Alternative analogies for the curvature of Space-Time

[geordief]

I have been searching for the "author" of the rubber sheet analogy for the curvature of Space Time and the question has arisen as to whether there exist better analogies.

I can well imagine that all analogies are likely to be flawed and misleading to one degree or another but perhaps they can be educational provided (as is not the case with me personally) the mathematical model is well embedded in the mind of the student.

Be that as it may ,I have come across this alternative analogy. Are there other ,better ones?

http://www.academia.edu/815814/Reflections_on_a_Variational_Principle_in_Space_Gravity_and_Light

on page 6 they give an analogy whereby spacetime is denser around massive bodies.

quote:

"The explanation of displacement of light-source location, whether star or quasar,as a gravitational lens in a gravitational potential field certainly improves on the rubbersheet analogy. Spacetime localized, that is, compressed by a massive body, such asthe sun, can be visualized as becoming denser around the body. In gravitational redshift, to pass through the increased denseness, a light ray contracts in height thus lengthening its frequency until it passes out of the lens"

Edited October 11, 2016 by geordief

[swansont]

That analogy suggests that there is a physical medium, as with refraction of light through a lens.

[geordief]

That analogy suggests that there is a physical medium, as with refraction of light through a lens.

Don't all analogies rely on misrepresenting the actual situation? Will the best analogy just have the best (=most overlookable) flaw ?

 

When it comes to 4D representations is it possible to design a computer graphic that will show how it would look like from a 5th dimension?

 

Or would that be pointless since ,as I have read Spacetime has intrinsic curvature and so (if I understand right) can not be embedded in an external dimension anyway ?

 

By the way , those posited extra dimensions (in String Theory?) do they only apply at the quantum level?

[dimreepr]

Don't all analogies rely on misrepresenting the actual situation? Will the best analogy just have the best (=most overlookable) flaw ?

 

When it comes to 4D representations is it possible to design a computer graphic that will show how it would look like from a 5th dimension?

 

Or would that be pointless since ,as I have read Spacetime has intrinsic curvature and so (if I understand right) can not be embedded in an external dimension anyway ?

 

By the way , those posited extra dimensions (in String Theory?) do they only apply at the quantum level?

 

 

Lies to children, springs to mind whenever an analogy is used; we build understanding through false premises that lead to less false premises.

[swansont]

I don't know that they rely on misrepresentation. There will always be shortcomings, but these aren't intentional, and are things you need to be aware of so you avoid misconceptions.

[geordief]

I don't know that they rely on misrepresentation. There will always be shortcomings, but these aren't intentional, and are things you need to be aware of so you avoid misconceptions.

Yes ,sorry I didn't mean "misrepresentations" in a judgemental way -I was trying to be objective but used the language ambiguously.

[dimreepr]

I don't know that they rely on misrepresentation. There will always be shortcomings, but these aren't intentional, and are things you need to be aware of so you avoid misconceptions.

 

 

I merely suggest an analogy is just part of the story, intended to lead one down the path to understanding.

Edited October 11, 2016 by dimreepr

[studiot]

OK geordief, since I prompted this thread I had better post.

 

I am going to reduce the discussion to one dimension for ease.

That is instead of a rubber sheet I am using a rubber band.

 

If I stretch the band horizontally between points A and B and ignore gravity for the moment I have the situation in the left hand of fig1.

The band is one dimensional and lies along the x axis.

 

So distances along the band are the same as distances along the x axis.

 

However (and this is the key to the discussion) I am only interested in points that are on the rubber band.

 

This becomes important as soon as I place a weight or force on the band to stretch it since we now have the situation in the right hand of fig1.

 

The band stretches and is now longer than before. The distance along the band no longer equals distance along the x axis.

 

So I say the band is 'curved', with a certain radius of curvature measured from a centre of curvature.

But in order to do this I must introduce a second axis, the y axis.

 

So I am now referring to things which are not on the band, and I repeat

 

I am only interested in points that are on the rubber band.

 

So we talk about the set S as all the points along the band and nowhere else. I have shown them taken in sequence, starting from A.

 

This sounds all very well and pedantic for the rubber band since we have an available existing y dimension,to work in.

But when we scale up to 3 (or 4) D we have never seen any evidence for the necessary extra dimension to work this way.

 

The band is a one dimensional manifold.

 

So I need a model that has this characteristic of only working with the points in the manifold of space(time).

 

 

So let us look for an example in real life to act as my model.

 

In fig2 I have drawn a typical surveying problem.

 

Construct a curved kerbline at the junction of one highway with another.

A builder might simply place a centre peg and swing a suitable arc from it, but what if the centre was inside a building?

In this case the centre does exist, but is inaccessible.

 

So our builder must stay on the kerbline and can only access points that are on it.

This is how the model offers greater similarity than the rubber band.

