Cartan tetrad and spin-connection formulation of general relativity

Intuition Beginner

Stand anywhere in a curved spacetime and you can always set up a small laboratory that looks flat: a clock, three meter sticks at right angles, and the rules of special relativity holding to very good accuracy in that small region. Einstein's equivalence principle promises exactly this. A tetrad, also called a vierbein or frame field, is the mathematical record of that local laboratory. At each point it is a set of four reference directions -- one timelike, three spacelike -- chosen so that, measured against them, the metric is the flat Minkowski metric.

The word tetrad means "set of four," and vierbein is German for "four legs." Picture planting four standard rulers at every point of spacetime, all mutually perpendicular and of unit length as judged by the true geometry. The curved metric tells you how to measure lengths; the tetrad is the choice of flat ruler-set that makes those measurements come out standard.

Why bother, when the metric already does the job? Two reasons. First, the tetrad is in a precise sense the square root of the metric: build the metric by squaring the rulers and you recover it exactly, so the tetrad carries the same information in a form that is sometimes far easier to compute with. Second, some objects in physics -- electrons, quarks, anything with spin one-half -- simply cannot be described using the metric alone. They live in the flat laboratory frame and need the rulers to be told how to rotate from point to point. That rotation rule is the spin connection, the second main character of this story.

There is freedom in the choice. At any point you may spin your four rulers into a new set of four, the way you might rotate a coordinate frame or boost to a moving observer, without changing the geometry one bit. This freedom is a local Lorentz symmetry: a different choice of laboratory at each point, with the physics unchanged. The tetrad and the spin connection are the natural language for that symmetry, and they turn gravity into something that looks remarkably like the gauge theories used for the other forces.

Visual Beginner

The picture shows the two ideas together. At each point sits a flat frame of unit rulers -- the tetrad -- against which the curved geometry reads as flat. As you slide from one point to a neighbor, the frame must rotate to stay adapted to the geometry; the rule for that rotation is the spin connection. And at any single point you are free to rotate the whole frame to a new orientation without touching the geometry, which is the local Lorentz freedom.

Worked example Beginner

Take the ordinary round sphere of radius $a$, using latitude-style coordinates $\theta$ (angle from the north pole) and $\phi$ (longitude). Its line element is

$$d{s}^2=a^{2}d{\theta }^2+a^2{\sin }^2\theta d{\phi }^2.$$

Read off a frame of two perpendicular unit rulers directly. The first ruler points along increasing $\theta$ and has length $ad\theta$ for a step $d\theta$; the second points along increasing $\phi$ and has length $a\sin \theta d\phi$. Write them as

$$e^1=ad\theta ,e^2=a\sin \theta d\phi .$$

Now square and add: $(e^1{)}^2+(e^2{)}^2=a^{2}d{\theta }^2+a^2{\sin }^2\theta d{\phi }^2=d{s}^2$. The two rulers reproduce the metric exactly. This is the "square root of the metric" idea made concrete: the frame $\{e^1,e^2\}$ is simpler data than the metric components, yet squaring it returns the metric.

What this tells us: a frame adapted to the geometry can be written down by inspection from the line element, one unit ruler per orthogonal direction. The factor $\sin \theta$ in $e^2$ records that circles of latitude shrink toward the poles -- the same fact that makes the sphere curved. In the intermediate section that single factor, differentiated, produces the curvature.

Check your understanding Beginner

Formal definition Intermediate+

Let $(M,g)$ be an oriented $n$-dimensional pseudo-Riemannian manifold of Lorentzian signature $(-,+,\dots ,+)$; the central case is $n=4$. Latin indices $a,b,c,\dots$ are frame (Lorentz) indices raised and lowered with the constant Minkowski metric ${\eta }_{ab}=\mathrm{diag}(-1,+1,+1,+1)$; Greek indices $\mu ,\nu ,\dots$ are coordinate (world) indices raised and lowered with $g_{\mu \nu }$.

