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I've been thinking about doing something involving bubbles of stopped (already answered here), slowed down or sped up time.

What would that look like for different speeds?

Is there any way of calculating the refraction based on the speed of time inside?

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    $\begingroup$ I don't have enough physics doctorates to know for sure, but you may be able to use gravitational time dilation equations to figure out the mass of an object that would cause the slower/faster time rate you want, and use that mass in some gravitational lens equations to figure out what the light deflection might look like. $\endgroup$ – Giter Jul 20 '18 at 12:49
  • $\begingroup$ Your question is a bit nebulous... can you clarify what you mean with "refraction based on the speed of time inside"? $\endgroup$ – L.Dutch Jul 20 '18 at 13:08
  • $\begingroup$ Hello, Laurin Schmidt, and welcome to Worldbuilding.SE! Please take our tour and visit the help center to learn more about the way the site works. Have a nice day! $\endgroup$ – Gryphon Jul 20 '18 at 13:27
  • $\begingroup$ @L.Dutch basically, if there would be refraction (light CAN pass through slowed time bubbles) i want to know what it would look like for say 1/2 the outside speed and 1/100th the outside speed etc. $\endgroup$ – laundmo Jul 20 '18 at 18:24
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Refraction is already calculated by change in speed of light through a medium: $n = {c/v}$ with n being the index of refraction c being the speed of light before the transition and v being the speed of light after the transition.

If inside time goes at 1/10th the rate light moves 1/10th as fast, the bubble would have a index of refraction of 10 (well above known materials) with a critical angle of about 10 degrees, so you could pretty much only look straight into it. With a bigger time difference even less light would escape the slow time.

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I imagine depending on which laws of physics you want to retain and which you want to drop, there are some pretty deep and counterintuitive results. Since that's too hard let's do something simple.

Objects near a large mass (black hole) seem to have their time slowed when viewed very far from the large mass. The same sucking action of gravity causes light traveling away from the large mass to lose energy and become redshifted.

If your slow-time bubbles use a similar principle (a localised area of high gravity pulling equally in all directions) the answer is they look redder depending on how powerful they are. Super powerful bubbles have the frequency lowered so much they become invisible as the reflected light is turned into radio waves. So those bubbles appear completely black.

There is no such thing as a reverse black hole so this has no physical basis -- but it makes narrative sense the fast-time bubbles have their stuff look bluer depending on their power. Again really powerful bubbles have their light turned into radiation which is invisible. Again the bubbles appear black.

Likewise, if you're in a slow bubble stuff outside appears bluer and if the bubble is really powerful the outside world is just a massive black void surrounding the bubble.

It's also a good narrative device if the slow bubble is completely black. If there's a fight going on inside we can't see what's going on and we can't help without hopping in yourself and becoming trapped!

Calculating the redshift would depend on how the gravity in these bubbles. I suggest the gravity is very strong but pulls evenly in all directions and cancels itself out. As though you were at the hollow centre of an immensely thick and heavy dyson sphere: you are weightless and experience slow time.

The normal formula for gravitational dilation assumes light traveling away from a large mass with the normal sloped gravity well so cannot be used here. It's probably easier to estimate how powerful the bubble can get before it becomes black. Any takers!

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    $\begingroup$ Worth while noting that while slow time bubbles might be black in the visible spectrum any infrared radiation inside would get turned to visible radiation, so you could see inside in heat vision. Similarly if in the inside of a slow time bubble you could potentially get sunburn from the radiation blueshifted into UV. $\endgroup$ – Joe Bloggs Jul 20 '18 at 13:31
  • $\begingroup$ Another concern would be light emitted behind the slowed/sped region, which passes through it on its way to your eye. If you take into account the bending that would occur as it passes through a change in "time-rate", then the bubble would also distort light passing through it or near it in a manner similar to a black hole. If one decides not to take the light-bending into account, then you would be able to see things behind the bubble normally for a sped-up bubble, and normally but perhaps delayed for a slowed time bubble. $\endgroup$ – Qami Jul 20 '18 at 14:46
  • $\begingroup$ Oh yeah that didn't occur to me. I still think the bubbles would appear darker since most of the Sun's light is on the 'visible' spectrum. But you might still see the insides in heat vision or ultraviolet! $\endgroup$ – Daron Jul 26 '18 at 11:20

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