Schwarzschild Metric

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Yukterez
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Schwarzschild Metric

Beitragvon Yukterez » Do 12. Apr 2018, 22:51

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Bild This is the english version.   Bild Deutschsprachige Version auf schwarzschild.yukterez.net.
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Free fall into a black hole with v=-√(2/r), viewed from the perspective of the freefaller. Shadow angular diameter function: click
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Full panorama of the oberserver falling with the negative escape velocity v=-√(2/r) when he crosses the event horizon
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Red/blueshift profile for the observer falling with the negative escape velocity in the image above
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Shadow and horizon of a black hole, the observer is at a distance of r=50GM/c². Zoom out: [-], ray animation: play, raytracing code: click
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left: lightrays in flat Minkoswki space, right: Schwarzschild. Distance of the light source to the black hole: 20GM/c²
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Orbit with perihelion shift; initial conditions: r0=5, θ0=π/2, v0=vz0=vθ0=51/50·√((1/5)/(1-2/5)). Geodesic solver: geodesics.txt
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Metric tensor in Schwarzschild coordinates {t,r,θ,φ}; superscripted letters are not powers but indices:

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Finkelstein coordinates with the horizon penetrating coordinate time dť→dt+2dr/(r-rs)/c:

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Gullstrand Painlevé (Raindrop) coordinates, horizon penetrating coordinate time defined by freefallers from infinity, dť→dt+dr√(rs/r)/(1-rs/r)/c:

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Equations of motion in Schwarzschild coordinates:

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For a purely radial motion the equation of motion simplifies to

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τ is the proper time of the test particle, and t the coordinate time of an observer at infinty. To get the shell time of a stationary fiducional observer at r=R, τ gets divided by √(1-rs/r), where rs=2GM/c² is the Schwarzschildradius. The total time dilation is the product of the gravitational and the kinetic component. v⊥=v cos ζ (the transverse), and v∥=v sin ζ (the radial component of the local velocity). ζ is the vertical launch angle (because of the radial length contraction ζ at small r looks flatter when viewed from infinity).

Transformation between local and observed (shapirodelayed) velocities:

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With Pythagoras we get the total velocity:

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The orbital angular momentum

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and the total energy of the test particle in the frame of an observer at infinity

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are conserved. The rest, kinetic and potential energy (defined as the difference between local and total energy) are

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The required radial velocity to get from r0 to r1 is

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and the escape velocity from r0 to infinity

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The physical distance between r1 and r2 in the frame of the far away bookeeper is

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Distance from the horizon to the singularity in the frame of a freefaller falling in with the negative escape velocity in units of GM/c²:

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in Droste coordinates with grr=-1/(1-rs/r) and v=-√(rs/r) where γ=1/√(1-v²/c²) or in Raindrop coordinates with grr=-1 and v=0 where γ=1. In the frame of a freefaller starting from rest at an infinitesimal above the horizon the integrated distance approaches d=πGM/c².

For the simulatior codes and more images and animations see the german version and the article about the relativistic raytracer.
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Simon Tyran aka Симон Тыран @ minds || vk || wikipedia || stackexchange || wolframBild

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