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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.

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marked as duplicate by David Z Aug 25 '11 at 18:10

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

    
Actually, you had the definition of velocity correct before your edit, not after it (unless you are talking about four-velocity, but then your numeric values are probably wrong). –  David Z Aug 25 '11 at 18:05
    
By the way, you're asking basically the same thing this other question did, so I'm going to close this as a duplicate for now. If reading that other question and its answer isn't enough for you, you can edit this question to say what you are still confused about, and we can reopen it. –  David Z Aug 25 '11 at 18:09
    
Sorry, but your question is still pretty unclear. I'm especially confused by this part: "then Project the 4d vector on a hyperbolic 4d space created in relation to the sun mass. This should bend the straight line, making the earth turn around the sun." Projection reduces the number of dimensions of a vector, so I'm not sure how you would project a 4D vector on a 4D space. Besides, the spacetime surrounding a massive body is usually approximated with a Schwarzschild metric, which AFAIK is not considered hyperbolic in any sense. –  David Z Aug 26 '11 at 5:59
    
Also, I don't think you've said anything about why the possible duplicate question I linked is insufficient for your needs... –  David Z Aug 26 '11 at 6:01
    
I think you meant unclear, but anyway thx... yes is probably what I'm looking for. –  Pella86 Aug 26 '11 at 6:02
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1 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.

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That is what I don't want... :D but thx anyway for the answer :D you will get an up point if I could give them, because is really well explained ;) –  Pella86 Aug 25 '11 at 23:20
    
Hahaha... sorry, no prob. –  AdamRedwine Aug 26 '11 at 11:46
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