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# General Relativity Explained

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I find it hard to understand General Relativity when it is casually referred to as curvature in spacetime or as the sagging in a trampoline mat.

Would a better explanation for the novice be to say things fall because they are seeking out the place where time runs the slowest?

For instance, when explaining how light is deflected as it passes close to a star imagine it surrounded by voxels (3D blocks of space). Time runs slowest in the voxels close to the star.  Because the speed of light is constant in all frames of reference and as speed = distance /time the voxels close to the star would appear to be smaller to the photons of light.  Consequently the quickest route for the path of light would be to travel close the the star.  This neatly explains the deflection.  Do you agree?

Does gravity dictate the flow of time or does time itself define gravity?  Thanks

Edited by scuddyx
typo fixed

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Personally I think the trampoline mat is a useful analogy when applied correctly so that the limitations are also explained. For instance it is not very complex to build one, letting novices do som hands on experimentation,. Hopefully that triggers curiosity and a drive to learn more. Really understanding GR is not possible using any analogy as far as I know, that requires studying the math of the models.

31 minutes ago, scuddyx said:

Would a better explanation for the novice be to say things fall because they are seeking out the place where time runs the slowest?

I'm not convinced, some questions:
If a novice have some trouble with the sagging mat analogy, it may be tricky to grasp the concept of voxels that vary in size as seen from different frame of reference?

32 minutes ago, scuddyx said:

distance /time the voxels close to the star would appear to be smaller to the photons of light.

If a novice have already grasped special relativity they would maybe know that photons are not a valid frame of reference in relativity. "would appear to be smaller to the photons" seems to suggest the opposite?
In the rubber mat analogy analogy objects close enough together will collide and stay together. I do not see from the voxel explanation how that works. Zero relative velocity relative to an object means that all voxels appear to have same size? Does that seem to imply zero gravity?

Edited by Ghideon
clarification and grammar

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45 minutes ago, scuddyx said:

I find it hard to understand General Relativity when it is casually referred to as curvature in spacetime or as the sagging in a trampoline mat.

Would a better explanation for the novice be to say things fall because they are seeking out the place where time runs the slowest?

No, probably not. You would be using an effect of GR to explain GR. Not to mention implying a causal connection that doesn’t exist.

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I think the trampoline explanation is really bad because there are two types of curvature intrinsic curvature and extrinsic curvature.

The curvature introduced by GR is intrinsic curvature which means that in order to detect it does not require leaving the space (manifold) the observer is in.
In more mathematical terms the space does not need to be embedded in a space of higher dimensions for a radius of curvature to exist.

On the other hand trampoline curvature is extrinsic curvature, which requires embedding in a space of higher dimension.
This means that the trampoline is initially a plane and pulled down into a curved shape in a third dimension at right angles to the original plane.

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10 hours ago, scuddyx said:

I find it hard to understand General Relativity when it is casually referred to as curvature in spacetime or as the sagging in a trampoline mat.

Would a better explanation for the novice be to say things fall because they are seeking out the place where time runs the slowest?

For instance, when explaining how light is deflected as it passes close to a star imagine it surrounded by voxels (3D blocks of space). Time runs slowest in the voxels close to the star.  Because the speed of light is constant in all frames of reference and as speed = distance /time the voxels close to the star would appear to be smaller to the photons of light.  Consequently the quickest route for the path of light would be to travel close the the star.  This neatly explains the deflection.  Do you agree?

Does gravity dictate the flow of time or does time itself define gravity?  Thanks

If time runs slower closer you are to a massive object - is it not correct to say matter will move in the direction of decreasing time?

This is easier to understand than saying 'spacetime' is curved.

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1 minute ago, scuddyx said:

If time runs slower closer you are to a massive object - is it not correct to say matter will move in the direction of decreasing time?

It might do that, but you said “seek” which has certain implication

1 minute ago, scuddyx said:

This is easier to understand than saying 'spacetime' is curved.

That’s subjective. Easier for you. But if it’s wrong, then it doesn’t matter so much if it’s easier to understand

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Thanks for replying to my posts.

