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Last modified: January 20, 2022 ✓

Lorentz Invariance
------------------

A quantity that remains unchanged by a Lorentz 
transformation is said to be Lorentz invariant.  
This implies that the laws of physics are 
independent of the orientation or the boost 
velocity of the reference frame through space.  
Lorentz invariance is achieved if the laws of 
physics are cast in terms of 4-vector inner 
products.  Let's see this for a rotation of 
cartesian axes.

If we rotate a set of cartesian axes so that 
x -> x' and y -> y' then from Pythagoras we 
can show that for any given point:

x'2 + y'2 = x2 + y2.  

The quantity x2 + y2 is invariant under the 
rotation.

For spacetime under a Lorentz transformation 
(t -> t' and x -> x'), the invariant quantity 
is instead given by:

x'2 + y'2 + z'2 - ct'2 = x2 + y2 + z2 - ct2 

Proof:  

We will consider only one spacial dimension 
and perform a Lorentz transformation of the 
2-vector (t,x) - ordinarily it would be a 
4-vector:

X = γ(x - vt)

T = γ(t - vx/c2)

γ = 1/√(1 - v2/c2)

s2 = -c2T2 + X2

   = -γ2c2(t - vx/c2)2 + γ2(x - vt)2

   = -γ2{c2t2 - 2xvt + v2x2/c2} + γ2{x2 - 2xvt + v2t2}

   = -γ2{c2t2 - 2xvt + v2x2/c2 - x2 + 2xvt - v2t2}

   = -γ2{c2t2 + v2x2/c2 - x2 - v2t2}
   
   = -γ2{(c2 - v2)t2 + (v2/c2 - 1)x2}

   = -γ2{(1 - v2/c2)c2t2 - (1 - v2/c2)x2}

    = -c2t2 + x2 - the invariant interval

We can also construct the invariant interval ds2 
by taking the dot product of the covariant and 
contravariant components of a 4-vector.  A 4-vector 
by itself is not invariant under a Lorentz 
transformation.

dxμ = (cdt,dx1,dx2,dx3) - contravariant

dxμ = (-cdt,dx1,dx2,dx3) - covariant 

ds2 = Σdxμdxμ 

    = dxμdxμ 

    = -c2(dt)2 + (dx)2/c2 + (dy)2/c2 + (dz)2/c2

The result is invariant under rotations in space 
and a Lorentz transformation.  This can also be 
written as:

ds2 = dxμdxμ = ημνdxμdxν where ημν is the MINKOWSKI 
METRIC.  

       -       -
      |-1 0 0 0 |
ημν = | 0 1 0 0 |
      | 0 0 1 0 |
      | 0 0 0 1 |
       -       -

Note, the time-coordinate has different sign than 
the space coordinates.  This simple change of sign 
leads to completely new hyperbolic geometry that 
is different from conventional Euclidean geometry.

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