# How do flat earth dwellers explain pi

## Gravitation: From the elevator to the space-time curvature

### What is actually gravity in Einstein's theory?

#### An article by Markus Pössel

What is gravity Einstein's general theory of relativity has an unusual answer to this question, which this specialization topic is about: Partly gravity is an illusion, partly associated with a property called "curvature". Overall, gravity is an aspect of the geometry of space and time.

### Freefall

An important feature of the gravitational force is that it drops all bodies (more precisely: all test bodies) at the same speed. A spring and a cannonball fall to the ground with exactly the same acceleration (at least in a vacuum container in which there are no air friction forces).

Conversely, this means that it is impossible for a researcher who is locked in a windowless cabin to determine whether the cabin is in empty space, far away from all sources of gravity, or in free fall in a constant gravitational field. In both situations, the researcher floats weightlessly in the cabin.

We know this situation from the terrestrial space travelers, who ultimately by no means move away from the sphere of influence of the earth's or solar gravity, but are in special varieties of free fall, for example in an orbit around the earth. The following photo, for example, shows the science astronaut C. Michael Foale on board the international space station ISS with two free-floating grapefruits.

Astronaut C. Michael Foale on the ISS

Under these circumstances, it should be possible to describe the physics in the free-falling cabin with the laws of special relativity - the same laws that apply to observers floating in free space far from all sources of gravity. (This statement is also known as the principle of equivalence; further information can be found in the advanced topic Cabin, Gravity and Rocket: The Principle of Equivalence.)

So is gravity just an apparent force? For the almost constant force of gravity that we are used to on the earth's surface, one could argue like this, because let's turn the tables and imagine the situation outlined here:

Various observers in space

Let us assume that we belonged to the crew of the space station, which can be seen on the right, and which is freely floating through space far away from all sources of gravity. Now come to see an observer on one left rocket-propelled Space station therefore, which is accelerated at 9.81 meters per second square. An observer on the accelerated space station feels just as heavy with this acceleration value as we do on the earth's surface: The acceleration due to gravity with which objects close to the earth's surface fall to the ground is 9.81 meters per second square.

Just as there is for a person on earth, for this observer there is a very clear above and below - below is where all bodies fall, and above is the opposite direction. If he looks up in his rocket-powered station, the observer sees the other space stations "falling" towards the floor of his own space station as fast as we see objects falling towards the ground. And yet in this situation there is nowhere gravity at work. The observers in the free-floating, unaccelerated space stations agree: The fact that the observer in question is viewing the world from an accelerated frame of reference is to blame for the fact that it looks to him as if all objects are "falling to the ground" , and its “top” is the direction in which the rocket is being accelerated. As soon as the rocket engine is switched off, the spook is over and no more objects “fall” down.

Is it the same with gravity on Earth? Is it ultimately just a consequence of the unnaturally accelerated frame of reference from which we observe the world - and disappears as soon as we switch to a freely falling frame of reference?

### What's left of gravity

In fact, the earth's gravity cannot be completely eliminated even when transitioning into a freely falling frame of reference (it cannot be "transformed away", as physicists would say).

To see this, we look at a free-falling but gigantic cabin with two huge balls floating in it. The following animation shows what happens when this cabin falls towards earth:

The effect here is that bodies that fall on earth do not fall in one and the same direction (“down”), but in one and the same direction Point in space to, namely the center of the earth. Even from the point of view of the observer falling freely with the cabin, the earth's gravity has a slight effect. This observer does not notice the fall of the balls down, because he himself falls in his cabin. On the other hand, he sees very well that the two balls are getting a little closer.

The reason for getting closer is that the gravity acting on the left ball has a slightly different direction than that on the right. The difference between the two forces, a so-called one, is responsible for the approach Tidal force. (It is named after the slight difference in force with which the gravitational force of the moon acts on the earth, on the oceans facing the moon and those facing away from it - this difference in force is responsible for the tides, for ebb and flow.)

The effects of gravity become even more drastic when a cabin occupant extends his observations to those bodies that are on the other side of the earth. Sure, the bodies falling right next to him behave as if there was no gravity. But the bodies on the other side of the earth even fall towards such an observer with twice the usual acceleration due to gravity!

All of this shows the differences to the situation of an accelerated observer in gravitation-free space treated in the last section. Such an observer only had to change his reference system - switch off the rocket engine of his space station - and what he had previously perceived as constant "gravity" disappeared. The earth's gravity, on the other hand, cannot completely disappear through the transition to a freely falling frame of reference. In a small, free-falling chamber, in which we only follow what is happening for a short time, the difference to the gravitation-free space is negligibly small. But the larger the space-time area under consideration, the larger our elevator car, the longer we observe, the clearer the tidal forces become - the “remainder of gravity” that can also be detected in free fall.

