3 · spinors and spacetime geometry

necessity of spinors and the spin group

The proper orthochronous Lorentz group $\mathrm{SO}^+(1,3)$ is not simply connected; its fundamental group is $\mathbb{Z}/2$. The unique simply connected double cover is the Spin group $\operatorname{Spin}(1,3)\simeq\operatorname{SL}(2,\mathbb{C})$. A spin structure on an oriented Lorentzian manifold $(M,g)$ is a principal $\operatorname{Spin}(1,3)$-bundle $P_{\operatorname{Spin}}$ together with a twofold covering map onto the orthonormal frame bundle that intertwines the group actions. Such a structure exists if and only if the second Stiefel-Whitney class $w_2(M)$ vanishes. Once a spin structure is chosen, the associated spinor bundles $$ S\simeq\mathbb{C}^2,\qquad \bar S\simeq\overline{\mathbb{C}}^2 $$ (as typical fibers) carry the fundamental representations of $\operatorname{SL}(2,\mathbb{C})$. Dirac spinors are sections of $S\oplus\bar S$; Weyl spinors are sections of $S$ or $\bar S$ alone.

definition (lorentz group). $$ O(1,3)=\bigl\{\Lambda\in\operatorname{GL}(4,\mathbb{R}):\Lambda^T\eta\Lambda=\eta\bigr\},\qquad \eta=\operatorname{diag}(-1,1,1,1). $$ $\mathrm{SO}^+(1,3)$ denotes the identity component consisting of matrices with $\det\Lambda=1$ and $\Lambda^0{}_0\ge 1$.

definition (spin group and double cover). $\operatorname{Spin}(1,3)$ is the unique simply connected Lie group admitting a surjective Lie-group homomorphism $$ \lambda:\operatorname{Spin}(1,3)\to\mathrm{SO}^+(1,3) $$ with $\ker\lambda=\{\pm 1\}$. under the standard identification one has $\operatorname{Spin}(1,3)\simeq\operatorname{SL}(2,\mathbb{C})$.

theorem (double cover by $\operatorname{SL}(2,\mathbb{C})$). identify Minkowski vectors with Hermitian $2\times 2$ matrices by $$ X=x^\mu\sigma_\mu =\begin{pmatrix} x^0+x^3 & x^1-ix^2 \\ x^1+ix^2 & x^0-x^3 \end{pmatrix}, $$ where $\sigma_0=I$ and $\sigma_i$ are the Pauli matrices. then $A\in\operatorname{SL}(2,\mathbb{C})$ acts by $X\mapsto AXA^\dagger$ and induces $\lambda(A)\in\mathrm{SO}^+(1,3)$ with $\ker\lambda=\{\pm I\}$.

proof. a direct computation gives $\det X=-(x^0)^2+(x^1)^2+(x^2)^2+(x^3)^2$ (up to the overall sign fixed by the metric convention), so $\det X$ reproduces the Minkowski quadratic form. conjugation $X\mapsto AXA^\dagger$ with $\det A=1$ preserves the Hermitian character of $X$ and preserves $\det X$, hence induces a linear isometry of Minkowski space. the map $A\mapsto\lambda(A)$ is a continuous homomorphism, and $\lambda(\pm I)=\mathrm{id}$. surjectivity onto $\mathrm{SO}^+(1,3)$ follows because boosts and spatial rotations are generated by the standard one-parameter subgroups of $\operatorname{SL}(2,\mathbb{C})$ under this action. if $\lambda(A)=I$ then $AXA^\dagger=X$ for all Hermitian $X$, so $A$ commutes with all such $X$ and Schur's lemma in this representation forces $A=\pm I$. thus $\lambda$ is a twofold covering. $\square$

definition (spin structure). let $P_{\mathrm{SO}}\to M$ be the principal $\mathrm{SO}^+(1,3)$-bundle of oriented orthonormal frames of an oriented Lorentzian manifold $(M,g)$. a spin structure is a principal $\operatorname{Spin}(1,3)$-bundle $P_{\operatorname{Spin}}\to M$ together with a twofold covering $P_{\operatorname{Spin}}\to P_{\mathrm{SO}}$ that intertwines the free right actions via $\lambda:\operatorname{Spin}(1,3)\to\mathrm{SO}^+(1,3)$.

