Lorentz Transformations

In Physics, when there is a coordinate transformation between 2 separate coordinate frames that are moving at a constant velocity which is relative to each other it is known as Lorentz Transformation. The name of the transformation comes from a Dutch physicist Hendrik Lorentz.

There are two frames of reference which are:

  1. Inertial Frames – Motion with a constant velocity
  2. Non-Inertial Frames – Rotational motion with constant angular velocity, acceleration in paths which are curved

Lorentz transformation is only related to transformation in the inertial frames usually in the context of special relativity. This transformation is a type of linear transformation in which mapping occurs between 2 modules that include vector spaces. In linear transformation, the operations of scalar multiplication and additions are preserved. This transformation has a number of instinctive features such as the observer that is moving at different velocity may measure elapsed times, different distances and ordering of events but the condition that needs to be followed is that the speed of light should be same in all the inertial frames.

Lorentz transformation can also include rotation of space, a rotation which is free of this transformation is called Lorentz Boost. The spacetime interval which occurs between any two events is preserved by this transformation.

Mathematical Representation

In the reference frame ‘F’ which is stationary, the coordinates defined are x, y, z, t. In another reference frame F’ which moves at a velocity v which relative to F and the observer defines coordinates in this moving reference frames as x’ , y’ , z’ , t’. In both the reference frames the coordinate axis are parallel and they remain mutually perpendicular. The relative motion is along the xx’ axes. At t = t’ = 0, the origins in both reference frames are same (x,y,z) = (x’,y’,z’) = (0,0,0)

If the events x, y, z, t are recorded in reference frames F then in F’ these coordinates have the following value:

\(x^{‘}=\frac{x-vt}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\)

\(t^{‘}=\frac{t-\frac{vx}{c^{2}}}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\)

\(y’=y\)

\(z’=z\)

\(\gamma =\frac{1}{\sqrt{1-\frac{v^{2}}{c^{2}}}}\)

Here lowercase gamma is the Lorentz factor

Then the above equation becomes

\(x’=\gamma (x-vt)\)

\(t’=\gamma (t-\frac{vx}{c^{2}})\)

y’ = y

z’ = z

Physical Implications

The invariance of the speed of light is one of the critical requirements for Lorentz Transformation, this fact is also used in the derivation of this equation.

There are 3 unintuitive but correct predictions of Lorentz transformations are:

  1. Length Contraction
  2. Time Dilation
  3. Relativity of Simultaneity

Practise This Question

A transformer has 1500 turns in the primary coil and 1125 turns in the secondary coil. If the voltage in the primary coil is 200 V, then the voltage in the secondary coil is