7 Transformation of Energy and Momentum

The generic transformation equations for energy and momentum depend indirectly on the equation for relativistic addition of velocities. Because a new one replaces this equation, it is necessary to rework the transformation equations for energy and momentum as well.

Figure 9 depicts an object moving with velocity $\mathbf{W}$ of magnitude $w$ relative to frame $x$ and velocity $\mathbf{U}$ of magnitude $u$ relative to frame $x'$ .

Figure 9: Generic transformation of energy and momentum in three reference frames with rotated dimensional axes.

(please refer also to Fig. 3 and the definitions given there)

• $E=\gamma (w)m_0 c^2$ is the energy of an object that moves with velocity $\mathbf{W}$ of magnitude $w$ relative to frame $x$ and measured in frame $x$ .
• $E'=\gamma (u)m_0 c^2$ is the energy of that same object moving with velocity $\mathbf{U}$ of magnitude $u$ relative to frame $x'$ and measured from frame $x'$ .
• Frame $x'$ moves with velocity $\mathbf{V}$ of magnitude $v$ relative to frame $x$ .
• $\gamma (u)=1 \left/ \sqrt{1-u^2/c^2} \right.$
• $\gamma (v)=1 \left/ \sqrt{1-v^2/c^2} \right.$
• $\gamma (w)=1 \left/ \sqrt{1-w^2/c^2} \right.$

For energy this leads to a generic transformation equation

$$E/E'=\gamma (w)/ \gamma (u)$$ (31)

which can be written in different forms using Eq. (19). With $u=0$ this reduces to the classical form:

$$E/E'=\gamma (v)$$ (32)

For momentum a generic transformation equation is

$$p/p'=wE/uE'$$ (33)

where:

• $p'=\gamma (u)m_0 u$ is the momentum of the object as measured from frame $x'$ .
• $p=\gamma (w)m_0 w$ is the momentum of the object as measured from frame $x$ .