The Positive Mass Theorem

Abstract

The important positive mass theorem roughly says that—in contrast to Newtonian gravity theory—it is impossible to construct an object out of ordinary matter, i.e., matter with positive local energy density, whose total energy (including gravitational contributions) is negative. In this chapter we will give essentially E. Witten’s proof of the positive energy theorem, which makes crucial use of spinor fields. To be complete, we develop the necessary tools on spinors in GR in an appendix to this chapter. It is very remarkable that spinors have turned out to be so useful in simplifying the proof of an entirely classical property of GR. We add some remarks on the Penrose inequality, which can be regarded as a sharpening of the positive energy theorem for black holes.

Appendix: Spin Structures and Spinor Analysis in General Relativity

In the following treatment of the bare essentials we assume that the reader is familiar with spinor analysis in SR, and also with the theory of connections in principle fibre bundles (see [41, 42]).

1.1 9.5.1 Spinor Algebra

We begin by reviewing those parts of spinor algebra that we shall need, mainly to fix various conventions and our notation. For detailed treatments see [34] and [40].

Let (ℝ⁴,〈⋅,⋅〉) be the Minkowski vector space with the non-degenerate symmetric bilinear form

$$ \langle x,y\rangle=\eta_{\mu\nu} x^\mu y^\nu=x^T\eta y ,\quad \eta =(\eta_{\mu\nu})= \operatorname{diag}(-1,1,1,1) . $$

(9.49)

(Note that x,y∈ℝ⁴ are regarded as column matrices.) The 1-component of the homogeneous Lorentz group—the orthochronous Lorentz group—is denoted by $L_{+}^{\uparrow}$. Its universal covering group is SL(2,ℂ). The two-fold covering homomorphism $\lambda:\mathit{SL}(2,\mathbb{C})\longrightarrow L_{+}^{\uparrow}$ is determined as follows:

$$ \underline{\lambda(L)x}=L\underline{x}L^\dagger , $$

(9.50)

where $\underline{x}$ denotes for each x∈ℝ⁴ the hermitian 2×2 matrix

(9.51)

(Here σ _k =σ ^k are the Pauli matrices, and L ^† denotes the hermitian conjugate of L.) From

(9.52)

it follows that

$$ \operatorname{det} \underline{x}=-\langle x,x\rangle . $$

(9.53)

Using this it is easy to see that the assignment L↦λ(L) is a homomorphism from SL(2,ℂ) into $L_{+}^{\uparrow}$. One can show that the image is all of $L_{+}^{\uparrow}$ (see [34] or [40]).

The 6-dimensional Lie algebra of SL(2,ℂ) is the following subalgebra of the general linear algebra gl(2,ℂ)

$$ \mathit{sl}(2,\mathbb{C})= \bigl\lbrace M\in \mathit{gl}(2,\mathbb{C}) \mid \operatorname{Tr} M=0 \bigr\rbrace . $$

(9.54)

A useful basis of sl(2,ℂ) is

$$ \biggl\lbrace\frac{1}{2 i}\sigma_j, \frac{1}{2}\sigma_j\ (j=1,2,3) \biggr\rbrace . $$

(9.55)

The Lie algebra of $L_{+}^{\uparrow}$ is

$$ \mathit{so}(1,3)= \bigl\lbrace M\in \mathit{gl}(4,\mathbb{R}) \mid \eta M+M^T\eta =0 \bigr\rbrace . $$

(9.56)

From (9.50) we obtain for the induced isomorphism λ _∗:sl(2,ℂ)⟶so(1,3) the formula

$$ \underline{\lambda_\ast(M)x}=M\underline{x}+ \underline{x}M^\dagger , $$

(9.57)

where M∈sl(2,ℂ). The reader is invited to show that the images of the basis (9.55) generate the 1-parameter subgroups of rotations about the j-th axis, respectively, the special Lorentz transformations along the j-th axis.

There are two types of fundamental representations of SL(2,ℂ) in terms of which all other non-trivial finite dimensional irreducible representations can be constructed (using tensor products and symmetrizations). The (fundamental) spinor representation, denoted by D ^(1/2,0) is simply L↦L, while the cospinor representation D ^(0,1/2) is defined by L↦(L ^†)⁻¹. Two-component spinors ξ, transforming with L are called right-handed, whereas two-component spinors η, transforming with (L ^†)⁻¹ are called left-handed.