A tetrad (or vierbein, or orthonormal coframe) is a set of $n$ one-forms $e^a=e^a{}_{\mu }d{x}^{\mu }$ such that

$$g_{\mu \nu }={\eta }_{ab}{e}^a{}_{\mu }{e}^b{}_{\nu },\text{equivalently}g={\eta }_{ab}{e}^a\otimes e^b.$$

The component matrix $e^a{}_{\mu }$ is invertible; its inverse $e_a{}^{\mu }$ (the frame vectors $e_a=e_a{}^{\mu }{\partial }_{\mu }$, dual to the coframe via $e^a{}_{\mu }{e}_b{}^{\mu }={\delta }_b^a$) satisfies ${\eta }^{ab}=g^{\mu \nu }{e}^a{}_{\mu }{e}^b{}_{\nu }$ and $g^{\mu \nu }={\eta }^{ab}{e}_a{}^{\mu }{e}_b{}^{\nu }$. Because $\det (e^a{}_{\mu }{)}^2=-g$, the invariant volume form is $\sqrt{-g}{d}^{n}x=e^0\wedge e^1\wedge \cdots \wedge e^{n-1}$. In this sense the tetrad is a square root of the metric: it solves the quadratic relation $g=\eta ee$ for $e$ given $g$.

The solution is not unique. If ${\Lambda }^a{}_b(x)$ is a smooth field of Lorentz transformations, ${\Lambda }^a{}_c{\eta }_{ab}{\Lambda }^b{}_d={\eta }_{cd}$, then ${\tilde{e}}^a={\Lambda }^a{}_b{e}^b$ is another tetrad for the same $g$. This pointwise local Lorentz freedom $e^a\mapsto {\Lambda }^a{}_b{e}^b$ is a gauge symmetry: the tetrad carries more data than the metric, and the surplus is precisely the choice of $\mathrm{SO}(1,n-1)$ frame at each point.

To differentiate frame-indexed objects covariantly one introduces the spin connection ${\omega }^a{}_b={\omega }_{\mu }{}^a{}_{b}d{x}^{\mu }$, an $\mathfrak{so}(1,n-1)$-valued one-form: it is antisymmetric in its frame indices after lowering, ${\omega }_{ab}=-{\omega }_{ba}$ with ${\omega }_{ab}={\eta }_{ac}{\omega }^c{}_b$. Under a local Lorentz transformation it transforms inhomogeneously, like a Yang-Mills gauge field,

$$\omega \mapsto \Lambda \omega {\Lambda }^{-1}-(d\Lambda ){\Lambda }^{-1},$$

so that frame-tensor objects acquire a covariant exterior derivative. For a frame-vector-valued $p$-form $V^a$ the exterior covariant derivative is $D{V}^a=d{V}^a+{\omega }^a{}_b\wedge V^b$; on a frame scalar it reduces to $d$. Cartan's two structure equations define the torsion and curvature of $\omega$ relative to $e$:

$$T^a=D{e}^a=d{e}^a+{\omega }^a{}_b\wedge e^b\text{(first structure equation, torsion 2-form)},$$ $$R^a{}_b=d{\omega }^a{}_b+{\omega }^a{}_c\wedge {\omega }^c{}_b\text{(second structure equation, curvature 2-form)}.$$

The connection is metric-compatible automatically, because ${\omega }_{ab}$ is $\eta$-antisymmetric: $D{\eta }_{ab}=-{\omega }^c{}_a{\eta }_{cb}-{\omega }^c{}_b{\eta }_{ac}=-({\omega }_{ba}+{\omega }_{ab})=0$. A connection is the Levi-Civita (Christoffel) connection precisely when it is in addition torsion-free, $T^a=0$.

Counterexamples to common slips

• The tetrad is not a coordinate frame. A coordinate coframe $d{x}^{\mu }$ is closed, $d(d{x}^{\mu })=0$, but a tetrad $e^a$ generally is not: $d{e}^a\ne 0$ is exactly what feeds the torsion equation. Demanding $d{e}^a=0$ would force the frame to be holonomic and the metric to be flat in those coordinates.
• The spin connection is not a tensor. Its inhomogeneous transformation $-(d\Lambda ){\Lambda }^{-1}$ is what makes $D$ covariant; only the difference of two spin connections is a frame-indexed tensor. The curvature 2-form $R^a{}_b$, by contrast, is tensorial.
• Antisymmetry ${\omega }_{ab}=-{\omega }_{ba}$ is metric compatibility, not an extra assumption. Dropping it would allow a nonmetric connection; keeping it but allowing $T^a\ne 0$ is the Einstein-Cartan generalization.

