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Curvature in space-time is shown as a "fabric"


pmourad
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Most images about general relativity and curvature of space time are show as the image below, where space-time is portrait as a "fabric". We see a grid that bends where earth is placed.  I wonder though, if this fabric should be portrait in 3D as well. In the image below earth is "sitting" over this fabric, but isn't this fabric all around earth?  If this is so, then space-time is being bent not only below earth, but all around it (view last image to see what I mean).

 

Screen Shot 2021-09-12 at 14.48.05.png

 

Screen Shot 2021-09-12 at 14.59.10.png

Edited by pmourad
Added a better image to explain my question.
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15 minutes ago, pmourad said:

Most images about general relativity and curvature of space time are show as the image below, where space-time is portrait as a "fabric". We see a grid that bends where earth is placed.  I wonder though, if this fabric should be portrait in 3D as well. In the image below earth is "sitting" over this fabric, but isn't this fabric all around earth?  If this is so, then space-time is being bent not only below earth, but all around it (view last image to see what I mean).

 

Yes and I wish they wouldn't do that as it is a bad analogy.

Yes the Earth appears sitting on the fabric pulling it down as if it were a heavy ball on a trampoline.

That is the trouble, a heavy ball on a trampoline pulls the fabric down into what ?

Space where there is no 'fabric' whatever that is supposed to mean. Where did that space come from ?

And the pulling it down  - well a heavy ball only pulls down the trampoline because the earth is 'underneath it' pulling on the ball with gravity.

And the lines on the fabric are stretched and distorted.

But distorted from what ? Aren't they supposed to be the correct gridlines of the fabric ? How can they be distorted ?

+1 for recognising it.

 

Here is a better one, soor no pretty dagram at the moment.

Say you are a road runner that can only travel along the road.

Now say you are standing at 5 West Street on a grid pattern of roads where the sides are completely built up with buildings.

And say you want to get to 5 North street.

Well you can't cut through the corner of the buildings, you have to go first along West street to the intersection of North Street and West Street.
And then you have to go up North Street until you reach number 5.

These are the 'rules' of this grid pattern of points.

Furthermore this route is the shortest possible route for a road runner.

This is an example in 2D, that works without invoking the 3rd dimension at all.

 

Now suppose we scale this up to 3D.
Again we have an arrangement of point, just now in 3D.
And we have rules either of travel between these points or equivalently the way these points are laid out.
This time there is no need to invoke a 4th dimension (ignoring time for this).

 

The (mathematical) rules are pretty complicated, but that basically how General Relativity works.

 

 

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Yes it is a poor analogy as you stated Studiot 

I think the idea behind this analogy was to try and get people who have no scientific knowledge or education to attempt to conceptualise how gravity works in the GR model.

Once you start realising, or rather delving into deeper understanding of GR gravity, it soon becomes apparent that this image analogy is misleading. I guess, since it is a difficult one to imagine, each individual will have their own concept of how to visualise it, some more accurate than others. 

I try to steer clear of this analogy, using or thinking about it, as it does impose on my conception, even though I know it's inaccurate. 

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I agree with both of you, @Baryon and @Studiot. I am not a scientist, just an enthusiast on astrophysics and I've been reading about Einstein's relativity and even though the first image did help me understand the idea, it also made me wonder.  As a matter of fact, here's another question if you don't mind.

In the first image, the earth is "sitting" on top of this fabric, therefore bending it. Now that I know that this fabric is all around the earth, the fabric on top of the earth, would it be bending the fabric towards the earth or would it be bending it upwards?  

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The mass on a trampoline shows how paths of objects bend in a curved space. studiot, I don't see how your model shows that. The curved sheet doesn't technically need gravity to show this; distort the sheet some other way, and run a "straight" line of tape over the curve and the path will bend (analogous to a null geodesic).

