Repost of question, hopefully with a little more clarity: I'm writing a scene involving a portal generating a time dilation field in the Ponyville town square.

1. There is a "rift" P_rift in the Ponyville town square.
2. The rift has an apparent event horizon at radius (r_pony < r_rift < r_{pickup truck}).
3. The rift is not spinning.
4. Light and time --but not gravity-- are bent as if there were a point mass equal to M_rift
5. Dr. Stone is a pony observer X feet away from P_rift who experiences 15 < T_{Dr. Stone} < 30 minutes.
6. Canterlot (and therefore the sun) experiences T_Canterlot such that mid-afternoon advances to evening a few days after Starlight's graduation, assumed to be a week or two before the Summer Sun Celebration.

Does Dr. Stone see an Einstein's ring around the portal?
Does Dr. Stone see anything else that might clue him into non-Euclidean (Ponification needed) geometries?

• Bad Dragon brought up increased light. The specifics of the story have the sun blocked out, and visibility is limited to the battle field. I may add slightly red/blue shifted colors as a clue for Dr. Stone.
• platogkrone brought up strain from different time rates within the body. A very valid concern. A safety zone will need to be considered. Possible points of failure are the circulatory system and the nervous system.

Edit: updated OP with more recent version of problem definition and possible effects seen.

6998258 The inside of the affected area would be insanely bright. Ponies inside would receive 5 hours of light in just 30 minutes. That means the light would be 10 times brighter. You can use this if you make it nighttime and they see as clearly as if it were day.

The light generated within the affected area would have to be super bright in order to be seen on the outside. If a pony charged their horn, it would be bearly detectable from the outside because 30 minutes of that light would be slowly seeped into 5 hours.

As for the lensing, it mostly depends on how it works. If there's a sharp barrier where the effect ceases, The light coming from small angles might even reflect off of it, because time dilation could act as a hard surface.

The light that would come through would curve toward the center. The whole dome would act as a lens.

If there was light directly on the other side of the time dilation dome, you'd see it clearly. Everything else would be jumbled as if you looked through a translucent globe. It would probably be mostly blue due to the sky color.

If there were ponies inside, you'd see them magnified.

Around the area of effect, there would be nothing different. Perhaps the items closest to it would give off a bit more bluish color due to the portal curving the blue skylight into them (like a river reflecting the sky on objects).

If I were you, I wouldn't throw everything into the mix. Time dilation is enough. You don't need to throw gravity and space-time curvature into the mix.

6998340

Ponies inside would receive 5 hours of light in just 30 minutes. That means the light would be 10 times brighter.

Good one. Visibility is down and there is cloud cover of magical density, so sunlight isn't an issue.

The light that would come through would curve toward the center. The whole dome would act as a lens.

This is the detail I want to clue in my observer to the fabric of spacetime being warped.

If I were you, I wouldn't throw everything into the mix. Time dilation is enough. You don't need to throw gravity and space-time curvature into the mix.

I've gotten this a lot. I'm trying to have fun learning about this effect at a mathematical level and hopefully encourage any readers to look up the effect in real life.

I'm looking for the an explanation of the equations that's slightly above what's easily understandable to the public. Enough to nod to actual science without kowtowing to it.

Why wouldn't physical matter be affected? Space-time curvature that's that extreme generally would. However, there is an option I'll explore at the end, but first what I know a bit more about. Side-note: I took a course in GR and have a PhD in Physics so I have a general idea. Loop in Pineta https://www.fimfiction.net/user/71233/Pineta (can you @ people here?), who is similarly positioned, and maybe PM us if you want something that is more in-depth but takes longer. Also ask him for a second opinion. I'd suggest PsiStarPsi as well, but he's not on FIMfiction and I don't know if he'd want me giving out his contact details, but I can ask if you want.

Anyway, The ordinary gravitational lens effect works by bending light from objects on one side of the object to a point 180-ish degrees on the other side (it's a little offset, which is how the effect was found). For instance, here is a graviational lens effect from something almost-but-not-quite directly behind the object:

What you are talking about is closer to a Warp Bubble, though of course a true warp bubble has the interior and exterior being causally disconnected and no light can pass either way. More on that in a moment.

You are combining 2 effects: time dilation near a massive, dense object (typically a black hole, though neutron stars are also dense enough for the effect to be noticeable) and gravitational lensing, which occurs AROUND dense objects.

What I think you want to do is to put town square as if it were near a black hole, and thus experiencing time dilation, but for reasons presumably magical, there is not an actual event horizon in it. I did this calculation once for the "World enough and Time" episode of Doctor Who.

So what would actually be seen is akin to either just outside a black hole's event horizon looking directly outward, or outside looking at that point. If that is your model, there are a few points to note here.

1) Light sources coming in at oblique angles will, as Bad Dragon noted, deflect AROUND the anomaly, limiting the field of view from the inside-looking-out.

