scholarly journals Kaluza–Klein towers for real vector fields in flat space

2011 ◽  
Vol 38 (9) ◽  
pp. 095004
Author(s):  
Fernand Grard ◽  
Jean Nuyts
Keyword(s):  
Vector Fields ◽  
Flat Space ◽  
Real Vector ◽  
Kaluza Klein

scholarly journals Complex Maxwell and Einstein Fields

1974 ◽  
Vol 64 ◽  
pp. 105-105
Author(s):  
Ezra T. Newman
Keyword(s):  
Maxwell Equations ◽  
Vector Fields ◽  
World Line ◽  
Flat Space ◽  
Null Vector ◽  
Einstein Equations ◽  
Maxwell Field ◽  
Real Solution ◽  
Complex Solution ◽  
Maxwell Fields

We consider the class of regular (in a certain precise sense) null vector fields, lμ which have the following properties; they are (1) tangent to geodesics, (2) diverging, (3) shear free, (4) twist (or curl) free. It is well known that the vacuum Einstein fields whose principle null vector field (pnvf) satisfies (1)–(4) are the Robinson-Trautman (1962) (RT) metrics and those which satisfy (1)–(3) are the algebraically special twisting metrics, (Kerr, 1963). To understand these metrics better we ask for those Maxwell fields (in flat space) whose pnvf also satisfy conditions (1)–(4) and (1)–(3). It can be shown that (1)–(4) imply (and are implied by) that the Maxwell field is a Lienard-Wiechart (LW) field. (This establishes the analogy between the RT metrics and the LW fields.) Conditions (1)–(3) imply that the Maxwell field is a complex LW field. (We mean by this that if the Maxwell equations are complexified (Newman, 1973) (in complex Minkowski space) then the real solution in question is induced from the complex solution which is associated with a charged particle moving along an arbitrary complex world line.) Finally it can be shown that the Einstein equations can be complexified and that the algebraically special twisting metrics can be interpreted as if they had a point source moving in the complex manifold and are thus analogous to the complex LW fields.

ON THE PHYSICAL INTERPRETATION OF ASYMPTOTICALLY FLAT GRAVITATIONAL FIELDS

2008 ◽  
Vol 17 (13n14) ◽  
pp. 2599-2606
Author(s):  
CARLOS KOZAMEH ◽  
EZRA T. NEWMAN ◽  
GILBERTO SILVA-ORTIGOZA
Keyword(s):  
Vector Fields ◽  
Center Of Mass ◽  
Flat Space ◽  
Einstein Equations ◽  
Position Vector ◽  
Approximate Symmetry ◽  
Asymptotically Flat ◽  
Gravitational Fields ◽  
Maxwell Fields ◽  
Physical Information

A problem in general relativity is how to extract physical information from solutions to the Einstein equations. Most often information is found from special conditions, e.g., special vector fields, symmetries or approximate symmetries. Our concern is with asymptotically flat space–times with approximate symmetry: the BMS group. For these spaces the Bondi four-momentum vector and its evolution, found at infinity, describes the total energy–momentum and the energy–momentum radiated. By generalizing the simple idea of the transformation of (electromagnetic) dipoles under a translation, we define (analogous to center of charge) the center of mass for asymptotically flat Einstein–Maxwell fields. This gives kinematical meaning to the Bondi four-momentum, i.e., the four-momentum and its evolution is described in terms of a center of mass position vector, its velocity and spin-vector. From dynamical arguments, a unique (for our approximation) total angular momentum and evolution equation in the form of a conservation law is found.

scholarly journals Global Hölder Estimates via Morrey Norms for Hypoelliptic Operators with Drift

2017 ◽  
Vol 2017 ◽  
pp. 1-6
Author(s):  
Yuexia Hou ◽  
Pengcheng Niu
Keyword(s):  
Vector Fields ◽  
Integral Operators ◽  
Morrey Spaces ◽  
Homogeneous Group ◽  
Hypoelliptic Operator ◽  
Real Vector ◽  
Constant Matrix ◽  
Hypoelliptic Operators ◽  
Hölder Estimates

Suppose thatX0,X1,…,Xmare left invariant real vector fields on the homogeneous groupGwithX0being homogeneous of degree two andX1,…,Xmhomogeneous of degree one. In the paper we study the hypoelliptic operator with drift of the kindL=∑i,j=1maijXiXj+a0X0,wherea0≠0and(aij)is a constant matrix satisfying the elliptic condition onRm. By proving the boundedness of two integral operators on the Morrey spaces with two weights, we obtain global Hölder estimates forL.

