8. From special relativity to general relativity

“Space-time tells matter how to move; matter tells space-time how to bend” John Archibald Wheeler¹

flowchart TD
A0(Teoria della Relatività Generale)
    A0 --> A1([le leggi fisiche sono le stesse  per tutti gli osservatori in tutti i  sistemi di riferimento])
    A1 --> A2([i sistemi di riferimento in moto accelerato tra di loro sono equivalenti])
    A2 --> A3([ i corpi in assenza di altre forze seguono i percorsi più brevi nello spazio tempo])
    A0 --> B1([non sono possibili azioni istantanee a distanza a velocità superiore a quella della luce])
    B1 --> B2([piccole perturbazioni del campo gravitazionale si propagano come onde alla velocità della luce])
    A0 --> C1([la massa inerziale è equivalente alla massa gravitazionale])
    C1 --> C2([la gravità è indistinguibile da una accelerazione del sistema di riferimento. ])
    C2 --> C3([il campo gravitazionale curva lo spaziotempo])
    C3 --> D1([<em> Lo spaziotempo dice alla materia come muoversi;<br/> la materia dice allo spaziotempo come curvarsi </em> <strong>J.Wheeler</strong>])
    A3 --> D1

General relativity

About ten years after the development of special relativity Einstein came up with a new theory, general relativity, to extend the principle of relativity, and the covariance requirement, to observers in accelerated, rotating and so on reference systems.

In the new general theory:

• physical laws are the same for all observers in all reference systems
• reference systems in accelerated, nonrectilinear uniform motion are also equivalent to each other
• instantaneous actions at a distance faster than the speed of light are never possible
• inertial mass is equivalent to gravitational mass even though they have different definitions
• gravity is indistinguishable from an acceleration of the reference system.
• the gravitational field of a mass curves spacetime
• bodies in the absence of other forces follow the locally shortest paths in curved spacetime, the geodesic lines
• small perturbations of the gravitational field propagate as waves at the speed of light
• Notes:
  1. Inertial mass is the ratio of force to acceleration in the second law of dynamics.
  2. Gravitational mass is that which appears in the law of universal gravitation.

Einstein had to modify even Newton’s law of gravity to meet these principles.

With a purely geometric theory, in which a force of gravity does not act instantaneously at a distance, but: “Space-time tells matter how to move; matter tells space-time how to bend” (Wheeler) that is, matter simply moves on the geodesic lines of curved space.

If gravitational fields are weak and static, the two theories of gravitation, by Einstein and Newton, obviously give the same results

Einstein started with two brilliant insights:

• A ray of light takes eight and a half minutes to travel from the Sun to Earth. But if there is a limiting speed, and no interaction can propagate over distance at a higher speed, then Newton’s law of gravity, with instantaneous action at any distance, does not is realistic
• Galileo understood that an observer in a reference system/laboratory moving with rectilinear and uniform motion does not perceive motion but thinks it is at rest; motion is relative. But an observer in a reference system/laboratory in free fall in a gravitational field (a re-entering spaceship, an elevator unhooked from the ropes) does not perceive the force of gravity, which is therefore indistinguishable from an acceleration of the reference system, is an apparent force.

The basic ideas are not complicated, but to specify them well requires advanced mathematical tools that were not well known at the time, especially outside the best Italian universities.

The differential, non-Euclidean geometry of curved spaces at any number of dimensions had been created by a genius, Bernhard Riemann, who died prematurely of tuberculosis ( while in Italy on Lake Maggiore).

The great importance of his work for the progress of mathematics and science was well understood only by the great Italian school of mathematics-Gregorio Ricci-Curbastro, Tullio Levi-Civita, Luigi Bianchi, Eugenio Beltrami and others. But almost ignored in other universities around the world, where the old Euclidean geometry, formulated more than two thousand years earlier, reigned supreme.

Einstein was stuck in his studies because of these mathematical difficulties for a decade or so, until his mathematical friend Marcel Grossmann thoroughly explained to him the work on tensor calculus by Ricci and Levi-Civita, with whom he then made direct contact.

The end result is the relativistic gravitational field equations, a complicated nonlinear system of ten partial differential derivative equations. Einstein thought that they could not be solved exactly in any case, but a few months after their publication he received a very important letter from Karl Schwarzschild.

Schwarzschild was a brilliant German astrophysicist, who died prematurely in the slaughterhouse of World War I, sent to the Russian-German front contracted an incurable autoimmune disease, was sent home and died soon after. While fighting in the trenches he had solved Einstein’s equations in one particular, very important case, a simplified model of a planet or star: a neutral, nonrotating spherical mass.

Only half a century later would the conditions of nonrotation and zero electric charge (Kerr-Newman solution) be removed, retaining only that of spherical symmetry, for a more general and realistic model (most black holes are believed to be rotating and neutral).