![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/us+br.png)
![Bild](http://org.yukterez.net/de+at.png)
![Bild](http://org.yukterez.net/strich.gif)
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/falling.into.a.schwarzschild.black.hole.gif)
Free fall into a black hole with v=-c√(rₛ/r), viewed from the perspective of the freefaller. Shadow angular diameter function: click
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/schwarzschild.horizon.crossing.raytraced.png)
Full panorama of the oberserver falling with the negative escape velocity v=-c√(rₛ/r) when he crosses the event horizon
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/schwarzschild.horizon.crossing.redshift.png)
Red/blueshift profile for the observer falling with the negative escape velocity in the image above
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](https://org.yukterez.net/schwarzschild.aberration.radial.tangential.png)
Aberration in the eyes of three different observers at the same position r=6GM/c² with different velocity vectors
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://cloud.yukterez.net/schwarzschild.earth.758.gif)
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
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://www.yukterez.net/org/relativistic.raytracer/shapiro.gif)
left: lightrays in flat Minkoswki space, right: Schwarzschild. Distance of the light source to the black hole: 20GM/c²
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://cloud.yukterez.net/relativistic.raytracer/sqrt27schwarzschild.gif)
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)
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/kerr.Lz0.a0000.gif)
Orbit with perihelion shift; initial conditions: r₀=5, θ₀=π/2, v₀=vz₀=vθ₀=51/50·√((1/5)/(1-2/5)).
![Bild](http://org.yukterez.net/strich758.gif)
![Bild](http://org.yukterez.net/strich.gif)
▥ Metric tensor in Schwarzschild coordinates {t,r,θ,φ}; superscripted letters are not powers but indices:
![Bild](http://org.yukterez.net/schwarzschild.latex/51.gif)
▦ Ingoing Finkelstein coordinates with the horizon penetrating coordinate time dt→dť+dr(rs/r)/(1-rs/r)/c:
![Bild](http://org.yukterez.net/schwarzschild.latex/52.gif)
▧ Raindrop aka Gullstrand Painlevé coordinates, coordinate time defined by freefallers from infinity, dt→dτ+dr√(rs/r)/(1-rs/r)/c:
![Bild](http://org.yukterez.net/schwarzschild.latex/53.gif)
Equations of motion in Schwarzschild coordinates:
![Bild](http://org.yukterez.net/schwarzschild.latex/11.gif)
![Bild](http://org.yukterez.net/schwarzschild.latex/12.gif)
![Bild](http://org.yukterez.net/schwarzschild.latex/18.gif)
For a purely radial motion the equation of motion simplifies to
![Bild](http://org.yukterez.net/schwarzschild.latex/54.gif)
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:
![Bild](http://org.yukterez.net/schwarzschild.latex/5.gif)
With Pythagoras we get the total velocity:
![Bild](http://org.yukterez.net/schwarzschild.latex/6.gif)
The orbital angular momentum
![Bild](http://org.yukterez.net/schwarzschild.latex/26.gif)
and the total energy of the test particle in the frame of an observer at infinity
![Bild](http://org.yukterez.net/schwarzschild.latex/25.gif)
are conserved. The rest, kinetic and potential energy (defined as the difference between local and total energy) are
![Bild](http://cache.yukterez.net/schwarzschild.latex/9.gif)
The required radial velocity to get from r₀ to r₁ is
![Bild](http://org.yukterez.net/schwarzschild.latex/27.gif)
and the escape velocity from r₀ to infinity
![Bild](http://org.yukterez.net/schwarzschild.latex/28.gif)
Circular orbit velocity, at the photon sphere at r=3rₛ/2 it is the speed of light:
![Bild](http://org.yukterez.net/schwarzschild.latex/50.gif)
Free fall time from rest at r₀ to r (proper time):
![Bild](http://org.yukterez.net/schwarzschild.latex/47.gif)
coordinate time for the free fall from r₀ to r:
![Bild](http://org.yukterez.net/schwarzschild.latex/55.gif)
The physical distance between r₁ and r₂ in the frame of the far away bookeeper is
![Bild](http://org.yukterez.net/schwarzschild.latex/24.gif)
Distance from the horizon to the singularity in the frame of a freefaller falling in with the negative escape velocity:
![Bild](http://org.yukterez.net/schwarzschild.latex/49.png)
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
![Bild](http://f.yukterez.net/einstein.equations/files/z/sdicon.png)
![Bild](http://org.yukterez.net/schwarzschild.latex/51.gif)
▦ Ingoing Finkelstein coordinates with the horizon penetrating coordinate time dt→dť+dr(rs/r)/(1-rs/r)/c:
![Bild](http://f.yukterez.net/einstein.equations/files/z/eficon.png)
![Bild](http://org.yukterez.net/schwarzschild.latex/52.gif)
▧ Raindrop aka Gullstrand Painlevé coordinates, coordinate time defined by freefallers from infinity, dt→dτ+dr√(rs/r)/(1-rs/r)/c:
![Bild](http://f.yukterez.net/einstein.equations/files/z/gpicon.png)
![Bild](http://org.yukterez.net/schwarzschild.latex/53.gif)
Equations of motion in Schwarzschild coordinates:
![Bild](http://org.yukterez.net/schwarzschild.latex/11.gif)
![Bild](http://org.yukterez.net/schwarzschild.latex/12.gif)
![Bild](http://org.yukterez.net/schwarzschild.latex/18.gif)
For a purely radial motion the equation of motion simplifies to
![Bild](http://org.yukterez.net/schwarzschild.latex/54.gif)
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:
![Bild](http://org.yukterez.net/schwarzschild.latex/5.gif)
With Pythagoras we get the total velocity:
![Bild](http://org.yukterez.net/schwarzschild.latex/6.gif)
The orbital angular momentum
![Bild](http://org.yukterez.net/schwarzschild.latex/26.gif)
and the total energy of the test particle in the frame of an observer at infinity
![Bild](http://org.yukterez.net/schwarzschild.latex/25.gif)
are conserved. The rest, kinetic and potential energy (defined as the difference between local and total energy) are
![Bild](http://cache.yukterez.net/schwarzschild.latex/9.gif)
The required radial velocity to get from r₀ to r₁ is
![Bild](http://org.yukterez.net/schwarzschild.latex/27.gif)
and the escape velocity from r₀ to infinity
![Bild](http://org.yukterez.net/schwarzschild.latex/28.gif)
Circular orbit velocity, at the photon sphere at r=3rₛ/2 it is the speed of light:
![Bild](http://org.yukterez.net/schwarzschild.latex/50.gif)
Free fall time from rest at r₀ to r (proper time):
![Bild](http://org.yukterez.net/schwarzschild.latex/47.gif)
coordinate time for the free fall from r₀ to r:
![Bild](http://org.yukterez.net/schwarzschild.latex/55.gif)
The physical distance between r₁ and r₂ in the frame of the far away bookeeper is
![Bild](http://org.yukterez.net/schwarzschild.latex/24.gif)
Distance from the horizon to the singularity in the frame of a freefaller falling in with the negative escape velocity:
![Bild](http://org.yukterez.net/schwarzschild.latex/49.png)
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