note-math

Subspace $span (𝑣, 𝑤) ↪ ℝ^{𝑝, 𝑞}$

The following are equivalent

$dim (span (𝑣, 𝑤)) = 2$
$𝑣, 𝑤$ not co-linear
$𝑣 \land 𝑤 \neq 0$

if $span (𝑣, 𝑤) ≃ ℝ^{1, 1}$ , possible

2 time(-like)
Example

$𝑣 = (\begin{matrix} 1 \\ 0 \end{matrix}), 𝑤 = (\begin{matrix} 𝑎 \\ 𝑏 \end{matrix})$ , where $0 < | 𝑏 | < | 𝑎 |$
${⟨ 𝑤 ⟩}^{2} = 𝑎^{2} - 𝑏^{2} > 0$
Can linearly generate $(\begin{matrix} 0 \\ 1 \end{matrix})$
1 time, 1 space
Example $𝑣 = (\begin{matrix} 1 \\ 0 \end{matrix}), 𝑤 = (\begin{matrix} 0 \\ 1 \end{matrix})$
1 time, 1 null
Example $𝑣 = (\begin{matrix} 1 \\ 0 \end{matrix}), 𝑤 = (\begin{matrix} 𝑎 \\ 𝑎 \end{matrix})$
2 null
Example $ℝ^{1, 1}, 𝑣 = (\begin{matrix} 𝑎 \\ 𝑎 \end{matrix}), 𝑤 = (\begin{matrix} 𝑎 \\ - 𝑎 \end{matrix})$ . Note that $𝑣 \cdot 𝑤 = 2 𝑎^{2} \neq 0$ . signature $(1, 1)$
2 space.
Example
$𝑣 = (\begin{matrix} 0 \\ 1 \end{matrix}), 𝑤 = (\begin{matrix} 𝑎 \\ 𝑏 \end{matrix})$ , where $0 < | 𝑎 | < | 𝑏 |$
other cases (symmetry of time $⟷$ space)

Consider the general $ℝ^{1, 𝑛}$ in $span (𝑣, 𝑤)$

signature-of-2d-subspace-of-spacetime_(tag) Prop The possible signature of Minkowski $(1, 𝑛)$ in $dim = 2$ 's $span (𝑣, 𝑤)$ is

$1, 1$
$0, 2$
$0, 1$

Prop time-like is only orthogonal to space-like

let $𝑣$ time-like. Using orthogonal decomposition, let $𝑣 = 𝑣_{0}, 𝑤 = 𝑤_{0} + 𝑤$ then $⟨ 𝑣, 𝑤 ⟩ = 𝑣_{0} 𝑤_{0} = 0 ⟹ 𝑤_{0} = 0$ ==> $𝑤$ space-like

Prop light-like $𝑣$ is not orthogonal to

time-like
light-like other than collinear with itself $𝑘 𝑣$ metric-cannot-distinguish-colinear-light-like_(tag)

Proof (ref-7, (ref-9, p.13))

Take an orthogonal decomposition according to the situation $ℝ^{1, 𝑛} = ℝ_{time} \oplus ℝ_{space}^{𝑛}$

\begin{matrix} 𝑣 & = & 𝑣_{0} + 𝑣 \\ 𝑤 & = & 𝑤_{0} + 𝑤 \end{matrix}

$𝑤$ time-like ==> let $𝑤 = 𝑤_{0}$ ==> $⟨ 𝑣, 𝑤 ⟩ = 𝑣_{0} 𝑤_{0} \neq 0$
$𝑤$ light-like

\begin{matrix} {⟨ 𝑣 ⟩}^{2} & = & 0 & ⟹ & 𝑣_{0}^{2} & = & {⟨ 𝑣 ⟩}^{2} \\ {⟨ 𝑤 ⟩}^{2} & = & 0 & ⟹ & 𝑤_{0}^{2} & = & {⟨ 𝑤 ⟩}^{2} \\ ⟨ 𝑣, 𝑤 ⟩ & = & 0 & ⟹ & 𝑣_{0} 𝑤_{0} & = & ⟨ 𝑣, 𝑤 ⟩ \end{matrix}

