Solving the HUGE Issue with Lorentz Transformations and Lightning Strikes

[JMart12]

TL;DR

I have been working on this problem for days, and I can easily do the math, but I really can’t seem to conceptualize it. Please help!

In the Earth’s reference frame, a tree is at x=0km and a pole is at x=20km. A person stands at x=0 (stationary relative to the Earth), and at t=10 microseconds, this person witnesses two simultaneous lightning strikes. One of these strikes hits the tree he is standing under, and the other hits the pole at x=20km.

Now let’s say that a rocket is traveling at 0.5c relative to the Earth (and therefore Bob) in the positive x direction. If we synchronize the clocks of Bob and the Rocket, so x=x’=0 at t=t’=0, then when did the rocket view each of the lightning strikes occurring? (End Problem)

The math is just a little bit of algebra, and when you do this you get that the rocket views the lightning striking the tree at about 11 microseconds rather than the 10 microseconds that Bob measured. This makes perfect sense because the rocket is moving away from the tree.

Here’s what doesn’t make sense, the rocket views the lightning striking the pole at -27 microseconds. I understand that the rocket is moving towards the pole, so the time should decrease, but the time shouldn’t be negative here.

The reason for this is NOT because negative times can’t happen. If we go back to Bob’s frame, and you view the time it would take for the light to Reach Bob’s eye from the pole, it is about 60 microseconds or so. This means that the original strike, relative to the pole, would have occurred at about -50 microseconds.

Here’s the problem, we shouldn’t be getting a negative time in this case, because that would infer that the rocket is closer to the pole than Bob. I say this because we originally synchronized Bob and the rocket by saying that x=x’=0 at t=t’=0. Since this is true, if we have a negative t’, x’ will also have to be negative, meaning that the rocket is further away than Bob is, and this is significant because then the light would travel and hit Bob before it hits the rocket.

This is the huge dilemma that I have been having for days, so will someone please help me. My professor that taught me Special Relativity said that he isn’t able to understand why this is happening, the graduate students in my lab don’t understand why this is happening, and I definitely don’t.

Please let me know if I need to clarify anything further.

 

[JMart12]

If you are having trouble visualizing this problem, I got it from this quick YouTube video, which actually draws it out:

 

[Ibix]

I'm confused. You say the lightning strikes happen at 10$\mu$s in Bob's frame, but then you say one of them happened at about -50$\mu$s. Which is it?

It's worth noting that you may be confused: the times inserted into the Lorentz transforms are the times things actually happen according to that frame, not when you receive light from them. Assuming that you inserted t=10$\mu$s into the Lorentz transforms (if my arithmetic is correct this gives t'=-37$\mu$s, not the -27 you quote) then you are asserting that, in Bob's frame, the lightning strikes were simultaneous. Assuming he survived the strike on his own tree, he will receive the light from the other strike at around t=77$\mu$s. So he does not see the strikes simultaneously, but concludes (after subtracting the time for light to travel from the flagpole) that the strikes were simultaneous.

 

[PeterDonis]

at t=10 microseconds, this person witnesses two simultaneous lightning strikes

Note that this statement, taken literally, is false. The person "witnesses" (as in "sees with his eyes") the strike on the tree he's standing next to at t = 10 microseconds. But he doesn't witness the other strike until the light from it gets to him, and it has to travel 20 km to do that. That will take about 67 microseconds, so he won't witness the other strike until t = 77 microseconds. He calculates that, since he saw the second strike at t = 77 microseconds and it had to travel 20 km at the speed of light, that it took place, in the coordinates of his rest frame, at t = 10 microseconds.

when did the rocket view each of the lightning strikes occurring?

When the light from them reaches him. You used the word "view", which also means "see".

