Symmetric Energy-momentum Tensor Research Articles

A note on generalized weakly ℋ-symmetric manifolds and relativistic applications

In this work, generalized weakly [Formula: see text]-symmetric space-times (GWHS)[Formula: see text] are investigated, where [Formula: see text] is any symmetric [Formula: see text] tensor. It is proved that, in a nontrivial (GWHS)[Formula: see text] space-time, the tensor [Formula: see text] has a perfect fluid form. Accordingly, sufficient conditions for a nontrivial generalized weakly Ricci symmetric space-time (GWRS)[Formula: see text] to be either an Einstein space-time or a perfect fluid space-time are obtained. Also, conditions for space-times admitting either a generalized weakly symmetric energy-momentum tensor or a generalized weakly symmetric [Formula: see text] tensor to be Einstein or perfect fluid space-times are provided.

Energy–momentum tensor in the scalar diquark model

We study in detail the energy–momentum tensor of the scalar diquark model at the one-loop level using two different regularization methods. We extract the perturbative expressions for the gravitational form factors and we check explicitly that all the Poincaré sum rules are satisfied. We confirm in particular that, like in quantum electrodynamics, the symmetric energy–momentum tensor is finite. Finally, we discuss the results for the trace of the energy–momentum tensor and its relation to the mass of the system.

Some remarks on relativistic fluids of divergence type

Relativistic fluids of divergence type constitute a special class of fluids whose states are determined from the knowledge of a scalar generating function χ and a dissipation tensor i.e. a second rank, symmetric and traceless tensor. Both depend upon fourteen variables, known in the literature as Lagrange multipliers, and these multipliers satisfy a symmetric, quasilinear, first order system determined by . For particular choices of the generating function χ, this system can be symmetric-hyperbolic and causal. We show in this work that this characteristic property of fluids of divergence type originates in the overdetermined nature of their dynamical equations. Combining their overdetermined nature with the work of Friedrichs on overdetermined system of conservation laws with more recent work by Boillat, Ruggeri and coworkers, we prove that fluids of divergence type are locally determined by a vector potential depending upon the Lagrange multipliers. For the case where the dynamical variables describing fluid states contain a symmetric energy momentum tensor, this vector potential is the gradient of a scalar field and this field is precisely the generating function χ introduced by Pennisi and independently by Geroch and Lindblom. Examples of scalar generating functions χ are discussed where this system is a symmetric hyperbolic and causal system in an open vicinity of equilibrium states.

The Disappearance and Reappearance of Potential Energy in Classical and Quantum Electrodynamics

In electrostatics, we can use either potential energy or field energy to ensure conservation of energy. In electrodynamics, the former option is unavailable. To ensure conservation of energy, we must attribute energy to the electromagnetic field and, in particular, to electromagnetic radiation. If we adopt the standard energy density for the electromagnetic field, then potential energy seems to disappear. However, a closer look at electrodynamics shows that this conclusion actually depends on the kind of matter being considered. Although we cannot get by without attributing energy to the electromagnetic field, matter may still have electromagnetic potential energy. Indeed, if we take the matter to be represented by the Dirac field (in a classical precursor to quantum electrodynamics), then it will possess potential energy (as can be seen by examining the symmetric energy-momentum tensor of the Dirac field). Thus, potential energy reappears. Upon field quantization, the potential energy of the Dirac field becomes an interaction term in the Hamiltonian operator of quantum electrodynamics.

Parametrization of transition energy-momentum tensor form factors

While the Lorentz structures of the N→Δ transition matrix element of the vector current are well-known, that of the symmetric energy-momentum tensor current is unknown. In this letter, we first reproduce the Lorentz structures of the 12±→12±, 12∓→12±, and 12∓→32± transition matrix elements of the vector current. We also repeat the parametrizations of the symmetric energy-momentum tensor current for the 12±→12± and 12∓→12± transitions. We then consider both 12±→32± and 12∓→32± matrix elements of the symmetric energy-momentum tensor current (say, N→Δ transition) for the first time. We found that there are five independent conserved and four independent non-conserved energy-momentum tensor form factors.

