I Time Dilation Observation: What Does Earth Observe?

[Punit D]

I'm wondering if anyone can answer this question for me.

Suppose we send a spaceship to the moon at 299,792,457 m/s (1m/s less than c). That's 99.9999997% of the speed of light. Plugging the numbers into the time dilation formula, we get that the 1.2822 seconds that it takes for the spaceship to reach the moon is equal to 15,698.4954 seconds on earth, or 3 hours and 24 mins roughly.

My question is, what exactly do we see here from Earth? Suppose an observer with a telescope is watching the spaceship take off and reach it's destination. We already know the spaceship will take about 1.28 seconds to reach the moon, yet the observer holds the telescope in his hand and watches the spaceship for 3 hours and 24 mins? I'm confused here.

 

[phyzguy]

You have it backwards. An observer on Earth will see the spaceship reach the moon in 1.28 seconds, but to an observer on the spaceship, only roughly 10 microseconds will elapse before it reaches the moon.

 

[Ibix]

Further to what @phyzguy said, an observer on the spaceship will see the distance to the moon length contracted to around 10μls, so the moon reaching her in around 10μs in that frame makes sense.

 

[tionis]

The question is: would the observer back on Earth be able to observe anything with his telescope? Wouldn't the craft be too redshifted to be seen?

 

[Nugatory]

The question is: would the observer back on Earth be able to observe anything with his telescope? Wouldn't the craft be too redshifted to be seen?

We could design a strobe light that emits a flash of hard x-ray radiation once every microsecond (as measured by an observer at rest relative to the strobe). That's effectively a clock that ticks once every microsecond. The radiation would be red-shifted down to visible wavelengths relative to the Earth so the earth-bound observer could see and count the flashes, thereby reading the clock.

Using the frame in which the ship is at rest and the moon is racing towards it at almost the speed of light and the Earth is moving away at that speed, it takes the ship about 10 microseconds to make its three-kilometer journey (Just three kilometers? Yes, there's some serious length contraction at these speeds) and the strobe flashes ten times.

The earth-bound observer counts these ten flashes emitted during the 1.28 second duration of the journey (using the frame in which the Earth and the moon are at rest and the ship is moving at almost the speed of light). That is one tick for every 128 milliseconds, so he correctly concludes that the clock is ticking way slow, and that's time dilation.

(Note that because of light travel time, the flash emitted as the ship arrives at the moon will not be received on Earth until 1.28 seconds after the arrival, which is 2.56 seconds after the departure. Although the ship clock is ticking once every 128 milliseconds according to Earth guy, this is a calculated result. The flashes arrive at Earth 256 milliseconds apart, and only after allowing for light travel time does it become clear that the emissions happened 128 milliseconds apart using the frame in which the Earth and the moon are at rest)

 

[Punit D]

My bad. I had it backward the whole time. I feel like an idiot now lol.

Thank you for answering my questions. Great answers!