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In Special Relativity, two events are said to be causally related or in causal contact if a signal could travel from one event to the other. Note that this does not require that an actual signal is transmitted. Indeed, this notion is valid in vacuum. Assume that in a given frame the spatial distance $\Delta r$ and time difference $\Delta t$ are such that $$\... 0 I think something that can make it more intuitive is to think about spacetime as a euclidean space with an imaginary coordinate. So if we think about a 4d space with coordinates (x,y,z,w) then we can substitute in w=ict. So if$$(x_1-x_2)^2+(y_1-y_2)^2+(z_1-z_2)^2+(w_1-w_2)^2=(x'_1-x'_2)^2+(y'_1-y'_2)^2+(z'_1-z'_2)^2+(w'_1-w'_2)^2$$then$$(x_1-x_2)^2+(...

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Although it is the accepted answer, I find it difficult to agree with Dale's perspective. I agree that if $s^2=0$ then it is intuitive why it must be invariant to translations, rotations and Lorenz transformations, and why any other $s^2$ is invariant under purely spatial transformations. However, I consider the last part, of "well ok then $s^2$ is ...

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You have two great answers, but you might find it interesting to know that it was once common for spacetime in SR to be described with an imaginary time axis. That allowed people to consider that it was a straightforward Cartesian arrangement, where the calculation of a length was through the usual Pythagorean method of taking the square root of the squares ...

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The intuition is not too difficult to build. For convenience, I will write $\Delta x$ instead of $x_1-x_2$. First, we know from the first postulate that the speed of light is invariant. If we write $$\Delta x^2 + \Delta y^2 + \Delta z^2 = c^2 \Delta t^2$$ this is the equation of a sphere of radius $c \ \Delta t^2$. In other words, this is a flash of light ...

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The general concept is that anything that starts at a time and place and ends at another time and place, will have the same $s$ no matter who watches it, no matter how they are moving. Maybe think of a rocket igniting and flying and blowing up. We might disagree about where and when started and about where and when exploded. But not about $s$ between the ...

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