η > 0), this implies that |ψ|2 = 0, and henceψ = 0. A similar argument shows that kerD∗ is also trivial if η < n− 1.


For more information concerning elliptic regularity, the theory of Fredholm operators (whichwe are, implicitly, using above), maximum principles, see, e.g., [7, 12, 13].

Remark 3.17. In the case where the metric and connection are smooth, which we will generallyassume, then elliptic regularity (see, e.g., [12, Chapter 6] or [13]) implies that if ϕ ∈W 1,p

−η−1(∆) ∩C∞(∆), then D−1ϕ ∈W 2,p

−η (∆) ∩ C∞(∆).

Theorem 3.18. Let (M,g) be asymptotically flat of order τ ≥ n−22 with non-negative scalar

curvature s ∈ L1(M). Let ψ0,lNl=1 be constant spinors defined in the asymptotic ends MlN

l=1.Then there exists a unique, smooth spinor field ψ ∈ C∞(∆) on M with the properties that

(i) Dψ = 0.(ii) ψ − ψ0,l ∈W 2,p

−τ (∆) in each end Ml.

Proof. In each end l, fix a smooth function 0 ≤ βR,l ≤ 1 that is identically 1 in the end Ml,3R and0 inside Ml,2R. Let ψ0 ∈ C(M ;∆) be the spinor

ψ0 =k∑

k=1

ψ0,lβR,l.

Since dψ0 = 0, we have

Dψ0 =n∑

i=1

ei ·

∂eiψ0 −18

n∑j,k=1

Γijk [e·, ek·]ψ0

∼ 0 + Γψ0

∈W 1,p−τ−1(∆),

since Γ ∼ ∂g ∈ W 1,p−τ−1(∆) and ψ0 is smooth and asymptotically constant. Therefore Dψ0 ∈

W 1,p−τ−1(∆) so, by Proposition 3.16, we deduce that there exists a unique ψ1 ∈ W 2,p

−τ (∆) with theproperty that

Dψ1 = −Dψ0.

Since ψ0 is smooth, elliptic regularity implies that ψ1 is also smooth. The spinor field ψ := ψ0+ψ1

then has the required properties.

Theorem 3.19. Let (M,g) asymptotically flat of order τ ≥ n−22 with non-negative scalar curva-

ture s ∈ L1(M). Let ψ0,lNl=1 and ψ be as in Theorem 3.18. Then

0 ≤∫

Σ

|∇ψ|2 +

s

4|ψ|2

µ = 4π

N∑l=1

El 〈ψ0,l, ψ0,l〉 . (3.6)

Lecture 9Proof. For the sake of simplicity, we assume, for the moment, that M only has one end, M1.We then use the integration by parts formula (3.5) applied to the region Σ ⊂ M interior to theasymptotic sphere, Sr, of radius r > R in M1. We need to work out the asymptotics of theboundary term (3.5) on Sr as r →∞.

Since Dψ = 0, we have∫Σ

|∇ψ|2 +

s

4|ψ|2

µ =

n∑i=1

∫Sr

〈ψ,∇iψ〉ei µ,

Since the left-hand-side is explicitly real, we can rewrite this in the form∫Σ

|∇ψ|2 +

s

4|ψ|2

µ = Re

n∑

i=1

∫Sr

〈ψ,∇iψ〉ei µ

,

= Ren∑

i=1

∫Sr

[〈ψ0,∇iψ0〉+ 〈ψ0,∇iψ1〉+ 〈ψ1,∇iψ0〉+ 〈ψ1,∇iψ1〉] ei µ

(3.7)

22 JAMES D.E. GRANT

Furthermore, since ψ1 = o(ρ−τ ), ∇ψ1 = o(ρ−τ−1), and ∇ψ0 = o(ρ−τ−1), the third and fourthterms are o(ρ−2τ−1) as r →∞. As such, when integrated over Sr, they give contributions of orderrn−1 · o(ρ−2τ−1) = o(rn−2τ−2). Since τ ≥ n−2

2 , it follows that n− 2τ − 2 ≤ 0, so the contributionsof the third and fourth terms tend to zero as r →∞.

