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Last modified: January 26, 2018

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 pf physics are cast in 
terms of 4-vector dot 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- t'2 = x2 + y2 + z2 - t2 

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)

Put c = 1 for simplicity, γ = 1/√(1 - v2)

dτ2 = -dT2 + dX2

    = -γ2(t - vx)2 + γ2(x - vt)2

    = -γ2{t2 - 2xvt + v2x2} + γ2{x2 - 2xvt + v2t2}

    = -γ2{t2 - 2xvt + v2x2 - x2 + 2xvt - v2t2}

    = -γ2{(1 - v2)t2 + (v2 - 1)x2}

    = -γ2{(1 - v2)t2 - (1 - v2)x2}

    = -dt2 + dx2

dτ2 = -dt2 + (dx/c)2 - the invariant interval

 t
  |         /
  |        /|  <- not necessarily linear
  |       / dt
  |      /  |
  |     /-dx-
  |    /
  |
  | 
   -------------------------- x

Construct the invariant interval dτ2 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 = (dx0,dx1,dx2,dx3) - contravariant

dxμ = (-dx0,dx1,dx2,dx3) - covariant 

dτ2 = Σdxμdxμ 

    = dxμdxμ 

    = dx0dx0 + dx1dx1 + dx2dx2 + dx3dx3

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

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

dτ2 = 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 means that in four-dimensional 
spacetime, one coordinate is different from the others 
and influences the distance differently. This simple 
change of sign leads to completely new hyperbolic 
geometry that is different from conventional Euclidean 
geometry.

In GR the Minkowski Metric is replaced by the METRIC 
TENSOR, grs, where,

grs = δmn(∂xm/∂yr)(∂xn/∂ys)

so

(ds)2 = grsdxrdxs


4-velocity
---------

 uμuμ = -γ2(c2 - v2)

      = -γ2c2(1 - v2/c2)

      = -c2

Thus, the velocity of light is invariant under a Lorentz 
transform.

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