Acceleration in a space time diagram

[DiracPool]

I was trying to draw a space-time diagram representing two objects traveling through spacetime, me and the planet Earth. At first I thought I would just draw two straight lines heading upwards some distance, x, apart, seeing as my position relative to the center of the Earth would not be changing over time. I then realized, though, that I am constantly being accelerated outwards from the center of the Earth at 9.8 m/s. Therefore, my space-time diagram should show a curved trajectory to the right representing such acceleration.

So this is my quandary. Can anyone help me reconcile this? What does this space-time diagram look like?

 

[tommyxu3]

I think the acceleration will just impact the trajectory in the graph like it does in the original x-t diagram, for it's also a kind of illustration of its position with respect to the time.

 

[pervect]

The approach that immediately comes to mind is to using the space-time embedding diagram due to Marolf, http://arxiv.org/abs/gr-qc/9806123

There may be other approaches depending on what it is you want to portray.

Marolf's approach is to represent the space-time of a black hole (which is the space-time of a strongly gravitating body) via embedding the 2-d diagram as the surface of a particular 3d object. Then you could draw the appropriate (non-geodesic) lines for someone at rest on the surface of the Earth.

If you're not too concerned with some of the more subtle details, though, you can use a less complicated surface. AT has a nice video of the process, I'm having a bit of a problem tracking down the link for you at the moment though.

For some purposes you could even draw it the way you described as two parallel lines, but straight lines on such a diagram wouldn't represent geodesic paths. This is rather similar to the way that a horizontal line of constant latitude on the Earth doesn't represent the shortest distance between two points, which must be a great circle.

 

[PeterDonis]

What does this space-time diagram look like?

It depends on what you want to emphasize. The underlying issue here is that, in curved spacetime, there is no way to draw a spacetime diagram that has all the properties that you are used to from SR. In particular, you can't draw a diagram that has both of the following properties: (a) all straight worldlines are unaccelerated, and (b) distances between neighboring straight worldlines are constant.

Your initial thought, that we would just draw Earth's worldline as a straight line and yours as another straight line, is fine if what you want to preserve is property (b)--i.e., you want the diagram to reflect the fact that you are stationary with respect to the Earth. But, as you saw, that diagram lacks property (a)--your worldline is a straight line in the diagram, but it's accelerated.

You could also draw a diagram in which property (a) held, i.e., straight worldlines were unaccelerated. (Such a diagram actually could not cover the entire Earth, as I'll discuss in the next paragraph, but we'll ignore that complication for a second.) In such a diagram, your worldline would not be straight: it would be curving away from the Earth, just as the worldline of an accelerating rocket in flat spacetime does in a standard SR spacetime diagram. But also, in such a diagram, property (b) would not hold; you are accelerating upward, but you're not moving away from the Earth, even though the diagram makes it look like you are.

There is actually another problem with the second diagram, the one I described in the last paragraph. As I mentioned just now, it can't cover the entire Earth. That's because, in curved spacetime, unaccelerated worldlines that start out parallel can end up crossing! For example, suppose there were two tunnels running all the way through the Earth, with endpoints separated by some fairly short distance at the surface. Consider two rocks momentarily held at rest at the endpoints of the two tunnels, and then released into free fall. The rock's worldlines would be straight in the second type of diagram (because they're in free fall). And they would start out parallel, because the rocks are initially at rest relative to each other. But both rocks would fall through the center of the Earth, so their worldlines would cross there. There is no way to represent this on a spacetime diagram where unaccelerated worldlines are always straight, because straight lines that are initially parallel in such a diagram will have to stay parallel forever, so they can never cross.

What I've just described is one way of showing, independently of any specific choice of coordinates, that spacetime near the Earth is curved: initially parallel straight lines not staying parallel is the definition of "curvature" in geometry. And drawing a diagram of any curved surface is going to present similar issues to the ones you've discovered. There is no way to "fix" them; you just have to accept that there is no one "spacetime diagram" that can represent all of the properties of curved surfaces in an intuitive way. You just have to pick the kind of diagram that works best for each particular problem.

 

[A.T.]

I then realized, though, that I am constantly being accelerated outwards from the center of the Earth at 9.8 m/s. Therefore, my space-time diagram should show a curved trajectory to the right representing such acceleration.

Locally you can use curvilinear coordinates to describe the non-inertial frame of the surface, like shown here in B1, where the lines of constant radial position (your world-line) are curved, as opposed to the shown red free faller world lines, which are straight:

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

• 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.
  
• 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.

The problem is that you cannot show both, the surface and the center of the Earth in a flat diagram like B1, because the intrinsic curvature shown in C. You need a curved diagram like here:

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

If you check the "fixed positon" box, you get the world line of you standing at constant height, which is curved (non-geodesic), in contrast to the geodesic free faller world line you get when the box is unchecked. Note that there is no difference between the world lines at the center of the spherical mass (initial position = 0), meaning that the center is free falling.

 

[Dale]

I was trying to draw a space-time diagram representing two objects traveling through spacetime, me and the planet Earth. At first I thought I would just draw two straight lines heading upwards some distance, x, apart, seeing as my position relative to the center of the Earth would not be changing over time. I then realized, though, that I am constantly being accelerated outwards from the center of the Earth at 9.8 m/s. Therefore, my space-time diagram should show a curved trajectory to the right representing such acceleration.

So this is my quandary. Can anyone help me reconcile this? What does this space-time diagram look like?

This is correct. But you will need a curved piece of paper for it to work. I mean intrinsically curved like a globe, not extrinsically curved like a newspaper.

See here:
https://www.physicsforums.com/insights/understanding-general-relativity-view-gravity-earth/

 

[DiracPool]

This is correct. But you will need a curved piece of paper for it to work. I mean intrinsically curved like a globe, not extrinsically curved like a newspaper.

Thanks, by the way, congrats on 20,000 posts!

 

[Dale]

Thanks, by the way, congrats on 20,000 posts!

Hooray! Thanks, now 20,001 and counting.