To calculate the contributions of the other terms, we need a little more information aboutthe asymptotics of the connection. We can, asymptotically, take an orthonormal coframe εi =dxi + 1

2

∑j aijdx

j + o(ρ−2τ ). (Note that the o(ρ−2τ ) terms is of the form a2, and therefore itsderivative is o(ρ−2τ − 1).) To get the components of the spin connection, we consider the exteriorderivative of the coframe

dεi =12

∑j,k

∂kaijdxk ∧ dxj + o(ρ−2τ−1)

=12

∑j,k

(∂kaij − ∂iajk) dxk ∧ dxj + o(ρ−2τ−1)

= −12

∑j,k

(∂kaij − ∂iajk) εj ∧ εk + o(ρ−2τ−1)

= −∑

k

12

∑j

(∂kaij − ∂iajk) εj + o(ρ−2τ−1)

∧ εk

≡ −∑

k

Γik ∧ εk.

Hence14

Γik =12

∑j

(∂kaij − ∂iajk) εj + o(ρ−2τ−1) =12

∑j

(∂kgij − ∂igjk) εj + o(ρ−2τ−1).

If we now consider the first term in (3.7) then, since ∂ψ0 = 0, we have

Re〈ψ0,∇iψ0〉 = −18

∑j,k

12

(∂kgij − ∂jgik) Re〈ψ0, [ej ·, ek·]ψ0〉+ o(ρ−2τ−1)

= −18

∑j,k

(∂kgij) Re〈ψ0, [ej ·, ek·]ψ0〉+ o(ρ−2τ−1)

Using the fact that the inner product 〈·, ·〉 is Hermitian, and the ei· are skew-Hermitian withrespect to it, we deduce that

〈ψ0, [ej ·, ek·]ψ0〉 = 〈[ej ·, ek·]ψ0, ψ0〉= 〈ej · ek · ψ0, ψ0〉 − 〈ek · ej · ψ0, ψ0〉= −〈ek · ψ0, ej · ψ0〉+ 〈ej · ψ0, ek · ψ0〉= 〈ψ0, ek · ej · ψ0〉 − 〈ψ0, ej · ek · ψ0〉= −〈ψ0, [ej ·, ek·]ψ0〉.

Therefore the first term in Re〈ψ0,∇iψ0〉 is zero, so Re〈ψ0,∇iψ0〉 = o(ρ−2τ−1). Therefore, by thesame argument as previously, the first term in (3.7) also vanishes as r →∞.

Lecture 10We are therefore left with the second term in (3.7). We first define the operator

Li := ∇i + ei ·D =12

∑j

[ei·, ej ·]∇j .

14Note that, since the Γik that we have constructed is explicitly skew-symmetric under interchange of i and k,it follows that it is the (unique) spin-connection defined by skew-symmetry and the first Cartan structure equation.


We define the (n− 2) form

α :=∑i,j

〈[ei·, ej ·]ψ0, ψ1〉 ei ej µ.

Calculating in our frame with ∇eiej |p = 0, then we deduce that, at p, we have

dα =

∑i,j

d [〈[ei·, ej ·]ψ0, ψ1〉] ∧ (ei ej µ)

=∑i,j,k

(∇k [〈[ei·, ej ·]ψ0, ψ1〉]) εk ∧ (ei ej µ)

= 2∑i,j

(∇i [〈[ei·, ej ·]ψ0, ψ1〉]) (ej µ)

= 2∑i,j

([〈[ei·, ej ·]∇iψ0, ψ1〉+ 〈[ei·, ej ·]ψ0,∇iψ1〉]) (ej µ)

= −4∑

j

(〈Ljψ0, ψ1〉 − 〈ψ0, Ljψ1〉

)ej µ.