Figuring out when the light rays reach the rocket is easy in Bob's frame. The rocket is traveling at 0.15 km per microsecond in the positive x direction, starting from x = 0 at t = 0. The strike on the tree starts at x = 0 at t = 10 microseconds and travels 0.3 km per microsecond in the positive x direction. The other strike starts at x = 20 km at t = 10 microseconds and travels at 0.3 km per microsecond in the negative x direction. So it's simple algebra to figure out where the two light rays meet the rocket:

The light ray from the first strike meets the rocket at x = 3 km, t = 20 microseconds.

The light ray from the second strike meets the rocket at x = 7.67 km, t = 51.1 microseconds.

Once we have these values, it's easy to Lorentz transform them into the rocket frame. The rocket frame coordinates are: x' = 0, t' = 17.3 microseconds, and x' = 0, t' = 44.3 microseconds. Notice that x' = 0 for both, as we expect because both events take place at the rocket, and the t' values are just the t values divided by the gamma factor for v = 0.5c.

Now of course that's not the question you actually meant to ask. The question you actually meant to ask is, what are the x', t' coordinates of the lightning strikes themselves, i.e., the events where they strike, not where the rocket sees the light rays from them. This is just Lorentz transforming their x, t coordinates to the primed frame, which again is easy:

The lightning strikes the tree at x' = - 1.73 km, t' = 11.5 microseconds.

The lightning strikes the pole at x' = 21.3 km, t' = - 26.7 microseconds.

Now let's see if these values make sense with the times we got above for when the light rays from the strikes reach the rocket. The light from the strike at the tree travels in the positive x direction at 0.3 km per microsecond. It has to travel 1.73 km to get to the rocket at x' = 0; this will take 5.8 microseconds. It starts at t' = 11.5 microseconds, so it reaches the rocket at t' = 11.5 + 5.8 = 17.3 microseconds, just as we saw above.

The light from the strike at the pole travels in the negative x direction at 0.3 km per microsecond. It has to travel 21.3 km to get to the rocket at x' = 0; this will take 71 microseconds. It starts at t' = - 26.7 microseconds, so it reaches the rocket at t' = - 26.7 + 71 = 44.3 microseconds, just as we saw above.

Several of the above numbers appear to be different from yours, so I suggest you read through the above several times to understand how the numbers were derived, and then re-think the scenario.

 

[JMart12]

I'm confused. You say the lightning strikes happen at 10$\mu$s in Bob's frame, but then you say one of them happened at about -50$\mu$s. Which is it?

It's worth noting that you may be confused: the times inserted into the Lorentz transforms are the times things actually happen according to that frame, not when you receive light from them. Assuming that you inserted t=10$\mu$s into the Lorentz transforms (if my arithmetic is correct this gives t'=-37$\mu$s, not the -27 you quote) then you are asserting that, in Bob's frame, the lightning strikes were simultaneous. Assuming he survived the strike on his own tree, he will receive the light from the other strike at around t=77$\mu$s. So he does not see the strikes simultaneously, but concludes (after subtracting the time for light to travel from the flagpole) that the strikes were simultaneous.

Thank you so much! I’m sorry for the confusion on the original post. What I was trying to say was that relative to Bob the second lightning strike would have occurred at t=10 microseconds, but if you subtract the time it takes for light to travel the 20km from the pole to Bob then, relative to the pole, the time the second lightning strike occurred was at something like -50 microseconds (I don’t remember the correct number).

This was all a bunch of bologna, because as you clearly stated, the person isn’t actually viewing these events as simultaneous. This person is already taking into account the time it would have taken for light to travel, and I think that has been my main problem all along.

Again thank you so much for your post, it really means a lot that someone I don’t even know would take the time to help me with a problem.

 

[JMart12]

Note that this statement, taken literally, is false. The person "witnesses" (as in "sees with his eyes") the strike on the tree he's standing next to at t = 10 microseconds. But he doesn't witness the other strike until the light from it gets to him, and it has to travel 20 km to do that. That will take about 67 microseconds, so he won't witness the other strike until t = 77 microseconds. He calculates that, since he saw the second strike at t = 77 microseconds and it had to travel 20 km at the speed of light, that it took place, in the coordinates of his rest frame, at t = 10 microseconds.
When the light from them reaches him. You used the word "view", which also means "see".

