Canonical Energy-momentum Tensor Research Articles

Side-jump induced spin-orbit interaction of chiral fluids from kinetic theory

We apply the Wigner-function approach and chiral kinetic theory to investigate the angular momentum and polarization of chiral fluids composed of Weyl fermions with background electric/magnetic fields and vorticity. It is found that the quantum corrections in Wigner functions give rise to nonzero anti-symmetric components in the canonical energy-momentum tensors, which are responsible for the spin-orbit interaction. In global equilibrium, conservation of the canonical angular momentum reveals the cancellation between the orbital component stemming from side jumps with nonzero vorticity and the spin component in the presence of an axial chemical potential. We further analyze the conservation laws near local equilibrium. It turns out that the canonical angular momentum is no longer conserved even in the absence of background fields due to the presence of a local torque coming from the spin-orbit interaction involving temperature/chemical-potential gradients, which is implicitly led by collisions.

On the \u2018simple\u2019 form of the gravitational action and the self-interacting graviton

The so-called ΓΓ-form of the gravitational Lagrangian, long known to provide its most compact expression as well as the most efficient generation of the graviton vertices, is taken as the starting point for discussing General Relativity as a theory of the self-interacting graviton. A straightforward but general method of converting to a covariant formulation by the introduction of a reference metric is given. It is used to recast the Einstein field equation as the equation of motion of a spin-2 particle interacting with the canonical energy-momentum tensor symmetrized by the standard Belinfante method applicable to any field carrying nonzero spin. This represents the graviton field equation in a form complying with the precepts of standard field theory. It is then shown how representations based on other, at face value completely unrelated definitions of energy-momentum (pseudo)tensors are all related by the addition of appropriate superpotential terms. Specifically, the superpotentials are explicitly constructed which connect to: i) the common definition consisting simply of the nonlinear part of the Einstein tensor; ii) the Landau-Lifshitz definition.

The energy–momentum tensor(s) in classical gauge theories

We give an introduction to, and review of, the energy–momentum tensors in classical gauge field theories in Minkowski space, and to some extent also in curved space–time. For the canonical energy–momentum tensor of non-Abelian gauge fields and of matter fields coupled to such fields, we present a new and simple improvement procedure based on gauge invariance for constructing a gauge invariant, symmetric energy–momentum tensor. The relationship with the Einstein–Hilbert tensor following from the coupling to a gravitational field is also discussed.

Energy, momentum and angular momentum conservations in de Sitter gravity

In de Sitter (dS) gravity, where gravity is a gauge field introduced to realize the local dS invariance of the matter field, two kinds of conservation laws are derived. The first kind is a differential equation for a dS-covariant current, which unites the canonical energy-momentum (EM) and angular momentum (AM) tensors. The second kind presents a dS-invariant current which is conserved in the sense that its torsion-free divergence vanishes. The dS-invariant current unites the total (matter plus gravity) EM and AM currents. It is well known that the AM current contains an inherent part, called the spin current. Here it is shown that the EM tensor also contains an inherent part, which might be observed by its contribution to the deviation of the dust particle’s world line from a geodesic. All the results are compared to the ordinary Lorentz gravity.

Einstein׳s physical strategy, energy conservation, symmetries, and stability: “But Grossmann & I believed that the conservation laws were not satisfied”

Recent work on the history of General Relativity by Renn et al. shows that Einstein found his field equations partly by a physical strategy including the Newtonian limit, the electromagnetic analogy, and energy conservation. Such themes are similar to those later used by particle physicists. How do Einstein׳s physical strategy and the particle physics derivations compare? What energy–momentum complex(es) did he use and why? Did Einstein tie conservation to symmetries, and if so, to which? How did his work relate to emerging knowledge (1911–1914) of the canonical energy–momentum tensor and its translation-induced conservation? After initially using energy–momentum tensors hand-crafted from the gravitational field equations, Einstein used an identity from his assumed linear coordinate covariance xμ′=Mνμxν to relate it to the canonical tensor. Usually he avoided using matter Euler–Lagrange equations and so was not well positioned to use or reinvent the Herglotz–Mie–Born understanding that the canonical tensor was conserved due to translation symmetries, a result with roots in Lagrange, Hamilton and Jacobi. Whereas Mie and Born were concerned about the canonical tensor׳s asymmetry, Einstein did not need to worry because his Entwurf Lagrangian is modeled not so much on Maxwell׳s theory (which avoids negative-energies but gets an asymmetric canonical tensor as a result) as on a scalar theory (the Newtonian limit). Einstein׳s theory thus has a symmetric canonical energy–momentum tensor. But as a result, it also has 3 negative-energy field degrees of freedom (later called “ghosts” in particle physics). Thus the Entwurf theory fails a 1920s–1930s a priori particle physics stability test with antecedents in Lagrange׳s and Dirichlet׳s stability work; one might anticipate possible gravitational instability.This critique of the Entwurf theory can be compared with Einstein׳s 1915 critique of his Entwurf theory for not admitting rotating coordinates and not getting Mercury׳s perihelion right. One can live with absolute rotation but cannot live with instability.Particle physics also can be useful in the historiography of gravity and space–time, both in assessing the growth of objective knowledge and in suggesting novel lines of inquiry to see whether and how Einstein faced the substantially mathematical issues later encountered in particle physics. This topic can be a useful case study in the history of science on recently reconsidered questions of presentism, whiggism and the like. Future work will show how the history of General Relativity, especially Noether׳s work, sheds light on particle physics.