We prove that $𝑤_{0} \cdot 𝑣 = 𝑣_{0} \cdot 𝑤$

\begin{matrix} 𝑤_{0} \cdot 𝑣 - 𝑣_{0} \cdot 𝑤 \\ = & 𝑤_{0} \cdot 𝑣 - 𝑣_{0} \cdot 𝑤 \\ \in & ℝ_{space}^{𝑛} \end{matrix}

but

\begin{matrix} {⟨ 𝑤_{0} \cdot 𝑣 - 𝑣_{0} \cdot 𝑤 ⟩}^{2} \\ = & 𝑤_{0}^{2} {⟨ 𝑣 ⟩}^{2} - 2 𝑣_{0} 𝑤_{0} \cdot ⟨ 𝑣, 𝑤 ⟩ + 𝑣_{0}^{2} {⟨ 𝑤 ⟩}^{2} \\ = & 0 \end{matrix}

space-like but length zero, so $𝑤_{0} \cdot 𝑣 - 𝑣_{0} \cdot 𝑤 = 0$

==> $𝑤_{0} \cdot 𝑣 - 𝑣_{0} \cdot 𝑤 = 0$

Prop The signature of the two-dimensional subspace of $ℝ^{1, 𝑛}$ cannot be $1, 0$ or $0, 0$

Proof Use the previous theorem

Prop The signature of $span (𝑣, 𝑤)$ expanded by two non-collinear time-like $𝑣, 𝑤$ is $1, 1$

Proof Generate an orthogonal basis of $span (𝑣, 𝑤)$ with one of them as the initial basis, but the signature cannot be $1, 0$ , so it can only be $1, 1$

The projection of $𝑣$ is ${𝑘 𝑣 \in ℝ^{1, 3} : 𝑘 \in ℝ} \subset cone$

Prop let ${⟨ 𝑣 ⟩}^{2} = 0$ , let $𝑤$ time-like or light-like with $𝑣, 𝑤$ non-collinear. Then $span (𝑣, 𝑤) ⊄ cone$

Proof

Known $⟨ 𝑣, 𝑤 ⟩ \neq 0$

On the light cone, it is equivalent to solving the quadratic equation for the variable $𝑏$ : $0 = {(𝑎 𝑣 + 𝑏 𝑤)}^{2} = 𝑎 𝑏 \cdot ⟨ 𝑣, 𝑤 ⟩ + 𝑏^{2} \cdot {⟨ 𝑤 ⟩}^{2} = 𝑏 (𝑏 \cdot {⟨ 𝑤 ⟩}^{2} + 𝑎 \cdot ⟨ 𝑣, 𝑤 ⟩)$

$𝑏 \neq 0 ⟹ 𝑏 = {\begin{matrix} - \frac{𝑎 \cdot ⟨ 𝑣, 𝑤 ⟩}{{⟨ 𝑤 ⟩}^{2}} & if {⟨ 𝑤 ⟩}^{2} \neq 0 \\ ℝ & if {⟨ 𝑤 ⟩}^{2} = 0 \end{matrix}$

Prop The signature of the span $span (𝑣, 𝑤)$ of two non-collinear light-like $𝑣, 𝑤$ in $ℝ^{1, 𝑛}$ is $1, 1$ or $0, 1$

Proof $ℝ^{0, 2}$ Euclidean has no light-like, so there is no other possibility

Example

$ℝ^{1, 1}$ 's $(\begin{matrix} 1 \\ \pm 1 \end{matrix})$
$ℝ^{1, 2}$ 's $(\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}), (\begin{matrix} 1 \\ 0 \\ 1 \end{matrix})$ . Subtracting gives an orthogonal basis $(\begin{matrix} 1 \\ 1 \\ 0 \end{matrix}), (\begin{matrix} 0 \\ 1 \\ - 1 \end{matrix})$

simultaneity-relativity_(tag) Simultaneity in relativity

use orthogonal basis continuation

in $1, 𝑛$ , the orthogonal complement of a $dim = 𝑛$ space-like subspace is a $dim = 1$ time-like subspace

( $span (𝑣, 𝑣^{'}) ≃ ℝ^{2}$ space-like <==> there exists a time-like $𝑤$ that is simultaneously orthogonal to $𝑣, 𝑣^{'}$ )
( $span (𝑣, 𝑣^{'}) ≃ ℝ^{1, 1}$ not space-like <==> there does not exist a time-like $𝑤$ that is simultaneously orthogonal to space-like $𝑣, 𝑣^{'}$ )

Intuition: Different space-like subspaces $𝑆, 𝑆^{'}$ cannot be calculated using compatible time calculation methods or the time-like orthogonal complements of $𝑆, 𝑆^{'}$ are not the same

use $ℝ^{1, 𝑛}$ orthogonal decomposition

$𝑣 = 𝑣_{0} + 𝑣$

$⟨ 𝑣, 𝑤 ⟩ = 𝑣_{0} 𝑤_{0} - ⟨ 𝑣, 𝑤 ⟩$

${⟨ 𝑣 ⟩}^{2}, {⟨ 𝑤 ⟩}^{2} \geq 0$

Categorical discussion of $sign (𝑣_{0} 𝑤_{0})$ . The sign of the product of time components of the inner product determines the sign of the inner product

\begin{matrix} sign (𝑣_{0} 𝑤_{0}) & = & sign (𝑣_{0} 𝑤_{0} - ⟨ 𝑣, 𝑤 ⟩) \\ = & sign (⟨ 𝑣, 𝑤 ⟩) \end{matrix}

