A physical theory that endeavors to describe how one or more fields in physics interact with matter, all through the elegant (or perhaps, merely less complicated) lens of field equations , without bothering with those pesky effects of quantization – that, in essence, is a classical field theory. Theories that insist on dragging quantum mechanics into the conversation are, predictably, dubbed quantum field theories . In the grand scheme of things, when someone mentions ‘classical field theory,’ they’re usually referring specifically to electromagnetism and gravitation , which happen to be two of the fundamental forces of nature. Because, apparently, we needed more than just a vague wave of the hand to explain why things fall and magnets stick.

One can conceptualize a physical field as a rather meticulous assignment: a specific physical quantity is attributed to every single point in space and at every given moment in time . Consider, for a moment, a weather forecast. The wind velocity across an entire country over the course of a day isn’t just a single number; it’s a dynamic tapestry. It’s meticulously described by attaching a vector to each infinitesimal point in that space. Each of these vectors, in its own unassuming way, points in the direction the air is currently moving at that precise location. Thus, the collective assembly of all these wind vectors, spanning an entire geographical area at a particular instant, forms what we call a vector field . And as the day inevitably grinds on, these vectors, much like our fleeting hopes, shift and change their directions in accordance with the capricious whims of the wind.

The earliest forays into field theories, namely Newtonian gravitation and Maxwell’s equations governing electromagnetic fields, emerged from the crucible of classical physics, long before the disruptive arrival of relativity theory in 1905. Naturally, these venerable theories had to undergo a rather significant overhaul to align themselves with the new, inconvenient truths of relativity. Consequently, classical field theories are now typically sorted into two neat little categories: non-relativistic and relativistic. Most modern expressions of field theories, for those who appreciate mathematical precision, are articulated using the sophisticated language of tensor calculus . A more recent, and perhaps more esoteric, alternative mathematical formalism describes classical fields as mere sections of abstract mathematical objects known as fiber bundles . Because, why use one layer of abstraction when you can have several?

Part of a series on Classical mechanics

F = dp/dt Second law of motion

• History
  • The chronicle of how humanity slowly, painfully, began to understand motion and forces. Or, more accurately, how a few brilliant minds dragged the rest of us along.
• Timeline
  • A chronological ledger of discoveries, primarily serving to remind us how much less we knew yesterday.
• Textbooks
  • Volumes filled with the accumulated wisdom (and occasional errors) of centuries. Required reading for anyone who wishes to truly suffer.

Branches

• Applied
  • Where the abstract theories are forced to confront the messy reality of the physical world. Often involves engineers.
• Celestial
  • The elegant, intricate dance of planets and stars, governed by forces unseen. Or, rather, seen only when someone bothers to look up.
• Continuum
  • Dealing with materials as continuous, unbroken entities, ignoring the inconvenient truth of atoms. A useful lie.
• Dynamics
  • The study of motion and the forces that instigate it. Because things rarely just sit still.
• Field theory
  • The very subject we’re currently dissecting. A framework for understanding influence across space and time.
• Kinematics
  • Pure motion, stripped of its causal forces. The “how” without the “why.”
• Kinetics
  • The forces that cause motion. The “why” behind the “how.”
• Statics
  • When nothing moves. A state of temporary, fragile equilibrium. Much like my patience.
• Statistical mechanics
  • Bridging the gap between the microscopic chaos and macroscopic order. Or, trying to, at least.

Fundamentals

• Acceleration
  • The rate at which velocity changes. A measure of how quickly things stop being predictable.
• Angular momentum
  • The rotational equivalent of linear momentum. Because spinning is also a form of motion.
• Couple
  • Two forces, equal and opposite, causing rotation without translation. A perfectly balanced, yet dynamic, relationship.
• D’Alembert’s principle
  • Turning dynamics into statics by introducing fictitious forces. A clever mathematical trick.
• Energy
  • The capacity to do work . The currency of the universe.
  • kinetic
    • Energy of motion. The universe’s way of saying, “look, I’m doing something!”
  • potential
    • Stored energy, waiting for its moment to be unleashed. Like a coiled spring, or a deeply held grudge.
• Force
  • That which causes acceleration. The fundamental push or pull.
• Frame of reference
  • The perspective from which you observe reality. Crucial, because everyone thinks their perspective is the only one.
• Inertial frame of reference
  • A frame where Newton’s laws hold true, without the need for additional, fabricated forces. A place of relative calm.
• Impulse
  • A sudden change in momentum. The universe’s way of delivering a swift kick.
• Inertia / Moment of inertia
  • Resistance to changes in motion, both linear and rotational. The universe’s inherent laziness.
• Mass
  • A measure of an object’s inertia. How much ‘stuff’ there is, and how much it resists moving.
• Mechanical power
  • The rate at which work is done. How quickly you can achieve something, or burn out.
• Mechanical work
  • Force applied over a distance. The effort expended, often for disappointing results.
• Moment
  • The turning effect of a force. Leverage, in its purest form.
• Momentum
  • Mass in motion. A quantity conserved, thankfully, or the universe would be even more chaotic.
• Space
  • The three-dimensional stage upon which all this drama unfolds. Vast, empty, and largely indifferent.
• Speed
  • How fast something is moving, without regard for direction. A scalar quantity, much like my enthusiasm.
• Time
  • The relentless, unidirectional flow of existence. Or, as some prefer, merely another dimension.
• Torque
  • The rotational equivalent of force. What makes things twist and turn.
• Velocity
  • Speed with direction. A vector, providing a complete picture of motion.
• Virtual work
  • Imaginary work done by real forces. A mathematical convenience to simplify complex systems.

