# Indefinite
orthogonal groups: The indefinite/pseudo orthogonal group, the split
orthogonal group, the Lorentz group

### Indefinite
(pseduo) and split orthognal groups \(O(n,m,F)\)

The bilinear form is:

\(u^TMv\)

The transformations which preserve this are:

\(P^TMP=M\)

#### The metric

If the metric is:

\(M=\begin{bmatrix}-1 & 0 & 0 &
0\\0 & 1 & 0 & 0\\0 & 0 & 1 & 1\\0 & 0 &
0 & 1\end{bmatrix}\)

Then we have the indefinite orthogonal group \(O(3,1)\)

#### The split orthogonal group

Where \(n=m\) we have the split
orthogonal group.

\(O(n,n,F)\)

#### Signatures

### The Lorentz group

The Lorentz group is the \(O(1,3)\)
group.

#### Symmetries of the Lorentz
group

We can do the usual \(3\) rotations,
however there are additional \(3\)
symmetries, making the Lorzentz group \(6\)-dimensional.

These are the Lorentz boosts.

A symmetry has:

\(t'^2 - x'2 - y'^2 - z'^2
= t^2 - x^2 - y^2 - z^2\)

We consider the case where we just boost on \(x\), so \(y =
y'\) and \(z = z'\).

\(t'^2 - x'2 = t^2 -
x^2\)

Or with \(c\):

\(c^2t'^2 - x'2 = ct^2 -
x^2\)

#### New

\(s^2 = t^2 - x^2 - y^z - z^2\)

\(s'^2 = t'^2 - x'^2 - y'^2
- z'^2\)

\(ds^2 = s'^2 - s^2\)

\(ds^2 = (t'^2 - x'^2 - y'^2 -
z'^2) - (t^2 - x^2 - y^z - z^2)\)

\(ds^2 = (t'^2 - t^2) - (x'^2 -
x^2) - (y'^2 - y^2) - (z'^2 - z^2)\)

\(ds^2 = dt^2 - dx^2 - dy^2 -
dz^2\)

boost: \(s^2 = c^2t^2 - x^2 - y^z -
z^2\)

we want new t and x where distance is same \(c^2t'^2 - x'^2 - y^z - z^2 = c^2t^2 - x^2
- y^z -z^2\) \(c^2t'^2 - x'^2 =
c^2t^2 - x^2\)

We know that both transformations are linear [WHY??], therefore \(x' = Ax + Bt\) \(t' = Cx+Dt\)

we transform to \(x' = 0\). so
\(Ax + Bt = 0\)

We define \(v = \dfrac{x}{t}\)

So: \(x = vt\)

We can plug these in: \(Avt + Bt =
0\) \(Av + B = 0\) \(\dfrac{A}{B} = -v\)