Observer Worldlines and Local Tetrads

An observer in curved spacetime is specified first by motion, not by coordinates. A future-directed timelike worldline identifies where the observer is, while an orthonormal tetrad along that worldline identifies the time axis and the three directions carried by the apparatus. Tensor components in this tetrad are local readings; coordinate components generally are not.

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.

Worldlines and carried tetrads

Let $\gamma:\tau\mapsto x^\mu(\tau)$ be a smooth future-directed timelike worldline, parametrized by proper time. Its unit tangent is

$$u^\mu=\frac{dx^\mu}{d\tau}, \qquad u^\mu u_\mu=-1.$$

At each event of $\gamma$, the observer’s instantaneous rest space is the subspace orthogonal to $u^\mu$. Choose three spacelike unit vectors $e_{\hat\imath}{}^\mu$ spanning that space and set $e_{\hat0}{}^\mu=u^\mu$. The tetrad obeys

$$g_{\mu\nu}e_{\hat a}{}^\mu e_{\hat b}{}^\nu =\eta_{\hat a\hat b}, \qquad \eta_{\hat a\hat b}=\operatorname{diag}(-1,1,1,1).$$

The dual coframe $\theta^{\hat a}{}_{\mu}$ is defined by $\theta^{\hat a}{}_{\mu}e_{\hat b}{}^\mu=\delta^{\hat a}{}_{\hat b}$. In particular, $\theta^{\hat0}{}_{\mu}=-u_\mu$. This construction is pointwise: it assumes an idealized, effectively pointlike apparatus in a time-oriented Lorentzian spacetime. It does not yet say how the spatial axes at different proper times are related.

Orthonormal tetrads and their local Lorentz freedom are developed explicitly in Carroll 1997, Ch. 3, pp. 88–91. Once the time leg is fixed to a particular $u^\mu$, the remaining freedom is a spatial rotation of the three apparatus axes. A Lorentz boost changes the time leg and therefore describes a different observer at that event.

From tensors to instrument readings

For a vector $V^\mu$, the tetrad components are

$$V^{\hat0}=-u_\mu V^\mu, \qquad V^{\hat\imath}=e^{\hat\imath}{}_{\mu}V^\mu.$$

They reconstruct the geometric vector through

$$V^\mu=V^{\hat0}u^\mu+V^{\hat\imath}e_{\hat\imath}{}^\mu.$$

The temporal reading depends on the observer velocity. The three spatial readings also depend on how that observer orients the apparatus. A coordinate change alters the displayed components of $V^\mu$, $u^\mu$, and $e_{\hat a}{}^\mu$ together, leaving each contraction unchanged.

A simple chart illustrates the distinction. If $g_{0i}=0$, $g_{00}=-N^2$, and an observer remains at fixed spatial coordinates, then $u^\mu=N^{-1}(\partial_t)^\mu$. The measured temporal component is $V^{\hat0}=NV^0$, not the coordinate component $V^0$. With nonzero $g_{0i}$, the relation also mixes temporal and spatial coordinate components. Normalizing a coordinate basis vector is therefore necessary but not always sufficient; the whole tetrad must be orthonormal.

The same rule applies to higher-rank tensors. For example,

$$T_{\hat a\hat b} =T_{\mu\nu}e_{\hat a}{}^\mu e_{\hat b}{}^\nu$$

are the components of a covariant rank-two tensor read in the tetrad. Conversely, the dual coframe reconstructs it as $T_{\mu\nu}=\theta^{\hat a}{}_{\mu}\theta^{\hat b}{}_{\nu}T_{\hat a\hat b}$. This reversibility matters: a tetrad changes the component description without discarding tensor information. For stress–energy, an observer measures local energy density

$$\rho_{(u)}=T_{\mu\nu}u^\mu u^\nu$$

and the energy-momentum current $j^\mu=-T^\mu{}_{\nu}u^\nu$. Its spatial projection is the energy flux seen by that observer. These identifications, including the fact that vectors at different events cannot be contracted without an additional comparison rule, are stated in Reall 2013, § 5.2, pp. 51–53, PDF.

Reconstruction and covariance checks

The rest-space projector

$$h^\mu{}_{\nu}=\delta^\mu{}_{\nu}+u^\mu u_\nu$$

provides a basis-free check. Every vector decomposes uniquely as

$$V^\mu=(-u_\nu V^\nu)u^\mu+h^\mu{}_{\nu}V^\nu.$$

Expanding the spatial term in $e_{\hat\imath}{}^\mu$ reproduces the tetrad formula above. Contracting with $u_\mu$ checks the minus sign, and applying $h^\mu{}_{\nu}$ twice verifies that it is a projector.

Under a coordinate transformation, the contracted tetrad readings are unchanged numbers. Under a local Lorentz change of tetrad, they transform as Lorentz components while the reconstructed tensor is unchanged. For two tetrads sharing the same time leg, this reduces to an ordinary rotation of the spatial component lists, so quantities such as $\delta_{\hat\imath\hat\jmath}V^{\hat\imath}V^{\hat\jmath}$ remain fixed. These two transformations answer different questions: coordinate covariance changes the description, whereas changing the tetrad changes the physical observer or the orientation of the apparatus.

In particular, two observers meeting at one event may assign different energies and spatial momenta to the same particle, yet both reconstruct the same four-momentum. Their tetrads are related by a Lorentz transformation, and the invariant mass shell supplies a direct consistency check. This local comparison needs no transport because every vector involved belongs to the same tangent space.

What one local frame does not determine

A tetrad at one event does not define simultaneity, distance, or relative velocity for distant events. Extending the frame along $\gamma$ requires a transport law; extending it away from $\gamma$ requires a coordinate or signaling prescription. Different choices can agree on the worldline and disagree elsewhere.

The point-observer approximation also suppresses detector size, response time, internal stresses, and tidal variation across the apparatus. It is reliable only when those scales are small compared with the relevant acceleration, curvature, and signal-variation scales. Proper-time accumulation, radar assignment, and gyroscope transport therefore require separate constructions rather than being hidden inside the word “observer.”

Exercise

At one event a diagonal chart has $ds^2=-N^2dt^2+A^2dx^2$ in the $t$–$x$ plane, with $N,A>0$. A static observer uses $e_{\hat0}=N^{-1}\partial_t$ and $e_{\hat1}=A^{-1}\partial_x$. For $V=V^t\partial_t+V^x\partial_x$, compute the two tetrad readings and verify the norm.

Solution

The observer has $u^\mu=(N^{-1},0)$ and $u_\mu=(-N,0)$, so

$$V^{\hat0}=-u_\mu V^\mu=NV^t, \qquad V^{\hat1}=e^{\hat1}{}_{\mu}V^\mu=AV^x.$$

Therefore

$$-(V^{\hat0})^2+(V^{\hat1})^2 =-N^2(V^t)^2+A^2(V^x)^2 =g_{\mu\nu}V^\mu V^\nu.$$

The coordinate components and tetrad readings differ, but both descriptions give the same scalar norm.

References

• Carroll, Sean M. “Lecture Notes on General Relativity”, arXiv:gr-qc/9712019v1, 1997, Ch. 3, pp. 88–91.
• Reall, Harvey S. General Relativity, PDF, Part III lecture notes, University of Cambridge, 2013, §§ 3.2 and 5.2, pp. 34–35 and 51–53.