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research32 min studyLesson 844 in Physics

Lapse, shift, and coordinate gauge conditions

Calculate local slicing changes and distinguish gauge-characteristic speeds from physical light propagation.

Before you begin
On this page

The question for today

What you’ll understand

  • Apply harmonic and one-plus-log lapse equations with the declared curvature sign.
  • Distinguish gauge speeds from null-cone speeds.
  • Explain why a gauge driver must be analyzed with its coupled evolution formulation.

Einstein's equations do not uniquely prescribe the coordinate labels used in an evolution. Numerical relativity chooses lapse and shift to keep those labels useful as the geometry changes. A poor gauge can stretch a grid, produce a coordinate singularity, or make a valid formulation hard to resolve. A good coordinate choice is therefore part of a successful calculation, while its apparent motion must remain separate from physical signals and invariant observables.

Specify the lapse equation and its advection

Retain signature minus-plus-plus-plus and extrinsic curvature defined with a negative normal Lie derivative. A useful slicing family evolves lapse with the derivative along the time vector after subtracting shift advection. Its right-hand side contains minus lapse squared, a prescribed positive function f of lapse, and the trace K. Setting f one gives harmonic slicing; choosing f equal to two divided by lapse gives the common advective one-plus-log condition.

The name one-plus-log comes from a special zero-shift integration relating lapse to the logarithm of the spatial metric determinant. It is not a general instruction to recompute lapse from that determinant in every shifted evolution. Advective and nonadvective versions differ, and initial lapse remains an additional coordinate choice. The gauge equation must also specify regularity and boundary data; its compact formula does not guarantee smooth coordinates indefinitely.

(∂t−βi∂i)α=−α2f(α)K,f=1 harmonic,f=2/α one-plus-log(\partial_t-\beta^i\partial_i)\alpha=-\alpha^2f(\alpha)K,\qquad f=1\ \text{harmonic},\qquad f=2/\alpha\ \text{one-plus-log}
Lapse and f are dimensionless, while K and the lapse time derivative have inverse-length units. The displayed family is a coordinate condition, not an extra Einstein constraint.↔ Scroll wide equations horizontally.

Compare characteristic coordinate speeds

Freezing coefficients and combining the lapse equation with the principal K evolution, which contains minus the spatial Laplacian of lapse, gives a gauge wave equation. Its speeds relative to the normal are proportional to the square root of f. Along a Cartesian direction with diagonal spatial metric, coordinate speeds contain the lapse divided by the square root of that metric component and the negative shift offset. Physical null rays instead use f one. When f exceeds one, a gauge characteristic can move faster in coordinates than the physical light characteristic.

This behavior does not transmit an observable message outside the spacetime light cone. It describes how coordinate information propagates within the chosen PDE system. It does matter numerically: a timestep based only on physical light speed can violate stability requirements when gauge or constraint modes are faster. The full characteristic analysis depends on the coupled formulation, gauge drivers, reduction variables, and background state. A slicing-family estimate alone cannot certify strong hyperbolicity of an arbitrary implementation.

vgauge±=−βx±αf/γxx,vlight±=−βx±α/γxxv_{\rm gauge}^{\pm}=-\beta^x\pm\alpha\sqrt{f/\gamma_{xx}},\qquad v_{\rm light}^{\pm}=-\beta^x\pm\alpha/\sqrt{\gamma_{xx}}
These are local one-direction frozen-coefficient speeds for the stated slicing analysis. They are coordinate velocities, not measured relative speeds of material particles.↔ Scroll wide equations horizontally.

Choose shift drivers and diagnostics deliberately

Shift conditions control spatial label motion and can counter severe coordinate stretching. Gamma-driver conditions, often paired with one-plus-log lapse in moving-puncture work, evolve auxiliary fields using conformal connection information. Driver parameters introduce additional coordinate scales and characteristic behavior. They are not universal constants of general relativity. Different advection choices or damping parameters can change coordinate tracks even when converged physical waveforms agree.

