Determining if events are causally connected.

Space-time is the four dimensional continuum of our universe. The three dimensions of space and the dimension of time form this continuum. An interval in space-time is found using 4D Pythagoras and is defined as:

ΔS2 = cΔt2 – Δx2 – Δy2 – Δz2

If the space-time interval ΔS between two events is:

  • Less than zero then the separation between the events is called Space Like. These events cannot be causally connected because even light could not travel fast enough to get from event 1 to event 2 so event 1 can’t have influenced event 2.
  • More than zero then the separation between the events is called Time Like. These events can be causally connected because it is possible to travel from event 1 to event 2 travelling at less than the speed of light and therefore someone or something at event 1 could influence event 2.
  • Equal to zero then the separation between the events is called Light Like. These events could be causally connected because something travelling at the speed of light could travel from event 1 to event 2.

Three events, A, B and C are measured in the same inertial frame. Their 2-D space-time (y=z=0) coordinates are: A (x = 4m, t = 10-8s), B (x = 1m, t = 10-8s) and C (x = 6m, t = 2×10-8s). Determine which, if any, of these events are causally connected.

A and B

Δx = 3 m and Δt = 0 s

So

ΔS2 = c2 × (0)2 – (3)2 – (0)2 – (0)2 = -9 < 0

This means the separation is space-like and so they are not causally connected.

A and C

Δx = 2 m and Δt = 10-8 s

So

ΔS2 = c2 × (10-8)2 – (2)2 – (0)2 – (0)2 = 9 – 4 = 5 > 0

This means the separation is time-like and so they are causally connected.

B and C

Δx = 5 m and Δt = 10-8 s

So

ΔS2 = c2 × (10-8)2 – (5)2 – (0)2 – (0)2 = 9 – 25 = -16 < 0

This means the separation is space-like and so they are not causally connected.

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