Proton gravitational form factors and mechanical properties in a light-front quark-diquark model

We present our recent calculation of the gravitational form factors (GFFs) of proton using the light-front quark-diquark model constructed by the soft-wall AdS/QCD. We extract the four quark GFFs by calculating the matrix elements of the symmetric energy momentum tensor. Using the D-term we calculate the pressure and shear distributions of quarks inside the proton in the impact parameter space. The GFFs, A(Q2) and B(Q2) are found to be consistent with the lattice QCD, while the qualitative behavior of the D-term form factor is in agreement with the extracted data from the deeply virtual Compton scattering (DVCS) experiments at JLab, the lattice QCD, and the predictions of different phenomenological models.

Multivector Fields of Noether Symmetries in the Lagrangian Formalism and Belinfante Tensor

Elsewhere, we gave the explicit expressions of the multivectors fields associated to infinitesimal symmetries which gave rise to Noether currents for classical field theories and relativistic mechanic using the Second Order Partial Differential Equation SOPDE condition for the Poincar\'e-Cartan form.\\ The main objective of this paper is to reformulate the multivector fields associated to translational and rotational symmetries of the gauge fields in particular those of the electromagnetic field which gave rise to symmetrical and invariant gauge energy-momentum tensor and the orbital angular momentum. The spin angular momentum appears however because of the internal symmetry inside the fiber.

Spin hydrodynamics and symmetric energy-momentum tensors – A current induced by the spin vorticity –

We discuss a puzzle in relativistic spin hydrodynamics; in the previous formulation the spin source from the antisymmetric part of the canonical energy-momentum tensor (EMT) is crucial. The Belinfante improved EMT is pseudo-gauge transformed from the canonical EMT and is usually a physically sensible choice especially when gauge fields are coupled as in magnetohydrodynamics, but the Belinfante EMT has no antisymmetric part. We find that pseudo-transformed entropy currents are physically inequivalent in nonequilibrium situations. We also identify a current induced by the spin vorticity read from the Belinfante symmetric EMT.

Canonical Noether and the energy–momentum non-uniqueness problem in linearized gravity

Recent research has highlighted the non-uniqueness problem of energy–momentum tensors in linearized gravity; many different tensors are published in the literature, yet for particular calculations a unique expression is required. It has been shown that (A) none of these spin-2 energy–momentum tensors are gauge invariant and (B) the Noether and Hilbert energy–momentum tensors are not, in general, equivalent; therefore uniqueness criteria is difficult to specify. Conventional wisdom states that the various published energy–momentum tensors for linearized gravity can be derived from the canonical Noether energy–momentum tensor of spin-2 Fierz-Pauli theory by adding ad-hoc ‘improvement’ terms (the divergence of a superpotential and terms proportional to the equations of motion), that these superpotentials are in some way unique or physically significant, and that this implies some meaningful connection to the Noether procedure. To explore this question of uniqueness, we consider the most general possible energy–momentum tensor for linearized gravity with free coefficients using the Fock method. We express this most general energy–momentum tensor as the canonical Noether tensor, supplemented by the divergence of a general superpotential plus all possible terms proportional to the equations of motion. We then derive systems of equations which we solve in order to prove several key results for spin-2 Fierz–Pauli theory, most notably that there are infinitely many conserved energy–momentum tensors derivable from the ‘improvement’ method, and there are infinitely many conserved symmetric energy–momentum tensors that follow from specifying the Belinfante superpotential alone. This disproves several recent claims that the Belinfante tensor is uniquely associated to the Hilbert tensor in spin-2 Fierz–Pauli theory. We give two new energy–momentum tensors of this form. Most importantly, since there are infinitely many energy–momentum tensors of this form, no meaningful or unique connection to Noether’s first theorem can be claimed by application of the canonical Noether ‘improvement’ method.

Double-logarithmic nonlinear electrodynamics

A new model of nonlinear electrodynamics is introduced and investigated. The theory carries one dimensionful parameter β as in Born-Infeld electrodynamics. It is shown that the dual symmetry and dilatation (scale) symmetry are broken in the proposed model. The electric field of a point-like charge is derived for this model, showing that it is non-singular at the origin. Using this electric field, the static electric energy of a point-like charge is calculated. In the presence of an external magnetic field, the theory shows a phenomenon known as vacuum birefringence. The refraction index of two polarizations, parallel and perpendicular to the external magnetic induction field, are calculated. The canonical and symmetrical Belinfante energy-momentum tensors are obtained. Using the causality and unitarity principles, the regions where the theory becomes causal and unitary are found.

