The Principle of Relativity · Albert Einstein
[P. C. M.]
Notes 13 and 14. We have denoted the four-vector ω by the matrix | ω₁ ω₂ ω₃ ω₄ |. It is then at once seen that [=ω] denotes the reciprocal matrix | ω₁ | | ω₂ | | ω₃ | | ω₄ | It is now evident that while ω¹ = ωA, [=ω]¹ = A⁻¹[=ω] [ω, _s_] The vector-product of the four-vector ω and _s_ may be represented by the combination [ω_s_] = [=ω]_s_ - _ṡ_ω It is now easy to verify the formula _f_¹ = A⁻¹_f_A. Supposing for the sake of simplicity that _f_ represents the vector-product of two four-vectors ω, _s_, we have _f¹_ = [ω¹_s¹_] = [[=ω]¹_s¹_ - [=_s_]^1ω^1] = [A⁻¹ [=ω]_s_A - A⁻¹_s_[=ω]A] = A⁻¹[[=ω]_s_ - _s_[=ω]]A = A⁻¹_f_A. Now remembering that generally _f_ = ρφ + ρ*φ*. Where ρ, ρ* are scalar quantities, φ, φ* are two mutually perpendicular unit planes, there is no difficulty in seeming that _f_^1 = A⁻¹_f_A. Note 15. The vector product (_w__f_). (P. 36). This represents the vector product of a four-vector and a six-vector. Now as combinations of this type are of frequent occurrence in this paper, it will be better to form an idea of their geometrical meaning. The following is taken from the above mentioned paper of Sommerfeld.
B_{_x_} = (B_{_x x_}, B_{_x y_}, B_{_x z_}) } B_{_y_} = (B_{_y x_}, B_{_y y_}, B_{_y z_}) } (B_{_i k_} = - B_{_k i_}, B_{_i i_} = 0). B_{_z_} = (B_{_z x_}, B_{_z y_}, B_{_z z_}) } Since B_{_j j_} is zero, B_{_j_} is perpendicular to the _j_-axis. The _j_-component of the vector-product of A and B is equivalent to the scalar product of A and B_{_j_}, _i.e._, (A B_{_j_},) = A_{_x_} B_{_j x_} + A_{_y_} B_{_j y_} + A_{_z_} B_{_j z_}. We see easily that this coincides with the usual rule for the vector-product; _e. g._, for _j_ = _x_. (AB_{_x_}) = A_{_y_} B_{_x_ _y_} - A_{_z_} B_{_z_ _x_}. Correspondingly let us define in the four-dimensional case the product (P_f_) of any four-vector P and the six-vector _f_. The _j_-component (_j_ = _x_, _y_, _z_, or _l_) is given by (P_f__{_j_}) = P_{_x_}_f__{_j_ _x_} + P_{_y_}_f__{_j_ _y_} + P_{_w_}_f__{_j_ _z_} + P_{_z_}_f__{_j_ _l_} Each one of these components is obtained as the scalar product of P, and the vector _f__{_j_} which is perpendicular to j-axis, and is obtained from _f_ by the rule _f__{_j_} = [(_f__{_j_ _x_}, _f__{_j_ _y_}, _f__{_j_ _z_}, _f__{_j_ _l_}) _f__{_j_ _j_} = 0.] We can also find out here the geometrical significance of vectors of the third type, when _f_ = φ, _i.e._, _f_ represents only one plane.
Text from Project Gutenberg, public domain in the United States.