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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?

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    $\begingroup$ Congratulations for posting the question number 20000! $\endgroup$ – L.Dutch - Reinstate Monica Feb 1 at 15:19
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    $\begingroup$ I think this is like asking how thick/viscuoud an atmosphere would still allow you to breath, right? I'm curious to learn that too. $\endgroup$ – Renan Feb 1 at 15:34
  • $\begingroup$ I believe the effect is basically the same as increasing the viscousness of the atmosphere, yes, although I'm not positive, and there are a lot of people a lot smarter than me here who I hope decide to answer and tell me whether I'm right or wrong in that assumption. $\endgroup$ – Gryphon - Reinstate Monica Feb 1 at 15:36
  • $\begingroup$ While i agree the breathing is acutally the more important question, would air resistence not play an equal factor in struggling to move forward? time slowing down would mean that the time for the air to "get out your way" and you run through it would also reduce. theorectically it would becine similar to the effect of traveling closer to the speed of light? and therefore the obligatory XKCD: what-if.xkcd.com/1 $\endgroup$ – Blade Wraith Feb 1 at 15:45
  • $\begingroup$ @BladeWraith That's an excellent point, but this question is about the maximum time distortion that can be undergone without suffocating. I may, in the future, decide to ask another question about how quickly one could move forward under that much distortion. Trying to combine the two would most likely make the question too broad. Also, the problem wouldn't be anywhere near the magnitude of the relativistic baseball, although the cause would be similar. $\endgroup$ – Gryphon - Reinstate Monica Feb 1 at 15:47
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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 there 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.

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    $\begingroup$ I like that this also applies a really neat limitation to the superhero in question, in that he or she can only go for so long at these super speeds before having to slow down again. $\endgroup$ – MrSpudtastic Feb 1 at 17:02
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    $\begingroup$ Fortunately though your body can detect elevated CO2 levels in the air (it's basically our anti-suffocation reflex and is what triggers the need to breath when you hold your breath), so you would actually realize that you are suffocating as it happens. Not an expert here, but I'm pretty sure the rising CO2 levels would make you feel like you can't catch your breath even though you are actually breathing, so you should have plenty of warning before you pass out. $\endgroup$ – conman Feb 1 at 19:14
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    $\begingroup$ i believe this would only be a problem whe you're standing still. when you're moving, your time aura affects different air around you all the time, you aren't dragging your air bubble wih you. $\endgroup$ – ths Feb 1 at 19:19
  • $\begingroup$ @ths, but that's the problem - "you aren't dragging your air bubble with you." That's a statement only the OP can make with certainty. The rest of us are making assumptions. Makes for a challenging question, though, doesn't it? $\endgroup$ – JBH Feb 1 at 19:51
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    $\begingroup$ @JBH No, it is not clear (for me, anyway). IMO, it would be better: "I am assuming 100% opacity (or close) in the time bubble in this entire answer", and I guess it would be more clear to do so right in the beginning. In that paragraph, you just told, "oh.. there are some fields (in science fiction) which pass, and others which don't", but didn't you stated which one your answer was about -- that kept me assuming 0%. Well, and, IMHO, it would make much more sense if it were 0% opacity (or close). Unfortunately, no answer explored that.. I am thinking in writing one :D. $\endgroup$ – Physicist137 Feb 8 at 21:38
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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.

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    $\begingroup$ +1! This is hilarious. Thermal problems didn't even cross my mind. And running makes it all so much worse! $\endgroup$ – JBH Feb 1 at 17:25
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    $\begingroup$ @JBH - And, by the same token, the speed-up effects can be notable. In "The Long Arm of Gil Hamilton", Larry Niven explores the effect of exactly such a time bubble. The inventor uses a (admittedly powerful) flashlight as a death ray. $\endgroup$ – WhatRoughBeast Feb 1 at 17:34
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    $\begingroup$ I think you would actually have the opposite effect. Per @JBH's answer, you've effectively isolated a small bit of atmosphere from the rest of the world. Yes, this would result in net radiative cooling, but I believe that will be an inefficient process. Instead, your body will continue to heat the air trapped with you via standard convective heating much faster than it can cool radiatively. Basically, you're a heat lamp under a blanket. I don't know how fast you would actually heat up the air around you but especially while running it might be quite fast. $\endgroup$ – conman Feb 1 at 18:34
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    $\begingroup$ @conman makes an interesting point. I'd like to differ. The air temperature is also effectively lowered - but this relative to the square of velocity of the air molecules. I might be somewhat wrong there, but if I'm reading this (pages.mtu.edu/~suits/SpeedofSound.html) right, the air temperature would be about 75K (-200 C or -325 F) (this is assuming 2x speed). That's pretty much instant frostbite, at best. $\endgroup$ – Spitemaster Feb 1 at 18:48
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    $\begingroup$ @Spitemaster from the sounds of it there is air near him that acts normally, and therefore would be "regular" temperature. The air outside that would act like cold air (for the reasons everyone is mentioning) but would also have a decreased level of interaction with the "normal" air around him. As a result I think the air outside would act like it is colder but also act like it is at lower density/pressure, and therefore it would cool ineffectively. $\endgroup$ – conman Feb 1 at 18:57
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Short answer: suffocation won't be an issue

