Dynamics of extended bodies in general relativity - II. Moments of the charge-current vector

1970 ◽  
Vol 319 (1539) ◽  
pp. 509-547 ◽  
Keyword(s):  
Tensor Field ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
The Body ◽  
Total Charge ◽  
Subsequent Paper ◽  
Multipole Moments ◽  
Current Vector ◽  
Charge Current ◽  
Infinite Set

The problem is considered of defining multipole moments for a tensor field given on a curved spacetime, with the aim of applying this to the energy-momentum tensor and charge-current vector of an extended body. Consequently, it is assumed that the support of the tensor field is bounded in spacelike directions. A definition is proposed for ‘a set of multipole moments’ of such a tensor field relative to an arbitrary bitensor propagator. This definition is not fully determinate, but any such set of moments completely determines the original tensor field. By imposing additional conditions on the moments in two different ways, two uniquely determined sets of moments are obtained for a vector field J α . The first set, the complete moments , always exists and agrees with moments defined less explicitly by Mathisson. If V α J α = 0, as is the case for the charge-current vector, these moments are interrelated by an infinite set of corresponding restrictions. The second set, the reduced moments , exists if and only if V α J α = 0. These avoid such an infinite set of interrelations, there being instead only one such restriction, the constancy of the total charge of the body. The energy-momentum tensor will be treated in a subsequent paper.

Dynamics of extended bodies in general relativity III. Equations of motion

1974 ◽  
Vol 277 (1264) ◽  
pp. 59-119 ◽  
Keyword(s):  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
High Order ◽  
Total Momentum ◽  
The Body ◽  
Full Description ◽  
Total Charge ◽  
Current Vector ◽  
Charge Current

A study is made of the motion of an extended body in arbitrary gravitational and electromagnetic fields. In a previous paper it was shown how to construct a set of reduced multipole moments of the charge-current vector for such a body. This is now extended to a corresponding treatment of the energy-momentum tensor. It is shown that, taken together, these two sets of moments have the following three properties. First, they provide a full description of the body, in that they determine completely the energy-momentum tensor and charge-current vector from which they are constructed. Secondly, they include the total charge, total momentum vector and total angular momentum (spin) tensor of the body. Thirdly, the only restrictions on the moments, apart from certain symmetry and orthogonality conditions, are the equations of motion for the total momentum and spin, and the conservation of total charge. The time dependence of the higher moments is arbitrary, since the process of reduction used to construct the moments has eliminated those contributions to these moments whose behaviour is determinate. The uniqueness of the chosen set of moments is investigated, leading to the discovery of a set of properties which is sufficient to characterize them uniquely. The equations of motion are first obtained in an exact form. Under certain conditions, the contributions from the moments of sufficiently high order are seen to be negligible. It is then convenient to make the multipole , in which these high order terms are omitted. When this is done, further simplifications can be made to the equations of motion. It is shown that they take an especially simple form if use is made of the extension operator of Veblen & Thomas. This is closely related to repeated covariant differentiation, but is more useful than that for present purposes. By its use, an explicit form is given for the equations of motion to any desired multipole order. It is shown that they agree with the corresponding Newtonian equations in the appropriate limit.

The interaction between a rotating, spherical shell of matter and a central black hole

1978 ◽  
Vol 362 (1711) ◽  
pp. 469-492
Keyword(s):  
Black Hole ◽  
Spherical Shell ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Leading Order ◽  
Multipole Moments ◽  
Matter Distribution ◽  
Central Black Hole ◽  
Magnetic Components ◽  
Axis Of Rotation