Yes the analogy has problems. The mass represents mass, but the curvature is not representative of spacetime curvature, which I think is 0 at the center of the mass?* Showing the Earth resting on the sheet incorrectly suggests that it's the volume of matter displaces spacetime. I think this fails Einstein's "as simple as possible, but no simpler" criterion. Instead of saying "fabric", it could be called a manifold made up of events---would that stop people from asking what it's made of? Maybe the rest of the analogy could be fixed by labeling things similarly abstractly, instead of using concrete things like an Earth. But I don't know how you'd label it because I don't know what the curvature of the trampoline is actually meant to represent. Is it gravitational potential? Or is it just a toy example of an abstract curved space? I think the trampoline model could be set up and described differently, "no simpler than possible", so that it would both be clearer what it's meant to show, and not suggest other things. At the very least, I feel it should make people think something like "curved spacetime bends the paths of objects" and not "gravity pulls on the fabric of space" or whatever.

Speaking of space vs. spacetime, the inclusion of time in the curvature is what makes masses at rest gravitate toward each other (is that right? along with constancy of 4-velocity magnitude?) but I don't see how that could be represented on a curved sheet.

 

* Edit: now I'm confused because the curvature of the trampoline is also zero under the center of the mass, so maybe it does fairly represent curvature?

Edited by md65536
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Studiot's example considers the space-time interval ( actually a 2d interval in his example ), but you are right md65536, it doesn't illustrate the geodesic deviation ( I have to stop saying curvature because of the implications ).

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31 minutes ago, md65536 said:

The mass on a trampoline shows how paths of objects bend in a curved space. studiot, I don't see how your model shows that. The curved sheet doesn't technically need gravity to show this; distort the sheet some other way, and run a "straight" line of tape over the curve and the path will bend (analogous to a null geodesic).

Yes the analogy has problems. The mass represents mass, but the curvature is not representative of spacetime curvature, which I think is 0 at the center of the mass? Showing the Earth resting on the sheet incorrectly suggests that it's the volume of matter displaces spacetime. I think this fails Einstein's "as simple as possible, but no simpler" criterion. Instead of saying "fabric", it could be called a manifold made up of events---would that stop people from asking what it's made of? Maybe the rest of the analogy could be fixed by labeling things similarly abstractly, instead of using concrete things like an Earth. But I don't know how you'd label it because I don't know what the curvature of the trampoline is actually meant to represent. Is it gravitational potential? Or is it just a toy example of an abstract curved space? I think the trampoline model could be set up and described differently, "no simpler than possible", so that it would both be clearer what it's meant to show, and not suggest other things. At the very least, I feel it should make people think something like "curved spacetime bends the paths of objects" and not "gravity pulls on the fabric of space" or whatever.

Speaking of space vs. spacetime, the inclusion of time in the curvature is what makes masses at rest gravitate toward each other (is that right? along with constancy of 4-velocity magnitude?) but I don't see how that could be represented on a curved sheet.

These misunderstandings show why I dislike this analogy.

1)

31 minutes ago, md65536 said:

but the curvature is not representative of spacetime curvature,

 

If it is meant to represent paths of objects (which is a very vague and ill defined statement) not the curvature of the fabric of space why is the picture so named the curvature of (the fabric of) space ? It can't be spacetime as there is no time axis.

2)

If it is a representation of a path of a real object then why does that object not follow the direct line towards the COG of the Earth ?

3)

The top picture implies that we start with an two dimensional model grid (whatever the grid represents), flat and empty of matter.
We then place some matter (eg the Earth) onto this grid and it is displaced somehow into the third dimension by 'bending of a grid', whether the one in the picture or another grid.

Now extend that up one dimension so we have a 3D grid, that implies, of necessity a 4D universe for this displacement to occur.

4)

if you prefer start with a one dimensional line. If this line is one dimensional it cannot be curved, the term cannot be invoked and maintain the one dimensionality of the universe containing that line. In differential geometry in order to give rho and kappa maning you must invoke further orthoganal derivatives.

Land surveyors are familiar with this one dimensional version under the name 'through chainage', which works basically as I have tried to describe it for pmourad as I seem to recall said they have only basic high school algebra.

5)

31 minutes ago, md65536 said:

The curved sheet doesn't technically need gravity to show this; distort the sheet some other way, and run a "straight" line of tape over the curve and the path will bend (analogous to a null geodesic).

