I'm trying to understand the causal structure of Minkowskian spacetime and I was wondering whether something can be said about the relation between the classification of events and curves.(adsbygoogle = window.adsbygoogle || []).push({});

To clarify: for Minkowskian inner product [itex]\eta[/itex] with signature (-+++), two events [itex]p[/itex] and [itex]q[/itex] can be timelike ([itex]\eta(\vec{pq},\vec{pq})<0[/itex]), spacelike ([itex]\eta(\vec{pq},\vec{pq})>0[/itex]) or lightlike related ([itex]\eta(\vec{pq},\vec{pq})=0[/itex]). A curve [itex]w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}[/itex] is timelike/spacelike/lightlike when its tangent vectors [itex]w'(t)[/itex] are. The question is now: can any type of events be connected by any type of curve?

A secondary question: can the velocity (tangent vector) of a curve in Minkowskian spacetime be defined as below?

[tex]

w \colon I\subset\mathbb{R}\to C\subset M_{p}^{4}\colon t\mapsto w_{v}(t)+o

[/tex]

[tex]

w_{v} \colon I\subset\mathbb{R}\to C\subset M_{v}^{4}\colon t\mapsto w_{v}(t)

[/tex]

[tex]

w'(t) =\lim_{h\to 0}\frac{(w_{v}(t+h)+o)-(w_{v}(t)+o)}{h}=w_{v}'(t)

[/tex]

where [itex]M_{p}^{4}[/itex] point space, [itex]o\in M_{p}^{4}[/itex] and [itex]M_{v}^{4}[/itex] vector space with Minkowskian inner product [itex]\eta[/itex] (i.e. inner product but weaken positive-definite to non-degenerate).

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# Classification of events and curves in Minkowskian spacetime

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