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For a nondegenerate orthonormal coframe in spacetime $E^a$ and independent antisymmetric Lorentz connection 1-form, the action $\int {ϵ}_{abc}(R^{ab}\wedge E^c+\lambda E^a\wedge E^b\wedge E^c)$ has independent coframe and connection variations. Since $\delta R^{ab}=D\delta {\omega }^{ab}$, integration by parts yields a connection equation ${ϵ}_{abc}{T}^c=0$. The three variations of the cubic coframe term give the coframe equation ${ϵ}_{abc}(R^{ab}+3\lambda E^a\wedge E^b)=0$. In three dimensions these are exactly the equations displayed above. The solution has constant sectional curvature $K=-3\lambda$. Omitting the factor of three changes the cosmological normalization.

An orthonormal coframe in spacetime can be rotated by a position-dependent Lorentz matrix without changing the spacetime metric tensor. Its spin connection transforms as ${\omega }^{\prime }=\Lambda \omega {\Lambda }^{-1}-d\Lambda {\Lambda }^{-1}$. A spinor field transforms through a chosen spin lift ${\psi }^{\prime }=S(\Lambda )\psi$, not through the vector matrix $\Lambda$. With $S^{-1}{\gamma }^{A}S={\Lambda }^A{}_B{\gamma }^B$, an infinitesimal parameter gives $S=1+\frac{1}{4}{\alpha }_{AB}{\gamma }^{AB}+O({\alpha }^2)$, where ${\gamma }^{AB}=\frac{1}{2}[{\gamma }^A,{\gamma }^B]$. Consistent global transition lifts require a spin structure; a $2\pi$ spatial rotation lifts to minus the identity on spinors.

In an orthonormal coframe in spacetime, antisymmetry applies after lowering the first index with the Lorentzian frame metric. Thus mixed time–space connection 1-forms are equal under index interchange, while mixed spatial forms are opposite. The same rule holds for curvature 2-forms. Confusing lowered antisymmetry with ordinary mixed-index antisymmetry changes the curvature signs.

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.

Use coordinate indices $\mu ,\nu$ and Lorentz-frame indices $A,B$, and take ${\eta }_{AB}=\mathrm{diag}(-1,1,1,1)$. Locally choose four independent one-forms $e^A=e^A{}_{\mu }d{x}^{\mu }$ satisfyingOne construction begins with a smooth timelike vector field in a sufficiently small neighborhood, normalizes it, and applies the Gram-Schmidt process to its spacelike orthogonal complement. The dual orthonormal coframe in spacetime gives the vierbein; equivalently its matrix factorizes the metric as $g=e^T\eta e$. The inverse matrix gives the frame $e_A=e_A{}^{\mu }{\partial }_{\mu }$ with $e_A{}^{\mu }{e}^B{}_{\mu }={\delta }_A^B$. This is a local construction: a global frame is not guaranteed by a metric, and a globally defined spinor field requires a compatible spin structure.

The metric does not determine a unique vierbein. A local Lorentz transformation changes the coframe components byThe coordinate index is unchanged in this frame transformation. Restrict to the proper orthochronous Lorentz group for its connected spin lift; orientation-reversing transformations require further structure and are not part of this local spin-group discussion.

Let constant gamma matrices obey the Clifford algebra $\{{\gamma }^A,{\gamma }^B\}=2{\eta }^{AB}$, and write ${\gamma }^{AB}=\frac{1}{2}[{\gamma }^A,{\gamma }^B]$. A spinor field transforms by the spin lift of the Lorentz transformation, not by its vector matrix:For $\Lambda =1+\alpha +O({\alpha }^2)$, with ${\alpha }_{AB}=-{\alpha }_{BA}$, the Clifford algebra identityshows explicitly thatIndeed the commutator of ${\gamma }^C$ with this infinitesimal spin matrix is ${\alpha }^C{}_B{\gamma }^B$. A finite transformation connected to the identity is obtained by exponentiating these generators along a path. Its lift has two possible signs, so spinors are not ordinary Lorentz vectors: a $2\pi$ spatial rotation acts by $-1$, and $4\pi$ by $+1$. Consistent transition lifts define the spin structure; the metric alone does not select it.