Therefore, by Stokes’ theorem, and the fact that Dψ1 = −Dψ0, the second term in (3.7) is

Re∑

i

∫Sr

〈ψ0, (Li − ei ·D)ψ1〉ei µ = Re∑

i

∫Sr

[〈Liψ0, ψ1〉+ 〈ψ0, ei ·Dψ0〉] ei µ. (3.8)

As before, 〈Liψ0, ψ1〉 = o(ρ−2τ−1), so the first term in the integral gives zero contribution asr →∞. Meanwhile the asymptotic form of the connection gives

ei ·Dψ0 = −18

∑j,k,l

(∂kglj + o(ρ−2τ−1)

)ei · el · [ej ·, ek·]ψ0

= −18

∑j,k,l

(∂kglj + o(ρ−2τ−1)

)ei · el · (2ej · ek + 2δjk) · ψ0

= −14

∑j,k

(∂jgkj − ∂kgjj + o(ρ−2τ−1)

)ei · ek · ψ0

Writing ei · ek· = 12 [ei·, ek·] − δik and noting, as before, that [ei·, ek·] is skew, we see that (3.8)

becomes14

∑i,j

∫Sr

(∂jgij − ∂igjj + o(ρ−2τ−1)

)|ψ0|2ei µ.

Putting this into (3.7) and letting r →∞ gives Witten’s formula∫M

(|∇ψ|2 +

s

4|ψ|2

)µ =

14

limr→∞

∑i,j

∫Sr

(∂jgij − ∂igjj)) |ψ0|2ei µ = 4πE|ψ0|2.

Finally, if M has more than one end, M1, . . . ,MN , then we apply the above argument on theinterior of the set bounded by the sphere of radius r in each end. Our integral formula thenbecomes ∫

M

|∇ψ|2 +

s

4|ψ|2

µ = Re

N∑

l=1

n∑i=1

∫Sr,l

〈ψ,∇iψ〉ei µ

= 4πN∑

l=1

El|ψ0,l|2.

The arguments then proceeds as above, giving (3.6).

Lemma 3.20. Let ψ and ψi be smooth spinor fields with ∇ψ = 0 and ∇ψi = 0 for each i.

24 JAMES D.E. GRANT

(a) If limψ(x) = 0, where this limit is taken along some path in one asymptotic end Ml, thenψ = 0.

(b) If ψi are linearly independent in some end Ml, then they are linearly independent ev-erywhere on M .

Proof. (a) If ∇ψ = 0, then we have

∇|ψ|2 = ∇ (〈ψ,ψ〉) = 〈∇ψ,ψ〉+ 〈ψ,∇ψ〉 = 0,

so |ψ|2 is constant. Since, by assumption, limψ(x) = 0, it follows that |ψ|2 = 0, so ψ = 0.

(b) Suppose that there are constants ci such that ψ =∑ciψi vanishes at some point x0 ∈ Σ.

Since ∇ψ = 0 we can repeat the above argument to conclude that ψ(x) = 0, for any x ∈M . Thiscontradicts the assumption that the ψi are linearly independent in Ml.

Proof of the Riemannian Positive Mass Theorem. Let ψ0,lNl=1 be constant spinors on the as-

ymptotic ends of M with ψl = 0 on each end except Ml. Theorem 3.18 then gives a field spinor ψthat satisfies Dψ = 0 and that asymptotically approaches ψ0,l in each end Ml. Substituting sucha ψ into (3.6) then shows that El ≥ 0. (For example, taking ψ0,1 6= 0, ψ0,l = 0 for l ≥ 2 impliesthat E1 ≥ 0, and so on.)

Now suppose that the energy of some end, say M1, is equal to zero. Choose a basis of constantspinors ψar

a=1 on the region M1, where r := rank∆, and let ψa be the solutions of Dψa = 0with ψa → ψ

ain M1, and ψa → 0 in Ml for l ≥ 2. (Such ψa exist, by Theorem 3.18.) Our integral

equality (3.6) then implies that, for a = 1, . . . , r∫M

|∇ψa|2 +

s

4|ψa|2

µ = 0.

Since s ≥ 0, this implies that ∇ψa = 0. Therefore

∇|ψa|2 = ∇ (〈ψa, ψa〉) = 〈∇ψa, ψa〉+ 〈ψa,∇ψa〉 = 0,

so |ψa|2 is constant. Since ψa → 0 on each Ml, for l = 2, . . . , N , this implies that ψa = 0 if N ≥ 2.If N ≥ 2, this gives a contradiction to out theorem asserting the existence of non-trivial solutionsof Dψ = 0 with constant boundary conditions. As such, we must have N = 1, so M1 is the onlyend of M .