Energy-momentum tensors in linearized Einstein’s theory and massive gravity: The question of uniqueness

The question of the uniqueness of energy-momentum tensors in the linearized general relativity and in the linear massive gravity is analyzed without using variational techniques. We start from a natural ansatz for the form of the tensor (for example, that it is a linear combination of the terms quadratic in the first derivatives), and require it to be conserved as a consequence of field equations. In the case of the linear gravity in a general gauge we find a four-parametric system of conserved second-rank tensors which contains a unique symmetric tensor. This turns out to be the linearized Landau-Lifshitz pseudotensor employed often in full general relativity. We elucidate the relation of the four-parametric system to the expression proposed recently by Butcher et al. "on physical grounds" in harmonic gauge, and we show that the results coincide in the case of high-frequency waves in vacuum after a suitable averaging. In the massive gravity we show how one can arrive at the expression which coincides with the "generalized linear symmetric Landau-Lifshitz" tensor. However, there exists another uniquely given simpler symmetric tensor which can be obtained by adding the divergence of a suitable superpotential to the canonical energy-momentum tensor following from the Fierz-Pauli action. In contrast to the symmetric tensor derived by the Belinfante procedure which involves the second derivatives of the field variables, this expression contains only the field and its first derivatives. It is simpler than the generalized Landau-Lifshitz tensor but both yield the same total quantities since they differ by the divergence of a superpotential. We also discuss the role of the gauge conditions in the proofs of the uniqueness. In the Appendix, the use of the symbolic tensor manipulation software Cadabra is briefly described.

The gauge-invariant canonical energy-momentum tensor

The canonical energy-momentum tensor is often considered as a purely academic object because of its gauge dependence. However, it has recently been realized that canonical quantities can in fact be defined in a gauge-invariant way provided that strict locality is abandoned, the non-local aspect being dictacted in high-energy physics by the factorization theorems. Using the general techniques for the parametrization of non-local parton correlators, we provide for the first time a complete parametrization of the energy-momentum tensor (generalizing the purely local parametrizations of Ji and Bakker-Leader-Trueman used for the kinetic energy-momentum tensor) and identify explicitly the parts accessible from measurable two-parton distribution functions (TMDs and GPDs). As by-products, we confirm the absence of model-independent relations between TMDs and parton orbital angular momentum, recover in a much simpler way the Burkardt sum rule and derive three similar new sum rules expressing the conservation of transverse momentum.

On the Definition of Energy for a Continuum, Its Conservation Laws, and the Energy-Momentum Tensor

We review the energy concept in the case of a continuum or a system of fields. First, we analyze the emergence of a true local conservation equation for the energy of a continuous medium, taking the example of an isentropic continuum in Newtonian gravity. Next, we consider a continuum or a system of fields in special relativity: we recall that the conservation of the energy-momentum tensor contains two local conservation equations of the same kind as before. We show that both of these equations depend on the reference frame and that, however, they can be given a rigorous meaning. Then, we review the definitions of the canonical and Hilbert energy-momentum tensors from a Lagrangian through the principle of stationary action in general space-time. Using relatively elementary mathematics, we prove precise results regarding the definition of the Hilbert tensor field, its uniqueness, and its tensoriality. We recall the meaning of its covariant conservation equation. We end with a proof of uniqueness of the energy density and flux, when both depend polynomially on the fields.

Higher derivative extensions of 3d Chern–Simons models: conservation laws and stability

We consider the class of higher derivative $3d$ vector field models with the field equation operator being a polynomial of the Chern-Simons operator. For $n$-th order theory of this type, we provide a general receipt for constructing $n$-parameter family of conserved second rank tensors. The family includes the canonical energy-momentum tensor, which is unbounded, while there are bounded conserved tensors that provide classical stability of the system for certain combinations of the parameters in the Lagrangian. We also demonstrate the examples of consistent interactions which are compatible with the requirement of stability.

