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Choose a local orthonormal coframe $e^A$ with $g={\eta }_{AB}{e}^A{e}^B$. The Levi-Civita connection is represented by the connection 1-forms satisfyingThe gamma matrices satisfy the Clifford algebra relations $\{{\gamma }^A,{\gamma }^B\}=2{\eta }^{AB}I$. The matrices $\frac{1}{4}[{\gamma }^A,{\gamma }^B]$ represent the infinitesimal orthogonal transformations on spinor fields; lifting the frame connection to this representation defines the spinor covariant derivativeBoth frame indices are summed, so each antisymmetric pair occurs twice. Under a local Lorentz transformation with spin lift $S$, the spin connection transforms as ${\Omega }_{\mu }\mapsto S{\Omega }_{\mu }{S}^{-1}-({\partial }_{\mu }S)S^{-1}$ and $ϵ\mapsto Sϵ$; consequently ${\nabla }_{\mu }ϵ\mapsto S{\nabla }_{\mu }ϵ$. This establishes covariance. Global construction requires a compatible spin structure (and the appropriate orientation and time orientation); the formulas themselves work on a local orthonormal frame patch.

The supplied gamma matrices have squares $-I,I,I$ and anticommute pairwise, so $\eta =\mathrm{diag}(-1,1,1)$. Work on a connected regular patch with $r>0$ and $V\ne 0$. Since the metric depends only on $V^2$, we may use its positive smooth square root $V>0$. TakeTheir exterior derivatives are $d{e}^0=-V^{\prime }{e}^0\wedge e^1$, $d{e}^1=0$, and $d{e}^2=(V/r)e^1\wedge e^2$. Solving Cartan's first structure equation and imposing Lorentzian connection-form antisymmetry givesIn particular ${\omega }_{01}=-V{V}^{\prime }dt$ and ${\omega }_{12}=-Vd\theta$. The parallel spinor equations are thereforeDifferentiate the angular equation with respect to $r$ and use ${\partial }_rϵ=0$. This gives $V^{\prime }{\gamma }^1{\gamma }^2ϵ=0$. Since $({\gamma }^1{\gamma }^2{)}^2=-I$, this matrix is invertible. A parallel spinor that vanishes at any point vanishes everywhere on the connected domain, by uniqueness of parallel transport. Thus a nonzero parallel spinor implies $V^{\prime }=0$ throughout the patch. There are no nonzero parallel spinors unless $V$ is constant. The zero spinor field is always a solution, so “nonzero” is essential in interpreting the printed assertion. A smooth original $V$ of either sign is constant on such a patch exactly when $|V|$ is constant; $V=0$ is excluded because the given metric is not defined there.

Write the constant as $c=|V|>0$, and introduce $T=ct$, $\rho =r/c$, and $\varphi =c\theta$. The metric becomesA circle at small $\rho$ has circumference $cP\rho$ if $\theta$ has period $P$, whereas its proper radius is $\rho$. Smoothness requires circumference divided by radius to tend to $2\pi$, not a multiple of $2\pi$. Conversely, when $cP=2\pi$, the coordinates $X=\rho \cos \varphi$, $Y=\rho \sin \varphi$ exhibit a smooth flat spatial disk. HenceFor a prescribed period $2\pi$, this requires $c=1$; otherwise there is a conical singularity at the apex.

For constant $V=c$, the time and radial equations make the components independent of $t,r$. The angular equation has all solutionswhere ${ϵ}_0$ is an arbitrary constant two-component spinor field and ${\gamma }^1{\gamma }^2=-{\gamma }^0$. There are two independent real solutions for real spinor fields, or two independent complex solutions if complex spinors are used. At the smooth angular period, these polar-frame components change sign. This is exactly the polar-frame parallel spinor transformation under the spin lift of a full rotation: the rotating polar orthonormal frame does not extend through the origin. The spin structure extending over the disk uses this antiperiodic polar trivialization; in a Cartesian orthonormal frame the same parallel spinors have constant components and extend smoothly. Requiring periodic components in the rotating frame would select the other spin structure on the punctured disk, which does not extend over its center.

Treat the orthonormal coframe in spacetime and the Lorentz connection 1-form as independent variables, with nondegenerate coframe and ${\omega }_{ab}=-{\omega }_{ba}$. Fix the curvature conventionThe second expression is Cartan's first structure equation; the first is Cartan's second structure equation. Use compactly supported variations, or boundary data that make the variation's surface term vanish.

For a connection variation, $\delta R^{ab}=D\delta {\omega }^{ab}$, where $D$ is the covariant exterior derivative. Since $D{ϵ}_{abc}=0$, its graded product rule givesFor a coframe variation, relabeling the dummy indices makes the three variations of the cubic term identical. HenceThe independent variations therefore give ${ϵ}_{abc}{T}^c=0$ and ${ϵ}_{abc}(R^{ab}+3\lambda E^a\wedge E^b)=0$. In three dimensions the alternating symbol identifies an antisymmetric index pair with a single index, so these equations are equivalent toThese are the equations of first-order three-dimensional gravity. The first selects the Levi-Civita connection; the second fixes constant sectional curvature $K=-3\lambda$. The factor of three follows from varying the coefficient exactly as printed.

