How to imagine distorted space

[Neitrino]

Gents,

I'am trying to imagine/visualize how does the distorted Space look like.

But geometry of Schwarzschild solution reminds us the Geometry on the sphere. So if in Schwarzschild geometry I want to look at a triangle lying in an equatorial plane, I just look at it on the sphere. so That's why in textbooks distortion of space-time is represented as a plastic surface pushed by finger.
If to put a BH on a flat plane, I was thinking that distorted space time will look like in images shown bellow, but geometry on such "plane" is not the same as Schwarzschild's one .(so was I wrong?) So it seems it is mandatory to think about distortion in higher dimensional space when thinking about gravity and it's not possible to imagine how does gravity distort two dimensional plane without deforming in 3rd dimension?

 

[DaveC426913]

No, it bends the other way around. See attached pics.

Think about what happens when space is distorted by gravity i.e. gravitational lensing. An object directly behind a massive object would normally not be seen - the light rays emanating from it would be blocked by the object in front. (Top pic in diagram.)

But the distorting effect of the massive object in front causes light rays to bend around it, so that a light ray that was behing it is now actually pointing right at us. (bottom pic in diagram)

So, the net effect of distorted 3-D is that it "grows" what's at the centre, as in the brick wall image.

 

[JesseM]

Think about what happens when space is distorted by gravity i.e. gravitational lensing. An object directly behind a massive object would normally not be seen - the light rays emanating from it would be blocked by the object in front. (Top pic in diagram.)

But the distorting effect of the massive object in front causes light rays to bend around it, so that a light ray that was behing it is now actually pointing right at us. (bottom pic in diagram)

So, the net effect of distorted 3-D is that it "grows" what's at the centre, as in the brick wall image.

What's the exact connection between the distortion of space in the Schwarzschild solution and the path taken by light rays? In general relativity, objects don't take the shortest path through curved space, they take the path through curved spacetime with the greatest proper time (which should be zero in the case of a path between events with a lightlike separation). But in flat spacetime, the longest path through spacetime is also the shortest path through space...is the same true for other static solutions to the GR equations, like the Schwarzschild solution? Or would it be wrong to explain gravitational lensing in terms of light rays taking the shortest path through the curved space around a black hole?

 

[Mortimer]

Since in the Schwarzschild geometry the radial distance is $$dR=\frac{dr}{\sqrt{1-2m/r}}$$, an observer at infinity would see a large $$dR$$ being cramped into a small $$dr$$. The tangential distance however is equal to that of a normal Euclidean sphere. So an object near $$2m$$ would seem crushed in the radial direction only.

 

[pervect]

I'd suggest taking a look at

http://casa.colorado.edu/~ajsh/schwp.html

and more specifically the description of the embedding diagram

http://casa.colorado.edu/~ajsh/schwp.html

Embedding diagram

The Schwarzschild geometry is illustrated in the embedding diagram at the top of the page, which shows a 2-dimensional representation of the 3-dimensional spatial geometry at a particular instant of universal time t. One should imagine that objects are confined to move only on the 2-dimensional surface. Each circle actually represents a sphere, of circumference 2 pi r. According to the Schwarzschild metric, the proper radial distance, the actual distance measured by an observer at rest at radius r, between two spheres separated by an interval dr of circumferential radius r is (1 - rs/r)-1/2 dr, which is larger than the radial interval dr expected in a flat, Euclidean geometry. Thus the geometry is `stretched' in the radial direction, as shown in the embedding diagram.

Outside the horizon, the lines in the embedding diagram are `space-like': they would be measured by some actual observer (in this case an observer at rest in the Schwarzschild geometry) as being intervals of space at some instant of the observer's proper time...

 

[Neitrino]

Gents,
To imagine the gravity in two dimensional space (plane), one pushes it (this palstic plane) and deforms it into 3rd dimension (like trivial examples in any textbook on GR).

Is it possible/how to imagine gravity working in plane... and not to deform it in 3rd dimension?

 

[Zanket]

Sure. The use of the 3rd dimension in the "examples in any textbook on GR" is just a way to impart the information (the degree to which gravity is working) about each coordinate in the plane. That info could be alternatively imparted by the use of different colors, like how temperature is imparted on a thermal map. Or by the use of numbers, like how elevation is imparted on a road map. Or...you get the idea.

 

[kleinwolf]

Yes...moreover you cannot anyhow create a space-time with negative metric coefficients by curving a higher dimensional space (with euclidean metric)...

But there is a nice way how to imagine a distorded space (R^3) (not space-time)...

A distorded space is a 3-dimensional manifold embedded in 4 dimensional space...however I cannot imagine 4-dimensional space...

But a kind of "inverse" of a distorded space can be seen as a time-dependent transformation (distortion) of usual space...(hence an animation of space transformation if you want).

 

[Neitrino]

So if a can imagine gravity working in plane... and not to deform it in 3rd dimension (just to apply may be some streching/compressing of my grid elements) can i say that my PICTURE IS OK? (attached bellow)

In this picture I tried to represent space as a grid beeing distorted (this distortion is obtained when gravitational centre attracts=distorts inward that grid )gravitational force.

Gents,

I'm confused with this and i badly need your advises.
Thks

 

[Zanket]

Can't speak for accuracy, but in general that looks fine to me. Looks like a top-down view of the 3D "examples in any textbook on GR". The degree of distortion of the grid at any given point represents the 3rd dimension value of the 3D example at that point.

If you want to improve it, I suggest using polar coordinates and a color or grayscale gradient that changes radially (e.g. darkens toward the center). The 3D example is probably still better but is misleading unless it's carefully explained.

 

[Boffin]

How about viewing a distorted space continuum as a solid object with variable densities? So the density around large objects such as the suns and planets would be greater, and inbetween lesser?

Or what about thinking of the space continuum as white and the closer one's gets to large objects the darker grey the space continuum becomes?

These are all just imperfect representations of the real thing. Anyone have any other ideas? Neitrino asked the original question awhile ago but I'd be interested in more ideas.