### Paper and ball

Remarkably, this dual nature of gravity - partly an apparent force, partly a tidal force - has an equivalent in pure mathematics, more precisely: in the geometry of distorted or curved surfaces.

The simplest two-dimensional surface is the flat plane - the infinitely extended analogue of a flat sheet of paper. The shortest connection between two points in the plane is a section along the straight line on which the two points lie. Geometric figures, such as triangles, can be put together from such straight lines. The relationships of plane geometry apply to them, as most readers should be familiar with from school - from the theorem of rays to the Pythagorean theorem and the fact that the sum of the angles in every triangle is 180 degrees:

Flat triangle; Angle sum

But a perfect plane is only the simplest case of a surface. More general surfaces, such as a spherical surface, a saddle surface, or the gently undulating surface that remains when the water has withdrawn from the Wadden Sea at low tide, no longer have the same geometric properties as a plane. An example of a surface that is not flat but curved is the surface of a sphere. There are no straight lines in a spherical surface, at most the straightest possible curves. Such straightest possible curves in a surface are Geodesics called. In the case of the sphere, the geodesics are the so-called great circles (for example the equator). The shortest connection between two points on the spherical surface leads along the great circle, along the geodesic on which the two points lie.

The following figure shows a sphere and, in green, a triangle formed by geodesic segments:

Sphere with a geodesic triangle

The two right angles at the equator (red line) alone add up to 180 degrees. The sum of the angles of the entire triangle, to which the angle at the apex of the triangle at the north pole is added, is thus significantly greater than 180 degrees. In fact, with the help of this excess angle, an exact measure for the curvature of the sphere (and thus the deviation of its geometry from that of the plane) can be defined.

### Where is the bend?

Despite the differences between the different geometries - here the plane, there curved surfaces - the following applies in general: If you look at a tiny surface section of a sphere with a magnifying glass, for example, it can hardly be distinguished from a small region of a flat sheet of paper. We take advantage of this fact almost every day - we create and read a city map that shows only a tiny section of the earth's surface as if the city surface had the same geometry as the flat sheet of paper on which the map was printed:

This approach is quite successful, although in reality our cities are not located on a gigantic level, but on the surface of a gigantic sphere. Only with larger surface regions does it become noticeable that the sphere is curved; the more extensive the region, the more noticeable the curvature.

The same applies to any smooth, curved surface: a tiny section of this surface is almost indistinguishable from a plane section. This is exactly analogous to the properties of gravity, which were dealt with in the last two sections: in a very small space-time region, for example the cabin of an observer who falls freely for a certain period of time, almost nothing is noticeable by gravity. The interior of the cabin cannot be distinguished from a section from the gravitation-free space-time of the special theory of relativity. Only in a larger space-time excerpt, for example a gigantic cabin, does it become apparent that the space-time observed deviates from the gravitation-free, "flat" space-time.

Einstein took this analogy seriously, and actually managed to work out a geometric theory of gravity.

### The geometry of gravity

In flat spacetime, the gravitation-free spacetime of the special theory of relativity, bodies that are not acted on by external forces move on “spacetime lines”: they run at constant speed along straight paths.

Let us now bring gravity into play, for example by placing a massive ball in our space. In Newton's picture of gravity, the sphere exerts a force on the small test bodies surrounding it, which deflects these test bodies from their spacetime lines by bending their trajectories towards the central mass and accelerating the test bodies in this direction.

In Einstein's geometric theory of gravity, on the other hand, this mass causes a Distortion of spacetime: If the gravitation-free spacetime (that of the special theory of relativity) was flat, the spacetime is curved in the presence of this mass. In this curved spacetime there are no longer any spacetime lines, just as there are no lines on the surface of a sphere. There are only geodesics, the straightest possible space-time orbits. Test bodies in the vicinity of the central mass follow the straightest possible paths in the space-time curved by the mass. Gravitation does not divert test objects from their straight paths - it distorts space and time and thus redefines what it means to move on the straightest possible path.

Einstein's universe is thus a constant dance in which matter and space-time influence each other: A given arrangement of matter distorts the geometry of space-time; the geometry of spacetime determines how matter moves on. The space-time geometry also changes according to the slightly changed configuration of matter due to the movement, this changed geometry influences the further movement of the matter now somewhat differently than before, and so on, and so on.

So what is gravity in Einstein's universe? Generally speaking, any distortion of the geometry of space and time. More precisely, gravity has two sides. Part of the gravitation is an artifact, depending on the reference system of the respective observer. This part of the gravitation can be made to disappear if one goes over to a freely falling frame of reference. The gravitational effects we can observe here on earth - essentially: bodies falling to the ground - belong to this type of "relative gravitation". The rest of the gravitation, one could call it “intrinsic gravitation”, is noticeable through tidal forces and is directly related to a very specific geometric property: the curvature of space-time.