Electromagnetic Field Except For Its Complexion

(a) Given a non-zero symmetric 4x4 tensor 7""" which has zero trace T= 0 and whose square is a multiple, A/4/(8m)2, of the unit matrix, show that, according as this multiple is zero ("null case") or positive, the tensor can be transformed to the form (20.60) or (20.59) by a suitable rotation in 3-space or by a suitable choice of local inertial frame, respectively.

(b) In the generic (non-null) case in the frame in question, show that T"" is the Maxwell tensor of the "extremal electromagnetic field" f with components

Show that it is also the Maxwell tensor of the "dual extremal field" with components

(c) Recalling that the duality operation * applied twice to an antisymmetric second-rank tensor (2-form) in four-dimensional space leads back to the negative of that tensor, show that the operator e'a ("duality rotation") has the value e'a = (cos a) + (sin a)*. (20.64)

(d) Show that the most general electromagnetic field which will reproduce the non-null tensor T11" in the frame in question, and therefore in any coordinate system, is

(e) Derive a corresponding result for the null case. [The field F^ defined in the one frame and therefore in every coordinate system by (d) and (e) is known as the "Maxwell square root" of J"*; is known as the "extremal Maxwell square root" of 7"""; and the angle a is called the "complexion of the electromagnetic field." See Misner and Wheeler (1957); see also Boxes 20.1 and 20.2, adapted from that paper.]

Box 20.1 CONTRAST BETWEEN PROPER LORENTZ TRANSFORMATION AND DUALITY ROTATION

Quantity

General proper Lorentz transformation

Components of the Maxwell stress-energy tensor or the "Maxwell square" of the field F

The combination [(E2 - B-)2 + (2E-Bf\ = [(E2 + B2)2 - (IE x B)2]

Duality rotation

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