My previous post was a description of the shape of spacetime around the Earth. I framed the discussion by asking what happens when I drop a ball from rest above the surface of the Earth. Spacetime is curved. And the ball takes the straightest possible path through spacetime. So what does that look like? Last time I generated a representation of the spacetime to illustrate.

However, I generated some confusion by claiming that it “should be obvious” that the straightest possible path is curved towards or away from the Earth. When a textbook author says “the proof is trivial” usually what they mean is that they don’t want to go through the work of writing a proof. The same is true here, I didn’t want to generate a picture with the path of the ball in it.  Since this was confusing however, I apologize. And to make it up to you, I’ve plotted the path of the ball, shown in figure 1.

Note that it approaches a straight line. That’s because as it accelerates it’s approaching the speed of light (we are neglecting air resistance and exaggerating the distance from the surface of the Earth to make that happen). The path of the ball is curved—it curves with the surface, after all. But it’s as straight as it possibly can be. And that’s what makes it a geodesic.

Note also that the speed of light is a straight line that’s wider than 45 degrees. I told you last time that in Minkowski space light travels at 45 degree angles. However, to make the curvature of the spacetime visible, I stretched out lengths radially (the direction of the red arrow) a bit. So actually light cones in this plot are wider. I didn’t think this would be visible when I made the plot before, but it’s quite clear if you include the geodesics. So I apologize for that slight misrepresentation last time.

I’ve updated the previous post to include this plot. So this week’s post is only for those of you who read the last post.