Formulations

• Newton’s laws of motion
  • The bedrock of classical mechanics. Simple, elegant, and surprisingly robust for how long we’ve known them.
• Analytical mechanics
  • A more abstract, often more powerful, approach to classical mechanics, using energy and generalized coordinates.
  • Lagrangian mechanics
    • Focuses on the difference between kinetic and potential energy. A path of least action, which frankly, sounds like a good life philosophy.
  • Hamiltonian mechanics
    • Another formulation, emphasizing canonical coordinates and momentum. Even more abstract, for those who enjoy such things.
  • Routhian mechanics
    • A hybrid approach, for when you can’t quite commit to either Lagrangian or Hamiltonian.
  • Hamilton–Jacobi equation
    • A single, first-order partial differential equation that encapsulates classical dynamics. Impressive, if you’re into that sort of thing.
  • Appell’s equation of motion
    • An alternative formulation based on accelerations. Because sometimes, you just want to look at things differently.
  • Koopman–von Neumann mechanics
    • A quantum-like formulation of classical mechanics. Because everything eventually tries to be quantum.

Core topics

• Damping
  • The dissipation of energy in oscillatory systems. All good things must come to an end, even vibrations.
• Displacement
  • The change in position of an object. A vector quantity, because direction matters.
• Equations of motion
  • Mathematical descriptions of how physical systems move over time. The universe’s instruction manual, if you can decipher it.
• Euler’s laws of motion
  • Extending Newton’s laws to rigid bodies. Because not everything is a point mass.
• Fictitious force
  • Forces that appear only in non-inertial frames. A consequence of choosing the wrong perspective.
• Friction
  • The resistance to relative motion between surfaces. The universe’s constant, irritating drag.
• Harmonic oscillator
  • A system that, when displaced, experiences a restoring force proportional to the displacement. Found everywhere, because the universe loves to oscillate.
• Inertial / Non-inertial reference frame
  • The fundamental distinction in how we observe motion. One is simple, the other requires more mental acrobatics.
• Motion (linear )
  • The fundamental act of changing position. The simplest form of movement.
• Newton’s law of universal gravitation
  • The force that binds us all, quite literally. Or, at least, binds masses.
• Newton’s laws of motion
  • Still fundamental, even after all these years.
• Relative velocity
  • Velocity as observed from a different moving frame. Everything is relative, isn’t it?
• Rigid body
  • An idealized object that doesn’t deform. A convenient fiction for simplifying calculations.
  • dynamics
    • How these idealized, unyielding objects move.
  • Euler’s equations
    • Governing the rotational motion of rigid bodies. More fun with tensors.
• Simple harmonic motion
  • The simplest form of oscillation. Pure, unadulterated back-and-forth.
• Vibration
  • Rapid oscillatory motion. The universe constantly humming, sometimes annoyingly.

Rotation

• Circular motion
  • Movement along a circular path. Simple, yet surprisingly complex to describe fully.
• Rotating reference frame
  • Observing the world from a spinning perspective. Prepare for fictitious forces.
• Centripetal force
  • The force that pulls an object towards the center of a circular path. Keeps things from flying off into the void.
• Centrifugal force
  • A fictitious force, experienced in a rotating frame, pushing objects away from the center. A consequence of inertia.
  • reactive
    • The actual reaction force to the centripetal force.
• Coriolis force
  • Another fictitious force, acting perpendicular to motion in a rotating frame. Responsible for hurricanes and other inconvenient truths.
• Pendulum
  • A classic example of oscillating motion. Simple, elegant, and endlessly studied.
• Tangential speed
  • The linear speed of an object moving along a curved path.
• Rotational frequency
  • How many rotations per unit of time. How fast something is truly spinning.
• Angular acceleration / displacement / frequency / velocity
  • The rotational equivalents of their linear counterparts. Because everything has to spin.

Scientists

• Kepler

  • Mapped the heavens and gave us laws of planetary motion. A tireless observer.

• Galileo

  • Challenged dogma, observed the universe, and laid foundations for modern science. A true troublemaker.

• Huygens

  • Optics, mechanics, timekeeping. A polymath who understood the wave nature of light before it was cool.

• Newton

  • The giant whose shoulders we stand on. Or, at least, he thought he was. Invented calculus to solve his own problems.

