Accelerated Coordinate Systems

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An accelerated coordinate chart can fail because its map loses rank, because its chosen simultaneity surfaces intersect, or because its fixed-coordinate curves cease to be timelike. None of these implies that Minkowski spacetime is singular. Coordinates are labels; a reference frame additionally selects physical observers, clocks, and axes.

Required background. Special Relativity provides Minkowski causal geometry, four-vectors, and mass-shell kinematics; Relativistic Observers and Measurement provides observer rest spaces and invariant decompositions of measured quantities. Helpful background. Calculus and Linear Algebra provides multivariable differentiation, linear maps, inner products, and basis changes; Ordinary Differential Equations provides initial-value problems, linearization, and asymptotic solution methods.

The metric of an accelerated frame

Let $z^\mu(\tau)$ be a reference worldline with four-velocity $u^\mu$ , four-acceleration $a^\mu=D u^\mu/d\tau$ , and a Fermi–Walker transported spatial triad $e_{\hat i}{}^\mu$ . A natural local map from coordinates $(\tau,\xi^{\hat i})$ to inertial Cartesian coordinates is

X^\mu(\tau,\boldsymbol\xi) =z^\mu(\tau)+\xi^{\hat i}e_{\hat i}{}^\mu(\tau).

At fixed $\tau$ , $\boldsymbol\xi$ labels the hyperplane orthogonal to the reference observer’s velocity. This is a convention for distant simultaneity, not a measurement made at one event. Radar coordinates, which use emission and reception times of light signals, are another operationally meaningful extension and need not agree far from the worldline; Pauri and Vallisneri 2000, §§2–3, printed pp. 2–5 compare these constructions.

In the inertial Cartesian components used by the map, Fermi–Walker transport gives

\frac{de_{\hat i}{}^\mu}{d\tau} =u^\mu a_{\hat i}, \qquad a_{\hat i}=a_\mu e_{\hat i}{}^\mu.

Therefore

dX^\mu =\left(1+a_{\hat i}\xi^{\hat i}\right)u^\mu d\tau +e_{\hat i}{}^\mu d\xi^{\hat i}.

Using $u^2=-1$ , $u\mathbin{\cdot}e_{\hat i}=0$ , and $e_{\hat i}\mathbin{\cdot}e_{\hat j}=\delta_{\hat i\hat j}$ yields the exact flat metric in this local chart:

\boxed{ds^2 =-\left(1+a_{\hat i}(\tau)\xi^{\hat i}\right)^2d\tau^2 +\delta_{\hat i\hat j}d\xi^{\hat i}d\xi^{\hat j}.}

The map, tetrad transport, and locality restriction are displayed in Gourgoulhon 2010, PDF slides 15–19. The derivation above additionally makes the exact metric coefficients explicit.

The coefficient

N(\tau,\boldsymbol\xi)=1+a_{\hat i}(\tau)\xi^{\hat i}

relates coordinate time to proper time for a fixed- $\boldsymbol\xi$ clock. On the connected component containing the reference worldline, choose the common future orientation $N\gt0$ ; then $d\sigma=N\,d\tau$ . A position-dependent clock rate has appeared in flat spacetime solely because the clock network accelerates.

Where the construction stops

The coordinate basis vector $\partial X^\mu/\partial\tau=Nu^\mu$ vanishes when $N=0$ , so the Jacobian loses rank there. Before reaching that surface, simultaneity hyperplanes belonging to different values of $\tau$ may already intersect, assigning multiple coordinate labels to one event. A physical chart built outward from the reference worldline must therefore stay one-to-one and inside its future-oriented $N\gt0$ component.

Even a nonsingular coordinate transformation does not guarantee a physical observer network. The curves of fixed spatial coordinates must be timelike. Rotating coordinates illustrate the distinction sharply: their transformation remains algebraically invertible beyond the light cylinder, but fixed rotating positions would move faster than light.

All components of the Riemann tensor remain zero wherever this chart is regular, because the metric is a coordinate transform of $\eta_{\mu\nu}$ . Nonzero Christoffel symbols and coordinate forces are not curvature. An explicit intersection of extended-tetrad simultaneity surfaces is shown and discussed in Pauri and Vallisneri 2000, §2, printed pp. 2–3.

A one-dimensional accelerated lattice

The induced spatial metric of the fixed-coordinate congruence is $\delta_{\hat i\hat j}$ and is independent of $\tau$ . Consequently its comoving distances do not expand or shear: this nonrotating congruence is Born-rigid wherever the map is regular and injective. This conclusion is special to the construction; a freely prescribed rotating Born-rigid body is much more restricted.

Suppose now that the acceleration always points along $e_{\hat1}$ , with signed magnitude $a(\tau)$ . At fixed $\xi$ on the $N\gt0$ component, the grid observer has $d\sigma=N\,d\tau$ and shares the instantaneous velocity $u^\mu(\tau)$ . Because $d/d\sigma=N^{-1}d/d\tau$ , its four-acceleration and magnitude are

A_{\rm grid}^{\mu}=\frac{a^\mu}{N}, \qquad \alpha(\tau,\xi)=\frac{|a(\tau)|}{1+a(\tau)\xi}.

Thus an extended, instantaneously rigid lattice cannot give every point the same proper acceleration. The acceleration profile is part of the frame choice, not an attribute of the coordinate label alone.

Transfer task: clock and acceleration comparison

At one instant, a reference observer has $a\gt0$ along $+\xi$ . A second grid clock is a proper distance $L\gt0$ ahead on the same $\tau=\text{constant}$ slice, so $\xi=L$ . Compare the two clock rates and accelerometer readings.

Solution

At the reference worldline, $N(0)=1$ and $\alpha(0)=a$ . At the forward clock,

\frac{d\sigma_L}{d\tau}=1+aL, \qquad \alpha_L=\frac{a}{1+aL}.

The forward clock accumulates more proper time per unit reference proper time and requires less proper acceleration. Their product satisfies $(d\sigma_L/d\tau)\alpha_L=a$ . For $aL\ll1$ , the fractional clock-rate difference is $aL$ , while $\alpha_L\simeq a(1-aL)$ . Reversing the direction of $L$ reverses the first-order signs. At $L=-1/a$ , the chart degenerates; no material lattice element can be placed there within this construction.

Uniform Acceleration and Rindler Wedges now turns this local boundary into an exactly solvable causal geometry.

References

Éric Gourgoulhon, Special relativity from an accelerated observer perspective (PDF), printed slides 15–19, Observatoire de Paris, 2010.
M. Pauri and M. Vallisneri, “Märzke–Wheeler Coordinates for Accelerated Observers in Special Relativity,” Foundations of Physics Letters 13(5), 401–425, 2000, doi:10.1023/A:1007861914639; Open PDF, §§2–3, printed pp. 2–5.