${⟨ 𝑣 ⟩}^{2}, {⟨ 𝑤 ⟩}^{2} \leq 0$

Categorical discussion of $sign (⟨ 𝑣, 𝑤 ⟩)$ . The sign of the product of space components of the inner product determines the sign of the inner product

\begin{matrix} sign (- ⟨ 𝑣, 𝑤 ⟩) & = & sign (𝑣_{0} 𝑤_{0} - ⟨ 𝑣, 𝑤 ⟩) \\ = & sign (⟨ 𝑣, 𝑤 ⟩) \end{matrix}

in Euclidean, we have #link(<quadratic-form-inequality-Euclidean>)[inner product inequality] $| ⟨ 𝑣, 𝑤 ⟩ |^{2} \leq | 𝑣 | | 𝑤 |$ ==> #link(<triangle-inequality-Euclidean>)[triangle inequality] $| 𝑣 + 𝑤 | \leq | 𝑣 | + | 𝑤 |$

in signature $𝑝, 𝑞$ quadratic form, this is generally not true

Derive the $𝑝, 𝑞$ quadratic form #link(<tensor-induced-quadratic-form>)[derived] to alternating second-order linear

${⟨ 𝑣 \land 𝑤 ⟩}^{2} = det (\begin{matrix} {⟨ 𝑣 ⟩}^{2} & ⟨ 𝑣, 𝑤 ⟩ \\ 𝑤 𝑣 & {⟨ 𝑤 ⟩}^{2} \end{matrix}) = {⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} - {⟨ 𝑣, 𝑤 ⟩}^{2}$

quadratic-form-inequality-Minkowski_(tag) Inner product inequality

in $ℝ^{1, 𝑛}$ , let $𝑣, 𝑤$ not co-linear, so $dim (span (𝑣, 𝑤)) = 2$

$ℝ^{1, 𝑛}$ quadratic form restricted to $span (𝑣, 𝑤)$ , signature

$1, 1$ ==> ${⟨ 𝑣 \land 𝑤 ⟩}^{2} = {⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} - {⟨ 𝑣, 𝑤 ⟩}^{2} < 0$ ==> ${⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} < {⟨ 𝑣, 𝑤 ⟩}^{2}$
$0, 2$ ==> ${⟨ 𝑣 \land 𝑤 ⟩}^{2} = {⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} - {⟨ 𝑣, 𝑤 ⟩}^{2} > 0$ ==> ${⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} > {⟨ 𝑣, 𝑤 ⟩}^{2}$

Proof

$dim (span (𝑣, 𝑤)) = 2$ ==> $dim (⋀^{2} span (𝑣, 𝑤)) = (\binom{2}{2}) = 1$

$span (𝑣, 𝑤)$ quadratic form derived to $⋀^{2} span (𝑣, 𝑤)$

signature

$1, 1$ of $span (𝑣, 𝑤)$ ==> $(- 1)$ of $⋀^{2} span (𝑣, 𝑤)$

Proof

$1, 1$ of $span (𝑣, 𝑤)$ orthogonal basis $𝑒_{0}, 𝑒_{1}$ , $𝑒_{0}^{2} = 1, 𝑒_{1}^{2} = - 1$ ==> $⋀^{2} span (𝑣, 𝑤)$ orthogonal basis $𝑒_{0} \land 𝑒_{1}$ , ${(𝑒_{0} \land 𝑒_{1})}^{2} = 𝑒_{0}^{2} \cdot 𝑒_{1}^{2} = - 1$

==> ${⟨ 𝑣 \land 𝑤 ⟩}^{2} < 0$ , i.e. inner product inequality
$0, 2$ of $span (𝑣, 𝑤)$ ==> $(+ 1)$ of $⋀^{2} span (𝑣, 𝑤)$

==> ${⟨ 𝑣 \land 𝑤 ⟩}^{2} > 0$

triangel-inequality-Minkowski_(tag) Triangle inequality

${⟨ 𝑣 + 𝑤 ⟩}^{2} = {⟨ 𝑣 ⟩}^{2} + 2 ⟨ 𝑣, 𝑤 ⟩ + {⟨ 𝑤 ⟩}^{2}$

2 time

${⟨ 𝑣 ⟩}^{2} > 0$ , $| 𝑣 | ≔ {({⟨ 𝑣 ⟩}^{2})}^{\frac{1}{2}}$

$⟨ 𝑣, 𝑤 ⟩ > 0$ ==> $| 𝑣 + 𝑤 | > | 𝑣 | + | 𝑤 |$
$⟨ 𝑣, 𝑤 ⟩ < 0$ ==> ${⟨ 𝑣 + 𝑤 ⟩}^{2} < {(| 𝑣 | - | 𝑤 |)}^{2}$

