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:

ΔS^{2} = cΔt^{2} – Δx^{2} – Δy^{2} – Δz^{2}

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^{-8}s), B (x = 1m, t = 10^{-8}s) and C (x = 6m, t = 2×10^{-8}s). Determine which, if any, of these events are causally connected.

#### A and B

Δx = 3 m and Δt = 0 s

So

ΔS^{2} = c^{2} × (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

ΔS^{2} = c^{2} × (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

ΔS^{2} = c^{2} × (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.