• Horrocks

  • A young prodigy who refined Kepler’s tables and predicted a transit of Venus. Briefly shone brightly.

• Halley

  • Best known for his comet, but also a significant figure in celestial mechanics.

• Maupertuis

  • Formulated the principle of least action. The universe, apparently, is lazy.

• Daniel Bernoulli

  • Contributed to fluid dynamics and probability. His principle explains why planes fly, among other things.

• Johann Bernoulli

  • A fierce mathematician, brother of Daniel, and rival of Newton and Leibniz. Loved a good intellectual brawl.

• Euler

  • A titan of mathematics, whose work touched almost every field. If you see a formula, Euler probably had a hand in it.

• d’Alembert

  • Mathematician, physicist, philosopher. His principle simplified dynamics.

• Clairaut

  • Made crucial contributions to celestial mechanics, particularly the shape of the Earth.

• Lagrange

  • Developed analytical mechanics, bringing elegance and power to the field.

• Laplace

  • A French polymath who contributed to astronomy, mathematics, and statistics. Known for his daemon.

• Poisson

  • His equation describes potentials in the presence of sources. Also known for his bracket.

• Hamilton

  • Irish mathematician who developed Hamiltonian mechanics and quaternions. A visionary, if a bit esoteric.

• Jacobi

  • German mathematician, further developed analytical mechanics and elliptic functions.

• Cauchy

  • French mathematician, rigorous in analysis, prolific in his output.

• Routh

  • English mathematician, known for his work on rigid body dynamics and stability.

• Liouville

  • French mathematician, known for Liouville’s theorem in Hamiltonian mechanics.

• Appell

  • French mathematician, known for Appell’s equations of motion.

• Gibbs

  • American theoretical physicist, chemist, and mathematician. Pioneer of statistical mechanics and vector calculus.

• Koopman

  • American mathematician, contributed to the Koopman–von Neumann formulation.

• von Neumann

  • Hungarian-American polymath, contributed to quantum mechanics, game theory, computing. A mind that probably didn’t suffer fools.

• Physics portal

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History

The very notion of a “field” – a space permeated by some influence – wasn’t always obvious. It took a rather imaginative mind, that of Michael Faraday , to first coin the term and introduce the concept of “lines of force” as a way to visualize and explain the otherwise invisible dance of electric and magnetic phenomena. One can almost picture the blank stares he must have received. It wasn’t until 1851 that Lord Kelvin , with his characteristic rigor, took Faraday’s somewhat intuitive ideas and formalized the concept of a field, extending its mathematical description to various other domains of physics. Because abstract concepts, it seems, always need a proper mathematical cage before they’re taken seriously.

Non-relativistic field theories

Some of the most straightforward and, dare I say, least complicated physical fields one encounters are those rather unadorned vector force fields. Historically, the moment fields truly began to command attention and be taken seriously was with Faraday’s evocative lines of force , which provided an intuitive, if not entirely rigorous, way to describe the electric field . Following this, the venerable gravitational field was, quite naturally, described in a remarkably similar fashion. Because if it works for one invisible influence, it must work for another.

Newtonian gravitation

The inaugural field theory concerning gravity was, of course, Newton’s theory of gravitation . In this foundational framework, the mutual interaction between any two masses was posited to adhere strictly to an inverse square law . This relatively simple yet profoundly powerful description proved exceptionally useful for accurately predicting the complex, orbital dance of planets around the Sun, a feat that, at the time, must have seemed nothing short of miraculous.

Any body possessing mass, let’s call it M, inherently generates a gravitational field g. This field, in its essence, describes the sphere of influence M exerts upon any other massive bodies unfortunate enough to be within its reach. To quantify this gravitational field generated by M at a specific point r in space, one typically determines the force F that M would exert on a minuscule, hypothetical test mass m strategically placed at r. This force F is then divided by m, yielding the field strength:

${\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}.}$

The crucial stipulation here, often overlooked by the less discerning, is that m must be considerably smaller than M. This ensures that the intrusive presence of our tiny test mass m has a utterly negligible impact on the overall behavior of the much larger M. Because we wouldn’t want our measurement tool to fundamentally alter the thing we’re trying to measure, would we?

According to Newton’s law of universal gravitation , the force F(r) is precisely given by:

${\displaystyle \mathbf {F} (\mathbf {r} )=-{\frac {GMm}{r^{2}}}{\hat {\mathbf {r} }},}$

where ${\displaystyle {\hat {\mathbf {r} }}}$ is a unit vector that, with unwavering precision, points along the direct line connecting M to m, and G is the universally acknowledged gravitational constant . Therefore, with a bit of algebraic tidying up, the gravitational field of M is elegantly expressed as:

${\displaystyle \mathbf {g} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{m}}=-{\frac {GM}{r^{2}}}{\hat {\mathbf {r} }}.}$

The rather striking experimental observation that inertial mass and gravitational mass are, to an almost uncanny degree of accuracy, identical, leads us down a rather significant path. It suggests that the gravitational field strength is, in fact, indistinguishable from the acceleration experienced by a particle within that field. This profound equivalence is not merely a curious coincidence; it serves as the fundamental cornerstone of the equivalence principle , which in turn, rather dramatically, ushers us into the realm of general relativity .

