One day, you suddenly get superpowers! Whoo-hoo, you can speed up or slow down the flow of time for yourself. This means you can run faster than a jet, right? So you decide to test it out. You go out to some deserted spot and start speeding up. You start running, and as you go faster, you start to notice a problem. It's getting really hard to breathe, and harder and harder to run. It's like you're running through water or something. Then you realize that since you're speeding up, the rest of the world seems to be slowing down. And since force is measured in $kg cdot m/s^2$ (also known as Newtons), and you're decreasing the length of a second for everything else, you're increasing the force required to draw air into your lungs and move around by the square of whatever time-distortion factor you are at. So the question is, assuming that the time-field extends a few inches out from your body and gradually ramps up to the full effect right at your skin/the entrance to your mouth/nose, how much can you speed up time without suffocating? And is there any training you can do to improve this speed?

You're dead fairly quickly in this scenario

And it's cool that you've brought this to our attention.

The force required to move air in and out of your lungs isn't the problem. The air in the pocket immediately surrounding your body is in the same time frame as your lungs. Therefore, breathing is quite normal.

The real problem is the exchange of atmosphere through the time differential. How much differential can their be, before the buildup of CO2 inside your time bubble kills you?

And the worst thing about that is while your superhero can train him/her/itself for high-percentage CO2 breathing, such as hiking at high altitudes a lot... your superhero can't do a thing about how slowly the molecules move between the time frames.

Which depends completely on how you define that transition in your story

I've enjoyed fiction that suggests things can move between time streams (e.g., the movie interstellar where time was affected by the gravity of a black hole) and where they can't (e.g. Star Trek NG "Time's Arrow").

Unfortunately, that means asking us how quickly those molecules can move through the differential is Too Story-Based. If the nature of your superpower permits no movement through the differential, even with training (and due to how little air is kept within inches of the body), your superperson has minutes at any time differential before suffering cerebral hypoxia... and only a few minutes after that before dying. If you do permit movement through the differential, then you need to tell us what the equation is defining the movement.

However, there is a bit of comical coolness here

Did I say he'd die? Heh... not really. What'd he'd do is black out, and I assume his superpower would shut off along with his consciousness. Until learning his limits, he'd seem to speed up and then collapse and skid across the pavement, suffering substantial road rash.

JBH has an excellent answer. I thought I'd add a different issue: freezing to death.

You exist in thermal equilibrium with your environment: the amount of heat you export through radiation/convection/conduction is equal to the amount you absorb, and the net allows you to maintain your body temperature. Since you are generating heat internally, your body temperature is greater than your surroundings (most of the time).

Let's assume that you are operating at twice "normal" rate. Ignore conduction. What happens? Your surroundings are radiatively cold, since from your point of view they are radiating with half the power they normally do. Generally speaking, objects radiate at a rate proportional to the fourth power of their temperature (where temperature is in degrees Kelvin). Room temperature is about 300 degrees Kelvin. So your surroundings are effectively at 252 degrees K, or about -54 degrees F. Furthermore, convection will not supply much heat for the same reason it doesn't supply much oxygen - especially if you are not moving.

From the point of view of the rest of the world, you have become extremely hot, and will continue to be so while you freeze solid.

Short answer: suffocation won't be an issue

Even without breath, your hero can do a lot. Freediving, where people don't breathe at all, have records involving considerable physical activity of around 20+ minutes (current record for simply holding breath in a pool is over 24 mins). Everyone can learn to handle longer breath holding, to some degree, and your hero has a really good motive.

Hypoxia and CO2 toxicity would only arise somewhat gradually and probably not be too harmful (accidents and bad luck aside) because any moderate-to-severe effect would, as a side effect, presumably remove the problem.

They also have many ways to mitigate the issue of suffocation. It would be quite simple to make some kind of slim body-shaped CO2 scrubber/rebreather unit, and/or also an oxygen supply/oxygen concentrator unit if needed, that fits in the effect space, which would provide long term breathing help.

But the maths of durability under "ordinary" breathing is fun. So let's have a go.....

Using Maths!

Focussing just on suffocation (not heat, cold, momentum, etc):

Unbreathed air comprises ~79% nitrogen, 21% oxygen, 0.03% CO2. Exhaled air has the same nitrogen but closer to 16% oxygen/5% CO2. An adult at rest breathes around 10000 - 15000 L of air per hour (1 2). CO2 becomes uncomfortable, then disabling, then toxic, at lowish concentrations however, and that's regardless of the oxygen level in the air. This page suggests that

• 0.1% CO2 = headache


• 1% CO2 = hot clammy fatigue concentration + "jelly legs"


• 2% CO2 = 50% faster breathing (roughly one breath every 2 secs not 3 secs, at rest), headache after some hours, tired.