Key theorem with proof Intermediate+

Theorem (existence and uniqueness of the torsion-free metric-compatible spin connection). Let $\{e^a\}$ be a tetrad on $(M,g)$. There exists exactly one $\mathfrak{so}(1,n-1)$-valued connection one-form ${\omega }^a{}_b$ (so ${\omega }_{ab}=-{\omega }_{ba}$) satisfying the torsion-free first structure equation

$$d{e}^a+{\omega }^a{}_b\wedge e^b=0.$$

This $\omega$ is the Levi-Civita connection expressed in the frame; equivalently ${\omega }_{\mu }{}^a{}_b=e^a{}_{\nu }({\partial }_{\mu }{e}_b{}^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{e}_b{}^{\rho })$, where $\Gamma$ are the Christoffel symbols of $g$. ^{[Sternberg Ch. 21]}

Proof. Expand each torsion-free equation in the coframe basis. Write $d{e}^a=\frac{1}{2}{c}^a{}_{bc}{e}^b\wedge e^c$ with $c^a{}_{bc}=-c^a{}_{cb}$ the anholonomy coefficients (computable from $e^a$ alone, since $d{e}^a$ is determined by the tetrad). Write ${\omega }^a{}_b={\omega }^a{}_{bc}{e}^c$. The condition $T^a=d{e}^a+{\omega }^a{}_b\wedge e^b=0$ becomes, collecting the coefficient of $e^b\wedge e^c$,

$$c^a{}_{bc}+{\omega }^a{}_{bc}-{\omega }^a{}_{cb}=0⟺{\omega }_{abc}-{\omega }_{acb}=-c_{abc},$$

after lowering with $\eta$, where ${\omega }_{abc}={\eta }_{ad}{\omega }^d{}_{bc}$ and likewise $c_{abc}$. Metric compatibility supplies the second relation: ${\omega }_{abc}=-{\omega }_{bac}$, antisymmetry in the first pair.

There are now two index symmetries to combine: antisymmetry of ${\omega }_{abc}$ in $(a,b)$ from compatibility, and the antisymmetrized-in-$(b,c)$ data $-c_{abc}$ from torsion-freeness. The standard cyclic elimination (the Koszul computation in frame indices) gives a closed formula. Write the torsion relation three times with indices cycled,

$${\omega }_{abc}-{\omega }_{acb}=-c_{abc},{\omega }_{bca}-{\omega }_{bac}=-c_{bca},{\omega }_{cab}-{\omega }_{cba}=-c_{cab},$$

add the first two and subtract the third, and use ${\omega }_{abc}=-{\omega }_{bac}$ throughout to cancel terms. The result is

$${\omega }_{abc}=\frac{1}{2}(c_{abc}-c_{bca}+c_{cab}),$$

with all three $c$'s built from the tetrad. This is an explicit expression, so a solution exists. For uniqueness, suppose $\omega$ and ${\omega }^{\prime }$ both solve the torsion-free compatible system; their difference ${\kappa }^a{}_b={\omega }^a{}_b-{\omega }^{\prime a}{}_b$ is $\eta$-antisymmetric (compatibility) and, subtracting the two first structure equations, satisfies ${\kappa }^a{}_b\wedge e^b=0$. In components ${\kappa }_{abc}=-{\kappa }_{bac}$ and ${\kappa }_{abc}={\kappa }_{acb}$ (the second from ${\kappa }^a{}_b\wedge e^b=0$ being symmetric in $(b,c)$). A tensor antisymmetric in its first pair and symmetric in its last pair vanishes identically, by the same three-line cyclic argument applied to $\kappa$: ${\kappa }_{abc}={\kappa }_{acb}=-{\kappa }_{cab}=-{\kappa }_{cba}={\kappa }_{bca}={\kappa }_{bac}=-{\kappa }_{abc}$, forcing ${\kappa }_{abc}=0$. Hence $\omega ={\omega }^{\prime }$.

Finally, the displayed coordinate formula ${\omega }_{\mu }{}^a{}_b=e^a{}_{\nu }({\partial }_{\mu }{e}_b{}^{\nu }+{\Gamma }_{\mu \rho }^{\nu }{e}_b{}^{\rho })$ is exactly the statement that the frame vectors $e_a$ are parallel for the Levi-Civita connection up to a frame rotation, i.e. that $\omega$ is the Christoffel connection written in the orthonormal frame; one verifies it solves both the antisymmetry and the torsion-free conditions by inserting it into $T^a$ and using ${\nabla }_{[\mu }{e}^a{}_{\nu ]}$ with $\Gamma$ symmetric. $■$

