Thus “time” from A to B as measured in S′, is given by _l_/W′ = _l_(1 - W_v_/_c²_)/(W - _v_) = _t′_ (1) Now if _v_ is less than _c_, then W being greater than _c_ (by hypothesis) W is greater than _v_, _i.e._, W > _v_. Let W = _c_ + μ and _v_ = _c_ - λ. Then W_v_ = (_c_ + μ)(_c_ - λ) = _c²_ + (μ + λ)_c_ - μλ. Now we can always choose _v_ in such a way that W_v_ is greater than _c²_, since W_v_ is > _c²_ if (μ + λ)_c_ - μλ is > 0, that is, if μ + λ > μλ/_c_; which can always be satisfied by a suitable choice of λ. Thus for W > _c_ we can always choose λ in such a way as to make W_v_ > _c²_, _i.e._, λ - W_v_/_c²_ negative. But W - _v_ is always positive. Hence with W > _c_, we can always make _t′_, the time from A to B in equation (1) “negative.” That is, the signal starting from A will reach B (as observed in system S′) in less than no time. Thus the effect will be perceived before the cause commences to act, _i.e._, the future will precede the past. Which is absurd. Hence we conclude that W > _c_ is an impossibility, there can be no velocity greater than that of light. It is _conceptually_ possible to imagine velocities greater than that of light, but such velocities cannot occur in reality. Velocities greater than _c_, will not produce any effect. Causal effect of any physical type can never travel with a velocity greater than that of light. [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.