theorem (obstruction to a spin structure). an oriented Lorentzian (or Riemannian) manifold admits a spin structure if and only if the second Stiefel-Whitney class $w_2(TM)\in H^2(M;\mathbb{Z}/2)$ vanishes. when a spin structure exists, the set of isomorphism classes of spin structures is a torsor over $H^1(M;\mathbb{Z}/2)$.

proof. the short exact sequence of Lie groups $$ 1\to\mathbb{Z}/2\to\operatorname{Spin}(1,3)\xrightarrow{\lambda}\mathrm{SO}^+(1,3)\to 1 $$ induces a long exact sequence in nonabelian cohomology for the sheaves of smooth group-valued maps. the primary obstruction to lifting the classifying cocycle of $P_{\mathrm{SO}}$ is a class in $H^2(M;\mathbb{Z}/2)$, identified with $w_2(TM)$ by the definition of Stiefel-Whitney classes via the orthogonal frame bundle. when the obstruction vanishes, different lifts differ by cohomology classes in $H^1(M;\mathbb{Z}/2)$, which act simply transitively on the set of spin structures. $\square$

definition (spinor and dirac bundles). given a spin structure, the associated bundles for the fundamental representation of $\operatorname{SL}(2,\mathbb{C})$ on $\mathbb{C}^2$ and its conjugate yield the Weyl spinor bundles $S$ and $\bar S$. the Dirac spinor bundle is $S\oplus\bar S$. a Weyl (resp. Dirac) spinor field is a smooth section of $S$ or $\bar S$ (resp. of $S\oplus\bar S$).

remark (why tensors fail for fermions). a continuous family of rotations by angle $t\in[0,2\pi]$ is a loop in $\mathrm{SO}(3)$ that is nontrivial in $\pi_1$, and lifts to a path in $\operatorname{Spin}(3)\simeq\operatorname{SU}(2)$ from $I$ to $-I$. spin-$1/2$ fields change sign under this lift; no tensor field on $TM$ can reproduce that projective representation of the Lorentz algebra while remaining single-valued on the frame bundle alone.

abstract-index formalism, infeld-van der waerden symbols, and null flags

Penrose's abstract-index notation provides a coordinate-free yet index-explicit calculus for spinors. Lower-case Greek indices $\mu,\nu,\dots$ label world-tensors, while unprimed and primed capital indices $A,B,\dots$ and $A',B',\dots$ label spinor and conjugate-spinor indices respectively. The Infeld-van der Waerden symbols $\sigma^\mu{}_{AA'}$ realize the isomorphism $$ T_p M\otimes\mathbb{C}\simeq S_p\otimes\bar S_p $$ and convert between world-vectors and Hermitian spinors: $$ v^\mu\leftrightarrow v^{AA'}=\sigma^\mu{}_{BB'}v^{BB'}. $$

definition (infeld-van der waerden symbols). a soldering form (soldering map) $\sigma^\mu{}_{AA'}$ is a bundle isomorphism $$ \sigma:S\otimes\bar S\xrightarrow{\sim} TM\otimes\mathbb{C} $$ intertwining the $\operatorname{Spin}(1,3)$ and $\mathrm{SO}^+(1,3)$ actions via $\lambda$, Hermitian in the sense that real vectors correspond to Hermitian spinors $v^{AA'}=\overline{v^{A'A}}$.

theorem (vector-spinor dictionary). the map $v^\mu\mapsto v^{AA'}=\sigma^\mu{}_{BB'} v^{BB'}$ (with inverse $v^{AA'}\mapsto v^\mu=\sigma_\mu{}^{AA'}v^{AA'}$ after lowering with $g$ and the spinor metric as appropriate) is a linear isomorphism $T_p M\otimes\mathbb{C}\simeq S_p\otimes\bar S_p$ intertwining the Lorentz and spin actions. moreover $$ g_{\mu\nu}v^\mu w^\nu =\varepsilon_{AB}\varepsilon_{A'B'}v^{AA'}w^{BB'} $$ for a suitably normalized spin metric $\varepsilon_{AB}$.

proof. on Minkowski space the identity is the explicit Pauli correspondence of the double-cover theorem: $X=v^\mu\sigma_\mu$ is Hermitian precisely when $v$ is real, and $\operatorname{tr}(XY^\dagger)$ recovers $2\eta(v,w)$ up to fixed convention. parallel transport and the soldering condition $\nabla\sigma=0$ for the spin connection (below) extend the pointwise algebra isomorphism from each tangent space to a global bundle isomorphism on a spin manifold. $\square$

A nonvanishing spinor $\xi^A$ determines a real null vector $\xi^A\bar\xi^{A'}$ and a flag plane orthogonal to it; the pair $(\xi^A,\text{flag plane})$ is called a null flag. The totality of null flags at a point reproduces the celestial sphere. The spinor space is totally reflexive: the dual of the dual recovers the original space, ensuring that all tensorial operations constructed from spinors remain globally consistent once a continuous choice of spin structure is fixed.

definition (null flag). for $\xi^A\neq 0$, the flagpole is the null vector $\ell^\mu\leftrightarrow\xi^A\bar\xi^{A'}$, and the flag plane is the two-plane of real bivectors generated by $\ell$ and directions obtained by varying the phase of $\xi$ (equivalently, the real part of $\xi^A\eta^{A'}$ for $\eta$ not proportional to $\xi$). the pair $(\xi^A,\text{flag plane})$ is a null flag.

theorem (null flags and the celestial sphere). the projectivization $\mathbb{P}(S_p) \simeq\mathbb{CP}^1$ is in natural bijection with the celestial sphere of future null directions at $p$. each projective class $[\xi^A]$ determines the ray of the flagpole $\xi^A\bar\xi^{A'}$, and every future null direction arises uniquely this way up to scale.

proof. a future null vector $k$ has vanishing Minkowski norm and positive energy component in a time orientation. under the Hermitian correspondence it is (up to scale) a pure tensor $\xi\otimes\bar\xi$ with $\xi\neq 0$. different scales and overall complex phases of $\xi$ that leave $k$ unchanged act by $\xi\mapsto\lambda\xi$ for $\lambda\in\mathbb{C}^\times$, which is precisely passage to $\mathbb{P}(S_p)$. the map is bijective by construction of the soldering isomorphism. $\square$

remark (reflexivity). the spinor metric $\varepsilon_{AB}=-\varepsilon_{BA}$ (unique up to scale once the complex volume form is fixed) identifies $S\simeq S^*$ by $\xi_A=\varepsilon_{AB}\xi^B$, so dualization is intrinsic. abstract indices keep this isomorphism notationally invisible while preserving tensorial type.

spinor covariant derivatives and einstein-cartan theory

A spin connection is obtained by lifting the Levi-Civita connection of $g$ to the spin bundle. When torsion is admitted, the connection acquires an independent contorsion tensor and the geometry becomes that of Einstein-Cartan-Sciama-Kibble theory. The spinor covariant derivative $\nabla_{AA'}$ then satisfies $$ \nabla_{AA'}\varepsilon_{BC}=0,\qquad \nabla_{AA'}\sigma^\mu{}_{BB'}=0 $$ (with torsion terms appearing in the commutators). The curvature of the spin connection decomposes into the Weyl spinor $\Psi_{ABCD}$, the Ricci spinor $\Phi_{ABA'B'}$, and the scalar curvature. The Bel-Robinson tensor, a completely symmetric, trace-free, positive-energy quadratic in the Weyl curvature, admits the elegant spinor expression $$ T_{AA'BB'CC'DD'}=\Psi_{ABCD}\bar\Psi_{A'B'C'D'}. $$ Its divergence vanishes whenever the Bianchi identities hold, furnishing a quasilocal measure of gravitational energy flux.

definition (spin connection). the unique lift of a metric-compatible affine connection on $TM$ to a connection on $P_{\operatorname{Spin}}$ (equivalently on $S$) is the spin connection. for Levi-Civita one obtains the torsion-free spin connection $\nabla^S$. components of the spinor covariant derivative are written $\nabla_{AA'}$ in the Infeld-van der Waerden soldering.

theorem (uniqueness of the metric spin connection). given a spin structure and a pseudo-Riemannian metric $g$, there is a unique connection $\nabla$ on $S$ such that $\nabla\varepsilon=0$, the induced connection on $TM$ via $\sigma$ is the Levi-Civita connection of $g$, and $\nabla\sigma=0$.

proof. the orthonormal frame bundle carries the unique Levi-Civita connection form $\omega_{\mathrm{LC}}$. because $\lambda_*:\mathfrak{spin}(1,3)\to\mathfrak{so}(1,3)$ is a Lie algebra isomorphism, there is a unique pullback connection form $\omega_{\operatorname{Spin}}$ on $P_{\operatorname{Spin}}$ with $\lambda_*\omega_{\operatorname{Spin}}=\omega_{\mathrm{LC}}$ under the double cover $P_{\operatorname{Spin}}\to P_{\mathrm{SO}}$. the associated connection on $S$ automatically preserves $\varepsilon$ (structure group in $\operatorname{SL}(2,\mathbb{C})$) and solders to Levi-Civita. any other connection with the same properties would differ by a tensorial one-form annihilated by these constraints, hence vanishing. $\square$

definition (contorsion and einstein-cartan connection). if the affine connection $\nabla$ on $TM$ is metric-compatible but has torsion $T$, one writes $\nabla=\nabla^{\mathrm{LC}}+K$ with contorsion tensor $K$ determined by $T$. the same $K$ lifts to the spin bundle. Einstein-Cartan-Sciama-Kibble theory couples $K$ algebraically to fermionic spin density.

theorem (spinor decomposition of curvature). the curvature of the spin connection decomposes uniquely into irreducible spinor parts: a totally symmetric Weyl spinor $\Psi_{ABCD}=\Psi_{(ABCD)}$, a Hermitian Ricci spinor $\Phi_{ABA'B'}$, and a real scalar curvature piece. the algebraic Weyl tensor of $g$ is recovered from $\Psi_{ABCD}$ via the soldering map.

proof. the complexified bivector space $\Lambda^2 T_p M\otimes\mathbb{C}$ is identified with $\operatorname{Sym}^2 S^*\oplus\operatorname{Sym}^2\bar S^*$ under the soldering isomorphism. the Riemann tensor, after imposition of the algebraic symmetries of chapter 2, is an element of $\operatorname{Sym}^2(\Lambda^2)^*$ and therefore decomposes into the indicated irreducible $\operatorname{SL}(2,\mathbb{C})$ representations. the totally trace-free self-dual part is $\Psi_{ABCD}$; contractions produce $\Phi_{ABA'B'}$ and the scalar. $\square$

theorem (bel-robinson and bianchi). define $T_{AA'BB'CC'DD'}=\Psi_{ABCD}\bar\Psi_{A'B'C'D'}$. then $T$ is totally symmetric, trace-free on any pair of unprimed (resp. primed) indices, and satisfies the dominant energy condition in vacuum whenever $\Psi$ is the Weyl spinor of $g$. moreover, if the Bianchi identities for the spin connection hold, then $\nabla^{AA'}T_{AA'BB'CC'DD'}=0$ after the vacuum Bianchi constraints on $\Psi$ are used.

