Spacetime Geodesics at Sea Level & Zoomed Out

[bland]

TL;DR Summary

Trying to visualise the spacetime field at Earth falling body level in the context of a larger grid.

I suppose that that a spacetime geodesic of an object falling on Earth would a appear as straight line. But what I'd like to see is a whole bunch of relevant geodesics that would represent falling bodies all around the Earth such that one could zoom out and so see these straight line geodesics in the context of the whole Earth.

Maybe someone has done this somewhere already. But I can't find it.

 

[Ibix]

From your description I'm not sure what it is you actually want. Generally, geodesics are structures in curved 4d spacetime, and there isn't a straightforward way to represent that.

You could restrict yourself to just motion in one plane, which is enough given that the Earth's gravitational field is more or less spherically symmetric. But there's still the problem of curvature, meaning that what geodesics look like will depend on the coordinates you choose to use. In the obvious choice - Schwarzschild coordinates - the spatial parts will basically be the familiar conic sections of Newtonian gravity. You'd need a much stronger gravitational field to see any visible relativistic effects.

Note that geodesics will not look like straight lines in the above approach. I don't think any approach will do that, since geodesics may cross multiple times and you can't represent that with straight lines on a Euclidean plane.

 

[bland]

Yeah I get it's tricky to visualise probably due to the 4 dimension represented on 2d or faux 3d. However to make my idea that I'm wanting to see clearer. It's as if someone wanted to see the local field of a magnet, well you'd use iron filings on paper and maybe photograph that and then say clean up the lines into a clean vector but we all know what that looks like.

And the gravitation field must be able to be represented somehow in an analogous way rather than just the single line I've seen around like this or the Earth on a trampoline thing. Both unsatisfying.

I'm more like Faraday than Maxwell in this regard.

Edit: I found this mathematical explanation that would probably do if I could understand any of it. Right at the bottom though is what appears to be a diagram that if expanded on a bit looks like it would show me what I want to see. But I'm not sure.

 

[Ibix]

Magnetic field lines aren't anything to do with geodesics, not even by analogy. They are the integral curves of the field. It's trivial to produce them for Newtonian gravity, but not very exciting for the Earth - the field lines just point radially inwards. However, I'm not sure that there is an equivalent construction for general relativistic gravity. It's a tensor field, not a vector field, and it provides the notions of space and time that underlie the idea of drawing lines in space.

There is Flamm's paraboloid (you'll need to scroll down). This is a surface which has the same relationship between radial and tangential distance as space (defined by hovering observers) around a spherically symmetric mass. It doesn't show the curvature in timelike planes, which is what's actually important for understanding gravitational motion near Earth.

You might also want to check out this thread, which includes several approaches to visualisation.

 

[Dale]

Summary: Trying to visualise the spacetime field at Earth falling body level in the context of a larger grid.

But what I'd like to see is a whole bunch of relevant geodesics that would represent falling bodies all around the Earth such that one could zoom out and so see these straight line geodesics in the context of the whole Earth.

That isn’t possible to draw. If you want to draw a curved manifold it can only be 2D. What you are asking for would require at least 3D. You could draw a surface representing the r and t directions and show the geodesics in that space, but that is it. You cannot add either theta or phi.

Forum member @A.T. has some excellent diagrams of this sort showing geodesics in a plot on a curved surface representing t and r. This allows you to understand geodesics along a single radial line.