Finally, assuming that M has only one end, and E1 = 0, we want to show that the metric g isflat. Letting ψa be as in the previous paragraph, then we know that ∇ψa = 0, a = 1, . . . , r. Byassumption, the ψa are linearly independent on M1. If they were not linearly independent on M ,this would mean the existence of p ∈M and constants a1, . . . , ar with

a1ψ1(p) + · · ·+ arψ

r(p) = 0.

However, since ∇ψa = 0, we deduce that ∇ (∑r

a=1 aaψa) = 0. By the same argument as in

the previous paragraph, we then deduce that | (∑r

a=1 aaψa) |2 = | (

∑ra=1 aaψ

a) |2∣∣p

= 0, so∑ra=1 aaψ

a = 0. Therefore the ψa would be linearly dependent on M1, contradicting our hy-pothesis. Therefore the ψa are linearly independent on M . Furthermore, ∇ψa = 0, so in a localframe ei of M ,

0 =(∇ei∇ej −∇ej∇ei −∇[ei,ej ]

)ψa = R(ei, ej)ψa

for i, j = 1, . . . , n. Since ψa are a basis of ∆ (and, technically speaking, because ∆ comes froma faithful representation of the spin group) this implies that R = 0. From the definition of R interms of the curvature tensor, it follows that this implies that the Riemann curvature tensor, R,is also zero. Hence, g is flat.


3.1. Modifications required to prove the Positive Energy Theorem [Non-examinable].The proof of the full Positive Energy Theorem on a Lorentzian manifold is, in its basic idea,very similar to the proof of the Riemannian Positive Mass Theorem. There are some importantcomplications that should be mentioned, though. We now summarise the modifications requiredin the case where M is a four-dimensional Lorentzian manifold.

First of all, we need to consider spin-bundles and Dirac operators on Lorentzian manifolds. Inthe Lorentzian case, the compatibility conditions of the Hermitian inner product are modified:

(v · s1, s2) = (s1,v · s2), ∀v ∈ TpM, ∀s1, s2 ∈ ∆p.

Given a space-like hyper-surface, Σ ⊂M with unit time-like normal vector n, which we assumeto be future-directed, then we consider the restriction of the spin-bundle ∆ → M to a vectorbundle ∆|Σ → Σ. (We will also denote this bundle by ∆ → Σ when no confusion can occur.)Sections of the bundle ∆ → Σ are called Dirac spinors along Σ. The bundle ∆|Σ inherits theinner product (·, ·) from ∆. There is also a second Hermitian inner product on ∆|Σ defined by

〈φ, ψ〉 := (n · φ, ψ) .

The most important point is that the inner product 〈·, ·〉 is definite, and (without loss of generality)may be chosen positive-definite.

The connection, ∇, on ∆ → M is compatible with the inner product (, ). On restriction toΣ, this determines a connection on the bundle ∆ → Σ that is compatible with the inner product(, ), which we also denote this by ∇. In general, this connection is not compatible with the innerproduct 〈, 〉, since ∇n 6= 0.

We now define a Dirac operator acting on sections of ∆ → Σ using the connection ∇ inheritedfrom the four-manifold M . We call this the hypersurface Dirac operator and denote it by D .Intrinsically, D is the composition

Γ(∆) ∇→ Γ(T ∗Σ⊗∆) c→ Γ(∆),

where c is Clifford multiplication. We adopt a local orthonormal frame eα3α=0 for M , where wetake e0 ≡ n. In this case, ei3i=1 becomes a local orthonormal frame for Σ along Σ. In terms ofsuch an orthonormal frame, we have

Dψ =3∑

i=1

ei · ∇eiψ

for ψ ∈ Γ(∆).Denoting the dual orthonormal coframe by εα, we define the volume form15 µ = ε1 ∧ ε2 ∧ ε3

on Σ. Using the volume µ, we may define the L2 inner product

〈ϕ,ψ〉L2 :=∫

Σ

〈ϕ,ψ〉µ, ϕ, ψ ∈ C∞(Σ,∆).

