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arxiv: 2108.07786 · v2 · submitted 2021-07-04 · ⚛️ physics.class-ph

Demystifying the Lagrangians of special relativity

Gerd Wagner , Matthew W. Guthrie This is my paper

Pith reviewed 2026-05-24 12:48 UTC · model grok-4.3

classification ⚛️ physics.class-ph
keywords special relativityLagrangian mechanicsLorentz transformationspacetime intervalelectromagnetic fieldsmass-energy equivalenceparticle dynamicsfield Lagrangians
0
0 comments X

The pith

Combining the Lagrangian approach to mechanics with the postulates of special relativity derives the invariance of spacetime intervals, the Lorentz transformation, the relativistic particle Lagrangian, and E=mc², while showing that the Lagr

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper establishes that special relativity integrates directly into the Lagrangian formalism of mechanics by starting from the standard variational principle and the two postulates of special relativity. It derives the invariance of the spacetime interval and the form of the Lorentz transformation as direct consequences. From there it constructs the Lagrangian for a relativistic free particle, obtains the transformation law for electromagnetic potentials, recovers E=mc², and proves that the electromagnetic field Lagrangian and its derived equations of motion are unchanged under Lorentz transformations. A reader would care because this places special relativity inside the same mathematical structure used for classical mechanics and field theory rather than treating it as a separate subject. The result supplies explicit Lagrangians whose invariance properties follow automatically from the postulates.

Core claim

By combining the Lagrangian approach to mechanics and the postulates of special relativity, the invariance of any spacetime interval and the Lorentz transformation are derived. This framework then yields the Lagrangian of a relativistic free particle, the transformation law of the electromagnetic potentials, Einstein's mass-energy equivalence law E=mc², and the explicit demonstration that the Lagrangians and equations of motion for the electric and magnetic fields remain invariant under Lorentz transformations.

What carries the argument

The action principle applied to a relativistic Lagrangian density, which enforces invariance of the spacetime interval and produces the Lorentz transformation as a symmetry of the variational equations.

If this is right

  • The mass-energy equivalence E=mc² is obtained directly from the relativistic free-particle Lagrangian.
  • The electromagnetic potentials acquire a definite transformation law under Lorentz transformations.
  • The equations of motion derived from the electromagnetic field Lagrangian remain the same in every inertial frame.
  • Special relativity appears as a natural extension of the variational methods already used in non-relativistic mechanics.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same variational construction could be applied to other relativistic systems, such as charged particles in external fields, to obtain their equations of motion without separate postulates.
  • Teaching sequences that introduce Lagrangians first and then add the relativity postulates would derive time dilation and length contraction as consequences rather than as independent statements.
  • The invariance proof for the field Lagrangian supplies a template that could be tested numerically by discretizing the action on a spacetime lattice and checking numerical covariance.

Load-bearing premise

The Lagrangian formulation of special relativity follows logically by combining the Lagrangian approach to mechanics and the postulates of special relativity.

What would settle it

An explicit calculation in which the variation of the electromagnetic field Lagrangian fails to produce Maxwell's equations whose form is preserved under a Lorentz boost.

read the original abstract

Special relativity beyond its basic treatment can be inaccessible, in particular because introductory physics courses typically view special relativity as decontextualized from the rest of physics. We seek to place special relativity back in its physics context, and to make the subject approachable. The Lagrangian formulation of special relativity follows logically by combining the Lagrangian approach to mechanics and the postulates of special relativity. In this paper, we derive and explicate some of the most important results of how the Lagrangian formalism and Lagrangians themselves behave in the context of special relativity. We derive two foundations of special relativity: the invariance of any spacetime interval, and the Lorentz transformation. We then develop the Lagrangian formulation of relativistic particle dynamics, including the transformation law of the electromagnetic potentials, the Lagrangian of a relativistic free particle, and Einstein's mass-energy equivalence law ($E=mc^2$). We include a discussion of relativistic field Lagrangians and their transformation properties, showing that the Lagrangians and the equations of motion for the electric and magnetic fields are indeed invariant under Lorentz transformations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

0 major / 3 minor

Summary. The paper claims that combining the Lagrangian formalism of mechanics with the two postulates of special relativity yields derivations of the invariance of the spacetime interval and the Lorentz transformations; it then constructs the Lagrangian for a relativistic free particle (including the transformation properties of the electromagnetic potentials), derives E=mc², and shows that the Lagrangians and equations of motion for the electric and magnetic fields are invariant under Lorentz transformations.