1 time, 1 null

${⟨ 𝑤 ⟩}^{2} = 0$ ==> ${⟨ 𝑣 + 𝑤 ⟩}^{2} = {⟨ 𝑣 ⟩}^{2} + 2 ⟨ 𝑣, 𝑤 ⟩$

$⟨ 𝑣, 𝑤 ⟩ > 0$ ==> ${⟨ 𝑣 + 𝑤 ⟩}^{2} > {⟨ 𝑣 ⟩}^{2}$
$⟨ 𝑣, 𝑤 ⟩ < 0$ ==> ${⟨ 𝑣 + 𝑤 ⟩}^{2} < {⟨ 𝑣 ⟩}^{2}$

Proof of 2 time-like

${⟨ 𝑣 ⟩}^{2}, {⟨ 𝑤 ⟩}^{2} > 0$

$| 𝑣 | ≔ {({⟨ 𝑣 ⟩}^{2})}^{\frac{1}{2}}$

$⟨ 𝑣, 𝑤 ⟩ > 0$

${⟨ 𝑣 + 𝑤 ⟩}^{2} > 0$

use #link(<quadratic-form-inequality-Minkowski>)[] ${⟨ 𝑣 ⟩}^{2} {⟨ 𝑤 ⟩}^{2} - {⟨ 𝑣, 𝑤 ⟩}^{2} = {⟨ 𝑣 \land 𝑤 ⟩}^{2} < 0$

==> $| 𝑣 | | 𝑤 | < ⟨ 𝑣, 𝑤 ⟩$

==>

\begin{matrix} {⟨ 𝑣 + 𝑤 ⟩}^{2} & > & {⟨ 𝑣 ⟩}^{2} + 2 | 𝑣 | | 𝑤 | + {⟨ 𝑤 ⟩}^{2} \\ = & {(| 𝑣 | + | 𝑤 |)}^{2} \end{matrix}

==>

| 𝑣 + 𝑤 | > | 𝑣 | + | 𝑤 |

$⟨ 𝑣, 𝑤 ⟩ < 0$

$⟨ 𝑣, 𝑤 ⟩ < 0$

==> $- | 𝑣 | | 𝑤 | > ⟨ 𝑣, 𝑤 ⟩$

==>

\begin{matrix} {⟨ 𝑣 + 𝑤 ⟩}^{2} & < & {⟨ 𝑣 ⟩}^{2} - 2 | 𝑣 | | 𝑤 | + {⟨ 𝑤 ⟩}^{2} \\ = & {(| 𝑣 | - | 𝑤 |)}^{2} \end{matrix}

$sign {⟨ 𝑣 + 𝑤 ⟩}^{2}$ is uncertain

Example let $𝑣 = (\begin{matrix} 1 \\ 0 \end{matrix})$ . let $𝑤$ past time-like

$𝑤 = (\begin{matrix} - 1 \\ 0 \end{matrix}) ⟹ {⟨ 𝑣 + 𝑤 ⟩}^{2} = 0$
$𝑤 = (\begin{matrix} - 1 \\ \frac{1}{2} \end{matrix}) ⟹ {⟨ 𝑣 + 𝑤 ⟩}^{2} = - \frac{1}{4}$
$𝑤 = (\begin{matrix} - \frac{1}{2} \\ 0 \end{matrix}) ⟹ {⟨ 𝑣 + 𝑤 ⟩}^{2} = \frac{1}{4}$

Euclidean space can already discuss different convergence directions e.g. whether the sequence $\frac{𝑥_{𝑛}}{| 𝑥_{𝑛} |}$ converges to $𝕊^{𝑛 - 1}$ . Spiral-like things do not converge in direction space

Euclidean space converges to a point in all $𝕊^{𝑛 - 1}$ directions <==> converges to a point uniformly in all directions, by compactness of $𝕊^{𝑛 - 1}, {ℝ ℙ}^{𝑛 - 1}$

Minkowski space direction space $ℚ^{1, 𝑛} (\pm 1)$ is non compact. Although we have not yet defined the net of $ℚ^{1, 𝑛} (\pm 1)$

