I'm designing a character that can slow time in an area around him/her. The living things in that area perceive time normally, so they see this character and the things out of that area moving very fast.
In movies if time is slowed, it will also make things fall slower. How would slowing time affect gravity? Will the character be able to run on the walls, like Quicksilver, even if they aren't super-fast like him? They just slow time.
It'll not go well for your character
From your description we can infer a few things. One is that your own character is unaffected. The other is that everything else in the area is affected.
First let's disregard the character. The time slow is dangerous. Imagine standing just outside the zone and grabbing a hose. Squirting some water in will slow that water, causing the faster running water behind it to run into it. This'll create a lot of pressure on the water as it tries to escape. Many forces from outside should approach carefully and not too much. Wind, a river, water in pipes and more can seriously mess up/explode, especially if the time stops being slowed.
Gravity is tricky, but from what I understand it'll not be like the hose. Normally gravity is instrumental to time dilation, so it's difficult to infer what will happen if a time slowing where gravity isn't instrumental. From my limited knowledge it wouldn't be like the hose. The gravity is constant, regardless of the speed of time. So the gravity of the earth will be constant and not increase like the hose. In other words, slowing time down to half will not increase the gravity to twice as much.
Now the character. She/he has to have an aura around him/her, or get a very bad time. Clothes and even the air would imprison them. If the character would try to move without this aura he/she would feel the friction of moving x tines as fast. In addition, they would barely be able to draw a simple breath. If a normal time aura extends outwards they still should keep moving to prevent circulation of the breathed air and getting CO2 poisoning. Fresh air is important.
But moving creates many risks. A door partly inside the personal aura will have the same problems as things from the outside of the time zone trying to get in. He/she opens the door and is likely to break it, as they will apply force like a lever against the door. She/he is applying in a half time zone basically twice the force in a time frame on the part of the door outside the aura. Now imagine items flying around with a tremendous force and coming into the aura, speeding up and striking the person. Not a good time.
As for gravity for the character it seems clear. The character will have the same force applied, regardless of time. Someone might correct me on the gravity part still (and they're welcome to). Yet it is dangerous any other way. Lower gravity can mean moving requires a fully different way of handling, while if the same amount of force from gravity is applied over a shorter time to the people in the time dialation they would experience twice gravity with a half reduction.
But walking on walls is never practical. If you push yourself off the surface, you need to have a way to get down again. In a hamster wheel it's fine and dandy, but if the hamster wheel is sideways so you have no gravity pulling you towards the surface you push off from it'll be near impossible to stay there. Of course they'll be able to do some forms, as you can see people do great gymnastics from (slanted) walls, but moving like Quicksilver is unthinkable. But as the time isn't slowed for the character I see no reason why she/he wouldn't just fall flat on their face.
Conclusion
In my examples I often was referring to slowing down time by half, making the character twice as fast. Quicksilver goes many, many times that, as he can outrun bullets with ease. With increase of the time slowing, the problems get progressively worse. Although above are some major simplifications of the problems, you can assume that slowing down is very dangerous and should be done very carefully, or have some extra magical rules that make it work.
So much of this is dependent on frame of reference and also how the flow of time is being disrupted.
For our current understanding of time there are basically two ways to change the flow of time. But it should be noted that both those ways of changing the flow of time do not change that rate for the subject they only change it for the observer. The two classic examples of time dilation are traveling at near light speeds and falling into a black hole.
For traveling at near light speed say you have a person on a ship and a person on a planet. The person on the ship travels at 90% of light speed going to another planet and returning and is gone for 5 years of planet time, but according to the clock/person near light speed only about 2.1 years have passed. If the observer was able to see the person on the ship they would see them moving very quickly and vice versa, but the person on the ship experiences time as normal. You basically get the same effect for a person near a black hole or other massive body, the deeper a gravity well you are in the faster you appear to move to an outside observer.
The rate at which things fall is determined by the acceleration that they are undergoing. Lets say our spaceship is accelerating at 9.81m/s (earth gravity equivalent acceleration) so if you drop something on the ship it will fall in the direction of acceleration at the same rate things would on earth. But to an outside observer it would appear to fall much more quickly because the Seconds(ship) and the Seconds(observer) are different.
The super fast thing you need to magic real quick because once you start going fast enough you can fuse air molecules and cause fusion which while useful is also a bit hard on the neighborhood.
But if you are only looking to go like 5 or 10x the speed of a normal human (still well within the normal speeds of things we deal with) what you need to constantly be asking yourself is what is within the time dilated area of effect and what is outside of it. Objects falling outside the time dilated area will appear to fall slower than objects within it.
What is breaking my brain is lets say our hero has timespace dilating suit on so their frame of reference moves 10x slower than that of the surrounding environment. If they jumps out of a first floor window I think they the only fall at 9.81m/s to an outside observer because the only thing accelerating them is gravity, but once they are on the ground running their legs can go 10x normal speed(to an outside observer) so presumably 10x faster. So they go at normal speed falling but hyper speed running. But maybe I just need another cup of coffee.
Frames of reference get kind of confusing, I remember from undergrad getting really confused doing the unit analysis because you had to keep track of the seconds differently for each frame of reference because time stop being a constant.
