Explore Black Hole Orbits with My Kerr Orbit Simulator on YouTube

[Yukterez]

Do you have any suggestions to assist in understanding how to calculate the x/y/z force on each particle for each frame using your linked equations?

If you use Mathematica you can automatically differentiate r'[t], θ'[t] and ψ'[t] (which are defined in the differential equation) with

Evaluate[r''[t] /. sol][[1]]
Evaluate[θ''[t] /. sol][[1]]
Evaluate[ψ''[t] /. sol][[1]]

to get the second (proper) time derivative of the motion. The transformation from r, θ, ψ components to x, y, z components is

x = r Sin[θ] Cos[ψ])
y = r Sin[θ] Sin[ψ]
z = r Cos[θ]

 

[pervect]

If you use Mathematica you can automatically differentiate r'[t], θ'[t] and ψ'[t] (which are defined in the differential equation) with

Evaluate[r''[t] /. sol][[1]]
Evaluate[θ''[t] /. sol][[1]]
Evaluate[ψ''[t] /. sol][[1]]

to get the second (proper) time derivative of the motion. The transformation from r, θ, ψ components to x, y, z components is

x = r Sin[θ] Cos[ψ])
y = r Sin[θ] Sin[ψ]
z = r Cos[θ]

I assume r, theta, and psi are Schwarzschild coordinates? The difficulty with constructing x,y, and z in the manner you did above from r, theta, and psi is that the speed of light is not isotropic in Schwarzschild coordinates, i.e. it depends on direction. Thus with these relationships, you have the speed of light being different in the x direction, the y direction, and the z direction. This means that the distance scales are not the same - if you consider the distance light travels in one "tick" of proper time (or, if you prefer, one "tick" of coordinate time), the distance the light moves depends on the direction, i.e. the distance is different in the x, y, and z directions.

This is rather incompatible with the standard SI definition of the meter if you attempt to interpret your coordinate values as having physical significance (i.e. representing distances). The proportionality between a change in the coordinate value and the change in distance depends on which coordinate you pick.

The usual solution to this is to use an alternate coordinate system, the isotropic coordinate system, which you can find the metric for in The "Alternate coordinate" section of https://en.wikipedia.org/wiki/Schwarzschild_metric. It's rather painful to compute things in isotropic coordinates, though it can be done, the equations are are lot more complicated. It would probably be easiest to compute the answer in Schwarzschild coordinates, and convert the answer into isotropic coordinates. Then you could convert that answer into x,y, and z, and the results would be much more physically significant.

I don't recall the transformation equations offhand, however, and I didn't notice them in the wiki article I quoted.

 

[Yukterez]

Thus with these relationships, you have the speed of light being different in the x direction, the y direction, and the z direction.

In Schwarzschild the factor for the transversal direction is √(1-rs/r) and for the radial direction (1-rs/r) without the square (because you have time dilation and radial length expansion of the same magnitude, also see this link). In Kerr you replace the Schwarzschild radius rs=2 with rk=1+√(1-a²Cos²[θ]) where a is the spin parameter.

 

[Yukterez]

I have a little trouble following your linked equations

I see I have a bug in the .txt-File code and mixed up some r0 and r(t) in the formula for the digit velocity display (the simulation is ok, but the digit output below the animation needs a repair). I'll fix that tomorrow, until then you can use the equations of motion from my animation code, but if you want to display the numbers transform from r', θ' & ψ' to vr, vθ & vψ like in my last posting above, where all r are r(t).

 

[Yukterez]

The bug in the code is fixed now, the transformation from proper to coordinate velocity should give the right numers now. If you use the kerr.txt force refresh (ctrl f5) to make sure to get the updated version.

I'll put the transformation in Latex (with G, M and c set to 1):
to transform from the derivatives in the differential equation to the local velocity components use

$\dot{r} = \frac{v_r (1-r_k/r)^{(3/2)}}{\sqrt{1-v^2}}$

$\dot{\theta} = \frac{v_{\theta}}{r \sqrt{1-v^2}}$

$\dot{\psi} = \frac{v_{\psi} }{r \sqrt{1-v^2}}$

The $v_{\psi}$ component is observed faster by

$d_{\psi} = \pm \frac{2 \alpha r^2 }{ \Sigma}$

with

$r_k = \sqrt{1-a^2 \cos^2 (\theta )}+1$

because of frame dragging, where

$\Sigma =\left(\alpha ^2+r^2\right)^2-\alpha ^2 \sin ^2(\theta ) \Delta$

and

$\Delta =r^{2}-2r+\alpha ^{2}$

After solving for $v_r$, $v_{\theta}$, $v_{\psi}$ and $d_{v_{\psi}}$ you can use Pythagoras for the total velocity $v$.

 

[Drakkith]

I'll post again and ask a moderator to delete the previous post:

Done.

 

[Yukterez]

I do step by step animations using Newtons law calculation an any number of particles between each step
Do you have any suggestions to assist in understanding how to calculate the x/y/z force on each particle for each frame using your linked equations?