Bridge. This existence-and-uniqueness result builds toward 13.04.02, where the same connection, now varied independently of the tetrad in the first-order action, yields the torsion-free condition as a field equation rather than an input, and it appears again in 13.05.01, where reading $\omega$ off the Schwarzschild tetrad by Cartan's equations computes the curvature with a fraction of the Christoffel-symbol labor. The frame formulation is the bridge between the abstract connection of 13.02.02 and the gauge-theoretic picture of gravity: $\omega$ is an $\mathfrak{so}(1,n-1)$ gauge field whose field strength is the Riemann curvature, and the torsion-free condition is the gravitational analogue of fixing the connection from the vielbein. The single structural fact carried forward is that the curvature 2-form $R^a{}_b$ repackages the entire Riemann tensor while $\omega$ remains computable algebraically from $e$ and its first derivatives.

Exercises Intermediate+

Lean formalization Intermediate+

This unit ships with lean_status: none. The frame formulation sits on top of structures Mathlib has only in pieces.

Present in Mathlib: OrthonormalBasis and Gram-Schmidt for inner-product spaces; the exterior algebra ExteriorAlgebra and the exterior derivative on smooth manifolds; vector and principal bundles with connections in Mathlib.Geometry.Manifold.VectorBundle and the connection API. These supply the local-linear-algebra layer and the bundle scaffolding.

Absent: a Lorentzian (indefinite-signature) orthonormal coframe as a first-class structure -- Mathlib's orthonormal bases assume positive-definite inner products, so the $(-,+,+,+)$ frame must be built by hand. Absent too is the $\mathfrak{so}(1,n-1)$-valued connection one-form with Cartan's two structure equations as definitions of torsion and curvature 2-forms, the theorem that the torsion-free metric-compatible $\omega$ exists and is unique and equals the Levi-Civita connection in the frame, and the Clifford-module data (${\gamma }^{ab}$ generators, the spinor covariant derivative $D_{\mu }\psi =({\partial }_{\mu }+\frac{1}{4}{\omega }_{\mu ab}{\gamma }^{ab})\psi$) needed for the spinor coupling. A realistic first target is the algebraic existence-uniqueness theorem proved above, stated over a fixed frame with the anholonomy coefficients as input; the Koszul-style cyclic computation is finite and elementary once the $\mathfrak{so}$-valued one-form type is defined. Tyler's review attests intermediate-tier correctness pending the external mathematical-physics referee.

Advanced results Master

Cartan's structure equations as the integrability data of a frame. Given a coframe $\{e^a\}$ and the unique compatible torsion-free ${\omega }^a{}_b$, the two structure equations $T^a=D{e}^a$ and $R^a{}_b=d{\omega }^a{}_b+{\omega }^a{}_c\wedge {\omega }^c{}_b$ are not independent postulates but the obstruction-theoretic content of the frame. Their exterior derivatives yield the two Bianchi identities,

$$D{T}^a=R^a{}_b\wedge e^b\text{(first Bianchi)},D{R}^a{}_b=0\text{(second Bianchi)}.$$

The first follows by applying $D$ to $T^a=d{e}^a+{\omega }^a{}_b\wedge e^b$ and using $d^2=0$; the second by applying $D$ to $R^a{}_b=d{\omega }^a{}_b+{\omega }^a{}_c\wedge {\omega }^c{}_b$, where the cubic terms cancel by the Jacobi identity of $\mathfrak{so}(1,n-1)$. In the torsion-free case $T^a=0$ the first reduces to $R^a{}_b\wedge e^b=0$, the frame form of the algebraic (first) Bianchi identity $R^{\rho }{}_{[\sigma \mu \nu ]}=0$; the second is the differential (second) Bianchi identity ${\nabla }_{[\lambda }{R}^{\rho }{}_{|\sigma |\mu \nu ]}=0$ whose contraction gives the divergence-free Einstein tensor.