The #link(<net>)[] of Minkowski space needs to be sufficiently far from the light cone ${⟨ 𝑣 ⟩}^{2} = 0$

let ${ℍ 𝕪}^{𝑛} (time / space) ≔ {𝑥 \in ℝ^{1, 𝑛} : 𝑥^{2} = \pm 1}$

For convergent timelike directions, they can be separated

Future: $𝑣 \in {ℍ 𝕪}^{𝑛} (time, future) = {𝑥 \in ℝ^{1, 𝑛} : 𝑥^{2} = 1, 𝑥_{0} > 0}$
Past: $𝑣 \in {ℍ 𝕪}^{𝑛} (time, past) = {𝑥 \in ℝ^{1, 𝑛} : 𝑥^{2} = 1, 𝑥_{0} < 0}$
Mixed: ${ℍ 𝕪}^{𝑛} (time)$ quotient away the two leaves $\pm 𝑣$ , becoming a projective space type direction space

in $ℝ^{1, 1}$

let $ℍ 𝕪 ≔ {ℍ 𝕪}^{1} (time, future) = {(𝑡, 𝑥) \in ℝ^{1, 1} : 𝑡^{2} - 𝑥^{2} = 1, 𝑡 > 0}$

hyperbolic-complex_(tag) Hyperbolic complex number. cf. #link(<split-complex-number>)[]

$(𝑥, 𝑦) ≃ 𝑥 + 𝑦 i_{split} = 𝑥 𝟙 + 𝑦 i_{split}$

$𝟙 \cdot i_{split} = i_{split} \cdot 𝟙 = i_{split}$
${i_{split}}^{2} = 𝟙$
$(𝑥_{1} + 𝑦_{1} i_{split}) \cdot (𝑥_{2} + 𝑦_{2} i_{split})$ expand according to the distributive law

hyperbolic-exp_(tag)

$exph 𝑧 ≔ \sum_{𝑛 \in ℕ} \frac{1}{𝑛!} 𝑧^{𝑛}$

use binomial

$exph (𝑧 + 𝑤) = (exph 𝑧) (exph 𝑤)$
$exph (𝑡 + i_{split} 𝑥) = exph (𝑡) exph (i_{split} 𝑥)$
$exph (ϕ i_{split}) = cosh ϕ + (sinh ϕ) i_{split} \in ℍ 𝕪$ , $ϕ \in ℝ$ . by ${cosh}^{2} - {sinh}^{2} = 1$

polor-coordinate-hyperbolic_(tag)

Hyperbolic polar coordinates $𝑣 = | 𝑣 | exph (ϕ i_{split})$ , $| 𝑣 | = {(𝑣 𝑣^{*})}^{\frac{1}{2}} = {⟨ 𝑣 ⟩}^{2^{\frac{1}{2}}}$ , $ϕ \in ℝ$ . $ϕ$ can come from $ℍ 𝕪$ geodesic length parameter. Also known as hyperbolic angle hyperbolic-angle_(tag)

Polar coordinates are the decomposition of distance and direction

$| 𝑣 |$ is not the geodesic length of $ℍ 𝕪$ , but the length of $𝑣 \in ℝ^{1, 1}$

hyperbolic-isom_(tag)

group isomorphism (compare with the case of complex numbers)

$ℝ$
$ℍ 𝕪$
$U (1, ℂ_{split})$
$SO (1, 1)$

$exph ((ϕ + 𝜓) i_{split}) = exph (ϕ i_{split}) exph (𝜓 i_{split})$

$ϕ ⇝ sinh ϕ = \frac{1}{2} (𝑒^{ϕ} - 𝑒^{- ϕ})$ monotonically increasing

Solving the quadratic equation $𝑥 = \frac{1}{2} (𝑒^{ϕ} - \frac{1}{𝑒^{ϕ}}) ⟺ {(𝑒^{ϕ})}^{2} - 2 𝑥 𝑒^{ϕ} - 1$ yields the inverse mapping

$ϕ = {sinh}^{- 1} (𝑥) = log (𝑥^{2} + {(𝑥^{2} + 1)}^{\frac{1}{2}})$

inverse $argh : ℍ 𝕪 \to ℝ$

$argh (𝑡 + 𝑥 i_{split}) = log (𝑥^{2} + {(𝑥^{2} + 1)}^{\frac{1}{2}})$

Question Similar to how $ℂ$ uses stereographic projection and $\tan^{- 1}$ , $ℂ_{split}$ uses #link(<stereographic-projective-hyperbolic>)[hyperbolic projection] and ${tanh}^{- 1}$ to handle hyperbolic angles or geodesic length mapping $argh$

The geodesic coordinates of $ℍ 𝕪$ are $exph (i_{split} ϕ), ϕ \in ℝ$

Notation conflict. Geodesic coordinates are also usually denoted as $exp$ , but not defined using $i, i_{split}$ algebra

