Show that \(E^{2}-p^{2} c^{2}\) for a particle is invariant under Lorentz transformations.

Lorentz transformations for energy and momentum can be derived from the four-momentum conservation law. The four-momentum of a particle can be represented as \(P^{\mu} = (E/c, \boldsymbol{p})\), where \(E\) is the energy of the particle, \(\boldsymbol{p}\) is the three-momentum, and \(c\) is the speed of light. In a system where the relative motion between the two frames is along the x-axis, the Lorentz transformation for energy and momentum are given by: \[ E' = \frac{E - v p_x}{\sqrt{1 - v^2/c^2}} \] \[ p'_x = \frac{p_x - v E/c^2}{\sqrt{1 - v^2/c^2}} \] \[ p'_y = p_y \] \[ p'_z = p_z \] Here, the primed quantities are the energy and momentum components as measured in the moving frame, and the unprimed quantities are the energy and momentum components as measured in the stationary frame.

The expression to be proven invariant is \(E^{2}-p^{2}c^{2}\). We will now substitute the Lorentz-transformed values for energy and momentum into this expression: \[ (E')^2 - (p'_x c)^2 - (p'_y c)^2 - (p'_z c)^2 \]

Replace the Lorentz-transformed values of energy and momentum into the expression: \[ \left(\frac{E - v p_x}{\sqrt{1 - v^2/c^2}}\right)^2 - \left(\frac{p_x - v E/c^2}{\sqrt{1 - v^2/c^2}}\right)^2 c^2 - (p_y c)^2 - (p_z c)^2 \] Now, we can simplify this expression by carefully expanding the squares and using the identity \((1 - v^2/c^2) = \gamma^{-2}\) where \(\gamma\) is the relativistic factor: \[ \frac{(E - v p_x)^2 - c^2(p_x - v E/c^2)^2}{1 - v^2/c^2} - (p_y c)^2 - (p_z c)^2 \] By further expansions and rearrangement, the expression becomes: \[ \frac{E^2 - 2 E v p_x + v^2 p_x^2 - c^2 (p_x^2 - 2p_x v E/c^2 + v^2 E^2/c^4)}{1 - v^2/c^2} - (p_y c)^2 - (p_z c)^2 \] \[ \frac{E^2 (1 - v^2/c^2) - p_x^2c^2(1 - v^2/c^2)}{(1 - v^2/c^2)} - (p_y c)^2 - (p_z c)^2 \] Now, we can cancel \((1 - v^2/c^2)\) from the numerator and denominator: \[ E^2 - p_x^2 c^2 - p_y^2c^2 - p_z^2 c^2 \] This expression simplifies to the original expression: \(E^2 - p^2 c^2\).

We demonstrated that when the energy and momentum values are transformed using the Lorentz transformations, the expression \(E^2 - p^2c^2\) remains unchanged, proving its invariance under Lorentz transformations.