Running away from a beam of light

[dog6880]

Ok tell me if i understand this correctly:

Object A is emitting light
Object B is moving away from it at half the speed of light (constant speed)
Object C is stationary

Object B and C are 5 light years away from Object A (B just passes C as the light is initially emitted)

the light will reach both Object B and C at the same time even though 5 years later when it does arrive at object B object C is 2.5 light years away from Object B. (i assume this is only from the perspective of Object B cause the light is not really there yet)

I hope i got it wrong cause that's crazy. Why did i pick up this damn book?

 

[JesseM]

Ok tell me if i understand this correctly:

Object A is emitting light
Object B is moving away from it at half the speed of light (constant speed)
Object C is stationary

Object B and C are 5 light years away from Object A (B just passes C as the light is initially emitted)

the light will reach both Object B and C at the same time even though 5 years later when it does arrive at object B object C is 2.5 light years away from Object B. (i assume this is only from the perspective of Object B cause the light is not really there yet)

I hope i got it wrong cause that's crazy. Why did i pick up this damn book?

"Stationary" doesn't have any absolute meaning in relativity, every object is stationary in its own rest frame. But perhaps you just meant that object C is stationary with respect to object A. In this case you have to consider the issue of length contraction, which says that if the distance between A and C is 5 light years in their own rest frame, in the rest frame of object B the distance between A and C is shorter by a factor of \sqrt{1 - v^2/c^2}, in this case 0.866, so the distance from A to C in B's frame is only 5*0.866 = 4.33 light years. You also have to consider the issue of the relativity of simultaneity, which says that if the event of B and C being next to each other is simultaneous with the event of A emitting the light according to the B/C rest frame, then these two events are not simultaneous in B's frame, instead the event of A emitting the light doesn't happen until 2.887 years have passed since B passed C according to this frame (I calculated this using the Lorentz transformation, which I can explain if you want), so B is now at a distance of 4.33 + 0.5*2.887 = 5.77 light years from A. So the light will take 5.77 more years to reach B in this frame, meaning it reaches B at a total of 2.887 + 5.77 = 8.66 years after B passed C, in B's frame.

Meanwhile, back in the rest frame of A and C, we calculate that 10 years after B passed C, B is now 10 light years away from A and the light is also 10 light years away from A, so that's how long it takes the light to reach B in this frame. But to predict what B's clock reads when the light catches up with it according to the A/C rest frame, we have to take into account the issue of time dilation, which says that B's clock is running slow by a factor of 0.866 in this frame. So, 10 years after B passed C in this frame, B's clock will only show 8.66 years having passed since B passed C, which agrees with what we found above in B's rest frame.

 

[dog6880]

uh so in object B's point of view, due to time dilation, the light reaches him in the same amount of time as if he never moved at all?

is using letters in math like using numbers in spelling? such as mate=m8. never mind me there is a reason i never followed the sciences in college.

I seem to understand ideas and theories when an author like Stephan hawking writes, but give me equations and my ignorance in mathematics gets the better of me. I will find the passage in the book i am reading that leads me to ask my initial question.

 

[JesseM]

uh so in object B's point of view, due to time dilation, the light reaches him in the same amount of time as if he never moved at all?

Yes, if you look at the numbers I gave, in B's frame A is 5.77 light years away when the signal is set off (which itself happens 2.887 years after B passes C in B's frame), and B receives the signal 5.77 years later in this frame.

I seem to understand ideas and theories when an author like Stephan hawking writes, but give me equations and my ignorance in mathematics gets the better of me. I will find the passage in the book i am reading that leads me to ask my initial question.

I tried to avoid giving equations in my answer (except for the length contraction factor), just the actual numbers for times and distances...did you have problems keeping track of which number represented what? Here's a little chart:

In the rest frame of A and C:
Distance between A and C: 5 light years
Time between event of B passing C and event of A sending signal: 0 years
Distance between A and C at the moment A sends signal: 5 light years
Time between A sending signal and signal reaching C: 5 years
Time between A sending signal and signal reaching B: 10 years
Time elapsed on B's clock between event of B passing C and event of signal reaching B: 8.66 years

In the rest frame of B:
Distance between A and C: 4.33 light years
Time between event of B passing C and event of A sending signal: 2.887 years
Distance between A and B at the moment A sends signal: 5.77 light years
Time between A sending signal and signal reaching B: 5.77 years
Time elapsed on B's clock between event of B passing C and event of signal reaching B: 8.66 years