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They must therefore be relations among Lorentz scalars, 4-vectors, 4-tensors, etc.
defined by their transformation properties under the Lorentz group in ways
analogous to the familiar specification of tensors of a given rank under three-
dimensional rotations. We are thus led to consider briefly the mathematical
structure of a space-time whose norm is defined by (11.59). We begin by
summarizing the elements of tensor analysis in a non-Euclidean vector space.
The space-time ...
A contravariant tensor of rank two F°s consists of 16 quantities that transform
according to ? y A convariant tensor of rank two Gff transforms as , , G'«^lt>G* (
1L64) and the mixed second rank tensor Hp° transforms as ^ H'V^H', * (11.65)
The generalization to contravariant, covariant, or mixed tensors of arbitrary rank
should be obvious from these examples. The inner or scalar product of two
vectors is defined as the product of the components of a covariant and a
contravariant vector, B A ...
(c) For macroscopic media, E, B form the field tensor F"" and D, H the tensor G"8.
What further invariants can be formed? What are their explicit expressions in
terms of the 3-vector fields? 11.13 In a certain reference frame a static, uniform,
electric field Eo is parallel to the x axis, and a static, uniform, magnetic induction
B„ = 2E„ lies in the x-y plane, making an angle 6 with the x axis. Determine the
relative velocity of a reference frame in which the electric and magnetic fields are
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Introduction and Survey
Introduction to Electrostatics
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