Question

Are the pairs of the following events spacelike, timelike or lightlike separated?: (a) Your friend at the other end of the room snapping his fingers and your seeing it, (b) your friend at the other end of the room snapping his fingers and your hearing it, (c) two friends at two ends of the room snapping their fingers simultaneously according to an observer standing halfway between them. You must explain your reasoning!

Answer 1

Let's first define what is spacelike, timelike, and lightlike event. If we have two events and , then spacetime interval between them is:

We know that

Therefore, from equation (1) and (2), we have:

In equation (3), is comparing the distance travelled by light between the events to their spatial separation.

Based on this:

1. If , this implies i.e. this spatial separation is less than the distance light travelled, and the interval is timelike.
2. If , this implies i.e. this spatial separation is equal to the distance light travelled, and the interval is lightlike.
3. If , this implies i.e. this spatial separation is greater than the distance light travelled, and the interval is spacelike.

Now, for our calculation let's consider the spatial coordinates of you and your friends be and your friend be , and let the observer in third event, be situated at point  .

Now, let's calculate spatial interval for:

Event 1 (Solution of a):

Spactime coordinate of event is

,

Where is the distance between the two friends.

From above,

And

Putting value of and into equation (3):

Hence the event is lightlike.

Event 2 (Solution of b):

Spactime coordinate of event is

,

Now, for calculating , we have speed of sound in air, and distance .

Again putting the value of and into equation (3):

Thus, , hence the event is timelike.

Event 3 (Solution of c):

For this situation, observer is situated halfway between the two friends at .

We are taking two times here, since the observer is observing the both the friends simultaneously.

Now, difference between you to observer

Now, difference between your friend to observer

represents time taken by light to reach from you to observer.

represents time taken by light to reach from your friend's to observer.

Spatial interval between you and observer by putting the values of and into equation (3):

Hence, the event is lightlike.

Spatial interval between your friend and observer by putting the values of and into equation (3):

​​​​​​​

Hence, the event is lightlike.