# Relativity of Simultaneity Question

1. Sep 22, 2010

### jowens1988

1. The problem statement, all variables and given/known data
The space and time coordinates for pairs of events are (ct,x,y)1 = (0.3,0.5,0.6) meters and (ct,x,y)2 = (0.4,0.7,0.9) meters. Could there be a causal connection between these two events? Is there a frame in which the two events would be recorded as simultaneous? If so, what is this frame?

2. Relevant equations
s2 = (ct)2 - (x)2 - (y)2

3. The attempt at a solution
I understand that if s2 is negative, which in this case I get that it is, then the events cannot be causally related. But I tried to use a different method of solution, and I was just curious if it is correct:

I plot the x and y coordinates given in the problem statement in the x-y plane, and take the ct coordinates to be a third dimension, the height above the x-y plane. I find the distance between the points given in the x-y plane, which I get in this case to be:

d = ((0.7-0.5)2 - (0.9-0.6)2)1/2 = 0.360555m

Then, I want to find the slope of the line connecting these points (including the ct coordinate, so it is a line in a plane that contains the line that connects the points in the x-y plane, with height value ct), which I get to be

slope = (ct2 - ct1)/(0.360555) = 0.27735 < 1 => you would have to travel faster than the speed of light for these two events to be causally related.

Now, I want to see if there exists a frame S' with some velocity v in which these two events can be considered simultaneous:

Or, mathematically: delta t' = 0

I make a new coordinate system, in which I keep the ct coordinate, but I combine the x and y coordinates, where (x,y)1 = z1 = 0 and (x,y)2 = z1 = 0.360555m (the distance between the two x-y coordinates.)

Given that t' = gamma/c(ct-beta*z1) from the Lorentz transformation, i can derive that the necessary v =

-c2(t2 - t1)/(z1-z1)

Which I get to be greater than the speed of light, so they cannot be simultaneous.

Is this thinking correct? Is there an easier way to do this?

If two events aren't causally connected, can they ever be simultaneous?

Thank you!

Last edited: Sep 22, 2010
2. Sep 22, 2010

### vela

Staff Emeritus
cΔt=c(t2-t1) is less than z2, right? So you can find a β<1 such that cΔt = β z2. I think you just made an algebra error solving for the speed.

3. Sep 22, 2010

### jowens1988

Okay, that's right. I re-worked the algebra, and I can find such a β.

So, the moral of the story is that even if two events aren't causally related, they CAN still be simultaneous?

4. Sep 22, 2010

### vela

Staff Emeritus
Right. They're mutually exclusive, actually. If two events are causally related, they're timelike or lightlike separated, so they can't be simultaneous. Similarly, if two events are simultaneous, they are spacelike separated, so they can't be causally related.