Geodesic coordinates are Riemman isomorphic or Euclidean isomorphic

$𝐴 \subset ℍ 𝕪$ compact <==> $\frac{1}{i_{split}} logh 𝐴 \subset ℝ$ compact

Hyperbolic polar coordinates $ℝ^{1, 1} (time, future) ≃ ℝ (\geq 0) \times ℍ 𝕪$

\begin{matrix} ℝ (\geq 0) \times ℝ & ⟶ & ℂ_{split} & ⟶ & ℝ^{1, 1} \\ (𝑟, ϕ) & ⟿ & 𝑟 exph (ϕ i_{split}) & ⟿ & 𝑟 (cosh ϕ, sinh ϕ) \end{matrix}

net structure of $0 \in ℝ^{1, 1}$

Distance $𝑟 = | 𝑧 |$ , direction space $ℍ 𝕪$ or its projection $ℍ 𝕪 ℙ$ , geodesic length $ϕ$ are all $SO (1, 1)$ invariant. $SO (1, 1)$ is the isometry group of $ℍ 𝕪$

Define (time,future) #link(<net>)[net] far away from the light cone ${⟨ 𝑣 ⟩}^{2} = 0$

$[0, 𝑟] \times [ϕ - 𝑅, ϕ + 𝑅]$ . $𝑅$ as the geodesic sphere radius

or product net struct of distance space $ℝ_{\geq 0}$ and direction space $ℍ 𝕪$

Limit method

$𝑟 \to 0$ distance continuous
$𝑅 \to 0$ direction continuous

in $ℝ^{1, 1}$ , time-like and space-like are basically symmetric, so space like net is similar

$𝑓 : ℝ^{1, 1} \to ℝ^{1, 1}$ ((time,future),(time,future)) continuous at $𝑓 (0) = 0$ :=

in hyperbolic polar coordinates

\begin{matrix} \forall 𝜀, Ε > 0 \\ \exists 𝛿, Δ > 0 \\ \forall 𝑟 < 𝛿, 𝑅 < Δ \\ (| 𝑓 | < 𝜀) \land (| argh (𝑓) | < Ε) \end{matrix}

Generalized to higher dimensions

The tangent space of the quadratic manifold $ℚ^{𝑝, 𝑞} (\pm 1)$ can be defined as the (affine) subspace orthogonal to the radial direction

The definition of geodesics of the quadric surface $ℚ^{𝑝, 𝑞} (\pm 1)$ does not require manifold techniques, only use geodesic as secant line of the cross-section span by (radial + tangent) + embedded $ℍ 𝕪$ and its geodesic length. Question Is there a better and more intuitive definition?

${ℍ 𝕪}^{𝑛} (time) = ℚ^{1, 𝑛} (1)$ type

${ℍ 𝕪}^{𝑛} (space) = ℚ^{1, 𝑛} (- 1)$ type

geodesic-of-quadratic-manifold_(tag) ${ℍ 𝕪}^{𝑛} (time)$ geodesic

let $𝑣 \in {ℍ 𝕪}^{𝑛} (time) = ℚ^{1, 𝑛} (1) = {𝑥_{0}^{2} - (𝑥_{1}^{2} + \dots + 𝑥_{𝑛}^{2}) = 1}$

Orthogonal complement space $𝑣^{⟂} ≃ ℝ^{𝑛}$ , $𝑛$ dimensional spacelike

Affine space $𝑣 + 𝑣^{⟂}$ as tangent space of ${ℍ 𝕪}^{𝑛} (time)$

let $𝑤 \in 𝑣^{⟂}$ , $| 𝑤 | = 1$

$span (𝑣, 𝑤)$ is a two-dimensional subspace, signature $1, 1$

$span (𝑣, 𝑤) ≃ ℝ^{1, 1}$ , intersecting with ${ℍ 𝕪}^{𝑛} (time)$ to get an embedded $ℍ 𝕪$

Obtain the geodesic of base point $𝑣$ in the direction $𝑤$

ϕ ⇝ 𝑣 cosh (ϕ) + 𝑤 sinh (ϕ)

${ℍ 𝕪}^{𝑛} (time)$ geodesic sphere

𝔹 (𝑣, 𝑅) = {𝑣 cosh (ϕ) + 𝑤 sinh (ϕ) \in {ℍ 𝕪}^{𝑛} (time) : 𝑤 \in 𝑣^{⟂}, | 𝑤 | = 1, ϕ \leq 𝑅}

where $𝑣^{⟂} ≃ ℝ^{𝑛}, {| 𝑤 | = 1} ≃ 𝕊^{𝑛 - 1}$

(time,future)-like net struct of $ℝ^{1, 𝑛}$

Hyperbolic polar coordinates as the product net struct of distance space $ℝ_{\geq 0}$ and direction space ${ℍ 𝕪}^{𝑛} (time)$

$[0, 𝑟] \times 𝔹 (𝑣, 𝑅)$

Limit method: $𝑟 \to 0$ , $𝑅 \to 0$ . or distance continuous + direction continuous

$𝑓 : ℝ^{1, 𝑛} \to ℝ^{1, 𝑛}$ (time,future),(time,future) continuous at $𝑓 (0) = 0$ :=

in hyperbolic polar coordinates (time,future)

\begin{matrix} \forall 𝜀, Ε > 0 \\ \exists 𝛿, Δ > 0 \\ \forall 𝑟 < 𝛿, 𝑅 < Δ \\ (| 𝑓 | < 𝜀) \land (| argh (𝑓) | < Ε) \end{matrix}

let $𝑣 \in {ℍ 𝕪}^{𝑛} (space) = ℚ^{1, 𝑛} (- 1) = {𝑥_{0}^{2} - (𝑥_{1}^{2} + \dots + 𝑥_{𝑛}^{2}) = - 1}$

Orthogonal complement space $𝑣^{⟂} ≃ ℝ^{1, 𝑛 - 1}$

Affine space $𝑣 + 𝑣^{⟂}$ as tangent space of ${ℍ 𝕪}^{𝑛} (space)$

let $𝑤 \in 𝑣^{⟂}$ , $| 𝑤 | = 1$

$𝑤$ timelike

$span (𝑣, 𝑤)$ signature $1, 1$

$span (𝑣, 𝑤) ≃ ℝ^{1, 1}$ intersects with ${ℍ 𝕪}^{𝑛} (space)$ to obtain an embedded $ℍ 𝕪$

Obtain the geodesic line with base point $𝑣$ and direction $𝑤$

ϕ ⇝ 𝑤 cosh (ϕ) + 𝑣 sinh (ϕ)

$𝑤$ spacelike

$span (𝑣, 𝑤)$ signature $0, 2$

$span (𝑣, 𝑤) ≃ ℝ^{2}$ intersects with ${ℍ 𝕪}^{𝑛} (space)$ to obtain an embedded $𝕊$

Obtain the geodesic line with base point $𝑣$ and direction $𝑤$

ϕ ⇝ 𝑤 cos (ϕ) + 𝑣 sin (ϕ)

${ℍ 𝕪}^{𝑛} (space)$ is not a Euclidean type metric manifold, so the concept of geodesic ball needs to be modified

spacelike direction space ${ℍ 𝕪}^{𝑛} (space)$ 's geodesic coordinates $𝑣^{⟂} ≃ ℝ^{1, 𝑛 - 1}$ , based on dimension induction, using $ℝ^{1, 𝑛 - 1}$ 's net struct, obtain ${ℍ 𝕪}^{𝑛} (space)$ 's local net struct

Since the net is a product type decomposition, it will probably decompose into multiple one-dimensional radii as we induct, which is called multi-radius-geodesic-ball_(tag) . Will the order of decomposition affect it?

Then try to define the space-like net struct of $ℝ^{1, 𝑛}$ using hyperbolic polar coordinates i.e. the product net struct of distance and direction

Then we can define $𝑓 : ℝ^{1, 𝑛} \to ℝ^{1, 𝑛}$ (space,space)-like continuous at $𝑓 (0) = 0$ , or simply spacelike continuous

The case of $(𝑝, 𝑞)$ signature should be similar

The timelike net and spacelike net of $ℝ^{1, 𝑛}$ are not equivalent

$𝑓 : ℝ^{1, 𝑛} \to ℝ^{1, 𝑛}$ Minkowski continuity is defined as timelike continuous and spacelike continuous

Minkowski homeomorphic is defined as $𝑓, 𝑓^{- 1}$ are both Minkowski continuous

all $𝑓 \in SO (1, 𝑛)$ are continuous and homeomorphic

$Lin (ℝ^{1, 𝑛} \to ℝ^{1, 𝑛})$ General linear functions may not be Minkowski continuous

Geodesic coordinates or hyperbolic polar coordinates are locally Minkowski homeomorphic or locally Euclidean homeomorphic by definition

${ℍ 𝕪}^{𝑛} (time)$ is a Riemman manifold, const negative curvature

${ℍ 𝕪}^{𝑛} (space)$ is a Lorentz manifold, const positive curvature

${ℍ 𝕪}^{𝑛} (space)$ alias de Sitter space

hyperbolic-cosine-formula_(tag) Hyperbolic cosine formula

let $𝑣, 𝑤 \in ℍ 𝕪$

let $𝑣 = exph (ϕ i_{split}), 𝑤 = exph (𝜓 i_{split})$

\begin{matrix} ⟨ 𝑣, 𝑤 ⟩ & = & Re (𝑣 \cdot 𝑤^{*}) \\ = & Re (exph ((ϕ - 𝜓) i_{split})) \\ = & cosh (ϕ - 𝜓) \end{matrix}