For the metric connection, take the Levi-Civita connection and determine the spin connection from the tetrad postulateIt is equivalently determined by $d{e}^A+{\omega }^A{}_B\wedge e^B=0$ and ${\omega }_{\mu AB}=-{\omega }_{\mu BA}$. Explicitly, in terms of the inverse frame,Thus both the coordinate connection and the Lorentz connection 1-form are determined by the metric and chosen frame; no independent torsion has been supplied.

An ordinary derivative fails covariance because ${\partial }_{\mu }(S\psi )=S{\partial }_{\mu }\psi +({\partial }_{\mu }S)\psi$. Seek ${\nabla }_{\mu }\psi =({\partial }_{\mu }+{\Omega }_{\mu })\psi$. Covariance requiresThe coframe's connection obeys ${\omega }^{\prime }=\Lambda \omega {\Lambda }^{-1}-d\Lambda {\Lambda }^{-1}$. Applying the spin representation of the same Lorentz algebra, whose infinitesimal generator was just determined, suppliesThis proves the transformation law rather than choosing the coefficient arbitrarily. It is the spinor covariant derivative induced by the spin connection. With curved matrices ${\gamma }^{\nu }(x)=e_A{}^{\nu }{\gamma }^A$, the same coefficient and tetrad postulate giveso differentiation is compatible with Clifford multiplication and $\{{\gamma }^{\mu }(x),{\gamma }^{\nu }(x)\}=2g^{\mu \nu }(x)$.

Finally, expand the commutator of these derivatives:The spin representation preserves the Lorentz algebra bracket. Consequently $d\Omega +\Omega \wedge \Omega =\frac{1}{4}{R}_{AB}{\gamma }^{AB}$, where $R^A{}_B=d{\omega }^A{}_B+{\omega }^A{}_C\wedge {\omega }^C{}_B$. In components the spinor curvature identity isFor the full second derivative, ${\nabla }_{\mu }({\nabla }_{\nu }\psi )$ also includes $-{\Gamma }_{\mu \nu }^{\rho }{\nabla }_{\rho }\psi$, because its first derivative has a covector index. This term cancels on antisymmetrization for the torsion-free connection. With unit-weight brackets, $X_{[\mu \nu ]}=(X_{\mu \nu }-X_{\nu \mu })/2$, the requested expression is thereforeThis convention distinguishes the antisymmetrized second derivative from the full commutator by its factor of two. In a noncoordinate orthonormal frame, an operator commutator must additionally subtract the derivative along the frame-vector commutator; using the full tensorial second derivative incorporates that correction.

Put $A,B\in \{1,2\}$ and write $h_{AB}=H_{AB}{e}^{ik\cdot x}$, where $H_{11}=H_{+}$, $H_{22}=-H_{+}$ and $H_{12}=H_{21}=H_{\times }$. Physical fields are the real parts. Use $c=1$ here, and take the initial separation at equal time, so $L^0=0$. For propagation in the positive $x^3$ direction one may set $k_{\mu }=(-\omega ,0,0,\omega )$; only $k_0^2={\omega }^2$ enters the answer.

The linearized Levi-Civita connection in this transverse-traceless gauge has the potentially nonzero componentswith all others zero to this order. In particular ${\Gamma }^{\mu }{}_{00}=0$, so a particle initially at rest has fixed spatial coordinates in transverse-traceless gauge, and $t$ equals its proper time. Expanding the Riemann curvature tensor givesThe plus sign in the last expression agrees with $R(T,S)T$ in the geodesic deviation convention above.