To compare gauges, hold the intended physical initial data fixed, identify corresponding physical events or asymptotic times, and then compare invariant diagnostics. Coordinate separation and coordinate merger time alone are insufficient. A local Euler update of lapse with K held fixed is a transparent teaching calculation, but it does not simultaneously evolve the spatial metric, curvature, or constraints. Explicitly limiting such a calculation prevents a numerical gauge illustration from being mistaken for an independent spacetime solution.

αn+1=αn−2αnKΔtfor a frozen-K, zero-shift Euler one-plus-log toy step\alpha_{n+1}=\alpha_n-2\alpha_nK\Delta t\qquad\text{for a frozen-K, zero-shift Euler one-plus-log toy step}
This finite update is first order in time and only approximates the declared local lapse equation. In a full evolution K is a dynamical field and shift advection must be discretized.↔ Scroll wide equations horizontally.

Work it through

Two local slicing updates

At one point set shift zero, K=0.5/LK=0.5/L, and α=0.8\alpha=0.8. Hold K fixed during a toy Euler step Δt=0.1L\Delta t=0.1L. Compare the updated lapse for one-plus-log and harmonic slicing.

  1. For one-plus-log the lapse derivative is −2αK=−0.8/L-2\alpha K=-0.8/L. The Euler decrement is 0.08, so the updated lapse is 0.72.

  2. For harmonic slicing the derivative is −α2K=−0.32/L-\alpha^2K=-0.32/L. The decrement is 0.032 and the updated lapse is 0.768. Both results use the same K sign and the same initial lapse.

  3. The different answers are different local coordinate prescriptions, not different matter sources or an invariant gravitational acceleration. Holding K fixed and using one Euler step leaves integration error and omits the coupled geometric evolution.

The result

The stipulated local Euler updates are lapse 0.72 for one-plus-log and 0.768 for harmonic slicing.

The living notebook

Explore the model

Interactive lab

Interactive lab · function

Linear and quadratic dependence on lapse

Reset and set coefficient one, position 0.5; predict output 0.25. Compare a fresh Reset run at coefficient one, position one; predict output one. This checks the quadratic dependence appearing in harmonic slicing, while one-plus-log cancels one lapse factor through f. The lab does not evolve lapse or shift, impose constraints, or analyze a coupled hyperbolic gauge system.

Linear and quadratic dependence on lapseThis lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Current readings: y at chosen x: 0.608; Tangent slope · 2ax: 1.35; Signed area · ax³/3: 0.182. All parameters are adjustable with the labeled controls below.−2.4−1.201.22.4−9−4.504.59xy(x, ax²)
y = ax²TangentSigned integral
y at chosen x
0.608
Tangent slope · 2ax
1.35
Signed area · ax³/3
0.182
t = 0 s

This lab uses only y = ax². Playback traces the curve within the fixed integration interval from 0 to x; it does not estimate numerical error or change the chosen point. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forThe slicing function changes algebraic dependence, but a quadratic plot is not a spacetime evolution.

Pause & reason

Check your understanding

A one-plus-log gauge characteristic is faster in coordinates than the local light characteristic. What is the appropriate interpretation?

Your turn

Try it yourself

Use the stated frozen-coefficient speed formulas with α=0.5\alpha=0.5, βx=0.2\beta^x=0.2, γxx=1\gamma_{xx}=1, and one-plus-log f. Find both gauge speeds and both physical light speeds, and explain why their difference matters for a timestep estimate.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

First evaluate f equal to two divided by lapse. Apply the same negative shift offset to both pairs.

Compare with the solution

Here f is four, so the normal-coordinate gauge contribution is 0.54=10.5\sqrt4=1. The gauge speeds are 0.8 and minus 1.2. The light contribution is 0.5, giving light speeds 0.3 and minus 0.7. The fastest absolute gauge speed exceeds the fastest absolute light speed in this local model. A timestep restriction using only 0.7 would miss that gauge characteristic; the actual bound still requires the discretization and all coupled characteristic fields.

Keep the idea

Take it with you

  • Gauge equations fix coordinate freedom rather than adding physical polarizations.
  • Advection and extrinsic-curvature signs affect lapse updates.
  • Gauge and constraint characteristics can control a stable timestep.

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