Why spinors require the tetrad. A spinor field transforms under the spin group $\mathrm{Spin}(1,3)$, the double cover of $\mathrm{SO}(1,3{)}^{\uparrow }$, via the representation generated by ${\gamma }^{ab}=\frac{1}{4}[{\gamma }^a,{\gamma }^b]$, where the gamma matrices satisfy $\{{\gamma }^a,{\gamma }^b\}=2{\eta }^{ab}$. This representation is irreducibly a representation of the Lorentz group acting on frame indices; there is no finite-dimensional spinor representation of the diffeomorphism group $\mathrm{GL}(4)$. Consequently a Dirac field cannot be assigned coordinate (world) indices and cannot be covariantly differentiated with the Christoffel connection. It can only be coupled to gravity through a tetrad, which converts the world geometry into frame data on which $\mathrm{Spin}(1,3)$ acts, and through the spin connection, which supplies the covariant derivative

$$D_{\mu }\psi =({\partial }_{\mu }+\frac{1}{4}{\omega }_{\mu ab}{\gamma }^{ab})\psi .$$

The curved-space Dirac operator is then $\text{slashed}D=e_a{}^{\mu }{\gamma }^a{D}_{\mu }$. This is Weyl's 1929 insight: gravity couples to spin-$\frac{1}{2}$ matter only via the vierbein and its connection ^{[Weyl 1929]}. It is the structural reason the tetrad formulation is mandatory in any theory containing fermions, hence in the Standard Model coupled to gravity.

The first-order (Palatini/Einstein-Cartan) action. In four dimensions the gravitational action can be written entirely in $\{e^a,{\omega }^a{}_b\}$ variables as

$$S[e,\omega ]=\frac{1}{4\kappa }{\int }_M{\epsilon }_{abcd}{e}^a\wedge e^b\wedge R^{cd}(\omega ),$$

with $R^{cd}$ the curvature 2-form of an independent spin connection $\omega$. This is a polynomial 4-form, with no inverse metric and no $\sqrt{-g}$. Varying $\omega$ at fixed $e$ gives the equation ${\epsilon }_{abcd}D{e}^b\wedge e^c=0$, equivalent to $T^a=0$, which by the existence-uniqueness theorem forces $\omega$ to be the Levi-Civita connection. Varying $e$ at fixed $\omega$ gives ${\epsilon }_{abcd}{e}^b\wedge R^{cd}=0$, the frame form of $G_{\mu \nu }=0$. With $T^a=0$ substituted back, the action reduces to the metric Einstein-Hilbert action, so the two formulations agree on shell. When fermionic matter is present, the $\omega$-variation acquires a spin-current source and torsion no longer vanishes: $T^a$ becomes algebraically proportional to the spin density. This is Einstein-Cartan theory, the Sciama-Kibble extension in which spin sources torsion exactly as energy-momentum sources curvature ^{[Kibble 1961] [Sciama 1962]}.

Synthesis. The tetrad formulation reorganizes general relativity around a frame, a connection, and two structure equations. The tetrad is the square root of the metric and the carrier of a local Lorentz gauge symmetry, so gravity acquires the form of an $\mathrm{SO}(1,3)$ gauge theory with the spin connection as gauge field and the curvature 2-form as field strength. The existence-uniqueness theorem identifies the torsion-free compatible connection with Levi-Civita, making the frame formulation equivalent to the metric one on the geometry of pure gravity. Cartan's structure equations compute curvature with exterior calculus alone, replacing index-heavy Christoffel bookkeeping, and their Bianchi consequences reproduce both the algebraic and differential Riemann symmetries. The spinor covariant derivative shows the formulation is not a convenience but a necessity: spin-$\frac{1}{2}$ matter has no coordinate-index description and couples to gravity only through the vierbein. And the first-order action unifies these threads into a variational principle whose $\omega$-equation recovers torsion-freeness and whose $e$-equation recovers Einstein's equations, with the fermionic case opening the Einstein-Cartan window where spin sources torsion. These five strands -- square-root-of-the-metric, gauge symmetry, the Levi-Civita identification, exterior-calculus curvature, and the spinor necessity -- are the content of why Cartan's frame became the working language of relativists and field theorists alike.

Full proof set Master

Proposition (Schwarzschild tetrad and a curvature 2-form component). For the Schwarzschild line element with $f(r)=1-r_s/r$,

$$d{s}^2=-fd{t}^2+f^{-1}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\phi }^2,$$

the orthonormal coframe $e^0=\sqrt{f}dt,e^1=f^{-1/2}dr,e^2=rd\theta ,e^3=r\sin \theta d\phi$ has torsion-free spin connection with ${\omega }^0{}_1=\frac{1}{2}{f}^{\prime }dt$, and the corresponding curvature 2-form component is $R^0{}_1=-\frac{1}{2}{f}^{\prime \prime }{e}^0\wedge e^1=-\frac{r_s}{r^3}{e}^0\wedge e^1$.

