Revisions to Computing Trajectory Equations of Kerr Geodesics

added 120 characters in body; edited tags

Source Link

edited Jun 5, 2019 at 5:17

8.7k
7
41
94

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab.

The trajectories look like:

I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^2);
dx(2) = sqrt(O / Sigma^2);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^2);
dx(2) = sqrt(O / Sigma^2);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab.

The trajectories look like:

I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^2);
dx(2) = sqrt(O / Sigma^2);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

edited body

Source Link

edited Jun 2, 2019 at 6:50

almost

1
2

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^4Sigma^2);
dx(2) = sqrt(O / Sigma^4Sigma^2);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^4);
dx(2) = sqrt(O / Sigma^4);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^2);
dx(2) = sqrt(O / Sigma^2);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

added 5 characters in body

Source Link

edited May 31, 2019 at 16:58

nicoguaro ♦

8.6k
6
26
51

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard range guttaRunge-Kutta method. Here'sHere it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^4);
dx(2) = sqrt(O / Sigma^4);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I'mI am a little confused by the R- and Q-term, because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I don'tdo not want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard range gutta method. Here's my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^4);
dx(2) = sqrt(O / Sigma^4);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I'm a little confused by the R- and Q-term, because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I don't want to use this approach.

I want to numerically solve the trajectory equations of a Kerr geodesic given by wikipedia in Matlab. I implemented the equations and solved it with the standard Runge-Kutta method. Here it is my code:

% Initial parameter (r, theta, phi, t)
r       = x(1);
theta   = x(2);
phi     = x(3);
t       = x(4);

% Constants of motion
mu = const(1);
E = const(2);
L = const(3);
Q = const(4);
 
% Abbreviations
Sigma = r^2 + a^2 * cos(theta)^2;
Delta = r^2 - 2 * M * r + a^2;
K = Q + (L - a*E)^2;
X = a^2*(E^2 + mu) - L^2 - Q;
R = (E^2 + mu)*r^4 - 2*M*mu*r^3 + X*r^2 + 2*M*K*r -a^2*Q;
O = Q + cos(theta)^2*(a^2*(E^2 + mu) - L^2/sin(theta)^2);

% ODE from O Neill Chapter 4.2.2 Geodesics
dx(1) = sqrt(R / Sigma^4);
dx(2) = sqrt(O / Sigma^4);
dx(3) = ((L - a*sin(theta)^2*E) / sin(theta)^2 + a*((r^2 + a^2)*E - L*a)/(Delta) ) / Sigma^2;
dx(4) = (a*(L - a*E*sin(theta)^2) + (r^2 + a^2)*((r^2 + a^2)*E - L*a)/(Delta)) / Sigma^2;

I am a little confused by the R- and Q-term because there is a square root. How can I handle the signs of these in the calculations?

I know there exists an alternative form of the equations with the components of momenta, but I do not want to use this approach.

Source Link

asked May 31, 2019 at 9:20

almost

1
2

Loading

Stack Exchange Network

Return to Question