A method due to Chrzanowski, involving horizon multipole moments, is applied to the problem of a black hole perturbed by an enclosing, distant, spinning, spherical shell of matter. The hole, of mass M and angular momentum J = aM , is at the centre of the shell, their respective axes of rotation differing by an angle ξ. The matter-distribution on the shell is axisymmetric about its axis of rotation, but otherwise arbitrary, except that the total mass of the shell is small in comparison with M . The energy-momentum tensor of such a shell has been previously found by Bass & Pirani. Using their expression, we calculate the spin-down law for the black hole, correct to leading order in the inverse of the shell’s radius, and to second order in its angular velocity. The solution may be expressed in terms of the ‘electric’ and ‘magnetic’ components E αβ and B αβ of the Weyl tensor C ijkl , as calculated at the centre of the shell, in the absence of the black hole. For, denoting by J ∥ and J ⊥ the components of J parallel and perpendicular, respectively, to the direction of spin of the shell, we have always d J ∥ /d t = 0 and 1/ J ⊥ d J ⊥ /d t =–4/15 M 3 ( E αβ E αβ + B αβ B αβ ) (1–3/4ã 2 +15/4ã 2 sin 2 ξ), where ã = a / M . This law is of theoretical interest. It shows points both of similarity to, and of difference from, the known laws describing the response of a black hole to (uniform) scalar and electromagnetic fields.

The Maxwell equations

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Electromagnetic Field ◽  
Angular Momentum ◽  
Variational Principle ◽  
Maxwell Equations ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Field Energy ◽  
Total Charge ◽  
Faraday’S Law ◽  
System Energy

This chapter presents Maxwell equations determining the electromagnetic field created by an ensemble of charges. It also derives these equations from the variational principle. The chapter studies the equation’s invariances: gauge invariance and invariance under Poincaré transformations. These allow us to derive the conservation laws for the total charge of the system and also for the system energy, momentum, and angular momentum. To begin, the chapter introduces the first group of Maxwell equations: Gauss’s law of magnetism, and Faraday’s law of induction. It then discusses current and charge conservation, a second set of Maxwell equations, and finally the field–energy momentum tensor.

scholarly journals CONSERVATION LAWS FOR THE CLASSICAL TODA FIELD THEORIES

1993 ◽  
Vol 08 (35) ◽  
pp. 3377-3385 ◽  
Author(s):  
ERLING G. B. HOHLER ◽  
KÅRE OLAUSSEN
Keyword(s):  
Conservation Laws ◽  
Energy Momentum Tensor ◽  
Momentum Tensor ◽  
Field Theories ◽  
Conformal Theory ◽  
Infinite Set ◽  
Different Origin

Some explicit calculations of the conservation laws for classical (affine) Toda field theories, and some generlizations of these models are performed. We show that there is a huge class of generalized models which have an infinite set of conservation laws, with their integrated charges being in involution. Amongst these models we find that only the Am, and [Formula: see text] Toda field theories admit such conservation laws for spin-3. The explicit calculations of spin-4 and spin-5 conservation laws in the (affine) Toda models we presented. Our perhaps most interesting finding is that there exist conservation laws in the Am, models (m≥4) which have a different origin than the exponents of the corresponding affine theory or the energy-momentum tensor of a conformal theory.

Conserved currents of the Klein–Gordon field

1981 ◽  
Vol 90 (3) ◽  
pp. 507-515 ◽  
Author(s):  
T. J. Gordon
Keyword(s):  
Field Theory ◽  
Energy Momentum Tensor ◽  
Differential Forms ◽  
Momentum Tensor ◽  
Mathematical Background ◽  
Poincaré Lemma ◽  
Klein Gordon ◽  
Countably Infinite ◽  
Infinite Set ◽  
Conserved Currents

AbstractA method is presented whereby all locally defined conserved currents of the Klein-Gordon field are found. The mathematical background to the method includes a generalization of the Poincaré lemma of the calculus of exterior differential forms. It is found that the only conserved currents are essentially a countably infinite set of functions, bilinear in the field, together with a single current in the case where the mass is zero. The usual energy-momentum tensor is included amongst these functions. The method does not depend on the use of any canonical formulation of the field theory.

scholarly journals EFFECTS OF GAUGING ON SYMPLECTIC STRUCTURE, THE HOPF TERM COUPLED TO THE CP1 MODEL, AND FRACTIONAL SPIN

1999 ◽  
Vol 14 (10) ◽  
pp. 1561-1590 ◽  
Author(s):  
B. CHAKRABORTY ◽  
A. S. MAJUMDAR
Keyword(s):  
Symplectic Structure ◽  
Energy Momentum Tensor ◽  
Time Derivative ◽  
Momentum Tensor ◽  
Total Charge ◽  
Dynamical Structure ◽  
Independent Manner ◽  
Classical Level ◽  
Fractional Spin ◽  
Chern Simons

