- #1

KDP

- 24

- 7

- TL;DR Summary
- Need a general equation for relativistic hyperbolic motion in a 2D plane for arbitrary initial parallel and transverse velocities, to complete a real time simulation I am coding for acceleration of extended 2D objects in a plane.

Hi. I'm working on coding a 2+1D simulation. The simulator moves 2D objects in a plane in real time, demonstrating Wigner rotation and other relativistic scenarios. So far I have created the algorithms for active and passive Lorentz transformation of extended 2D objects and for relativistic velocity addition in any direction in the plane. Now I need an algorithm for the general case of Born rigid acceleration in a 2D plane. Wikipedia gives the special case of acceleration in the x direction with a non-zero initial velocity in the parallel x direction, which can be written like this:

$$ X(T)~ = ~\frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( V_X ~ G_{X} + \alpha ~ T \right)^2 } - G_{X} \right), ~~~~~~~~_{(Eq1a)}$$

$$ V(T) ~=~ \frac{ V_X ~ G_X + \alpha ~ T }{ \sqrt{ 1+\left( V_X ~ G_X + \alpha ~ T \right)^2 } } , ~~~~~~~~_{(Eq1b)}$$

where [itex] \alpha[/itex] is the constant proper acceleration in the x direction, [itex] V_X [/itex] is the initial velocity in the x direction, [itex]G_{X} [/itex] is the gamma factor of the initial parallel velocity and using units such that the speed of light is unity.

For motion in the 2D plane, my program needs to cope with initial transverse velocities. After some trial and error I came up with this equation to handle initial transverse velocity:

$$ X(T) ~=~ \frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( \alpha ~T~/~G_Y \right)^2 } - 1 \right), ~~~~~~~~_{(Eq2)}$$

$$ V(T) ~=~ \frac{ \alpha ~ T~/~G_Y }{ \sqrt{ 1+\left( \alpha ~ T~ / ~G_Y \right)^2 }}~*~\frac{1}{G_Y} , ~~~~~~~~_{(Eq2b)}$$

where [itex]G_{Y} [/itex] is the gamma factor of the initial transverse velocity in the Y direction. This equation works well if the initial X velocity is zero.

Now if the equations are combined like this:

$$ X (T)~=~ \frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( V_X ~ G_{X} + ~ \alpha ~ T~/~G_Y \right)^2 } - G_{X} \right), ~~~~~~~~_{(Eq3a)}$$

$$ V(T) ~=~ \frac{ V_X ~ G_X + \alpha ~ T ~/~G_Y }{ \sqrt{ 1+\left( V_X ~ G_X + \alpha ~ T~/~G_Y \right)^2 } } ~*~\frac{1}{G_Y}, ~~~~~~~~_{(Eq3b)}$$

I do not get the desired result. The final equations will only work correctly if either the initial parallel x velocity or the initial transverse y velocity are zero. Can anyone derive or locate the the fully general equation for hyperbolic motion in a 2D plane, so I can continue with my coding project? I have come up with some ad hock equations and fudge factors for limited special cases, but the whole thing is becoming messy and inaccurate. I have spent weeks on coding this simulator, but I seem to have reached a dead end. This is a shame as I think it would be very useful to anyone interested in relativity. I would very much appreciate any help with this.

I would be happy with an equation that is limited to acceleration only in the x direction (It is always possible to make sure this is the case by rotating the whole frame before the calculation and rotating everything back before displaying on the screen).

$$ X(T)~ = ~\frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( V_X ~ G_{X} + \alpha ~ T \right)^2 } - G_{X} \right), ~~~~~~~~_{(Eq1a)}$$

$$ V(T) ~=~ \frac{ V_X ~ G_X + \alpha ~ T }{ \sqrt{ 1+\left( V_X ~ G_X + \alpha ~ T \right)^2 } } , ~~~~~~~~_{(Eq1b)}$$

where [itex] \alpha[/itex] is the constant proper acceleration in the x direction, [itex] V_X [/itex] is the initial velocity in the x direction, [itex]G_{X} [/itex] is the gamma factor of the initial parallel velocity and using units such that the speed of light is unity.

For motion in the 2D plane, my program needs to cope with initial transverse velocities. After some trial and error I came up with this equation to handle initial transverse velocity:

$$ X(T) ~=~ \frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( \alpha ~T~/~G_Y \right)^2 } - 1 \right), ~~~~~~~~_{(Eq2)}$$

$$ V(T) ~=~ \frac{ \alpha ~ T~/~G_Y }{ \sqrt{ 1+\left( \alpha ~ T~ / ~G_Y \right)^2 }}~*~\frac{1}{G_Y} , ~~~~~~~~_{(Eq2b)}$$

where [itex]G_{Y} [/itex] is the gamma factor of the initial transverse velocity in the Y direction. This equation works well if the initial X velocity is zero.

Now if the equations are combined like this:

$$ X (T)~=~ \frac{ 1 }{\alpha } \left( \sqrt{ 1+\left( V_X ~ G_{X} + ~ \alpha ~ T~/~G_Y \right)^2 } - G_{X} \right), ~~~~~~~~_{(Eq3a)}$$

$$ V(T) ~=~ \frac{ V_X ~ G_X + \alpha ~ T ~/~G_Y }{ \sqrt{ 1+\left( V_X ~ G_X + \alpha ~ T~/~G_Y \right)^2 } } ~*~\frac{1}{G_Y}, ~~~~~~~~_{(Eq3b)}$$

I do not get the desired result. The final equations will only work correctly if either the initial parallel x velocity or the initial transverse y velocity are zero. Can anyone derive or locate the the fully general equation for hyperbolic motion in a 2D plane, so I can continue with my coding project? I have come up with some ad hock equations and fudge factors for limited special cases, but the whole thing is becoming messy and inaccurate. I have spent weeks on coding this simulator, but I seem to have reached a dead end. This is a shame as I think it would be very useful to anyone interested in relativity. I would very much appreciate any help with this.

I would be happy with an equation that is limited to acceleration only in the x direction (It is always possible to make sure this is the case by rotating the whole frame before the calculation and rotating everything back before displaying on the screen).

Last edited: