Fundamental Equation of Special Theory of Relativity

Consider two observers in two different inertial frames \(S\) and \(S'\). Frame \(S\) is at rest and \(S'\) is moving with uniform velocity \(v\) along the x-axis with respect to frame \(S\). Suppose at \(t = 0\), the origins of two frames coincide.

Both the observers observe the same event. The position and time of the event observed by \(S\) is denoted by \((x,y,z,t)\) and the position and time of the event observed by \(S'\) is denoted by \((x',y',z',t')\).

Consider a wave of light starts from \(O\) and \(O^{'}\) at \(t = 0\) with speed \(c\). Let the wave reaches a point P after time \(t\) from \(O\) and takes the time \(t^{'}\) to reach at P from point \(O^{'}\).

Then the distance covered by light ray from point \(O\) to point \(P\):

\[|OP| = ct\]

\[\Longrightarrow \sqrt{x^{2} + y^{2} + z^{2}} = ct\]

\[\Longrightarrow x^{2} + y^{2} + z^{2} = c^{2}t^{2}\]

\[\Longrightarrow x^{2} + y^{2} + z^{2} - c^{2}t^{2} = 0\ \ - - - - - \ \ \ \ (1)\]

And the distance covered by light ray from point \(O'\) to point P:

\[|O'P| = ct'\]

\[\Longrightarrow \sqrt{{x'}^{2} + {y'}^{2} + {z'}^{2}} = ct'\]

\(\Longrightarrow {x'}^{2} + {y'}^{2} + {z'}^{2} = c^{2}{t'}^{2}\)

\[\Longrightarrow {x^{'}}^{2} + {y^{'}}^{2} + {z^{'}}^{2} - c^{2}{t^{'}}^{2} = 0\ \ \ - - - - - \ \ \ \ (2)

Comparing these equations, we get:

\[x^{2} + y^{2} + z^{2} - c^{2}t^{2} = {x^{'}}^{2} + {y^{'}}^{2} + {z^{'}}^{2} - c^{2}{t^{'}}^{2}\ \ - - - - - \ \ \ (3)

This is the fundamental equation of special theory of relativity given by Einstein in 1905.

Galilean Transformations Doesn't Satisfy the Fundamental Equation of Relativity

Applying the values of \(x',y',z',t'\) from Galilean Transformation in Fundamental Equation of Relativity

\[x^{2} + y^{2} + z^{2} - c^{2}t^{2} = (x - vt)^{2} + y^{2} + z^{2} - c^{2}t^{2}\]

\[\Longrightarrow x^{2} = (x - vt)^{2}\]

This is clearly impossible until \(t = 0\). Hence Galilean Transformation fail to satisfy Fundamental Equation of Relativity.

Lorentz Transformations Satisfy the Fundamental Equation of Relativity

Hence we need such transformations which satisfy Fundamental Equation of Special Theory of Relativity. Such transformations are called Lorentz Transformation. These are given below:

\[\begin{Bmatrix} x’ = \gamma\ (x - vt)\ \\ y’ = y\ \\ z’ = z \\ t^{'} = \ \gamma\left( t - \frac{vx}{c^{2}} \right) \\ \end{Bmatrix}\]

Derivation of Lorentz Transformation

Consider two inertial frames of reference \(S\) and \(S’\). Frame \(S\) is at rest and \(S’\) is moving with velocity \(v\) in the direction of increasing \(x\).

As relative velocity \(v\) is along \(x-axis\). Moreover, \(x-axis\) and \(x’-axis\) coincide at \(t = 0\), so: \(y’ = y\) and \(z’ = z\). So the fundamental equation of special theory of relativity \(x^{2} + y^{2} + z^{2} - c^{2}t^{2} = {x’}^{2} + {y’}^{2} + {z’}^{2} - c^{2}{t’}^{2}\) becomes: