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The coordinate basis vector $K={\partial }_v$ has constant components $K^a={\delta }_v^a$. Its norm isso it is a nonzero null vector everywhere. The Lie derivative of the metric tensor isThus $\boxed{K={\partial }_v\text{ is a null Killing vector field}.}$ In fact it is a parallel vector field: $K_b=g_{bv}={\delta }_b^u$ and the Christoffel symbols found below give ${\nabla }_a{K}_b=-{\Gamma }^u{}_{ab}=0$. This stronger property is characteristic of this plane-fronted gravitational wave.

On a bosonic background, a Killing spinor is a nonzero spinor parameter that makes all fermionic supersymmetry variations vanish. It identifies an unbroken supersymmetry of the background; the number of independent solutions counts preserved supercharges. In simple ungauged minimal four-dimensional supergravity, with the gravitino set to zero, its equation is ${\nabla }_{\mu }ϵ=0$ and the bosonic field equations are $R_{\mu \nu }=0$. In theories with additional fields or a cosmological deformation, the Killing-spinor equation instead uses the corresponding modified connection.

Use commuting spinors to form ordinary geometric bilinears, and orient the Dirac current to the future with $\overline{ϵ}=-{ϵ}^{†}{\gamma }^{\hat{0}}$ in mostly-plus signature. Since the spinor and gamma matrices are parallel,Thus $K$ is actually parallel, and in particular is a Killing vector field. In a local orthonormal frame,For every spatial unit vector $u$, $u_i{\alpha }^i$ is Hermitian and squares to one. Its eigenvalues are $\pm 1$, so $|u_i{K}^i|\le K^0$. Taking $u$ along the spatial current givesA nonzero parallel spinor cannot vanish at an isolated point, so this causal Dirac current is nonzero everywhere on a connected solution. Its squared norm is constant: it cannot alternate between timelike and null regions.

If $K$ is timelike, normalize its constant norm. Its parallel dual one-form is closed, and its orthogonal distribution is integrable. Adapted local coordinates put the metric in the product formThe time independence follows because $K$ is Killing, and the unit lapse and zero cross terms follow from parallelism. The vacuum equations give $R_{ij}(h)=0$. In three dimensions the Riemann tensor is completely determined by the Ricci tensor and its scalar contraction, so $h$ is flat. ThereforeConstant spinors in its flat frame provide all local Killing spinors. Global identifications are possible only when they preserve a compatible spin structure and the required spinors.

If $K$ is null, its parallelism first gives a Brinkmann coordinate system with $K={\partial }_v$. The parallel-spinor condition further restricts the local spin holonomy to the stabilizer of that spinor. In four dimensions, for a nonzero Weyl component represented by $(1,0{)}^T$, its stabilizer in $SL(2,\mathbb{C})$ is the upper-triangular group $\left(\begin{array}{cc}1& b\\ 0& 1\end{array}\right)$, $b\in \mathbb{C}$: the two real null rotations, with no transverse rotation. Consequently the transverse frame is flat and the allowed curvature is of the wave type $R_{uiuj}$. Using the coordinate freedom to remove transverse cross terms gives the local plane-fronted gravitational wave formHere $R_{uiuj}=-H_{ij}/2$ and the only potentially nonzero Ricci component is $R_{uu}=-(H_{xx}+H_{yy})/2$, so the last equation is exactly the vacuum equation. Conversely, in the coframe $e^{+}=du$, $e^{-}=dv-Hdu/2$, $e^i=d{x}^i$, the nonzero spin-connection matrices are proportional to $H_i{\gamma }^{+i}du$. Constant spinors with ${\gamma }^{+}ϵ=0$ are therefore parallel. Every harmonic transverse profile, with arbitrary $u$ dependence, gives such a supersymmetric vacuum wave; the flat profile is included. This verifies both the form and sufficiency in the parallel-spinor classification of vacuum four-geometries.

There is a reality-condition subtlety in interpreting the branches. A single commuting four-dimensional Majorana spinor has a null current of a commuting Majorana spinor: its two-component representation gives $|{\xi }^{†}\mathbf{\sigma }\xi {|}^2=({\xi }^{†}\xi {)}^2$. A timelike current can instead come from a Dirac Killing spinor or a sum of independent real-spinor currents. The conditional timelike classification above remains valid, but does not assert a generic timelike current for a single real Majorana parameter.

