Tensors for Physics

372 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations 18.1.3 Lorentz Transformation Matrix Components of the 4-vector ...

18.1 Lorentz Transformation 373 Lik:= ⎛ ⎜ ⎜ ⎝ γ 00 −βγ 0100 0010 −βγ 00 γ ⎞ ⎟ ⎟ ⎠. (18.17) 18.1.5 General Lorentz Transformation ...

374 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations (T′)ik=LinLkmTnm. (18.19) Equations of physics, properly formula ...

18.2 Lorentz-Vectors and Lorentz-Tensors 375 Thus the 4-velocity, divided byc, is a 4-dimensional unit vector. As a consequence, ...

376 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations 18.2.3 Differential Operators, Plane Waves The 4D generalization ...

18.2 Lorentz-Vectors and Lorentz-Tensors 377 yields KnKn=−k^2 +ω^2 /c^2 = 0. (18.32) This 4D version of the dispersion relation ...

378 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations to the simpler assumption made by Lorentz and Einstein: they- an ...

18.3 The 4D-Epsilon Tensor 379 18.3.2 Products of Two Epsilon Tensors Formulas for the product of two 4D epsilon tensors, simila ...

380 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations This implies, e.g.A ̃^12 =(A 34 −A 43 )/ 2 =A 34 ,A ̃^13 =−(A 24 ...

18.4 Maxwell Equations in 4D-Formulation 381 18.4 Maxwell Equations in 4D-Formulation 18.4.1 Electric Flux Density and Continuit ...

382 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations 18.4.3 Field Tensor Derived from the 4-Potential In the 3D formu ...

18.4 Maxwell Equations in 4D-Formulation 383 18.4.4 The Homogeneous Maxwell Equations From the definitionFik=∂kΦi−∂iΦk,cf.(18.55 ...

384 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations The inhomogeneous Maxwell equations (7.56), viz. ∇μDμ=ρ, εμνλ∇νH ...

18.4 Maxwell Equations in 4D-Formulation 385 As before, the abbreviationγ =( 1 −β^2 )−^1 /^2 is used. In terms of the pertaining ...

386 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations δS:= ∫ δLd^4 x= 0. (18.72) It is understood thatδΦis zero at the ...

18.5 Force Density and Stress Tensor 387 f^4 =jνEν/c=ρvνEν/c=vνkν/c. (18.75) By analogy to the 3D description, where the force d ...

388 18 From 3D to 4D: Lorentz Transformation, Maxwell Equations 18.3 Exercise: Derivation of the 4D Stress Tensor 18.4 Exercise: ...

Appendix Exercises: Answers and Solutions Exercise Chapter 1 1.1 Complex Numbers as 2D Vectors(p.6) Convince yourself that the c ...

390 Appendix: Exercises... respectively. The corresponding angles are exactly 45◦for the angle betweenaand b, and≈ 70. 5 ◦and≈ 3 ...

Appendix: Exercises... 391 yields the same matrix as in (A.4). Thus one has also AasyμλAasyμλ= 1 2 . Due to AμνAμν= 1 , AμνAνμ= ...