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I present a family of foliations of Schwarzschild-Droste spacetime based on radially freefalling observers. These are contrasted with the usual "static" foliation based on hypersurfaces of Schwarzschild coordinate t=const. These different conventions result in different splittings of spacetime into "space" and "time". Hence those properties which depend on the splitting, such as spatial distance or simultaneity, are distinct between the two conventions. For instance, it is only observers at fixed r-coordinate who measure radial distance by the familiar expression (1-2M/r)^(1/2)dr, whereas the falling observers measure differently. The curvature of 3-space is commonly depicted by Flamm's paraboloid but this corresponds to static observers, whereas the 3-space of the falling observers has a cone shape. The statement that distant observers determine infinite time for an object to pass the horizon is valid under the static slicing plus redshift factor, but under the alternate slicing the far away time is only finite. My motivation is conceptual and pedagogical, and may have applications for quantum fields on curved spacetime. See current version here: http://colinscosmos.com/2017/08/04/time-slicings-of-black-holes-poster/

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I investigate the mechanics of a relativistic string in an arbitrary stationary spacetime. Suppose a reel is anchored to a fixed location in space (meaning its 4-velocity is parallel to a chosen timelike Killing vector). Now suppose a long cable/string composed of ordinary matter has one end wound up on the reel, with the other end extended through space. Then in certain spacetimes, tension will result on the cable due to its own mass. An intuitive example is the weight of a cable hanging from a tree branch; another example is de Sitter space with the cable particles comoving in de Sitter's original "static" coordinate system. Such mechanisms have been proposed to test the possibility of mining energy from the expansion of the universe (Davies; Harrison). Cables have also been considered for black holes, to lower an object to mechanically extract its rest mass-energy (Penrose), or test the laws of thermodynamics (Geroch; Bekenstein) including Hawking radiation. Investigations of the rope in the static case (Gibbons; Redmount; Unruh & Wald) show a "redshift" of force transmitted along the cable. I extend the analysis to a moving cable, albeit a motion with time symmetry.

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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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