Deriving the Einstein-Hilbert Equation from First Principles

J. Philippe Blankert – blankertbooks.com

14 March 2025

Introduction

The Einstein-Hilbert action serves as the mathematical cornerstone of Einstein’s theory of general relativity, elegantly describing how gravity arises from the curvature of spacetime, a curvature caused by the presence of mass and energy. Developed independently by Albert Einstein and mathematician David Hilbert, this approach represents a profound unification of fundamental physics concepts into a single, elegant equation. This article will provide a detailed, step-by-step derivation of the Einstein-Hilbert action and the resulting field equations, aimed at readers with a basic understanding of physics but seeking a deeper dive into the subject.

Fundamental Concepts

Action Principle

The action principle, also known as the principle of least action, is a cornerstone of physics. It dictates that physical systems evolve in a way that minimizes the action, denoted by $S$ . The action is a measure of energy over time, with dimensions of Joule-seconds (J·s). Mathematically, it is expressed as the integral of the Lagrangian over time:

$S = \int L \, dt$

Here, $L$ represents the Lagrangian, which is the difference between the kinetic energy $T$ and the potential energy $V$ of the system:

$L = T - V$

The action principle is not just a mathematical curiosity; it has deep physical implications. It suggests that nature is inherently “economical” in its processes, favoring paths that require the least “effort” in terms of action. This principle underlies much of classical mechanics, electromagnetism, and even quantum mechanics.

In the context of general relativity, the action principle is crucial for determining how spacetime itself evolves in response to the presence of mass and energy. We seek the spacetime geometry that minimizes the action, leading us to Einstein’s field equations.

Mathematical Foundations

Metric Tensor

In general relativity, the curvature of spacetime is described by a mathematical object called the metric tensor, denoted by $g_{\mu\nu}$ . This tensor is a generalization of the concepts of vectors and scalars and is essential for understanding relationships in multiple dimensions. It provides a way to measure distances and angles in curved spacetime, which is fundamentally different from the flat spacetime of Newtonian physics.

The metric tensor is a 4×4 matrix, where the indices $\mu$ and $\nu$ each range from 0 to 3 (or 1 to 4, depending on convention), representing the four dimensions of spacetime: one time dimension and three spatial dimensions. The components of the metric tensor encode how spacetime intervals are measured at each point. For example, in special relativity, the metric tensor for flat spacetime (Minkowski spacetime) is a diagonal matrix with entries (-1, 1, 1, 1) or (1, -1, -1, -1), depending on the convention used. This simple form reflects the homogeneity and isotropy of flat spacetime.

In general relativity, the metric tensor is not constant but varies from point to point, depending on the curvature of spacetime. This variation is what gives rise to the effects we perceive as gravity. The metric tensor, therefore, is not just a mathematical tool but a fundamental physical field that determines the geometry of spacetime and how objects move within it.

Ricci Scalar

The Ricci scalar, denoted by $R$ , is a single number that encapsulates the curvature of spacetime at a point. It is derived from the Ricci tensor $R_{\mu\nu}$ , which is a more complex object that describes the curvature in a particular direction. The Ricci scalar is obtained by contracting the Ricci tensor with the metric tensor:

$R = g^{\mu\nu} R_{\mu\nu}$

The Ricci scalar provides a way to quantify the overall curvature of spacetime. A positive Ricci scalar indicates that spacetime is curved in a way that objects tend to converge, while a negative Ricci scalar indicates divergence. A zero Ricci scalar implies that spacetime is flat (at that particular point).

The Ricci scalar plays a central role in the Einstein-Hilbert action, as it is the term that encodes the curvature of spacetime. By including it in the action, we ensure that the theory of general relativity describes how spacetime curvature affects the dynamics of physical systems.