On a static coordinate patch where $f>0$, choose the Lorentz frame metric ${\eta }_{ab}=\mathrm{diag}(-1,1,1)$ and coframePut $A=f^{\prime }/(2\sqrt{f})$ and $B=\sqrt{f}/r$. Exterior differentiation gives $d{E}^0=-A{E}^0\wedge E^1$, $d{E}^1=0$, and $d{E}^2=B{E}^1\wedge E^2$. Solving the equation of vanishing torsion forms givesHere antisymmetry is in the two raised or two lowered frame indices; mixed time-space components have equal, rather than opposite, signs. This is the Lorentzian connection-form antisymmetry needed to retain the curvature signs.

Using Cartan's second structure equation, the nonzero curvature 2-forms areFor example, $d(A{E}^0)=-(\sqrt{f}{A}^{\prime }+A^2)E^0\wedge E^1=-f^{\prime \prime }{E}^0\wedge E^1/2$, while the $02$ form is the product ${\omega }^0{}_1\wedge {\omega }^{12}=-AB{E}^0\wedge E^2$. Thus the field equations require $f^{\prime \prime }/2=3\lambda$ and $f^{\prime }/(2r)=3\lambda$. Integrating the latter gives $f=C+3\lambda r^2$; the former is then satisfied automatically. The origin condition fixes $C=1$, yieldingAs a sign and normalization check, the Ricci tensor is $R_{\mu \nu }=-6\lambda g_{\mu \nu }$ and the scalar curvature is $R=-18\lambda$. The coframe calculation applies initially in the region with $f>0$; if $\lambda <0$, $f$ vanishes at $r=1/\sqrt{-3\lambda }$, delimiting that static chart without changing the derived function.

The displayed $e^a$ are one-forms, so they form an orthonormal coframe, dual to the orthonormal frame $E_0=A^{-1}{\partial }_t$, $E_1=A^{-1}{\partial }_x$, $E_2=A^{-1}{\partial }_y$, $E_3={\partial }_z$. Use ${\eta }_{ab}=\mathrm{diag}(-1,1,1,1)$ and abbreviateFor $i=0,1,2$, $d{e}^i=a{e}^3\wedge e^i$, while $d{e}^3=0$. The Levi-Civita connection is fixed by Cartan's first structure equation together with Lorentzian connection-form antisymmetry, ${\omega }_{ab}=-{\omega }_{ba}$, where ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$. Its nonzero connection 1-forms areIn particular, the time-index pair has the same sign with one index raised; incorrectly imposing ${\omega }^a{}_b=-{\omega }^b{}_a$ would lose the Lorentzian sign. All diagonal and all other entries vanish.

Apply Cartan's second structure equation. For $i=0,1,2$,For distinct $i,j$ in $\{0,1,2\}$, putting $s_0=-1$ and $s_1=s_2=1$ gives ${\Theta }^i{}_j={\omega }^i{}_3\wedge {\omega }^3{}_j=-s_j{a}^2{e}^i\wedge e^j$. Thus a complete independent set of curvature 2-forms isAll diagonal entries vanish. The remaining entries follow from ${\Theta }_{ab}=-{\Theta }_{ba}$: ${\Theta }^j{}_i=-s_i{s}_j{\Theta }^i{}_j$ for $i,j\le 2$, and ${\Theta }^3{}_i=-s_i{\Theta }^i{}_3$. These relations and the displayed six entries specify all sixteen curvature 2-forms. This computation is Cartan curvature of a planar warped spacetime.

Treat the printed $e^{\mu }$ as an orthonormal coframe, with frame metric $\eta =\mathrm{diag}(-1,1,1,1)$. Work on a patch $z\ne 0$ and $F=1-\alpha z^3>0$, so the given real coframe exists. Define $f=\sqrt{F}$ and $A=z{f}^{\prime }-f$. The exterior derivatives areThe connection 1-forms satisfying Cartan's first structure equation and metric compatibility areAll other forms vanish. With a Lorentzian frame, it is the lowered forms ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$ that are antisymmetric; the two mixed time–space forms are equal. These displayed forms solve the torsion-free structure equation, and uniqueness of the Levi-Civita connection identifies them as the required connection.

Apply Cartan's second structure equation. For example,The remaining derivatives give ${\Theta }^1{}_3=Af{e}^1\wedge e^3$ and ${\Theta }^1{}_2=-f^2{e}^1\wedge e^2$, with analogous forms for index 2. Put $q=\alpha z^3$ and $B=1+q/2$. From $f^2=1-q$,Therefore all six independent curvature 2-forms areThe other six are fixed by ${\Theta }^b{}_a=-{\eta }_{aa}{\eta }_{bb}{\Theta }^a{}_b$ for $a\ne b$, with no sum: equal for time–space pairs and opposite for spatial pairs. All diagonal forms are zero. The Lorentzian connection-form antisymmetry is essential to these signs.