For a discrete assembly of masses, let’s denote them as M_i, each patiently situated at its own point, r_i, the total gravitational field at an arbitrary point r due to this collection of masses is simply the vector sum of their individual contributions:

${\displaystyle \mathbf {g} (\mathbf {r} )=-G\sum {i}{\frac {M{i}(\mathbf {r} -\mathbf {r_{i}} )}{|\mathbf {r} -\mathbf {r} _{i}|^{3}}},,}$

Should we, however, be dealing with a more amorphous, continuous mass distribution, conveniently represented by a mass density ρ, then the discrete summation above is, as one might expect, gracefully supplanted by an integral:

${\displaystyle \mathbf {g} (\mathbf {r} )=-G\iiint _{V}{\frac {\rho (\mathbf {x} )d^{3}\mathbf {x} (\mathbf {r} -\mathbf {x} )}{|\mathbf {r} -\mathbf {x} |^{3}}},,}$

A quick, yet vital, observation: the direction of this field, always, points from the observation position r directly toward the location of the masses r_i. This inherent attractive quality is precisely guaranteed by that rather unassuming minus sign. In essence, it’s the universe’s way of stating, unequivocally, that all masses, without exception, are drawn to one another.

In its integral manifestation, Gauss’s law for gravity offers a powerful statement about the flux of the gravitational field through any closed surface:

${\displaystyle \iint \mathbf {g} \cdot d\mathbf {S} =-4\pi GM}$

while its more localized, differential form concisely expresses the divergence of the gravitational field at any point:

${\displaystyle \nabla \cdot \mathbf {g} =-4\pi G\rho _{m}}$

Consequently, the gravitational field g can be, with a certain mathematical elegance, expressed in terms of the gradient of a gravitational potential φ (r):

${\displaystyle \mathbf {g} (\mathbf {r} )=-\nabla \phi (\mathbf {r} ).}$

This particular formulation isn’t merely a mathematical convenience; it’s a direct consequence of the rather significant fact that the gravitational force F is, fundamentally, a conservative field . Meaning, the path you take doesn’t matter, only the start and end points. A comforting thought, perhaps.

Electromagnetism

Electrostatics

Main article: Electrostatics

A charged test particle with charge q finds itself experiencing a force F that, rather conveniently, depends solely on its own charge. We can, in a strikingly similar manner to our gravitational endeavors, describe the electric field E that is generated by some source charge Q. This relationship is defined such that F = qE:

${\displaystyle \mathbf {E} (\mathbf {r} )={\frac {\mathbf {F} (\mathbf {r} )}{q}}.}$

Armed with this definition and the venerable Coulomb’s law , the electric field emanating from a solitary charged particle can be precisely expressed as:

${\displaystyle \mathbf {E} ={\frac {1}{4\pi \varepsilon _{0}}}{\frac {Q}{r^{2}}}{\hat {\mathbf {r} }},.}$

Much like its gravitational counterpart, the electric field is also inherently conservative , and thus, with satisfying consistency, it can be represented as the gradient of a scalar potential, V (r):

${\displaystyle \mathbf {E} (\mathbf {r} )=-\nabla V(\mathbf {r} ),.}$

Gauss’s law for electricity, in its integral form, quantifies the total electric flux piercing through any closed surface:

${\displaystyle \iint \mathbf {E} \cdot d\mathbf {S} ={\frac {Q}{\varepsilon _{0}}}}$

while its differential sibling provides a localized statement about the divergence of the electric field:

${\displaystyle \nabla \cdot \mathbf {E} ={\frac {\rho _{e}}{\varepsilon _{0}}},.}$

Magnetostatics

Main article: Magnetostatics

A steady, unwavering current I, gracefully flowing along a designated path ℓ, will invariably exert a force on any nearby charged particles. This force, however, is quantitatively and qualitatively distinct from the purely electric field force we’ve just discussed. The force F exerted by I on a nearby charge q possessing a velocity v is given by the Lorentz force law:

${\displaystyle \mathbf {F} (\mathbf {r} )=q\mathbf {v} \times \mathbf {B} (\mathbf {r} ),}$

where B(r) represents the magnetic field . This magnetic field, a product of the current I, is itself determined through the rather intricate Biot–Savart law :

${\displaystyle \mathbf {B} (\mathbf {r} )={\frac {\mu _{0}I}{4\pi }}\int {\frac {d{\boldsymbol {\ell }}\times d{\hat {\mathbf {r} }}}{r^{2}}}.}$

Unlike the electric field, the magnetic field is generally not conservative . Consequently, it cannot, in most circumstances, be simply expressed as the gradient of a scalar potential. However, it can, with a different kind of mathematical elegance, be articulated in terms of a vector potential , A(r):

${\displaystyle \mathbf {B} (\mathbf {r} )=\nabla \times \mathbf {A} (\mathbf {r} )}$

Gauss’s law for magnetism, in its integral form, makes a rather definitive statement about magnetic flux:

${\displaystyle \iint \mathbf {B} \cdot d\mathbf {S} =0,}$

while its differential counterpart reinforces this point with unwavering precision:

${\displaystyle \nabla \cdot \mathbf {B} =0.}$

The profound physical interpretation of this seemingly simple equation is that there are no magnetic monopoles . The universe, it seems, prefers its magnetic charges to come in inseparable pairs. A neat little symmetry, if you bother to notice.