2) Note well: This means that although 10x the light intersects the edge of the anomaly per unit time, not all of that light actually enters. This means the brightness, as viewed from the inside, depends largely on the angle of the sun. While usually during the day, atmospheric scattering (which also makes the sky blue) renders the surface of Earth the same brightness so long as the sun is above the horizon (roughly anyway), in this scenario only nearly-direct rays would get through. Here's where I get tripped up without actually trying to run the numbers. My intuition is that it would be brighter than normal at noon, but the same or dimmer the rest of the time. However, I can't swear to that.

3) Light rays that do make it in would tend to converge slightly, making the object appear closer/larger than it actually is.

4) Looking from the outside in is a bit trickier as you don't have the 'hard' cut-off you have with a black hole, but in the black hole case, the light rays that make it out are again, those closer to direct, and those will tend to diverge, making the object look faint and far away.

Now that we've covered the case I know, let's cover something I know a bit less about (though I am trying to learn more in my free time). What you are describing, where those inside are not affected, is closer to a 'warp bubble': A thin region of severely warped space-time surrounding a 'normal' interior. Actually the 'normal' interior here would need to be different than the exterior, to maintain the time dilation effect, which is not usually a desireable property in a warp drive (don't want to find everyone back home having aged 50 years!), but is geometrically possible: the inner one should be 'flat' to the first derivative, but not to the second, I think. Again, can't swear to that. Without spending considerable time both figuring out what the hey that space-time would look like and deriving the mass-energy distribution from there, I'm not sure if either step is even possible. 'Magic' can justify the mass-energy distribution normally being impossible (if it is), but the other is trickier.

This isn't the sort of thing one usually tries to do with GR, so I don't immediately know how or what the consequences would be.

...Anyway, hopefully that helped at least a little.

6998511
Thanks for your well-structured response. Most of the places I've asked have been more focused on slamming me for the impossibility of the situation than on actually trying to help with the math.

I was already thinking of a "warp bubble" type effect if too many factors didn't play out nicely. I want to have time flowing more slowly the deeper normal ponies get to the singularity after the battle starts. (Sorry if I'm seeking advice in 3-5 locations and miss the full context) You're also the second person to recommend Pineta; I think I will try a DM.

I am also looking to hoofwave the increase of air pressure such a field would cause.

6998565
Ohhhhh, you want it to increase the further in you go? Well then the weird semi-flat bubble (or whatever) isn't needed. It can function essentially like a black hole that just doesn't go to infinity. In that case there is no singularity (if there was one, the event horizon would have to exist, which would just be a black hole at that point). I don't think that would be a possible matter distribution (though again I can't be sure without crunching the numbers: it's not guaranteed to exist by the GR equations), but that's what magic is for.

Hmmm... didn't think about the air pressure issue. If you want it to increase as you go in, you effectively have a gravity well, which would drag air in. The way "World Enough and Time" solved that was by being in a spaceship: the atmosphere was artificially maintained, so it wasn't an issue. Since Equestria's weather is controlled, maybe it's not too much of a stretch to say that something similar is happening here?

Incidentally, in order to produce that severe an effect over that small a space (technically, over an interval), the tidal forces would tear anycreature apart. Hence why I was thinking of the warp bubble: That way you only die when you try to cross the boundary.

Two questions, out of curiosity. Who else recommended Pineta? Might be someone I know. Incidentally, his speciality is particle physics, not GR (as my speciality is radiation cascades and their effects, not GR), but he's also better at explaining things simply than I am. :|
Second, where else were you asking? Did you make it clear it was for a Science Fiction/Fantasy thing?

6998579
I was on the Fimfiction Discord server, a Discord server aimed at academics, and the Writers' Group Discord server -- where I got the recommendation from someone who I'll have to look up when I have a better Internet interface. Each time, I geared the pitch towards the audience and my present understanding of the situation: At about 50 yards away from the portal, mid-afternoon turns to evening in Canterlot in about 15-30 minutes. Each time, I tried to say magic of some sort would handwave the effects I would handwave.

The bit at the middle is a portal between dimensions spewing an invading army. They might need some natural resistance to the time field. All in all: the detail wasn't in the first draft, but my total rewrite of the chapter has grown into it, particularly when I did a very rough estimate of the number of Unicorns living in Ponyville (one day after "Celestial Advice") who can cast a shield spell. (About 10, I figured)

Edit: hawthornbunny sent me his way.
I take Friday sundown to Saturday sundown off from working on writing/robotics blog, so I won't be responding to related messages during that time, but I would still like to pick it up afterword. I'm in the Pacific time zone.

6998763
Feynman estimate. Check your auto-correct. :P

Huh, did not know Fimfiction had a discord server. That a writer's group from fimfiction or another one? (just curious, don't answer if you don't want).

Ah, so it's an Einstein-Rosen bridge but without the usual assumption of stabilization extending out to the very edge of it. That works a little better in terms of equations, though the end would be in danger of pinching off. Anyway, that means you can use this handy image as a reference: https://en.wikipedia.org/wiki/Wormhole#/media/File:Wurmloch.jpg
Though in this one the distorted region around the wormhole is relatively small, whereas you're extending the region by making the fall-off more gradual.

