#### CarlB

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My Java applet gravity simulator http://www.gaugegravity.com/testapplet/SweetGravity.html [Broken]

draws beautiful orbits, however the GR simulation is very badly broken as one can tell when comparing it with Newton at long distances. The source code is at:

http://www.gaugegravity.com/testapplet/SwGrav_Top.java [Broken]

Test bodies are required to all lie in a plane. The Newtonian physics is produced by writing the x and y accelerations in terms of the phase space parameters of the test body, that is, in terms of its (x,y) position and its (Vx,Vy) velocity. With Newton, the acceleration has no dependence on velocity of course.

The GR "physics" is produced by a complicated calculation that follows the artificial potential that is discussed with respect to the Schwarzschild orbits in MTW and in various places all over the web. These methods use energy and angular momentum as constants of the motion to describe the orbits but they are less than optimal when used in an integration form.

I've seen a computer method for computing GR orbits that should work better, but I am hesitating to implement it because it is based on polar coordinates. This means that the calculation has to use trig functions to convert back and forth and these are slow.

So I'm wondering how hard it is to put the Schwarzschild orbits into phase space form in Cartesian coordinates. Here's the basic plan:

(1) Write the Schwarzschild metric in Cartesian coordinates.

(2) Write the proper length of a path as an integral over coordinate time.

(3) Vary the path and use the Euler-Lagarange equation to determine a pair of 2nd order differential equations that the orbits solve.

(4) Find [tex]d^2x/dt^2[/tex] and [tex]d^2y/dt^2[/tex] from the two 2nd order DEs.

Now for the poll: Do you think that this can be done reasonably easily? Part of the reason I am working on it is that I want to get a better idea on how these gravity theories differ one from another. I will post calculations as I make them, if convenient.

Carl

draws beautiful orbits, however the GR simulation is very badly broken as one can tell when comparing it with Newton at long distances. The source code is at:

http://www.gaugegravity.com/testapplet/SwGrav_Top.java [Broken]

Test bodies are required to all lie in a plane. The Newtonian physics is produced by writing the x and y accelerations in terms of the phase space parameters of the test body, that is, in terms of its (x,y) position and its (Vx,Vy) velocity. With Newton, the acceleration has no dependence on velocity of course.

The GR "physics" is produced by a complicated calculation that follows the artificial potential that is discussed with respect to the Schwarzschild orbits in MTW and in various places all over the web. These methods use energy and angular momentum as constants of the motion to describe the orbits but they are less than optimal when used in an integration form.

I've seen a computer method for computing GR orbits that should work better, but I am hesitating to implement it because it is based on polar coordinates. This means that the calculation has to use trig functions to convert back and forth and these are slow.

So I'm wondering how hard it is to put the Schwarzschild orbits into phase space form in Cartesian coordinates. Here's the basic plan:

(1) Write the Schwarzschild metric in Cartesian coordinates.

(2) Write the proper length of a path as an integral over coordinate time.

(3) Vary the path and use the Euler-Lagarange equation to determine a pair of 2nd order differential equations that the orbits solve.

(4) Find [tex]d^2x/dt^2[/tex] and [tex]d^2y/dt^2[/tex] from the two 2nd order DEs.

Now for the poll: Do you think that this can be done reasonably easily? Part of the reason I am working on it is that I want to get a better idea on how these gravity theories differ one from another. I will post calculations as I make them, if convenient.

Carl

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