Electrodynamics

Main article: Electrodynamics

In the more general, and frankly, more realistic scenario where both a charge density ρ (r, t) and a current density J(r, t) are present, one must contend with the simultaneous existence of both an electric and a magnetic field. Crucially, both of these fields will, more often than not, vary with time. Their intricate interplay and evolution are governed by Maxwell’s equations , a beautifully concise set of differential equations that directly link the electric field E and the magnetic field B to the sources: electric charge density ρ (charge per unit volume) and current density J (electric current per unit area). A truly elegant summary of how electricity and magnetism intertwine.

Alternatively, one can choose to describe the entire system using the scalar and vector potentials, V and A, respectively. A set of integral equations, famously known as retarded potentials , provides a pathway to calculate V and A directly from ρ and J. (This, of course, is contingent on the correct choice of gauge . φ and A are not uniquely determined by ρ and J; rather, they are only determined up to some scalar function f (r, t) known as the gauge. The retarded potential formalism requires one to choose the Lorenz gauge .) From these potentials, the electric and magnetic fields are then meticulously determined via the following fundamental relations:

${\displaystyle \mathbf {E} =-\nabla V-{\frac {\partial \mathbf {A} }{\partial t}}}$

${\displaystyle \mathbf {B} =\nabla \times \mathbf {A} .}$

These equations reveal a deeper truth: the electric and magnetic fields are not entirely independent entities, but rather different manifestations of a single, unified electromagnetic field , their interrelationship made explicit by the time-varying nature of the potentials.

Continuum mechanics

Fluid dynamics

Main article: Fluid dynamics

In the fascinating, often chaotic, realm of fluid dynamics , one encounters fields of pressure, density, and flow rate. These fields are not merely arbitrary constructs; they are intimately connected by fundamental conservation laws governing energy and momentum. The mass continuity equation , for instance, serves as a direct representation of the conservation of mass within the fluid:

${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \mathbf {u} )=0}$

And then, there are the infamous Navier–Stokes equations , which, in their formidable complexity, encapsulate the conservation of momentum within the fluid. These equations are derived by applying Newton’s laws to the fluid continuum itself:

${\displaystyle {\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} +p\mathbf {I} )=\nabla \cdot {\boldsymbol {\tau }}+\rho \mathbf {b} }$

This formidable equation holds true provided that the density ρ, the pressure p, the deviatoric stress tensor τ of the fluid, and any external body forces b are all known quantities. The primary goal, the elusive prize, is to solve for the velocity field u, which is itself a vector field describing the motion of every fluid parcel. A truly messy business, fluids.

Other examples

In the year 1839, a rather ambitious James MacCullagh presented his own set of field equations. These were designed to describe the phenomena of reflection and refraction in his work titled “An essay toward a dynamical theory of crystalline reflection and refraction.” A commendable effort to bring order to optical chaos.

Potential theory

The term “potential theory ” itself derives from a rather significant belief held in 19th-century physics: that the fundamental forces of nature, in their elegant simplicity, could all be derived from scalar potentials that obediently satisfied Laplace’s equation . Poisson , ever the diligent mathematician, tackled the thorny question of the stability of planetary orbits . This problem had, admittedly, already been addressed by Lagrange to the first degree of approximation, using perturbation forces. Nevertheless, Poisson went further, deriving what is now famously known as Poisson’s equation , named, quite rightly, after him. The general form of this ubiquitous equation is:

${\displaystyle \nabla ^{2}\phi =\sigma }$

where σ represents a source function (often interpreted as a density, a quantity per unit volume) and φ is the scalar potential that one endeavors to solve for.

In the context of Newtonian gravitation , masses are unequivocally identified as the sources of the gravitational field. Consequently, gravitational field lines are understood to terminate at objects possessing mass. Similarly, in electrostatics, charges serve as both the sources and sinks of electric fields: positive charges are conceptualized as emanating electric field lines, while these lines ultimately converge and terminate at negative charges. These fundamental field concepts are further elucidated and beautifully illustrated by the general divergence theorem , specifically in the forms of Gauss’s laws for gravity and electricity. For the cases where gravity and electromagnetism are time-independent – a rather convenient simplification – the fields can be expressed as gradients of their corresponding potentials:

${\displaystyle \mathbf {g} =-\nabla \phi _{g},,\quad \mathbf {E} =-\nabla \phi _{e}}$

Substituting these expressions back into Gauss’s law for each respective case yields:

${\displaystyle \nabla ^{2}\phi _{g}=4\pi G\rho _{g},,\quad \nabla ^{2}\phi {e}=4\pi k{e}\rho _{e}=-{\rho _{e} \over \varepsilon _{0}}}$

Here, ρ_g denotes the mass density , ρ_e the charge density , G the venerable gravitational constant , and k_e = 1/4πε₀ the electric force constant.

