Visualising curved space

[NotPrinceHamlet]

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-Tensor-Calculus-Relativity-Cosmology/dp/0486425401/ref=wl_it_dp?ie=UTF8&coliid=IIGXSNHN0Q7CK&colid=24ZKNZ3CHQBTP

[omnivorr]

stare into M C Escher pictures for long periods ? :)

[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?

[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.



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]

:) 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]

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]

"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]

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

[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.

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]

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]

[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]

"....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]

"....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]

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]

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!)

[sjastro]

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.




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

Regards

Steven

[Archy]

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]

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]

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.



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]

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

[xelasnave]

I found this thread very interesting.

It seems silly for me to offer a comment given my lack of grip on the concept space time... I realise there are specific math implications... however the question was...........

"Has anyone managed to find a mental trick to assist in visualising a curved space?"

So I offer how I "see" space in a geometric visualisation..up and down to me are irrelevant so the problem with the ant is lost on me...

Still I feel space can be looked at like a barometric map..but in 3d.. start with a visualisation of a 3d grid...make the cubes any size... but all the same size..this is unbent space... as you get closer to mass the grid becomes smaller...making greater gravity in effect...but as everything is moving none of the grid lines remain straight and across the expanses of space would probably resemble more smoke drifting in a light breeze... the complexity of the various curvatures would see one having trouble then holding to a regular grid but one that is distorted greatly...

Also for who ever can field this one....IF Earth does not curve space how does GR give us our gravity???

alex
alex

[sjastro]

Good question Alex.

In the absence of mass (and gravity), space is perfectly flat.

The strength of the gravitational field as a function of mass density determines the degree of space time curving.

The Earth's field strength is not strong enough for any significant space time curving.

It's no coincidence that for weak gravitational fields GR equations reduce to Newtonian equations. Hence Newtonian physics becomes a very good approximation for GR for weak fields.

Steven

[Archy]

Not to the point: the contradiction in your statements are self evident

[Archy]

Sure, I don't ague that mass doesn't affect space-time, I ask again how does it effect space-time.

[Archy]

The Earth is massive enough to curve space-time: if it were not, there would be no gravity.
Since we do experience gravity, my first statement "We all live in curved space so we don't notice it" has to be modified to:
We all live in curved space time and experience it as gravity.

[sjastro]

Ever heard of Einstein's principle of equivalence on which GR is founded? Evidently not.

We experience gravity in the gravitational field.
We can also experience the same gravity by being accelerated at g in a gravity free field. A gravity free field is flat space.

Hence we can experience gravity in flat space which contradicts your post.

It is how curved space time acts on bodies and light that would otherwise be travelling in a straight line in flat space that reveals it's presence.

As mentioned in a previous thread the perhelion advance of a planet is one way. The advance of the moons orbit is explained by classical means (ie tides) and is not due to space time curving.

The Earth's field is not strong enough for the effects of space time curving
to be observed.
The moons orbit is satisfactorily explained by Newtonian physics.

They are the facts!

On a final note this is my last response to your posts. It's quite obvious where this thread is heading.

[Archy]

Re: "Ever heard of Einstein's principle of equivalence on which GR is founded": Yes I have: the Wikipedia says: "In the physics (http://en.wikipedia.org/wiki/Physics) of relativity (http://en.wikipedia.org/wiki/Theory_of_relativity), the equivalence principle refers to several related concepts dealing with the equivalence of gravitational (http://en.wikipedia.org/wiki/Mass#Gravitational_mass) and inertial mass (http://en.wikipedia.org/wiki/Mass#Inertial_mass), and to Albert Einstein's (http://en.wikipedia.org/wiki/Albert_Einstein) assertion that the gravitational "force" as experienced locally while standing on a massive body (such as the Earth) is actually the same as the pseudo-force (http://en.wikipedia.org/wiki/Fictitious_force) experienced by an observer in a non-inertial (http://en.wikipedia.org/wiki/Inertial_frame_of_reference) (accelerated) frame of reference."
It might be noted that it is apparent from your posts in this thread that you do not agree with Wikipedia that the Earth is a massive body.

