Lorentz Transformations

Lorentz transformation (L.T.) is the rotation in Minkowski space. Minkowskispace is the mathematical representation of the space time inEinstein's theory of relativity. Unlike the Euclidean space,Minkowski space treats space and time on a different footing and the spacetime interval between the two events would be the same in all frames ofreference.

If an event is observed in an inertial frame S at thecoordinates (x, y, z,t) (Fig.A.1) and in another inertial frame at thecoordinates and the frame is moving with a constant velocityv with respect to S in the directionof x, then the measurements in S andS’ are related by

where

The L.H.S. of these equations describe the transformation of the coordinatesfrom S to S’ and theR.H.S. of the equations describe the transfer of coordinates fromS’ to S.Equation (A.1) may also be written as

The invariant length element squared ds2 is defined as

where is a contravariant vector and is a covariant vector. The metric tensoris defined as

One may rewrite Eq. (A.2) as

Where is the Lorentz transformation matrix. Any set of quantities which havefour components and transform like Eq. (A.5) under L.T. form a four vector,that is,

with the properties

For an infinitesimal may also be written as The tensor generates theinfinitesimal L.T. and it is considered to be a quantity that is smallerthan unity. The major contributor is (unit matrix) and generatesinfinitesimal boost or infinitesimal rotation or both. This can beunderstood as demonstrated in the next section.

Rotation of coordinates around the zaxis in 3D

If we consider the rotation around the z-axis by an angleθ, then

In the matrix form, Eq. (A.7) can be written as

For an infinitesimal rotation, that is,

where represents an infinitesimal rotation if. Equation (A.7) can berewritten as

The matrix representing the infinitesimal change is antisymmetric.Reconsidering Eq. (A.2),

When and Now and is given by

For an infinitesimal boost or rotation, that is,

Therefore, Eq. (A.11) implies that

Now,

The second term on the R.H.S. of Eq. (A.13) can be neglected as it is theproduct of two infinitesimal numbers, that is,

Since should be an invariant quantity,

that is, Eq. (A.14) shows antisymmetry. Therefore, the generator of an L.T.must be an antisymmetric matrix.