6998777
FimFiction Discord, yes
Writers' group, yes.
hawthornbunny recommended, as posted in my edit.
I also tried asking on the SweetieBot project Discord server, but got no response.
I was on mobile, with little ability to check spelling.

The anomaly leads to a sort of dimension between dimensions, possibly even passing through Limbo.
Also in case you missed my edit, I'm taking sundown Friday to sundown Saturday off from writing/unrelated blog stuff, but I would like to continue discussion afterwords. (Pacific Time)

6998784
Aight, well probably best to PM me if you want to know more. Not sure how much more I can add at this point, though.

I'm pretty sure entering a time-dilated field would cause massive damage to a living being, as the front end suddenly experiences much less time relative to the back end. That could quite easily crush and/or tear living tissue and other materials.

6999467
I'm not sure you understand the geometry I'm going for, but that's a good point nonetheless. I think you're imagining a discrete boundary like stepping down stairs, but I'm imagining a bowling ball in a trampoline. I'd imagine there would be some amount of tolerance for differences in time flow within the body. My guess is that the circulatory system would be the most vulnerable.

Unfortunately, since HiE is involved, hoofwaving it like that would bring up questions about magical centers in humans that I don't want to answer right now. I'd need to research the body's tolerance for a temporal differential and establish a safety zone for normal humans/ponies (HiE is involved). If the zone is more than a few yards, I'll need to hoofwave it for ponies by saying their bodies are temporally synchronized to their own magical center, located in the heart ("Magical Mystery Cure"). I don't want to answer any potential questions about humans' magical centers, so the human character will need to stay in the safety zone. If the safety zone is too big for the story, I may need to look into a sudden death zone and reevaluate.

Thank you for your contribution.

Sorry, HiE?

6999883
Human in Equestria. It's strongly associated with one of the story tags for when you have literal humans involved with your plot and they aren't EQG. I'd imagine standing upright would be easier on the time gradient, but the brain fires off at light speed, so timings within the nervous system might be another early point of failure.

6999929
Well it's not light speed, though still quite fast. It maxes out at near c, but for impulses travelling from the brain to somewhere else, it's actually quite a lot slower: https://hypertextbook.com/facts/2002/DavidParizh.shtml It's about 100 m/s in the brain, which is order 10 cm in size (0.1 m), so our scale time is 10^-3 s (1 ms) over 1 decimeter, whereas the bubble has a gradient of something like 20 (seconds in per 1 second out)/100 (meter radius) = 1/5 seconds/meter, or 1000/50 = 200 ms/decimeter, much larger and thus enough to cause problems. That is going to at least cause disorientation, at least initially, and probably worse. The question, and it may not be one that's answerable with current knowledge, is whether the brain's well-known ability to adjust to various situations would be able to even things out eventually, such as by staggering signals.

In point of fact, if impulses WERE travelling at c, I wouldn't expect there to be any problems, as the speed of light is constant regardless of the frame of reference. However, because they travel slower, but very fast, we have a potential issue.

An ordinary black hole neatly avoids this problem for all but the smallest black holes, since there are no stationary observers past the event horizon: https://physics.stackexchange.com/questions/187917/thought-experiment-would-you-notice-if-you-fell-into-a-black-hole, and in many cases you'd need to be that close to the singularity for the time gradient to be fast enough to be noticeable to the one falling in (though it would be to one stationed some distance away). However, you've constructed an artificial situation where we have time dilation but also stationary observers. I'm not sure without actually doing the math (which would take too long), but I thiiiiink that if you were moving, the brain wouldn't be able to adapt and coordination would break apart. I think autonomous systems would still work (though I don't know enough neuro-biology to be sure), but little else.
However, if you were still for awhile, I think the brain would eventually adapt and so long as you didn't move closer in or further out, you could function in some capacity. Again, couldn't swear to that. It's plausible enough for fiction, though.

Just had a thought: because we've created a situation where only the time-like dimension is curved, light may not follow the usual 'einstein ring' pattern. In fact, because we're in an atmosphere, light doesn't move at speed c exactly, and so may not follow null geodesics.

This means my earlier thoughts on light scattering off the surface may not be correct. Instead, you could calculate the bending (and I'm not going to do so, as it would take awhile) by treating the bubble as if the time discrepancy were altering the frequency of the light. Then you go through the usual method of deriving an index of refraction, but whereas that changes wavelength, you're changing frequency. This would bend the light, but in a different way than normal refraction. It's an interesting problem, but it would take some time to sit down and calculate. Thankfully this situation renders it solveable using basic geometry and algebra, so you should be able to handle it on your own from there.

7000021
I may have to just say Equestrian nervous systems have a natural resistance to changes in timeflow; I have some Pegasi getting pretty close.

7000027
That's a good one. I think I may need to just drop the Einstein's ring all together and focus more on the stuff Bad Dragon and platogkrone brought up. Also, it looks like FimFiction now has a collective 7 million+ messages sent.