Special Relativity and Blue Shift question

[Astro_Will]

Earlier today I was on youtube when I saw this comment

Consider you are sitting on a planet 100 light years away from Earth and point a very powerful telescope directly at it. You see how the Earth looked like 100 years ago since light from "today" hasn't reached the planet I am sitting on yet. Now I start moving towards Earth with a speed close to the speed of light while still pointing the telescope at earth, Would everybody run like crazy and skyscrapers rise very fast? Would time speed up from my point of view?

I went to respond to this saying that the light would blue shift out of the visible spectrum and you wouldn't be able to see anything. But then I thought about time dilation from relativity and wondered if the slowing down of time would be significant enough to keep the view of Earth inside the visible spectrum. Or does the Doppler effect not apply here for some reason. Or if I am just misunderstanding relativity all together.

I know that no matter what you wouldn't be able to see Earth in that much detail but that's not the point. Would time dilation keep you inside the visible spectrum (but still slightly blue shifted)? Assuming you were able to see the full spectrum would things appear significantly sped up or would your time slowing down make them seem to move only slightly faster? Does time dilation even apply here?

 

[Austin0]

Earlier today I was on youtube when I saw this comment
I went to respond to this saying that the light would blue shift out of the visible spectrum and you wouldn't be able to see anything. But then I thought about time dilation from relativity and wondered if the slowing down of time would be significant enough to keep the view of Earth inside the visible spectrum. Or does the Doppler effect not apply here for some reason. Or if I am just misunderstanding relativity all together.

I know that no matter what you wouldn't be able to see Earth in that much detail but that's not the point. Would time dilation keep you inside the visible spectrum (but still slightly blue shifted)? Assuming you were able to see the full spectrum would things appear significantly sped up or would your time slowing down make them seem to move only slightly faster? Does time dilation even apply here?

-I think there is no getting around the blue shift. But ignoring that ,yes things would appear relativistically doppler shifted. I.e. Sped up if you could see them.Just as any sequence of signals would be.

 

[PeterDonis]

Austin0 is correct; relativistic Doppler shift would cause the incoming light signals from Earth to appear "speeded up" to you.

Think of it this way: suppose you start 100 light years away from Earth at this instant (in the Earth's rest frame) and travel to Earth at 0.5c (half the speed of light). That means you will arrive on Earth 200 years from this instant, Earth time.

But when you started, you were seeing light from Earth 100 years before this instant; so in the 200 years (Earth time) that you travel, 300 years' worth of light signals from Earth will reach you (because when you reach Earth, of course you are seeing light signals from Earth instantly, with no time delay).

Plus, since you are traveling at 0.5c, less than 200 years pass by your clock during the journey. It turns out that only a little over 173 years will pass for you during the journey. So in 173 years of elapsed time by your clock, you will receive 300 years' worth of light signals from Earth. So you will be seeing events on Earth "speeded up" by a factor of 300/173 during your trip.

 

[Astro_Will]

Thank you both that was incredibly easy to understand and incredibly interesting to think about.

 

[PeterDonis]

Thank you both that was incredibly easy to understand and incredibly interesting to think about.

Glad it helped. Just one additional comment: 300/173 is the relativistic Doppler shift factor at a relative velocity of 0.5c--actually if you do the precise math the Doppler shift factor is sqrt(3), and the elapsed time by the traveler's clock is 100 * sqrt(3) years, so we have the "speed-up" ratio 300 / (100 * sqrt(3)) = 3 / sqrt(3) = sqrt(3), exactly equal to the Doppler shift factor. It's easy to show that this holds generally, so as Austin0 said, the "speed-up" applies to any signals received, and it applies at any relative velocity.

 

[ghwellsjr]

...the elapsed time by the traveler's clock is 100 * sqrt(3) years...

I don't understand this. If someone asked me how much time elapsed on a clock that traveled at 0.5c for 200 years in the Earth's frame, I would have said to divide 200 by gamma which is 200/1.1547 = 173.2 which is the same answer you got but not the same calculation. What is the 100 and why do you multiply it by √3?

 

[PeterDonis]

I don't understand this. If someone asked me how much time elapsed on a clock that traveled at 0.5c for 200 years in the Earth's frame, I would have said to divide 200 by gamma which is 200/1.1547 = 173.2 which is the same answer you got but not the same calculation. What is the 100 and why do you multiply it by √3?

I didn't, it was just a way of expressing the answer that makes the Doppler calculation more obvious. I actually got the answer the way you said. Note, though, that if you write out how you got the 1.1547 factor, you'll see that it's 2/sqrt(3), so you can convert 200/1.1547 to 100 * sqrt(3) in one line of algebra.