metric tensor - Causality and how it fits in with relativity

I was talking to my teacher the other day about Einstein's spacetime and there's one thing he couldn't explain about the nature of Cause. I may be being stupid or just unable to comprehend, thanks for any replies.

According to Einstein and relativity, two observers will agree on what things happened but not necessarily on the chronological order in which they happen. I understand how this radically alters our view of time into something that isn't the same or experienced equally by all. What I don't understand is how this fits in (or doesn't) with cause. If event A causes Event B, but we're saying that two people could experience them in a different order, how can event B happen before event A which caused it.

I'm intrigued?

Answer

It's a bit more complicated than that. Given any two events, there is a quantity, called the interval (also 'spacetime interval' or 'invariant interval'), denoted $\Delta s^2$, and which equals $\Delta s^2=c^2\Delta t^2-\Delta \mathbf r^2$, which determines how the two events can relate to each other causally.

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  If $\Delta s^2>0$, then we say $A$ and $B$ are "timelike separated" (or lightlike separated if $\Delta s^2=0$). In this case all observers will agree that (say) $A$ happened before $B$, and $A$ can causally influence $B$.

  
  


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  If $\Delta s^2<0$, then we say $A$ and $B$ are "spacelike separated". In this case $A$ and $B$ are causally disconnected, and neither can influence the other. Different observers will disagree on their temporal order, and in fact you can always find observers for whom $A$ happened before $B$, $A$ happened after $B$, and $A$ happened at the same time as $B$.

  
  



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  Finally, is $\Delta s^2=0$, then we say that $A$ and $B$ are "lightlike separated", or that the interval between them is "null". This is identical to timelike separations: all observers will agree that (say) $A$ happened before $B$, and $A$ can causally influence $B$; moreover, a light ray emitted at $A$ in the direction of $B$ will reach that position at the exact instant that $B$ is happening, and it will do so in all frames of reference.

  
  

  The set of all events $B$ which are at lightlike separations from $A$ is called the light cone of $A$, and it separates space in three regions: the interior, with timelike separations, itself split into the causal future and the causal past of $A$, and the exterior, with spacelike separations, which contains all events that are causally disconnected from $A$, and which are simultaneous with it in some frame of reference.

  
  

Thus, as you succinctly put it,

if $A$ and $B$ are linked (one causes the other), then they have to be timelike [or lightlike] separated and all observers will agree on their temporal order.