Visualising curved space

[NotPrinceHamlet (Graham)]

Hi all,

Has anyone managed to find a mental trick to assist in visualising a curved space? I find it pretty difficult, as it takes a mammoth effort of will not to superimpose a euclidian geometry over the top of my so called curved space.

My thought experiment is the classic surface of a sphere with a triangle with three 90 degree angles... I find it difficult to change my thinking to 'see' the lines of the triangle as one dimensional and not 2 dimensional in euclidian space. So to my cartesian mind, its not 'really' a triangle.

My ultimate objective is to finally get a grip on the basic maths of general relativity. I've been reading through an excellent NASA primer on tensors, which I studied at university in Applied maths - somehow they managed to remove all practical application from them - so as a result I've completely forgotten the math.

This book is on my wish list too http://www.amazon.com/Introduction-T...=24ZKNZ3CHQBTP

[omnivorr]

stare into M C Escher pictures for long periods ?

[NotPrinceHamlet (Graham)]

Say that I'm an ant that lives on this curved sphere.

I walk up to the vertex of the triangle at the north pole and I measure it - I find it to be 90 degrees - right?

I walk to the other two and I measure 90 degrees. Totalling 180 degrees.

Would I perceive that I was upsidedown at the south pole as compared to the north pole? I don't think so, as the space itself is curved, but if I was on a sphere in euclidian space, then I would perceive myself as upsidedown on the south pole.

So according to my ant brain, this triangle is flat, but with 270 degrees in it.

Does this sound right?

[sjastro]

A useful visual aid used in GR is the Flamm paraboloid.

http://en.wikipedia.org/wiki/File:Flamm.jpg

A triangle projected onto to the surface of the paraboloid, shows all those nasty non-Euclidean effects such as angles not adding up to 180 degrees.

Originally Posted by NotPrinceHamlet

Say that I'm an ant that lives on this curved sphere.

I walk up to the vertex of the triangle at the north pole and I measure it - I find it to be 90 degrees - right?

I walk to the other two and I measure 90 degrees. Totalling 180 degrees.

Would I perceive that I was upsidedown at the south pole as compared to the north pole? I don't think so, as the space itself is curved, but if I was on a sphere in euclidian space, then I would perceive myself as upsidedown on the south pole.

So according to my ant brain, this triangle is flat, but with 270 degrees in it.

Does this sound right?

Yes that's true if the ant and sphere are embedded in 3-D space, it will perceive an "upside down".

If on the other hand the ant is a 2-D organism on the surface of the sphere it is constrained in 2-D curved space. The triangle will be flat as the ant won't perceive the curvature of the sphere. The angles however will not total 180 degrees.

If you master tensor analysis you'll find the maths used in GR quite straightforward although the calculations can be rather laborious.

Regards

Steven

[Archy (George)]

We all live in curved space so we don't notice it.
I don't know about an ant's perception, but when I travel to the Northern Hemisphere I don't notice that I am upside down. Is the reference to Euclidian space a reference to plane geometry?

[sjastro]

Originally Posted by Archy

We all live in curved space so we don't notice it.
I don't know about an ant's perception, but when I travel to the Northern Hemisphere I don't notice that I am upside down. Is the reference to Euclidian space a reference to plane geometry?

Hello George,

In this context Euclidian space refers to degrees of freedom.
So in 3-D Euclidian space we can measure length, width and height dimensions or a combination of the three.
In 2-D space we can measure length and width dimensions or a combination of the two.

Using your plane analogy, whether you are upside down or not, can be defined by your position in space (instead of the Earth's surface) by using 3-D Euclidian coordinates.

Steven

[Archy (George)]

"Using your plane analogy, whether you are upside down or not, can be defined by your position in space (instead of the Earth's surface) by using 3-D Euclidian coordinates."

I've read Euclid, but can't recall a reference to coordinates.
Ptolemy has coordinates (spherical at that)
Plane coordinates come with Descartes hence Cartesian coordinates.
Although by carefully choosing the origin (0,0,0) position in space can be defined so that there is an up and a down, it is also possible in every case to have a position in 3D or nD space ( where n is two or more) such that there is no up or down, although there will generally be a "higher"or "lower".

In reference to the ant going from the North pole to the South pole, I can't see that the ant's perception will be different to mine. When I travel to the UK I do not see that i am upside down. Do You?

[NotPrinceHamlet (Graham)]

What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.

However, if the space is actually curved around, then I think the situation is different.

This is doing my head in!

I've come across a couple of websites now that have said that time is not the 4th dimension that Einstein refers to... I always thought that it is. The sites are saying that the 4th dimension is actually a spacial dimension.

E.g.

http://www.cs.unm.edu/~joel/NonEuclid/space.html

Some people say that Time is the 4th dimension. This is, in a sense, true. However, time is not the "different direction" of which I was speaking. If we want to count time as a dimension, then we should say that we live in a four dimensional space-time that is curved into the 5th dimension! So where is this "different direction"?

[sjastro]

Originally Posted by Archy

"Using your plane analogy, whether you are upside down or not, can be defined by your position in space (instead of the Earth's surface) by using 3-D Euclidian coordinates."

I've read Euclid, but can't recall a reference to coordinates.
Ptolemy has coordinates (spherical at that)
Plane coordinates come with Descartes hence Cartesian coordinates.
Although by carefully choosing the origin (0,0,0) position in space can be defined so that there is an up and a down, it is also possible in every case to have a position in 3D or nD space ( where n is two or more) such that there is no up or down, although there will generally be a "higher"or "lower".

In reference to the ant going from the North pole to the South pole, I can't see that the ant's perception will be different to mine. When I travel to the UK I do not see that i am upside down. Do You?

Since the Earth is not sufficently massive to cause space-time to curve, the surrounding space is flat and the geometry must be Euclidean.

With this in mind one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection.

If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down.

In the Earth's frame of reference, an observer on the ground can definitely perceive the Earth's curvature. The ancient Greeks were aware of it buy noticing that ships disappeared below the horizon.

Steven

[sjastro]

Originally Posted by NotPrinceHamlet

I've come across a couple of websites now that have said that time is not the 4th dimension that Einstein refers to... I always thought that it is. The sites are saying that the 4th dimension is actually a spacial dimension.

E.g.

http://www.cs.unm.edu/~joel/NonEuclid/space.html

Time is not a spatial variable.

I think the confusion arises from Special Relativity.

The metric for distance between 2 points in 3-D flat space is
ds2=dx2+dy2+dz2.

In space-time the metric is
ds2=c2dt2-dx2-dy2-dz2 (c is the speed of light.)

ds2 is still a spatial property, but since t is now in the equation it is easy to think of t as spatial.

Steven

[Archy (George)]

[quote=sjastro;432572]Since the Earth is not sufficently massive to cause space-time to curve, the surrounding space is flat and the geometry must be Euclidean.

Is there a critical mass that makes space curved? the closer one gets to a centre of mass no matter how small, the more curved space-time gets. The further one gets from a centre of mass no matter how large, the "flatter" space-time gets.

[Archy (George)]

"....one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection

If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down. "..

You have defined a special set of conditions, namely a frame of reference with the origin 0,0,0 at the earth.

Consider the universe: there are a very large number (some would say infinite) locations for the origin of a frame of reference. Those locations within the earth, are by comparison an almost infinitesimally small fraction. Choosing the centre of the earth as the origin is a very special case. In the general case my statement is true: there is no up or down.



Steven[/quote]

[quote=NotPrinceHamlet;432570]What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.

[Archy (George)]

"....one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection

If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down. "..

You have defined a special set of conditions, namely a frame of reference with the origin 0,0,0 at the earth.

Consider the universe: there are a very large number (some would say infinite) locations for the origin of a frame of reference. Those locations that are within the earth, are by comparison an almost infinitesimally small fraction. Choosing the centre of the earth as the origin, as you have done, is a very special case. In the general case my statement is true: there is no up or down.

Steven[/quote]

[quote=NotPrinceHamlet;432570]What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.

You have chosen a special type of scale : it is Cartesian.
But it is not the only way to view things. For instance, maps of the earth use various projections.

The Mercator projection has a non uniform scale for latitude.