Figuring out when the light rays reach the rocket is easy in Bob's frame. The rocket is traveling at 0.15 km per microsecond in the positive x direction, starting from x = 0 at t = 0. The strike on the tree starts at x = 0 at t = 10 microseconds and travels 0.3 km per microsecond in the positive x direction. The other strike starts at x = 20 km at t = 10 microseconds and travels at 0.3 km per microsecond in the negative x direction. So it's simple algebra to figure out where the two light rays meet the rocket:

The light ray from the first strike meets the rocket at x = 3 km, t = 20 microseconds.

The light ray from the second strike meets the rocket at x = 7.67 km, t = 51.1 microseconds.

Once we have these values, it's easy to Lorentz transform them into the rocket frame. The rocket frame coordinates are: x' = 0, t' = 17.3 microseconds, and x' = 0, t' = 44.3 microseconds. Notice that x' = 0 for both, as we expect because both events take place at the rocket, and the t' values are just the t values divided by the gamma factor for v = 0.5c.

Now of course that's not the question you actually meant to ask. The question you actually meant to ask is, what are the x', t' coordinates of the lightning strikes themselves, i.e., the events where they strike, not where the rocket sees the light rays from them. This is just Lorentz transforming their x, t coordinates to the primed frame, which again is easy:

The lightning strikes the tree at x' = - 1.73 km, t' = 11.5 microseconds.

The lightning strikes the pole at x' = 21.3 km, t' = - 26.7 microseconds.

Now let's see if these values make sense with the times we got above for when the light rays from the strikes reach the rocket. The light from the strike at the tree travels in the positive x direction at 0.3 km per microsecond. It has to travel 1.73 km to get to the rocket at x' = 0; this will take 5.8 microseconds. It starts at t' = 11.5 microseconds, so it reaches the rocket at t' = 11.5 + 5.8 = 17.3 microseconds, just as we saw above.

The light from the strike at the pole travels in the negative x direction at 0.3 km per microsecond. It has to travel 21.3 km to get to the rocket at x' = 0; this will take 71 microseconds. It starts at t' = - 26.7 microseconds, so it reaches the rocket at t' = - 26.7 + 71 = 44.3 microseconds, just as we saw above.

Several of the above numbers appear to be different from yours, so I suggest you read through the above several times to understand how the numbers were derived, and then re-think the scenario.

Thank you so much for this comment. This explains the entire situation perfectly, and I know feel that I have a much better grasp on this. I’m sorry for my numbers being off, I was approximating what I remembered some of these to be, and it was late at night, so that is completely my fault.

The main misconception that I was having, as you pointed out, is that I assumed that the Bob actually saw the lightning strikes at the exact same time. I didn’t realize that they weren’t literally simultaneous relative to Bob.

Again, thank you so much for this comment.

 

[Pencilvester]

I assumed that the Bob actually saw the lightning strikes at the exact same time. I didn’t realize that they weren’t literally simultaneous relative to Bob.

If you plan to continue to study special relativity, it’s important to note that when it’s said that two events are simultaneous in a certain frame, that means that unless the events are equidistant from the observer to which that frame belongs (as measured by that observer) then he will not see them occurring at the same time. All of the interesting results of SR reveal themselves after the travel time of light is accounted for.

 

[JMart12]

If you plan to continue to study special relativity, it’s important to note that when it’s said that two events are simultaneous in a certain frame, that means that unless the events are equidistant from the observer to which that frame belongs (as measured by that observer) then he will not see them occurring at the same time. All of the interesting results of SR reveal themselves after the travel time of light is accounted for.

Thanks so much. I’ll definitely have to keep this in mind!