Gravity & Spacetime According to GR: Exploring a Planet's Path

[daniel_i_l]

According to GR objects (for simplification I'll only talk about the small "planet" attracted to the big star) are attracted cause the shortest path the planet can follow through ST (it's always going at light speed) when it's close to a planet is a path in the direction of the planet.
Lets imagine that there're 2 spatial dimensions and one time. Now the star and planets would be circles, and if we look at ST as a whole the star would be a cylinder. My question is, wouldn't the cylinder warp ST evenly all along it's length? If so, let's say the planet comes into the picture after the star (as usually happens), that would mean that it's at the middle of the length of the cylinder and if ST there is all curved by the same amount then shouldn't the shortest path be straight up?
I'm sure I have some fatal flaw here, could someone tell me were it is?
Thanks.

 

[MeJennifer]

According to GR objects (for simplification I'll only talk about the small "planet" attracted to the big star) are attracted cause the shortest path the planet can follow through ST (it's always going at light speed) when it's close to a planet is a path in the direction of the planet.

Not sure what you mean by going at light speed. An object with mass cannot travel at light speed.

Lets imagine that there're 2 spatial dimensions and one time. Now the star and planets would be circles, and if we look at ST as a whole the star would be a cylinder.

Well normally points with light cones are used and for SR that is sufficient. Of course in reality they are really expanding spheres, but one cannot represent that on paper.

But to represent GR space-time warping one would need to imagine that the lightcones are on a piece of clay that can be twisted and stretched.

So taking these things into account your model becomes a bit too simple (no representation of null lines and hence no Minkowski representation of space and time, circles instead of points). But anyway let me try to go along with your reasoning.

My question is, wouldn't the cylinder warp ST evenly all along it's length? If so, let's say the planet comes into the picture after the star (as usually happens), that would mean that it's at the middle of the length of the cylinder and if ST there is all curved by the same amount then shouldn't the shortest path be straight up?
I'm sure I have some fatal flaw here, could someone tell me were it is?
Thanks.

Well if I understand you correctly, and in your (flawed) model, you assume that ST is the time part extending the circle in a cilinder, am I right? If so, then that is not the case. Space-time is the combination of the 3 spatial dimensions and a time dimension in a particular relationship, not just as another axis. That is why this kind of space is called a Minkowski space.

But that is not the whole story, because in your case there is mass involved, then once you got the Minkowski space you got to imagine that that 4-dimensional space can be warped by mass.

But anyway, when the planet approches the star the location would be closer and closer to the circle right. Then in would intercept at a point of the star's world line (e.g. the cilinder). So in your model (after you read all the ifs and buts), then if the planet it stuck near the star it's world line would be very close to the star and staight up as well.

Hope this helps

 

[coalquay404]

I haven't had a chance to think about the original question but I do have one small observation:

It's *MINKOWKSI* space, not Minkowsky.

 

[Daverz]

Free falling objects follow extremal paths in spacetime, not the shortest paths.

 

[baryon]

But, daverz, remember that we are talking in GR terms. The shortest path that a free falling object will follow inside a gravitational field a curve. If you stood atop the Sears Tower and your friend was standing at the same height but twenty feet away and you both drop tennis balls at the same time, you would notice the balls do not fall straight down but toward a poiint equally distant between them. The paths of the two balls would curve toward each other.

 

[baryon]

I do agree with MeJenn however. Dan, you are saying that a circle expanded in the third dimension would be a cylinder. That's incorrect because we use different formulas to find the area of each shape.

 

[Jorrie]

But, daverz, remember that we are talking in GR terms. The shortest path that a free falling object will follow inside a gravitational field a curve. If you stood atop the Sears Tower and your friend was standing at the same height but twenty feet away and you both drop tennis balls at the same time, you would notice the balls do not fall straight down but toward a poiint equally distant between them. The paths of the two balls would curve toward each other.

Are you sure the spatial paths of the two balls will be curves? Are they not falling in straight lines towards the center of gravity of the Earth (ignoring Earth’s rotation)? If you are talking ST paths, surely they will be curved relative to a reference frame stationary relative to Earth's surface. But, falling at the same speed, they still do not curve towards each other.

 

[MeJennifer]

Are you sure the spatial paths of the two balls will be curves? Are they not falling in straight lines towards the center of gravity of the Earth (ignoring Earth’s rotation)? If you are talking ST paths, surely they will be curved relative to a reference frame stationary relative to Earth's surface. But, falling at the same speed, they still do not curve towards each other.

It all depends on the coordinate system you are using.
But the rate that the two balls are approaching each other while falling towards the center of gravity is not linear.

 

[Jorrie]

It all depends on the coordinate system you are using.
But the rate that the two balls are approaching each other while falling towards the center of gravity is not linear.

Ok, agreed. So spatially the paths are not curved in a reference system with Earth's centre (of mass) permanently at rest - they are straight. The balls approach each other faster and faster. I suppose this translates to a 'curved ST paths' of the one ball in the inertial frame of the other, correct?

 

[selfAdjoint]

Ok, agreed. So spatially the paths are not curved in a reference system with Earth's centre (of mass) permanently at rest - they are straight. The balls approach each other faster and faster. I suppose this translates to a 'curved ST paths' of the one ball in the inertial frame of the other, correct?

What the paths are geodesics in is spacetime, not 3-D space as you imagine. So an accelerated straight-line 3-D motion toward the center of mass is curved in spacetime, no?

 

[Jorrie]

What the paths are geodesics in is spacetime, not 3-D space as you imagine. So an accelerated straight-line 3-D motion toward the center of mass is curved in spacetime, no?

But I thought this was approximately, precisely what I said! Spatially the paths are straight, but both balls follow spacetime paths with identical curvature. The issue was, IMO, the balls are also accelerating towards each other. Hence my statement: "I suppose this translates to a 'curved ST path' of the one ball in the inertial frame of the other, correct?" Note ST paths, not space paths.

 

[MeJennifer]

Spatially the paths are straight, but both balls follow spacetime paths with identical curvature. The issue was, IMO, the balls are also accelerating towards each other. Hence my statement: "I suppose this translates to a 'curved ST path' of the one ball in the inertial frame of the other, correct?" Note ST paths, not space paths.

In the general theory of relativity a 3D space region in a gravitational field (except for in the center, or inside hollow masses and things like that) is no longer Euclidean. The case is comparable to a rapid rotating disk.

This is also the case for light. It's speed is no longer constant in a gravitational field but instead varies with position.

 

[Jorrie]

In the general theory of relativity a 3D space region in a gravitational field (except for in the center, or inside hollow masses and things like that) is no longer Euclidean. The case is comparable to a rapid rotating disk.

This is also the case for light. It's speed is no longer constant in a gravitational field but instead varies with position.

Thanks for the reply, MeJ, but my question is not answered, so I will rephrase it. The two balls of the OP fall straight towards the center of the hypothetically non-rotating Earth. They are accelerating downwards and also towards each other. Now choose one of the balls as the reference frame. Is the spacetime path of the other ball curved in this (free falling) reference frame?

 

[pervect]

Thanks for the reply, MeJ, but my question is not answered, so I will rephrase it. The two balls of the OP fall straight towards the center of the hypothetically non-rotating Earth. They are accelerating downwards and also towards each other. Now choose one of the balls as the reference frame. Is the spacetime path of the other ball curved in this (free falling) reference frame?

It's hard to understand exacatly what you mean by curved here with regards to paths.

Generally speaking, one says that both paths are geodesics. This statement is independent of the coordinate system used.

 

[Jorrie]

It's hard to understand exacatly what you mean by curved here with regards to paths.

Generally speaking, one says that both paths are geodesics. This statement is independent of the coordinate system used.

Thanks pervect. I suppose one can then conclude that both balls follows spacetime geodesics that can be projected to cross at the center of the (assumed non-rotating) Earth.

 

[MeJennifer]

Thanks pervect. I suppose one can then conclude that both balls follows spacetime geodesics that can be projected to cross at the center of the (assumed non-rotating) Earth.

Actually they do not cross, unless you are thinking of things like wormholes.

 

[Jorrie]

Actually they do not cross, unless you are thinking of things like wormholes.

No I was not thinking of wormholes. But then, since pervect pointed out that one should generally not equate geodesics with spacetime paths, I’m not sure how to think about spacetime paths anymore! Maybe one should rather refer to them by the standard term wordlines. Now worldliness can cross, do they not?

 

[pervect]

Thanks pervect. I suppose one can then conclude that both balls follows spacetime geodesics that can be projected to cross at the center of the (assumed non-rotating) Earth.

Yes. If there are no external forces on the balls, they are following geodesics.

A rather standard comparison is the behavior of two geodesics on the surface of the Earth - two great circles. They intersect at a point, deviate for a bit, and then come back together. The apparent relative acceleration of geodesics (in this case apparently towards one another) is given by the geodesic deviation equation.

see for example
http://math.ucr.edu/home/baez/gr/geodesic.deviation.html

 

[MeJennifer]

But then, since pervect pointed out that one should generally not equate geodesics with spacetime paths, I’m not sure how to think about spacetime paths anymore!

All geodesics are spacetime paths but not all spacetime paths are geodesics!

Now worldliness can cross, do they not?

Yes they can, but generally not two objects who move towards the center of a gravitational field such as one around a planet.

The geodesics of the two balls most certainly converge. But they converge to a point in spacetime, they don't cross, unless you include exotic solutions. If they would really meet at the center, which they obviously do not in our example, since we assume the planet must have some volume, they would simply end there. In other words there would be no future for the wordlines, time for them has simply ended.

 

[Jorrie]

All geodesics are spacetime paths but not all spacetime paths are geodesics!


Yes they can, but generally not two objects who move towards the center of a gravitational field such as one around a planet.

Nicely summarized, pervect and MeJ.

I understand that if objects are falling towards the center, their geodesics stop somewhere, i.e. at the surface or in the extreme at the central singularity of a BH.
If they are on other geodesics (e.g. orbits), then geodesics can work like pervect said above: approaching, crossing, deviating, provided they do not crash into each other…

Thanks