Einstein's Universe: Physical Foundations of the Theory of Relativistic Gravitation

From Newton's Universal Gravitation to Einstein's Geometry

Gravitation, as understood before Einstein, was described by Newton's law of universal gravitation, a force acting instantaneously at a distance. The revolution introduced by Albert Einstein in the early 20th century radically transformed our physical understanding of gravitation, linking it not to a classical force but to the very geometry of spacetime.

Before Gravitation: Theoretical Foundations of Special Relativity

In 1905, Einstein published the theory of special relativity, which challenged the classical notions of absolute space and time. It is based on two postulates:

• The laws of physics are the same in all inertial frames of reference.
• The speed of light in a vacuum is constant and independent of the motion of the source or the observer.

In this framework, spatial and temporal coordinates merge into a four-dimensional continuum called Minkowski spacetime, equipped with a pseudo-Euclidean metric defined by:

$$ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2$$

This metric preserves the invariant interval \(ds\) between two events, thus structuring the reference framework in which particles and fields evolve.

N.B.: The four-dimensional continuum called Minkowski spacetime is a fundamental concept introduced by Albert Einstein's special relativity and formalized by the mathematician Hermann Minkowski in 1908. It corresponds to a geometric framework unifying space and time into a single mathematical structure with four dimensions: three of space \((x, y, z)\) and one of time \(t\).

N.B.: A pseudo-Euclidean metric is a distance measurement law where certain dimensions (such as time) have an opposite sign to others (the spatial dimensions). This reflects the intrinsic nature of the relativistic spacetime continuum, where distances can be negative, zero, or positive, depending on their physical meaning.

Source: Einstein Online - Max Planck Institute for Gravitational Physics, Living Reviews in Relativity, 2016

The Limits of Special Relativity in the Face of Gravitation

Special relativity only deals with inertial frames of reference, thus excluding accelerations and gravitational effects. However, gravitation acts precisely on the trajectory of bodies and can be interpreted as an acceleration. To integrate gravitation, Einstein formulates the principle of equivalence:

A uniformly accelerated reference frame is locally equivalent to a reference frame at rest in a gravitational field.

This fundamental idea paves the way for a geometric description of gravitation, distinct from the Newtonian concept of a force acting at a distance.

N.B.: The principle of equivalence states that gravitation and acceleration are locally indistinguishable. It underpins the idea that a gravitational field is equivalent to a geometric deformation of spacetime, laying the foundations for general relativity.

General Relativity: Gravitation as the Curvature of Spacetime

General relativity, published in 1915, extends special relativity to non-inertial frames of reference and proposes a theory of gravitation based on differential geometry. Spacetime becomes a dynamic object, whose metric \(g_{\mu\nu}\) depends on the distribution of matter and energy. The fundamental law is given by Einstein's equations, a system of nonlinear partial differential equations:

$$G_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}$$

• \(G_{\mu\nu}\) is the Einstein tensor, a function of the metric and its derivatives up to the second order, describing the local curvature of spacetime.
• \(T_{\mu\nu}\) is the energy-momentum tensor representing the density and flux of energy and momentum in spacetime.
• \(G\) is Newton's gravitational constant, \(c\) is the speed of light.

These equations thus express that matter-energy dictates the curvature of spacetime, and this curvature guides the motion of bodies; gravitation is no longer a force but a geometric manifestation.

Physical Implications and Applications

The physical consequences of general relativity are numerous and have been rigorously tested experimentally:

• Deflection of light by masses: Confirmed in 1919 by Eddington during an eclipse, this phenomenon shows that light follows curved geodesics in spacetime. It is a direct effect of the curvature induced by a mass.
• Precession of Mercury's perihelion: The shift in the closest point to the Sun in Mercury's orbit can only be explained by general relativity, with an excess of ~43 arcseconds per century.
• Gravitational time dilation: A clock placed in an intense gravitational field (e.g., near a star or at sea level) ticks more slowly than a clock at altitude or in space. This effect has been precisely measured by atomic clocks.
• Gravitational waves: Accelerated masses (e.g., merging black holes or neutron stars) generate perturbations in the metric that propagate at the speed of light. Directly detected by LIGO/Virgo since 2015.
• Gravitational lenses: Very compact masses (e.g., galaxy clusters, black holes) can bend light rays and act as optical lenses, distorting or multiplying the images of distant objects.
• Gravitational redshift: Light emitted from a source in a gravitational well is shifted towards longer wavelengths, a phenomenon confirmed in the Pound-Rebka experiments.
• Relativistic GPS: GPS satellites must correct their clocks for relativistic effects: time dilation due to velocity (special relativity) and altitude (general relativity), otherwise precision would rapidly degrade.
• Relativistic cosmology: General relativity allows for the dynamic modeling of the universe: Friedmann equations, Big Bang, accelerated expansion, dark energy, inflation, etc. It replaces Newtonian cosmology limited to small scales.
• Black holes: Exact solutions of Einstein's equations (e.g., Schwarzschild metric, Kerr metric), they predict compact objects with an event horizon. Confirmed by indirect observations (accretion, stellar dynamics) and direct observations (image of M87* by the EHT).
• Lense-Thirring effect (frame dragging): Rotating bodies drag spacetime around them. This effect has been confirmed by the Gravity Probe B satellite, illustrating the dynamic nature of spacetime geometry.

From Concept to Calculation

The complete physical understanding of Einstein's theory of gravitation is very complex, as it relies on a sophisticated and highly geometric mathematical structure that requires mastery of the following concepts:

• The pseudo-Riemannian geometry of four-dimensional spacetime.
• Tensor calculus: metric, Riemann curvature tensor, Ricci tensor, and Einstein tensor.
• The Lagrangian and variational formulation leading to Einstein's equations.
• The physical interpretation of exact solutions (e.g., Schwarzschild metric, Friedmann-Lemaître-Robertson-Walker metric).

This is why its complete understanding remains today reserved for a small circle of physicists trained in these tools, although its consequences (GPS, black holes, cosmology) are observable and experimentally confirmed.

Summary of the Physical Foundations of the Theory of Relativistic Gravitation

Einstein's theory of gravitation is based on a profound reformulation of the structure of the Universe, founded on several essential physical principles:

• Strong Equivalence Principle: Locally, a freely falling reference frame in a gravitational field is equivalent to an inertial reference frame without gravitation. This principle generalizes the weak equivalence principle and underpins the idea that gravitation can be locally eliminated by an adequate choice of reference frame.
• Geometrization of Gravitation: Rather than acting as a force, gravitation is a manifestation of the curvature of spacetime. The presence of matter and energy modifies the metric \(g_{\mu\nu}\), which determines the geometry and influences the dynamics of bodies and light rays.
• General Invariance: The physical laws, and in particular Einstein's equations, must be invariant under any differentiable coordinate transformation (diffeomorphisms). This ensures that the physical description is independent of the chosen reference system, thus affirming the intrinsic nature of the geometry of spacetime.
• Energy-Momentum Tensor as Source: Unlike Newtonian gravitation, where mass is the sole source, general relativity considers all forms of energy, momentum, and pressure encapsulated in the tensor \(T_{\mu\nu}\) as sources of curvature.

These foundations imply a dynamic universe where spacetime evolves according to the distribution of matter-energy, paving the way for a relativistic cosmology capable of explaining the expansion of the Universe, the Big Bang, and black holes as natural solutions to Einstein's equations.