Professor Jim Al-Khalili (theoretical physicist) at the end of his BBC programme “The Amazing World of Gravity” says:

﻿Why does the apple fall?  Hundreds of years of Scientific inquiry investigating this single action have led us to completely redefine the way we think about the very nature of space and time.  And now I’ve been presented with this extraordinary proposition that somehow, in some profound way, the apple falls because it's seeking out the place where Time runs the slowest.

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1 hour ago, scuddyx said:

Thanks for replying to my posts.

Professor Jim Al-Khalili (theoretical physicist) at the end of his BBC programme “The Amazing World of Gravity” says:

﻿Why does the apple fall?  Hundreds of years of Scientific inquiry investigating this single action have led us to completely redefine the way we think about the very nature of space and time.  And now I’ve been presented with this extraordinary proposition that somehow, in some profound way, the apple falls because it's seeking out the place where Time runs the slowest.

Even professors of Physics have their off days.

I am not a fan of JAK as a celebrity physicist, but he was trying to express the calculus of variations as applied to geodesics in three words.

He could equally have said that the apple whilst attached to the tree is moving with the rest of the universe.
Once detached it does not move at all.
It is just that the rest of the universe moves past it the other way.

A good question is not

why does the apple fall ?

but

Why does it move at all ?

Hint ask a poet not a physicist.

Edited by studiot

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I wasn’t aware that BBC programs are peer-reviewed

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On 1/20/2020 at 11:50 AM, studiot said:

Even professors of Physics have their off days.

I am not a fan of JAK as a celebrity physicist, but he was trying to express the calculus of variations as applied to geodesics in three words.

He could equally have said that the apple whilst attached to the tree is moving with the rest of the universe.
Once detached it does not move at all.
It is just that the rest of the universe moves past it the other way.

A good question is not

why does the apple fall ?

but

Why does it move at all ?

Hint ask a poet not a physicist.

"Why does it move at all?"

Is this a question that General Relativity answers or one waiting for an answer?

Thanks for your replies.

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1 hour ago, scuddyx said:

"Why does it move at all?"

Is this a question that General Relativity answers or one waiting for an answer?

Thanks for your replies.

It's what Newton answered (to him, gravity was a force)

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scuddyx, see this video. Much closer to what GR says than the rubber sheet analogy.

On 1/18/2020 at 6:18 PM, studiot said:

I think the trampoline explanation is really bad because there are two types of curvature intrinsic curvature and extrinsic curvature.

Are these two types of curvature, or are these different ways to describe curvature? E.g. one can describe the curvature of the surface of a 3 dimensional object in 3 dimensional space in both ways, no?

And my understanding of the 'curvature' of the universe is that it is as if it is curved. If I remember well, Mordred said that GR only says how worldlines of free moving objects in the universe look like. In some book I read that one can formulate GR without the use of curved spacetime, which for some kind of problems seems to make the calculations easier (I think this was in the book about black holes of Kip Thorne).

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45 minutes ago, Eise said:

Are these two types of curvature, or are these different ways to describe curvature? E.g. one can describe the curvature of the surface of a 3 dimensional object in 3 dimensional space in both ways, no?

For intrinsic curvature that is correct.
An example of intrinsic curvature would be the old Atari 'Asteroids' game.
The left side of the screen is identified with the right hand side, and the top with the bottom, such that when your spaceship disappears from one side of the screen it reappears on the other. The screen would be an example of a 'flat' 2dimensional torus, as opposed to the surface of a 'regular' torus which looks like a doughnut, and requires a 3rd dimension to visualize.

Extrinsic curvature actually requires an external 'embedding' dimension. IOW a 4th dimension to describe 3dimensional curvature.
An example of extrinsic curvature is the rubber sheet analogy of GR, and it is precisely why it is problematic.
The flat 2dimensional sheet actually bends into an external ( 3rd, embedding ) dimension, when a mass is placed on it.

I don't really consider the space-time of GR to be 'bendable' ( whatever that means ).
But GR allows me to assign a co-ordinate system to that space-time, and the altered spacing of the co-ordinate gridlines in the presence of mass-energy ( or energy-momentum ) is the equivalent of 4dimensional 'curvature'. These co-ordinate gridlines are essentially equipotential field lines of the gravity field.