Let V be the space of right-handed spinors ξ=(ξ ^A). We then have for ξ∈V

$$ L:\xi\longmapsto L\xi ,\qquad L:\xi^A\longmapsto L_B^A \xi^B . $$

(9.58)

Since $\operatorname{det} L=1$, the following symplectic form on V

$$ s\bigl(\xi,\xi'\bigr)=\xi^T\varepsilon \xi'=\xi^A\varepsilon_{AB} \xi'^{B} , $$

(9.59)

with

(9.60)

is invariant under SL(2,ℂ). This means that L ^T εL=ε.

It will turn out to be useful to introduce the following operation on 2×2 complex matrices A:

$$ A\longmapsto\hat{A}=\varepsilon\bar{A} \varepsilon^{-1} , $$

(9.61)

where $\bar{A}$ denotes the complex conjugate matrix. From the previous remark it follows

$$ \hat{L}=\bigl(L^\dagger\bigr)^{-1}\in \mathit{SL}(2, \mathbb{C}) $$

(9.62)

for L∈SL(2,ℂ). The cospinor representation is thus $L\longmapsto\hat{L}$.

The symplectic form (9.59) on V can be used to lower spinor indices

$$ \xi_A:=\xi^B \varepsilon_{BA} . $$

(9.63)

The inverse of this is

$$ \xi^A=\varepsilon^{AB}\xi_B ,\quad \bigl(\varepsilon^{AB}\bigr)=\varepsilon . $$

(9.64)

Clearly, $\xi^{A} \xi'_{A}$ is invariant under SL(2,ℂ) for ξ,ξ′∈V.

Let $\dot{V}$ be the space of 2-component left-handed spinors. Following van der Waerden, we denote the components of $\eta\in\dot{V}$ with lower dotted indices $\eta_{\dot{A}}$. Obviously, the symplectic form

$$ s\bigl(\eta,\eta'\bigr)=\eta_{\dot{A}} \varepsilon^{\dot{A} \dot{B}} \eta_{\dot{B}} ,\quad \bigl( \varepsilon^{\dot{A} \dot{B}}\bigr)=\bigl(\varepsilon^{AB}\bigr) , $$

(9.65)

on $\dot{V}$ is also SL(2,ℂ) invariant. We can thus raise lower indices

$$ \eta^{\dot{A}}=\varepsilon^{\dot{A} \dot{B}}\eta_{\dot{B}} . $$

(9.66)

The inverse is, as for right-handed spinors,

$$ \eta_{\dot{A}}=\eta^{\dot{B}}\varepsilon_{\dot{B} \dot{A}} ,\quad (\varepsilon_{\dot{A} \dot{B}})=(\varepsilon_{AB})=\varepsilon . $$

(9.67)

From (9.61) it follows that $\eta^{\dot{A}}$ transforms with $\bar{L}$:

$$ \eta^{\dot{A}}\longmapsto\overline{L_B^A} \eta^{\dot{B}} . $$

(9.68)

The representations $L\longmapsto\hat{L}$ and $L\longmapsto\bar{L}$ are, of course, equivalent.

It can be shown (see Chap. 2 of [34]) that all finite dimensional irreducible representations of SL(2,ℂ) (apart from the trivial one) are given by

$$ \bigl(D^{(1/2,0)} \bigr)^{\otimes_sm}\otimes \bigl(D^{(0,1/2)} \bigr)^{\otimes_sn} , $$

(9.69)

where ⊗ _s m denotes the m-fold symmetric tensor product.

Dirac spinors transform according to the reducible representation

(9.70)

(If space reflections are added, the representation becomes irreducible.)

Equation (9.50) shows that the matrix elements of $\underline{x}$ transform as a spinor with one upper undotted plus one upper dotted index. Therefore, we write

$$ \underline{x}= \bigl(x^{A\dot{B}} \bigr) . $$

(9.71)

Thus, a vector x ^μ can be regarded as a spinor $x^{A\dot{B}}$. Equation (9.71) tells us that the matrix elements of σ _μ should be written as $\sigma_{\mu}=(\sigma_{\mu}^{A\dot{B}})$. The relation between x ^μ and $x^{A\dot{B}}$ is then given by

$$ x^{A\dot{B}}=x^\mu\sigma_\mu^{A\dot{B}} . $$

(9.72)

Lowering the indices in this equation or in (9.71) can be expressed as

$$ (x_{A\dot{B}} )= (\underline{\hat{x}} )^T . $$

(9.73)

We define σ ^μ by . Note that , and also the important identity

(9.74)

In particular,

(9.75)

In SR one uses the differential operators

(9.76a)

(9.76b)

