Geometry in General Relativity: Round Marble on a Box

In summary, you can describe your setup by using a local frame that is close to flat, and then using GR to calculate the deviations from flatness.
  • #1
berra
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Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!
 
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  • #2
  • #3
A.T. said:
It sounds like you are asking more generally about Euclidean vs. non-Euclidean geometry:
http://en.wikipedia.org/wiki/Non-Euclidean_geometry
I am asking how you go from what you see with your eyes or a measurement device that is to my understanding the best representation of a time-foliation of spacetime to making a mathematical model of it in spacetime.

I will read the wiki page now and see if it addresses my question.

Nope, I think rectangles have right angles so the wiki page doesn't help. I am asking how you interpolate your measurements into a bunch of positions in spacetime. Does one assume that the 3D subspace is euclidian and that all points came from the same time? Or something else?
 
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  • #4
One relates geodesics to motion. That gives the entire thing. Or, as stated in _Gravitation_ by Misner, Thorne, and Wheeler, time is defined to make motion look simple.
 
  • #5
berra said:
Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!

Take a look at http://www.eftaylor.com/pub/chapter2.pdf if you get a chance, an excerpt from Taylor and Wheeler's book "Exploring black holes". They have an example there of how you could measure the shape of a rowboat by driving nails into it, and precisely measuring the distances between nails with strings running along the surface of the rowboat.

The mathematical object in General Relativity that gives you distances between points (and also the more general space-time equivalent, the Lorentz interval) is called a metric.

Einstein's field equations are basically equations that the metric must satisfy, given the matter distribution. So the short answer to your question is that you define the shapes by the metric, which can be regarded as a tool that gives you the distance between every pair of events on the space-time manifold, which are analogous to the nails that you drove into the hull of the rowboat.
 
  • #6
berra said:
Say you have a box with a round marble on it. Classically I would have no problem describing that, but how is it done in general relativity? Do one just assume that the marble is round and that the box is ... boxy... in the 3D subspace at a certain foil of time? (Foliation is the correct term right?) Or does one iteratively perturb/deform the (materials) geometry, until Einsteins laws with the stress energy tensor etc is satisfied? I'm asking because I have a gut feeling that a round marble isn't round in general relativity. And also because every single textbook I have read have been void of examples.

Thanks!

You do not need GR to describe your setup. According to GR we live in a locally minkowskian geometry that is 'carried' with us as we go through time ( and possibly space).

Your local frame in your lab is as close enough to Euclidean/Minkowski over a large enough volume, to use those models.

The deviation of the local coordinates from flatness is calculable ( actual numbers !).
 
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FAQ: Geometry in General Relativity: Round Marble on a Box

What is the significance of a round marble on a box in General Relativity?

The round marble on a box is a commonly used thought experiment to demonstrate the concept of curved space in General Relativity. It helps to visualize how the presence of mass and energy can cause spacetime to curve, affecting the paths of objects in space.

How does the shape of the box affect the marble's trajectory in General Relativity?

The shape of the box does not have a significant effect on the marble's trajectory in General Relativity. The box serves as a reference frame for the marble's motion, but the curvature of spacetime is determined by the distribution of mass and energy in the universe.

Can the marble's motion on the box be described by Euclidean geometry?

No, the motion of the marble on the box cannot be described by Euclidean geometry. In General Relativity, spacetime is curved and non-Euclidean, meaning that the laws of Euclidean geometry do not apply. Instead, the marble's motion must be described using the mathematics of curved spacetime.

How does General Relativity explain the movement of the marble on the box?

In General Relativity, the marble's movement is explained by the curvature of spacetime caused by the presence of mass and energy. The marble follows the path of least resistance on this curved surface, which is known as a geodesic. This explains the observed motion of objects in the universe, such as the orbit of planets around stars.

Why is it important to understand the concept of curved space in General Relativity?

Understanding the concept of curved space in General Relativity is important because it is a fundamental aspect of our understanding of the universe. It helps to explain the behavior of objects in space, such as the motion of planets and the bending of light around massive objects. It also provides the foundation for our current theories of gravity and has implications for concepts such as black holes and the expansion of the universe.

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