Significance. If the derivations hold without hidden assumptions, the work offers a self-contained route that embeds special relativity within the Lagrangian framework from the outset, which could aid pedagogy and conceptual unification. The explicit verification of Lorentz invariance for both particle and field Lagrangians is a standard result but is here obtained in a single logical sequence starting from the postulates.

minor comments (3)
  1. The abstract states that the spacetime-interval invariance and Lorentz transformation are derived from the Lagrangian approach plus the two postulates, but the manuscript should clarify in §2 or §3 whether the action principle is introduced before or after the postulates, to avoid any appearance of circularity in the choice of the interval as the invariant quantity.
  2. In the particle-dynamics section, the transformation law for the electromagnetic potentials is presented; the manuscript should add an explicit statement of the gauge freedom retained after the Lorentz transformation to make the derivation of the relativistic particle Lagrangian fully transparent.
  3. The field-Lagrangian invariance proof (final section) relies on the standard electromagnetic Lagrangian density; a brief remark on why the Proca or other massive-vector extensions are outside scope would help readers understand the generality of the invariance result.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive summary of the manuscript, recognition of its pedagogical value, and recommendation of minor revision. No major comments appear in the report.

Circularity Check

0 steps flagged

Derivation self-contained from postulates and Lagrangian formalism

full rationale

The paper explicitly starts from the two standard postulates of special relativity together with the Lagrangian formalism of mechanics. It derives spacetime-interval invariance and the Lorentz transformation as outputs rather than inputs. Particle and field Lagrangians are then constructed from those derived foundations, and their Lorentz invariance is demonstrated rather than presupposed. No equations reduce by construction to fitted parameters, self-citations, or ansatzes smuggled from prior work by the same authors. The chain is therefore independent of the target results.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The central claims rest on the two postulates of special relativity and the applicability of the Lagrangian formalism; no free parameters, new entities, or ad-hoc axioms are introduced in the abstract.

axioms (2)
  • domain assumption Postulates of special relativity (constancy of speed of light and equivalence of inertial frames)
    Invoked to derive invariance of spacetime interval and Lorentz transformations.
  • domain assumption Lagrangian mechanics can be applied directly to relativistic systems
    Central premise that allows the combination of Lagrangian formalism with relativity postulates.

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Reference graph

Works this paper leans on

60 extracted references · 60 canonical work pages · 5 internal anchors

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    Non-relativistic setup We observe a particle from within two IRFs T and ¯T . The axes of the frames point in the same direction, and ¯T moves with velocity v along T ’s x-axis such that their classical Galilean transformation is given by: ( ¯t ¯x ) = ( 1 0 −v 1 ) ( t x ) , ¯y =y, ¯z =z, (14) where t, x, y, z are the coordinates of the particle in T , and ...

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    ( 4) becomes c2¯t2 − ¯x2 − ¯y2 − ¯z2 =c2t2 − x2 − y2 − z2

    Ansatz for a transformation compatible with invariance o f the spacetime interval Because at t = ¯t = 0, the two reference frames T and ¯T are on top of each other, Eq. ( 4) becomes c2¯t2 − ¯x2 − ¯y2 − ¯z2 =c2t2 − x2 − y2 − z2. (15) As an ansatz for a coordinate transformation between the two re ference frames which is consistent with Eq. ( 15), we write:...