For the final question use constant sectional curvature $-1/a^2$, so $R_{\mu \nu }=-3g_{\mu \nu }/a^2$. The spinor curvature identity givesDefine the Anti-de Sitter Killing-spinor connection ${\mathcal{D}}_{\mu }={\nabla }_{\mu }+{\gamma }_{\mu }/(2a)$. Since ${\nabla }_{\mu }{\gamma }_{\nu }=0$ and the connection is torsion-free,Thus the modified connection is flat. On a simply connected region, transport any initial spinor with this connection; flatness makes the result path independent and supplies actual nonzero solutions, not merely a necessary integrability condition. Hencehas a full local space of solutions on Anti-de Sitter spacetime, and globally on its universal cover. Global quotients require compatible spinor monodromy. These are Killing spinors of the cosmological deformation, not parallel-spinor vacua of the undeformed zero-cosmological-constant theory.

Using the affine parameter and the normalization requested for the geodesic Lagrangian,The coordinate $v$ is cyclic. Its Euler-Lagrange equation gives $\ddot{u}=0$. The transverse coordinate equations give $\ddot{x}+H_x{\dot{u}}^2=0$ and $\ddot{y}+H_y{\dot{u}}^2=0$. For the remaining coordinate,Its Euler-Lagrange equation, after using $\ddot{u}=0$, isCompare with ${\ddot{x}}^a+{\Gamma }_{bc}^a{\dot{x}}^b{\dot{x}}^c=0$. With indices $(1,2,3,4)=(u,v,x,y)$, the complete list of nonzero Christoffel symbols, including symmetry of the lower indices, isAll contracted symbols ${\Gamma }_{ad}^d$ vanish. For $R_{11}$, the first derivative term in the supplied Ricci tensor formula is ${\partial }_v{H}_u+{\partial }_x{H}_x+{\partial }_y{H}_y=H_{xx}+H_{yy}$, since $H$ is independent of $v$. The second derivative term vanishes because the trace symbols vanish. The third term vanishes for the same reason, and every possible product in the last term requires a zero symbol with upper index $1$ or a lower $2$, so it also vanishes. ThusThe Vacuum Einstein equations with zero cosmological constant require $R_{ab}=0$, and therefore $\boxed{H_{xx}+H_{yy}=0}$. The wave profile of this plane-fronted gravitational wave is harmonic in the transverse plane for each fixed $u$.

Put $u=t-z$. Wherever $|u|>U$, the profile $A(u)$ vanishes and the displayed metric is exactly the Minkowski metric. Thus the two flat regions in the $(t,z)$-plane are the half-planes $t-z>U$ and $t-z<-U$, separated by the stripAt fixed $t$ this strip is $t-U\le z\le t+U$, so it has longitudinal width $2U$. A surface $u=\text{constant}$ obeys $z=t-u$ and travels in the positive $z$ direction at the speed of light. The curvature is confined to that moving strip, identifying the solution as a gravitational-wave pulse represented by a plane-fronted gravitational wave.

For the given metric, direct substitution of the stated connection coefficients into the curvature formula leavesThe inverse metric has $g^{uu}=0$, and every other Ricci component vanishes, soThus the full, nonlinear vacuum Einstein equation is precisely the transverse Laplace equation from a plane-fronted gravitational wave:The dependence on $u$ is unrestricted.

For the plane-fronted gravitational wave $d{s}^2=2dudv+d{\mathbf{x}}^2+{\mathbf{x}}^{T}M(u)\mathbf{x}d{u}^2$, synchronized neighboring geodesics parametrized by $u$ have transverse separation $\mathbf{\xi }$ satisfying the displayed equation. Along the central timelike geodesic $(u,v,\mathbf{x})=(\tau ,-\tau /2,0)$ the coordinate basis is parallel transport, so this is also the transverse Jacobi field equation in a transported frame. Vacuum requires $\mathrm{tr}M=0$, producing equal and opposite instantaneous transverse tidal acceleration eigenvalues. The ordinary coordinate equation ${\ddot{\xi }}^v=0$ holds on that central reference, but generally fails on off-axis reference curves.