We couple the Hopf term to the relativistic CP 1 model and carry out the Hamiltonian analysis at the classical level in a gauge-independent manner. The symplectic structure of the model given by the set of Dirac brackets among the phase space variables is found to be the same as that of the pure CP 1 model. This symplectic structure is shown to be inherited from the global SU(2)-invariant S3 model, and undergoes no modification upon gauging the U(1) subgroup, except for the appearance of an additional first class constraint generating U(1) gauge transformation. We then address the question of fractional spin as imparted by the Hopf term at the classical level. For that we construct the expression of angular momentum through both the symmetric energy–momentum tensor and Noether's prescription. The two expressions agree for the model, indicating that no fractional spin is imparted by this term at the classical level in an approach which is different from the earlier analyses carried out in the literature. We next show that the model has to be altered à la Bowick et al., by using an identity (which is not a constraint as it involves a time derivative and thus changes the basic dynamical structure of the model) valid in the radiation gauge in order to yield the fractional spin given in terms of the soliton number. Finally, by making the gauge field of the CP 1 model dynamical by adding the Chern–Simons term, we find that the model ceases to be a CP 1 model, as is the case with its nonrelativistic counterpart. This model is also shown to reveal the existence of "anomalous" spin. This is, however, given in terms of the total charge of the system, rather than any soliton number.

scholarly journals Chiral hydrodynamics in strong external magnetic fields

2021 ◽  
Vol 2021 (4) ◽  
Author(s):  
Martin Ammon ◽  
Sebastian Grieninger ◽  
Juan Hernandez ◽  
Matthias Kaminski ◽  
Roshan Koirala ◽  
...  
Keyword(s):  
Energy Momentum Tensor ◽  
Transport Coefficients ◽  
Momentum Tensor ◽  
Effective Field ◽  
Quantum Fluids ◽  
Hall Conductivity ◽  
Hydrodynamic Description ◽  
Strongly Coupled ◽  
Charge Current ◽  
Transport Effects

Abstract We construct the general hydrodynamic description of (3+1)-dimensional chiral charged (quantum) fluids subject to a strong external magnetic field with effective field theory methods. We determine the constitutive equations for the energy-momentum tensor and the axial charge current, in part from a generating functional. Furthermore, we derive the Kubo formulas which relate two-point functions of the energy-momentum tensor and charge current to 27 transport coefficients: 8 independent thermodynamic, 4 independent non-dissipative hydrodynamic, and 10 independent dissipative hydrodynamic transport coefficients. Five Onsager relations render 5 more transport coefficients dependent. We uncover four novel transport effects, which are encoded in what we call the shear-induced conductivity, the two expansion-induced longitudinal conductivities and the shear-induced Hall conductivity. Remarkably, the shear-induced Hall conductivity constitutes a novel non-dissipative transport effect. As a demonstration, we compute all transport coefficients explicitly in a strongly coupled quantum fluid via holography.

CASIMIR EFFECT FOR THE ROBIN SURFACES IN STATIC ROBERTSON–WALKER SPACE–TIME

2011 ◽  
Vol 20 (02) ◽  
pp. 161-168 ◽  
Author(s):  
MOHAMMAD R. SETARE ◽  
M. DEHGHANI
Keyword(s):  
Scalar Field ◽  
Energy Momentum Tensor ◽  
Casimir Effect ◽  
Vacuum Expectation ◽  
Robin Boundary Condition ◽  
Momentum Tensor ◽  
Space Time ◽  
Conformally Invariant ◽  
Time Space ◽  
Robin Boundary

We investigate the energy–momentum tensor for a massless conformally coupled scalar field in the region between two curved surfaces in k = -1 static Robertson–Walker space–time. We assume that the scalar field satisfies the Robin boundary condition on the surfaces. Robertson–Walker space–time space is conformally related to Rindler space; as a result we can obtain vacuum expectation values of the energy–momentum tensor for a conformally invariant field in Robertson–Walker space–time space from the corresponding Rindler counterpart by the conformal transformation.

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