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    ( 15) and ansatz ( 16) result in the following equation [( a d b e ) ( ct x )] T ( 1 0 0 − 1 ) [( a d b e ) ( ct x )] = ( ct x )T ( 1 0 0 − 1 ) ( ct x )

    Finding the matrix elements of the ansatz Because ¯y =y, ¯z =z, Eq. ( 15) and ansatz ( 16) result in the following equation [( a d b e ) ( ct x )] T ( 1 0 0 − 1 ) [( a d b e ) ( ct x )] = ( ct x )T ( 1 0 0 − 1 ) ( ct x ) . (17) Since t and x are arbitrary the only way for this equation to be true is ( a b d e ) ( 1 0 0 − 1 ) ( a d b e ) = ( 1 0 0 − 1 ) (1...

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    To do so, we consider Eq

    Calculation of b We would now like to give a physical interpretation of b. To do so, we consider Eq. ( 15) for the special case where the particle moves with the origin of reference frame ¯T . We first use ¯y =y, and ¯z =z to simplify Eq. ( 15) to c2¯t2 − ¯x2 =c2t2 − x2. (32) Since the particle resides at the origin of ¯T , ¯x is zero which simplifies Eq. (...

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    ( 31), we first consider √ 1 +b2 = √ 1 + v2/c 2 1 − v2 c2 = √ 1 − v2/c 2 +v2/c 2 1 − v2 c2 = 1√ 1 − v2 c2

    Lorentz transformation In preparation to use our representation of b into Eq. ( 31), we first consider √ 1 +b2 = √ 1 + v2/c 2 1 − v2 c2 = √ 1 − v2/c 2 +v2/c 2 1 − v2 c2 = 1√ 1 − v2 c2 . (42) 7 Using this, Eq. ( 31) turns into ( ¯t ¯x ) =    1 √ 1− v2 c2 − v/c 2 √ 1− v2 c2 − v√ 1− v2 c2 1√ 1− v2 c2    ( t x ) . (43) Eq. (

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    is called the Lorentz transformation

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    4-velocity

    Interpretation and generalization This result is valid with rigor for the IRF ¯T in which the particle is at rest at the IRF’s origin. Even if the particle is accelerated along the x axis, for small time intervals there always exists such an IRF. Those IRFs are called the particle’s momentary rest IRFs. If we look at Eq. ( 43) in its full 4 dimensional fo...

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    In this section we take the coordinates to be of the form ( ct,x 1,x 2,x 3)

    we gave the Lorentz transformation for differences of the coor dinates (t,x,y,z ). In this section we take the coordinates to be of the form ( ct,x 1,x 2,x 3). This will change the Lorentz transformation slightly: it will require reverting what we did in Eq. ( 28) to Eq. ( 31). For the rest of this paper we will be interested in the Lorentz transformation ...

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    4-vector

    is a direct consequence of Eq. ( 4); it is the formal definition of a Lorentz transformation, i.e. any 4 × 4 matrix that fulfills this equation is a Lorentz transformation of ∆X. With these definitions, the transformation of ∆ X by Λ is given by matrix-vector multiplication Λ (∆X). Up to now we know two 4-component objects that transform like ∆X. Those are ∆...

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    Energy conservation The Lagrangian formalism for particle physics, as described in Ref. [ 1], allows us to derive energy conservation from analyzing how the action S behaves under infinitesimal time translations. By doing so, we will find a definition for the energy of any physical system that is described by a partic le Lagrangian. To begin, we consider the...

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    The Lagrangian that corresponds to the Lorentz f orce is given by LL = −e(φ − A ·v), (76) where φ and A are respectively the electric and magnetic potentials as discussed in Ref

    The Lorentz force and its Lagrangian The Lorentz force on a particle with charge e in Cartesian coordinates is given by FL =eE +ev × B, (75) where v is the particle’s velocity and E and B are the electric and magnetic fields at the particle’s coordinates, respectively. The Lagrangian that corresponds to the Lorentz f orce is given by LL = −e(φ − A ·v), (76...

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    Invariance of the relativistic particle Euler-Lagrange equations a. Preliminaries and notations The invariance of the Euler-Lagrange equations under wide ranges of transformations has always been our crucial argument to accept the Lagrangian formalism. We already used this argument to introduce and rectify the Lagrangian formalism for classical mechanics ...