 

We are talking about 'distortion' due to gravity alone. Surely you are not suggesting some additional force ?

 

 

Edited by studiot
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2 hours ago, studiot said:

These misunderstandings show why I dislike this analogy. [...]

1) It is meant to represent the curvature of a space (just not real space), and the paths of objects through that space. What I meant was that I don't think the sheet's curvature is intended to actually model real spacetime curvature. I don't think they curve in the same way. But that's fine, it is an analogy of real spacetime curvature, not a model of it. My problem is that it is not clear how closely it is analogous, or even what properties exactly it is representing. Usually with a physical analogy you can clearly see the differences between the analogy and the real thing, and you don't confuse them. Here, the "fabric of spacetime" is such an elusive abstract thing that people see the sheet as a model of that "fabric".

2) I don't understand. It doesn't represent the real path of an object through spacetime (see (1)), yet a marble initially at rest on the trampoline does follow the line to the COG of the mass.

4) Why can't a 1D line curve? Can't you map one 1D space onto another different 1D space?

5) Again it's an analogy. You can't practically show the curvature of a sheet due to mass alone, the thing that causes the curvature in the sheet is only an analogy to the thing that causes curvature in spacetime.

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4 hours ago, md65536 said:

Yes the analogy has problems. The mass represents mass, but the curvature is not representative of spacetime curvature, which I think is 0 at the center of the mass?*

Not the point the analogy is addressing,  though. It’s showing the curvature away from the mass.

4 hours ago, md65536 said:

Showing the Earth resting on the sheet incorrectly suggests that it's the volume of matter displaces spacetime.

Possibly, but more mass would cause more distortion, so this is avoidable

4 hours ago, md65536 said:

I think this fails Einstein's "as simple as possible, but no simpler" criterion. Instead of saying "fabric", it could be called a manifold made up of events---would that stop people from asking what it's made of?

They’d be asking what you mean by manifold, which is the kind of thing you’re trying to avoid if you use an analogy.

 

4 hours ago, md65536 said:

Maybe the rest of the analogy could be fixed by labeling things similarly abstractly, instead of using concrete things like an Earth. But I don't know how you'd label it because I don't know what the curvature of the trampoline is actually meant to represent. Is it gravitational potential? Or is it just a toy example of an abstract curved space? I think the trampoline model could be set up and described differently, "no simpler than possible", so that it would both be clearer what it's meant to show, and not suggest other things. At the very least, I feel it should make people think something like "curved spacetime bends the paths of objects" and not "gravity pulls on the fabric of space" or whatever.

Demos I’ve seen use a smaller ball to show an orbit. Showing curvature by itself doesn’t really demonstrate anything

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1 hour ago, md65536 said:

2) I don't understand. It doesn't represent the real path of an object through spacetime (see (1)), yet a marble initially at rest on the trampoline does follow the line to the COG of the mass.

Yes of course a marble would roll down the fabric of the trampoline.

But the fabric of the trampoline is woven so there is a mesh with fine holes.

This is fine so long as the holes are much smaller than the marble.

But what would happen if we rolled successively smaller and smaller marbles until the marble was smaller than the holes ?

1 hour ago, md65536 said:

1) It is meant to represent the curvature of a space (just not real space), and the paths of objects through that space. What I meant was that I don't think the sheet's curvature is intended to actually model real spacetime curvature. I don't think they curve in the same way. But that's fine, it is an analogy of real spacetime curvature, not a model of it. My problem is that it is not clear how closely it is analogous, or even what properties exactly it is representing. Usually with a physical analogy you can clearly see the differences between the analogy and the real thing, and you don't confuse them. Here, the "fabric of spacetime" is such an elusive abstract thing that people see the sheet as a model of that "fabric".

If this is so then you can sketch the real curvature of space in the 1D and 2D cases.

I look forward to your sketches.

1 hour ago, md65536 said:

4) Why can't a 1D line curve? Can't you map one 1D space onto another different 1D space?

Again let us see a sketch, employing only a single axis.

It is only possible if you have a non linear function that maps points on one (part of) the x axis to another part of the x axis.

