## H6    Gravitation and the Curvature of Space

We have already seen in G4 that the metric of space in the vicinity of a gravitational mass no longer obeys the laws of Euclid. For example, local yardsticks measure the diameter of the earth as slightly larger than its circumference divided by π. The effect in a weak gravitational field such as that of the earth is again very small, but in the vicinity of the sun it is nowadays easily experimentally demonstrated.

Consider again the diagram in G4. The dimple, or as Epstein says, the bump can help uncover the behavior of these metrics in an additional dimension that is not related to the z-direction. Epstein gives [15-165ff] detailed instructions on how you yourself can build such a model of a plane through the center of the sun (i.e., the ecliptic). The continuous changes in curvature are ignored for simplicity’s sake and the whole bump is represented by a cone:

This simple model shows qualitatively all of the effects which according to the GTR have their source in the non-Euclidean metric of space!

I trust the reader can craft such a cone themselves with no additional instructions. Keep in mind that it must be possible to ‘open’ the cone and spread it flat on a table and then afterwards again reform it into a cone. Therefore it is preferable to use the milky variety of scotch tape rather than the clear ...

With this model we are investigating – following always [15-165ff] Epstein - what effect such a space bump has on the path of a light beam (further applications will follow in the next section I). It is truly awesome how Epstein using such simple means, accurately and clearly shows the effects of space curvature!

A beam of light approaches our space bump. Which path will it follow, when it enters the field of the non-Euclidean metric?

The transition from the diagram’s flat surface to the curved cone is elegant: Remember that the bump rises softly from the flat surface and that the edge does not really exist! Flatten the cone locally just a bit (dotted circle) and extend the linear beam inside the dashed circle.

The beam will continue to spread out in a 'straight' line. But what does this mean on the surface of a cone? You can easily answer this if you uncoil the cone onto the diagram’s flat surface. Extend the small, straight piece you already have inside the dashed circle of the cone mantle until the beam of light again leaves the cone.

Return the cone to its 3D shape. Put it in exactly the same position as it already had in Figure 2. Thus we have – as viewed from above - the conditions of a distorted geometry of space produced near a large mass. We will now see which path the light beam follows around the center of this mass.

Exactly as in step 2 we now construct a further path. Press the cone in the dotted circle flat, and extend the direction of the path at the cones edge into the plane - done!

Light particles in a gravitational field fall (like everything else) in the direction of the mass center due to the curvature of space-time. Since this is independent of the mass of the falling particle, one was able to calculate this effect even before one knew the inertial mass of a light particle. The 27-year-old Johan Soldner submitted in 1803 an essay based on Newton's theory in which he calculated the deflection of a light beam passing the edge of the sun. The result of his calculation was an angle of 0.875 arc sec. This is precisely the value Einstein obtained in 1911 when he tested his newly conceived theory. In 1916 the 'completed' GTR provided a prediction of 1.75 arc sec, i.e., twice the value. This angle was now large enough that the possibility existed to clearly measure it from a photograph of stars in the sun’s vicinity during a solar eclipse. This was achieved in 1919 by two excursions of the ‘Royal Astronomers' under the direction of Arthur Stanley Eddington.

It was space curvature that was still lacking in the theory of 1911. Both curvatures (space-time and space-space) contribute almost the same and double the total effect. So a difference to the predictions of Newton's theory arose and the result of the measurement was in favor of the GTR and against Newton. Two theories may be structured differently – but if they make the same predictions for all phenomena, it cannot be decided through an experiment which is preferable. Fundamentally an experiment can only disprove a theory - never prove it.

There is however a remarkable difference between the curvature of space-time and that of space: The spatial curvature only affects objects that are moving through space, it has no influence on objects at rest! In particular, the curvature of space cannot cause an object to begin falling. But if it falls then it affects its trajectory. That the apple begins to fall is attributable to the curvature of space-time alone.

Epstein offers an interesting analogy [15-174, footnote]: It is like the effect of electric and magnetic fields on charged particles. The electrostatic force is there and works, regardless of the speed of the particle. The magnetic field has no effect on a charge at rest; the Lorentz force is proportional to velocity. Here, unfortunately, the analogy stops: The effect of spatial curvature on the path of falling objects does not depend on their velocity, it is important only that they have a path through the area to follow. However, velocity has a big impact on how long the curvature of space-time has an effect. Here the analogy fits again, since this holds for a charged particle in an electric field as well.