Physical demonstration of the Curvature of SpaceTime

[geordief]

 

 

Not really. It doesn't matter if you measure in furlongs or millimetres, or base 10 or base 17. As long as you are consistent between measuring radius and area.

 

 

Yes (if you mean what I think you mean). If the relationship is no longer A = 4 pi r² then that represents curved geometry.

Thanks . Apologies for going off topic. I don't think this detour goes anywhere further really.

.

[studiot]

Since no one is interested in my contribution perhaps I should bow out of this thread.

 

However on the way I would ask you to consider exactly what is meant by intrinsic and extrinsic and explain how a universe with the same laws as ours, but comprising a single point can support an intrinsic curvature.

 

Furthermore it is worth observing that a single expression of curvature is only appropriate in 2 dimensions.

 

In 3 dimensions there are two independent curvature expressions involved.

 

Can you extrapolate to 4 dimensions?

[MigL]

This is not a relativity problem, and it can be easily described and visualized by reducing the dimensionality.

 

Consider intrinsically flat, Euclidian space. if a 'Flatlander' living on this surface draws a circle he will have no problem establishing that the circumference increases linearly with the radius ( and as elementary geometry teaches us ).

 

Now consider a 'Flatlander' living on a positive, intrinsically curved surface ( a sphere ). As he draws his circles to determine the relationship between radius and circumference, he finds that the circumference does NOT increase linearly with the radius, but starts falling behind.

 

Similarly a 'Flatlander' living on negative, intrinsic curvature ( saddle shape, or a stretched piece of clothing that does not lie flat anymore because there is too much material near the edges ), will, upon performing the same exercise, measure the circumference increasing faster than linearly WRT radius.

 

It is a simple task to extend this mathematically to 3 or 4 dimensions, but you can no longer visualize the effect.

And note that we only considered measurements made on the 2D surface, Studiot, so this is in intrinsic effect and has no need for embedding

[geordief]

Since no one is interested in my contribution perhaps I should bow out of this thread.

 

However on the way I would ask you to consider exactly what is meant by intrinsic and extrinsic and explain how a universe with the same laws as ours, but comprising a single point can support an intrinsic curvature.

 

Furthermore it is worth observing that a single expression of curvature is only appropriate in 2 dimensions.

 

In 3 dimensions there are two independent curvature expressions involved.

 

Can you extrapolate to 4 dimensions?

I was interested in your earlier post but I couldn't understand why you thought Feynman's " simplification was not totally 'right' either" as you put it in post#8

 

I admit my first instinct was to suppose it was a measurement that had actually been done whilst ,in fact this is to be viewed as a "thought experiment" albeit one that is not 100% impossible.

 

But I don't think that was what you were getting at.

 

When imatfaal picked you up on that point in post#12 I thought that I might learn something from your exchange of views but I did not and assumed that the point was "over my head"

 

Your latest post sounds interesting but perhaps the same "over my head" point applies

[Tim88]

 

It is nothing to do with compression. The surface area of a sphere is 4 pi r² where r is the radius. That is a Euclidean measurement - ie when there is no intrinsic curvature to the geometry of space; the mass of the earth causes an intrinsic positive curvature and the euclidean ratios no longer quite apply.

 

The easiest way to think of an analogy is in an extrinsic curvature situation - a curved 2d surface in our 3d world; that is to say the surface of a sphere. Draw a large circle on the ground, go to the centre of the circle and measure the distance to the circumference by walking a tape measure from the centre to the line, then measure around the circumference. We would expect the ratio to be Circumference = 2 pi r; but it is not. In this example it is clear that it cannot be because the "radius" we have walked from the centre to the circumference is curved; we can easily see this as the curvature is extrinsic (ie it is a curved object embedded in our 3d space).

 

In Feynman's example the background geometry itself is curved - there is an intrinsic curvature but it still screws up ratios and expectations which are made on the basis of flat euclidean geometry. The amount that the radius is excessive compared to flat geometry (if the sphere is constant density) is simple and directly proportional to the mass

[..]