Derivation from First Principles

Einstein-Hilbert Action

The Einstein-Hilbert action is the mathematical expression that forms the basis of general relativity:

$S = \frac{c^4}{16 \pi G} \int R \sqrt{-g} \, d^4x$

Let’s break down the components of this equation:

$S$ : This is the action, as discussed earlier. It is a functional that takes the metric tensor as its input and returns a number.
$c$ : The speed of light in vacuum. It is a fundamental constant of nature that appears in many areas of physics, including relativity.
$G$ : The gravitational constant, also known as Newton’s gravitational constant. It quantifies the strength of the gravitational force.
$R$ : The Ricci scalar, representing the curvature of spacetime.
$\sqrt{-g}$ : The square root of the negative of the determinant of the metric tensor. This term represents the volume element in spacetime. In curved spacetime, the volume element is not constant, and $\sqrt{-g}$ accounts for this variation.
$d^4x$ : An infinitesimal four-dimensional spacetime volume element. It represents a tiny “chunk” of spacetime.
$\frac{c^4}{16 \pi G}$ : This factor is a constant that ensures the correct units and scaling of the equation. It relates the curvature of spacetime to the distribution of mass and energy.

The Einstein-Hilbert action essentially states that the dynamics of spacetime are determined by the integral of the Ricci scalar over all of spacetime, weighted by the appropriate factors. The principle of least action then dictates that the spacetime geometry that minimizes this action is the one that physically occurs.

Variation Principle

To derive Einstein’s field equations, we apply the variation principle to the Einstein-Hilbert action. This involves finding the metric tensor $g_{\mu\nu}$ that makes the action stationary, meaning that small changes in the metric tensor do not change the value of the action to first order. Mathematically, this is expressed as:

$\delta S = 0$

where $\delta S$ represents the variation of the action.

The variation is taken with respect to the metric tensor $g^{\mu\nu}$ . This process involves using the tools of variational calculus, which is a branch of mathematics that deals with finding functions that optimize certain quantities. In this case, we are seeking the function (the metric tensor) that optimizes the action.

The variation of the Einstein-Hilbert action is a complex calculation, but it leads to the following expression:

$\delta S = \frac{c^4}{16 \pi G} \int \left( R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \right) \sqrt{-g} \, \delta g^{\mu\nu} \, d^4x = 0$

This equation states that the variation of the action is equal to the integral of a certain expression involving the Ricci tensor $R_{\mu\nu}$ , the Ricci scalar $R$ , and the metric tensor $g_{\mu\nu}$ , multiplied by the variation of the metric tensor $\delta g^{\mu\nu}$ and the volume element $\sqrt{-g} \, d^4x$ .

Step-by-Step Derivation of Einstein Field Equations

Here is a more detailed breakdown of the derivation of Einstein’s field equations from the Einstein-Hilbert action:

Step 1: Starting with the Einstein-Hilbert Action

We begin with the explicit form of the Einstein-Hilbert action:

$S = \frac{c^4}{16 \pi G} \int R \sqrt{-g} \, d^4x$

This is the action that governs the dynamics of spacetime in the absence of matter and energy.

Step 2: Variational Calculus

Applying the variation $\delta$ to the action with respect to the inverse metric tensor $g^{\mu\nu}$ , we get:

$\delta S = \frac{c^4}{16 \pi G} \int \left( R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} \right) \sqrt{-g} \, \delta g^{\mu\nu} \, d^4x$

This step involves using the rules of variational calculus and tensor algebra. The calculation is quite involved and requires careful manipulation of the metric tensor, the Ricci tensor, and the Ricci scalar. It also involves considering the variation of the determinant of the metric tensor.

The key idea here is that we are looking for how the action changes when we make small changes to the metric tensor. This will tell us which metric tensor corresponds to the physical spacetime.

Step 3: Requiring Minimization (Stationary Action)

Since $\delta g^{\mu\nu}$ is arbitrary, the integrand in the above equation must vanish for the integral to be zero for all possible variations. This leads to:

$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = 0$

This equation is known as Einstein’s field equations in vacuum. It describes the curvature of spacetime in the absence of matter and energy. It tells us that even in empty space, spacetime can be curved, and this curvature is determined by the relationship between the Ricci tensor and the Ricci scalar.

Step 4: Incorporating Matter-Energy

To account for the presence of matter and energy, we introduce the energy-momentum tensor $T_{\mu\nu}$ . This tensor describes the distribution of energy and momentum in spacetime. It is a source term that tells spacetime how to curve.

