The Principle of Relativity · Albert Einstein
[M. N. S.]
Note 12. Light-velocity as a maximum. Page 23, and Electro-dynamics of Moving Bodies, p. 17. Putting _v_ = _c_ - _x_, and _w_ = _c_ - λ, we get V = (2_c_ - (_x_ + λ))/(1 + (_c_ - _x_)(_c_ - λ)/_c²_) = (2_c_ - (_x_ + λ))/(_c²_ + _c²_ - (_x_ + λ)_c_ + _x_λ/_c²_) = _c_ (2_c_ - (_x_ + λ))/(2_c_ - (_x_ + λ) + _x_λ/_c_) Thus _v_ lt; _c_, so long as | _x_λ | > 0. Thus the velocity of light is the absolute maximum velocity. We shall now see the consequences of admitting a velocity W > _c_. Let A and B be separated by distance _l_, and let velocity of a “signal” in the system S be W > _c_. Let the (observing) system S′ have velocity +_v_ with respect to the system S. Then velocity of signal with respect to system S′ is given by W′ = (W - _v_)/(1 - W_v_/_c²_) 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.
Ideas discussed in this chapter
Text from Project Gutenberg, public domain in the United States.