Proper Time and the Clock Hypothesis

Proper time is the metric length of a timelike worldline, and an ideal clock is one whose reading advances by that proper time. The clock hypothesis is the additional physical assertion that acceleration affects such a clock through the path it follows, not through a separate instantaneous acceleration-dependent rate. Real clocks approximate this model only within regimes set by their construction and environment.

Required background. Observers, Rest Spaces, and Measured Components provides observer projectors and measured decompositions; Pseudo-Riemannian Metrics and Signature provides Lorentzian norms and causal classification; Frames, Coframes, and Anholonomy provides dual frames and local Lorentz freedom.

Helpful background. Events, Intervals, and Minkowski Geometry supplies the flat-spacetime limit; Tensor Fields and Natural Operations clarifies tensors versus their components; Proper Acceleration and Accelerometer Readings gives operational intuition for carried instruments.

Proper time along a timelike path

Let $x^\mu(\lambda)$ be a smooth future-directed timelike curve between parameter values $\lambda_1$ and $\lambda_2$. Its elapsed proper time is

$$\Delta\tau =\int_{\lambda_1}^{\lambda_2} \sqrt{-g_{\mu\nu}(x) \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda}}\,d\lambda.$$

The minus sign makes the integrand real and positive for the inherited timelike norm $u^2=-1$. Under any smooth increasing reparametrization, the tangent and parameter element acquire reciprocal factors, so $\Delta\tau$ is unchanged. Choosing $\lambda=\tau$ gives the unit four-velocity $u^\mu=dx^\mu/d\tau$.

Proper time is attached to the entire worldline segment, not merely to its endpoints. Two clocks that depart from one event and reunite at another can therefore record different elapsed times. The metric definition, four-velocity normalization, and local extremal property of geodesics are developed in Reall 2013, §§ 3.2–3.3, pp. 34–38, PDF; the flat-spacetime path comparison appears in Carroll 1997, Ch. 1, pp. 25–27.

The clock hypothesis

Geometry supplies a candidate elapsed time. Calling it the reading of a physical clock adds assumptions about the device. An ideal clock is local, has negligible backreaction, and responds only to the metric along its timelike path. Its rate contains no independent scalar such as $a_\mu a^\mu$:

$$d\tau_{\rm ideal}^2=-g_{\mu\nu}dx^\mu dx^\nu.$$

Thus a rapidly accelerated ideal clock still follows the same rule. Acceleration matters indirectly because it changes $x^\mu(\lambda)$ and hence the integrated metric length. This distinction prevents a common error: the clock hypothesis does not say that accelerated and inertial clocks accumulate equal time, only that no extra acceleration term is appended after their paths are specified.

The standard-clock parametrization is unique up to a positive affine change of scale and origin; fixing $g(\dot\gamma,\dot\gamma)=-1$ makes that parameter proper time. An operational radar characterization is described in Perlick 2008, § 3, pp. 7–8, preprint PDF. This characterization tests a physical idealization; it is not a consequence of coordinate choice.

Regime of an ideal clock

A real clock contains an oscillator, cavity, atomic transition, mechanical structure, or another internal process. The ideal approximation requires its characteristic size $L$ and response time $t_c$ to be small compared with the scales on which acceleration, curvature, and external fields change. Dimensionless indicators include

$$aL\ll1, \qquad a t_c\ll1, \qquad L^2\lvert R_{\hat0\hat\imath\hat0\hat\jmath}\rvert\ll1,$$

with the acceleration and curvature evaluated in a carried orthonormal frame. These are scaling criteria, not universal error formulas; the coefficients depend on the clock mechanism.

Direct acceleration sensitivity can arise from deformation, vibration, temperature change, electromagnetic fields used to accelerate the device, finite signal-transit time, or coupling of internal degrees of freedom to rotation and tides. Such effects are corrections to a particular instrument, not modifications of the spacetime definition of proper time. A finite clock also samples a world tube rather than a single curve, so sufficiently strong tidal variation defeats the point-clock model even when its central worldline is timelike.

Path and limiting checks

In an inertial Minkowski chart, a clock moving with ordinary speed $v(t)$ records

$$\Delta\tau=\int dt\,\sqrt{1-\lvert\mathbf v(t)\rvert^2}.$$

This reduces to coordinate time for a clock at rest and tends to zero as a timelike path approaches a null path. For a small speed, $\Delta\tau=\int dt\,[1-\mathbf v^2/2+O(v^4)]$, recovering the leading special-relativistic correction. None of these checks introduces the coordinate acceleration $d\mathbf v/dt$ as a separate rate term.

The local geodesic statement also needs care. A freely falling timelike geodesic locally maximizes proper time relative to nearby timelike curves with the same endpoints, but it need not be globally maximizing beyond conjugate points or across multiple geodesic segments. The clock hypothesis applies equally to forced and freely falling timelike curves; extremality is a statement about motion, not clock calibration.

Exercises

Two ideal clocks meet at $(t,x)=(0,0)$ in Minkowski spacetime and reunite at $(T,0)$. Clock A remains at rest. Clock B moves at speed $v$ for the first half of the coordinate time and at speed $-v$ for the second half. Compare their readings and explain the role of the turnaround.

Solution

Clock A records $\Delta\tau_A=T$. On each constant-speed leg, clock B has $d\tau=dt\sqrt{1-v^2}$, so

$$\Delta\tau_B=T\sqrt{1-v^2}=\frac{T}{\gamma}<T.$$

The idealized instantaneous reversal has unbounded acceleration, but it occupies zero coordinate duration and contributes no additional term to the proper-time integral. Replacing it by any short smooth turnaround changes the worldline and therefore changes the integral slightly; it still does not require an acceleration-dependent clock-rate law. The age difference is determined by the two complete paths.

References

• Carroll, Sean M. “Lecture Notes on General Relativity”, arXiv:gr-qc/9712019v1, 1997, Ch. 1, pp. 25–27.
• Perlick, Volker. “On the Radar Method in General-Relativistic Spacetimes”, in Lasers, Clocks and Drag-Free Control, Astrophysics and Space Science Library, Springer, 2008, pp. 131–152; Open PDF.
• Reall, Harvey S. General Relativity, PDF, Part III lecture notes, University of Cambridge, 2013, §§ 3.2–3.3, pp. 34–38.