proof. total symmetry and vanishing traces are purely algebraic consequences of $\Psi_{ABCD}=\Psi_{(ABCD)}$ and of the spinor contractions with $\varepsilon^{AB}$. positivity of the associated energy-momentum functional on observers is the standard Bel-Robinson estimate (the Hermitian form induced by $\Psi\otimes\bar\Psi$ on self-dual bivectors is positive semidefinite). for the divergence, the second Bianchi identity in spinor form reduces, in vacuum, to $\nabla^{A'A}\Psi_{ABCD}=0$; contracting against $\bar\Psi$ and using symmetry produces vanishing of $\nabla^{AA'}T_{AA'\cdots}$. $\square$

remark (quasilocal energy flux). the Bel-Robinson tensor is not itself a gravitational stress-energy tensor in the sense of a variational Noether current, but its conservation and positivity make it a standard quasilocal surrogate for gravitational energy flux in the analysis of the Einstein equation.

compacted spin-coefficient formalism and spin-weighted harmonics

The Newman-Penrose null-tetrad formalism and its compacted spin-coefficient version reduce the spinor covariant derivative to a set of twelve complex scalar operators $(\mathrm{thorn},\mathrm{eth},\dots)$ and twelve spin coefficients. Cartan's method of moving frames yields the same structure equations in the language of exterior forms. On the celestial two-sphere the operators $\mathrm{eth}$ and $\bar{\mathrm{eth}}$ become the spin-weighted raising and lowering operators; their eigenfunctions are the spin-weighted spherical harmonics ${}_s Y_{\ell m}$. These harmonics provide a complete basis for the expansion of zero-rest-mass fields of arbitrary spin and are indispensable for the asymptotic analysis of radiation.

definition (null tetrad). a Newman-Penrose null tetrad is a local frame $(\ell,n,m,\bar m)$ of null vectors with $$ g(\ell,n)=-1,\qquad g(m,\bar m)=1, $$ and all other inner products zero (up to standard signature conventions). equivalently one chooses spinor dyads $(o^A,\iota^A)$ with $\varepsilon_{AB}o^A\iota^B=1$ and sets $$ \ell\leftrightarrow o^A\bar o^{A'},\quad n\leftrightarrow\iota^A\bar\iota^{A'},\quad m\leftrightarrow o^A\bar\iota^{A'}. $$

definition (spin coefficients). the twelve complex spin coefficients are the directional covariant derivatives of the dyad spinors (or equivalently of the null tetrad) along the tetrad legs: $\kappa,\sigma,\rho,\tau,\varepsilon,\beta,\pi,\mu,\gamma,\lambda,\alpha,\nu$ in Newman-Penrose notation. the compacted operators $\mathrm{thorn}$, $\mathrm{eth}$, and their primed analogs are the components of $\nabla_{AA'}$ projected onto the dyad.

theorem (structure equations as cartan equations). the first and second Cartan structure equations for the orthonormal (or null) coframe, $$ d\theta^a+\omega^a{}_b\wedge\theta^b=T^a,\qquad d\omega^a{}_b+\omega^a{}_c\wedge\omega^c{}_b=\Omega^a{}_b, $$ are equivalent, under the soldering map, to the Newman-Penrose commutation relations and curvature equations for the spin coefficients and the Weyl/Ricci spinors.

proof. the connection one-forms $\omega^a{}_b$ encode exactly the same information as the spin coefficients once a null tetrad is dualized to a coframe. exterior differentiation of the coframe produces torsion (vanishing for Levi-Civita) and identifies $d\theta+\omega\wedge\theta=0$ with the Newman-Penrose metric identities among spin coefficients. exterior differentiation of $\omega$ produces the curvature two-forms, whose self-dual parts are linear in $\Psi_{ABCD}$ and $\Phi_{ABA'B'}$. matching components with the tetrad decomposition yields the Newman-Penrose curvature equations. $\square$

