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### S7. Lorentz Transformation

Review:

• Galilean relativity states that, velocity is a relative concept and that every frame, which is in the constant uniform motion, is equally valid frame to perform physics experiment.
• Galilean transformation tries to mathematically sum up the Galilean relativity. It worked fine until the advent of the Maxwell’s Theory of electromagnetic waves.
• Trying to link Maxwell theory with Galilean relativity posed a great problem; people came up with the concept of the luminiferous ether which permeated thorough space. However, this theory had an inherent difficult in its construction. So,
• Einstein in 1905 resolved the problem by completely abandoning the concept of ether and postulating two simple yet powerful postulates.”

1.  Laws of physics are the same in all inertial frame of reference moving with uniform relative velocity.
2.  Velocity of light is constant in all inertial frame of reference irrespective of the relative motion.

What did this imply?

Ans: There was no absolute standard of rest. There exists a relativity principle that applies to all of physics, but it's not Galilean relativity. Laws of mechanics must be revised. Perhaps laws of electrodynamics can survive intact. A new kind of transformation must take precedence over the Galilean transformation, the Lorentz Transformation.

Suppose there are two reference frames (systems) designated by $S$ and ${S}'$ such that the co-ordinate axes are parallel (as in figure). In S, we have the co-ordinates $(x,y,z,t)$ and in ${S}'$ we have the co-ordinates $({x}', {y}',{z}',{t}')$. ${S}'$ is moving with respect to $S$ with velocity $v$ (as measured in $S$) in the $x$ direction. The planes $y=0$ and $z=0$ always coincide with the planes ${y}’=0$ and ${z}’=0$, and we can set the zero points on the clocks so that $t={t}’=0$ at the instant when $x={x}’$.

One can clearly see that the position of the ${S}'$ frame as seen from the $S$ changes as the time passes and vice-versa. We then perform the Lorentz transform the coordinate of the ${S}'$ frame so that the mechanical laws as well as that of the electromagnetic laws of physics seems exactly as same a that observed in the $S$ frame of reference.

It is very important to note that the, Lorentz transformation is derived with two reasonable assumptions (i.e. space is both homogeneous and isotropic): that is, the laws of physics do not change from one place to another, or with orientation of the coordinate frames. Isotropy of space implies that rotation should have no effect on experiment performed. The assumption of homogeneity implies that any transformation from one inertial frame to another is linear.

The lorentz Transformation is given by the following set of Equations,

This then is the required Lorentz Transformation. Note that the inverse transformation has the positive sign in between. The derivation of the result and its subsequent understanding will be done in the next note.