Bowling ball rubber sheet analogy

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Everyone got that memo? We are not allowed to use diagrams to visualize space-time anymore! :rolleyes:

If something moves along a line in space, you can omit 2-space dimensions, and visualize a 2D-spacetime. Works fine for me, and you don't have to use it, if you don't like it.
Are you reading your own posts? You just said it was NOT some curved "thing". Things people can visualize, non-things people cannot visualize.

Are you reading carefully? Omitting 2 space dimensions reduces the problem to an analogy. Just like putting a stick figure on a sphere. It's just an analogy. If it were the "real thing" it wouldn't be an analogy and we'd be able to visualize "the real thing".

Go ahead, show us 'a' 4D space-time?
 

A.T.

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Are you reading carefully? Omitting 2 space dimensions reduces the problem to an analogy. Just like putting a stick figure on a sphere. It's just an analogy. If it were the "real thing" it wouldn't be an analogy and we'd be able to visualize "the real thing".
I guess you just disagree with my usage of the term "visualization", and would like me to use "analogy" or "geometric interpretation" instead?
 
I guess you just disagree with my usage of the term "visualization", and would like me to use "analogy" or "geometric interpretation" instead?
The only time a visualization equals understanding is if you can visualize what you are trying to understand. If you are trying to understand GR, and GR is based on 4D entities, then you cannot visualize nor by extension understand GR through visualization. When we visualize a stick figure on a rubber sheet with depressions in it and stick figures "living" on/in it we are not understanding GR. We are understanding a theory about Flatland at *best*.
 

A.T.

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If you are trying to understand GR, and GR is based on 4D entities, then you cannot visualize nor by extension understand GR through visualization.
GR is based on geometric concepts like geodesics and intrinsic curvature, which can be easily visualized and understood on 2D manifolds. The extension to 4D is not necessary for simple examples like movement along one space dimension, which are sufficient to explain how geodesics and intrinsic curvature imply what we perceive as mass attraction.
 
GR is based on geometric concepts like geodesics and intrinsic curvature, which can be easily visualized and understood on 2D manifolds. The extension to 4D is not necessary for simple examples like movement along one space dimension, which are sufficient to explain how geodesics and intrinsic curvature imply what we perceive as mass attraction.
Is GR based on 4D objects or not?
 

A.T.

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GR is based on geometric concepts like geodesics and intrinsic curvature, which can be easily visualized and understood on 2D manifolds. The extension to 4D is not necessary for simple examples like movement along one space dimension, which are sufficient to explain how geodesics and intrinsic curvature imply what we perceive as mass attraction.
Is GR based on 4D objects or not?
In general yes, but for specific cases 2D is sufficient, which allows visualization.
 
In general yes, but for specific cases 2D is sufficient, which allows visualization.
Is the universe, or anything that exists, 2D?
 
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A.T.

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Is the universe, or anything that exists, 2D?
Depends on the meaning of 'exists', but I'm not interested in philosophical discussions.
 

DrGreg

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Is GR based on 4D objects or not?
Of course, GR is 4D, but there are some situations where all the relevant information is contained within a 2D cross-section -- in the same way that some aspects of Euclidean geometry can be understood in two dimensions only.

I have studied N-dimensional vector spaces, not only when N is finite and greater than 3, but even when N is infinite. And I used my geometrical intuition to help me. Concepts like orthogonality and "length" still make sense in higher dimensions -- as coordinate-independent concepts -- and a geometrical picture can help you cope with such concepts. (For example, to find the distance between a point and a hyperplane, drop a perpendicular.) Of course, you have to back up your intuition with rigorous calculation, but the geometrical picture is still a big help in getting to grips with the subject. The trick, which comes with experience, is to know which concepts generalise to higher dimensions and which don't.

Remember GR is a model of reality, it is not reality itself. In circumstances where a 2D model gives the same answer as a 4D model (because the other two dimensions are irrelevant), by all means use the simpler 2D model which is easier to visualise.
 
Depends on the meaning of 'exists', but I'm not interested in philosophical discussions.
the scientific definition: shape and location

Done, as far as science is concerned. In philosophy they can go on and on all they want.

DrGreg just reiterated what I said, that visualizations are merely models we use to help guide our intuition. Sometimes they extend and sometimes they don't. But the fact is you can't visualize 'a' 4D "object" anymore than you can visualize 'a' 1D "object". DrGreg mentioned calculations, the argument here, in this thread, is about visual analogies. I'm saying you cannot understand GR through visualization or any visual analogy because you cannot visualize a single "object" of GR.
 
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A.T.

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DrGreg just reiterated what I said, that visualizations are merely models we use to help guide our intuition.
No, he said that GR itself is a model of reality, and not only the visualizations. And I fully agree with that. He also agrees that 2D models in GR are sufficient for some cases, which is exactly what I have been saying, and you have been disputing.

I'm saying you cannot understand GR through visualization or any visual analogy, because you cannot visualize a single "object" of GR.
Of course I can. The worldline of a point particle that is moving along a line in space has no extension in the other two space-dimensions. It can be easliy visualized in a 2D diagram.
 
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Of course I can. The worldline of a point particle that is moving along a line in space has no extension in the other two space-dimensions. It can be easliy visualized in a 2D diagram.
You fooled yourself. All objects of GR, whatever their "extent" in any direction, are embedded in a "4D space" without exception. Indeed, it is the "space" in which an object is embedded that distinguishes one geometric theory from another! Therefore, to dispose of the "4D space" in which the "object" moves is to dispose of whatever theory you were talking about in the first place.
 