For the separation measured in a parallel-propagated orthonormal frame, write $S^{\hat{\mu }}=L^{\hat{\mu }}+{\xi }^{\hat{\mu }}$. The frame removes the connection terms from the time derivative, so the geodesic deviation equation reduces to ${\ddot{\xi }}^{\hat{A}}=R^{\hat{A}}{}_{\hat{0}\hat{0}\hat{B}}{L}^{\hat{B}}={\ddot{h}}_{AB}{L}^{\hat{B}}/2$ at first order. The observable transverse relative acceleration isThese are the plus polarization and cross polarization tidal patterns; there is no longitudinal acceleration. The curvature is evaluated along the reference worldline, and replacing $S$ by $L$ on the right changes only second-order terms. This is the infinitesimal-separation approximation intrinsic to geodesic deviation.

There is an important component distinction if the symbols $L^{\mu }+{\xi }^{\mu }$ are instead interpreted literally in the transverse-traceless gauge coordinate basis. Expanding the left side, rather than replacing a covariant derivative by an ordinary derivative, givesThus in TT coordinatesWith zero initial coordinate displacement and velocity, ${\xi }_{\mathrm{coord}}=0$ throughout. This does not remove the physical gravitational wave strain. To first order an orthonormal coframe in spacetime along the reference worldline is ${\vartheta }^{\hat{A}}=({\delta }^A{}_B+h_{AB}/2)d{x}^B$; its dual frame is parallel propagated because ${\partial }_0({\delta }^A{}_B-h_{AB}/2)+{\Gamma }^A{}_{0B}=0$. Henceafter choosing the initial flat-frame reference. The TT coordinate separation and measured separation distinction accounts exactly for the apparently different accelerations. In a region initially free of the wave, the measured displacement is ${\xi }^{\hat{A}}=h_{AB}{L}^{\hat{B}}/2$ with the corresponding initial conditions.

Choose signature $(-,+,+,+)$, with $e^0$ timelike. The orthonormal coframe in spacetime reconstructs the Lorentzian metric asBecause $A$ and $B$ are nowhere zero on the domain, the coframe is invertible and the metric nondegenerate; either sign of each function is allowed. In retarded and advanced null coordinates $u=t-z$, $v=t+z$ the metric is $-dudv+A(u{)}^{2}d{x}^2+B(u{)}^{2}d{y}^2$, a diagonal example of Rosen coordinates for a plane gravitational wave. Multiplication by $-1$ would give the opposite overall signature convention; all subsequent signs here use the displayed one.

At each point, apply the indefinite-signature version of the Gram-Schmidt process to any local coframe, choosing one timelike and three spacelike one-forms. This produces an orthonormal coframe in spacetime $\{{\theta }^a\}$ satisfyingIt is unique only up to a local Lorentz transformation. The torsion-free connection 1-forms are then determined by Cartan's first structure equation and metric compatibility,Cartan's second structure equation gives the curvature 2-formsand contraction gives the Ricci tensor $R_{bd}=R^a{}_{bad}$.

For the displayed metric, takeThen $d{\theta }^0=0$ and $d{\theta }^i=-{\theta }^0\wedge {\theta }^i$. The nonzero connection forms are thereforewith ${\omega }^i{}_j=0$. A second application of the structure equations givesEquivalently,so this is a constant sectional curvature spacetime with $K=1$. In four dimensions,The Vacuum Einstein equations with a cosmological constant are $G_{ab}+\Lambda g_{ab}=0$, and henceThe metric is the contracting flat slicing of de Sitter spacetime with curvature radius one.

An orthonormal frame in spacetime is a local basis $\{e_a\}$ satisfying $g(e_a,e_b)={\eta }_{ab}$. The metric determines it only up to a point-dependent Lorentz transformation, with orientation and time orientation optionally restricting the component of the Lorentz group. The orthonormal coframe in spacetime is defined byand the metric is

A zweibein is a two-dimensional orthonormal coframe in spacetime: its components obey $h_{\mu \nu }={\eta }_{AB}{e}^A{}_{\mu }{e}^B{}_{\nu }$. It converts worldsheet coordinate indices into local Lorentz-frame indices and defines curved gamma matrices ${\gamma }^{\mu }=e_A{}^{\mu }{\gamma }^A$. The associated spin connection supplies the covariant derivative of worldsheet spinors. In flat conformal gauge the coframe can be constant.