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Please bear with me because I'm noob at physics. Numbers don't have to be calculated, these are just examples, I'm just asking conceptual stuff.

  1. Just double check: Objects traveling at $0.99c$ in circles like in LHC would still experience time dilation (i.e. for the physicist it'll be $1\,\mathrm{s}$ but for the object will be ~$7\,\mathrm{s}$. For simplicity, I used the classic equation which is $1\over\sqrt{1-(v^2/c^2)}$ - this is acceptable, right?

  2. Obama is traveling at $300\,\mathrm{km/h}$ in his plane. A Russian spy put spycam in his plane and he saw Obama is looking at a video of him through a spycam CIA unknowingly put in the Russian spy's apartment. Let's say the lag is less than total time difference, will the Russian spy see himself at a fraction of his current speed in Obama's TV? (I know $300\,\mathrm{km/h}$ and stationary isn't much different but due to the fact that the spy is slower, will he see himself at $99.99999999$...% speed)

  3. What if we can somehow put some kind of "webcam" inside the object that move at $0.99$c, and it can somehow send data back faster than $0.99$c, will we see ourselves move at $\frac{1}{7} speed? (not sure how since I'm a physics noob but hopefully u get the idea. Lifi? I heard entanglement can do this too)

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Yes and we know this because it's been done - sort of.

When you talk about seeing this turns out to be complicated because you have to take into account the time the light takes to reach your eye. If you have some object moving at relativistic speed then its distance from you is changing rapidly and this affects the time the light reflected off the object takes to reach your eye. This has some unexpected effects - for example it means you don't see fast moving objects compressed due to Lorentz contraction. Instead you see them rotated.

So exactly what you'd see is complicated, but you would see the fast moving object moving at a different rate to a similar stationary object. Having said that, as far as I know no-one has ever managed to record such an image. The practical problems of accelerating large objects to speeds near $c$ are currently intractable.

But we have done the experiment using gravitational time dilation rather than time dilation due to high speed. The Pound-Rebka experiment observed radiation emitted by a radioactive iron isotope and found the radiation was red shifted. That means the oscillating iron nucleus was oscillating more slowly when viewed from a height where time was running faster.

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