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    ( 101), the last summand vanishes and we are left with δS = ∫ ϕ (¯t2) ϕ (¯t1)   ∂L ∂f − d dϕ   ∂L ∂ ( df dϕ )     δf dϕ

    into δS = ∫ ϕ (¯t2) ϕ (¯t1) ∂L ∂f δf − d dϕ   ∂L ∂ ( df dϕ )   δf dϕ +   ∂L ∂ ( df dϕ )δf   ϕ (¯t2) ϕ (¯t1) (103) Because of the boundary condition defined in Eq. ( 101), the last summand vanishes and we are left with δS = ∫ ϕ (¯t2) ϕ (¯t1)   ∂L ∂f − d dϕ   ∂L ∂ ( df dϕ )     δf dϕ. (104) Also, because of the relation in Eq. ( 100), δf is a...

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    (114) 17 In an observer IRF with time t and where the particle’s velocity is v =v(t), we may write Eq

    Heuristic argument A common heuristic argument to derive the results of equations ( 112) and (113) is to claim that, for special relativity, the action integral must be taken over proper time τ instead of classical time t: S = ∫ τ2 τ1 L dτ. (114) 17 In an observer IRF with time t and where the particle’s velocity is v =v(t), we may write Eq. ( 114) in the...

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    Principle of Relativity

    Einstein ’s original first postulate Einstein’s original first postulate is different from the one we discuss ed in the end of section IV B 1 f. In his first paper on relativity [ 18], Einstein writes: “... suggest that the phenomena of electrodynamics as well as of me chanics possess no properties corresponding to the idea of absolute rest. They suggest rath...

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    ( 76), the Lagrangian of the Lorentz force is given by LL = −e(φ − A ·v) = −e(φ − A ·v)√ 1 − v2 c2 √ 1 − v2 c2

    Application to the Lorentz force law According to Eq. ( 76), the Lagrangian of the Lorentz force is given by LL = −e(φ − A ·v) = −e(φ − A ·v)√ 1 − v2 c2 √ 1 − v2 c2. (116) Section IV B 1 f tells us that the Lorentz force will be the same in two IRFs ¯T and T when −e( ¯φ − ¯A ·¯v)√ 1 − ¯v2 c2 = −e(φ − A ·v)√ 1 − v2 c2 , (117) where • ¯φ and ¯A denote the e...

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    Energy of the relativistic free particle ( E = mc2) Using Eq. ( 72), the energy of the free particle can be calculated from Eq. ( 128) as follows: E = ∂L ∂vv − L (129) = −mc2 1 2 √ 1 − v2 c2 ( − 2v c2 ) ·v − ( −mc2 √ 1 − v2 c2 ) (130) =mc2   v2/c 2 √ 1 − v2 c2 + √ 1 − v2 c2   (131) =mc2   v2/c 2 + 1 − v2/c 2 √ 1 − v2 c2   (132) = mc2 √ 1 − v2 c2 ....

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    The equations of motion of the free relativistic particle From the result of Eq. ( 128), the equations of motion for the free relativistic particle are given by 0 = d dt ∂ ∂vi ( −mc2 √ 1 − v2 c2 ) − ∂ ∂xi ( −mc2 √ 1 − v2 c2 ) ⇐ ⇒ 0 = d dt ∂ ∂vi ( −mc2 √ 1 − v2 c2 ) ⇐ ⇒ 0 = d dt ∂L ∂vi for i ∈ { 1, 2, 3}. (134) 20 We start with ∂L ∂vi = −mc2 1 2 √ 1 − v2 c...

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    (140) The equations of motion of the relativistic particle with charge e in an electromagnetic field can be calculated from Eq

    The Lagrangian and equations of motion of a relativistic p article in an electromagnetic field Using the results from sections IV B 4 and IV B 5, we can write down the Lagrangian of the relativistic particle with chargee in an electromagnetic field: L = −mc2 √ 1 − v2 c2 − e(φ − A ·v) = ( −mc2 − e (φ c,A 1,A 2,A 3 ) gu ) √ 1 − v2 c2. (140) The equations of m...

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