The light-front gauge-invariant energy-momentum tensor

We provide for the first time a complete parametrization for the matrix elements of the generic asymmetric, non-local and gauge-invariant canonical energy-momentum tensor, generalizing therefore former works on the symmetric, local and gauge-invariant kinetic energy-momentum tensor also known as the Belinfante-Rosenfeld energy-momentum tensor. We discuss in detail the various constraints imposed by non-locality, linear and angular momentum conservation. We also derive the relations with two-parton generalized and transverse-momentum dependent distributions, clarifying what can be learned from the latter. In particular, we show explicitly that two-parton transverse-momentum dependent distributions cannot provide any model-independent information about the parton orbital angular momentum. On the way, we recover the Burkardt sum rule and obtain similar new sum rules for higher-twist distributions.

The truth about the energy-momentum tensor and pseudotensor

The operational and canonical definitions of an energy-momentum tensor (EMT) are considered as well as the tensor and nontensor conservation laws. It is shown that the canonical EMT contradicts the experiments and the operational definition, the Belinfante-Rosenfeld procedure worsens the situation, and the nontensor “conservation laws” are meaningless. A definition of the 4-momentum of a system demands a translator since integration of vectors is meaningless. The mass of a fluid sphere is calculated. It is shown that, according to the standard energy-momentum pseudotensor, the mass-energy of a gravitational field is positive. This contradicts the idea of a negative gravitational energy and discredits the pseudotensor. And what is more, integral 4-pseudovectors are meaningless in general since reference frames for their components are not determined even for coordinates which are Minkowskian at infinity.

On the energy-momentum and spin tensors in the Riemann–Cartan space

General classical theories of material fields in an arbitrary Riemann-Cartan space are considered. For these theories, with the help of equations of balance, new non-trivially generalized, manifestly generally covariant expressions for canonical energy-momentum and spin tensors are constructed in the cases when a Lagrangian contains (a) an arbitrary set of tensorial material fields and their covariant derivatives up to the second order, as well as (b) the curvature tensor and (c) the torsion tensor with its covariant derivatives up to the second order. A non-trivial manifestly generally covariant generalization of the Belinfante symmetrization procedure, suitable for an arbitrary Riemann-Cartan space, is carried out. A covariant symmetrized energy-momentum tensor is constructed in a general form.

Covariant differential identities and conservation laws in metric-torsion theories of gravitation. II. Manifestly generally covariant theories

The present paper continues the work of Lompay and Petrov [J. Math. Phys. 54, 062504 (2013)] where manifestly covariant differential identities and conserved quantities in generally covariant metric-torsion theories of gravity of the most general type have been constructed. Here, we study these theories presented more concretely, setting that their Lagrangians \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}$\end{document}L are manifestly generally covariant scalars: algebraic functions of contractions of tensor functions and their covariant derivatives. It is assumed that Lagrangians depend on metric tensor g, curvature tensor R, torsion tensor T and its first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\mathbf {T}$\end{document}∇T and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\mathbf {T}$\end{document}∇∇T covariant derivatives, besides, on an arbitrary set of other tensor (matter) fields \documentclass[12pt]{minimal}\begin{document}$\bm {\varphi }$\end{document}φ and their first \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}\bm {\varphi }$\end{document}∇φ and second \documentclass[12pt]{minimal}\begin{document}${\bm {\nabla }}{\bm {\nabla }}\bm {\varphi }$\end{document}∇∇φ covariant derivatives: \documentclass[12pt]{minimal}\begin{document}$\mathscr {L}= \mathscr {L}(\mathbf {g},\mathbf {R}; \;\mathbf {T},{\bm {\nabla }}\mathbf {T},{\bm {\nabla }}{\bm {\nabla }}\mathbf {T}; \;\bm {\varphi },{\bm {\nabla }}\bm {\varphi },{\bm {\nabla }}{\bm {\nabla }}\bm {\varphi })$\end{document}L=L(g,R;T,∇T,∇∇T;φ,∇φ,∇∇φ). Thus, both the standard minimal coupling with the Riemann-Cartan geometry and non-minimal coupling with the curvature and torsion tensors are considered. The studies and results are as follow: (a) A physical interpretation of the Noether and Klein identities is examined. It was found that they are the basis for constructing equations of balance of energy-momentum tensors of various types (canonical, metrical, and Belinfante symmetrized). The equations of balance are presented. (b) Using the generalized equations of balance, new (generalized) manifestly generally covariant expressions for canonical energy-momentum and spin tensors of the matter fields are constructed. In the cases, when the matter Lagrangian contains both the higher derivatives and non-minimal coupling with curvature and torsion, such generalizations are non-trivial. (c) The Belinfante procedure is generalized for an arbitrary Riemann-Cartan space. (d) A more convenient in applications generalized expression for the canonical superpotential is obtained. (e) A total system of equations for the gravitational fields and matter sources are presented in the form more naturally generalizing the Einstein-Cartan equations with matter. This result, being a one of the more important results itself, is to be also a basis for constructing physically sensible conservation laws and their applications.