This is exactly the type of relation I described to those new philosophers who have recently joined.

1 hour ago, md65536 said:

5) Again it's an analogy. You can't practically show the curvature of a sheet due to mass alone, the thing that causes the curvature in the sheet is only an analogy to the thing that causes curvature in spacetime.

Which is why it is a bad analogy.

It is allegedly gravity that gauses the 'curvature' in both cases.

Nothing else.

Just gravity.

Yet your description of the rolling  marble has normal reactive forces acting.

 

42 minutes ago, StringJunky said:

Do you really need to know stress energy tensors before you can visualize curved spacetime properly?

It's not the stress - energy tensor you need but the Riemann tensor, if you must use tensors.

But nobody mentioned them.

In another thread Markus mentioned an alternative to curvature - torsion.
This model (analogy) does not suffer from the difficulties of R .

But these are way above the level this thread is pitched at.

@pmourad

Do you understand the idea of 'length of arc' of a curved line, that is distance as measured along the line (not the mathematical formulae involved) ?

This is the same as what I meant by land surveyors' through chainage.

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Most analogies have limitations, we all know that. They are used to offer an explanation as to what is basically happening. Invariably when getting down to the nitty gritty, faults, limitations will always be found. Still when I first became familiar with the rubber sheet/bowling ball analogy in the early to mid fifties as a hairy arse kid, It was a revelation to me. It was like someone turning on a light in a darkened room. It did its job for me. Since then of course, as I have learnt more, read more reputable books, the limitations are far more obvious. 

Anyway I did find a paper on the same subject with an evaluation of this commonly used analogy.

https://upcommons.upc.edu/bitstream/handle/2117/176999/Fisidabo_Paper_vfinale.pdf;jsessionid=4185CF6A53188D7F069FE5F69750A73D?sequence=1

Evaluating the rubber sheet spacetime analogy by studying ball movement in a bent trampoline:

Abstract.

A usual qualitative analogy used to explain gravitation in general relativity is comparing spacetime warping by massive objects with deformation in a rubber sheet. Motivated by this analogy, which identifies planet orbits with trajectories of rolling objects on the rubber sheet, the movement of a small ball in a trampoline bent because of the presence of a heavy mass in its center is studied. It is concluded that the similarities between how masses move under warped spacetime and under a warped trampoline are only qualitative, and later some analogy flaws are outlined, which can be useful for general relativity teaching. Since the “relativistic model” does not match the ball movement in the experimental conditions, two models based on classical mechanics are presented to describe it. The models are implemented computationally and parameters of such models are optimized to match experimental trajectories. In the case of the most complex of these two models, the high accuracy between optimized and observed trajectories implies that the model is able to explain the experiment behaviour.

 

 Conclusions

The two main questions this article addressed were evaluating the rubber-sheet analogy and predicting the orbit of balls rolling on a warped trampoline, by using in both cases data collected from the experiment performed in Tibidabo park. Firstly, no way of extending the analogy accurately beyond the rubber sheet being a visualization of the spacetime warping caused by masses has been found. It can not be considered as a complete spacetime where bodies follow geodesics, neither the orbits a ball follows on a warped trampoline are compatible with Newtonian and GR planetary orbits. The alternative model proposed by K.Thorne in [9], in which the rubber sheet only represents a slice of space instead of the full spacetime would help overcome some of the analogy deficits but would still not suffice to describe the apsidal precession observed in ball trajectories. Regarding the second question, two different physical models were proposed and compared to describe the phenomenon. Computational simulations of both the trampoline and the ball movement concluded that the proposed rolling sphere model can predict accurately the experimental results, outperforming the other model proposed, the point particle model.

:::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::::......................................

We know analogies have limitations, so when explaining it to someone, that point needs to be emphasised.

The same applies with pop science docos. They generally give a basic rundown on a situation or scenario, and obviously with limitations also. The relevant point is that those that are interested and/or attracted to that scenario, will probably go out of their way to do further research and learn more,including getting down to the real nitty gritty. What I'm trying to say, is that analogies do play a necessary part in educating, as do pop sci docos.