Similar to the discussion of the square of the Dirac operator on a Riemannian manifold, wehave the following result.

Proposition 3.21. Let D∗ and ∇∗ be the formal adjoints of D and ∇ under the inner product〈·, ·〉L2 . Then we have the Weitzenbock formula

D∗D = D2 = ∇∗∇+ R (3.9)

where

R =14

(s+ 2Ric(e0, e0)− 2

3∑i=1

Ric(e0, ei)c(e0)c(ei)

)∈ End(S).

15Strictly speaking, what we require is µ := i∗`ε1 ∧ ε2 ∧ ε3

´∈ Ω3(Σ), where i : Σ→ M is the embedding of Σ

as a submanifold of M . Since εi, i = 1, 2, 3 are defined up to an SO(3) transformation, and such transformationsdo not change µ, it follows that µ is uniquely defined once e0 is fixed.

26 JAMES D.E. GRANT

The integral form of (3.9) is∫Σ

(|∇ψ|2+ < ψ,R · ψ > −|Dψ|2

)µ =

12

3∑i,j=1

∫Σ

d[⟨ψ, [ei·, ej ·] · ∇ej

ψ⟩ei µ

]=

12

3∑i,j=1

∫∂Σ

⟨ψ, [ei·, ej ·] · ∇ejψ

⟩ei µ

for any Dirac spinor ψ along Σ.

Remark 3.22. The formal adjoint ∇∗ is now (at a point p ∈ Σ) given by

∇∗(

3∑i=1

εi ⊗ si

)= −

3∑i=1

∇eisi −

3∑i,j=1

kije0 · ei · sj .

in a local frame with

∇eie0|p =

3∑j=1

kijej

and all other derivatives vanishing at p.

Remark 3.23. The endomorphism R may be rewritten in terms of the energy-momentum tensorT using the Einstein equations. We have

R = 4π

(T00 +

3∑i=1

T0ie0 · ei·

).

The dominant energy condition then implies that

R ≥ 4π

T00 −

(3∑

i=1

T 20i

)1/2 ≥ 0.

Theorem 3.24. Let (M,g) be Lorentzian manifold with any matter present satisfying the domi-nant energy condition, and Σ a complete, asymptotically flat, space-like hypersurface. Let ψ0,lN

l=1

be constant spinors defined in the asymptotic ends ΣlNl=1. Then there exists a unique, smooth

spinor ψ on Σ that satisfies(i) Dψ = 0.(ii) ψ − ψ0,l ∈W 2,p

−1 (∆) in each end Σl.Moreover, we have the following identity:∫

Σ

〈∇ψ,∇ψ〉+ 〈ψ,R · ψ〉µ = 4πN∑

l=1

(El 〈ψ0,l, ψ0,l〉+

3∑k=1

pl,k 〈ψ0,l, ∂0 · ∂k · ψ0,l〉

).

Here ∂a is the standard coordinate basis of T (R3,1), viewing Σl ⊂ R3 ⊂ R3,1 in each end.

Remark 3.25. The proof of the first part of this theorem is essentially the same as for the Rie-mannian Positive Mass Theorem, with non-negativity of R playing the role of non-negative scalarcurvature in the proof of existence of an inverse for D . The extrinsic curvature terms in theintegral equality arise from extra terms in the asymptotics of the connection due to the fact thatwe are working on a submanifold Σ, rather than the manifold M , and therefore need to considercomponents of the connection in the normal directions as well as the spatial directions.

The proof of the Positive Energy Theorem now follows from the above integral equality in thesame way as the Riemannian Positive Mass Theorem, along with some use of Clifford multiplicationidentities.


Part 2. Appendices [Non-examinable]

Appendix A. Justification for definition of E and P

For your peace of mind, we give here a brief derivation of the formulae (3.1) for the energy andmomentum of an asymptotically flat space-like hypersurface. For more details, see, e.g., [10].

Let (M,g) be a Lorentzian manifold, with the metric g satisfying the Einstein equations

G[g] := Ric[g]− 12s[g]g = 8πT.