 

Knowing the characteristics of the curve (equivalent to having an expression for the value of curvature in GR),

The builder can start at point A and calculate the 'deflection angle', he must turn around to get to point P which is also on the curve

 

This is shown in fig3.

 

So only the points on the curve and properties of those points (eg chord length AP) which are known or can be calculated are used.

 

Obviously he needs mathematical expressions connecting points that are in the manifold as does the relativist for his work.

 

The distance, s, along the kebline is known as through-chainage to the surveyor.

Edited October 12, 2016 by studiot

[swansont]

Yes ,sorry I didn't mean "misrepresentations" in a judgemental way -I was trying to be objective but used the language ambiguously.

 

 

My main objection was the use of "rely on", as if the imperfection of an analogy was intentional.

[geordief]

OK geordief, since I prompted this thread I had better post.

 

I am going to reduce the discussion to one dimension for ease.

That is instead of a rubber sheet I am using a rubber band.

 

 

 

Can try and dig a bit deeper into your analogy? What ,in spacetime would the points A and P (well the set of points along the rubber band ) correspond to?

 

My first thought would be events or potential events in spacetime. Would that be correct?

 

A concatenation of (potential) events ? Or am I badly misunderstanding the GR/SR model ?

 

Can my (mis?)understanding be shown up by your surveyor's analogy? Or is this an area your analogy doesn't "reach" ?

 

Are points A and P simply 2 points in space along the line taken by a body (or a beam of light) connecting 2 other points in space (or space-time depending on terminology)?

Edited October 12, 2016 by geordief

[studiot]

 

Can try and dig a bit deeper into your analogy? What ,in spacetime would the points A and P (well the set of points along the rubber band ) correspond to?

 

My first thought would be events or potential events in spacetime. Would that be correct?

 

A concatenation of (potential) events ? Or am I badly misunderstanding the GR/SR model ?

 

Can my (mis?)understanding be shown up by your surveyor's analogy? Or is this an area your analogy doesn't "reach" ?

 

Are points A and P simply 2 points in space along the line taken by a body (or a beam of light) connecting 2 other points in space (or space-time depending on terminology)?

 

 

I introduced the word 'manifold' and I think you have been party to discussions as to the meaning of this term before.

 

Briefly a manifold is a set of connected points or objects or members.

(There are some other considerations that do not need to concern us here)

 

So you are hinting at this connectedness in your use of 'concatenation'.

 

Connected means that you can get fom one point to the next by travelling only through points in the manifold.

It further means there is some (known) relationship between the points such as an equation.

 

Again we have talked about this before when I mention constrains in reference to a relativity thread.

 

If you can get from one point to the next (following the instructions contained in the constraint equations), you have a path or line or track in the manifold.

 

The surveyor is following his constraint instructions to provide a curved bellmouth at the road junction.

The analogy is for understanding the idea that he can only use other points in his manifold, he cannot refer to external points which may as well be imaginary or not exist for him.

The analogy is not mathematically the same as the actual expressions of curvature are different. The surveyor usually works to constant curvature, although there are occasions to use variable it involves a lot more calculation work.

'Curvature' in Relativity is of the variable type.

[geordief]

Thanks

as always I appreciate your (and others' ) forbearance -I sometimes wonder about the concatenation involved in my own mentation

No I do like that analogy (if analogy it is) where the surveyor works out the curvature without benefit of elements outside the set of points in the line .(I may have seen other examples of this -figure 3 in http://www.cco.caltech.edu/~kip/scripts/PubScans/BlackHoles-Thorne-Starmus.pdf ,for example **)

Can I extrapolate from your kerbline analogy to suppose that in a 3d+1 space-time environment we can calculate curvature (or perhaps something closely related to it) by first considering the line joining 2 extremely neighbouring points in the space-time manifold and then another line connecting one of these two points with a third point ?

Would the "angle" (perhaps a group of angles?) between these 2 lines show the curvature of space in the local area?? Or would it at least have a close mathematical relation ship to the local curvature ?

** the ant walking up and out of the black hole

Edited October 12, 2016 by geordief

[MigL]

All models are, by necessity, incomplete.

GR may be the best mathematical model we have, but it also has some 'misrepresentations', as it gives non-sensical results at certain limits. Newtonian gravity is even more crude.

Analogies are simply non-mathematical models; And even more crude !

And, just like GR, they only make sense when applied in their areas of validity.

[Tim88]

 

I have been searching for the "author" of the rubber sheet analogy for the curvature of Space Time and the question has arisen as to whether there exist better analogies.

I can well imagine that all analogies are likely to be flawed and misleading to one degree or another but perhaps they can be educational provided (as is not the case with me personally) the mathematical model is well embedded in the mind of the student.

