Relativistic Momentum and Relativistic form of Newton's Equation:

Relativistic momentum of a particle moving at a velocity u is defined with respect to the proper time t' (the time measured by an observer at rest w.r.t the clock)

p = m₀(dx/dt'),

          = m₀(dx/dt)(dt/dt').

Thus,                                                  p = m₀u/(1-u²/c²)^½ = g m₀u,          

where,            

g = 1/(1-u²/c²)^½.

g is also called the Lorentz factor.

m₀ is the 'rest mass' or the 'proper mass' (i.e. the mass measured by an observer at rest w.r.t the mass). The above equation can also be written in terms of the relativistic mass m (mass due to motion), where

m=gm₀.

One can clearly see that the relativistic mass m is greater (equals when the object is at rest) than the rest mass, m₀ as g >1 (convince yourself).

Justify the statement: "Moving objects are heavier than when they are at rest."

In terms of the relativistic mass the relativistic momentum is p=mu.

The relativistic force, F on the particle with momentum p is defined as

F=dp/dt = d(gm₀u)/dt = m₀[gdu/dt + u dg/dt].

Substituting for dg/dt = u/[c²(1-u²/c²)^3/2] we get after some algebraic steps,

F=m₀(du/dt)/(1-u²/c²)^3/2 .   

The above force equation is the relativistic form of Newton's equation F=m₀(du/dt).               

Relativistic Energy:

The  kinetic energy K of a particle moving at a velocity u is,

K = m₀c²/(1-u²/c²)^1/2 - m₀c²,

= gm₀c² - m₀c².

Check that in the limit u<<c, we get back the Newtonian expression for the kinetic energy, i.e. K= m₀u²/2.

The total relativistic energy E is defined as:

E = gm₀c² = K + m₀c².

m₀c² is defined as the rest energy (E₀) of the particle. The total energy E is thus the sum of two terms, the kinetic energy and the rest energy. The above equation suggests that even for a particle at rest, the contribution from the rest energy is enormous (e.g. for electrons,  m₀c² = 0.511MeV ).

The total energy E can also be expressed in terms of the momentum p as:

E² = p²c² + (m₀c²)².

Note that the quantity (E² - p²c²) is equal to a frame independent constant (m₀²c⁴) which means that this quantity is the same for all inertial frames.

For massless particles like photons, m₀=0, thus E=pc. Note, that the rest mass of photons are zero, not the relativistic mass  (why? Convince yourself from the expression of relativistic mass in terms of the rest mass), thus photons do have relativistic momentum.

Momentum and Energy Conservation:

Remember that both relativistic momentum and energy are conserved quantities for all inertial observers (for observers in any inertial reference frames).

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