Energy-Momentum Relocalization, Surface Terms, and Massless Poles in Axial Current Matrix Elements

The energy-momentum relocalization in classical and quantum theory is addressed with specific impact on non-perturbative QCD and hadronic structure. The relocalization is manifested in the existence of canonical and symmetric (Belinfante and Hilbert) energy momentum tensors (EMT). The latter describes the interactions of hadrons with classical gravity and inertia. Canonical EMT, in turn, is naturally emerging due to the translation invariance symmetry and appears when spin structure of hadrons is considered. Its relation to symmetric Hilbert and Belinfante EMTs requires the possibility to neglect the contribution of boundary terms for the classical fields. For the case of quantum fields this property corresponds to the absence of zero-momentum poles of matrix element of the axial current dual to the spin density. This property is satisfied for quarks manifesting the symmetry counterpart of UA(1) problem and may be violated for gluons due to QCD ghost pole.

Gravitational transition form factors of N(1535)→N

We employ the quark part of the symmetric energy-momentum tensor current to calculate the transition gravitational form factors of the $N(1535) \rightarrow N$ by means of the light cone QCD sum rule formalism. In numerical analysis, we use two different sets of the shape parameters in the distribution amplitudes of the $ N(1535) $ baryon and the general form of the nucleon's interpolating current. It is seen that the momentum squared dependence of the gravitational form factors can be well described by the p-pole fit function. The results obtained by using two sets of parameters are found to be quite different from each other and the $N(1535) \rightarrow N$ transition gravitational form factors depend highly on the shape parameters of the distribution amplitudes of the $N(1535)$ state that parametrize relative orbital angular momentum of the constituent quarks.

Spacetimes with different forms of energy–momentum tensor

The object of the present paper is to characterize spacetimes with different types of energy–momentum tensor. At first we consider spacetimes with pseudo symmetric energy–momentum tensor T. We obtain a necessary and sufficient condition for a spacetime with pseudo symmetric energy–momentum tensor to be a pseudo Ricci symmetric spacetime. Next we consider the spacetimes with Codazzi type of energy–momentum tensor and several interesting results are pointed out. Moreover, some results related to perfect fluid spacetimes with different forms of energy–momentum tensors have been obtained. We study spacetimes with quadratic Killing energy–momentum tensor T and show that a GRW spacetime with quadratic Killing energy–momentum tensor is an Einstein space. Finally, we have considered general relativistic spacetimes with semisymmetric energy–momentum tensor and obtained some important results.

Noether currents of locally equivalent symmetries

Local symmetry transformations play an important role for establishing the existence and form of a conserved (Noether) current in systems with a global continuous symmetry. We explain how this fact leads to the existence of linear relations between Noether currents of distinct global symmetries that coincide on the local level, thus generalizing the well-known relationship between momentum and angular momentum . As a byproduct, we find a natural interpretation for the discrepancy between the canonical and metric energy–momentum (EM) tensors in theories of particles with spin. A symmetric EM tensor can thus be obtained from the Noether procedure without adding any ad hoc corrections or imposing additional constraints such as gauge invariance in Maxwell’s electrodynamics.

Possible solution of Abraham–Minkowski controversy by generalizing the principle of invariance of light speed

Abraham–Minkowski controversy has been discussed from both classical and quantum levels. Here, we shall show that, assuming to generalize the principle of invariance of light speed from vacuum to lossless linear isotropic homogeneous media (LLIHM), a new type of covariant formulations of classical electrodynamics for LLIHM may be derived, which are similar to that for vacuum, and thus, Minkowski form of electromagnetic wave momentum density is obtained from a diagonally symmetric energy–momentum tensor. On the other hand, the momentum–energy relation is modified due to generalization of principle of invariance of light speed, and then, only the Minkowski momentum is obtained by applying wave–particle duality of a photon. Our work reveals close relationship between the principle of invariance of light speed and the Abraham–Minkowski controversy, which may be helpful to deeply understand Abraham–Minkowski controversy.

