Kerr Metric

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Kerr Metric

Beitragvon Yukterez » Do 12. Apr 2018, 22:24

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Bild This is the english version.   Bild Deutschsprachige Version auf kerr.yukterez.net und Yukipedia.
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Shadow and surfaces of a spinning black hole (a=1), click to enlarge (png). Zoom out: [-], Contours: ƒ, Raytracing Code:
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Shadow and surfaces of a spinning black hole (a=0.99), Animation parameter: polar angle (θ=1°..90°). Slower:
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Accretion disk with inner radius ri=isco and outer radius ra=7 around a BH with a=0.95, observer at r=10, θ=70°
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The same black hole like in the example above, but with the observer at r=100. For more details click on the images.
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Retrograde orbit of a particle around a spinning black hole (a=0.95), coordinates: cartesian
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Here we use natural units of G=M=c=1, so lengths are in GM/c² and times in GM/c³. The metric signature is time-positive (+,-,-,-). a is the spin parameter (for black holes 0≤a≤M), M the mass equivalent of the total energy of the black hole, and Mirr its irreducible mass:

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Shorthand terms:

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Covariant metric coeffizients:

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Contravariant components (superscripted letters are not powers, but indices):

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The dimensionless spin parameter is a=Jc/G/M². Transformation into cartesian coordinates:

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Line element in Boyer Lindquist coordinates:

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Metric tensor (t,r,θ,Ф):

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With a=0 Boyer Lindquist coordinates reduce to classical Schwarzschild coordinates.
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With the transformation:

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where T is a finkelsteinlike time coordinate (radially infalling photons move with dr/dt=1) and ψ the flattened azimuthal angle:

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the metric in Kerr Schild coordinates (T,r,θ,ψ) is:

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With a=0 Kerr Schild coordinates reduce to Eddington Finkelstein coordinates.
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In Doran coordinates (т,r,θ,φ) where the coordinate time is the proper time of local free fallers from infinity the metric tensor becomes

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With a=0 Doran coordinates reduce to Gullstrand Painlevé coordinates.
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Equations of motion in Boyer Lindquist coordinates
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Canonical four-momentum components:

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Coordinate time by proper time (dt/dτ):

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First proper time derivative of the radial coordinate (dr/dτ):

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Radial momentum derivative:

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Radial momentum:

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Derivative of the poloidial component of motion (dθ/dτ):

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Derivative of the poloidial angular momentum (dpθ/dτ):

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Axial angular momentum:

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Derivative of the axial component of motion (dФ/dτ):

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Axial angular momentum derivative (pФ/dτ):

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Longitudinal component of the angular momentum:

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Constant of motion, Carter's constant:

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Constant of motion, Carter k:

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Constant of motion, total energy:

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Constant of motion, axial angular momentum:

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Local 3-velocity component along the r-axis:

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Local 3-velocity component along the θ-axis:

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Local 3-velocity component along the Ф-axis:

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Local 3-velocity, total:

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For massive testparticles μ=-1 and for photons μ=-0. δ is the inclination angle. With α as the vertical launch anglel the components of the local velocity (relative to a ZAMO) are

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Shapirodelayed and frame dragged velocity as observed at infinity:

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The radial effective potential which defines the turning points is:

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Radial escape velocity:

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Frame-Dragging angular velocity oberserved at infinity (dФ/dt):

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Delayed Frame-Dragging transverse velocity at the equator of the outer horizon:

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with the horizons and ergospheres (solution for r at Δ=0 and gtt=0):

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r and θ dependend delayed Frame-Dragging transverse velocities:

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at the equatorialen plane at θ=π/2:

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r und θ dependend local Frame-Dragging transverse velocities (greater than c inside of the ergosphere):

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at the equatorialen plane at θ=π/2:

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Cartesian projection of the Frame-Dragging transverse velocity:

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at the equatorialen plane at θ=π/2:

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Gravitational time dilation component relative to a ZAMO (dt/dτ):

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Axial and coaxial radius of gyration:

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Axial and coaxial circumference:

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The innermost stable orbit (ISCO) is at

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with the shorthand terms

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For images and animations see the german version of this site.
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Simon Tyran aka Симон Тыран @ minds || vk || wikipedia || stackexchange || wolframBild

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