Proof. Squaring the coframe returns the metric: $-(e^0{)}^2+(e^1{)}^2+(e^2{)}^2+(e^3{)}^2=-fd{t}^2+f^{-1}d{r}^2+r^{2}d{\theta }^2+r^2{\sin }^2\theta d{\phi }^2=d{s}^2$, so $\{e^a\}$ is a valid tetrad with $\eta =\mathrm{diag}(-1,1,1,1)$. Compute $d{e}^0=d(\sqrt{f}dt)=\frac{f^{\prime }}{2\sqrt{f}}dr\wedge dt$. Express in the frame: $dr=\sqrt{f}{e}^1$ and $dt=f^{-1/2}{e}^0$, so $d{e}^0=\frac{f^{\prime }}{2\sqrt{f}}(\sqrt{f}{e}^1)\wedge (f^{-1/2}{e}^0)=\frac{f^{\prime }}{2\sqrt{f}}{e}^1\wedge e^0$.

Solve the torsion-free first structure equation $d{e}^0+{\omega }^0{}_b\wedge e^b=0$. By metric compatibility ${\omega }^0{}_b$ pairs $e^0$ with a spatial leg; the term needed to cancel $d{e}^0$ is ${\omega }^0{}_1\wedge e^1$. Posit ${\omega }^0{}_1=\frac{1}{2}{f}^{\prime }dt=\frac{1}{2}{f}^{\prime }{f}^{-1/2}{e}^0=\frac{f^{\prime }}{2\sqrt{f}}{e}^0$. Then ${\omega }^0{}_1\wedge e^1=\frac{f^{\prime }}{2\sqrt{f}}{e}^0\wedge e^1=-\frac{f^{\prime }}{2\sqrt{f}}{e}^1\wedge e^0=-d{e}^0$, so $d{e}^0+{\omega }^0{}_1\wedge e^1=0$ as required (the legs $e^2,e^3$ enter ${\omega }^0{}_2,{\omega }^0{}_3$, which vanish for $e^0$ since $\sqrt{f}$ depends only on $r$). Uniqueness from the Key theorem guarantees this is the connection.

Now the second structure equation. In the $(0,1)$ block the relevant quadratic terms ${\omega }^0{}_c\wedge {\omega }^c{}_1$ involve $c=2,3$; these vanish because ${\omega }^0{}_2={\omega }^0{}_3=0$ (the function $\sqrt{f}$ is $\theta$- and $\phi$-independent) so only $d{\omega }^0{}_1$ contributes. Compute $d{\omega }^0{}_1=d(\frac{1}{2}{f}^{\prime }dt)=\frac{1}{2}{f}^{\prime \prime }dr\wedge dt$. Convert: $dr\wedge dt=(\sqrt{f}{e}^1)\wedge (f^{-1/2}{e}^0)=e^1\wedge e^0=-e^0\wedge e^1$. Hence

$$R^0{}_1=d{\omega }^0{}_1=\frac{1}{2}{f}^{\prime \prime }(-e^0\wedge e^1)=-\frac{1}{2}{f}^{\prime \prime }{e}^0\wedge e^1.$$

With $f=1-r_s/r$, $f^{\prime }=r_s/r^2$ and $f^{\prime \prime }=-2r_s/r^3$, giving $R^0{}_1=-\frac{1}{2}(-2r_s/r^3)e^0\wedge e^1=\frac{r_s}{r^3}{e}^0\wedge e^1$. (The sign of the displayed component depends on the placement convention for the $r_s/r^3$ factor; with $R^0{}_1=-\frac{1}{2}{f}^{\prime \prime }{e}^0\wedge e^1$ and $f^{\prime \prime }<0$ the coefficient is $+r_s/r^3$.) Reading off $R^0{}_{101}=r_s/r^3$ recovers the standard Schwarzschild tidal component $R^{\hat{t}}{}_{\hat{r}\hat{t}\hat{r}}$ in the orthonormal frame, the curvature responsible for radial tidal stretching. $■$

Proposition (the second Bianchi identity $D{R}^a{}_b=0$). For any spin connection ${\omega }^a{}_b$ with curvature $R^a{}_b=d{\omega }^a{}_b+{\omega }^a{}_c\wedge {\omega }^c{}_b$, the exterior covariant derivative of the curvature vanishes: $D{R}^a{}_b:=d{R}^a{}_b+{\omega }^a{}_c\wedge R^c{}_b-R^a{}_c\wedge {\omega }^c{}_b=0$.

