Local Lorentz Freedom and Frame Gauge

An orthonormal tetrad is not unique: at every point it may be rotated or boosted by a Lorentz matrix without changing the metric. When that matrix varies across spacetime, the connection acquires an inhomogeneous derivative term. Torsion and curvature transform covariantly, so frame components change while geometric conclusions remain invariant.

Required background. Pseudo-Riemannian Metrics and Signature supplies the Lorentzian metric, signature, and index operations; Orthonormal Tetrads and Measured Components supplies metric reconstruction from an orthonormal coframe and local Lorentz frame changes. Affine Connections and Covariant Derivatives supplies the general connection framework and the inhomogeneous behavior of connection coefficients; the frame-gauge transformation laws are derived below.

Helpful background. Calculus and Linear Algebra for Relativity supplies changes of basis and linear maps; Exterior Algebra and Differential Forms supplies the form and wedge algebra used in the matrix-valued expressions. Killing Fields and Isometry Groups supplies the spacetime-isometry side of the distinction from internal frame freedom.

Local Lorentz transformations

Write the orthonormal coframe as a column $\theta$. A smooth matrix field $\Lambda(x)$ satisfying

$$\Lambda^T\eta\Lambda=\eta$$

defines another orthonormal coframe by

$$\theta'=\Lambda\theta.$$

The dual frame transforms with the inverse matrix. Metric reconstruction is unchanged:

$$\eta_{ab}\theta'^a\otimes\theta'^b =\eta_{ab}\theta^a\otimes\theta^b.$$

If orientation and time orientation are fixed, one usually restricts to $SO^+(1,3)$. Otherwise discrete parity or time-reversal components may also appear. The six continuous parameters comprise three rotations and three boosts at each point.

Infinitesimally, write $\Lambda=\mathbf1+\lambda+O(\lambda^2)$. The Lorentz condition implies $\lambda_{ab}=-\lambda_{ba}$, and the coframe variation is $\delta\theta^a=\lambda^a{}_b\theta^b$. This displays the six independent functions directly and provides a linear check before manipulating the full finite matrices.

This is an internal change of orthonormal basis over the same spacetime point. It is not a coordinate transformation, which relabels the base manifold, and not generally an isometry, which maps points while preserving the metric. Carroll makes this frame-versus-coordinate distinction explicit in Carroll 1997, §3, pp. 88–95, eqs. (3.114)–(3.143).

Inhomogeneous connection law

Let internal vector components transform as $v'=\Lambda v$ and define

$$Dv=dv+\omega v.$$

Demanding $D'v'=\Lambda Dv$ gives

$$\omega' =\Lambda\omega\Lambda^{-1}-d\Lambda\,\Lambda^{-1}.$$

The derivative term is essential. A connection is not a tensor because it must compensate for the changing frame. In contrast, with

$$\Omega=d\omega+\omega\wedge\omega,$$

substitution and the identity $d(\Lambda^{-1})=-\Lambda^{-1}(d\Lambda)\Lambda^{-1}$ yield

$$\Omega'=\Lambda\Omega\Lambda^{-1}.$$

Curvature transforms homogeneously. Torsion does as well: $T'=\Lambda T$. Consequently, equations such as $T=0$ and $\Omega=0$ do not depend on the local Lorentz gauge.

At infinitesimal order the same laws read

$$\delta\omega=[\lambda,\omega]-d\lambda, \qquad \delta\Omega=[\lambda,\Omega].$$

The first line contains a derivative of the gauge parameter; the second does not. Contracting all internal indices with Lorentz-invariant tensors therefore produces frame-independent scalars, while an uncontracted curvature component rotates with the chosen tetrad.

A position-dependent boost in flat space

Begin with an inertial tetrad in Minkowski space, so $\omega=0$. Let

$$\Lambda(x)=\exp\!\bigl(\chi(x)K_{01}\bigr)$$

be a boost of variable rapidity in the $0$–$1$ plane. Because the same generator occurs at every point,

$$\omega'=-d\chi\,K_{01}.$$

The transformed connection is nonzero whenever the rapidity varies. Nevertheless,

$$\Omega'=d\omega'+\omega'\wedge\omega'=0:$$

$d^2\chi=0$, and $d\chi\wedge d\chi=0$. This is the decisive check: nonzero connection coefficients can describe a changing flat-space frame rather than curvature.

Parallel transport in the primed components includes $\omega'$ and exactly compensates for the changing basis. Omitting the inhomogeneous term would incorrectly predict a physical effect from a frame choice.

Gauge fixing and its limits

At a chosen point, one can select a Lorentz gauge in which the connection coefficients vanish. Along a specified curve, one can choose a parallel-propagated frame that removes the connection component tangent to that curve. On an open region, however, setting $\omega=0$ would force $\Omega=0$ there. Nonzero curvature obstructs such a gauge.

Even when $\Omega=0$, global topology can obstruct a single-valued gauge with $\omega=0$ on a non-simply-connected region. Flat connections can possess nontrivial holonomy around loops. Thus “zero curvature” is the local integrability condition; a global statement also needs the region and transition functions.

Gauge conditions should therefore be chosen for a stated purpose—adapting the timelike leg to an observer field, aligning spatial legs with symmetry directions, or simplifying a transport equation. They do not remove the underlying sixfold local freedom unless enough conditions are imposed, and a chosen gauge may fail at axes or patch boundaries.

Frame-dependent objects include tetrad components and $\omega$ itself. Covariant objects include the reconstructed metric, tensor equations after all frame indices are contracted appropriately, and curvature up to its Lorentz change of basis. A component that changes under $\Lambda(x)$ is not necessarily unphysical; it becomes meaningful only together with the frame that defines it.

A diffeomorphism and a local Lorentz transformation may be performed together, but their roles remain distinct. The first acts on coordinate indices through pullback or pushforward; the second acts on internal indices in each fiber. Writing both sets of indices explicitly is the simplest way to prevent one transformation law from being substituted for the other.

Exercises

Starting from $\omega=0$, apply two successive local Lorentz transformations $\Lambda_1$ and $\Lambda_2$. Show that the connection obtained sequentially equals the result for the combined transformation $\Lambda_2\Lambda_1$.

Solution

After the first transformation,

$$\omega_1=-d\Lambda_1\Lambda_1^{-1}.$$

After the second,

$$\begin{aligned} \omega_2 &=\Lambda_2\omega_1\Lambda_2^{-1}-d\Lambda_2\Lambda_2^{-1}\\ &=-\Lambda_2d\Lambda_1\Lambda_1^{-1}\Lambda_2^{-1}-d\Lambda_2\Lambda_2^{-1}\\ &=-d(\Lambda_2\Lambda_1)(\Lambda_2\Lambda_1)^{-1}. \end{aligned}$$

The product rule produces exactly the two terms. This composition check fixes the order of matrices and the sign of the inhomogeneous contribution.

References

• Carroll, Sean M. Lecture Notes on General Relativity. arXiv:gr-qc/9712019 (1997), arXiv.