Relativity of Simultaneity Question

[jowens1988]

Homework Statement

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?

Homework Equations

s² = (ct)² - (x)² - (y)²

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!

 

[vela]

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.

 

[jowens1988]

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.

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?

 

[vela]

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.

 

[paranormal]

Your thinking is correct, and your approach is a valid way to solve this problem. Another way to approach this problem is to use the Lorentz transformation equations to find the time coordinates in a frame in which the events are simultaneous. This would involve setting the time coordinates (ct)1 and (ct)2 equal to each other and solving for the velocity v.

As for your last question, two events that are not causally connected can never be simultaneous in any frame. This is a consequence of the principle of relativity, which states that the laws of physics should be the same in all inertial frames of reference. If two events are not causally connected, there is no way for one to influence the other, and thus they cannot occur at the same time in any frame.