G3    Our Restriction to a Special Case

The (strong) equivalence principle was for Einstein both the starting point and the litmus test for any mathematical formulation of a theory of gravitation. Yet before 1911 he had made little progress. In 1912 he returned to Zurich from Prague to rejoin his friend and colleague Marcel Grossmann, who had meanwhile become a professor at the ETH (Eidgenössische Technische Hochschule = Swiss Federal Institute of Technology). He is reported to have beseeched Grossman: "you must help me or else I’ll go crazy."  [7-212]  Grossmann was quick to help, and Einstein milked mathematics as he had never done before. Soon they found the correct field equations - but rejected them because they thought the first approximations did not agree with Newton's theory. In the summer of 1915 (Einstein had already been in Berlin more than a year) he presented his work to David Hilbert and his people in Göttingen. On November 4th, 1915, he submitted to the Prussian Academy another essay from the series "On the general theory of relativity". A week later he was forced to make a retraction. On November 25th he brought this to an end and published the final version of his equations. Hilbert had already submitted a work on gravitation on November 20th, but it appeared in print only on March 31st, 1916. It also appeared to contain the correct equations. It nearly came to an unattractive controversy. Some still try incorrigibly to create a dispute, but since 1997 we know definitely that the plagiarism accusation can only apply to Hilbert (see [31-105f]).

In a letter to Arnold Sommerfeld Einstein wrote: “Think of my joy ... that the equations correctly predict the precessions of the perihelion of Mercury!” And to his friend Paul Ehrenfest: “I was stunned for several days by joyful excitement.” [32-216f, translation by Samuel Edelstein] Einstein was also completely exhausted and had to be looked after for several weeks.

Lecture notes on Differential Geometry by Marcel Grossmann. 
Einstein frequently played hooky from his mathematics lectures and depended
on the (beautiful) notes of his friend to prepare for exams

Einstein writes in his treatise for the Prussian Academy: “The magic of this theory will escape no one who grasps it, it is a true triumph of the methods founded by Gauss, Riemann, Christoffel, Ricci and Levi-Civita in general differential calculus.” [32-219, translation by Samuel Edelstein] Einstein's enthusiasm, as well as that of a few other 'insiders’, could not really be shared by most people (the author of this book included) since they were not versed in the mathematical 'tools’ that were used. Galileo wrote that the book of nature is written in the language of mathematics - but the math of Einstein’s theory is for most people an unreasonable demand. A. Herman writes in his very readable biography of Einstein:

“That made some contemporaries angry. The doctor and writer Alfred Döblin said he could understand Copernicus, Kepler and Galileo but the new theory - the ‘abominable doctrine of relativity’ – excludes him ‘and the immense quantity of all people, including the thoughtful and well educated from its insights’. The scientists of today with Einstein at the top had become a ‘brotherhood’, using ‘Masonic signs and communicating in a manner similar to spiritualists with their talking boards'.”    [32-220, translation by Samuel Edelstein]

As with the STR, in the course of several decades a number of individuals have opened wide the door to a qualitative understanding of the GTR (General Theory of Relativity) for educated non-specialists. We can even perform correct calculations for at least the most important special case, without first needing to complete several semesters of higher mathematics. For me, the two books [15] and [27] already mentioned were most stimulating.

Marcel Grossmann, Albert Einstein, Gustav Geissler and Marcel's brother Eugen 
during their time as students at the ETH.

Our restriction:

We only calculate the influence of a single, spherical, non-rotating mass with a rather weak gravitational field on its surface. We treat only the case of a spherically symmetric, weak gravitational field. Weak means: the escape velocity from the surface of the field-producing sphere will be much less than the speed of light.

Thus for a box, which is in free fall from a great distance toward the surface of the sphere, the classic expression is a very good approximation for the kinetic energy:
Ekin = 0.5 • m0 • v2.
In the diagram in section E4, we see that this applies to about v = c/3. The escape velocity of the Earth is only about 11.2 km/s, and on the sun's surface it is 618 km/s! These values are far below our limit of c/3 = 100,000 km/s, at which point our derivation becomes problematic. Our conditions are extremely well met almost everywhere in the universe except in the vicinity of very exotic objects like neutron stars or black holes. That is particularly true for any location in the gravitational ‘catchment area’ of our sun.

Strangely enough the formulas we are going to develop within the limits of our restriction and employing very elementary considerations will be the exact result even in the case of a strong gravitational field of a non-rotating massive spherical body!

Now let a small laboratory fall from far away along the x-axis toward the center of a spherical mass M:

Let the small laboratory have the initial speed given by  0.5 • m • v2 =  Ekin =  - Epot =  G • M • m / r  at the beginning of the observation.
Through conservation of energy this equation is met at each stage of the fall by increasingly smaller values of r and increasingly larger values of v. After division by m, we get

v2 =  2 • G • M / r  =  - 2 • Φ(r)      and      v2/ c2 =  2 • G • M / (c2 • r)  =  - 2 • Φ(r) / c2 =  2 • α / r  =  Rs / r

with the definitions   Φ(r) =  - G • M / r ,    α  =  G • M / c2 mm and      Rs =  2 • G • M / c2.

Φ(r) is the classical expression for the potential in a Newtonian gravitational field and Rs is the so-called Schwarzschild radius. For our ever-important radical expression, we have

After applying our special preconditions, the value of v2 / c2 and thus the value of 2 • α / r becomes very small. The square of α / r is therefore even much smaller, permitting the following reformulation:

Given the fact that Φ(r) = - (α / r) • c2, we can somewhat simplify our radical:

We want to once again satisfy ourselves that this approximation is very, very good: The strongest gravitational field in the solar system occurs at the surface of the sun. Check that the value there of α / r is about 2.1 • 10-6 !  We smuggled the square of this expression into our calculation above - something in the range of 4 • 10-12 ! This term is one million times smaller than the significant term 2 • α / r.

Our definitions of  Rs ,  α  and  Φ (r)  have also earned a red box:

Now we are prepared to consider the ramifications of the equivalence principle in our important special case and to derive how a gravitational field influences the speed of clocks and the length of yardsticks.

Finally, we want to remind ourselves of the assumptions we have made in the derivation of these formulas. We can formulate these in several different ways:

It must also be emphasized that the formulas derived above are only valid in the exterior of the sphere. Inside, the gravitational field loses strength and at its center – from symmetry reasons alone – is zero. This decrease is, according to Newton, linear. We will return to this in H5.