A GENERALIZATION OF THE ATIYAH–DUPONT VECTOR FIELDS THEORY

2002 ◽  
Vol 04 (04) ◽  
pp. 777-796 ◽  
Author(s):  
ZIZHOU TANG ◽  
WEIPING ZHANG
Keyword(s):  
Vector Bundle ◽  
Cross Sections ◽  
Vector Fields ◽  
Complex Structure ◽  
Index Theory ◽  
Index Theorem ◽  
Obstruction Theory ◽  
Real Vector ◽  
Complete Answer ◽  
Linearly Independent

To generalize the Hopf index theorem and the Atiyah–Dupont vector fields theory, one is interested in the following problem: for a real vector bundle E over a closed manifold M with rank E = dim M, whether there exist two linearly independent cross sections of E? We provide, among others, a complete answer to this problem when both E and M are orientable. It extends the corresponding results for E = TM of Thomas, Atiyah, and Atiyah–Dupont. Moreover we prove a vanishing result of a certain mod 2 index when the bundle E admits a complex structure. This vanishing result implies many known famous results as consequences. Ideas and methods from obstruction theory, K-theory and index theory are used in getting our results.

scholarly journals Flags in zero dimensional complete intersection algebras and indices of real vector fields

2007 ◽  
Vol 260 (1) ◽  
pp. 77-91 ◽  
Author(s):  
L. Giraldo ◽  
X. Gómez-Mont ◽  
P. Mardešić
Keyword(s):  
Complete Intersection ◽  
Vector Fields ◽  
Real Vector

scholarly journals A DISCRETIZED VERSION OF KALUZA–KLEIN THEORY WITH TORSION AND MASSIVE FIELDS

1996 ◽  
Vol 11 (13) ◽  
pp. 2403-2418 ◽  
Author(s):  
NGUYEN AI VIET ◽  
KAMESHWAR C. WALI
Keyword(s):  
Noncommutative Geometry ◽  
Vector Fields ◽  
Complex Structure ◽  
Scalar Fields ◽  
Type Model ◽  
Internal Space ◽  
Tensor Fields ◽  
Fifth Dimension ◽  
Klein Theory ◽  
Kaluza Klein

We consider an internal space of two discrete points in the fifth dimension of the Kaluza–Klein theory by using the formalism of noncommutative geometry — developed in a previous paper1 — of a spacetime supplemented by two discrete points. With the non-vanishing internal torsion two-form there are no constraints implied on the vielbeins. The theory contains a pair of tensor fields, a pair of vector fields and a pair of scalar fields. Using the generalized Cartan structure equation we are able to uniquely determine not only the Hermitian and metric-compatible connection one-forms, but also the nonvanishing internal torsion two-form in terms of vielbeins. The resulting action has a rich and complex structure, a particular feature being the existence of massive modes. Thus the nonvanishing internal torsion generates a Kaluza–Klein type model with zero and massive modes.

KALUZA–KLEIN–GORDON FIELD THEORY

1998 ◽  
Vol 07 (05) ◽  
pp. 737-747 ◽  
Author(s):  
HONGYA LIU ◽  
PAUL S. WESSON
Keyword(s):  
Field Theory ◽  
Scalar Particle ◽  
Flat Space ◽  
Momentum Tensor ◽  
Plane Wave Solution ◽  
Second Derivatives ◽  
Klein Gordon ◽  
Induced Matter ◽  
Derivatives Of ◽  
Kaluza Klein

We extend the induced-matter interpretation of Kaluza–Klein field theory to obtain the Klein–Gordon equation for a scalar particle. The motion of the particle is correctly recovered, and we give an exact plane-wave solution whose induced energy-momentum tensor depends only on first derivatives of the scalar field, as in flat-space quantum field theory. However, in general the energy and momenta of the particle also include terms in second derivatives, which should repay investigation.

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