How to bend 3d space embededd in a 4d enviornment? [duplicate]

Question

Possible Duplicate:
Calculating position in space assuming general relativity

I recently started to program a 4 dimensional visualization program. I have the 4 dimension space perspectively projected in the 3d space. Now I describe 3d sphere inside this space. I want to describe the movement of those 3d spheres. What I thought it will be cool is to apply the general relativity theory to it. As far as I know from the general relativity theory, is that masses bends the space around it.

What I think it means: let's imagine a 2d plane. if I put a mass on this 2d plane, the plane would be actually pinched in the 3d space. Parctically what I have to do is to find a function that describe this pinch, in function of the mass. like a Gauss distribution under the mass. Now this transformation will be applied to the moving objects, which I think would bend the trajectory of the objet to a curve.

What I want to do is to apply this to the 3d space: if you put a mass in the 3d space it will cause a pinch in the 4th dimension. So what I want is to find the transformation function which describe this pinch.

I think that what Zaslavsky is the actual answer, I din't got it at the beginning because I didn't tought it was actually what I was looking for but anyway:

$$\frac{\mathrm{d}^2x^\lambda}{\mathrm{d}t^2} + \Gamma^{\lambda}_{\mu\nu}\frac{\mathrm{d}x^\mu}{\mathrm{d}t}\frac{\mathrm{d}x^\nu}{\mathrm{d}t} = 0$$

The solution to this differential equation should give the pinch I mean, am I right?

PS: found by surfing random on the internet: Lorentz transformations are just hyperbolic rotations.

edit: I've edited the question, I hope, in a way is comprehensible.

Answer 1

Unless your program is specifically designed to address issues of relativity, I would highly recommend that you use the classical Newtonian mechanics.

Assuming the massive body is the center of a stationary reference frame (as you suggest), the smaller object experiences a force toward the origin of

$$F_g = G\frac{m M}{r^2}$$

where

$G = 6.67 \times 10^-11 \frac{m^3}{kg \cdot s}$

$m = $ mass of object

$M = $ mass of planet

$r = $ distance from center of planet to object

The acceleration due to this force is $a = \frac{d v}{d t} = F/m$. The nature of your question suggests that you are interested in working with discrete time steps. I would suggest calculating the position as normal and then adding the increase in velocity after the next time step.

Example:

At time $t = 0$, the object has position $\textbf{r}_0 = \langle x, y, z \rangle$ and velocity $\textbf{v}_0 = \langle \frac{d x}{d t}, \frac{dy}{dt}, \frac{dz}{dt} \rangle = \langle v_{x, 0}, v_{y, 0}, v_{z, 0} \rangle$.

At time $t$, the object has position $\textbf{r}_1 = \langle x + v_{x, 0}\cdot t, y + v_{y, 0} \cdot t, z + v_{z, 0} \cdot t \rangle$ and velocity of $\langle v_{x, 0}, v_{y, 0}, v_{z, 0} \rangle + \frac{G M}{(\Delta r)^2}$. At the next time step, you simply use the new velocities. This would, of course, be greatly simplified if you used a spherical coordinate system centered on the massive body.