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.
[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 |