Geodesics - Some help, please.

[RCopernicus]

We've all seen an image similar to this one:

This is displaying the projection of GR Geodesics onto 3-D space (well, 2D in the picture). I'm still working my way through the General Relativity texts, so I'm not yet able to do the calculation on my own. Can anyone give me a formula that I can poke into a simulation that takes the mass an input parameter and gives me the geodesic coordinate system projected onto the 2-D space? I'm basically looking to draw the concentric circles in the above picture (given a mass).

 

[jedishrfu]

Try this:

http://en.wikipedia.org/wiki/Gravity_well

 

[pervect]

We've all seen an image similar to this one:

<snip>

This is displaying the projection of GR Geodesics onto 3-D space (well, 2D in the picture). I'm still working my way through the General Relativity texts, so I'm not yet able to do the calculation on my own. Can anyone give me a formula that I can poke into a simulation that takes the mass an input parameter and gives me the geodesic coordinate system projected onto the 2-D space? I'm basically looking to draw the concentric circles in the above picture (given a mass).

I think what you are looking for is called an embedding diagram, in particular the diagram you show embeds the 2 dimensional equatorial plane of a massive body in the curved Schwarzschild space-time onto a 3d surface, as a visual aid to understanding the curved spatial geometry of the equatorial (r, theta) plane.

The exact formula can be found be found in the discussion of "Flamm's paraboliod" in the wiki article about the Schwarzschild metric, i.e. http://en.wikipedia.org/w/index.php?title=Schwarzschild_metric&oldid=628392899#Flamm.27s_paraboloid

 

[RCopernicus]

I think what you are looking for is called an embedding diagram, in particular the diagram you show embeds the 2 dimensional equatorial plane of a massive body in the curved Schwarzschild space-time onto a 3d surface, as a visual aid to understanding the curved spatial geometry of the equatorial (r, theta) plane.

The exact formula can be found be found in the discussion of "Flamm's paraboliod" in the wiki article about the Schwarzschild metric, i.e. http://en.wikipedia.org/w/index.php?title=Schwarzschild_metric&oldid=628392899#Flamm.27s_paraboloid

Thanks for the post. Very helpful.
Here's a follow up question: In the diagram above, we see a star that appears to be sitting in some sort of well. The coordinate system of space-time appears to curve into this well making it appear like the spaceship is sitting at a higher potential than the star at the bottom of the well. What isn't obvious is what dimension is pointing into the well? My first reflect was 'time', but upon more reflection, I think it's 'velocity' (inasmuch as the velocity is a projection of the geodesics on a 2-D surface). Is that a fair interpretation?

 

[pervect]

Thanks for the post. Very helpful.
Here's a follow up question: In the diagram above, we see a star that appears to be sitting in some sort of well. The coordinate system of space-time appears to curve into this well making it appear like the spaceship is sitting at a higher potential than the star at the bottom of the well. What isn't obvious is what dimension is pointing into the well? My first reflect was 'time', but upon more reflection, I think it's 'velocity' (inasmuch as the velocity is a projection of the geodesics on a 2-D surface). Is that a fair interpretation?

As far as I know the vertical dimension has no physical significance, and it's a matter of convention whether it's a well or a mountain. The diagram is a visual aid, really, it illustrates a hypothetical 3d surface that gives the correct distances between pairs of points in space. Note that the diagram presupposes a particular slicing of space-time (because it suppresses the t coordinate). The diagram illustrates that given this particular slice, space is curved.