Edited by MigL

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35 minutes ago, MigL said:

An example of intrinsic curvature would be the old Atari 'Asteroids' game.

Isn't that an example of a 2-D object with intrinsic curvature? (And for which we can make a 3-D model.)

My idea (hopefully), was even simpler, say the curvature of the surface of a ball (the ball is 3-D, but the surface is 2-D). Isn't it true that one can use both descriptions? In that case we see the 3-D space in which the 2-D surface of the ball is curved, so we can describe it extrinsically. The most important lesson about this in GR, that I thought I learned, is that one can describe curvature intrinsically, and therefore, because we cannot observe 4 dimensions, the question in what the 3-D universe is curved is a metaphysical question (and therefore physics can do without it). Therefore also my 'as if' in my previous posting:

1 hour ago, Eise said:

And my understanding of the 'curvature' of the universe is that it is as if it is curved.

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3 hours ago, swansont said:
4 hours ago, scuddyx said:

"Why does it move at all?"

Is this a question that General Relativity answers or one waiting for an answer?

Thanks for your replies.

It's what Newton answered (to him, gravity was a force)

Please sir, swansont sir, you've done it again.

Didn't you like my poem?

1 hour ago, Eise said:

Isn't that an example of a 2-D object with intrinsic curvature? (And for which we can make a 3-D model.)

My idea (hopefully), was even simpler, say the curvature of the surface of a ball (the ball is 3-D, but the surface is 2-D). Isn't it true that one can use both descriptions? In that case we see the 3-D space in which the 2-D surface of the ball is curved, so we can describe it extrinsically. The most important lesson about this in GR, that I thought I learned, is that one can describe curvature intrinsically, and therefore, because we cannot observe 4 dimensions, the question in what the 3-D universe is curved is a metaphysical question (and therefore physics can do without it). Therefore also my 'as if' in my previous posting:

Here is a good account

Edited by studiot

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19 minutes ago, studiot said:

Please sir, swansont sir, you've done it again.

Didn't you like my poem?

I'm a physicist, not a poet.

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8 minutes ago, swansont said:

I'm a physicist, not a poet.

Sorry, I realise that it was my mistake this time.

I thought you were attributing my question to scuddy to scuddy, but I see now he repeated it.

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15 hours ago, Eise said:

Are these two types of curvature, or are these different ways to describe curvature?

Under certain circumstances you can use either one to describe the same thing, but that is not always the case. As a counterexample, consider the (2D) surface of a (3D) cylinder - it is extrinsically curved, but intrinsically flat. So clearly, in this case these two descriptions are not equivalent.

In the case of GR, spacetime is not thought to be embedded into any higher dimensional space, so it has only intrinsic curvature. For arguments sake, it is possible to construct a mathematical model that embeds spacetime into something higher-dimensional, and then use extrinsic curvature to capture all the same information. The problem with this is that the embedding would have to have a lot of dimensions; I can’t actually remember the exact number, but I think it was 48. So I fail to really see the advantages in this, as it makes most of the maths very much more complicated than it already is in standard GR.

In general terms, any Riemann manifold can be embedded into a higher-dimensional Euclidean space in such a way that paths lengths are preserved. This is called the Nash embedding theorem.

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44 minutes ago, Markus Hanke said:

Under certain circumstances you can use either one to describe the same thing, but that is not always the case. As a counterexample, consider the (2D) surface of a (3D) cylinder - it is extrinsically curved, but intrinsically flat. So clearly, in this case these two descriptions are not equivalent.

I understand, but with some hesitation. Maybe it is because of my (stupid?) idea that intrinsic curvature is the way out for 'n-D' organisms to discover that they live in a curved space, because they cannot access the n+1th dimension to actually see it. But you are saying they cannot discover that they are living in a cylindrical universe based on its intrinsic curvature: the metrics of a flat universe and a cylinder universe are the same.

57 minutes ago, Markus Hanke said:

The problem with this is that the embedding would have to have a lot of dimensions; I can’t actually remember the exact number, but I think it was 48.