We have, for example, the identity

$$\partial_{A\dot{B}}\partial^{A\dot{B}}=\operatorname{Tr}(\underline{\partial} \underline{\hat{\partial}})=-2 \eta^{\mu\nu}\partial_\mu \partial_\nu , $$

i.e.

$$ \Box\equiv\eta^{\mu\nu}\partial_\mu \partial_\nu=-\frac{1}{2}\partial_{A\dot{B}} \partial^{A\dot{B}} . $$

(9.77)

Similarly, the divergence of a vector field u ^μ can be written as

$$ \partial_\mu u^\mu=-\frac{1}{2} \partial_{A\dot{B}}u^{A\dot{B}} . $$

(9.78)

In terms of $\partial_{A\dot{B}}$ the Weyl equations for fundamental spinor fields φ ^A and $\chi_{\dot{B}}$ read

$$ \partial_{A\dot{B}}\varphi^A=0 ,\qquad \partial^{A\dot{B}}\chi_{\dot{B}}=0 . $$

(9.79)

A Dirac spinor field ψ is composed of two Weyl spinor fields as

$$ \psi= \begin{pmatrix} \varphi^A\\ \chi_{\dot{B}} \end{pmatrix} , $$

(9.80)

and the Dirac equation is equivalent to the pair

$$ \underline{\hat{\partial}} \varphi=i m \chi ,\qquad \underline{ \partial } \chi=i m \varphi . $$

(9.81)

Introducing the Dirac matrices

(9.82)

this pair can be written in the traditional form

$$ \bigl(i \gamma^\mu\partial_\mu+m \bigr) \psi=0 . $$

(9.83)

1.2 9.5.2 Spinor Analysis in GR

The question is now, how to extend these elements of spinor analysis in SR to GR. In particular, we need a covariant derivative of spinor fields. To this end we introduce some basic concepts.

Let (M,g) be a Lorentz manifold, $P(M,L_{+}^{\uparrow})$ the principle fibre bundle of space and time oriented Lorentz frames with the proper orthochronous Lorentz group (1-component of the homogeneous Lorentz group) as structure group and $\lambda:\mathit{SL}(2,\mathbb{C})\longrightarrow L_{+}^{\uparrow}$ the universal covering homomorphism.

Definition 9.1

A spin structure on M is a principle bundle $\tilde{P}(M,\mathit{SL}(2,\mathbb{C}))$ on M with the structure group SL(2,ℂ), together with a 2 : 1 bundle morphism $\pi_{s}:\tilde{P}\longrightarrow P$, which is compatible with λ. Thus the following diagram is commutative:

Given a representation ρ of SL(2,ℂ) in a vector space F we can introduce the vector bundle E(F,SL(2,ℂ)) which is associated to $\tilde{P}(M,\mathit{SL}(2,\mathbb{C}))$ and has typical fibre F.

Definition 9.2

A Weyl spinor of type ρ over (M,g) is a section in the associated bundle π _E :E⟶M, i.e., a map ϕ:M⟶E such that π _E ∘ϕ=id _M .

We do not assume in what follows that the reader is familiar with associated bundles, because we shall use an equivalent definition of Weyl spinors, which makes only use of the principle bundle $\tilde{\pi}:\tilde{P}\longrightarrow M$. A Weyl spinor can also be viewed as a map $\tilde{\phi}:\tilde{P}\longrightarrow F$ which satisfies the following covariance condition

$$ \tilde{\phi}(p\cdot g)=\rho\bigl(g^{-1}\bigr)\tilde{\phi}(p) $$

(9.84)

for each $p\in\tilde{P}$ and g∈G.

The Levi-Civita connection form ω on $P(M,L_{+}^{\uparrow})$ induces a 1-form $\tilde{\omega}$ on $\tilde{P}$ with values in the Lie algebra sl(2,ℂ) by

$$ \tilde{\omega}=\lambda^{-1}_\ast \bigl( \pi^\ast_s\omega \bigr) , $$

(9.85)

where λ _∗:sl(2,ℂ)⟶so(1,3) is the Lie algebra isomorphism belonging to λ. It is a good exercise to verify that $\tilde{\omega}$ is a connection form on the spin bundle $\tilde{P}$. With it we can now define in the usual fashion a covariant derivative of Weyl spinor fields by

$$\nabla\tilde{\phi}:=(d\tilde{\phi})_{hor} , $$

where ‘hor’ denotes the horizontal component belonging to $\tilde{\omega}$. Explicitly, one has always

$$ \nabla\tilde{\phi}=d\tilde{\phi}+\rho_\ast(\tilde{\omega}) \tilde{\phi}. $$

(9.86)