Incidentally, this rather striking similarity between the two sets of equations arises directly from the profound structural resemblance between Newton’s law of gravitation and Coulomb’s law . A universe that, it seems, enjoys recycling its mathematical forms.

In the specific scenario where there is no source term – for instance, in a perfect vacuum, or when dealing with perfectly paired charges that cancel each other out – these potentials obediently satisfy Laplace’s equation :

${\displaystyle \nabla ^{2}\phi =0.}$

For any given distribution of mass (or, equivalently, charge), the potential can be expanded into a series of spherical harmonics . Each n^th term within this series can then be interpreted as a potential arising from the 2ⁿ-moments, a concept elegantly captured in the multipole expansion . For a great many practical applications and calculations, one often finds that only the monopole, dipole, and quadrupole terms are strictly necessary, simplifying the universe, if only slightly.

Relativistic field theory

Main article: Covariant classical field theory

Modern formulations of classical field theories, for those still paying attention, almost invariably demand Lorentz covariance . This isn’t merely a matter of aesthetic preference; it’s now universally recognized as a fundamental, inescapable aspect of nature itself. A field theory, in this more advanced context, is typically expressed with exquisite mathematical precision using Lagrangians . This Lagrangian is not just any function; it’s a carefully constructed entity that, when subjected to the profound dictates of an action principle , spontaneously gives rise to the field equations and, rather conveniently, a conservation law for the theory in question. The action itself is a Lorentz scalar , a quantity that remains invariant under Lorentz transformations, from which both the field equations and the inherent symmetries of the system can be derived with remarkable ease.

A quick, but essential, note for the mathematically inclined: throughout this discussion, we shall employ a system of units where the speed of light in vacuum is, for simplicity’s sake, set to 1. That is, c = 1. (This is equivalent to choosing units of distance and time as light-seconds and seconds or light-years and years. Choosing c = 1 allows us to simplify the equations. For instance, E = mc² reduces to E = m (since c² = 1, without keeping track of units). This reduces complexity of the expressions while keeping focus on the underlying principles. This “trick” must be taken into account when performing actual numerical calculations.) Because why complicate things with unnecessary constants when the underlying physics is already challenging enough?

Lagrangian dynamics

Main article: Lagrangian (field theory)

Given a field tensor, let’s denote it as ${\displaystyle \phi }$, one can meticulously construct a scalar quantity known as the Lagrangian density , represented as:

${\displaystyle {\mathcal {L}}(\phi ,\partial \phi ,\partial \partial \phi ,\ldots ,x)}$

This density is artfully assembled from ${\displaystyle \phi }$ itself and its various derivatives. From this Lagrangian density, the action functional, a cornerstone of variational principles, can then be constructed by the rather elegant process of integrating it over the entirety of spacetime:

${\displaystyle {\mathcal {S}}=\int {{\mathcal {L}}{\sqrt {-g}},\mathrm {d} ^{4}x}.}$

Where ${\displaystyle {\sqrt {-g}},\mathrm {d} ^{4}x}$ serves as the volume form in the potentially (or actually) curved expanse of spacetime, and ${\displaystyle (g\equiv \det(g_{\mu \nu }))}$ represents the determinant of the metric tensor .

Therefore, the Lagrangian itself, stripped of its density, is simply the integral of the Lagrangian density across all space.

Then, by rigorously enforcing the action principle – the universe’s inherent preference for paths of stationary action – the profound Euler–Lagrange equations are inevitably obtained:

${\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \phi }}={\frac {\partial {\mathcal {L}}}{\partial \phi }}-\partial _{\mu }\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu }\phi )}}\right)+\cdots +(-1)^{m}\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\left({\frac {\partial {\mathcal {L}}}{\partial (\partial _{\mu _{1}}\partial _{\mu _{2}}\cdots \partial _{\mu _{m-1}}\partial _{\mu _{m}}\phi )}}\right)=0.}$

These equations, though intimidating in appearance, are the very heart of the dynamics, dictating how the field ${\displaystyle \phi }$ evolves. A rather elegant, if slightly masochistic, way to describe reality.

Relativistic fields

Now, let’s delve into two of the most celebrated and well-understood Lorentz-covariant classical field theories. Because some fields simply refuse to be confined to non-relativistic approximations.

