Connection Effects versus Curvature

Connection coefficients describe how a chosen coordinate basis changes; curvature describes an invariant failure of parallel transport to be path independent. Apparent gravitational acceleration can therefore be created by accelerated or curvilinear coordinates in flat spacetime, and it can be removed at one event in a curved spacetime. Tidal response, infinitesimal holonomy, and the Riemann tensor distinguish the two cases.

Required background. We use the Riemann curvature tensor and the operational limits established in the equivalence-principles chapter.

Helpful background. Useful preparation includes Minkowski geometry, smooth tensor fields, frames and anholonomy, and affinely parametrized geodesics.

Why the connection is coordinate dependent

For the Levi-Civita connection,

$$\Gamma^\rho{}_{\mu\nu} =\frac12 g^{\rho\sigma} \left(\partial_\mu g_{\sigma\nu} +\partial_\nu g_{\sigma\mu} -\partial_\sigma g_{\mu\nu}\right).$$

Under a coordinate transformation, $\Gamma^\rho{}_{\mu\nu}$ acquires a term containing second derivatives of the coordinate map. It is not a tensor. The coordinate geodesic equation,

$$\frac{d^2x^\rho}{d\lambda^2} +\Gamma^\rho{}_{\mu\nu} \frac{dx^\mu}{d\lambda}\frac{dx^\nu}{d\lambda}=0,$$

accordingly divides the same geometric motion differently between coordinate second derivatives and connection terms in different charts. Reall derives this transformation behavior and the normal-coordinate cancellation explicitly Reall 2013, §§ 4.2 and 4.4, pp. 41–48, PDF.

This does not make the connection unphysical. It is indispensable for comparing vectors at neighboring events and for differentiating tensor fields. It means only that a numerical list of Christoffel symbols has no observer-independent meaning without its coordinates or frame.

Nonzero connection in flat spacetime

Consider the Rindler form of the Minkowski metric in one wedge,

$$ds^2=-\kappa^2\rho^2d\eta^2+d\rho^2+dy^2+dz^2, \qquad \rho>0.$$

The nonzero independent connection coefficients include

$$\Gamma^\rho{}_{\eta\eta}=\kappa^2\rho, \qquad \Gamma^\eta{}_{\eta\rho} =\Gamma^\eta{}_{\rho\eta}=\frac1\rho.$$

Observers at fixed $\rho,y,z$ are accelerated. Their nonzero proper acceleration and the coordinate acceleration of freely moving bodies can look gravitational, yet direct substitution into

$$R^\rho{}_{\sigma\mu\nu} =\partial_\mu\Gamma^\rho{}_{\nu\sigma} -\partial_\nu\Gamma^\rho{}_{\mu\sigma} +\Gamma^\rho{}_{\mu\alpha}\Gamma^\alpha{}_{\nu\sigma} -\Gamma^\rho{}_{\nu\alpha}\Gamma^\alpha{}_{\mu\sigma}$$

gives zero. For example,

$$R^\rho{}_{\eta\rho\eta} =\partial_\rho(\kappa^2\rho) -\Gamma^\rho{}_{\eta\eta}\Gamma^\eta{}_{\rho\eta} =\kappa^2-\kappa^2=0.$$

The derivative and quadratic connection terms cancel. Dropping either class would incorrectly diagnose curvature.

Vanishing connection in curved spacetime

At any event $p$ of a smooth torsion-free Lorentzian geometry, Riemann normal coordinates can be chosen so that

$$g_{\mu\nu}(p)=\eta_{\mu\nu}, \qquad \partial_\alpha g_{\mu\nu}(p)=0, \qquad \Gamma^\rho{}_{\mu\nu}(p)=0.$$

Nevertheless, first derivatives of the connection need not vanish. At the central event the curvature is precisely their antisymmetrized combination, because the $\Gamma\Gamma$ terms vanish there. A nonzero $R^\rho{}_{\sigma\mu\nu}(p)$ cannot be removed by another smooth coordinate choice: it is a tensor.

Carroll emphasizes the complementary roles of connection and curvature and relates curvature to parallel transport around infinitesimal loops Carroll 1997, Chapter 3, pp. 68–88, PDF. The comparison yields a decisive two-case test: Rindler coordinates have $\Gamma\ne0$ but $R=0$; normal coordinates in a curved spacetime can have $\Gamma(p)=0$ but $R(p)\ne0$.

What local instruments distinguish

An accelerometer measures the proper acceleration of its own worldline. It can read nonzero in flat spacetime when supported or driven, so it does not measure curvature by itself. A gradiometer compares the relative acceleration of separated freely falling probes. To leading order that response is linear in the Riemann tensor and the separation, making it a curvature measurement.

Comparing transport along two paths with common endpoints provides another discriminator: the leading mismatch is curvature times the oriented area between the paths. For gyroscopes, accelerations and frame alignment must first be accounted for; the closed loop is formed mathematically by reversing one transport map, not by sending a gyroscope backward in time. Both tidal and transport procedures require a finite separation or area. A single pointlike freely falling experiment can eliminate connection effects at its central event, exactly as the equivalence principle predicts.

Exercises

Verify the displayed Rindler curvature component and explain why the two nonzero terms must be kept together.

Solution

For $R^\rho{}_{\eta\rho\eta}$, stationarity makes the $\partial_\eta\Gamma$ term vanish. The remaining derivative is $\partial_\rho\Gamma^\rho{}_{\eta\eta}=\kappa^2$. The only nonzero quadratic contribution has the opposite sign:

$$-\Gamma^\rho{}_{\eta\eta}\Gamma^\eta{}_{\rho\eta} =-(\kappa^2\rho)(1/\rho)=-\kappa^2.$$

Their sum is zero. Connection coefficients can be nonzero solely because the basis varies; curvature tests the combination in which coordinate artifacts cancel.

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.