A.T.

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You fooled yourself. All objects of GR, whatever their "extent" in any direction, are embedded in a "4D space" without exception.
So everyone who uses 2D-space-time diagrams, even in classical mechanics, is fooling himself, by omitting the other two space-dimensions? Thanks for the enlightenment. :rolleyes:.

Indeed, it is the "space" in which an object is embedded that distinguishes one geometric theory from another! Therefore, to dispose of the "4D space" in which the "object" moves is to dispose of whatever theory you were talking about in the first place.
Omitting elements of a model, which don't affect the result in a particular case is not equal to "disposing the theory".
 
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Of course I can. The worldline of a point particle that is moving along a line in space has no extension in the other two space-dimensions. It can be easliy visualized in a 2D diagram.
I should also question whether you can visualize this "point particle". I believe they are defined as having "0 extent" i.e. they are 0D. This alone renders this supposed "visualization" meaningless.

So everyone who uses 2D-space-time diagrams, even in classical mechanics, is fooling himself, by omitting the other two space-dimensions? Thanks for the enlightenment. :rolleyes:.



Omitting elements of a model, which don't affect the result in a particular case is not equal to "disposing the theory".
If they are visualizing a 2D image but think they are visualizing a 3D image, then yes they are fooling themselves.

Omitting elements of a model may not affect the results quantitatively but we're not talking about that! We're talking about if we can visualize the physical components of the theory itself.

So far we have.

Point particles which look like this:

4D space-time that looks like this:
 

A.T.

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I should also question whether you can visualize this "point particle".
I was talking about the worldline of a point particle.

I believe they are defined as having "0 extent" i.e. they are 0D. This alone renders this supposed "visualization" meaningless.
Please, go explain to the mathematicians not to visualize points as dots anymore, because it makes their diagrams meaningless and they "are fooling themselves".

Omitting elements of a model may not affect the results quantitatively but we're not talking about that!
Well I don't know what you are talking about. But I am talking about visualizing those parts of the model that affect the results in a particular case. Why should I visualize irrelevant stuff?
 
I was talking about the worldline of a point particle.


Please, go explain to the mathematicians not to visualize points as dots anymore, because it makes their diagrams meaningless and they "are fooling themselves".


Well I don't know what you are talking about. But I am talking about visualizing those parts of the model that affect the results in a particular case. Why should I visualize irrelevant stuff?
The 4D space in which events occur in a theory is irrelevant to visualizing what is happening in the theory!?
 

A.T.

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Well I don't know what you are talking about. But I am talking about visualizing those parts of the model that affect the results in a particular case. Why should I visualize irrelevant stuff?
The 4D space in which events occur in a theory is irrelevant to visualizing what is happening in the theory!?
Some dimensions of the 4D spacetime are irrelevant in some cases. What is so hard to grasp about this for you? 2D diagrams in classical mechanics are also omitting one irrelevant space dimension:

Inclinedthrow.gif
 
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And are you visualizing an explanation involving 2 dimensions or 3?
 
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Is the universe, or anything that exists, 2D?
How about a holographic universe. Or a black hole event horizon.
 

A.T.

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The diagram in your post. Is this an illustration in 2D or 3D?
It is obviously a 2D diagram, that omits one space dimension. Is it therefore "meaningless" and using it is "fooling yourself"?
 
It is obviously a 2D diagram, that omits one space dimension. Is it therefore "meaningless" and using it is "fooling yourself"?
If you think you are visualizing a 3D image then, yes, you are fooling yourself. GR, first and foremost, takes place in 'a' 4D space-time. This is the central hypothesis. You cannot visualize the fundamental hypothesis of the theory and so only visualize the theory through analogy and metaphor.
 

Mentz114

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If you think you are visualizing a 3D image then, yes, you are fooling yourself. GR, first and foremost, takes place in 'a' 4D space-time. This is the central hypothesis. You cannot visualize the fundamental hypothesis of the theory and so only visualize the theory through analogy and metaphor.
Learn some maths. I can calculate the radial geodesics in a spherical spacetime in 2D - completely ignoring the angular dimensions and get correct results ( agreeing with experiment). Reducing the dimensions in a physically meaningful way is not 'visualising' or 'fooling' anyone.
 

A.T.

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It is obviously a 2D diagram, that omits one space dimension. Is it therefore "meaningless" and using it is "fooling yourself"?
If you think you are visualizing a 3D image then, yes, you are fooling yourself.
What I think is, that the 2D diagram visualizes the relevant space dimensions of the trajectories.
GR, first and foremost, takes place in 'a' 4D space-time. This is the central hypothesis.
4D space-time was already introduced Minkowski to visualize SR. And guess what, he used 2D diagrams too. The central hypothesis of GR is that the 4D space-time is curved and that free fallers have geodesic world lines. This are mathematical concepts which can be explained in lower dimensional spaces easily.
You cannot visualize the fundamental hypothesis of the theory and so only visualize the theory through analogy and metaphor.
But I don't want to visualize the hypothesis, just particular applications of it. People want to understand why apples are falling from trees. And in such particular cases irrelevant dimensions can be often omitted allowing visualization.
 
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