Lecture08_GR2 - 1 Examples Spacetime Geometries Black Holes...

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2 Examples: Spacetime Geometries Black Holes Morris-Thorne Wormholes The Global Positioning System Summary
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4 Event horizon r S Karl Schwarzschild 1873 - 1916
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5 Far from the black hole, centered at r = 0, the Schwarzschild metric approaches the geometry of flat spacetime We can therefore interpret the symbol t as the time measured by someone far from the hole ds 2 = c 2 (1 r s / r ) dt 2 dr 2 (1 r s / r ) r 2 d ϕ 2
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6 r S Consider an observer at a fixed point with radial coordinate r from the black hole, i.e., with dr = 0, and d φ = 0. The elapsed time at a fixed r is given by d τ = ds / c = dt 1 r s r We see that the elapsed time is less than the elapsed time for someone very far from the black hole
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8 In this spacetime, time is unwarped , but space is. What is the meaning of r ? The radial parameter r is defined so that r = C /2 π where C is the circumference of a circle around the wormhole
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9 The proper radial distance from the wormhole’s throat at r = a to another point r = b
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