Including the energy-momentum tensor, we obtain Einstein’s complete field equations:

$R_{\mu\nu} - \frac{1}{2} R g_{\mu\nu} = \frac{8 \pi G}{c^4} T_{\mu\nu}$

These equations are the cornerstone of general relativity. They describe how the curvature of spacetime (represented by the left-hand side) is related to the distribution of matter and energy (represented by the right-hand side). They tell us that matter and energy warp spacetime, causing it to curve, and this curvature is what we perceive as gravity.

Physical Interpretation

Einstein’s field equations have a profound physical interpretation. They demonstrate that the geometry of spacetime is not fixed but is dynamically determined by the distribution of matter and energy. This means that spacetime is not just a passive backdrop against which events occur but an active participant in the dynamics of the universe.

The equations show that gravity is not a force in the traditional Newtonian sense but rather a manifestation of the curvature of spacetime. Objects move along geodesics, which are the “straightest possible paths” in curved spacetime. These paths appear to us as curved trajectories due to the underlying curvature of spacetime.

In essence, Einstein’s field equations provide a geometric description of gravity, replacing Newton’s concept of a gravitational force acting at a distance.

Experimental Evidence

General relativity has been subjected to numerous experimental tests, and it has passed with flying colors. These tests have not only confirmed the theory’s predictions but also provided stunning visual evidence of the curvature of spacetime and the existence of phenomena that were previously unimaginable.

Gravitational Waves: Detected by LIGO

Gravitational waves are ripples in the fabric of spacetime. They are generated by the acceleration of massive objects, such as black holes or neutron stars, and propagate outward from their source at the speed of light. These waves cause tiny distortions in spacetime as they pass, stretching and squeezing objects in their path.

The detection of gravitational waves is a major triumph for general relativity. It provides direct evidence for the dynamic nature of spacetime and opens up a new window for observing the universe.

Detection by LIGO

The Laser Interferometer Gravitational-Wave Observatory (LIGO) is a scientific instrument designed to detect these minuscule spacetime distortions. LIGO consists of two identical detectors located thousands of kilometers apart. Each detector uses laser interferometry to measure changes in distance as small as one-thousandth the diameter of a proton.

LIGO works by splitting a laser beam into two perpendicular paths. The beams are reflected by mirrors and recombined. If a gravitational wave passes through, it will cause the lengths of the two paths to change slightly, which will be detected as an interference pattern in the recombined beam.

Historic Detection

On September 14, 2015, LIGO made the first direct detection of gravitational waves, known as GW150914. This event was caused by the merger of two black holes, each around 30 times the mass of the Sun, approximately 1.3 billion light-years away. The detected signal matched the predictions of general relativity with remarkable accuracy.

This detection confirmed a key prediction of general relativity and marked the beginning of gravitational wave astronomy.

Impact

The detection of gravitational waves has revolutionized our understanding of the universe. It provides a new way to observe cosmic events that are invisible to traditional telescopes, such as black hole mergers and neutron star collisions. Gravitational waves also allow scientists to test general relativity in the strong-field regime, where gravity is extremely intense.

The study of gravitational waves promises to reveal new insights into the nature of gravity, the evolution of the universe, and the fundamental laws of physics.

Gravitational Lensing: Observed as the Bending of Light Paths by Massive Objects

Gravitational lensing is another striking phenomenon predicted by general relativity. It occurs when the gravitational field of a massive object, such as a galaxy or galaxy cluster, bends and magnifies the light from objects behind it. This bending of light paths creates distorted, magnified, or multiple images of the background objects.

Gravitational lensing is a direct consequence of the curvature of spacetime. Since light follows the curvature of spacetime, its path is bent when it passes near a massive object. The amount of bending depends on the mass of the object and the distance of the light path from it.

Einstein’s Prediction

Einstein himself predicted that massive objects would bend the path of light. This prediction was first observationally confirmed during a solar eclipse in 1919. During the eclipse, astronomers observed that the positions of stars near the Sun were slightly shifted from their normal positions. This shift was precisely the amount predicted by general relativity, providing early and compelling experimental support for the theory.