Electromagnetism

Main articles: Electromagnetic field and Electromagnetism

Historically, the earliest (classical) field theories were those that meticulously described the electric and magnetic fields, initially treating them as separate, distinct entities. However, after countless experiments and persistent intellectual inquiry, it became unequivocally clear that these two fields were, in fact, inextricably linked. More profoundly, they were revealed to be merely two different facets of a single, unified field: the magnificent electromagnetic field . Maxwell’s seminal theory of electromagnetism provides a comprehensive description of the intricate interaction between charged matter and this overarching electromagnetic field. The initial formulation of this groundbreaking field theory employed simple vector fields to describe the electric and magnetic components. However, with the revolutionary arrival of special relativity , a far more complete and elegant formulation, utilizing tensor fields, was discovered. Instead of wrestling with two distinct vector fields, one could now employ a single, powerful tensor field that seamlessly encompassed both the electric and magnetic aspects.

The electromagnetic four-potential is precisely defined as A^a = (−φ, A), while the electromagnetic four-current is given by j^a = (−ρ, j). The electromagnetic field at any given point in spacetime is then meticulously described by the antisymmetric (0,2)-rank electromagnetic field tensor :

${\displaystyle F_{ab}=\partial {a}A{b}-\partial {b}A{a}.}$

The Lagrangian

To unravel the dynamics governing this field, one typically embarks on the quest to construct a scalar from the field itself. In the pristine emptiness of a vacuum, this yields the rather concise Lagrangian density:

${\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu {0}}}F^{ab}F{ab},.}$

We can then invoke the principles of gauge field theory to introduce the crucial interaction term, which expands our Lagrangian density to:

${\displaystyle {\mathcal {L}}=-{\frac {1}{4\mu {0}}}F^{ab}F{ab}-j^{a}A_{a},.}$

The equations

To extract the field equations from this Lagrangian, the electromagnetic tensor F within the Lagrangian density must be meticulously replaced by its definition in terms of the 4-potential A. It is this potential, A, that ultimately enters the Euler-Lagrange equations . The EM field F itself is not varied in the Euler-Lagrange equations. Thus, the Euler-Lagrange equations for the field A^a take the form:

${\displaystyle \partial {b}\left({\frac {\partial {\mathcal {L}}}{\partial \left(\partial {b}A{a}\right)}}\right)={\frac {\partial {\mathcal {L}}}{\partial A{a}}},.}$

Upon carefully evaluating the derivative of the Lagrangian density with respect to the field components:

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial A_{a}}}=\mu _{0}j^{a},,}$

and similarly, the derivatives with respect to the derivatives of the field components:

${\displaystyle {\frac {\partial {\mathcal {L}}}{\partial (\partial {b}A{a})}}=F^{ab},,}$

one ultimately, and quite satisfyingly, obtains Maxwell’s equations in vacuum. The source equations (which include Gauss’ law for electricity and the Maxwell-Ampère law ) are concisely expressed as:

${\displaystyle \partial _{b}F^{ab}=\mu _{0}j^{a},.}$

Meanwhile, the other two fundamental equations (Gauss’ law for magnetism and Faraday’s law of induction ) are not derived from varying the action with respect to A^a, but rather emerge from the intrinsic definition of F as the 4-curl of A. Or, to put it another way, they are a direct consequence of the Bianchi identity always holding true for the electromagnetic field tensor:

${\displaystyle 6F_{[ab,c]},=F_{ab,c}+F_{ca,b}+F_{bc,a}=0.}$

where the comma notation, for the uninitiated, simply indicates a partial derivative . A beautiful, if abstract, testament to the self-consistency of the universe.

Gravitation

Main article: Gravitation

Further information: General Relativity and Einstein field equation

Once Newtonian gravitation was, rather inconveniently, discovered to be fundamentally inconsistent with the dictates of special relativity , Albert Einstein was compelled to formulate a completely new and far more profound theory of gravitation, which he famously christened general relativity . This theory, in a stroke of genius, re-conceptualized gravitation not as a force acting across space, but as an intrinsic geometric phenomenon – specifically, the ‘curved spacetime ’ itself, a distortion caused by the presence of masses and energy. The gravitational field is no longer a force vector, but is mathematically represented by a sophisticated tensor field known as the metric tensor . The formidable Einstein field equations then precisely describe how this curvature of spacetime is brought into being by the distribution of mass and energy. Newtonian gravitation , once the undisputed champion, is now seen as merely a low-energy, weak-field approximation, having been definitively superseded by Einstein’s theory of general relativity . In this new paradigm, gravitation is elegantly understood as being due to the inherent curvature of spacetime , a curvature wrought by the presence of masses. The Einstein field equations, those rather iconic expressions:

${\displaystyle G_{ab}=\kappa T_{ab}}$

describe with remarkable precision how this curvature is produced by matter and radiation. Here, G_ab is the Einstein tensor , a geometric object that encapsulates the curvature of spacetime:

