One event occurs at the origin at t equal to zero, and a second events occurs at the point x=5m along the x-axis at time with ct=4m. Is the seperation between the two events lightlike, timelike, or spacelike? Could the first event cause the second event? Explain

The first event can't cause the second event, as there exist an frame of reference in which both happens at the same time, in different positions, so, if there were causally connected, it will imply an instant connection, this is faster than light.

Explanation:

We can define the separation between two events (using the + - - - signature) as :

[tex](\Delta s )^2 = (ct_2 - c t_1 )^2 - (x_2 - x_1)^2[/tex]

where the separation will be lightlike if is equal to zero, timelike if is positive and spacelike if is negative.

For our problem

[tex]c t_1 = 0[/tex]

[tex]x_1 = 0[/tex]

[tex]ct_2 = 4 \ m[/tex]

[tex]x_2 = 5 \ m[/tex]

[tex](\Delta s )^2 = (4 \ m - 0 )^2 - ( 5 \ m - 0)^2[/tex]

[tex](\Delta s )^2 = (4 \ m )^2 - ( 5 \ m 0)^2[/tex]

[tex](\Delta s )^2 = 16 \ m^ 2 - 25 \ m^2[/tex]

[tex](\Delta s )^2 = - 9\ m^2[/tex]

So the separation will be spacelike, and the first event can't cause the second event, as there exist an frame of reference in which both happens at the same time, in different positions, so, if there were causally connected, it will imply an instant connection, this is faster than light.