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Relativistic Mechanics

  • Lorentz contraction of length and slowing of a moving clock:
    $$l = l_0 \sqrt{1 - (v/c)^2}, \; \Delta t = \frac{\Delta t_0}{\sqrt{1 - (v/c)^2}},$$

    where $ l_0 $ is the proper length and $ \Delta t_0 $ is the proper time of the moving clock.

  • Lorentz transformation*:
    $$ x^\prime = \frac{x - Vt}{\sqrt{1 - (V/c)^2}}, \; y^\prime = y, \; t^\prime = \frac{t-x V/c^2}{1-(V/c)^2}.$$
  • Interval $ s_{12} $ is an invariant:
    $$s^2_{12} = c^2 t^2_{12} - l^2_{12} = inv,$$

    where $ t_{12} $ is the time interval between events 1 and 2, $ l_{12} $ is the distance between the points at which these events occurred.

  • Transformation of velocity*:
    $$v_x^\prime = \frac{v_x - V}{1-v_x V/c^2}, \; v_y^\prime = \frac{v_y \sqrt{1 - (V/c)^2}}{1 - v_x V/c^2}.$$
  • Relativistic mass and relativistic momentum:
    $$m = \frac{m_0}{\sqrt{1-(v/c)^2}}, \; \vec{p} = m \vec{v} = \frac{m_0 \vec{v}}{\sqrt{1-(v/c)^2}},$$

    where $ m_0 $ is the rest mass, or, simply, the mass.

  • Relativistic equation of dynamics for a particle:
    $$\frac{d\vec{p}}{dt} = \vec{F},$$

    where $ \vec{p} $ is the relativistic momentum of the particle.

  • Total and kinetic energies of a relativistic particle:
    $$E = mc^2 = m_0 c^2 + T, \; T = (m-m_0)c^2.$$
  • Relationship between the energy and momentum of a relativistic particle
    $$E^2 - p^2c^2 = m_0^2 c^4, \; pc = \sqrt{T(T+2m_0 c^2)}.$$
  • When considering the collisions of particles it helps to use the following invariant quantity:
    $$E^2 - p^2c^2 = m_0^2 c^4,$$

    where $ E $ and $ p $ are the total energy and momentum of the system prior to the collision, and $ m_0 $ is the rest mass of the particle (or the system) formed.

* The reference frame $\displaystyle K^\prime $ is assumed to move with a velocity $ V $ in the positive direction of the $ x $ axis of the frame $ K $, with the $\displaystyle x^\prime $ and $ x $ axes coinciding and the $\displaystyle y^\prime $ and $ y $ axes parallel.

Keywords: 
relativistics, Lorentz, Lorentz transformation, Lorentz contraction, interval, invariant
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