How do relativity explain elliptical orbits of planets?

[Fredrik]

I'd like to point out things that are relevant here:

1. Geodesics are defined as the straightest possible paths, not as the shortest (or longest).
2. It's the path through space-time that has to be a geodesic, not the path through space.

The straightest path connecting two events in space-time is the one that has the longest proper time, so I'd rather say that a geodesic is the longest path than the shortest. Of course, if you draw a space-time diagram representing an inertial frame in SR and draw a geodesic and a non-geodesic curve connecting two events, the geodesic is the shortest path on that piece of paper.

More importantly, the fact that an object in free fall is going to trace out the shortest/longest path between where it is and where it's going to be isn't sufficient to tell you where it's going to be. Let's play a game: You're moving around on the surface of a sphere and you're not allowed to change your speed. At the moment you happen to be at the equator moving north. If you are told that you have to go as straight as possible, you know you have to continue towards the north pole. If instead you're told that you have to take the shortest path, you don't know if you're allowed to make a sharp left turn right away. The rule "go as straight as possible" seems to be equivalent to "take the shortest path and don't make any sharp turns". The rule "don't make any sharp turns" is of course equivalent to "keep going as straight as possible", so we might as well just say that. The instruction to take the shortest/longest path seems completely redundant.

Because of this I prefer to say that a planet in orbit is going as straight as possible through space-time, instead of saying that it takes the shortest/longest path around the sun back to (almost) its original position. Saying that it takes the shortest/longest path leaves the question "...to where?" unanswered.

Let's move on to my second point. If we only talk about geodesics in space rather than in space-time it's impossible to understand how orbits can have different eccentricities. Consider a circle inside an ellipse such that they coincide at the two points on the ellipse that are the closest to the center. It's clear that no matter what the geometry of space is, at most one of them can be a geodesic. However, if we consider the paths through space-time of two objects in orbits that look like what I just described, we can no longer immediately rule out that they are both geodesics.

This is one of the reasons why a bowling ball on a rubber sheet is such a bad analogy. A better mental image is to imagine the sun and the planets as dots on a piece of paper, and imagine time as the "up" direction (out of the paper). The path of the sun is a line straight up. The paths of the planets are spirals around that straight line. Note that if we consider the spirals that correspond to the circle and the ellipse I described, their tangents aren't the same at the points in space where the circle touches the ellipse. They have different slopes relative to the paper. That's why we can't immediately rule out that they are both geodesics.

 

[niin]

I want to thanks everyone who responded.
I think, I will have to think a little more before i fully get it, but i feel i have a better understanding of the subject now.
Thanks for helping me out. =)

 

[Dale]

1. Geodesics are defined as the straightest possible paths, not as the shortest (or longest).

Yes, this is correct, but my understanding is that the two statements are essentially equivalent. You could just as easily define them as the longest path and then derive the fact that they are "straight". The Euclidean analogy is "the shortest distance between two points is a straight line".

If you define your spacetime path with an initial event and a direction (initial conditions) then you can follow the "straight line" rule to construct the geodesic. If you define your spacetime path with an initial and final event (boundary conditions) then you can use a variational approach based on the "longest distance rule" to find a geodesic that connects them. The only difference, other than initial v. boundary conditions, is that there may be more than one geodesic path between two events in a curved spacetime. Each one is a local maximum path length, but one may be a global maximum.

 

[Fredrik]

Agreed. My point is just that the description of a geodesic as the shortest/longest path is more appropriate when the endpoints are known, and the description as the straightest possible path is more appropriate when one endpoint and a velocity (tangent vector) is known. But that's pretty much what you just said, so I don't have to tell you that.