The Principle of Relativity · Albert Einstein
| K₁ K₂ K₃ K₄ |
In all these operations the operator lor plays the part of a space-time vector of the first kind. If _f_ represents a space-time vector of the second kind,—lor _f_ denotes a space-time vector of the first kind with the components ∂_f₁₂_/∂_x₂_ + ∂_f₁₃_/∂_x₃_ + ∂_f₁₄_/∂_x₄_, ∂_f₂₁_/∂_x₁_ + ∂_f₂₃_/∂_x₃_ + ∂_f₂₄_/∂_x₄_, ∂_f₃₁_/∂_x₁_ + ∂_f₃₂_/∂_x₂_ + ∂_f₃₄_/∂_x₄_, ∂_f₄₁_/∂_x₁_ + ∂_f₄₂_/∂_x₂_ + ∂_f₄₃_/∂_x₃_ So the system of differential equations (A) can be expressed in the concise form {A} lor f = -_s_, and the system (B) can be expressed in the form {B} log F* = 0. Referring back to the definition (67) for log _ṡ_, we find that the combinations lor ([=(lor _f_)=]), and lor ([=(lor F*)]) vanish identically, when _f_ and F* are alternating matrices. Accordingly it follows out of {A}, that (68) (∂_s₁_/∂_x₁_) + (∂_s₂_/∂_x₂_) + (∂_s₃_/∂_x₃_) + (∂_s₄_/∂_x₄_) = 0, while the relation (69) lor (lor F*) = 0, signifies that of the four equations in {B}, only three represent independent conditions. I shall now collect the results. Let ω denote the space-time vector of the first kind (_u_/√(1 - _u²_}), _i_/√(1 - _u²_)) (_u_ = velocity of matter), F the space-time vector of the second kind (M,-_i_E) (M = magnetic induction, E = Electric force, _f_ the space-time vector of the second kind (_m_,-_ie_) (_m_ = magnetic force, _e_ = Electric Induction.
Ideas discussed in this chapter
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