An Alternative Way to Present a Relativistic Lagrangian Definition

Revista Brasileira de Ensino de F´ısica, vol. 43, e20210075 (2021) Articles www.scielo.br/rbef c b DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0075 LicençaCreative Commons An alternative way to present a relativistic Lagrangian definition M.A. De Andrade1, C. Neves*1 1Universidade do Estado do Rio de Janeiro, Faculdade de Tecnologia, Resende, RJ, Brasil. Received on February 26, 2021. Revised on June 25, 2021. Accepted on June 28, 2021. As long we known, the non-relativistic Lagrangian definition, L = T − U with T and U as being, respectively, kinetic and potential energy, it is not valid at the relativistic level. This issue partially obstructs the college student’s comprehension of special theory of relativity from the Lagrangian formalism. Further, it is well known that the space-time world demands essentials principles of special theory of relativity: four-vectors and Lorentz invariance. In this work, we departure from these basics concepts of space-time world in order to get a relativistic Lagrangian definition parallel to the non-relativistic one and, as a consequence, this can help the college student to better understand special theory of relativity [1]. Keywords: Special Relativity, Lorentz Invariance, Four-vector in space-time, Lagrangian and Hamiltonian formalism. 1. Introduction how these concepts help to formulate the relativistic Lagrangian in a straightforward and transparent way It is well known that the Lagrangian formalism, as well in order to facilitate the college student to understand as, Hamiltonian one are important to describe classical special theory of relativity from the Lagrangian point systems, because these formalisms reveal many impor- of view. tant features, symmetries [2, 3] for example, and allow In order to get a self-contained work, it is organized to quantize them through some quantization processes: as follows: in section 2 the basics of four-vectors and canonical [4] and path integral [5] quantization methods the importance of Lorentz invariance are reviewed. for instance. However, when it is considered relativistic After that, departing from this review, the relativistic systems, the usual non-relativistic Lagrangian definition Lagrangian of the free particle is obtained in a very clean (L = T − U) is not extended to the relativistic one, and straightforward way. In section 3, the steps that as the Hamiltonian formalism is, i.e., H = T + U. led us to obtain the relativistic Lagrangian of the free It happens because, unlike Hamiltonian, Lagrangian is particle given in the end of the section 2 are extended defined in the configuration space. This issue embraces to explore the dynamics of a generic interacting system. some concepts that are difficulty to the college student In section 4, it is considered a particle in a background to understand. Among others things, this is a point that electromagnetic field in order to illustrate how this pre- we want to clarify in this work. viously alternative relativistic Lagrangian formulation The relativistic Lagrangian formalism is usually pre- is more effective than the usual one. Consequently, it sented, through fundamental books in college, in an is possible to explore a parallel between the relativistic opaque way due to some assumptions considered and it and non-relativistic Lagrangian definitions, indeed, it is is not possible to do a parallel between non-relativistic obtained a relativistic Lagrangian definition parallel to and relativistic Lagrangian definitions. Precisely, the L = T −U. At the end, some conclusions and discussions non-relativistic Lagrangian definition, L = T − U, is are presented. not valid at the relativistic level. Consequently, this obliterates the student to better comprehend the special theory of relativity and, indeed, some gaps arise in the 2. Basics on Four-Vectors of the Special student comprehension of the special theory of relativity Relativity in the Lagrangian framework. In order to fill this gap and to promote a better way to teach special theory We start this Section describing the notation used here. of relativity from the Lagrangian formalism, at college The Greek indices indicate the four-dimensional space- level, we start to discuss the concepts and importance of time with the coordinates xµ = x0, xi while the Latin four-vector and Lorentz invariance. After that, we show indices indicate the standard three-dimensional space with i = 1, 2, 3. So the x0 and xi refer to the timelike * Correspondence email address: cliff[email protected] and the spacelike components, respectively. Copyright by Sociedade Brasileira de F´ısica.Printed in Brazil. e20210075-2 An alternative way to present a relativistic