Re: "We experience gravity in the gravitational field.
We can also experience the same gravity by being accelerated at g in a gravity free field. A gravity free field is flat space.

Hence we can experience gravity in flat space which contradicts your post."

The last two statements are contradictory.

Re: "The Earth's field is not strong enough for the effects of space time curving
to be observed." Is incorrect: the GPS system has a correction algorithm to correct for changes in space-time.

[Archy]

To Steven
On a final note: You say space-time is affected by density, that the earth does not curve space-time and, I infer, that the precession of mercury shows the sun's effect on space-time.
How can you explain these views when the sun's density is 1.4 g/cm3 (Namowitz, Samuel N. Heath Earth Science. Canada: Heath, 1994: 379. "… Sun's density is 1.4 times that of water.)
and the Earth's density is 5.519 g/cm3 (Neff, Robert F. & Zitewitz, Paul W. Physics, Principles and Problems. New York: Glencoe, 1995: 159. "Mass of the Earth 5.979 × 1024 kg Radius of the Earth 6.3713 × 103 km)

The sun curves space-times more than the earth because it is more massive, not because it is denser.

[sjastro]

A non inertial frame of reference is accelerated in gravity free space.
The pseudo force is your gravity.
It simply confirms that the effects of gravity can occur in flat space for the reasons I have already given.



Changes to space-time doesn't make any sense.

Corrections to GPS systems involve the synchronization of Earth clocks and GPS satellite clocks.

GR predicts a clock at a higher gravitational potential (ie a clock on the Earths surface) will run slower than one at a lower potential (satellite clock).

This is an example of gravitational red shift which has nothing to do with the geometry of space-time. In fact the original concept of gravitational red shift was devised using SR which assumes that space is flat.

[sjastro]

The answer is obvious.
The Sun's density of 1.4 g/cm3 is an average density.

The density of the core is 150 g/cm3 which is why space time curves at the Sun. It has nothing to do with mass.

Stars of 15 solar masses or more require the thermonuclear reactions to keep their cores from collapsing under gravity.

If the core does collapse a supernova occurs and the core forms a super dense neutron star or black hole with the corresponding space time curvature.

The mass of a neutron star is much less than the original star but the density is millions of times greater than the average density of the original star.

[Archy]

An acceleration produces a real force, not a pseudo force.
Mass distorts space see: Slide 17 www.physics.ucsd.edu/~tmurphy/phys10/lectures/15_gen_rel.ppt. from which we see the statement "Mass tells space-time how to curve and Space tells mass how to move....Gravity is a property of the geometry of spacetime"
Since frequency is a function of inverse time (eg beats per second) a change in frequency, which is what is implied by a redshift, requires a change in time (since there are the same number of beats), which agrees with what I have said that the mass of the earth causes a change in space-time
Re the GPS see: Pogge, Richard W., “Real-World Relativity: The GPS Navigation System” where in the seventh para graph there is the statement:
"Further, the satellites are in orbits high above the Earth, where the curvature of spacetime due to the Earth's mass is less than it is at the Earth's surface. A prediction of General Relativity (http://www.astronomy.ohio-state.edu/%7Epogge/Ast162/Unit5/gr.html) is that clocks closer to a massive object will seem to tick more slowly than those located further away (see the Black Holes lecture (http://www.astronomy.ohio-state.edu/%7Epogge/Ast162/Unit3/blackholes.html)). As such, when viewed from the surface of the Earth, the clocks on the satellites appear to be ticking faster than identical clocks on the ground. A calculation using General Relativity predicts that the clocks in each GPS satellite should get ahead of ground-based clocks by 45 microseconds per day." (Bold and underlining are mine.)You, of course, are free to believe that the earth does not distort spacetime because it is not dense enough, or whatever else you wish.

