Relativistic kinematics

Special theory of relativity

An inertial fame is a reference frame in which all relative accelerations due to external forces are eliminated.


I. The laws of nature are the same in all inertial reference frames.
II. In vacuum, light propagates with respect to any inertial frame and in all directions with the universal speed c.  This speed is a constant of nature.

The Lorentz transformation

Consider two reference frames K and K’. 
Assume that the coordinate axes in the two frames are parallel and that the origins of the coordinates coincide at t = t’ = 0. 
Assume that K’ is moving with velocity vi with respect to K. 
The Lorentz transformation gives the coordinates of a space-time point (x0,x1,x2,x3) = (ct,x,y,z) in K in terms of its coordinates (x'0,x'1,x'2,x'3) = (ct',x',y', z') in K’ and vice versa.

β = v/c,  β = v/c , γ = (1 - β2).

Since 0 ≤ β ≤ 1, we may write β = tanh(B), where B is the boost parameter or the rapidity.
[tanh(u) = (eu - e-u)/(eu + e-u)].
Then γ = (1 - tanh2(B)) = cosh(B),  γβ = tanh(B)cosh(B) = sinh(B), and

reminiscent of a rotation. 
We define as a 4-vector any set of 4 quantities which transform like (x0,x1,x2,x3) under a Lorentz transformation; (a0,a1,a2,a3) is a 4-vector if
a0' = γ(a0β∙a),  a'|| =  γ( a||  – βa0),   a' = a.

The "dot product"
(a0,a)∙(b0,b) = a0b0 - ab
is invariant under a Lorentz transformation.

The 4-vector (x0,r) defines an event.  The space-time interval between two events is ds.
ds2 = c2dt2 - |dr|2.
In a reference frame in which two events have the same space coordinates dr = 0 and ds2 = c22,
where dτ = ds/c is the proper time interval.  It is a Lorentz invariant quantity.
dτ = dt/γ.

Important 4-vectors

4-velocity:  (u0,u) = (γc,γdr/dt) = (dx0/dτ,dr/dτ). 
(u0,u)∙(u0,u) = c2 is a Lorentz invariant scalar.
4-vector momentum:  (p0,p) = (mu0,mu) = (E/c,p) = (γmc,γmdr/dt).
(p0,p)∙(p0,p) = mc2 --> E2 = m2c4 + p2c2.

Transformation of velocities

A particle moves in K with velocity u = dr/dt.  K' moves with respect to K with velocity v.  The particle's velocity in K’, u' = dr'/dt',  is given by

u'|| = (u|| - v)/(1 - v∙u/c2),
u' =  u/(γ(1 - v∙u/c2)).

where parallel and perpendicular refer to the direction of the relative velocity v.
It is impossible to obtain speeds greater than c.

Relativistic dynamics

Relativistic collisions

In every reference frame energy and momentum are conserved in collisions between free particles.
For each component pμ of the 4-vector (p0, p1, p2, p3) we have
particles_in pμ = ∑particles_out pμ, or ∑i (pi)μ = ∑j (pj)μ,
where i denotes the particles going into the collision and j denotes the particles emerging from the collision.

For transformations between reference frames we have
(P0,P)∙(P0,P) = (P0',P')∙(P0',P').
Here  P0 = ∑particles p0  and  P = ∑particles p.
This is a consequence of the invariance of the dot product under a Lorentz transformation.

Assume we have a collection of initially free particles.  They interact with each other and possibly change into different particles.  After the interaction the new particles are free again.  The following rules always apply:
(a)  The "length" of the 4-vector (P0,P) is invariant when transforming from one one reference frame to another.
(b)  P0 and P are conserved in any reference frame.

The Doppler shift

The angular frequency ω of a sinusoidal electromagnetic wave with wave vector k (k = 2π/λ = ω/c) in a reference frame K is measured as ω' in a reference frame K' moving with uniform velocity v with respect to K.
ω' = γω(1 - (v/c)cosθ), where θ is the angle between the directions of k and v.
ω' = ω[(1 - v/c)/(1 + v/c)]½ if k and v are parallel to each other.
ω' = ω[(1 + v/c)/(1 - v/c)]½ if k and v are anti-parallel to each other.