Relativistic Lagrangian model of a nematic liquid crystal interacting with an electromagnetic field

We develop a relativistic variational model for a nematic liquid crystal interacting with an electromagnetic field. The constitutive relation for a general anisotropic uniaxial diamagnetic and dielectric medium is analyzed. We discuss light wave propagation in this moving uniaxial medium, for which the corresponding optical metrics are identified explicitly. A Lagrangian for the coupled system of a nematic liquid crystal and the electromagnetic field is constructed, from which a complete set of equations of motion for the system is derived. The canonical energy-momentum and spin tensors are systematically obtained. We compare our results with those within the nonrelativistic models. As an application of our general formalism, we discuss the so-called Abraham-Minkowski controversy on the momentum of light in a medium.

ENERGY-MOMENTUM TENSORS WITH WORLDLINE NUMERICS

We apply the worldline formalism and its numerical Monte-Carlo approach to computations of fluctuation induced energy-momentum tensors. For the case of a fluctuating Dirichlet scalar, we derive explicit worldline expressions for the components of the canonical energy-momentum tensor that are straightforwardly accessible to partly analytical and generally numerical evaluation. We present several simple proof-of-principle examples, demonstrating that efficient numerical evaluation is possible at low cost. Our methods can be applied to an investigation of positive-energy conditions.

Erratum to: General spherically symmetric elastic stars in relativity

As pointed out right after equation (BCV-8) in [1], the symmetric energy-momentum tensor taking part in the Einstein field equations is the negative of the so-called canonical energy-momentum tensor (see also the paragraph following equation (BCV-6) in that paper). Equation (BCV-33) in section 4 of [1] corresponds to the canonical energymomentum tensor, and therefore, the paragraph right before (BCV-33) should better start: In [4], the canonical energy-momentum..., and therefore,

Derivation of energy-momentum tensors in theories of micropolar hyperbolic thermoelasticity

In the framework of the classical field theory and using the theory of action variational symmetries, we consider the construction of canonical energy-momentum tensors for a coupled micropolar thermoelastic field taking account of the nonlocality of the Lagrangian density, which is typical of continuum micromechanics. We use the algorithms of group analysis to calculate the Noether currents and the energy-momentum tensors in three cases where the Lagrangian depends on the gradients of field variables of orders not exceeding 1, 2, and 3. In each of these cases, we present explicit formulas for the components of the canonical energy-momentum tensor. We construct the energy-momentum tensor for micropolar thermoelastic bodies in which the heat conduction process is characterized by a generalized heat equation of hyperbolic analytical type. In the equations of micropolar thermoelastic field, all possible restrictions on the microrotations are taken into account.

Maxwell–Chern–Simons Topologically Massive Gauge Fields in the First-Order Formalism

We find the canonical and Belinfante energy-momentum tensors and their nonzero traces. We note that the dilatation symmetry is broken and the divergence of the dilatation current is proportional to the topological mass of the gauge field. It was demonstrated that the gauge field possesses the `scale dimensionality' d=1/2. Maxwell - Chern - Simons topologically massive gauge field theory in 2+1 dimensions is formulated in the first-order formalism. It is shown that 6x6-matrices of the relativistic wave equation obey the Duffin - Kemmer - Petiau algebra. The Hermitianizing matrix of the relativistic wave equation is given. The projection operators extracting solutions of field equations for states with definite energy-momentum and spin are obtained. The 5x5-matrix Schrodinger form of the equation is derived after the exclusion of non-dynamical components, and the quantum-mechanical Hamiltonian is obtained. Projection operators extracting physical states in the Schrodinger picture are found.

Noether derivation of exact conservation laws for dissipationless reduced-fluid models

The energy-momentum conservation laws for general reduced-fluid (e.g., gyrofluid) models are derived by Noether method from a general reduced variational principle. The reduced canonical energy-momentum tensor (which is explicitly asymmetric and has the Minkowski form) exhibits polarization and magnetization effects associated with dynamical reduction. In particular, the asymmetry in the reduced canonical momentum-stress tensor produces a nonvanishing reduced intrinsic torque that can drive spontaneous toroidal rotation in axisymmetric tokamak plasmas.

Note on Dirac–Kähler massless fields

We obtain the canonical and symmetrical Belinfante energy-momentum tensors of Dirac--K\"{a}hler's fields. It is shown that the traces of the energy-momentum tensors are not equal to zero. We find the canonical and Belinfante dilatation currents which are not conserved, but a new conserved dilatation current is obtained. It is pointed out that the conformal symmetry is broken. The canonical quantization is performed and the propagator of the massless fields in the first-order formalism is found.