Special relativity: flashes of light

[whatisreality]

Homework Statement

In a frame of reference A lights are on the x-axis at x = D and x = -D, where D = 0.6 x10⁹. They flash simultaneously at t = 0.

There's also a frame of reference A' moving at v = 0.8c.

i) Where and when do the flashes happen in A'?

ii) Therefore when would observers at the origins of A and A' see the light?

Homework Equations

The Attempt at a Solution

i) Well, I have to add D to the location in x to find the location in x'. Then I have to add something else. But I'm pretty confused about what to add. As far as an observer in A' is concerned, I think the light has to travel x' = D + vt'? But I can't use that because I don't know t' or x', so that's probably wrong...
In A, both flashes happen at t = 0.6 x10⁹ / 3x10⁸ = 2 seconds. So I have t' as well:
t' = γt where γ is the Lorentz factor, so maybe I do have t' actually, I think it's 10/3.

Essentially, my question for the first part is whether x' = D + vt' is right.

And then the problem is that haven't I already worked out when the observer at A sees the light? It's at t =2. And for the observer at the origin of A', at t = 10/3?

 

[Chestermiller]

What does the Lorentz Transformation tell you about where and when the flashes occur in A'?

Chet

 

[whatisreality]

What does the Lorentz Transformation tell you about where and when the flashes occur in A'?

Chet

Using x' = γ(x - vt) , t=0, x' = -1 x 10⁹m for the bulb at -D, and 1 x 10⁹ for the bulb at D.

And as to when they happen - that should be at two different times. Using t' = γ( t - $\frac{vx}{c^2}$), and subbing in t = 0,
t' = -2.67 s for the bulb at +D, and 2.67 s for the bulb at -D. I think.

So if I'm in frame F', it looks like F is moving, doesn't it? And that means anything measured in F will appear shorter to me, and seem like it happened either later/ earlier, depending on the direction of movement?

 

[whatisreality]

And about the second part: I'm pretty sure the time at the origin of A is easy to calculate. It's just t = D/c and is the same for both bulbs.

As for at the origin of A', for the bulb at D: That person thinks the flash happened at t = 2.67, and that the light traveled for t = $\frac{1 \times 10^9}{c}$- 2.67 s, but I'm really not 100% sure about subtracting 2.67 seconds! Should I be adding instead? Or just doing nothing? Does that take account of the fact that F' is moving towards the bulb?

And for the bulb at -D: it travels for $\frac{1 \times 10^9}{c}$ + $\frac{x'}{v}$ + 2.67? Maybe? Struggling to get my head round this.

 

[Chestermiller]

Using x' = γ(x - vt) , t=0, x' = -1 x 10⁹m for the bulb at -D, and 1 x 10⁹ for the bulb at D.

And as to when they happen - that should be at two different times. Using t' = γ( t - $\frac{vx}{c^2}$), and subbing in t = 0,
t' = -2.67 s for the bulb at +D, and 2.67 s for the bulb at -D. I think.

So if I'm in frame F', it looks like F is moving, doesn't it? And that means anything measured in F will appear shorter to me, and seem like it happened either later/ earlier, depending on the direction of movement?

Forget about the F frame of reference for now. In the F' frame of reference, the two flashes occur at -1 x 10⁹m, + 2.67 s and 10⁹, -2.67 s. Since they are traveling at the speed of light in F', at what value of t' do each of these flashes reach x' = 0?

 

[whatisreality]

Forget about the F frame of reference for now. In the F' frame of reference, the two flashes occur at -1 x 10⁹m, + 2.67 s and 10⁹, -2.67 s. Since they are traveling at the speed of light in F', at what value of t' do each of these flashes reach x' = 0?

t' = $\frac{1 \times 10^9}{c}- 2.67$ for x' = D?

$\frac{1 \times 10^9}{c} + 2.67 + \frac{x'}{v}$ for x = -D. Possibly? Not sure I've got the adding/ subtracting of 2.67 the right way round. Or that I'm accounting for the movement of the frame correctly!

 

[Chestermiller]

t' = $\frac{1 \times 10^9}{c}- 2.67$ for x' = D?

$\frac{1 \times 10^9}{c} + 2.67 + \frac{x'}{v}$ for x = -D. Possibly? Not sure I've got the adding/ subtracting of 2.67 the right way round. Or that I'm accounting for the movement of the frame correctly!

Your answer for -2.67 is correct. Since frame F' doesn't know that it is moving, your answer for +2.67 is not correct. Lost the x'/v.

 

[whatisreality]

Your answer for -2.67 is correct. Since frame F' doesn't know that it is moving, your answer for +2.67 is not correct. Lost the x'/v.

F' thinks the light bulb at -D is moving away from it though, doesn't it? At the v = 0.8c assigned to F' relative to F? Although by that logic I should also have subtracted x'/v for x' = D as well. That's seriously confusing.

 

[whatisreality]

Or is the -2.67 term accounting for the movement?

 

[Chestermiller]

F' thinks the light bulb at -D is moving away from it though, doesn't it? At the v = 0.8c assigned to F' relative to F? Although by that logic I should also have subtracted x'/v for x' = D as well. That's seriously confusing.

Once the flash occurs at the indicated location and time, it doesn't matter what the light bulb does next. It is the light flash that is traveling toward x' =0 at the speed of light; the light bulb has nothing to do with this.

 

[whatisreality]

Once the flash occurs at the indicated location and time, it doesn't matter what the light bulb does next. It is the light flash that is traveling toward x' =0 at the speed of light; the light bulb has nothing to do with this.

Oh, I get it!

Thanks for wading through my messy first post and replying. I really appreciate it! :)