Strong and Weak Lensing

Gravitational lensing can be broadly categorized into two types: strong lensing and weak lensing.

Strong Lensing: Occurs when the lensing effect is so strong that it produces multiple images, arcs, or even complete rings of the background object. This typically happens when the lensing object is very massive, such as a galaxy cluster. Strong lensing provides spectacular visual evidence of the bending of light by gravity.
Weak Lensing: Results in subtle distortions of background galaxies. These distortions are usually too small to be seen by eye but can be statistically analyzed to map the distribution of dark matter in the universe. Dark matter is a mysterious substance that does not emit or absorb light but interacts gravitationally, making up a large portion of the mass of the universe.

Applications

Gravitational lensing has become a powerful tool in astronomy and cosmology. It is used to:

Study the distribution of dark matter.
Measure the masses of galaxies and clusters.
Detect exoplanets.
Magnify distant galaxies, allowing us to observe them in greater detail.

Gravitational lensing acts as a natural telescope, magnifying the light from distant objects and revealing details that would otherwise be invisible.

Black Hole Imaging: Directly Observed by the Event Horizon Telescope

Black holes are regions of spacetime where gravity is so strong that nothing, not even light, can escape. They are predicted by general relativity and are thought to exist at the centers of most galaxies.

Black Hole Imaging

Black hole imaging involves capturing an image of the immediate environment around a black hole, particularly the event horizon. The event horizon is the boundary beyond which nothing can escape the black hole’s gravitational pull.

Event Horizon Telescope (EHT)

The Event Horizon Telescope (EHT) is a global network of radio telescopes that work together to create an Earth-sized virtual telescope. This collaboration aims to image the event horizons of supermassive black holes with unprecedented angular resolution.

The EHT uses a technique called very-long-baseline interferometry (VLBI) to achieve its high resolution. VLBI combines the signals from multiple telescopes located thousands of kilometers apart, effectively creating a telescope with a diameter equal to the distance between the most distant telescopes.

Historic Image

In April 2019, the EHT released the first-ever image of a black hole, located at the center of the galaxy M87. The image shows a bright ring of emission surrounding the dark shadow of the event horizon.

Conclusion

Through the detailed derivation of the Einstein-Hilbert action and its subsequent application, this exposition has demonstrated the profound connection between the geometry of spacetime and the distribution of matter-energy. Einstein’s field equations, derived from this action, have been rigorously tested and supported by experimental evidence, including the detection of gravitational waves, the observation of gravitational lensing, and the direct imaging of a black hole’s event horizon. These findings have not only validated the core tenets of general relativity but also opened new avenues for exploring the universe and its fundamental laws.

However, despite these successes, challenges and open questions remain. The unification of general relativity with quantum mechanics remains a central problem in physics. Understanding the nature of dark matter and dark energy, which play significant roles in the large-scale structure and dynamics of the universe, requires further investigation within a general relativistic framework and beyond. Moreover, testing general relativity in even more extreme environments, such as the immediate vicinity of singularities, and searching for deviations from its predictions, are crucial for pushing the boundaries of our knowledge and potentially revealing new physics.

Scientific References

Einstein, A. (1915). Die Grundlage der allgemeinen Relativitätstheorie. Annalen der Physik, 49(7), 769–822.
Hilbert, D. (1915). Die Grundlagen der Physik. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1915, 395–407.
Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration). (2016). Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116(6), 061102.
Abbott, B. P. et al. (LIGO Scientific Collaboration and Virgo Collaboration). (2017). GW170817: Observation of Gravitational Waves from a Binary Neutron Star Inspiral. Physical Review Letters, 119(16), 161101.
Misner, C. W., Thorne, K. S., & Wheeler, J. A. (1973). Gravitation. W. H. Freeman.
Schneider, P. (2006). Extragalactic Astronomy and Cosmology: An Introduction. Springer Science & Business Media.
Event Horizon Telescope Collaboration. (2019). First M87 Event Horizon Telescope Results. I. The Shadow of the Supermassive Black Hole. The Astrophysical Journal Letters, 875(1), L1.