In this article, we describe and numerically implement a method for relativistic location in slightly curved but otherwise generic spacetimes. For terrestrial positioning in the context of Global Navigation Satellite Systems, our algorithm incorporates gravitational as well as tropospheric and ionospheric effects modeled by the Gordon metric. The algorithm is implemented in the squirrel.jl code, which employs a quasi-Newton Broyden algorithm in conjunction with automatic differentiation of numerical geodesics. Our work provides a practical solution to the relativistic location problem in a generic spacetime and consolidates relativistic and atmospheric effects in a single framework. Though optimization is not our primary focus, our implementation is already fast enough for practical use, establishing a position from five emission points in $<1\mathrm{s}$ on a desktop computer for reasonably simple spacetime geometries. In vacuum, our implementation can achieve submillimeter accuracy considering the Kerr metric with terrestrial parameters and submeter accuracy including tropospheric and ionospheric effects.

中文翻译：

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弯曲时空中的相对论定位算法

在本文中，我们描述并以数值方式实现了一种在略微弯曲但其他通用时空中的相对论定位方法。对于全球导航卫星系统背景下的地面定位，我们的算法结合了由 Gordon 度量建模的引力以及对流层和电离层效应。该算法在squirrel中实现。jl代码，它采用准牛顿布罗伊登算法与数值测地线的自动微分相结合。我们的工作为通用时空中的相对论定位问题提供了一个实用的解决方案，并将相对论和大气效应整合到一个框架中。虽然优化不是我们的主要关注点，但我们的实现已经足够快以供实际使用，从五个排放点建立一个位置$<1\mathrm{s}$在台式计算机上进行相当简单的时空几何。在真空中，考虑到具有地面参数的 Kerr 度量和包括对流层和电离层效应在内的亚米级精度，我们的实现可以实现亚毫米级精度。