Wow! And I thought one extra dimension would suffice... Obviously that is when one, like me, only has a very superficial understanding of differential geometry, and has to do with analogy arguments. (I am not @scuddyx, who talks about 'tensor force' as if he is understanding what he is talking about...).

Gasho,

Eise

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18 hours ago, Eise said:

In some book I read that one can formulate GR without the use of curved spacetime, which for some kind of problems seems to make the calculations easier (I think this was in the book about black holes of Kip Thorne).

Yes, it was in Thorne. He says that one can formulate GR in flat spacetime, where gravitation deforms lengths and durations, whereas the standard interpretation is that lengths and durations stay the same, but spacetime is deformed by the presence of matter/energy. The results are the same, but it seems the math is not completely, and so, dependent on the kind of problem, one can use the interpretation that makes the calculations easiest. And he explicitly says that the question if spacetime is really curved is a philosophical question that does not bother physicists.

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29 minutes ago, Eise said:

Yes, it was in Thorne. He says that one can formulate GR in flat spacetime, where gravitation deforms lengths and durations, whereas the standard interpretation is that lengths and durations stay the same, but spacetime is deformed by the presence of matter/energy. The results are the same, but it seems the math is not completely, and so, dependent on the kind of problem, one can use the interpretation that makes the calculations easiest. And he explicitly says that the question if spacetime is really curved is a philosophical question that does not bother physicists.

Actually the idea of  alternative viewpoints is very common and goes back a long way.

For example consider a rotating arm.
You could equally well model this as rotating the axes the other way. \
All you need are a couple of minus signs in the equations.

Or where is a point in space ?

X,Y,Z ?  or R, θ,φ or R,L, θ

But you also need some connecting equation or equations to establish relational conditions between the axes, eg direction cosines.

Again alternatively the (some or all of) the axes may be non linear as in log-linear and log-log plots.

there is a famous quote I can't quite remember, I think it may have been in 'Alice'

"There are many way to something or other, and each and every one of them is right"

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19 hours ago, Eise said:

the metrics of a flat universe and a cylinder universe are the same

Yes, an unrolled cylinder is flat.
As opposed to a hemisphere ( + curvature ), which lacks material near the edges to allow flattening ( the world map problem )
Or  a saddle shape (- curvature ), which has too much material near the edges to allow flattening.

And I remember reading the same in K Thorne's book, about the various ways to formulate GR.
But I've never seriously considered space-time as something that can 'bend' or curve.
However, manipulating the co-ordinates in that manner seems to fit extremely well with observation.
( by the way, what does "gasho" mean ? )

16 hours ago, studiot said:

There are many way to something or other

Wheuw !
I thought you were gonna say "There are many ways to skin a cat".
Those are fighting words to us cat-owners  .

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3 minutes ago, MigL said:

But I've never seriously considered space-time as something that can 'bend' or curve.
However, manipulating the co-ordinates in that manner seems to fit extremely well with observation.

Well that's good as spacetime isn't some fabric like substance lol. I always find it most accurate to think of it as the freefall paths that bend. Example the worldlines

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22 hours ago, Eise said:

But you are saying they cannot discover that they are living in a cylindrical universe based on its intrinsic curvature: the metrics of a flat universe and a cylinder universe are the same.

The metrics are not the same, but they are diffeomorphisms of one another. The crucial difference is that, on the surface of a cylinder, geodesics in one direction are closed curves - you can walk all around the cylinder, and end up again at the same spot where you started. On a flat sheet, there are no closed geodesics, they all extend to infinity, so no matter in what direction you walk, you will never get back to where you started. So based on this observation, they could probably figure out that they are indeed on a surface with the topology of a cylinder. But that alone does not allow them to conclude anything about whether or not that surface is embedded in a higher dimensional space.

The crucial point however is that, in both cases these world lines are locally straight everywhere. You can smoothly deform one “world” into the other while exactly preserving all path lengths and angles.

Edited by Markus Hanke

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55 minutes ago, Markus Hanke said:

The metrics are not the same, but they are diffeomorphisms of one another.

I think you mean the opposite: the spaces are not diffeomorphic, but they are locally isometric

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