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10 hours ago, StringJunky said:

Do you really need to know stress energy tensors before you can visualize curved spacetime properly?

As studiot already said, you only need to know about this tensor if you want to look at spacetime in the interior of an energy distribution, where it functions as the source term. In vacuum it is identically zero. Curvature itself is described by other tensors.

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9 hours ago, studiot said:

Everyone is entitled to thir opinion, including those that can't or won't back up their assertions with detail.

I already justified the opinion, your model doesn't show the paths of objects bending in a curved space, and the rubber sheet analogy does. But I don't see what your analogy is even trying to say. What I get from it is you're saying that objects can only travel along gridlines through space?

On 9/12/2021 at 12:32 PM, studiot said:

Well you can't cut through the corner of the buildings, you have to go first along West street to the intersection of North Street and West Street.

And then you have to go up North Street until you reach number 5.

These are the 'rules' of this grid pattern of points.

Are the streets representative of dimensions? Are they both spatial, or is one meant to be time? (I guess spatial, since you said "ignoring time", but then I don't understand MigL's comment that it "considers the space-time interval".) You can only travel in one dimension at a time, one "first" and then another? And that should give someone an idea of how gravity works?

Edited by md65536
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5 hours ago, md65536 said:

I already justified the opinion, your model doesn't show the paths of objects bending in a curved space, and the rubber sheet analogy does. But I don't see what your analogy is even trying to say. What I get from it is you're saying that objects can only travel along gridlines through space?

Are the streets representative of dimensions? Are they both spatial, or is one meant to be time? (I guess spatial, since you said "ignoring time", but then I don't understand MigL's comment that it "considers the space-time interval".) You can only travel in one dimension at a time, one "first" and then another? And that should give someone an idea of how gravity works?

No I don't think you did answer either my questions, nor those of swansont.

Anyway the OP picture 1 is purely Newtonian. GR is not involved.

The curvatature is in the values of Newtonian gravitational potential.

It is a depiction of a Newtonian gravity well.

There is a similar picture of the Schwarzchild solution in GR which with which it should not be confused.

On 9/12/2021 at 6:55 PM, pmourad said:

Most images about general relativity and curvature of space time are show as the image below, where space-time is portrait as a "fabric". We see a grid that bends where earth is placed.  I wonder though, if this fabric should be portrait in 3D as well. In the image below earth is "sitting" over this fabric, but isn't this fabric all around earth?  If this is so, then space-time is being bent not only below earth, but all around it (view last image to see what I mean).

 

Screen Shot 2021-09-12 at 14.48.05.png

 

 

@pmourad

This New Zealand teaching website offers a really good explanation and some interesting material to play with.

 

ITV_Image_map_ExploringPhysicsConceptsWi

https://www.sciencelearn.org.nz/image_maps/76-exploring-physics-concepts-with-a-gravity-well

 

ITV_Image_map_ExploringPhysicsConceptsWithA_GravityWell_BG-PLATE_FINAL_18Jun19.jpeg

Edited by studiot
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On 9/12/2021 at 12:32 PM, studiot said:

Here is a better one, soor no pretty dagram at the moment.

Say you are a road runner that can only travel along the road.

Now say you are standing at 5 West Street on a grid pattern of roads where the sides are completely built up with buildings.

And say you want to get to 5 North street.

Well you can't cut through the corner of the buildings, you have to go first along West street to the intersection of North Street and West Street.
And then you have to go up North Street until you reach number 5.

These are the 'rules' of this grid pattern of points.

Furthermore this route is the shortest possible route for a road runner.

This is an example in 2D, that works without invoking the 3rd dimension at all.

 

Now suppose we scale this up to 3D.
Again we have an arrangement of point, just now in 3D.
And we have rules either of travel between these points or equivalently the way these points are laid out.
This time there is no need to invoke a 4th dimension (ignoring time for this).

 

The (mathematical) rules are pretty complicated, but that basically how General Relativity works.

Can you please explain the statement that I bolded? You implied that I should back up assertions with detail, and when I looked closer at your analogy that you claim is a "better one", I find that it makes no sense at all. Can you explain what this is meant to show about GR, especially related to the topic of how curvature in space-time is shown?

