Nonsensical representation of gravity

[Simon Bridge]

Almost: the Newtonian trajectory is the projection of the 4D geodesic onto 3D space.

Also see the pic in Wikipedia's Spacetime article:

Two-dimensional analogy of spacetime distortion.
Matter changes the geometry of spacetime, this (curved) geometry being interpreted as gravity.
White lines do not represent the curvature of space but instead represent the coordinate
system imposed on the curved spacetime, which would be rectilinear in a flat spacetime.
... this would be a careful description of what the rubber-sheet models are trying to represent.

And yep - to come up with a representation of space-time curvature in 3D space that preserves important aspects of the GR treatment while being accessible to the layman or to resign ourselves to an endless repetition of the same old flawed artistic renderings and deal with the confusions that are so generated ... that is the question.

I tell you: Shakespear had it easy!

 

[Jimmy Snyder]

I'm trying to figure out how to graph the geodesics. For a mass, the exact paths would depend on the speed so this kind of 3-space sampling wouldn't be useful. Plotting the light-paths may be illustrative ... or maybe graphing the overall curvature at different points? Jimmy - help?

I think a grid of geodesics is best. The reason I say that is that geodesics are the straight lines in curved space and hence show the curvature of space. There is a formula for the geodesics, I don't know what it is though. With such a formula, you should be able to map them easily. The curvature of space is 4 dimensional curvature. For this reason, you would need to have several images for various speeds. You might consider making a movie with the first frame being the geodesics for low speed and the last frame being for light speed. thus using time as the 4th dimension. A novel idea, but I'm not sure it wouldn't be misleading.

Assume that the mass of the body that is traveling along the geodesic is small. The mass of the gravitating body also has an effect on the geodesics. In the case of a black hole, some of the geodesics probably have no counterparts in less exotic fields. You might consider making a movie of light geodesics starting with our sun and gradually increasing density up to a black hole. This might be a better idea than the movie I suggested above.

 

[Simon Bridge]

Let me know when you find an equation in a form I can use.
If I use those for a very small test mass won't that just give me the light-lines?

I'm also going to need a better renderer - it seems that octave now uses an OpenGL renderer and devs have yet to implement an alpha-mapping for it. Something I'm going to need for meshes with holes in them.

(may be possible with patches)
But we seem to have moved away from the original question and into how to do different representations - constructive but I think it may be for another thread.
I posted by big graphic because it seemed unfair to be telling people to come up with their own without doing it myself.

I suspect the original question has been answered.

 

[#Thomas#]

That is the exact opposite of what I have been saying. The path that an imaginary object would take while passing through the field is called a geodesic. I want the lines to represent geodesics.

It would seem we lack consences here. Although I respect your preferences in this matter, there must be way to represent an image to the enthusiast without causing too much confusion as it did in me, but on the other hand this could only be me, becuase in my language words such a "geodesics" can translate into multiple meanings which could all apply to this topic... i need to loose my habbit of allways using the most commonly used translation and consider all alternatives.

 

[Jimmy Snyder]

If I use those for a very small test mass won't that just give me the light-lines?

No, the geodesics do not depend on the test mass as long as that mass is small. It only depends on the speed. However, no test mass can travel at light speed.

To illustrate this dependency, remember that when Eddington photographed the eclipse in 1919, he found that starlight had been deflected 1.73 seconds of arc from their non-deflected positions. This indicates a very slight curvature for the geodesics of light (travelling, of course, at light speed) grazing the surface of the sun. However, the Earth itself also travels (at much less speed) in a geodesic 93 million miles from the surface of the sun and the curvature is so great as to bend the path into an ellipse. Also, consider the so-called asteroid belt where millions of fragments with a wide range of masses are all traveling basically along the same geodesic. That is because there is not such a wide range of speeds among the asteroids in the belt. We know that their range of speeds is small because if not then they would be colliding like crazy and showering the Earth with dinosaur killers on a regular basis.

I do have a pair of equations for geodesics, but I don't completely understand it. It comes from the book "The Einstein Theory of Relativity" by Lieber and Lieber. It also give the Newtonian equations for reference. Note that they give the geodesic in terms of polar coordinates r and \phi. This is compatible with what I have said before, namely, that you can draw a strictly two dimensional image. Spherical symmetry will take care of the rest. Also notice that the mass of the gravitating body, m, is used, but not the test mass.

Newton:
\frac{d^2u}{d\phi^2} + u = \frac{m}{h^2}
r^2\frac{d\phi}{dt} = h

Einstein:
\frac{d^2u}{d\phi^2} + u = \frac{m}{h^2} + 3mu^2
r^2\frac{d\phi}{ds} = h

Where h is a constant of integration, r is the radius, \phi is the angle of rotation, and u = 1/r. The authors point out that there is an extra term 3mu^2 in the first Einstein equation and that differentiation is with respect to proper time s for Einstein, but with repect to time t for Newton.

 

[#Thomas#]

No, the geodesics do not depend on the test mass as long as that mass is small. It only depends on the speed. However, no test mass can travel at light speed.

Actually recent experiments suggest that neutrinos can, in fact, move faster than light! Which was, in my humple oppinion, and adequate kick in the butt for those complacent CERN scientists... but this isn't what this topic is about...

Here, have a look at this:
http://www.nasa.gov/mpg/146898main_viz_shiftingall_21.320x240.mpg

Surely NASA would satisfy your strict standards!

 

[Simon Bridge]

Actually recent experiments suggest that neutrinos can, in fact, move faster than light!

Actually, recent experiments suggest that we need to look much harder at how neutrino experiments are done. But you are correct, there are other threads for that. In this thread, we are discussing traditional representations of Einstein space-time. After all, we could try for a representation using string theory too but that would also be off topic.

Surely NASA would satisfy your strict standards!

You mean the guys who crash spacecraft ? They also use misleading images as part of PR. (Hmmm ... can't seem to play the mpg.)

in my language words such a "geodesics" can translate into multiple meanings which could all apply to this topic

... same in English: context is everything. "Geodesic" is a term of the art[1] in general relativity where we have to describe paths in 4 dimensions. There is a lot of this in physics - "work" has several meanings too.

When people put up these pics they usually make some glib comment about showing the "shape of space time" or similar without actually defining their terms. Whenever you see this, you are not expected to take it seriously.

NASA, for example, produce a range of documents with a wide range of rigour. You will see the rubber-sheet pics where they are talking to Joe Public, a mass of equations when they are talking to scientists, and brightly colored drawings with baby-talk and smiley-faces for talking to Congressmen.

------------------------------------------
[1] i.e. jargon - this is a useful phrase I got off a lawyer.

 

[Simon Bridge]

Here's another, related, artistic impression that has misleading features. This time about the shape of the Universe.
https://www.physicsforums.com/showthread.php?t=298279
... shoud help illustrate what these sorts of drawings are supposed to be doing.

 

[#Thomas#]

Actually
You mean the guys who crash spacecraft ? They also use misleading images as part of PR. (Hmmm ... can't seem to play the mpg.)

Actually this isn't some artist's expression, this is crunch time of a supercomputer calculating how gravity would put 2 black holes together and release gravity wawes.