Locating Event E0 on a Space-Time Diagram & Speeds

[Ylle]

Hello...

I have a little problem I hope you guys can help me with.
I'll write down the problem-text first, so it will be easier to point out the problem - hopefully.

Here it goes:
Event E₀ has the following coordinates in some frame S:

x = 2 m
ct = 4 m

Locate E₀ on a space-time diagram for frame S. Draw the lines that define the three invariant regions of space-time for E₀. On the diagram, locate each of the follow events:

E₁: x = 0, ct = 1 m
E₂: x= 4 m, ct = 2 m
E₃: x = 4 m, ct = 5 m

Specify whether each event is on the light cone of E₀, in the past or the future of E₀, or in the elsewhere. If E is within one of the light cones, find the speed (relative to S) of a frame in which E occurs at the same place as E₀. If E is in the elsewhere of E₀, find the speed if a frame in which it is simultaneous with E₀.


First I draw my diagram, as you can see on the image, define the invariant regions (Past, future and 2x elsewhere), and locate the 3 events.

The next question is to specify where the events are happening.
I get:
E₁ = The past
E₂ = On the light cone
E₃ = In the elsewhere

The next question is to find the speed (relative to S) of a frame in which E occurs at the same place as E₀.

Then I draw the red line, and get the speed: (2/3)c


And that's what I got so far. The last part "If E is in the elsewhere of E₀, find the speed if a frame in which it is simultaneous with E₀." I cannot understand.
So maybe you can help me out ?


Thank you
- Ylle

 

[jbsteele]

For the last part, you need to find the speed in a frame where E3 is simultaneous with E0. To do this, draw a line parallel to the x-axis that passes through E3. This line is the worldline of a frame in which E3 is simultaneous with E0. Then, calculate the slope of the worldline, which will give you the speed relative to S.

 

[baba89]

Hello Ylle,

Thank you for explaining your problem and providing a diagram. It looks like you have correctly located the events and determined their positions relative to E0. To answer the last part of the problem, we need to understand the concept of simultaneity in special relativity.

In special relativity, simultaneity is relative. This means that two events that are simultaneous in one frame may not be simultaneous in another frame. In your diagram, event E0 and E3 are simultaneous in frame S, but in a different frame, they may not be simultaneous.

To find the speed of a frame in which E3 is simultaneous with E0, we can use the equation for time dilation: t' = t/γ, where t' is the time in the new frame, t is the time in the original frame, and γ is the Lorentz factor, given by γ = 1/√(1-v^2/c^2), where v is the relative velocity between the two frames and c is the speed of light.

In this case, we want E3 to be simultaneous with E0, so t' = t. We can rearrange the equation to solve for v, which gives us v = √(1-1/γ^2)c. Plugging in the values for E0 and E3, we get v = √(1-5/9)c or approximately -0.62c. This means that in a frame where E3 is simultaneous with E0, the relative velocity between the two frames is -0.62 times the speed of light.

I hope this helps clarify the concept of simultaneity and how to find the speed in different frames. Let me know if you have any further questions. Keep up the good work in your studies!

Best,