What is a negative spacetime curvature

How to bend space-time. Part III Negative curvature and a tip for eating pizza

So far in this series we have looked at curvatures that were analogous to the curvature of the earth's surface: angle sums in the triangle were greater than 180 °, circles had an excess radius. But there is also another possibility: the negative curvature.

If you are not that terribly interested in the exact details of the curvature, you can be content with briefly looking at the pictures (this will give you an idea of ​​what negative curvature is) and then jump directly to the warning below.

Another triangle
First, let's take another look at the triangle from the last part:

The distance between two degrees of latitude (here plotted for 15 ° each), i.e. in the vertical direction in the picture, was always the same; the distance between two degrees of longitude changed the further north you got, but again was not dependent on the degree of latitude (all horizontal Lines at the same height have the same value).

Next, let's make things a little more complicated: we also change the distance between the latitudes the further north we go, but in such a way that the distance increases:

The further you go “north” in the picture, the closer the longitudes still come together; but at the same time the distance between the latitudes is increasing. Here, too, with the same value of the latitude, all distances are the same, so the whole thing is still quite even.

In this case too, just like last time, you can search for the shortest connection between two points. For example, if you connect two points on the same latitude (as in the triangle on the map of the earth above), then the shortest path is again not the one that runs on the latitude. If we go a little north, the distance is shortened again, but on the other hand the detour to the north is also “more expensive” because the vertical distances in the picture increase further and further towards the north. Thankfully, there are formulas for the geodesics in this situation (see below), so that I finally managed (with a little gimmick with gnuplot) to draw a triangle similar to the one for the globe in this new map:

At first glance it looks very similar to the triangle on the globe, but the angles have all become a bit more acute. If you measure the sum of the angles correctly, you will see that it is less than 180 °.

The space of this map is also curved, but different from the sphere. One also speaks of a negative curvature. The sum of the angles is smaller than 180 °, not larger. To a geodesic there is not just a parallel through a neighboring point, but an infinite number.

But I admit: It's a bit difficult to see if you just look at the map. The curved surface shown by the map can, however, (with a few obstacles) be shown as a surface in three-dimensional space. This is what this area looks like:
It is the famous pseudosphere that our topology expert Thilo has already reported on several times, namely in his topology series Sequence IL, Sequence LIV and Sequence LV, because it allows a representation of the so-called hyperbolic geometry.

Here I have drawn a similar triangle as above on my map in the Wikipedia pseudospheric image:

Based on a picture by Claudio Rocchini - Own work, CC BY 2.5, Link

In case someone wants to know how I calculated the picture above: On Wolfram Alpha you can find a formula for the metrics of the pseudosphere as well as for the geodesics. The distance on a line above corresponds to the square root of the metric, since ds2= gμνdxμdxν is. I used the given formula for the geodesics to calculate for two given sets of coordinates (u1, v1) and (u2, v2) calculate the constants c and k in the geodesic formula and have represented the result in gnuplot. Finally, I redefined the coordinates as follows: Actually, the triangle connects the points (1.5, 0) (1.5, 1) and (1,1) in the Wolfram Alpha coordinate system. I multiplied the distances by 1000 and finally changed the axis labels so that the 1 becomes a 45 on the horizontal axis, a 0 on the vertical axis and a 45 from the 1.5, so that it is better for the map of the earth fits.