GPS in Space: Relativistic Distortions in XNAV Near a Black Hole
DOI:
https://doi.org/10.58445/rars.4040Keywords:
special relativity, general relativity, X-ray Pulsar Navigation (XNAV)Abstract
Special relativity describes the relative nature of space and time, while general relativity explains gravity not as a force, but as the curvature of spacetime caused by mass and energy. Spacecraft navigation systems like the Global Positioning System (GPS) and the new X-ray Pulsar Navigation (XNAV) concept both depend on precise timing to calculate position, meaning both are sensitive to relativistic effects. Near Earth, these effects are small and can be corrected as minor adjustments layered on top of an otherwise flat-space calculation. Near a black hole, however, that assumption no longer works. This paper investigates how gravitational redshift, Shapiro delay, and gravitational lensing would corrupt an XNAV system's ability to determine a spacecraft's position near a Schwarzschild black hole. Using Python-based simulations normalized to the Schwarzschild radius, we model three separate effects: the degradation of a received signal's frequency as a function of distance from the event horizon, the positional error introduced when Shapiro delay is left unaccounted for across a range of impact parameters, and the angular deflection of an incoming signal caused by gravitational lensing. Our results show that all three effects remain small and manageable at large distances, but increase sharply and nonlinearly once the signal's path nears the photon sphere, to the extent where an uncorrected system would mistake these relativistic distortions for genuine errors in the spacecraft's position. As for a lighter and more intuitive understanding of these results, we also generated a set of audio recordings that model how the pitch and speed of a transmitted signal would change with varying black hole mass and emitter distance. When put together, this paper seeks to show that deep space navigation must take relativistic corrections into serious consideration, and corrections would need to be built directly into the timing and position model from the start.
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