Relativistic Dynamics

In the previous chapter we have seen that a proper extension of the principle of relativity to electromagnetism necessarily implies that the correct transformation laws between two inertial frames are the Lorentz transformations.

1. 1.

   Here by classical mechanics we refer to the Newtonian theory.

2. 2.

   Here and in the rest of this chapter, when referring to the conservation of the total linear momentum of an isolated system of particles, we shall often omit to specify that we consider the total momentum and that the system is isolated, regarding this as understood.

3. 3.

   Note that \(\gamma (V)=\frac{1}{\sqrt{1-\frac{V^2}{c^2}}}\) is the relativistic factor associated with the motion of \(S'\) relative to S, while \({ \gamma }(v'_i)=\frac{1}{\sqrt{1-\frac{v_i^{\prime 2}}{c^2}}} \quad \text { and }\quad { \gamma }(u'_i)=\frac{1}{\sqrt{1-\frac{u_i^{\prime 2}}{c^2}}}\) are the relativistic factors depending on the velocities of each particle and relate the time \(dt'\) in \(S'\) to the proper times \(d\tau _i, d\tilde{\tau _i}\) referred to the rest-frames of the various particles, according to

   $$ \left\{ \begin{array}{ccc} d\tau _i &{}=&{}\sqrt{1-\frac{v_i^{\prime 2}}{c^2}}dt'\\ d\tilde{\tau _i} &{}=&{}\sqrt{1-\frac{u_i^{\prime 2}}{c^2}}\,dt'. \end{array}\right. $$
4. 4.

   Note that at order \(O(v^2/c^2)\) Eq. (2.21) can be written \(c^2\varDelta M(0)= (\frac{1}{2}m_1v_1^2+\frac{1}{2}m_2v_2^2)\).

5. 5.

   We recall that \(1\,\mathrm{MeV} = 10^6\,\mathrm{eV}\), where \(1\,\mathrm{eV}\) is the energy acquired by an electron (whose charge is \(\mathrm{e}\cong 1.6\times 10^{-19 }\,\mathrm{C}\)) crossing an electric potential difference of 1 V:

   $$ 1\,\mathrm{eV}=1.6\times 10^{-19}\,\, \text { J}.$$

   Another commonly used unit, when considering energy exchanges in atomic processes, is the atomic mass unit u, that is defined as 1 / 12 the rest mass \(M_C\) of the isotope \({}^{12}C\) of the carbon atom at rest; this unit is more or less the proton mass. Precisely we have: \(1\,u= 1.660 \, 538\, 782(83)\times 10^{-24}\ \mathrm{g}= \frac{1}{N_A}\,\mathrm{g}\), where \(N_A\) is the Avogadro number. Taking into account the equivalence mass-energy we also have

   $$ 1\,u =\frac{1}{12}\,M_C \simeq 931.494 \,\mathrm{MeV}/c^2. $$
6. 6.

   Note that if we had a chemical reaction instead of a nuclear one, involving just the electrons of two hydrogen atoms (\(H+H\rightarrow H_2\)) we would obtain an energy release of \(E\simeq 2\times 10^6\,\mathrm{J}\), which is eight orders of magnitude smaller.

7. 7.

   This does not happen however, because the nuclear fusion of hydrogen ceases when there is no more hydrogen, and after that new reactions and astrophysical phenomena begin to take place.

8. 8.

   As for \(\varDelta x^\mu \) we define \(p^0=E/c\) so that all the four components of \(p^\mu \) share the same physical dimension.

9. 9.

   Alternatively also the denominations Lorentzian or Minkowskian distance are used.

10. 10.

    So far the position of indices in vector components and matrices has been conventionally fixed. We shall give it a meaning in the next chapters.

11. 11.

    We observe that contracted indices, being summed over, can be denoted by arbitrary symbols, for example \(\varLambda ^{\mu }{}_{\nu }\,p^{\nu }\equiv \varLambda ^{\mu }{}_{\rho }\,p^{\rho }\).

Authors and Affiliations

1. Polytechnic University of Turin, Torino, Italy

   Riccardo D’Auria & Mario Trigiante

Corresponding author

Correspondence to Riccardo D’Auria .

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