Schwarzschild Metric

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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=-c√(rₛ/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=-c√(rₛ/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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Aberration in the eyes of three different observers at the same position r=6GM/c² with different velocity vectors
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Optical distortion of a sphere with r=1.0001rₛ, observer at R=17.5rₛ: due to gravitational lensing the back of the sphere is also visible
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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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Ray bundle with impact parameter b=√27≈5.2GM/c² (the apparent radius of the black hole from the perspective of the far away observer)
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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)).
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Metric tensor in Schwarzschild coordinates {t,r,θ,φ}; superscripted letters are not powers but indices:

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Ingoing Finkelstein coordinates with the horizon penetrating coordinate time dt→dť+dr(rs/r)/(1-rs/r)/c:

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Raindrop aka Gullstrand Painlevé coordinates, coordinate time defined by freefallers from infinity, dt→dτ+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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Circular orbit velocity, at the photon sphere at r=3rₛ/2 it is the speed of light:

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Free fall time from rest at r₀ to r (proper time):

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coordinate time for the free fall from r₀ to r:

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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:

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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. Other coordinates: see here
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by Simon Tyran, Vienna @ minds || gab || wikipedia || stackexchange || License: CC-BY 4. If images don't load: [ctrl]+[F5]Bild

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