Question about spacetime curvature

This maybe a simple question, but if Earth orbits the Sun due to the Sun's mass 'curving' spacetime, wouldn't we be moving closer to the sun? like if you spun a marble around within a bowl, it ends up in the center.

What am I missing here?

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Ibix
The marble is subject to friction which slows it down. It will "orbit" for longer on a smoother surface. The Earth is not subject to any serious friction, so it doesn't spiral in.

Thanks, that cleared it up. But I would have thought the marble slows down too due to gravity? or not, since the force the marble exerts while spinning in the bowl is greater than the earth's gravity? Am I correct?

WannabeNewton
Hi surprise! The marble analogy is a poorly formulated one that leads to misconceptions about what it means for gravitation to be modeled by space-time curvature. GR tells us that test particles in free fall (such as, to a good approximation, the Earth free falling due to the Sun) follow the 'geodesics' of the curved space-time. The geodesics of the curved geometry are, loosely put, the "straightest possible lines" one can have on that geometry. For example, on a 2 dimensional sphere, the geodesics are the great circles (and, in a similar spirit, the geodesics of the space-time around the Sun are to a good approximation those of the Schwarzschild space-time: http://en.wikipedia.org/wiki/Schwarzschild_metric). The trajectories of test particles free falling in the Sun's vicinity (e.g. the Earth) will follow paths in space-time given by the geodesics of this metric (after supplying initial conditions of course). See here for more: http://en.wikipedia.org/wiki/Schwarzschild_geodesics

The diagrams which try to model space-time curvature as "dents" in space-time are not accurate because we are actually talking about curvature in 4 dimensions which is extremely difficult to picture so one must settle for so-called 'embedding diagrams'.

Ibix
Try not to read too much into "ball on a curved surface" analogies for General Relativity.

If you could make a perfectly frictionless bowl, the marble would carry on "orbiting" forever (assuming you've picked your bowl very carefully to be shaped such that stable orbits are possible). It would speed up as it went down into the bowl, and slow down by the same amount as it climbed back out. But it will not spiral inwards.

In reality, you cannot make a frictionless bowl. Friction inevitably leads to the marble slowing down and spiralling in. But this is due to friction sapping energy, not an effect of gravity.

Hi surprise! The marble analogy is a poorly formulated one that leads to misconceptions about what it means for gravitation to be modeled by space-time curvature. GR tells us that test particles in free fall (such as, to a good approximation, the Earth free falling due to the Sun) follow the 'geodesics' of the curved space-time. The geodesics of the curved geometry are, loosely put, the "straightest possible lines" one can have on that geometry. For example, on a 2 dimensional sphere, the geodesics are the great circles (and, in a similar spirit, the geodesics of the space-time around the Sun are to a good approximation those of the Schwarzschild space-time: http://en.wikipedia.org/wiki/Schwarzschild_metric). The trajectories of test particles free falling in the Sun's vicinity (e.g. the Earth) will follow paths in space-time given by the geodesics of this metric (after supplying initial conditions of course). See here for more: http://en.wikipedia.org/wiki/Schwarzschild_geodesics

The diagrams which try to model space-time curvature as "dents" in space-time are not accurate because we are actually talking about curvature in 4 dimensions which is extremely difficult to picture so one must settle for so-called 'embedding diagrams'.
This straight line you refer to, does it exist in the 4th dimension (time)?

WannabeNewton
This straight line you refer to, does it exist in the 4th dimension (time)?
Yes these paths are paths in space-time. This is one of the most elegant aspects of GR!

Try not to read too much into "ball on a curved surface" analogies for General Relativity.

If you could make a perfectly frictionless bowl, the marble would carry on "orbiting" forever (assuming you've picked your bowl very carefully to be shaped such that stable orbits are possible). It would speed up as it went down into the bowl, and slow down by the same amount as it climbed back out. But it will not spiral inwards.

In reality, you cannot make a frictionless bowl. Friction inevitably leads to the marble slowing down and spiralling in. But this is due to friction sapping energy, not an effect of gravity.
I'll have to learn more about friction then, but thanks.

Yes these paths are paths in space-time. This is one of the most elegant aspects of GR!
Interesting. Do space-time geodesic's reduce entropy in 4-dimensional space?

Nugatory
Mentor
Interesting. Do space-time geodesic's reduce entropy in 4-dimensional space?
No, they have nothing to do with entropy.
Although we cannot (well, I cannot!) visualize a four-dimensional space, you can learn a far amount about space-time works by learning to draw and understand two-dimensional space-time diagrams. There are many examples in various posts in this forum.

WannabeNewton
Interesting. Do space-time geodesic's reduce entropy in 4-dimensional space?
I'm not sure I understand the question. Entropy is a state function of a given system involving the number of microstates associated with a macrostate. What system are we looking at here and what is it being compared to when asked if it is "reduced"? A thermally isolated system at equilibrium has maximized its entropy. By the way, if you have questions on entropy in GR, you should create a new thread on it since it is not related to your original question.

Based on the work of Jacobson [1–6], Padmanabhan [7– 12], and others, there are also good reasons to suspect a thermodynamic interpretation of the fully relativistic Einstein equations might be possible.
From the same source.

No, they have nothing to do with entropy.
Although we cannot (well, I cannot!) visualize a four-dimensional space, you can learn a far amount about space-time works by learning to draw and understand two-dimensional space-time diagrams. There are many examples in various posts in this forum.
Thanks for the advice. I'm currently self-teaching myself the subject along with thermodynamics and some of my own work.

WannabeNewton
I'll have to learn more about friction then, but thanks.
All Ibix is saying is that in the absence of friction, a particle can stay in a certain "orbit" for all time if the system is let to evolve on its own. The bowl analogy is not a good one for another reason: there are constraint forces on the particles so they are not in free fall. Their motion isn't geodesic so it isn't really representative of the motion of a freely falling particle in space-time. As long as the earth is in free fall around the Sun, there is nothing perturbing it significantly from its observed path around the Sun so as to make it spiral inwards (it has a stable orbit).

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Nugatory
Mentor
Sorry, I'm just trying to understand the logic of space-time geodesics. I also asked because there was a recent theory that gravity arises from the effects of entropy.
http://en.wikipedia.org/wiki/Entropic_gravity
You might want to try reading the "Talk" page for that article as well.

Alright, thanks.

Sorry, I'm just trying to understand the logic of space-time geodesics. I also asked because there was a recent theory that gravity arises from the effects of entropy.
http://en.wikipedia.org/wiki/Entropic_gravity
Read about SR to learn about the dimensions & intervals. Those are great concepts to know if poking about GR stuff, specifically geodesic paths, acceleration. That helped me know what gravity is a bit better, far from "understanding" though, you need to be able to speak math . Light decreasing in gravitational potential is a good scenario / thought experiment.

You maybe asking why energy density can be used to calculate gravity so accurately, I don't think the How part is known.

Stress-Energy Tensor

Cool. As I said, I'm self-teaching myself this stuff. It's getting intense.

Cool. As I said, I'm self-teaching myself this stuff. It's getting intense.
Be careful though, there are plenty of ideas about how gravity works, but you should really master special and general relativity [and by that I mean at the mathematical level, not just popular science treatment] before you try to understand the newer ideas like entropic gravity, or even formulate your own idea. Having said that, have you look at this?

http://johncarlosbaez.wordpress.com/2012/02/01/entropic-forces/

Yeah, thanks for the advice. I know, the maths looks tough, but I'll get it.