Now I see what you mean, but that needs some further explanation. You can compute the force acting on the test particle from the velocity derivative (after substracting the frame drag), but you don't use the force to compute the velocity. Normally with Newton or Schwarzschild you plug in your function for the accelerations r''(t), ψ''(t) & θ''(t) to equate the motion and get your velocities r'(t), ψ'(t) & θ'(t) and positions r(t), ψ(t) & θ(t). But in Kerr even when a particle has 0 velocity, the velocity in the equatorial direction - ψ'(t) - is not zero from begin with. So your test particle already starts with a velocity even when it has none (because of the frame dragging). Therefore under Kerr the equation of motion is not second order but first order, and therefore a little more complicated because you have to write the equation in terms of conserved quantities that go into the definions of r'(t), ψ'(t) & θ'(t). Therefore it is not possible to plug in Kerr corrections into a Newtonian simulator (with Schwarzschild on the other hand this is no problem, the equation of motion only differs by a small additional term). If you want to use Kerr you have to switch from second order derivative to first order derivative, the force is only on the output side but not on the input.

 

[Yukterez]

I had a numeric error in the last posting, because it was too late to edit I deleted and repost

This trajectory is a useful example to explain the numeric display below the animation, with focus on the last three numbers on the bottom right:

At radial coordinate r=3GM/c² we launch a test particle to a prograde orbit spiraling into the ergosphere of a rotating Kerr black hole.

When the test particle approaches the horizon at 1+√[1-a²],
it has a local velocity of almost c relative to a local probe which is locally at rest,
a delayed velocity of almost 0 relative to the same probe when described from the perspective of the observer at infinity, and
an observed velocity of 0.45c (that is the frame dragging velocity at the latitude where the particle nears the horizon, so we observe the locally stationary probe and the locally almost light speed fast particle to corotate with almost the same 0.45c).

First in proper time steps:

Now the same situation in coordinate time steps (the interval is as above 1/8GM/c³):

Comparison of proper speed versus coordinate velocity as viewed from different observers (left: particle, right: coordinate observer):

Initial settings: a=0.9, r0=3GM/c², v0=√(1/r0)=1/√(3)c, ψ0=0, θ0=π/2=90°, φ0=0, δ0=π/5=36°

The red tail length is 1/4 GM/c³ of proper time (so before the particle enters the horizon it gets infinitely long because it whirls up while freezing and corotating from the perspective of the coordinate observer). As one can see, the test particle plunges into the horizon in a finite proper time. But it also makes an infinite amount of revolutions with the black hole before it does so.

I heard that there is a coordinate system where one can transform this infinty away and follow the particle on its way behind the horizon, but as far as I am with Kerr right now I can only simulate until there and then I get an infinity in the spin rate when differentiating coordinates by proper time at the inner horizon (the particle proper time freezes while it is frame dragged around the horizon and corotating with a constant velocity, in this case 0.45c)

Code.txt

 

[edguy99]

At radial coordinate r=3GM/c² we launch a test particle to a prograde orbit spiraling into the ergosphere of a rotating Kerr black hole.

Thanks for the post. I would like to see the simulation of a pair of black holes orbiting one another (or a star orbiting a small black hole) where the mass of one object is affecting the position of the other object and you can expand it to more objects if you want (ie. http://www.animatedphysics.com/planets/moon_orbit.htm this animation calculates both the effect of Newtonian gravity of the Earth's pull on the moon as well as the moons pull on the earth, where both the Earth's position and the moons position changes between each frame due to the gravity pull of the moon). Is this doable with your equations? Specifically, I don't see anything that would represent the mass of the particle around the black hole? I could be not reading German properly of just significantly confused.

The only way I see to attempt to model a pair of black holes is through x/y/z timesteps, with calculations and movement of each object between timesteps. Perhaps there is an easier way to do this?

 

[Yukterez]

Is this doable with your equations?

With my Newtonian equations it is doable, but only in the nonrelativistic limit, see here. With Kerr's metric it is not doable, since

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a spherical event horizon.

So the mass of the test particle is assumed to be small in Kerr metric, just like in Schwarzschild metric. If you want to simulate full relativistic n-body you need to go for the BSSN formalism but I doubt that you can do that on a home computer, as far as I know supercomputers are busy for weeks if not months with that.

 

[Yukterez]

The old simulator had a bug, as one can see in the animations above the ergosphere was not displayed correctly in it's pumpkin-shape but as an ellipsoid. That was fixed in an update; also the numerical display was extended and the plot now shows not only the outer but also the inner horizon and ergosphere. So use the new code instead of the old one. Example: plunge orbit with v0=vz=0.975c from r0=3GM/c², a=0.998:

The initial conditions can be fed in in terms of the local 3-velocity (relative to a ZAMO) either vx, vy, vz or equivalently vr, vφ, vθ or in terms of the total Energy and angular momentum components E, Lz, pθ0 or alternatively Q (the carter constant). Index for the numerical display: