SpaceTime Graph

DEFINITION

Orthogonal Cartesian Reference Frame $E^{3}$ representation, obtained by applying Homeomorphism Map. WHERE World Line is drawn.
IF $3 D$ , called "Light Cone"

DIVISION

DIVSION - SECTOR

Subdivision based on Length $s$ FROM origin $O$

Name	Property	Causality w/ $O$	Simultaneity w/ $O$	Same place w/ $O$	Relationship w/ $O$	Note
Time-Like Interval	$$s^2{\color{orange}>}0$$	❌	✔	❌	"Intrinsic Space Separation"
Space-Like Interval	$$s^2{\color{orange}<}0$$	✔	❌	✔	"Intrinsic Time Separation"
Light-Like Interval	$$s^2{\color{orange}=}0$$	✔	❌	❌		=Bisectors

Light-LikeSpace-LikeTime-Like

DIVISION - TIME

Base on World Line's angle limitation to $θ < 45 °$

Name	Description
Absolute Future	Possible next coordinates of World Line passing in origin $O$
Present	Points that can be simultaneous w/ origin $O$
Absolute Past	Possible previous coordinates of World Line passing in origin $O$

Absolute PastPresentAbsolute FutureWorld Line

PROPERTIES

Let's focus on 2D

ORTHOGONALITY

2 @Four-Vector are $⊥$ WHEN there's $45 °$ symmetry → $\tan (\frac{u^{1}}{u^{0}}) = \tan (\frac{v^{0}}{v^{1}})$

Orthogonality in

M^{2}

$u^{μ} \cdot v_{μ} = u^{0} \cdot v^{0} - u^{1} \cdot v^{1} = 0$

Orthogonality in

E^{2}

$\vec{u} \cdot \vec{v} = u_{x} \cdot v_{x} + u_{y} \cdot v_{y} = 0$

Pasted image 20230528213234.png|150

MAGNITUDE

Magnitude in

M^{2}

$v = \sqrt{v^{μ} \cdot v_{μ}} = \sqrt{v^{0} \cdot v^{0} - v^{1} \cdot v^{1}}$

Magnitude in

E^{2}

$v = \sqrt{v_{x}^{2} - v_{y}^{2}}$

LENGTH

Imagine distance of $| s | = 1$
WHILE in $E^{2}$ is a Sphere.
In $M^{2}$ , CAUSE Interval $s \in C$ → $| s | = 1$ → ${\begin{cases} s^{2} = 1 & \to s = \pm 1 & \to (x^{0})^{2} - (x^{1})^{2} = 1 \\ s^{2} = - 1 & \to s = \pm i & \to (x^{0})^{2} - (x^{1})^{2} = - 1 \end{cases}$ → 2 Hyperbole Pasted image 20230528213615.png|150

QUESTIONS

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