More generally, if $\tilde{\phi}$ is a spinor-valued ‘horizontal’ p-form, the (exterior) covariant derivative is given by

$$ \nabla\tilde{\phi}=d\tilde{\phi}+\rho_\ast(\tilde{\omega}) \wedge\tilde{\phi}. $$

(9.87)

Relative to a local section $\tilde{\sigma}:U\longrightarrow\tilde{P}$ the pull-back of $\tilde{\omega}$ is

$$ \tilde{\sigma}^\ast\tilde{\omega}=\lambda^{-1}_\ast \bigl(\tilde{\sigma}^\ast \circ\pi_s^\ast\omega \bigr)=\lambda^{-1}_\ast\sigma^\ast(\omega) , $$

(9.88)

where $\sigma=\pi_{s}\circ\tilde{\sigma}:U\longrightarrow P$ is the local section of P on U belonging to $\tilde{\sigma}$. For the local representative $\phi_{U}=\tilde{\sigma}^{\ast}\tilde{\phi}:U\longrightarrow F$ of $\tilde{\phi}$ we obtain from (9.87)

$$ \nabla\phi_U=d\phi_U+\rho_\ast \bigl(\tilde{\sigma}^\ast\tilde{\omega}\bigr)\wedge\phi_U . $$

(9.89)

From now on we work with local sections and write for (9.88) and (9.89)

(9.90a)

(9.90b)

The section σ defines an orthonormal basis {θ ^μ} of 1-forms (tetrad belonging to σ).

Since ω is so(1,3)-valued, we can decompose it as follows

$$ \omega=\omega^\alpha_{\;\beta} \varSigma ^\beta_\alpha= \omega_{\alpha\beta }\varSigma ^{\alpha\beta} , $$

(9.91)

where Σ ^αβ=E ^αβ−E ^βα=−Σ ^βα, E ^αβ being the matrix with elements $(E^{\alpha\beta})_{\mu\nu}=\delta^{\alpha}_{\mu}\delta^{\beta}_{\nu}$. (The $\varSigma ^{\beta}_{\alpha}$ with α<β, form a basis of so(1,3).) A routine calculation (exercise) gives for $\tilde{\omega}$ in (9.90a) the following decomposition

$$ \tilde{\omega}=\omega_{\alpha\beta}\sigma^{\alpha\beta} ,\quad \sigma^{\alpha\beta}= -\frac{1}{8} \bigl(\sigma^\alpha\hat{\sigma}^\beta-\sigma^\beta\hat{\sigma}^\alpha \bigr) . $$

(9.92)

Recall that , where σ _k are the Pauli matrices, and σ ^μ=η ^μν σ _ν , $\hat{\sigma}^{\mu}=\eta^{\mu\nu}\hat{\sigma}_{\nu}$, with

(9.93)

A bar denotes complex conjugation, and η _μν is the flat metric $\operatorname{diag}(-1,1,1,1)$.

Note that $\omega^{\alpha}_{~\beta}$ in (9.91) and (9.92) are just the Levi-Civita connection forms relative to the tetrad {θ ^μ}.

Not every Lorentz manifold has a spin structure. We quote the following famous

Theorem 9.4

A space or time orientable pseudo-Riemannian manifold has a spin structure if and only if its second Stiefel–Whitney class vanishes.

R.P. Geroch (see [290]) has proven the following

Theorem 9.5

A space and time orientable open (non-compact) Lorentz manifold has a spin structure if and only if the manifold is parallelizable.

Spinors and spinor analysis in GR are treated very extensively in [136] and [289].

1.3 9.5.3 Exercises

Exercise 9.3

Compute the images of the basis (9.55) for sl(2,ℂ) under the Lie algebra homomorphism λ _∗.

Exercise 9.4

Let γ(x)=x ^μ γ _μ , x∈ℝ⁴, and show that for the representation (9.70) for Dirac spinors the Dirac matrices satisfy

$$\gamma \bigl(\lambda(L)x \bigr)=\rho(L)\gamma(x)\rho(L)^{-1} , $$

where L∈SL(2,ℂ).

Exercise 9.5

Show that the 1-form $\tilde{\omega}$ in (9.85) has all the defining properties of a connection form on the spinor bundle $\tilde{P}$.

Exercise 9.6

Prove (9.86) by evaluating both sides for horizontal and fundamental (vertical) vector fields.