${\displaystyle G_{ab},=R_{ab}-{\frac {1}{2}}Rg_{ab}}$

This tensor is, in turn, expressed in terms of the Ricci tensor R_ab and the Ricci scalar R = R^ab g_ab. On the right-hand side, T_ab is the stress–energy tensor , which describes the density and flux of energy and momentum, effectively telling spacetime where the “stuff” is. And κ = 8πG / c⁴ is a constant that links the geometry of spacetime to its contents. In the elegant absence of matter and radiation (including all sources), the vacuum field equations simplify to:

${\displaystyle G_{ab}=0}$

These equations can be derived with a certain austere beauty by varying the Einstein–Hilbert action :

${\displaystyle S=\int R{\sqrt {-g}},d^{4}x}$

with respect to the metric, where g is the determinant of the metric tensor g_ab. Solutions to these vacuum field equations are, quite logically, termed vacuum solutions . An intriguing alternative interpretation, famously proposed by Arthur Eddington , suggests that the Ricci scalar ${\displaystyle R}$ is the truly fundamental entity, while the stress–energy tensor ${\displaystyle T}$ is merely one aspect of ${\displaystyle R}$, and the constant ${\displaystyle \kappa }$ is simply forced upon us by our arbitrary choice of units. A perspective that shifts the focus from matter dictating geometry to geometry defining matter.

Further examples

Beyond the titans of electromagnetism and gravitation, other notable Lorentz-covariant classical field theories include:

• Klein-Gordon theory, which meticulously describes real or complex scalar fields.
• Dirac theory, providing a framework for a Dirac spinor field, a crucial step toward quantum electrodynamics.
• Yang–Mills theory , a more abstract and powerful theory for a non-abelian gauge field, forming the backbone of the Standard Model of particle physics.

Unification attempts

Main article: Classical unified field theories

The persistent human drive to find a single, overarching theory that explains everything led to numerous attempts to create a unified field theory based squarely on the principles of classical physics . These efforts are collectively known as classical unified field theories . During the tumultuous years nestled between the two World Wars, the ambitious idea of unifying gravity with electromagnetism became a fervent pursuit for a multitude of brilliant mathematicians and physicists. Among these dedicated minds were giants like Albert Einstein , Theodor Kaluza , Hermann Weyl , Arthur Eddington , Gustav Mie , and Ernst Reichenbacher. It seems the universe wasn’t quite unified enough for their liking.

Early attempts to construct such a theory primarily centered on the rather clever strategy of incorporating electromagnetic fields directly into the geometry of general relativity . A notable milestone occurred in 1918, when Hermann Weyl first proposed a geometrization of the electromagnetic field. The following year, 1919, saw Theodor Kaluza introduce the revolutionary idea of a five-dimensional approach. This audacious concept eventually blossomed into a comprehensive framework known as Kaluza-Klein Theory . This theory, in its bold ambition, sought to unify both gravitation and electromagnetism not in our familiar four dimensions, but within a more expansive, five-dimensional space-time .

Various avenues for extending the representational framework of a unified field theory were explored by Einstein and his contemporaries. These extensions generally fell into two broad categories. The first involved relaxing some of the stringent conditions initially imposed on the original formulations, allowing for greater flexibility. A prime example of this approach is the aforementioned Kaluza-Klein Theory , which relaxes the restriction to four-dimensional space-time by venturing into higher-dimensional representations. The second major option involved introducing additional mathematical objects into the existing theoretical framework. The most prominent illustration of this path emerged from the concept of the affine connection , which found its way into the theory of general relativity largely through the foundational work of Tullio Levi-Civita and Hermann Weyl .

However, the relentless march of scientific progress, particularly the further development of quantum field theory , irrevocably shifted the focus of the quest for a unified field theory. The search moved decisively from classical descriptions to the more fundamental quantum realm. Because of this paradigm shift, many theoretical physicists, with a collective sigh of intellectual resignation, eventually abandoned the pursuit of a purely classical unified field theory. The quantum field theory promised a grander unification, one that would encompass not only gravity and electromagnetism but also the two other, more elusive fundamental forces of nature : the strong and weak nuclear force , which operate at the infinitesimal, subatomic level. A continuous quest for a single, elegant answer to the universe’s many questions, even if it requires discarding perfectly good classical theories along the way.

See also

• Relativistic wave equations
• Quantum field theory
• Classical unified field theories
• Variational methods in general relativity
• Higgs field (classical)
• Lagrangian (field theory)
• Hamiltonian field theory
• Covariant Hamiltonian field theory

Notes

• ^ This is contingent on the correct choice of gauge . φ and A are not uniquely determined by ρ and J; rather, they are only determined up to some scalar function f (r, t) known as the gauge. The retarded potential formalism requires one to choose the Lorenz gauge .
• ^ This is equivalent to choosing units of distance and time as light-seconds and seconds or light-years and years. Choosing c = 1 allows us to simplify the equations. For instance, E = mc² reduces to E = m (since c² = 1, without keeping track of units). This reduces complexity of the expressions while keeping focus on the underlying principles. This “trick” must be taken into account when performing actual numerical calculations.