Energy Momentum Tensor in .NET

Maker PDF417 in .NET Energy Momentum Tensor

1-2-2 Energy Momentum Tensor
Decode PDF-417 2d Barcode In .NET
Using Barcode Control SDK for .NET Control to generate, create, read, scan barcode image in .NET framework applications.
PDF 417 Generation In .NET
Using Barcode printer for VS .NET Control to generate, create PDF-417 2d barcode image in VS .NET applications.
For an infinite system we assume a lagrangian depending on the space-time coordinates x only through fields and their gradients. Under a translation we therefore have
Read PDF-417 2d Barcode In VS .NET
Using Barcode reader for Visual Studio .NET Control to read, scan read, scan image in VS .NET applications.
Bar Code Encoder In Visual Studio .NET
Using Barcode creator for VS .NET Control to generate, create bar code image in .NET applications.
Read Bar Code In VS .NET
Using Barcode scanner for Visual Studio .NET Control to read, scan read, scan image in .NET applications.
Encoding PDF-417 2d Barcode In Visual C#.NET
Using Barcode maker for .NET framework Control to generate, create PDF 417 image in Visual Studio .NET applications.
+ a) == 2'[ CPi(X + a), O/lCPi(X + a)]
PDF417 Creation In .NET
Using Barcode encoder for ASP.NET Control to generate, create PDF417 image in ASP.NET applications.
Printing PDF417 In VB.NET
Using Barcode encoder for VS .NET Control to generate, create PDF417 image in Visual Studio .NET applications.
Barcode Creator In VS .NET
Using Barcode generator for .NET Control to generate, create barcode image in Visual Studio .NET applications.
Linear Encoder In .NET Framework
Using Barcode printer for .NET Control to generate, create Linear 1D Barcode image in VS .NET applications.
Consider an infinitesimal x-dependent transformation
Matrix 2D Barcode Printer In Visual Studio .NET
Using Barcode creation for VS .NET Control to generate, create Matrix 2D Barcode image in Visual Studio .NET applications.
Drawing 2/5 Standard In .NET Framework
Using Barcode generation for Visual Studio .NET Control to generate, create 2/5 Industrial image in VS .NET applications.
Data Matrix ECC200 Recognizer In Java
Using Barcode scanner for Java Control to read, scan read, scan image in Java applications.
Bar Code Generation In .NET
Using Barcode generator for ASP.NET Control to generate, create barcode image in ASP.NET applications.
E>a/l(x)o /lCPi(X)
EAN 13 Generation In None
Using Barcode creator for Office Excel Control to generate, create EAN13 image in Office Excel applications.
Create ANSI/AIM Code 128 In None
Using Barcode encoder for Microsoft Excel Control to generate, create Code 128C image in Microsoft Excel applications.
E>o /lCPi(X) = E>avovo /lCPi(X)
1D Drawer In Java
Using Barcode drawer for Java Control to generate, create Linear image in Java applications.
Creating UPC Code In None
Using Barcode creator for Software Control to generate, create UPCA image in Software applications.
+ a/l[ E>aV(x)]oVCPi(X)
Code 3 Of 9 Creator In Java
Using Barcode maker for BIRT reports Control to generate, create Code 39 image in Eclipse BIRT applications.
EAN128 Encoder In None
Using Barcode maker for Microsoft Word Control to generate, create EAN / UCC - 13 image in Office Word applications.
After an integration by parts the corresponding variation of the action is
d X{Ov2' -
O/l[O(~~i) OVCPi(X)]} E>aV(x)
We obtain in this way a generalization of our previous discussion of energy momentum in a completely local form. From the vanishing of M for arbitrary baV(x), we deduce that the energy momentum flow described by the canonical tensor (1-97) satisfies the conservation law (1-98) (From this derivation the two indices of llv playa dissimilar role.) It follows that the four quantities Pv corresponding to total energy (v = 0) and threemomentum (v = 1,2,3): (1-99) are time independent, since
ftv =
d 3 x ooeOV(x, t) = -
d 3 x itl Oieiv(X, t) = 0
provided that the fields vanish sufficiently rapidly for large arguments, i.e., no energy or momentum escapes at infinity. This result is a typical case of Noether's theorem. The latter states that to any continuous one-parameter set of invariances of the lagrangian is associated a local conserved current. Integrating the fourth component of this current over three-space generates a conserved "charge." Invariance of the lagrangian means in this geometrical context that we also allow the possible transformation of the space-time argument as appears, for instance, in (1-94). Furthermore, the space integral defining the charge can be performed on a space-like surface <J with a surface element d<J Il without affecting the result:
Pv =
It can, of course, happen that the lagrangian also depends explicitly on the coordinates x, in which case (1-94) is no longer valid and Eq. (1-98) is replaced by
(1-101) Such is the case if f is a sum of an invariant part f plus a coupling term to external sources ji(X) of the form f 1 = Li (MX)ji(X). The energy momentum v tensor gets a contribution from f plus an added term from f 1 and reads
(1-102) with
according to (1-101). This last equation can be rewritten as 0lleg v = 'Ij;(X)OvCPi
These two ways of expressing the local variation of energy and momentum differ according to whether the interaction energy - Sd 3 x 2 1 is included or not in the system. Let us apply this to the electromagnetic field coupled to an external conserved current j where the lagrangian is given by
2 = _!F2 - j
+ gllVjo A egv = - pIlP OV Ap + !gIlVF 2
e llV =
The tensor e is not gauge invariant even if j is zero! Under a transformation A---+A+ocp: (1-106)
In the absence of external sources the added term is a divergence and will not contribute to the value of the total energy momentum if the fields vanish at infinity. The energy momentum density is in principle measurable and, moreover, is coupled to the gravitational field. It is therefore very unpleasant to find a gaugedependent expression. In addition, the antisymmetric part of e llv is nonvanishing:
(1-107) We know that the lagrangian is not entirely determined by the equations of motion. Correspondingly, the energy momentum tensor admits some arbitrariness. Generally speaking, let kll(x) denote a conserved current. Then the charge K = Sd 3x kO(x, t) and the local conservation law are unchanged if kll goes into (1-108) provided that
0lll1k ll = 0
d 3x I1kO(x, t)
A solution to these constraints is given by (1-111) where k llP is antisymmetric and depends locally on the fields. Indeed, (1-109) is verified and so is (1-110), since
Jd 3XO p k OP = Jd3xtOikOi=0
Returning to the electromagnetic energy momentum tensor, we see that we can v add to the canonical a piece
where e"P,v is antisymmetric in (j1, p) and local in the fields. The discussion of the gauge dependence where
Copyright © . All rights reserved.