 

Einstein objected to that analogy as it could lead to the misunderstanding that objects are somehow "curved" in a fourth dimension. In my opinion, the easiest way to think about it is the way he thought about it: rulers are shortened in radial direction by the Earths gravitational field, but they are not shortened when laid along the surface of the Earth.

Ha ha . Do I get points for naivete ?

 

I realize that Newtonian mechanics falls/has fallen at one or more hurdles but is it out of the question to run a computer simulation where we model the Earth -or any test body as a group of plastic massive bodies that settle into a ball shape under the action of Newtonian gravity?

 

Is it possible to compare the results obtained by such a simulation with that obtained by another that ran a GR algorithm?

 

It that as daunting a project as physical measurement?

 

What about quantum computers ? Would they be up to the task?

 

I never tried to simulate that, but if you don't need extreme precision, the equations are rather simple (see the link above). So it should be not too difficult to run such a simulation on a laptop.

[studiot]

This is not a relativity problem, and it can be easily described and visualized by reducing the dimensionality.

 

Consider intrinsically flat, Euclidian space. if a 'Flatlander' living on this surface draws a circle he will have no problem establishing that the circumference increases linearly with the radius ( and as elementary geometry teaches us ).

 

Now consider a 'Flatlander' living on a positive, intrinsically curved surface ( a sphere ). As he draws his circles to determine the relationship between radius and circumference, he finds that the circumference does NOT increase linearly with the radius, but starts falling behind.

 

Similarly a 'Flatlander' living on negative, intrinsic curvature ( saddle shape, or a stretched piece of clothing that does not lie flat anymore because there is too much material near the edges ), will, upon performing the same exercise, measure the circumference increasing faster than linearly WRT radius.

 

It is a simple task to extend this mathematically to 3 or 4 dimensions, but you can no longer visualize the effect.

And note that we only considered measurements made on the 2D surface, Studiot, so this is in intrinsic effect and has no need for embedding

 

Thank you for you points migL

 

The point I was making is that

 

A point, a line, a plane, a ball surface or its topological equivalent are all proper subsets (or subspaces) of Rⁿ.

 

With the exception of a point each of these have proper sub-subsets of their own. A point has no proper subsets.

 

Curvature is defined by a calculus on a set, but you cannot have a calculus without proper subsets, so a(n isolated) point has no curvature of its own.

 

For instance the Kappa you are talking about in plane curves or the Kappa and Tau I referred to (though not by name) in space curves.

Edited November 21, 2016 by studiot

[geordief]

I hope this is a related question.

 

The phrase "curvature of space time " seems to cause a bit of befuddlement and I am wondering is it more to the point (and perhaps linguistically accurate) to think of this as a curvature of the co ordinate system employed by an observer to describe events in an accelerating or gravitational frame of reference so as to be easily understandable to another observer in his or her own frame of reference?

 

Can we replace the term "space" or "space time" with "co ordinate system" ** and basically avoid the whole area of debate around whether space actually curves or not?

 

By the way I am not clear whether an observer in an accelerating frame of reference sees the frame of reference of an inertially moving frame of reference as curved in a symmetrical way. to the way an observer views events in an "accelerating frame of reference"

 

** giving "curvature of the co ordinate system"

[Strange]

Can we replace the term "space" or "space time" with "co ordinate system" ** and basically avoid the whole area of debate around whether space actually curves or not?

 

 

I would say you go to the top of the class with that observation. You have just killed multiple threads on this in the Philosophy section!

[geordief]

Is it known why Einstein came up with the idea of curving the co ordinate system ? Was it a shot in the dark?

 

I mean nobody knows why mass-energy does this.Did he perhaps just notice all the curvature effects that show up in acceleration and treat the effects of gravitation in a similar way?

 

Was he already aware of deficiencies in Newtonian physics so that he had tests in mind whereby he could verify whether his model was more accurate?