Observables and Coordinate-Invariant Questions

A physically meaningful reading must survive a change of coordinates, but merely contracting all spacetime indices does not make it gauge invariant. A scalar evaluated at an unphysical coordinate label changes under active diffeomorphism gauge. Tetrad components can be coordinate-independent detector readings once their event and physical frame are specified. The missing ingredient is relational specification, not a particular notation for indices.

Required background. Tensor fields and pullbacks identify corresponding fields and events. Observer projections turn tensor components into physical readings.

Helpful background. Special relativity provides clock and frame examples. Spacetime symmetries, variational gauge directions, and the gravitational boundary problem extend the discussion to allowed boundary transformations and charges.

Three different invariance questions

First, an expression is coordinate covariant when its components transform consistently between charts. Second, a functional is gauge invariant when it has the same value on field configurations related by allowed active diffeomorphisms. Third, a reading may depend on a physical frame or reference system while remaining coordinate independent. These are distinct properties.

For example, the curvature scalar $R(p)$ is independent of coordinates once the event $p$ is geometrically fixed. But the expression “$R$ at $x^\mu=(1,0,0,0)$” uses a coordinate label to select the event. Under an active diffeomorphism, the scalar field is pulled back while the label is held fixed; its value generally changes. Scalar transformation character did not make the coordinate-anchored functional gauge invariant.

Tambornino explains why local scalar fields at fixed manifold labels are not generally diffeomorphism-invariant observables and why relations between dynamical fields supply the appropriate completion Tambornino 2012, § 1, pp. 1–3, PDF.

Relational events and clock readings

Suppose four scalar reference fields $X^A$, with $A=0,1,2,3$, identify an event by $X^A=\tau^A$. A sufficient local condition is that the four gradients $dX^A$ are linearly independent, so the inverse-function theorem supplies a unique event in a chosen neighborhood. Define

$$\mathcal O_R(\tau)=R(p)\quad\text{where}\quad X^A(p)=\tau^A.$$

Under an active diffeomorphism $\Phi$, both $R$ and $X^A$ are pulled back. If the original reference condition selects $p$, the transformed condition selects $p'=\Phi^{-1}(p)$, because $(\Phi^*X^A)(p')=X^A(p)=\tau^A$. Consequently $(\Phi^*R)(p')=R(p)$. This is a relational observable: curvature when physical rods and clocks have specified readings.

The construction can fail if the reference fields are degenerate, take the same values at several events, or significantly backreact in a model that treated them as ideal. Relational specification is not a magic global coordinate system; its domain and physical realization must be stated.

Proper time between two worldline coincidences gives a simpler example:

$$\Delta\tau=\int_{A}^{B}\sqrt{-g_{ab}dx^adx^b}.$$

If $A$ and $B$ are defined by detector interactions rather than coordinate times, the metric, worldline, and endpoints transform together and the integral is gauge invariant.

Tetrad components are legitimate readings

An observer carrying an orthonormal tetrad measures

$$T_{(a)(b)}=T_{\mu\nu}e_{(a)}{}^\mu e_{(b)}{}^\nu.$$

For each fixed pair of frame labels $(a),(b)$, this contraction is a spacetime scalar: the coordinate indices $\mu,\nu$ are fully contracted. The collection transforms under boosts and rotations of the tetrad, so a physically boosted or rotated apparatus generally reports different components. It nevertheless remains invariant under a passive coordinate change because tensor and tetrad components transform oppositely. Apparatus dependence is not coordinate dependence.

Energy $E=-p_au^a$ and the three measured momentum components $p_{(i)}=p_ae_{(i)}{}^a$ are all scalar contractions with the supplied apparatus. The momentum readings still carry spatial frame labels and rotate when the axes rotate. Neither scalarhood nor those labels decide the gauge question: the event, observer, and axes must all be specified relationally and transformed together.

Boundary observables and nonlocality

Boundary and asymptotic quantities provide another class. A total energy can be associated with a normalized asymptotic time translation and a surface integral. It is coordinate independent under transformations preserving the asymptotic structure, yet it is not a local scalar density in the interior. Wilson-loop-like holonomies and elapsed times are also extended functionals.

This nonlocality is natural. Diffeomorphism gauge moves manifold points, so an observable tied to one bare point cannot generally survive. Matter coincidences, boundaries, integrated curves, or dynamical reference fields anchor the question. Carroll’s active/passive discussion supplies the geometric basis for this conclusion Carroll 1997, Chapter 5, pp. 129–141, PDF.

A practical test

To assess a proposed observable, ask: What physical procedure selects the event or region? Which fields and apparatus are transformed under gauge? Which boundary frame or normalization is fixed? Does the result change under a passive chart? Does it remain constant along allowed active gauge orbits? These questions are more discriminating than asking whether the final formula contains contracted indices.

Exercises

Explain why $R(0,0,0,0)$ and “the curvature scalar when a specified clock reads noon at its intersection with a specified detector” have different gauge status.

Solution

The coordinate origin is not transported as a dynamical field under an active diffeomorphism, so the pulled-back scalar at that fixed label can change. In the second statement, the clock, detector, intersection event, and curvature field all transform together. The same relational coincidence is selected on every gauge-related representative, so the curvature value is unchanged within the protocol’s domain.

References

• Carroll, Sean M. “Lecture Notes on General Relativity.” NSF-ITP/97-147, arXiv:gr-qc/9712019v1 (1997). arXiv.
• Tambornino, Johannes. “Relational Observables in Gravity: a Review.” SIGMA 8 (2012): 017. DOI; arXiv.