Difference Between Lorentz Transformation and Galilean Transformation

Lorentz Transformation vs Galilean Transformation
 

A set of coordinate axes, which can be used to pin-point the position, orientation and other properties, is employed when describing the motion of an object. Such a coordinate system is called a frame of reference.

Since different observers may use different frames of references, there should be a way of transforming observations made by one frame of reference, to suit another frame of reference. Galilean Transformation and Lorentz Transformation are both such ways of transforming observations. But both can be used only for frames of references which are moving with constant velocities with respect to each other.

What is a Galilean Transformation?

Galilean Transformations are employed in Newtonian Physics. In Newtonian physics, it is assumed that there exists a universal entity called ‘time’ which is independent of the observer.

Assume that there are two frames of references S(x, y, z, t)and S’ (x’, y’, z’, t’)out of which S is at rest and S’ is moving with constant velocity v along the direction of the x-axis of the frame S. Now assume that an event occurs at the point P which at the space-time coordinate (x,y,z,t) with respect to the frame S. Then the Galilean transform gives the position of the event as observed by an observer in frame S’. Assume the space-time coordinate with respect to S’ is (x’,y’,z’,t’) then x’=x – vt, y’=y, z’=z and t’=t. This is the Galilean Transformation.

Differentiating these with respect to t’ the Galilean velocity transformation equations are obtained. If u = (u_x,u_y,u_z) is the velocity of an object as observed by an observer in S then the velocity of the same object as observed by an observer in S’ is given by u’= (u_x’,u_y’,u_z’)where  u_x’ = u_x– v,u_y’ = u_y andu_z’ = u_z. It is interesting to note that under Galilean transformations, the acceleration is invariant; i.e. the acceleration of an object is the observed to be the same by all the observers.

What is a Lorentz Transformation?

Lorentz Transformations are employed in the special relativity and relativistic dynamics. Galilean transformations do not predict accurate results when bodies move with speeds closer to the speed of light. Hence, Lorentz transformations are used when bodies travel at such speeds.

Now consider the two frames in the previous section. The Lorentz transformation equations for the two observers are x’=γ (x– vt), y’=y, z’=z and t’=γ(t – vx/c²) where c is the speed of light and γ = 1/√(1 – v²/c²). Observe that according to this transformation, there is no universal quantity as time, as it dependent on the observer’s speed. As a consequence of this, observers traveling at different speeds will measure different distances, different time intervals and observe different ordering of events.