Relativistic Kinematics

Special theory of relativity

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

Postulates

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.

.

b = v/c,  b = v/c , g = (1 - b2)-1/2.

Since 0 £ b £ 1, we may write b = tanh(B), where B is the boost parameter or the rapidity.
[tanh(u) = (eu - e-u)/(eu + e-u)].

Then g = (1 - tanh2(B))-1/2 = cosh(B),  gb = tanh(B)cos(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

,

or

.

The "dot product"

 

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 = c2dt2,

where dt = ds/c is the proper time interval.  It is a Lorentz invariant quantity. 

dt = dt/g.

Important 4-vectors

4-velocity:

,

a Lorentz invariant scalar.

4-vector momentum:

We define

.

Then

and

.

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^/(g(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.

The Doppler shift

The angular frequency w of a sinusoidal electromagnetic wave with wave vector k (k = 2p/l = w/c) in a reference frame K is measured as w' in a reference frame K' moving with uniform velocity v with respect to K.
w' = gw(1 - (v/c)cosq), where q is the angle between the directions of k and v.
w' = w[(1 - v/c)/(1 + v/c)]1/2 if k and v are parallel to each other.
w' = w[(1 + v/c)/(1 - v/c)]1/2 if k and v are anti-parallel to each other.