ES1 - Exact Solutions in Four and Higher Dimensions: Mathematical Aspects |
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Speaker |
MacLaurin, Colin |
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| Talk Title |
Clarifying spatial distance measurement |
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Abstract |
The theoretical measurement of spatial distance is surprisingly not well known. For Schwarzschild-Droste spacetime one paper claims a 10^77 light-year separation inside the horizon, while textbooks state an absolute-sounding radial distance but don't clarify which local observers measure it nor what others would determine. Despite consensus on Bell's spaceship paradox and Ehrenfest's rotating disc, some authors continue to resist. Still, it is reasonably well known that spatial distance should be measured orthogonal to the observer's worldline. I overview four theoretical tools to achieve this, assuming only local measurement, and show their equivalence: the spatial projector, Landau-Lifshitz radar metric, adapted coordinates, and orthonormal frames. Building on this consistent foundation, I derive expressions comparing an observer's ruler with some external standard, either a scalar field or another ruler. For a scalar such as a coordinate, take the differential (1-form) and contract it with the ruler vector. For another ruler, decompose the vector in frame, and compensate for Lorentz factor terms. These results are covariant once observer, ruler, and external standard are provided. |
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