definition (spin-weighted spherical harmonics). on the unit two-sphere, a function $\eta$ has spin weight $s$ if it transforms as $\eta\mapsto e^{is\psi}\eta$ under a rotation of the dyad by phase $\psi$. the operators $\mathrm{eth}$ and $\bar{\mathrm{eth}}$ raise and lower spin weight by one. the spin-weighted spherical harmonics ${}_s Y_{\ell m}$ are a complete orthonormal basis of weight-$s$ functions with $$ \bar{\mathrm{eth}}\,\mathrm{eth}\,{}_s Y_{\ell m}=-(\ell-s)(\ell+s+1)\,{}_s Y_{\ell m}. $$

theorem (completeness for fixed spin weight). for each integer or half-integer $s$ and each $\ell\ge|s|$, the set $\{{}_s Y_{\ell m}:|m|\le\ell\}$ is an orthonormal basis of $L^2(S^2)$ sections of the spin-weight-$s$ line bundle. in particular every smooth weight-$s$ field expands uniquely as $\sum_{\ell,m}a_{\ell m}\,{}_s Y_{\ell m}$.

proof. the round sphere carries the Hopf spin structure; weight-$s$ functions are sections of $L^{\otimes 2s}$ for the Hopf line bundle $L\to S^2$. the operators $\mathrm{eth},\bar{\mathrm{eth}}$ are (up to scale) the $\bar\partial$ and $\partial$ operators of that bundle, and the spherical Laplacian restricted to weight $s$ is $\bar{\mathrm{eth}}\,\mathrm{eth}$ shifted by constants. its eigenfunctions are obtained by applying $\mathrm{eth}^{s}$ (or $\bar{\mathrm{eth}}^{|s|}$) to ordinary spherical harmonics $Y_{\ell m}$ with $\ell\ge|s|$, which produces ${}_s Y_{\ell m}$. spectral theorem for the compact self-adjoint Laplacian gives completeness and orthogonality. $\square$

massless fields, conformal rescalings, and exact integrals

Zero-rest-mass free fields of spin $s$ are described by totally symmetric spinors $\phi_{A_1\dots A_{2s}}$ satisfying the massless field equation $$ \nabla^{A_1 A'}\phi_{A_1\dots A_{2s}}=0. $$ Under a conformal rescaling $g_{\mu\nu}\mapsto\Omega^2 g_{\mu\nu}$ the spinor field transforms with a definite conformal weight, allowing the equations to be transferred to any conformally related metric. The characteristic initial-value problem on a light cone admits an explicit integral solution: the value of the field at a point is recovered by integrating suitable derivatives of the initial data over the intersection of the past light cone with an initial null hypersurface. Specializing to spin $1$ yields the Kirchhoff-d'Adhemar formula for the electromagnetic field; the coupled Einstein-Maxwell system likewise possesses exact integral representations once the conformal factor and the Maxwell spinor are prescribed on the initial null surface.

definition (zero-rest-mass field of spin $s$). a smooth totally symmetric spinor field $\phi_{A_1\dots A_{2s}}=\phi_{(A_1\dots A_{2s})}$ satisfying $\nabla^{A_1 A'}\phi_{A_1\dots A_{2s}}=0$. for $s=1$ one recovers the anti-self-dual Maxwell spinor; for $s=2$ the linearized vacuum Weyl spinor equation.

theorem (conformal covariance). let $\tilde g=\Omega^2 g$ with $\Omega>0$ smooth, and let the corresponding spin structures be compatible. if $\phi_{A_1\dots A_{2s}}$ solves the massless equation for $g$, then $$ \tilde\phi_{A_1\dots A_{2s}} =\Omega^{-1}\phi_{A_1\dots A_{2s}} $$ solves the massless equation for $\tilde g$ (with the spinor covariant derivative of $\tilde g$), for the conformal weight appropriate to spin $s$ in four dimensions (equivalently $\tilde\phi=\Omega^{-1}\phi$ in the standard Penrose convention for the unprimed field).

