Relativistic Momentum and Energy

Similarly to the four-dimensional velocity m also the 4-momentum is defined in analogy to the nonrelativistic set-up, i.e.. 

Obviously, the relativistic momentum pi is also a Lorentz 4-vector, where we have introduced the relativistic (kinematic) 3-momentum 

It should be emphasized that the relativistic energy E does not equal mc, since the famous expression E = m v)c assumes a velocity-dependent mass m v) = jm. However, we do not adopt any concept of velocity-dependent mass (or relativistic mass) throughout this book. Mass m always denotes the rest mass, which is independent of the chosen frame of reference and therefore a fundamental property of the particle. 

The deeper reason for calling E and p the relativistic energy and momentum, respectively, is that they are conserved quantities in the following sense. If an observer in one specific inertial frame of reference IS sees that E and/or p are conserved throughout a reaction or process, so does any other observer in another frame of reference IS related to IS via a Lorentz transformation. This unique feature of energy and momentum directly follows from the 4-vector property of p since the change Ap in energy or momentum in IS is related to those in IS by 

The last equality holds since the Lorentz transformation A depends only on the relative motion between IS and IS and not on specific features of the system under investigation nor does it feature an explicit space-time dependence. If the change in energy and /or momentum vanishes in IS, it will therefore also vanish in IS. We mention without proof that energy E and momentum p are the only functions of velocity whose conservation is Lorentz invariant. Furthermore, if momentum p is conserved, so is the energy E. This is most easily seen by considering two inertial frames IS and IS with 

---