A null frame for spacetime positioning by means of pulsating sources

We introduce an operational approach to the use of pulsating sources, located at spatial infinity, for defining a relativistic positioning and navigation system, based on the use of four-dimensional bases of null four-vectors, in flat spacetime. As a prototypical case, we show how pulsars can be used to define such a positioning system. The reception of the pulses for a set of different sources whose positions in the sky and periods are assumed to be known allows the determination of the user’s coordinates and spacetime trajectory, in the reference frame where the sources are at rest. We describe our approach in flat Minkowski spacetime, and discuss the validity of this and other approximations we have considered.

💡 Research Summary

The paper proposes a relativistic positioning system that uses pulsating sources located effectively at spatial infinity—most notably pulsars—to define a four‑dimensional null frame in flat Minkowski space. The core idea is to treat each source’s light‑like world‑line as a null four‑vector (k^{(a)}{\mu}) (a = 1…4). Because a null vector satisfies (k^{(a)}{\mu}k^{(a)\mu}=0) and points in the direction of propagation, a set of four linearly independent null vectors spans the entire spacetime and can serve as a basis for coordinates.

Each pulsar emits pulses with a known period (T_{a}) and a known direction (\hat n_{a}) on the celestial sphere. The emitted wave‑four‑vector can be written as (k^{(a)}{\mu}= \omega{a}(1,\hat n_{a})) where (\omega_{a}=2\pi/T_{a}). When a receiver records the arrival times of successive pulses from each source, it obtains a sequence of proper‑time stamps (t^{(a)}{n}). The accumulated phase (\phi^{(a)}=n,2\pi) is related to the receiver’s spacetime position (x^{\mu}) through the scalar product (k^{(a)}{\mu}x^{\mu}= \phi^{(a)}). Thus, four such equations—one per source—form a linear system

\