Lagrangian definition We would like to do a brief comment about the metric Whenever the motion of the particle is taken in definition. Consider a N-dimensional space (V N ) with consideration, space-time comes into play, then we must no special structure in which tensors play a role are use the metric tensor ηµν . With the intervention of µ metric spaces, i.e., they possess a rule which assigns the metric tensor, an additional condition for Λ ν , or “distances” to pairs of neighboring points. In particular, for the corresponding transformation matrix Λ, can be one calls a space (pseudo-)Riemannian if there exists an found in a way to provide invariant scalar forms in invariant quadratic differential form, namely, four-dimensional space-time, like the distance in the three-dimensional space. With the aim to simplify the 2 a b ds = gabdx dx , (1) procedure, we can employ the usual notation in which the space-time metric tensor (η ) and its inverse (ηµν ) with a, b = 1, 2,...,N and the g’s are generally func- µν can be used to lower or raise indices. Thus, using the tions of position and are subject only to the following inverse of the metric tensor to write the four-differentials restriction: det(g ) 6= 0. As it is required that ds2 is ab on both sides of the Eq. (5) with their indices lowered, invariant, it follows that g must be a tensor, which ab as well as in dx0µ = ηµαdx0 , it can be verified that is called the metric tensor, while Eq. (1) is the metric. α In pseudo-Riemannian spaces such as Minkowski space, 0 −1 ρ dxν = (ηΛη )ν dxρ. (10) the square of a vector, given in Eq. (1), can be a positive 0 or a negative real number and Eq. (1) is rewritten as So that dxν is covariant four-differential, whose contraction with its contravariant partner dx0ν results in an 2 µ ν ds = ηµν dx dx , (2) invariant scalar quantity, the transformation matrix in Eq. (10) must be the same as that found for the four- with the metric tensor ηµν , that reflects the nature of derivative operator in Eq. (9), therefore the space-time, as well as its inverse ηµν , reads as ηΛη−1 = Λ−1 T or ΛT ηΛ = η. (11) µν ηµν = η = diagonal (+, −, −, −), (3) The transformation of the four-differential performed by µ and, consequently, it is get the following identity, the Λ ν , which satisfies the Eq. (11), is called a Lorentz transformation1. Therefore, it assures the invariance of the quadratic form, i.e., the invariance of the metric, µν µ η ηνλ = δλ , (4) namely: µ 0µ 0 µ with δλ as the Kronecker delta. dx dxµ = dx dxµ = invariant. (12) The transformation law of a contravariant tensor, in particular a component four-differential dxρ, from A generic contravariant four-vector must have the one coordinate system {xµ} to another, {x0µ}, is transformation matrix satisfying the Eq. (11) and its expressed by transformation law, as that of the four-differential in Eq. (5), then 0µ µ ρ dx = Λ ρdx , (5) 0µ µ ν A = Λ ν A . (13) where, by the chain rule of differentiation, we have A generic covariant four-vector must have the transfor- ∂x0µ mation matrix satisfying the Eq. (11) and its transfor- Λµ = . (6) mation law, as that of the four-derivative in Eq. (9), ρ ∂xρ 0 −1ρ By using, again, the chain rule of differentiation, we can Bµ = Λ µBρ. (14) take the following trivial identity In such a way that the contraction of a contravariant 0µ ρ 0µ ρ ∂x ∂x ∂x four-vector with other covariant one, and vice-versa, Λµ Λ−1 = = = δµ. (7) ρ ν ∂xρ ∂x0ν ∂x0ν ν results in a Lorentz invariant scalar: 0µ 0 ν From which, we can read A Bµ = A Bν . (15) ρ ∂x ρ Any four-vector can be split into timelike and spacelike = Λ−1 . (8) ∂x0ν ν components and the metric tensor can be used to lower the index as After the contraction of both sides of the Eq. (8) with the ∂ Aµ = A0, A and A = η Aν = A0, −A , (16) four-derivative operator , we get the transformation µ µν ∂xρ law of the four-derivative operator as being 1 Lorentz transformation encompasses a variety of transformations like spatial rotations [3], Lorentz boosts [6], discrete symmetries ∂ ρ ∂ ρ = Λ−1 or ∂0 = Λ−1T ∂ , (9) transformations, let it be said that not all Lorentz-invariant laws ∂x0ν ν ∂xρ ν ν ρ are expressible as relations between 4-vectors and scalars; some require 4-tensors and, in quantum mechanics, spinors transforma- where the superscript T means the transpose operation. tions, which is a representation of Lorentz-invariant laws [7]. Revista Brasileira de Ensino de F´ısica, vol. 43, e20210075, 2021 DOI: https://doi.org/10.1590/1806-9126-RBEF-2021-0075 Andrade and Neves e20210075-3 where A0 is the timelike component and A is the vector obtained with the help of Eq. (20), Eq. (21) and Eq. (22). assembled with the spacelike components Ai.

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