Be that as it may ,I have come across this alternative analogy. Are there other ,better ones?

http://www.academia.edu/815814/Reflections_on_a_Variational_Principle_in_Space_Gravity_and_Light

on page 6 they give an analogy whereby spacetime is denser around massive bodies.

quote:

"The explanation of displacement of light-source location, whether star or quasar,as a gravitational lens in a gravitational potential field certainly improves on the rubbersheet analogy. Spacetime localized, that is, compressed by a massive body, such asthe sun, can be visualized as becoming denser around the body. In gravitational redshift, to pass through the increased denseness, a light ray contracts in height thus lengthening its frequency until it passes out of the lens"

 

 

Sure that's not bad - it's indeed called "gravitational lensing". However a big caveat is that the effect is anisotropic.

 

Moreover I get the impression that they don't understand how to consistency apply it; for the phrase "thus lengthening its frequency" sounds rather blurred. Assuming a static, stationary case, frequency cannot and should not change.

Edited October 13, 2016 by Tim88

[studiot]

 

Tim88

 

on page 6 they give an analogy whereby spacetime is denser around massive bodies.

 

What do you understand denser to mean?

 

Density (of something) is that something per unit volume.

 

So what is that something, and where does it live?

Edited October 13, 2016 by studiot

[geordief]

 

What do you understand denser to mean?

 

Density (of something) is that something per unit volume.

 

So what is that something, and where does it live?

"on page 6 they give an analogy whereby spacetime is denser around massive bodies." was actually from my post#1 (the OP) ,studiot.

 

You seem to have formatted the quotes incorrectly and attributed it to Tim88.

 

Not sure if that changes the sense of your post or not....

[MigL]

That is one of the difficulties. The use of words and definitions which aren't meant to be used for such concepts.

The 'density' of space-time is the density , or spacing if you will, of a co-ordinate system.

It is like taking a sheet of graph paper and changing the spacing of the grid lines.

 

That 'something' is a mathematical concept , and as such, it doesn't need to be anything or live anywhere.

It just needs to be measurable.

[geordief]

That is one of the difficulties. The use of words and definitions which aren't meant to be used for such concepts.

The 'density' of space-time is the density , or spacing if you will, of a co-ordinate system.

It is like taking a sheet of graph paper and changing the spacing of the grid lines.

 

That 'something' is a mathematical concept , and as such, it doesn't need to be anything or live anywhere.

It just needs to be measurable.

Have analogies got any benefits at all? Or are they just like a straw a drowning man might grasp at when ,if he but knew it the water beneath had a rock to stand on. ?

 

Do they allow a student to glimpse a hazy outline of a theory or a model before he or she has got up close?

[michel123456]

Or are they just like a straw a drowning man might grasp at when ,if he but knew it the water beneath had a rock to stand on. ?

Good analogy.

[MigL]

All models allow you to do is make certain predictions ( within limits ).

That is their purpose.

 

But a model is NOT 'reality' ( whatever that may be ).

And there is certainly no one-to-one correspondence between every/all aspects of the model and the 'reality'.

It takes a little thinking to consider only the corresponding aspects between model and 'reality' so as to be able to make valid predictions and draw valid conclusions.

The non-corresponding aspects lead to non-sensical results, such as infinities, 'fabric' of space-time, and what pulls down the rubber sheet in the analogy.

[Eise]

I have been searching for the "author" of the rubber sheet analogy for the curvature of Space Time and the question has arisen as to whether there exist better analogies.

 

This?

 

https://www.youtube.com/watch?v=jlTVIMOix3I

[Mordred]

I liked that vid thanks for sharing.

[studiot]

The mistake in post#15 was entirely mine.

 

I copy/pasted it from the post immediately above it, but didn't look properly.

 

My apologies to Tim and Geordie and anyone else inconvenienced.

 

Thank you Geordie for putting me right so pleasantly.

 

 

The point I was making still stands however.

 

Let us take mass density

 

[math]\rho = \mathop {\lim }\limits_{\delta v \to 0} \frac{{\delta m}}{{\delta v}}[/math]

 

Since we are saying delta v tends to zero, it makes no sense to talk of compressed space or compressed anything else.

 

You can't compress below zero!

 

I did wonder if the article was referring to what is called tensor density, which is quite a different thing.

 

Perhaps Mordred will enlighten us since that is his field.

 

Finally if spacetime has any sort of 'density' then how can it not be physical?

[MigL]

As I said, maybe 'density' isn't the appropriate word to use.

But if the separation of the co=ordinate system which is space-time is altered in some areas as compared to others, somewhat like if the ruling on graph paper was closer together in some areas compared to others, what would we call it

[StringJunky]

Isn't the degree of curvature a function of the energy-density that causes it?

I always thought that spacetime curvature was an abstract mathematical construct that generated a curved graphical distribution of the gravitational field; not literally curved space but values that created mathematical curves graphically.

Edited October 13, 2016 by StringJunky