2 hours ago, studiot said:

No I don't think you did answer either my questions, nor those of swansont.

I don't see any questions asked by swansont. I agree with all the points he made.

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Spacetime itself is the volume and age of the Universe, Now. The only physical property of it is that it is measurable. Information.
 

If space can have any other physical attributes like (dark or normal) energy or mass, that would mean, that the evolution of space by time is directly responsible for the creation of energy and mass. 
 

For me time seems to be linear from t0-t(now), and any point of space is just an empty spot for something (wave…black hole) to exist in.
 

Every physical entity is constantly shifting within space. Our galaxy including 100 million different sized black holes, with 600km/s.
 

If something those black holes and huge suns should give a very strong curve to space, so when these black holes pass with 600 km/s, the curved space supposed to stay behind and we should see “glitches” in the path of the physical entities, filling up or passing by the previous spots of existence of those black holes.

How can Space, what has no physical attribute, be curved?
 

I just can’t see the fabric of space.
 

I see the information every point of it contains: 0

I can just see the space of fabrics (waves…everything) within space(time).

 

Edited by Conscious Energy
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6 hours ago, Conscious Energy said:

I just can’t see the fabric of space.

Yes - but you can see how test particles in free fall (ie only gravity acts on them) move in space as they age into the future. In particular, you can see what happens over time when test particles initially move in parallel.

6 hours ago, Conscious Energy said:

I see the information every point of it contains: 0

That’s because in a small enough local area, spacetime appears flat, just like the surface of the Earth looks flat if you only look at a small patch of it. Global curvature emerges from the way many of such small local patches are assembled.

To put it differently, a single point contains no information; but there’s information in how such points are related to one another. The physics are in the relationships, not the points themselves.

Edited by Markus Hanke
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On 9/15/2021 at 1:49 AM, Markus Hanke said:

but you can see how test particles in free fall (ie only gravity acts on them) move in space as they age into the future.

That is what I mean.
 

I see fabric (particles) act upon mass (gravity) in a local spacetime. 
 

I do not see how the elements of a zero set could give a path/route to anything, from as little as a wave, not to speak about a black hole, when space and time has no physically detectable attributes beyond data. 

Edited by Conscious Energy
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1 hour ago, Conscious Energy said:

I see fabric (particles) act upon mass (gravity) in a local spacetime. 

There isn’t really any kind of ‘action’ in the mechanistic sense of the word. It’s just that test particles and their world lines are themselves part of spacetime, so they cannot do anything other than follow its underlying geometry. There’s no duality of any kind. See below analogy for clarity.

1 hour ago, Conscious Energy said:

I do not see how the elements of a zero set could give a path/route to anything, from as little as a wave, not to speak about a black hole, when space and time has no physically detectable attributes beyond data. 

As I’ve mentioned in my last post, there is information, in the form of the metric which determines the relationship between points. So it isn’t a ‘zero set’. This is true even very far from any sources - even spacetime without gravity has geometric structure that is different from that of Euclidean space. This is (eg) why you can’t accelerate to the speed of light - the fundamental reason for this is geometric, so geometry has real measurable consequences.

It’s exactly like the calculus you learned at school - the derivative of a function is defined at a single point, yet gives you information about the slope of the entire function. That’s because what it really does is tell you about the relationship between neighbouring points on the graph of the function - how it changes from point to point. If you’re given just the (local) derivative, plus boundary conditions, you can reconstruct the entire function, even though any one single point of the function is just an (x,y) pair.

To give an analogy (!!!) - suppose you have two people starting out on different points along the equator, and flying north simultaneously at a constant altitude. When they start out on the equator, let them be - say - 1000miles apart. What happens? The further north they get, the smaller the distance between them becomes. Eventually they’ll meet at the pole. Why? There is no detectable ‘action’ or force between the two planes. Each plane starts off at 90 degree angle from the equator (so their trajectories are initially parallel), and they always fly straight (there’s never any detectable change in direction from their initial trajectory). Yet they approach one another. That’s because they are both confined to the surface of the Earth, which is a sphere; so they must follow its intrinsic geometry. The metric governing this has real, detectable consequences.

