Trampoline analogy for gravity

[arunshanker]

The trampoline analogy tries to explain gravity in terms of space time curvature
the orbit of objects around a massive object can be understood, but what about centre of gravity of the massive object, the images of trampoline is generally shown as seen from top where the massive object is making a curvature downwards when looking from the top, if that is the case how to explain the gravitational attraction of the objects on the surface of the massive object at the bottom side of the curvature

 

[A.T.]

The trampoline analogy tries to explain gravity in terms of space time curvature

Actually it just shows space curvature. There is no time dimension on that sheet.

the images of trampoline is generally shown as seen from top where the massive object is making a curvature downwards when looking from the top,

There is no down or top in space. The 4D space-time geometry is spherically symmetrical, but the trampoline just shows a selected 2D spatial slice of it.

if that is the case how to explain the gravitational attraction...

It doesn't explain attraction at all, because it doesn’t include the time dimension. These are better visualizations of the attraction:

 

[wabbit]

That second video is just the simplest, most concise, and accurate illustration I've seen of spacetime curvature in GR - much better then any rubber sheet analogy. Very nice.

 

[DrGreg]

That second video is just the simplest, most concise, and accurate illustration I've seen of spacetime curvature in GR - much better then any rubber sheet analogy. Very nice.

To be pedantic, that video doesn't quite cover spacetime curvature. It covers steps A, B₁ and B₂ below, but step C is needed to cover spacetime curvature.

This is my own non-animated way of looking at it:

• A. Two inertial particles, at rest relative to each other, in flat spacetime (i.e. no gravity), shown with inertial coordinates. Drawn as a red distance-time graph on a flat piece of paper with blue gridlines.
• B₁. The same particles in the same flat spacetime, but shown with non-inertial coordinates. Drawn as the same distance-time graph on an identical flat piece of paper except it has different gridlines.
  B₂. Take the flat piece of paper depicted in B₁, cut out the grid with some scissors, and wrap it round a cone. Nothing within the intrinsic geometry of the paper has changed by doing this, so B₂ shows exactly the same thing as B₁, just presented in a different way, showing how the red lines could be perceived as looking "curved" against a "straight" grid.
  
• C. Two free-falling particles, initially at rest relative to each other, in curved spacetime (i.e. with gravity), shown with non-inertial coordinates. This cannot be drawn to scale on a flat piece of paper; you have to draw it on a curved surface instead. Note how C looks rather similar to B₂. This is the equivalence principle in action: if you zoomed in very close to B₂ and C, you wouldn't notice any difference between them.

Note the diagrams above aren't entirely accurate because they are drawn with a locally-Euclidean geometry, when really they ought to be drawn with a locally-Lorentzian geometry. I've drawn it this way as an analogy to help visualise the concepts.

 

[wabbit]

True, but it still illustrates it : ) and it's very easy to grasp. I guess the second series you show would be "part II" to that first illustration's "part I".
Are these all from one website ?

Edit : and as to something equivalent with true Lorentzian metric, I don't recall seeing one, is that even possible ? If we look at a picture of a surface we are always going to interpret it as euclidian it would seem ?Edit : removed incorrect mention of 1D/2D
"Part I" refers to video # 2 in post # 2
"Part II" refers to pictures A B C in post # 4

 

[A.T.]

True, but it still illustrates it : ) and it's very easy to grasp. I guess the second series you show would be "part II (2+1D)" to that first illustration's "part I (1+1D)".

The two videos don't show intrinsic curvature, which is related to tidal effects, because they are negligible on small scale. Here is an applet that shows the global picture:

http://www.adamtoons.de/physics/gravitation.swf

Edit : and as to something equivalent with true Lorentzian metric, I don't recall seeing one, is that even possible ? If we look at a picture of a surface we are always going to interpret it as euclidian it would seem?

For visualization purposes you can work with space-propertime diagrams, which are Euclidean.

 

[DrGreg]

I guess the second series you show would be "part II (2+1D)" to that first illustration's "part I (1+1D)".

Sorry, I'm not clear exactly what you mean by "I" and "II" in this context.

 

[wabbit]

Thanks - for some reason I can't open that video, (tablet now) will try on computer later.
But C above (post 4) does show tidal effect/geodesic divergence, as far as i could tell that's how it differs from B2.
Also I don't understand

space-propertime diagrams, which are Euclidean

the issue is that the metric has null cones / negative intervals, which means any euclidean representation is wrong in the same way, as "visual distance" cannot match "lorentzian distance" ? - not at all to say the euclidean diagrams aren't useful, more that they're the best we can hope for but they cannot be exactly correct.

 

[A.T.]

Also I don't understand

If proper-time is on the axis, then coordinate time is the Euclidean line element:

http://www.adamtoons.de/physics/relativity.swf
http://www.adamtoons.de/physics/twins.swf
http://www.relativitet.se/Webtheses/lic.pdf (Chapter 6)

 

[wabbit]

Oh Thanks, looks like some good reading to do now.

 

[DrGreg]

Sorry just part 1 and part 2, seeing the second series as a tad more advanced than the first

Sorry, I'm not clear what you mean by that, either, or what you mean by "series" in this context.

 

[wabbit]

Corrected that post # 5 above, thanks DrGreg for pointing this out.

 

[arunshanker]

Thanks AT, Wabbit and DrGreg