Spatial Gauge and Shift Conditions

A shift condition controls how spatial coordinate labels move from one slice to the next. It can reduce coordinate shear, hold selected geometric features near fixed coordinate locations, or damp the motion of conformal connection variables, while the normal evolution remains $\alpha n^\mu$. Minimal distortion expresses this aim as an elliptic condition; Gamma-freezing and Gamma-driver choices express it through BSSN variables. Their equivalence and effectiveness are conditional on the formulation, reference connection, parameters, boundaries, and residual spatial symmetries.

Required background. The Einstein Equation supplies the covariant solution being represented, while Killing Fields and Isometry Groups supplies Lie transport and the conformal-Killing kernels of spatial gauge equations.

Helpful background. Hypersurface Normals and Tangential Projectors supplies the tangential projector and induced metric, and PDEs and Green Functions for Gravitation supplies elliptic boundary problems and characteristic gauge drivers.

Tangential coordinate motion

The evolution vector decomposes as

$$(\partial_t)^\mu=\alpha n^\mu+\beta^\mu, \qquad n_\mu\beta^\mu=0.$$

The foliation fixes $n^\mu$ and the lapse fixes the normal separation of neighboring slices. The shift $\beta^\mu$ identifies which point on the next slice receives the same spatial coordinate label. For every spatial tensor it appears through the advective derivative

$$\partial_\perp\equiv\partial_t-\mathcal L_\beta.$$

With $K_{ij}=-\gamma_i{}^\mu\gamma_j{}^\nu\nabla_\mu n_\nu$ and future unit normal $n^\mu$, in particular,

$$\partial_\perp\gamma_{ij}=-2\alpha K_{ij}.$$

Changing $\beta^i$ changes $\partial_t\gamma_{ij}$ by a spatial Lie derivative but does not change the normal derivative on the right-hand side. This is the precise sense in which a shift can reduce coordinate shear without changing normal evolution. Coupled lapse and shift drivers can still interact through their coefficients and advective terms, so their analysis cannot always be separated at the level of a full PDE system.

The coordinate threads generated by $\partial_t$ need not be timelike. Their norm is

$$g(\partial_t,\partial_t) =-\alpha^2+\gamma_{ij}\beta^i\beta^j.$$

A large shift can make those threads spacelike while every $t=\text{constant}$ hypersurface remains spacelike. This is coordinate motion, not superluminal transport of matter.

Minimal distortion

Split the extrinsic curvature into trace and trace-free parts,

$$K_{ij}=A_{ij}+\frac13\gamma_{ij}K,$$

and define the conformal Killing operator on a spatial vector field:

$$(\mathbb L\beta)_{ij} =D_i\beta_j+D_j\beta_i -\frac23\gamma_{ij}D_k\beta^k.$$

The trace-free coordinate-time velocity of the spatial metric is

$$u_{ij}\equiv(\partial_t\gamma_{ij})^{\mathrm{TF}} =-2\alpha A_{ij}+(\mathbb L\beta)_{ij}.$$

A minimal-distortion condition makes this velocity transverse,

$$D_j u^{ij}=0.$$

The shift therefore solves the vector elliptic equation

$$D_j(\mathbb L\beta)^{ij} =2D_j(\alpha A^{ij}).$$

This equation chooses the tangential relabeling whose trace-free metric motion has no longitudinal part in the selected spatial norm. It does not force $u_{ij}=0$: transverse-traceless geometric evolution remains.

Boundary data and kernels are indispensable. If $v^i$ is a conformal Killing field, then $(\mathbb Lv)_{ij}=0$, so $\beta^i+v^i$ solves the same differential equation. Inner boundaries, asymptotic frames, or an orthogonality condition must fix that residual freedom. On compact slices, solvability also requires the source to be orthogonal to the adjoint kernel.

Gamma-freezing and Gamma drivers

In the BSSN conformal split, write

$$\gamma_{ij}=e^{4\phi}\widetilde\gamma_{ij}, \qquad \det\widetilde\gamma=1$$

in the chosen coordinate volume element. The contracted conformal connection is commonly represented as

$$\widetilde\Gamma^i \equiv\widetilde\gamma^{jk} \widetilde\Gamma^i{}_{jk} =-\partial_j\widetilde\gamma^{ij}.$$

A Gamma-freezing shift aims for

$$\partial_t\widetilde\Gamma^i=0.$$

Substituting the BSSN evolution equation gives an elliptic equation whose principal spatial operator resembles the vector Laplacian in minimal distortion. The lower-order terms, conformal factors, constraint substitutions, and boundary data are not identical, so the two conditions should be compared through their actual equations rather than their names.

An elliptic solve can be replaced by a hyperbolic or damped Gamma driver. A common schematic form is

$$\partial_0\beta^i=\mu_S B^i, \qquad \partial_0B^i =\partial_0\widetilde\Gamma^i-\eta B^i,$$

where $\partial_0$ is either $\partial_t$ or an explicitly chosen advective derivative. The positive parameters $\mu_S$ and $\eta$ set gauge propagation and damping scales. Factors such as $3/4$, powers of the lapse, and advective terms differ among systems and change their characteristic fields. “The Gamma driver” is therefore a family, not one formulation-independent equation.

Gourgoulhon develops minimal-distortion, minimal-strain, and Gamma-freezing conditions in Gourgoulhon 2007, §§9.3–9.4, pp. 162–174, PDF and introduces the conformal connection variables in Gourgoulhon 2007, §10.4, pp. 181–188, PDF.

The contracted connection needs a reference

The expression $\widetilde\Gamma^i$ is not a spatial vector under a general coordinate transformation. In Cartesian-like coordinates with a fixed volume element this may be acceptable as an explicit coordinate gauge, but a covariant comparison requires a reference connection $\widehat\Gamma^i{}_{jk}$. The difference

$$\widetilde\Lambda^i \equiv\widetilde\gamma^{jk} \left( \widetilde\Gamma^i{}_{jk}-\widehat\Gamma^i{}_{jk} \right)$$

is a spatial vector. A curvilinear or reference-metric formulation should drive $\widetilde\Lambda^i$ or another declared tensorial target, not silently reuse the Cartesian formula for $\widetilde\Gamma^i$.

This distinction explains why the physical judgment is conditional. A Gamma-driver that behaves well in one BSSN coordinate implementation does not define a coordinate-independent optimal shift, and its continuum hyperbolicity must be rechecked when its variables, parameters, or advective terms change.

Comparison by geometric target

Condition	Quantity controlled	PDE character	Residual or failure mode
Prescribed shift	Chosen coordinate threads	None beyond supplied function	Can stretch or rotate coordinates without feedback
Minimal distortion	Longitudinal trace-free metric velocity	Elliptic	Conformal-Killing kernel and boundary dependence
Gamma freezing	Time derivative of a conformal connection target	Usually elliptic after substitution	Target is coordinate dependent without a reference connection
Gamma driver	Dynamical relaxation toward Gamma freezing	Hyperbolic or damped driver	Parameter-dependent gauge speeds, transients, and off-constraint behavior

None of these conditions removes physical transverse-traceless evolution. None guarantees that a coordinate map remains invertible, and none replaces the constraint equations.

A rotating-grid check

Write flat spacetime in cylindrical inertial coordinates $(T,R,\Phi,Z)$ and introduce a rigidly rotating angle

$$T=t, \qquad R=r, \qquad \Phi=\phi+\Omega t, \qquad Z=z.$$

Then

$$ds^2 =-\left(1-\Omega^2r^2\right)dt^2 +2\Omega r^2dt\,d\phi +dr^2+r^2d\phi^2+dz^2.$$

The lapse is $\alpha=1$, the shift is $\beta^\phi=\Omega$, the spatial metric is the ordinary Euclidean cylindrical metric, and $K_{ij}=0$. The vector $\partial_\phi$ is a spatial Killing field, so

$$\mathcal L_\beta\gamma_{ij}=0.$$

Rigid rotation lies in the minimal-distortion kernel. At $r>1/\lvert\Omega\rvert$, $\partial_t$ is spacelike, but the slices and normal evolution remain regular and flat. Boundary conditions that specify a nonrotating or corotating frame decide whether this residual shift is allowed.

For differential rotation, use the actual map $\Phi=\phi+\Omega(r)t$. Its differential is $d\Phi=d\phi+\Omega\,dt+t\Omega'\,dr$, so the spatial metric also acquires time-dependent cross terms: simply replacing $\Omega$ in the rigid-rotation line element would give a different geometry. At $t=0$ the spatial metric is Euclidean, but $\beta=\Omega(r)\partial_\phi$ is not a Killing field when $\Omega'\ne0$. Its Lie derivative generates coordinate shear while $K_{ij}$ remains zero. A distortion-minimizing equation detects that shear.

Common pitfalls

Shift is not physical velocity. It is the velocity of coordinate labels relative to the normal observers. Its magnitude is not bounded by the speed of light.

Gamma variables are not automatically vectors. Their convenient Cartesian formula depends on the conformal determinant and coordinate structure. A reference connection is required for a covariant curvilinear version.

Damping does not establish hyperbolicity. The parameter $\eta$ controls a lower-order driver term. Strong hyperbolicity depends on the full principal symbol of the coupled gauge–Einstein system.

Exercise: the kernel of minimal distortion

Let $v^i$ be a conformal Killing field of $(\Sigma,\gamma_{ij})$, so that $(\mathbb Lv)_{ij}=0$. Show that if $\beta^i$ solves the minimal-distortion equation, then $\beta^i+v^i$ also solves it. For the rotating flat-space example, identify $v^i$ and explain which boundary datum can remove the ambiguity.

Solution

Linearity of the conformal Killing operator gives

$$D_j\!\left[\mathbb L(\beta+v)\right]^{ij} =D_j(\mathbb L\beta)^{ij}+D_j(\mathbb Lv)^{ij} =D_j(\mathbb L\beta)^{ij}.$$

The source $2D_j(\alpha A^{ij})$ is unchanged, so $\beta+v$ is another solution. In Euclidean cylindrical space, $v=\partial_\phi$ generates rigid rotations and is a Killing field, hence also a conformal Killing field. Requiring the shift to approach zero in a chosen nonrotating asymptotic frame removes this mode; prescribing a corotating outer frame instead fixes its coefficient to the desired angular velocity.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.