proof. under $\tilde g=\Omega^2 g$ the Levi-Civita connection changes by terms linear in $d\log\Omega$. the induced spin connection differs from $\nabla$ by a one-form built from $\Upsilon_\mu=\partial_\mu\log\Omega$. contracting $\tilde\nabla^{A_1 A'}$ against a weight-adjusted spinor produces extra $\Upsilon$ terms that cancel precisely against the derivative of the conformal factor in $\tilde\phi=\Omega^{-1}\phi$, leaving $\tilde\nabla^{A_1 A'}\tilde\phi_{A_1\dots A_{2s}}=\Omega^{-2}\nabla^{A_1 A'}\phi_{A_1\dots A_{2s}}$ up to the fixed normalization of $\sigma^\mu{}_{AA'}$ under conformal rescaling. vacuum of the original equation implies vacuum of the rescaled equation. $\square$

theorem (characteristic integral representation, spin $1$). let $F$ be a smooth source-free Maxwell field on a globally hyperbolic region, written as a Maxwell spinor $\phi_{AB}$ with $F\leftrightarrow\phi_{AB}\varepsilon_{A'B'}+\mathrm{c.c.}$. for a point $p$ in the domain of dependence of an initial null hypersurface $\mathcal{N}$, the value $\phi_{AB}(p)$ is given by a Kirchhoff-d'Adhemar integral of $\phi$ and its derivatives over the cut $C_p=\mathcal{N}\cap\dot J^-(p)$, with kernel determined by the null geodesic generators of the past light cone of $p$.

proof. in flat space the retarded fundamental solution of the wave equation reduces Maxwell's equation for $\phi_{AB}$ (which implies $\square\phi_{AB}=0$ after using $\nabla^{A'A}\phi_{AB}=0$) to an integral over the past light cone. Stokes' theorem on the region bounded by that cone and $\mathcal{N}$ converts the volume integral of a total divergence into a surface integral over $C_p$, with the singular cone contribution isolated as the free-data Kirchhoff term. in curved vacuum (or Einstein-Maxwell) the same argument applies after conformal rescaling to a metric for which the light cone is shear-free in adapted coordinates (Penrose's conformal method); the weights of theorem (conformal covariance) track the measure factors on $C_p$. $\square$

Taken together, the spinor calculus supplies both the kinematic language required for fermionic classical fields and a powerful computational toolkit for the bosonic curvature quantities of general relativity. It converts the geometric structures of the preceding chapter into a form optimally adapted to the null geometry of spacetime and prepares the ground for the subsequent incorporation of global topological invariants and nonabelian gauge fields.

remark (to dynamics and topology). chapter 4 elevates spinors and tensors to fields with action principles; chapters 5 and 6 quantify the topological sectors that label spin and gauge structures globally.

exercises

exercise 1 (2pi lift). exhibit an explicit path $A(t)\in\operatorname{SU}(2)$, $t\in[0,2\pi]$, with $A(0)=I$ and $A(2\pi)=-I$, whose projection under $\lambda:\operatorname{SU}(2)\to\mathrm{SO}(3)$ is a family of rotations by angle $t$ about a fixed axis. conclude that a spinor changes sign under a full rotation of the frame.

exercise 2 (null flag). for $\xi^A=(1,0)$ in a standard dyad on Minkowski space, compute the flagpole $\ell^\mu$ and verify $g(\ell,\ell)=0$. describe the flag plane in an orthonormal frame.

exercise 3 (weyl spinor of maxwell). for a source-free Maxwell field with spinor $\phi_{AB}=\phi_{(AB)}$, define the analogous Bel-Robinson-like density $T_{AA'BB'}=\phi_{AB}\bar\phi_{A'B'}$. show that $\nabla^{AA'}T_{AA'BB'}=0$ whenever $\nabla^{A'A}\phi_{AB}=0$, by a short index computation.

exercise 4 (minkowski boost on null vectors). in $(1+1)$ Minkowski space with metric $-dt^2+dx^2$, write a boost of rapidity $\rho$ and show that it preserves each light ray through the origin setwise while acting nontrivially on non-null vectors. compute the action on the future unit hyperbola $t^2-x^2=1$.