How to Incorporate Metric Coefficients into Worldline Plots in Curved Spacetime?

[exmarine]

TL;DR Summary

How does one plot the world-lines for curved spacetime? That is such a useful tool for fixing the concepts for the Minkowski case that I would like to also use it to understand other cases.

Since it is nonlinear, the 3 leg lengths would be limited to differentials?

But how would the metric coefficients be incorporated into those leg lengths?

It seems like the leg differential lengths would have to vary inversely with the magnitudes of the metric coefficients? For example, near the horizon for the Schwarzschild case as the g_tt coefficient approaches zero, the component time differentials get very long as the proper clock slows down and counts fewer seconds? The radial differentials would get very short as g_rr gets very large and even light cannot get away?

Thanks.

 

[Ibix]

It depends which coordinate system you choose to use and how you choose to represent those coordinates. See the plethora of methodologies for drawing maps of the Earth - this problem is analogous.

That said, if you've got a symmetric spacetime you can often suppress a dimension or two and come up with something. Kruskal diagrams are a good map around Schwarzschild black holes, and Penrose diagrams are more abstract but can be drawn for more spacetimes.

 

[robphy]

Check out
https://www.spacetimetravel.org/sectormodels1/sectormodels1.html

I met these researchers at a conference last year. I’m looking into whether my light-clock diamonds can be incorporated into these sector models.

In the simplest curved cases, you could look at the de Sitter spacetimes using hyperboloids. My students were studying the clock effect on them.

 

[pervect]

Consider the curved surface of the earth. You can represent it on a flat plane with a map, but it won't be a map to scale.

The mathematical process of drawing maps of the curved surface of the Earth on a flat piece of paper is known as projection. There are several widely used schemes, for example Mercator projection.

If you use a globe, rather than a flat piece of paper, though, you can draw a map of the Earth on the surface of the globe (i.e. on a sphere) that is to scale.

Equivalently, there is no difficulty in drawing a space-time diagram in a curved space-time on a flat piece of paper. But it won't be to scale.

A certain amount of insight into GR can be gained by drawing space-time diagrams on curved surfaces, such as the sphere, though the technique is limited. I believe it is not generally possible to crate a curved 3d surface to draw a 2d space-time graph on for a general 1+1 space-time geometry "to scale", but certain special cases can be illustrated that way.

The metric of GR can be thought of as a tool to get actual distances and times from the distored map imposed on a curved space-time by a particular choice of coordinates.

 

[Ibix]

Check out
https://www.spacetimetravel.org/sectormodels1/sectormodels1.html

Marolf's paper, referenced by Zahn and Kraus, also looks interesting. In particular, figure 16 is a rather nice visualisation of the Schwarzschild coordinates breaking at the horizon. Edit: actually it's the wormhole in maximally extended Schwarzschild spacetime. Still a nice diagram...

 

[Nugatory]

A certain amount of insight into GR can be gained by drawing space-time diagrams on curved surfaces, such as the sphere, though the technique is limited.

For an example, @exmarine might want to take a look at Flamm's parabaloid (Google finds some good links, including the wikipedia article on the Schwarzschild spacetime).

 

[PAllen]

Look, world lines for SR drawn on paper using Minkowski coordinates already need to be interpreted using the metric. Visually longer intervals often have shorter invariant interval. There is really no difference for arbitrary world line in arbitrary coordinates for an arbitrary metric. You can just place orthogonal axes on a flat plane (even if the actual coordinates are not 4-orthogonal), draw world lines, then compute (using the metric expressed in said coordinates) where to place marks for proper time. For qualitative understanding, it is often useful to draw light cones, which can be read from any metric at any point trivially by setting the line element to zero.

 

[A.T.]

How does one plot the world-lines for curved spacetime? That is such a useful tool for fixing the concepts for the Minkowski case that I would like to also use it to understand other cases.

Check out the articles by Rickard Jonsson (the two theses at the very bottom provide a good overview):
http://www.relativitet.se/articles.html