[sjastro]

You change the subject matter when it's convenient for you.
This has nothing to do with the context of the equivalence principle.

"All fictitious forces are proportional to the mass of the object upon which they act, which is also true for gravity (http://en.wikipedia.org/wiki/Gravity).[17] (http://en.wikipedia.org/wiki/Fictitious_force#cite_note-16) This led Albert Einstein (http://en.wikipedia.org/wiki/Albert_Einstein) to wonder whether gravity was a fictitious force as well. He noted that a freefalling (http://en.wikipedia.org/wiki/Freefall) observer in a closed box would not be able to detect the force of gravity; hence, freefalling reference frames are equivalent to an inertial reference frame (the equivalence principle (http://en.wikipedia.org/wiki/Equivalence_principle)). Following up on this insight, Einstein was able to formulate a theory with gravity as a fictitious force; attributing the apparent acceleration of gravity to the curvature (http://en.wikipedia.org/wiki/Curvature) of spacetime (http://en.wikipedia.org/wiki/Spacetime). This idea underlies Einstein's theory of general relativity (http://en.wikipedia.org/wiki/General_relativity)."




And what do you think is the property of mass that causes this, it is density. It's why low mass high density Neutron stars cause space time to distort.



You haven't explained anything. There is no relationship between frequency and geometry.



Red shift (or time dilation due to gravity) is independant of space time geometry. It can occur in flat space or curved space and is dependant on the gravitational potential differences between earth and satellite clocks.

As I have already mentioned gravitational redshift has been mathematically formulated using flat space (and the equivalence principle).




I certainly did not state the Earth does not distort space time, but on the basis of observation the effects are not apparent.
And on that note this thread is concluded.

[Jay-qu]

For anyone wanting an actual visualisation of curved space here are some links with videos created by my General relativity lecturer at uni:

http://users.monash.edu.au/~leo/research/movies/gr/index.html
http://users.monash.edu.au/~leo/research/movies/sr/index.html

[Archy]

Rubbish: I have stayed on the subject.
If you think gravity is fictitious: go jump off a tall building.

Density is not a property of mass, but mass is a property of density.
Where do you get the idea that a neutron star has low mass: according to NASA a neutron star has the mass (http://imagine.gsfc.nasa.gov/docs/dict_jp.html#mass) of about 1.4 times that of our Sun.
Despite what you have asserted, the Earth does change the geometry of space-time. Certainly not to the extent of a neutron star, nor to the extent that our sun does, but nevertheless by a measurable and measured amount as I have already illustrated.

Get off: who are you to say the thread is concluded

[xelasnave]

Will the Gravity B probe mission show curvature or is it limited to showing frame dragging? and I cant recall the variance they look for to demonstrate framedragging but I do remember it being a very small number they expected to be working with... I had all this on my dead lappy but without it I can not be more specific unless I read the available material from the net...however one would think one could not show frame dragging unless you did not have a demonstratable curvature of space.

This is a wonderful thread please dont end it too early.
We must remember that strangly the subject of gravity can get folk going.. one would not think this could be the case but it has been my experience... and so we proceed knowing the subject can be as volitile as religion or politics.. and I am not sure that gravity is entirely issolated from either of those considerations.

alex

[sjastro]

Alex,

The gravity B probe will measure the geodetic effect (space-time curvature) and frame dragging.

The geodesic effect is one part in 10000 deviation from flat space.
The frame dragging effect is one part in a few trillion.

Putting the geodesic effect in perpsective, it doesn't require us to jettison Newtonian mechanics, space around Earth is still essentially flat.



I'm no longer contributing to the thread that shouldn't stop anyone else.

Regards

Steven

[xelasnave]

Thank you for your reply Steven
alex

[Archy]

This link may be of use re frame dragging: http://www.nasa.gov/vision/earth/lookingatearth/earth_drag.html

[xelasnave]

Thanks for the link George
alex