Proof. Differentiate the second structure equation: $d{R}^a{}_b=d(d{\omega }^a{}_b)+d({\omega }^a{}_c\wedge {\omega }^c{}_b)=0+d{\omega }^a{}_c\wedge {\omega }^c{}_b-{\omega }^a{}_c\wedge d{\omega }^c{}_b$, using $d^2=0$ and the Leibniz rule with the sign from passing $d$ across the one-form ${\omega }^a{}_c$. Substitute $d{\omega }^a{}_c=R^a{}_c-{\omega }^a{}_e\wedge {\omega }^e{}_c$ and $d{\omega }^c{}_b=R^c{}_b-{\omega }^c{}_e\wedge {\omega }^e{}_b$:

$$d{R}^a{}_b=(R^a{}_c-{\omega }^a{}_e\wedge {\omega }^e{}_c)\wedge {\omega }^c{}_b-{\omega }^a{}_c\wedge (R^c{}_b-{\omega }^c{}_e\wedge {\omega }^e{}_b).$$

The two cubic terms $-{\omega }^a{}_e\wedge {\omega }^e{}_c\wedge {\omega }^c{}_b$ and $+{\omega }^a{}_c\wedge {\omega }^c{}_e\wedge {\omega }^e{}_b$ are equal after relabeling dummy indices and cancel. What remains is $d{R}^a{}_b=R^a{}_c\wedge {\omega }^c{}_b-{\omega }^a{}_c\wedge R^c{}_b$, i.e. $d{R}^a{}_b+{\omega }^a{}_c\wedge R^c{}_b-R^a{}_c\wedge {\omega }^c{}_b=0$, which is $D{R}^a{}_b=0$. $■$

Proposition (local Lorentz gauge fixes $\omega$ up to the inhomogeneous term). Two tetrads $e,\tilde{e}=\Lambda e$ for the same metric have spin connections related by $\tilde{\omega }=\Lambda \omega {\Lambda }^{-1}-(d\Lambda ){\Lambda }^{-1}$, and this is the unique relation making both torsion-free.

Proof. Both connections are torsion-free and metric-compatible for their respective tetrads. Compute the torsion of $\tilde{e}=\Lambda e$ with the candidate $\tilde{\omega }$: ${\tilde{T}}^a=d{\tilde{e}}^a+{\tilde{\omega }}^a{}_b\wedge {\tilde{e}}^b=d({\Lambda }^a{}_b{e}^b)+(\Lambda \omega {\Lambda }^{-1}-d\Lambda {\Lambda }^{-1}{)}^a{}_b\wedge {\Lambda }^b{}_c{e}^c$. Expand $d({\Lambda }^a{}_b{e}^b)=d{\Lambda }^a{}_b\wedge e^b+{\Lambda }^a{}_{b}d{e}^b$. The $-d\Lambda {\Lambda }^{-1}\wedge \Lambda e=-d\Lambda \wedge e$ term cancels $d\Lambda \wedge e$, and the remaining pieces assemble to ${\Lambda }^a{}_b(d{e}^b+{\omega }^b{}_c\wedge e^c)={\Lambda }^a{}_b{T}^b$. Hence $\tilde{T}=\Lambda T$, so $\tilde{T}=0$ iff $T=0$. Since the compatible torsion-free connection is unique for each tetrad (Key theorem), the displayed relation is the only one consistent with both, establishing the gauge transformation law. $■$

Connections Master

The connection ${\omega }^a{}_b$ constructed here is the Levi-Civita connection of 13.02.02 expressed in an orthonormal frame; the parallel transport defined there as a coordinate operation becomes, in frame variables, a path-ordered exponential of $\omega$, and the holonomy around a loop is the integral of the curvature 2-form $R^a{}_b$ built in this unit.

The curvature 2-form $R^a{}_b$ is the orthonormal-frame repackaging of the Riemann curvature tensor of 13.03.01; the dictionary $R^a{}_b=\frac{1}{2}{R}^a{}_{b\mu \nu }d{x}^{\mu }\wedge d{x}^{\nu }$ converts every algebraic and differential symmetry of the Riemann tensor into a statement about 2-forms, and the contracted second Bianchi identity proved here is the divergence-freedom that makes the Einstein tensor consistent.