Let η be another metric on M that satisfies the Einstein equations without any source term

G[η] = Ric[η]− 12s[η]η = 0.

We then define the energy momentum tensor of g with respect to η as the linearisation

t = G′[η] · (g − η)−G[g],

where G′[η] is the derivative of the map g 7→ G[g] with respect to g evaluated at η. In the casewhere η is flat and we adopt local flat coordinates xa3a=0, then we find that the components ofG′[η] · (g − η) with respect to these local coordinates takes the form

(G′[η] · h)ab =12

−∑c,d

ηcd∂c∂dHab +∑c,d

ηcd∂a∂cHbd

+∑c,d

ηcd∂b∂cHad − ηab

∑c,d,e,f

ηcdηef∂c∂dHdf

,

where

Hab := hab −12

∑c,d

ηcdhcd

ηab.

We then consider the (0, 2) tensor field on M18π

(t + 8πT) ≡ 18π

G′[η] · a,

where we have defineda := g − η.

It follows from general theory that the above quantity is covariantly divergence-free with respectto η:

∇η · (t + 8πT) = 0.If ξ ∈ X(M) is a Killing vector (i.e. Lξg = 0), then the 1-form

p :=18π

(t + 8πT) (ξ, ·) ∈ Ω1(M)

is co-closedd (?p) = 0, (A.1)

where ? : Ωp(M) → Ωn−p(M) is the Hodge dual. Integrating this formula over a four-dimensionalsubset S ⊂M , we deduce that ∫

∂S

?p = 0.

In particular, the flux of p over the surface of S is zero. In appropriate circumstances, thiscondition may be interpreted as the conservation of energy or momentum.

It follows from (A.1) that (locally) there exists a two-form q ∈ Ω2(M) with the property that?p = d (?q). Integrating the three-form ?p over a three-dimensional set Σ, then we have∫

Σ

?p =∫

Σ

d (?q) =∫

∂Σ

?q. (A.2)

28 JAMES D.E. GRANT

It turns out that such a q may be constructed globally on M . In particular, if we choose η tobe the flat metric, and ξ to be a parallel vector field (with respect to η), then, again in localcoordinates,

qab =∑c,d,e

ηcd (∂dKabce) ξe, (A.3)

where

K(T,X,Y,Z) =1

16π(η(T,Z)H(X,Y) + η(X,Y)H(T,Z)

−η(T,Y)H(X,Z)− η(X,Z)H(T,Y)) . (A.4)

If we then consider the integral (A.2), where ∂Σ is a sphere of large radius r in the surfacet = constant, which we denote Sr, then we deduce that

P (ξ) :=∫

Sr

?q =∫

∂Σ

q0idSi.

Using the formula (A.3) for q, the formula (A.4) for K, and the fact that ξ is constant in thecoordinate system xα, we find that

limr→∞

P (ξ) = Eξ0 +3∑

i=1

piξi,

where

E =1

16π

3∑i=1

limr→∞

∫Sr

3∑j=1

∂jHij − ∂iH00

dΣi =1

16πlim

r→∞

∫Sr

3∑i,j=1

(∂jgij − ∂igjj) dΣi,

pi =1

16π

3∑k=1

limr→∞

∫Sr

−∂0Hik + δik

∂0H00 −3∑

j=1

∂jH0j

+ ∂kH0i

dΣk

=1

16πlim

r→∞

∫Sr

23∑

i=1

kik − δik

3∑j=1

kjj

dΣi.


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pp. 21–37. 2

Index

asymptotically flat, 14, 16asymptotically flat of order τ , 17

Cartan’s first structure equation, 4Cartan’s second structure equation, 4Clifford bundle, 7

Dirac operator, 7Dirac spinors along Σ, 25dominant energy condition, 14

Einstein equations, 14Ends of asymptotically flat hypersurface, 14energy-momentum tensor, 14

hypersurface Dirac operator, 25

orthogonal group, 3orthonormal coframe, 3

special orthogonal group, 3spin bundle, 11spin connection, 4

30

global lorentzian geometry ii - univie.ac.at

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