Higher derivative scalar quantum field theory in curved spacetime

We study a free scalar field $\phi$ in a fixed curved background spacetime subject to a higher derivative field equation of the form $F(\Box)\phi =0$, where $F$ is a polynomial of the form $F(\Box)= \prod_i (\Box-m_i^2)$ and all masses $m_i$ are distinct and real. Using an auxiliary field method to simplify the calculations, we obtain expressions for the Belinfante-Rosenfeld symmetric energy-momentum tensor and compare it with the canonical energy-momentum tensor when the background is Minkowski spacetime. We also obtain the conserved symplectic current necessary for quantisation and briefly discuss the issue of negative energy versus negative norm and its relation to Reflection Positivity in Euclidean treatments. We study, without assuming spherical symmetry, the possible existence of finite energy static solutions of the scalar equations, in static or stationary background geometries. Subject to various assumptions on the potential, we establish non-existence results including a no-scalar-hair theorem for static black holes. We consider Pais-Uhlenbeck field theories in a cosmological de Sitter background, and show how the Hubble friction may eliminate what would otherwise be unstable behaviour when interactions are included.

Gravitational form factors of a spin one particle

We define the form factors of the quark and gluon symmetric energy-momentum tensor (EMT). The static EMT is related to the spatial distributions of energy, spin, pressure and shear forces. They are obtained in the form of a multipole expansion. The relations between gravitational form factors and the generalised parton distributions are given.

Anomalous geometric spin Hall effect of light

The geometric spin Hall effect of light (GSHEL), similar to the spin Hall effect of light, is also a spin-dependent shift of the centroid of light beam's intensity (energy flux), but it is a purely geometric effect that does not depend on a particular light-matter interaction. In this paper, we discuss the GSHEL with respect to momentum instead of energy flux, and find out that in the case of the symmetric energy-momentum tensor, the shift of the centroid of momentum flux is double that of energy flux. Interestingly, for the canonical energy-momentum tensor, the centroid shift of momentum flux agrees with that of energy flux. If we consider the effect of orbital angular momentum, however, the centroid displacement of momentum flux is twice that of energy flux for both energy-momentum tensors. To tell which energy-momentum tensor of light field would be more "correct", we propose a experimental scheme to test the GSHEL of momentum flux through the mechanical effect of light.

Energy quantization for a nonlinear sigma model with critical gravitinos

We study some analytical and geometric properties of a two-dimensional nonlinear sigma model with gravitino which comes from supersymmetric string theory. When the action is critical w.r.t. variations of the various fields including the gravitino, there is a symmetric, traceless, and divergence-free energy-momentum tensor, which gives rise to a holomorphic quadratic differential. Using it we obtain a Pohozaev type identity and finally we can establish the energy identities along a weakly convergent sequence of fields with uniformly bounded energies.

Noether currents for the Teukolsky master equation

Conserved currents associated with the time translation and axial symmetries of the Kerr spacetime and with scaling symmetry are constructed for the Teukolsky master equation (TME). Three partly different approaches are taken, of which the third one applies only to the spacetime symmetries. The results yielded by the three approaches, which correspond to three variants of Noether’s theorem, are essentially the same, nevertheless. The construction includes the embedding of the TME into a larger system of equations, which admits a Lagrangian and turns out to consist of two TMEs with opposite spin weight. The currents thus involve two independent solutions of the TME with opposite spin weights. The first approach provides an example of the application of an extension of Noether’s theorem to nonvariational differential equations. This extension is also reviewed in general form. The variant of Noether’s theorem applied in the third approach is a generalization of the standard construction of conserved currents associated with spacetime symmetries in general relativity, in which the currents are obtained by the contraction of the symmetric energy–momentum tensor with the relevant Killing vector fields. Symmetries and conserved currents related to boundary conditions are introduced as well, and Noether’s theorem and its variant for nonvariational differential equations are extended to them. The extension of the latter variant is used to construct conserved currents related to the Sommerfeld boundary condition.