ABSTRACT
I place a metre-rod in the x′-axis of K′ in such a manner that one end (the beginning) coincides with the point x′ = 0, whilst the other end (the end of the rod) coincides with the point x′ = 1. What is the length of the metre-rod relative to the system K? In order to learn this, we need only ask where the beginning of the rod and the end of the rod lie with respect to K at a particular time t of the system K. By means of the first equation of the Lorentz transformation the values of these two points at the time t = 0 can be shown to be https://www.w3.org/1998/Math/MathML"> x ( beginning of rod ) = 0 1 − v 2 c 2 x ( end of rod ) = 1 ⋅ 1 − v 2 c 2 , https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315886749/e35f481d-8198-476b-a9ee-e779129551fb/content/math_8_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> the distance between the points being https://www.w3.org/1998/Math/MathML"> 1 − v 2 c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315886749/e35f481d-8198-476b-a9ee-e779129551fb/content/inequmath_9_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> .But the metre-rod is moving with the velocity v relative to K. It therefore follows that the length of a rigid metre-rod moving in the direction of its length with a velocity v i https://www.w3.org/1998/Math/MathML"> 1 − v 2 / c 2 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315886749/e35f481d-8198-476b-a9ee-e779129551fb/content/inequmath_10_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> of a metre. The rigid rod is thus shorter when in motion than when at rest, and the more quickly it is moving, the shorter is the rod. For the velocity v = c we should have https://www.w3.org/1998/Math/MathML"> 1 − v 2 / c 2 = 0 https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315886749/e35f481d-8198-476b-a9ee-e779129551fb/content/inequmath_11_B.tif" xmlns:xlink="https://www.w3.org/1999/xlink"/> and for still greater velocities the square-root becomes imaginary. From this we conclude that in the theory of relativity the velocity c plays the part of a limiting velocity, which can neither be reached nor exceeded by any real body.
