C2 - four vector form and tensors

the four components of a vector: $(x^{0}, x^{1}, x^{2}, x^{3}) = x^{μ}$ , making a '4-position', where $μ$ is an index that runs from 0 to 3
$x^{μ}$ is a contravariant component of the vector $\vec{x}$ , whose covariant component is $x_{μ}$
choices for $x^{μ} :$
- origin of all the axes
- orientation of the spatial axes, changeable by rotation
- change frame of reference via an LT with $v$ along $x$
rotation of a 3-position:

x^{' i} = R_{3 \times 3} x^{i}, R = (\begin{matrix} \cos θ & \sin θ & 0 \\ - \sin θ & \cos θ & 0 \\ 0 & 0 & 1 \end{matrix})

for a 4-position: $x^{' μ} = Λ_{ν}^{μ} x^{ν}, Λ_{ν}^{μ} = (\begin{matrix} 1 & 0 & 0 & 0 \\ 0 & ⋱ \\ 0 & R_{3 \times 3} \\ 0 & ⋱ \end{matrix})$
for change of reference:

\begin{matrix} x^{' μ} = Λ_{ν}^{μ} x^{ν} \\ Λ_{ν}^{μ} = (\begin{matrix} γ & - \frac{v}{c} γ & 0 & 0 \\ - \frac{v}{c} t & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) \end{matrix}

$Λ_{ν}^{μ}$ is a second-rank tensor with the indices $μ$ representing rows, and $ν$ representing columns
$x^{μ}$ is a first-rank tensor

Einstein summation convention

sum over repeated indices with 1 up and 1 down
eg: $x^{' 1} = \sum_{α = 0}^{3} λ_{α}^{1} x^{α} \equiv Λ_{α}^{1} x^{α}$

properties of 4-vectors

a 4-vector represents a physical quantity expressed in 4 coordinates
eg: position $x^{μ}$ , velocity $u^{μ}$ , force $f^{μ}$
it must respect the symmetry of spacetime
transform according to LT
if $a^{μ} = b^{μ}$ , all four components are equal

relationship between covariant and contravariant forms

the relationship is given by:

a_{μ} = g_{μ ν} a^{ν}

where, $g_{μ ν}$ is the metric tensor

for special relativity, spacetime is 'flat', and $g_{μ ν} = diag (1, - 1, - 1, - 1)$
this is called the positive time, negative space convention, used in this course
negative time, positive space is used by some communities
for a wave along $+ k :$

\exp i (ω t - \vec{k} \cdot \vec{r}) x_{μ} = (c t, - x, - y, - z) a_{μ} = η_{μ ν} a^{ν} : index lowering a^{μ} = η^{μ ν} a_{ν} : index raising

for flat space time:

η_{μ ν} = η^{μ ν} η_{μ λ} η^{λ ν} = I_{μ}^{ν} = δ_{μ}^{ν}

product of the four vectors:

a_{ν} a^{ν} = \sum_{ν = 0}^{3} a_{ν} a^{ν} = a_{0} a^{0} + a_{1} a^{1} + a_{2} a^{2} + a_{3} a^{3}

in every frame of reference, the product of four vectors is identical:

a_{μ}^{'} a^{' μ} = a_{μ} a^{μ} a^{' μ} = Λ_{ν}^{μ} a^{ν} a_{μ}^{'} = Λ_{μ}^{ν} a_{ν}

Λ_{ν}^{μ} = (\begin{matrix} γ & - \frac{v}{c} γ & 0 & 0 \\ - \frac{v}{c} t & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix}) Λ_{μ}^{ν} = (\begin{matrix} γ & \frac{v}{c} γ & 0 & 0 \\ \frac{v}{c} t & γ & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix})

\begin{aligned} a_{μ}^{'} a^{' μ} = Λ_{μ}^{ν} a_{ν} Λ_{λ}^{μ} a^{λ} & = a_{ν} a^{λ} Λ_{μ}^{ν} Λ_{κ}^{μ} = a_{ν} a^{λ} δ_{λ}^{ν} \\ ∴ a_{μ}^{'} a^{' μ} & = a_{ν} a^{ν} \end{aligned}

eg: $$a'{0}a'^{0} = aa^{0} \left(\gamma^{2} - \frac{v^{2}}{c^{2}}\gamma^{2}\right) = a_{0}a^{0}$$
hence, these are invariant quantities:

x_{μ} x^{μ} = (c t)^{2} - (x^{2} + y^{2} + z^{2}) = s^{2}

$s^{2}$ is called the interval

d s^{2} = (c d t)^{2} - (d x^{2} + d y^{2} + d z^{2})

this is the separation between events
if $d s^{2} > 0 :$ time-like interval
if $d s^{2} < 0 :$ space-like interval
if $d s^{2} = 0 :$ light-like interval

image: K Aainsqats
the cone has a gradient of $1$ , and anything on its surface has a light-like interval
anything inside the cone can be causally connected, anything outside it cannot