The following sequence of relativistic proofs shows the alternative path leading from classical physics to Special Relativity.
The relation E = mc² obtained from classical physics is the first link in the chain of relativistic proofs.
The reader can easily find that each proof makes use of the results of those that precede it.
In this way it is shown that, contrary to general belief, there is a connection between classical and relativistic mechanics.
Furthermore, it can be observed that the Theory of Relativity can be derived using a simpler and more intuitive method than the conventional one and without assuming any postulates.
The energy-mass equivalence E = mc²
The formula of the equivalence principle of mass and energy E = mc² can be proven from the relation of the momentum for light radiation p = E/c with the sole use of classical physics (see derivation).
The relativistic mass formula
The relation for the energy dE = mvdv + v²dm from Newton’s Second Law of Motion in connection with the equivalence principle E = mc² results in the relativistic mass formula and thus in the relativistic momentum (see derivation):
\[ m = \frac{m_0}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \quad ; \quad p = mv = \frac{m_{0}v}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \]
The relativistic kinetic and total energy
The relation for the energy dE = mvdv + v²dm from the Second Law of Motion in connection with E = mc² and with the relativistic mass formula results in the equation of the kinetic and the total energy of the physical body (see derivation):
\[ E_k = \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}} – m_0c^2 \quad ; \quad E_g = mc^2 = \frac{m_0c^2}{\sqrt{1-\frac{v^{2}}{c^{2}}}} \]
The Relativistic E-p-m Triangle
The relativistic E-p-m triangle illustrates in a clear way the relationships existing between energy, momentum, and mass (see derivation).
The relativistic addition of velocities
The formula for the relativistic addition of velocities can be derived from the relations of momentum and energy (see derivation):
\[ v_{12} = \frac{v_1 + v_2}{1 + \frac{v_1v_2}{c^2}} \]
The constancy of the speed of light
The theoretical proof of the constancy of the speed of light can be brought about from the formula of the relativistic addition of speeds (see derivation).
The relativistic length contraction and time dilation
The formulas of the relativistic length contraction and time dilation can be derived from the relation of the total energy (see derivation):
\[l^´ = l\sqrt{1-\frac{v^2}{c^2}}\quad ; \quad t^´ = t\sqrt{1-\frac{v^2}{c^2}}\]
The Lorentz transformations for space and time
The Lorentz transformations for space and time can be derived from the relation of the relativistic length contraction (see derivation):
\[x^´ =\frac{x-vt}{\sqrt{1-\frac{v^2}{c^2}}}\quad ; \quad t^´ =\frac{t-\frac{xv}{c^2}}{\sqrt{1-\frac{v^2}{c^2}}} \]
The relativistic Doppler effect
The relativistic formula of the frequency of electromagnetic radiation can be derived from the relations of momentum and energy (see derivation):
\[f_1=f\sqrt{\frac{c+v}{c-v}}\quad ; \quad f_2=f\sqrt{\frac{c-v}{c+v}}\]
The longitudinal and transversal acceleration
Newton’s Second Law of Motion in connection with the equivalence principle E=mc² and with the mass formula results in the relativistic formula of the longitudinal and the transversal acceleration (see derivation):
\[a_L=\frac{F_L}{m_0}\left( 1-\frac{v^2}{c^2} \right)^\frac{3}{2}\quad ; \quad a_T=\frac{F_T}{m_0}\left( 1-\frac{v^2}{c^2} \right)^\frac{1}{2}\]
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The book “Newton and Relativity” by Francesco Cester exposes a detailed treatment of this Sequence of Relativistic Proofs.
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