Why do geodesics of space-time cause elliptical orbits in our dimension?

[bluescience]

I have understood why and how geodesics are related to gravity. So is gravity still a force?

 

Also since the geodesics taken in the fourth dimension cause us to see planets taking orbits doe to gravity, what causes the elliptical shape of orbits?

 

So far with all the reasoning i seem to only think that the orbits should be circular.

 

Any enlightenment is awesome.

[mathematic]

Newton gravity (with sun much more massive than all the solar system members), all orbits are conic sections. Ellipses (including circles) are the only ones that last.

[StringJunky]

Newton gravity (with sun much more massive than all the solar system members), all orbits are conic sections. Ellipses (including circles) are the only ones that last.

Could it be that gravitationally captured objects will always form an elliptical orbit; the direction of the longest part of the elliptical curve is roughly the line of capture? Also, in the case of coaelescing matter around a planetary body, it will never do so in perfectly balanced way, leading to an elliptical orbit?

[bluescience]

My chain of thought is that if a geodesic drawn from one point to another on a sphere is laid out on a 2-d surface, than the line of the geodesic would be a curve. However when you think of it would not the shape of the curve depend on where that geodesic line lies on the sphere, so i think it is the same in the geodesic of the fourth dimension. I think the orbits are elliptical since the geodesic in the 4th dimension is shaped in the third dimension in such a way that it cause a elliptical orbit. But why always an ellipse? also if an object is taking the shortest distance in the fourth dimension, then why does the object keep on circling a more massive object continuously without ever moving forward?

[ajb]

You can examine explicitly geodesics on the Schwarzschild metric and see that there are a few classes of orbits that are allowed. We can recover the orbits as described in Newtonian theory.

[mathematic]

If you use Newton's gravity (inverse square law) and prescribe an initial position and velocity, the trajectory will be a conic section (using elementary calculus - - from Newton).

[bluescience]

If you use Newton's gravity (inverse square law) and prescribe an initial position and velocity, the trajectory will be a conic section (using elementary calculus - - from Newton).

so then an orbit will always have to be a conic section no matter what. And the exact measurements or shapes of these conic sections are determined by what exactly?

Edited April 2, 2016 by bluescience

[mathematic]

so then an orbit will always have to be a conic section no matter what. And the exact measurements or shapes of these conic sections are determined by what exactly?

Initial conditions. The conic section is exact for the two body problem. In the solar system there are perturbations due to other planets.

[michel123456]

You can examine explicitly geodesics on the Schwarzschild metric and see that there are a few classes of orbits that are allowed. We can recover the orbits as described in Newtonian theory.

But why is there an orbit in the first place?

Why not a line that make the 2 bodies crash together?

[Strange]

But why is there an orbit in the first place?

Why not a line that make the 2 bodies crash together?

 

Because the orbiting object started with some motion (which was not perfectly directed towards the centre of the thing it is orbiting). Those objects which didn't have enough speed will fall out of orbit. Those that had too much speed for a stable orbit will leave the system. The rest will find a stable orbit.

Edited April 3, 2016 by Strange

[ajb]

But why is there an orbit in the first place?

Why not a line that make the 2 bodies crash together?

There are such solutions to the geodesic equation. You can, for example have radially in falling objects. This is all highly dependent on the initial position and velocity.

[MigL]

You have two 'special' cases...

One is where the test mass is moving directly towards the gravitating body. This obviously results in 'straight' motion to a collision.

The second is where the test mass is moving tangentially to the gravitating body at exactly the right speed to result in a circular orbit.

 

All other ( and there are infinitely more than these two 'special' cases ) gravitational interactions will result in elliptical orbits ( including parabolic and hyperbolic ), because of momentum conservation. Any initial speed ( i.e. momentum ) going in, is regained going out.

 

And ALL are geodesic solutions.

[EdEarl]

Circles are perfect, and perfection is either rare or does not exist. Thus, few orbits will be a perfect circle.

[michel123456]

What is this initial speed coming from?

[EdEarl]

Everything in the Universe has been moving since the Big Bang.

[bluescience]

Thanks, you guys really cleared out that doubt for me. However I have not treaded into the mathematics of all these theories cause I have school and homework, so I do not have enough time to focus on extracurricular stuff. So for the most part I am trying to expand my knowledge horizontally, not vertically, if ya know what I mean. Thanks tho, even if you tried to help with intense math.