Active and Passive Diffeomorphisms

A passive coordinate change and an active diffeomorphism can produce the same component formula, but they answer different questions. Passive change relabels the same geometric fields and events. Active pullback constructs new fields on the same labeled manifold points. General relativity is covariant under both descriptions; whether an active transformation is gauge additionally depends on which fields and boundary data are transformed.

Required background. Smooth maps and tensor fields supply pullbacks and component transformations; Lie derivatives and Killing fields describe infinitesimal flows and exact metric symmetries.

Helpful background. Special-relativistic transformations provide familiar examples. Variational boundary potentials and the gravitational boundary problem explain why some boundary transformations represent physical symmetries.

Passive relabeling

Let $x^\mu$ and $x^{\mu'}$ be two coordinate charts on the same region. A scalar field $f$ has component representatives related by

$$f'(x')=f(x(x')).$$

A metric has

$$g_{\mu'\nu'}(x') =\frac{\partial x^\mu}{\partial x^{\mu'}} \frac{\partial x^\nu}{\partial x^{\nu'}}g_{\mu\nu}(x).$$

Nothing has moved. The event, scalar value there, and bilinear metric on its tangent space are the same; only their numerical labels and basis components change. A relational measurement such as $g_{ab}u^av^b$ remains identical after all components are transformed.

Reall introduces diffeomorphisms by first separating maps between manifolds from their coordinate representatives Reall 2013, §§ 7.1–7.2, pp. 67–76, PDF. This viewpoint prevents covariance from being mistaken for a physical motion of the fields.

Active pullback

Let $\Phi:\mathcal M\to\mathcal M$ be a diffeomorphism. Its pullback produces new fields

$$g' = \Phi^*g, \qquad f'=\Phi^*f=f\circ\Phi.$$

At a fixed manifold point $p$, the new scalar value is $f'(p)=f(\Phi(p))$, generally different from $f(p)$. The new metric at $p$ is obtained by applying the original metric at $\Phi(p)$ to the pushed-forward tangent vectors. This is an active comparison of field configurations using the manifold points as an identification map.

If $g$ and matter fields $\psi$ solve covariant field equations, then $\Phi^*g$ and $\Phi^*\psi$ solve them as well. Every coincidence relation is transported together. An apparatus-defined event and its measured scalar therefore carry to an isomorphic event with the same relational result.

For a one-parameter flow generated by $\xi^a$, choose the convention

$$\delta_\xi g_{ab}=\mathcal L_\xi g_{ab}=2\nabla_{(a}\xi_{b)}, \qquad \delta_\xi f=\mathcal L_\xi f=\xi^a\nabla_af.$$

Some sources use the inverse flow and an overall minus sign; equations must be translated as a set.

The same formula, different interpretation

Consider the real line with scalar $f(x)=x^2$ and translation $\Phi_a(x)=x+a$. Active pullback gives $f'(x)=(x+a)^2$ at the same coordinate label $x$. A passive coordinate $x'=x-a$ represents the original field as $f'(x')=(x'+a)^2$. The arrays have the same functional appearance, but the active reading compares two fields at one identified point while the passive reading describes one field with a new label.

The distinction is not determined by primes. It is determined by what is held fixed: geometric fields for passive change, or point identification for active change. Carroll illustrates this relation and its use in general covariance Carroll 1997, Chapter 5, pp. 129–141, PDF.

When active maps are gauge

An active diffeomorphism connected to the identity is a gauge redundancy when it transforms every dynamical field and acts trivially on the physical boundary, asymptotic, and external-reference data. Compactly supported transformations in the interior provide the standard example. Preserving the form of boundary conditions is not sufficient: an asymptotic time translation can preserve the allowed falloff while possessing an energy generator. The boundary variational problem determines which transformations are null gauge directions and which are physical symmetries. Transforming the metric while leaving a material reference system fixed can also change measurements.

Thus “diffeomorphism invariant” has at least three layers: tensor equations retain their form under passive charts; the solution space is mapped into itself by active pullbacks; and a subset of those active maps is quotiented as gauge after boundary conditions are declared.

Exercises

Under an active pullback, why must the detector worldline be transformed together with the metric before concluding that a measured proper time is unchanged?

Solution

Proper time is a relational functional of both metric and curve. Pulling back only $g$ while holding the curve fixed compares a different pairing and can change the integral. If $g'=\Phi^*g$, the corresponding curve is $\gamma'=\Phi^{-1}\circ\gamma$, including its interaction-defined endpoints. Then $g'(\dot\gamma',\dot\gamma')=g(\dot\gamma,\dot\gamma)$ along corresponding events, so the proper-time integrals agree. The inverse map on the curve is essential to the active-pullback convention.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.