There is no detectable information about this at any one point on the Earth’s surface. This is because the geometry concerns relationships between points, so what you do is take measurements of path lengths, areas, or angles. For example, you’ll find that the sum of the angles in a triangle on Earth’s surface is no longer exactly 180 degrees - it’s possible to directly measure this deviation. But you can’t do it at a single point, you need to measure across some distance. That’s because the effects of a non-flat metric are accumulative - mathematically, you integrate components of the metric to obtain path lengths. To put it differently, the metric defines an inner product of tangent vectors, so it’s a local object, but with global effects across the manifold.

Similar principles are true for curved spacetime as well. You can measure path lengths through spacetime pretty much directly (Shapiro delay, Pound-Rebka, gravitational wave detectors,...) and find that they differ from what you’d expect in a flat geometry. You can also directly measure angular distortions in the geometry, ie gyroscopic precessions, frame dragging etc. Gravitational light deflection is in effect a demonstration of the angle sum in a large triangle being different from 180 degrees close to a massive body. And so on. 

Edited by Markus Hanke
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2 hours ago, Markus Hanke said:

There isn’t really any kind of ‘action’ in the mechanistic sense of the word. It’s just that test particles and their world lines are themselves part of spacetime, so they cannot do anything other than follow its underlying geometry. There’s no duality of any kind. See below analogy for clarity.

As I’ve mentioned in my last post, there is information, in the form of the metric which determines the relationship between points. So it isn’t a ‘zero set’. This is true even very far from any sources - even spacetime without gravity has geometric structure that is different from that of Euclidean space. This is (eg) why you can’t accelerate to the speed of light - the fundamental reason for this is geometric, so geometry has real measurable consequences.

It’s exactly like the calculus you learned at school - the derivative of a function is defined at a single point, yet gives you information about the slope of the entire function. That’s because what it really does is tell you about the relationship between neighbouring points on the graph of the function - how it changes from point to point. If you’re given just the (local) derivative, plus boundary conditions, you can reconstruct the entire function, even though any one single point of the function is just an (x,y) pair.

To give an analogy (!!!) - suppose you have two people starting out on different points along the equator, and flying north simultaneously at a constant altitude. When they start out on the equator, let them be - say - 1000miles apart. What happens? The further north they get, the smaller the distance between them becomes. Eventually they’ll meet at the pole. Why? There is no detectable ‘action’ or force between the two planes. Each plane starts off at 90 degree angle from the equator (so their trajectories are initially parallel), and they always fly straight (there’s never any detectable change in direction from their initial trajectory). Yet they approach one another. That’s because they are both confined to the surface of the Earth, which is a sphere; so they must follow its intrinsic geometry. The metric governing this has real, detectable consequences.

There is no detectable information about this at any one point on the Earth’s surface. This is because the geometry concerns relationships between points, so what you do is take measurements of path lengths, areas, or angles. For example, you’ll find that the sum of the angles in a triangle on Earth’s surface is no longer exactly 180 degrees - it’s possible to directly measure this deviation. But you can’t do it at a single point, you need to measure across some distance. That’s because the effects of a non-flat metric are accumulative - mathematically, you integrate components of the metric to obtain path lengths. To put it differently, the metric defines an inner product of tangent vectors, so it’s a local object, but with global effects across the manifold.

Similar principles are true for curved spacetime as well. You can measure path lengths through spacetime pretty much directly (Shapiro delay, Pound-Rebka, gravitational wave detectors,...) and find that they differ from what you’d expect in a flat geometry. You can also directly measure angular distortions in the geometry, ie gyroscopic precessions, frame dragging etc. Gravitational light deflection is in effect a demonstration of the angle sum in a large triangle being different from 180 degrees close to a massive body. And so on. 

Is the geometry of spacetime universally applicable because the old geometry of space and time as measured from any one point has proved incapable of modeling events in other places and times to a sufficient degree of accuracy  under high speeds of relative motion?  (which spacetime can do except   under extreme conditions)

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