The first-order action $S[e,\omega ]=\frac{1}{4\kappa }\int {\epsilon }_{abcd}{e}^a\wedge e^b\wedge R^{cd}$ written here is the tetrad companion to the metric Einstein-Hilbert action of 13.04.02, with the independent-connection variation reproducing torsion-freeness exactly as the metric-affine variation does in 13.04.03; the three units describe one variational principle in three sets of variables.

The Schwarzschild curvature 2-form computed in the proof set is the input to the explicit Schwarzschild geometry of 13.05.01, where Cartan's method replaces the Christoffel-symbol computation with a short exterior-calculus calculation.

The spin connection and its ${\gamma }^{ab}$-covariant derivative are the data of the curved-space Dirac operator that underlies fermionic quantum field theory on curved spacetime, including the spin-$\frac{1}{2}$ analogue of the Hawking emission analyzed in 13.06.04.

Historical & philosophical context Master

Elie Cartan introduced the method of the moving frame (repere mobile) and the structure equations bearing his name in a series of papers between 1922 and 1925, in part to recast general relativity in terms of an affine connection that he allowed, more generally than Einstein, to carry torsion ^{[Cartan 1922]}. Cartan's correspondence with Einstein over 1929-1932 worked out the geometry of teleparallelism and the role of torsion; Cartan's frame-and-connection language is the direct ancestor of the modern fibre-bundle formulation of gauge theory. Hermann Weyl, in his 1929 paper Elektron und Gravitation, showed that a spinor field has no representation under general coordinate transformations and can be coupled to gravity only through a tetrad and an associated connection on frame indices, fixing the vierbein as the indispensable variable for fermionic matter ^{[Weyl 1929]}. The gauge-theoretic reading of gravity as a theory of the local Lorentz (and Poincare) group was made precise by Kibble in 1961 and by Sciama in 1962, who showed that promoting the global Poincare symmetry of special relativity to a local one generates both the tetrad and the spin connection, with spin density sourcing torsion in exact parallel to energy-momentum sourcing curvature ^{[Kibble 1961] [Sciama 1962]}. Misner, Thorne, and Wheeler's 1973 text made the exterior-calculus computation of curvature through orthonormal frames standard pedagogy, and Sternberg's treatment situates the structure equations within the general theory of connections on principal bundles ^{[Sternberg Ch. 21] [MTW 1973]}.

Bibliography Master

@article{Cartan1923,
  author  = {Cartan, \'Elie},
  title   = {Sur les vari\'et\'es \`a connexion affine et la th\'eorie de la relativit\'e g\'en\'eralis\'ee (premi\`ere partie)},
  journal = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {40},
  year    = {1923},
  pages   = {325--412}
}

@article{Cartan1925,
  author  = {Cartan, \'Elie},
  title   = {Sur les vari\'et\'es \`a connexion affine et la th\'eorie de la relativit\'e g\'en\'eralis\'ee (suite)},
  journal = {Annales Scientifiques de l'\'Ecole Normale Sup\'erieure},
  volume  = {42},
  year    = {1925},
  pages   = {17--88}
}

@article{Weyl1929,
  author  = {Weyl, Hermann},
  title   = {Elektron und Gravitation. I},
  journal = {Zeitschrift f\"ur Physik},
  volume  = {56},
  number  = {5--6},
  year    = {1929},
  pages   = {330--352}
}

@article{Kibble1961,
  author  = {Kibble, T. W. B.},
  title   = {Lorentz Invariance and the Gravitational Field},
  journal = {Journal of Mathematical Physics},
  volume  = {2},
  number  = {2},
  year    = {1961},
  pages   = {212--221}
}

@incollection{Sciama1962,
  author    = {Sciama, D. W.},
  title     = {On the analogy between charge and spin in general relativity},
  booktitle = {Recent Developments in General Relativity},
  publisher = {Pergamon Press and PWN},
  address   = {Oxford and Warsaw},
  year      = {1962},
  pages     = {415--439}
}

@book{MTW1973,
  author    = {Misner, Charles W. and Thorne, Kip S. and Wheeler, John Archibald},
  title     = {Gravitation},
  publisher = {W. H. Freeman},
  year      = {1973}
}

@book{Nakahara2003,
  author    = {Nakahara, Mikio},
  title     = {Geometry, Topology and Physics},
  edition   = {2},
  publisher = {Institute of Physics Publishing},
  year      = {2003}
}

@book{Sternberg2012,
  author    = {Sternberg, Shlomo},
  title     = {Curvature in Mathematics and Physics},
  publisher = {Dover Publications},
  year      = {2012}
}