• 3% CO2 = breathing doubles (panting), severe headache, dizzy, visual and hearing disturbances (sparks, low night vision), blood pressure up. "Extremely sluggish but not usually fatal"


• 4-5% CO2 = "immediately dangerous", in addition to above, 4x normal breathing, choking/"unable to breathe" feeling, unconscious <30 mins, extended exposure = possible permanent effects and risk of death.


• 5%+ CO2 = additionally: tinnitus, confusion, panting, impaired vision


• 10% CO2 - unconsciousness/death in minutes.

Your superhero will probably respond to these in a self limiting way - the less functional they are, the more likely it is they would drop superpower engagement due to distress or at worst, unconsciousness).

Oxygen deficient air (hypoxia) is also dangerous. This page suggests that:

• 15-19% affects thinking, coordination and judgement


• 12-15% causes poor coordination/judgement, fatigue on exertion, and emotional upset


• 10-12% causes "very poor" coordination/judgement, nausea/vomiting, possibly unconsciousness within minutes, impaired respiration, possible cardiac damage.


• <10% almost immediate unconsciousness, physical torpor, convulsions, death.

We can assume that suffocation involves 2 issues - whichever hits sooner, out of hypoxia (lack of oxygen) and CO2 toxicity.

We also need an idea of the volume of air and diffusion rate. Those are very handwavey, but let's suppose the effect extends about 6-8 inches (15-20cm).

• Humans have ~ 1.5 - 2 sq.m. of skin (Wikipedia) so the person has an air volume of ~ 225-400 L carried by the effect. Suppose that fading if the effect with distance means that they only get effective use of 60% of this volume, they effectively have an air space of 135-240 L of usable air. We'll also ignore diffusion around the body shape, and assume the air close to them can mix nicely.


• Breathing produces an equal volume but with 5% CO2 instead of negligible, so they produce about 5% x (10k - 15k) litres of CO2 in an hour, or about 500 - 750 L/hr, or 8.3 - 12.5 L/min. They use up oxygen at about 2.1k - 4.1k L/hr (10-15k breathed x 21% O2), or about 35At those rates, CO2 is likely to be by far the more serious problem.


• The CO2 leaches out and O2 leach in, at some rate - I'm not going to do the differential equations for partial gas pressures in a handwaved physics scenario, instead I'll ignore this for now, and see what time scale we get initially, without diffusion (worst case).


• I'm also going to assume they are relaxed and conserve energy (they know to do a bit, breathing slowly, and repeat once they catch breath, is better than doing a lot at once). So they use oxygen at near-resting rates.


• I'm also assuming they don't forcibly project exhaled air away, in order to obtain greater rates of fresh air.

On those assumptions, and simplifying a lot, it should be easy to graph how their oxygen and CO2 go, and roughly when it becomes difficult/dangerous because of either of these.

Problem..

But it's not that easy, I'm going to have to go think hard, first, about how to reconcile 2 wildly different figures:

• several sites say a person breathes 10k-15k air per hour


• but we also have about 20 breathes a minute at rest = 1200 breathes per hour, with a tidal volume of ~ 1/3 L, suggesting a tidal intake of 400L/hour. (In addition to 2.5-3 L residual volume).

Once I figure out what that's about, I'll try to finish this. But for now, this should still be useful enough to add anyhow.

Mitigation/breathing aids

The kicker is, they could mitigate both suffocation issues pretty easily. The effect extends a few inches, and that's plenty of space to fit a custom-made slim oxygen supply, oxygen concentrator unit, or CO2 rebreather/scrubber unit, if they needed to. Those things can be quite small, and can be designed flat, to fit against the skin within the effect's "few inches".

So after the first couple of unpleasant incidents, which are survivable bad luck/accidents aside), your hero learns they need this, and develops it or has it custom made, tests it, refines it, and then laughs happily next time.

Assuming that you don't suffocate or freeze to death...

You would be extremely loud, to the point of causing permanent damage to the hearing of everyone around you, breaking fragile items like glass and computers, and causing structural damage to any building that you enter. Maybe even killing everyone you walk near.

The surface of your time bubble will act as a sort of reversed event horizon for sound, where sound can get out, but it can't get in.

Every breath you take will make quiet noises that spread out at the speed of sound right to the edge of the time bubble, right to its own event horizon.

Every time you brush your clothes. Every footstep. Every heartbeat.

Each sound will add to the amplitude of sound that is already at the edge of your time bubble, just building up, compounding, and, while no time will have passed for our observers outside, from our hero's perspective, that sound is just waiting for us to release control of time, to have a moment of inattention, then BOOM, there's an explosion as the pure pressure, that behaves just like detonating a high explosive, complete with the overpressure damage to everything nearby.