Derivation of Lorentz transformation equations using orthogonal transformations
Let us consider two uniformly moving frames of reference where origins coincides at . let the source of light is fixed at unprimed frame of reference and emits a pulse of light. The observer fixed in the unprimed frame of reference will observe a spherical wave-front propagating with the speed of light , whose equation can be written as
Since speed of light is invariant according to special theory of relativity so the observer fixed in the primed frame of reference would also observe the light pulse propagating as the spherical wave-front from his own origin whose equation is
We now need appropriate transformation equations because here we note that time is no more scalar invariant. The transformations should be such that the light pulse viewed from both the frames of reference must be in form that wave pulse is propagating as simultaneously concentric spheres in both the systems. This reveals that
or in terms of ,, and where we have
Thus square of radius vector is invariant under a transformation of co-ordinates in four dimensional space. We can thus say that equation (4) represents the orthogonal transformation of vectors in four dimensional Minkowiski space. Thus Lorentz transformations can be represented as
is a linear transformation matrix. Now here frame is moving w.r.t. with velocity in the positive direction and all the axis of are parallel to that of . We will now obtain the matrix element of transformation between and for pure Lorentz transformation.
As the primed system is taken to be moving along axis , then and , being in a direction perpendicular to that system , motion will remain unaffected by the transformation i.e.,
as the components and are the components to undergo transformation so components and should not appear in the transformation matrix of and . Thus above equation 6 becomes
Therefore the transformation matrix is
Further since these matrix elements must obey the orthogonality condition that is,
it is if and it is if
putting , and first and second we get
Also with , and we get
Thus the orthogonality condition furnishes the above three equations connecting the four matrix elements.
The four unknown elements can be determined uniquely when the fourth relation between them is provided. We know that the origin of primed frame is is moving uniformly along axis; thus after time it’s co-ordinate will be , i.e.,
which, with matrix relation for gives
which when substituted with equation 10 gives
On solving equation 11 and equation 12 it is quite easy to show that
We can now write the Lorentz transformation as
equations 13 are known as Lorentz transformation equations. Lorentz transformation to real co-ordinate is possible when indicating that one can not have relative velocity greater than , the sped of light.