# Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year

## Abstract

**:**

## 1. Introduction

**g**in the old coordinates $\left\{x\right\}$, i.e.,

## 2. The Incompatibility of the Newtonian Theory of Gravitation with STR

**a**imparted on a test particle by a mass distribution of density ρ could be formally reformulated in the language of the differential equations governing a field-type state variable Φ, known as potential, through the Poisson equation [23]

## 3. The Equivalence Principle and Its Consequences

#### 3.1. The Equality of the Inertial and Gravitational Masses Raised to the Status of a Fundamental Principle of Nature

#### 3.2. Predictions of the Equivalence Principle

**g**by writing down the square of the spacetime line element ${\left(ds\right)}^{2}$ between two infinitely near events in arbitrary curvilinear coordinates ${x}^{\mu}$ as

**g**as the correct state variables playing the role of the Newtonian scalar potential Φ. Indeed, to a first-order level of approximation characterized by neglecting terms quadratic in $v/c$ and the squares of the deviations of the ${g}_{\mu \nu}$ from their STR values

**g**are not assigned independently of the matter-energy distributions, being determined by field equations.

**g**determines not only the gravitational field, but also the behaviour of clocks and measuring rods, i.e., the chronogeometry of the 4-dimensional spacetime which contains the geometry of the ordinary 3-dimensional space as a particular case. Such a fusion of two fields until then completely separated-metric and gravitation-should be regarded as a major result of GTR, allowing, in principle, to determine the gravitational field just from local measurements of distances and time intervals.

## 4. The Field Equations for the Metric Tensor and Their Physical Consequences

#### 4.1. The Field Equations

**g**; because of the required general covariance, $\mathsf{G}$ must be a tensor as well. The most general expression for it turned out to be

#### 4.2. First Predictions of the Theory and Confrontation with Observations

**g**. Schwarzschild [80] extended the validity of his solution also to the interior of a material body modelled as a sphere of incompressible fluid. Having in hand this exact solution of the Einstein field equations revolutionized the successive development of GTR. Indeed, instead of dealing only with small weak-field corrections to Newtonian gravity, as Einstein had initially imagined would be the case, fully nonlinear features of the theory such as gravitational collapse and singularity formation could be studied, as it became clear decades later. About the Schwarzschild solution, the Birkhoff’s Theorem [81] was proved in 1923. According to it, even without the assumption of staticity, the Schwarzschild metric is the unique vacuum solution endowed with spherically symmetry. As a consequence, the external field of a spherical body radially pulsating or radially imploding/exploding is not influenced at all by such modifications of its source.

#### 4.3. The General Approximate Solution by Einstein

#### 4.3.1. Gravitational Waves

#### 4.3.2. The Effect of Rotating Masses

#### 4.4. Black Holes and Other Physically Relevant Exact Solutions of the Field Equations

#### 4.4.1. The Reissner-Nordström Metric

#### 4.4.2. Black Holes

#### 4.4.3. The Kerr Metric

#### 4.4.4. The Kerr-Newman Metric

## 5. Application to Cosmology

#### 5.1. Difficulties of Newtonian Cosmologies

#### 5.2. Relativistic Cosmological Models

#### 5.2.1. The Static Einstein Model

#### 5.2.2. The de Sitter Model

#### 5.2.3. The Fridman-Lemaître-Robertson-Walker Expanding Models

**g**of standard expanding cosmologies is commonly named as Fridman-Lemaître-Robertson-Walker (FLRW) metric (see Section 5.4).

#### 5.3. The Einstein-de Sitter Model

#### 5.4. Some Peculiar Characteristics of the FLRW Models

## 6. Summary

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**MDPI and ACS Style**

Iorio, L.
Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year. *Universe* **2015**, *1*, 38-81.
https://doi.org/10.3390/universe1010038

**AMA Style**

Iorio L.
Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year. *Universe*. 2015; 1(1):38-81.
https://doi.org/10.3390/universe1010038

**Chicago/Turabian Style**

Iorio, Lorenzo.
2015. "Editorial for the Special Issue 100 Years of Chronogeometrodynamics: The Status of the Einstein’s Theory of Gravitation in Its Centennial Year" *Universe* 1, no. 1: 38-81.
https://doi.org/10.3390/universe1010038