let $𝑣, 𝑤$ future time-like. $| 𝑣 | ≔ {({⟨ 𝑣 ⟩}^{2})}^{\frac{1}{2}}$

$\frac{𝑣}{| 𝑣 |}, \frac{𝑤}{| 𝑤 |} \in ℍ 𝕪$

$\frac{⟨ 𝑣, 𝑤 ⟩}{| 𝑣 | | 𝑤 |} = cosh (ϕ - 𝜓)$

Cosine formula

\begin{matrix} {⟨ 𝑣 + 𝑤 ⟩}^{2} & = & {⟨ 𝑣 ⟩}^{2} + 2 ⟨ 𝑣, 𝑤 ⟩ + {⟨ 𝑤 ⟩}^{2} \\ = & {⟨ 𝑣 ⟩}^{2} + {⟨ 𝑤 ⟩}^{2} + 2 | 𝑣 | | 𝑤 | \frac{⟨ 𝑣, 𝑤 ⟩}{| 𝑣 | | 𝑤 |} \\ = & {⟨ 𝑣 ⟩}^{2} + {⟨ 𝑤 ⟩}^{2} + 2 | 𝑣 | | 𝑤 | cosh (ϕ - 𝜓) \end{matrix}

isom-top-hyperbolic-Euclidean_(tag)

$ℍ 𝕪$ in $ℂ_{split} = ℝ^{1, 1}$ limit structure under distance $≃$ geodesic distance $≃$ Euclidean $ℝ^{1}$

Proof

let $𝑣, 𝑤 \in ℍ 𝕪$ , $𝑣 = exph (ϕ i_{split}), 𝑤 = exph (𝜓 i_{split})$

\begin{matrix} {⟨ 𝑣 - 𝑤 ⟩}^{2} & = & {⟨ 𝑣 ⟩}^{2} + {⟨ 𝑤 ⟩}^{2} - 2 ⟨ 𝑣, 𝑤 ⟩ \\ = & 2 (1 - cosh (ϕ - 𝜓)) \\ \leq & 0 by cosh \geq 1 \end{matrix}

let $dist (𝑣, 𝑤) ≔ {(- {⟨ 𝑣 - 𝑤 ⟩}^{2})}^{\frac{1}{2}}$

use $cosh ϕ = 1 ⟺ ϕ = 0$

\begin{matrix} dist (𝑣, 𝑤) = 0 & ⟺ & {⟨ 𝑣 - 𝑤 ⟩}^{2} = 0 \\ ⟺ & ϕ = 𝜓 \\ ⟺ & 𝑣 = 𝑤 \end{matrix}

use $cosh ϕ = \frac{1}{2} (𝑒^{ϕ} + 𝑒^{- ϕ})$ continuity

\begin{matrix} \forall 𝜀 > 0, \exists 𝛿 > 0, \forall ϕ, 𝜓 \in ℝ \\ | ϕ - 𝜓 | < 𝛿 ⟹ dist (𝑣, 𝑤) < 𝜀 \end{matrix}

Generalize to ${ℍ 𝕪}^{𝑛} \subset ℝ^{1, 𝑛}$ , Euclidean $ℝ^{𝑛}$

Proof

use geodesic coordinates

similar to $ℝ^{1, 1}$ , try to prove

\begin{matrix} {⟨ 𝑣 - 𝑤 ⟩}^{2} & = & {⟨ 𝑣 ⟩}^{2} + {⟨ 𝑤 ⟩}^{2} - 2 ⟨ 𝑣, 𝑤 ⟩ \\ = & 2 (1 - cosh (| ϕ - 𝜓 |)) \\ \leq & 0 \end{matrix}

where

$ϕ, 𝜓$ are geodesic coordinates of $𝑣, 𝑤$
$| ϕ - 𝜓 |$ is the Euclid distance in geodesic coordinates

The base point of the stereographic projection of the sphere $𝕊^{𝑛}$ is on $𝕊^{𝑛}$ . More than two coordinate charts are needed to cover all of $𝕊^{𝑛}$

stereographic-projective-hyperbolic_(tag) time-like hyperboloid ${ℍ 𝕪}^{𝑛} (time)$ considers stereographic projection, with two base points on the two branches of the hyperboloid respectively, and the projection forms separate singular points in the direction of the light cone

space-like hyperboloid, use space-like base points to define hyperbolic projection, and the projection coordinate chart is a lower-dimensional Minkowski space

Should the transformation function be a Minkowski continuous homeomorphism?

Perform 3d plotting for the case of $ℝ^{1, 2}$ , and draw the light cone of the base point (note that the light cone is "vertical")

Even if the intuition of drawing may be difficult, the analytical calculation should not be difficult

$exph$ can be generalized to $ℍ^{'} ≃ ℝ^{2, 2}$ and $𝕆^{'} ≃ ℝ^{4, 4}$ ?