The Cartesian system is OK for small scale things, but it is not a particularly useful method of determining the locus of the shortest path from here to New York. Spherical geometry and specially constructed polar (nonlinear) grids give the result in seconds.

In other circumstances, a referential system might use imaginary numbers and scales eg the scale might be the square root of -1: I've used such a referential system to solve certain problems to do with x-ray crystallography.

In other circumstances one could use a spherical referential system. A triangle has 180 degrees in a plane system, but can have 180 to 540 degrees if drawn with a pole as the apex and the base on the equator.

One could also use a conical referential system: Australia's geological maps use such a system.

[sjastro]

[quote=Archy;432690]

Originally Posted by sjastro

Since the Earth is not sufficently massive to cause space-time to curve, the surrounding space is flat and the geometry must be Euclidean.

Is there a critical mass that makes space curved? the closer one gets to a centre of mass no matter how small, the more curved space-time gets. The further one gets from a centre of mass no matter how large, the "flatter" space-time gets.

Space time is effected by density not mass.

The density of the Sun's core causes barely perceptable space time curving such as the bending of star light.

As one moves away from the source, space becomes flatter as indicated by the perihelion advance of each planet's orbit. Mercury has the greatest value which progressively decreases for each planet as one moves outwards.

Space-time however is still extremely flat in the Sun's vicinity.

Steven

[NotPrinceHamlet (Graham)]

Originally Posted by sjastro

Time is not a spatial variable.

I think the confusion arises from Special Relativity.

The metric for distance between 2 points in 3-D flat space is
ds2=dx2+dy2+dz2.

In space-time the metric is
ds2=c2dt2-dx2-dy2-dz2 (c is the speed of light.)

ds2 is still a spatial property, but since t is now in the equation it is easy to think of t as spatial.

Steven

So that website I linked to that asserts

"If we want to count time as a dimension, then we should say that we live in a four dimensional space-time that is curved into the 5th dimension! So where is this "different direction"?"

is just plain wrong?

It looks like the author is trying to get a grip on the curve by basically stating that the curve exists into the fourth dimension and the projection of the four dimensional space onto the 3rd dimension is the flat space we all know. I think that this is incorrect (happy to be shown I'm wrong though - because its nice and easy to understand!)

[sjastro]

Originally Posted by Archy

"....one can have the origin at the centre of the Earth, and the Z-axis along the polar axis. A person standing at the North Pole is at position (0, 0, R), at the South Pole (0,0, -R) where R is the Earth's radius.
A coordinate transformation from (0,0,R) to (0,0, -R) is equivalent to an upside down reflection

If you are travelling in a plane at a constant velocity you are in an plane's inertial frame of reference not the Earth's frame of reference. Your position and velocity is relative to the plane's frame of reference.
If your upside down it means the plane has turned upside down. "..

You have defined a special set of conditions, namely a frame of reference with the origin 0,0,0 at the earth.

Consider the universe: there are a very large number (some would say infinite) locations for the origin of a frame of reference. Those locations that are within the earth, are by comparison an almost infinitesimally small fraction. Choosing the centre of the earth as the origin, as you have done, is a very special case. In the general case my statement is true: there is no up or down.

I'm using a localized geometry to describe the Earth nothing more.

Cosmologists use a localized geometry to describe the observable Universe and a global geometry to describe the entire Universe.

The geometry for the observable Universe is Euclidean. That's not to say that the space-time around every object in the Universe is flat.
Black Holes and Neutron stars are clear exceptions.

The local geometry of these objects needs to be treated separately.
It is perfectly valid to use a local geometry for any object.

Originally Posted by NotPrinceHamlet

What I mean by euclidean space is just flat 3d space, with all 3 axes orthogonal to each other and all uniformly scaled - then even if the ant doesn't realise that it is upside down at the south pole, relative to the north pole, it is.

Originally Posted by NotPrinceHamlet

You have chosen a special type of scale : it is Cartesian.
But it is not the only way to view things. For instance, maps of the earth use various projections.

The Mercator projection has a non uniform scale for latitude.

The Cartesian system is OK for small scale things, but it is not a particularly useful method of determining the locus of the shortest path from here to New York. Spherical geometry and specially constructed polar (nonlinear) grids give the result in seconds.

