Einstein's General Relativity vs Quantum Mechanics - Key Physics Concepts

We have examined the constraints that bind these three realities together at the microscopic level and how these constraints influence the behavior of light rays. In doing so, we have introduced some elementary concepts that have been outlined with the development of quantum mechanics.

At the opposite end of the atomic description provided by Quantum Mechanics is the description operated by a competing theory developed by Albert Einstein and known as the theory of General Relativity. The theory of General Relativity is in its essence very simple and intuitive and was born to describe the relationship between space, time and energy over large distances.

While the equations we have seen in previous chapters seem to describe the granular behavior of the space-time fabric, the theory we will now examine aims to describe its global behavior, describing its curves, ripples and singularities. These two visions therefore represent two distinct approaches used alternatively depending on convenience to describe atomic phenomena or spatial phenomena.

From an essential point of view, General Relativity consists of two fundamental assumptions or novelties that distinguish it from all previous theories:

1. (Principle of Equivalence) Gravity is resolved as a different Geometry of space and time

2. (Einstein's Field Equation) The metric, i.e. the Geometry of space-time, depends on the distribution of Energy

These are the two fundamental conceptual elements synthesized by Albert Einstein's physical genius and prefigured by the mathematical genius of William Clifford who led to the formulation of General Relativity. The first is a qualitative philosophical principle and expresses the equivalence between deformations of space-time geometry and gravitational forces. The second point is instead the key that will allow us to solve all quantitative problems numerically.

These two fundamental concepts are then synthesized by Einstein into a single fundamental equation called Einstein's Field Equation which we can write very simply:

$$G_{mn} = 8\pi T_{mn}$$

In Einstein's field equation, the first part $G_{mn}$ is called Einstein's curvature tensor and indicates the Geometry of space-time. It, through Einstein's principle of equivalence, represents, in effect, Gravity. On the other side of the equation we have instead the second ingredient $T_{mn}$ called the Energy tensor which contains information relating to the distribution of energy density. It contains all the information relating to energy in the form of matter, electromagnetic radiation and whatever else can be imagined.

In the middle there is the equality sign that marks the mutual connection of action and reaction between the two realities, i.e. between the geometry of space-time and the distribution of energy. Geometry, therefore, is seen in General Relativity as the resolving element that allows Energy to be reunited with Space and Time.

In summary, quoting John Wheeler, one of Einstein's most famous and important collaborators, we could summarize all of General Relativity in a single sentence:

"Matter tells space-time how to bend and space-time tells matter how to move."
— John A. Wheeler

Ultimately, General Relativity is nothing more than two simple concepts, but these simple concepts, as we will see, will allow us to understand much of the results of modern cosmology.

In-depth note: Commentary on Einstein's field equations (requires some knowledge of tensor calculus)

$$G_{mn} = 8\pi T_{mn}$$

$\nabla_m$ are symbols that mean variation according to various directions in a curvilinear space, the requirement to conserve energy $\nabla_m T^{mn} = 0$ led Einstein to formulate the tensor that now bears his name.