How the Lorentz Transform Preserves the Speed of Light

Introduction to the Lorentz Transformation

The Lorentz transformation is a fundamental concept in the theory of special relativity, introduced by Hendrik Lorentz and Henri Poincaré. These equations relate the coordinates of an event as measured in two different inertial frames of reference that are in relative motion. The Lorentz transformation is essential for understanding the invariance of the speed of light in all inertial frames, a principle postulated by Albert Einstein.

Formulation of the Lorentz Transformation

The Lorentz transformation equations for time and space in special relativity are given by:

x' γ (x - vt)

t' γ (t - vx/c^2)

where:

x and t are the position and time in the original frame

x' and t' are the position and time in the moving frame

v is the relative velocity between the two frames

c is the speed of light

γ 1/√(1 - v^2/c^2) is the Lorentz factor

Preservation of the Speed of Light

To demonstrate how the Lorentz transformation preserves the speed of light, consider a light signal emitted from the origin of both frames, where x 0 and t 0. In both frames, the light travels at speed c. Thus:

x ct in the original frame

Substituting x ct into the Lorentz transformation equations provides us with the expressions in the moving frame:

x' γ ct - vt γ tc - v

t' γ (t - vct/c^2) γ t (1 - v/c)

Now, to find the speed of light in the moving frame, we take the ratio of x' to t':

speed' x'/t' (γ tc - v) / (γ t (1 - v/c))

Simplifying this expression:

speed' c ? (1 - v/c) / (1 - v/c) c

Hence, the speed of light, x'/t' c, is preserved in the moving frame as well.

Conclusion

The Lorentz transformation ensures that all observers, regardless of their relative motion, measure the speed of light as c. This invariance is a cornerstone of Einstein's theory of special relativity and leads to many counterintuitive consequences of relativistic physics, such as time dilation and length contraction.