Time Dilation: Two Observers, Different Times

[nikeadidas]

Suppose there are two observers separated by distance of 2 light minutes. They have 2 identical clocks, which are synchronized. Suppose one of the observer shoots a missile at 3 PM. The other observer, who is able to see him, will observe that the missile has been shot at 3.02 PM. According to the first observer the missile hits the other observer at 3.04 PM (considering speed of missile to be 0.5c), while according to the observer who is hit by the missile, the missile will hit at 3.06 PM. If both the observers are at rest relative to each other, how do they measure different time for a common event?. Am I missing something out over here?

 

[TGlad]

Wouldn't the first observer see the missile hit at 3.06pm (4 minutes to arrive, 2 mintes for the light to get back)
And wouldn't the 2nd observer experience the hit at 3.04pm?
In this case they measure a different time because light takes a while to get from the event (missile hits 2nd observer) back to the first observer.

Of course, the 1st observer knows that the target is 2 light minutes away, so he will subtract 2 minutes from the time, so, if he is smart, he will know that the missile landed at 3.04pm.

 

[PAllen]

Suppose there are two observers separated by distance of 2 light minutes. They have 2 identical clocks, which are synchronized. Suppose one of the observer shoots a missile at 3 PM. The other observer, who is able to see him, will observe that the missile has been shot at 3.02 PM. According to the first observer the missile hits the other observer at 3.04 PM (considering speed of missile to be 0.5c), while according to the observer who is hit by the missile, the missile will hit at 3.06 PM. If both the observers are at rest relative to each other, how do they measure different time for a common event?. Am I missing something out over here?

It is true that the missile's victim won't see the launch until 3:02. However, in your scenario, they will see the missile hit them at 3:04, just like the launch observer. The issue is that the light travel time from the missile to the victim decreases from two minutes to zero as the missile approaches.

 

[PAllen]

Wouldn't the first observer see the missile hit at 3.06pm (4 minutes to arrive, 2 mintes for the light to get back)
And wouldn't the 2nd observer experience the hit at 3.04pm?
In this case they measure a different time because light takes a while to get from the event (missile hits 2nd observer) back to the first observer.

Of course, the 1st observer knows that the target is 2 light minutes away, so he will subtract 2 minutes from the time, so, if he is smart, he will know that the missile landed at 3.04pm.

Right, this makes explicit a point I glossed over - what the launching observer sees versus interprets.

 

[Alexroma]

My similar stupid question was answered in the "Astrophysics" section. I hope that would help:
https://www.physicsforums.com/showthread.php?t=542820

 

[nikeadidas]

@ alexroma
Thank you extremely for saving me a lot of trouble. I get the answer intuitively though. Can someone explain what the observer who's been hit will actually see. The journey of missile will be compressed for him, i.e he will watch the 4 secs journey in 2 secs, since the missile is approaching him.

 

[TGlad]

He'll see a missile coming towards him very quickly, then he'll see some blood and flying limbs.
The blue shift of the missile might be so much that the visible colours of the missile go out of visual range, and the infra red colours might come into visual range. He might visually see the heat of the missile.. before it goes boom.
Actually I lie, he won't see anything, the missile will be too small to see, then bam! it'll be in his face.