In other circumstances, a referential system might use imaginary numbers and scales eg the scale might be the square root of -1: I've used such a referential system to solve certain problems to do with x-ray crystallography.

In other circumstances one could use a spherical referential system. A triangle has 180 degrees in a plane system, but can have 180 to 540 degrees if drawn with a pole as the apex and the base on the equator.

One could also use a conical referential system: Australia's geological maps use such a system.

Remember this is a thread on General Relativity which involves the geometry of space not the geometry of bodies.

Regards

Steven

[Archy (George)]

RE: "Remember this is a thread on General Relativity which involves the geometry of space not the geometry of bodies.
Steven[/quote]"

The thread started with: "Has anyone managed to find a mental trick to assist in visualising a curved space." This is a general question, not one restricted to General Relativity.

Graham had a second line: "My thought experiment is the classic surface of a sphere with a triangle with three 90 degree angles... I find it difficult to change my thinking to 'see' the lines of the triangle as one dimensional and not 2 dimensional in euclidian space. So to my cartesian mind, its not 'really' a triangle."

Later he added: "Say that I'm an ant that lives on this curved sphere.
I walk up to the vertex of the triangle at the north pole and I measure it - I find it to be 90 degrees - right?
I walk to the other two and I measure 90 degrees. Totalling 180 degrees.
Would I perceive that I was upsidedown at the south pole as compared to the north pole? I don't think so, as the space itself is curved, but if I was on a sphere in euclidian space, then I would perceive myself as upsidedown on the south pole."

It was in that context that I commented:
"Although by carefully choosing the origin (0,0,0) position in space can be defined so that there is an up and a down, it is also possible in every case to have a position in 3D or nD space ( where n is two or more) such that there is no up or down, although there will generally be a "higher"or "lower".

In reference to the ant going from the North pole to the South pole, I can't see that the ant's perception will be different to mine. When I travel to the UK I do not see that i am upside down. Do You? "

[Archy (George)]

Steven
Re: "Space time is effected by density not mass." Steven[/quote]"
There are two contradictions:
First: Having said: "Since the Earth is not sufficently massive to cause space-time to curve.... " you then say it's not mass it's density.
Second: "Space time is effected by density not mass" is self contradictory as density is a function of mass.

Finally, where is the proof that density effects (ie causes) space-time?

[sjastro]

Originally Posted by Archy

Steven
Re: "Space time is effected by density not mass."
There are two contradictions:
First: Having said: Since the Earth is not sufficently massive to cause space-time to curve.... " you then say it's not mass it's density." you then say it's not mass it's density.
Second: "Space time is effected by density not mass" is self contradictory as density is a function of mass.

By mentioning the planet Earth I have automatically defined.

(a) It's mass
(b) It's diameter (or volume)

The statement "Since the Earth is not sufficently massive to cause space-time to curve.... " should therefore be self evident.

In other words the Earth is not sufficently massive for it's diameter (or volume) to cause space time to curve.

Density=mass/volume. If volume is constant, then density is purely a function of mass. Increasing mass increases the density which may cause space time to curve.

Hence there is no contradiction.

Finally, where is the proof that density effects (ie causes) space-time?

Neutron stars have solar masses around 1.3 -2.1 where as the progenitor stars before becoming supernovas are 15 solar masses or more.

Neutron stars can form gravitational lens due to space time curving. 15 solar mass stars do not. This is because the density of Neutron stars is very much greater.

Steven

[sjastro]

Originally Posted by NotPrinceHamlet

So that website I linked to that asserts is just plain wrong?

It looks like the author is trying to get a grip on the curve by basically stating that the curve exists into the fourth dimension and the projection of the four dimensional space onto the 3rd dimension is the flat space we all know. I think that this is incorrect (happy to be shown I'm wrong though - because its nice and easy to understand!)

The trouble with making time a spatial dimension (or like adding a fourth axis), is that a particle does not have the same degree of freedom in moving in the tx, ty, tz planes as in the spatial planes without the possibility of violating properties such as cause and effect, exceeding the speed of light etc.

The geometrical interpretation of space time is given by Minkowski diagrams. There is a lot of stuff on the Internet about this.

Steven