Even without breathing, 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 one's 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.

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  • $\begingroup$ I suspect that the 10-15K liters per hour should be liters per day. The tidal volume (1/3) seems low, but the breaths per hour seems high. $\endgroup$ – Sherwood Botsford Feb 6 at 14:25
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The following is a very long answer but it'll be worth it, if only because it comes to the opposite conclusion compared to the question and all answers so far.

To get a better idea of what's going to happen we are going to go throught the process of the trial-run.

To start off, you've realized that your time-field is basically the reverse of time-dilation that happens near black-holes or when you are moving at tremendous speed but without the negative gravity or kinetic energy required to get that time acceleration for yourself. A time dilation field would simulate something similar to what you are experiencing: The time outside the dilation field is accelerated compared to the time inside the dilation field and we know that matter is free to move inbetween those time barriers without problem. So you know the air will still diffuse into your field.

Now you've moved to the desert and want to start slow by just standing still while you activate your field, you never know what's going to happen right? So you activate your time acceleration field, accelerate yourself to twice the time and wait.

The first thing you notice is that you are getting warmer. While you receive less radiative heat from your surroundings and still generating the same radiative heat per time unit, you are also producing more heat and are somewhat insulated from the outside world. Radiative heat is one of the least efficient ways to lose heat (which is why space is such a good insulater despite it's cold, as you can mostly lose heat through radiation), so the netto gain of being insulated from the outside world makes you warmer for quite some time, but it's going to get colder if you wait long enough.

Compared to you the air particles at your skin don't accelerate at all, but compared to the outside world they'll be going twice as fast. They act as though they have twice the heat energy but without the actual kinetic energy! To get this into equilibrium, the air particles next to your skin will need to slow down to the speed of the particles outside of your field.

https://en.wikipedia.org/wiki/Kinetic_energy

The kinetic energy of the air particles next to your skin will drop. E=0,5*m*V^2. Regardless of the mass, any 50% loss of speed means a 25% of energy loss, so the air around you is about to get a lot colder! What's worse is that this is just a 2x time acceleration, at any multiple acceleration higher this difference will exponentially grow! But it's not going to be instant. If the air around you was actually twice as hot as the surroundings, it would simply diffuse away slowly. So it takes a while before you start feeling colder, especially since you are now generating more heat and losing less. The way your field acts also protects you: The air particles that bump into eachother will have barely any time difference even at the edges of your time-field. While the air particle that is faster in time is going to lose more energy to the other particle in the reaction, it won't be a lot more than it would normally as that difference is going to be so small when they touch.

But as you stand there in your slowly colder and colder environment you notice something worse: The air pressure is dropping around you. The time barrier makes it harder for your air particles to escape as they push against the "slowed" air particles that take a lot more energy to budge. Still, since your particles move faster compared to the rest you'll have more air particles "escape" your time accelerated zone and less return to it. As the pressure drops at higher time accelerations it'll keep dropping untill you might suffer from hypoxia. For a moment you panic and move your head... And sweet fresh normal-pressure air rushes in as you move the time-acceleration field over them. Something the other answers seem to be missing is that as long as you move you'll "push" normal pressure and normal temperature air into your time-field. And that air can easily be breathed in and out. Any CO2 and low-pressure cold air will be left behind your body, so suffocation won't be imminent! Yet.

To prevent hypoxia, you start to move at a leisurely pace while playing with the acceleration. What you immediately notice is that the gravity is still the same, but every movement you make has a lot more force behind it. Previously you might put 10 newton per second into moving a particular limb, but due to the time acceleration it looks to the outside world like you put in 20 newtons per second! Fortunately you won't be jumping a mile and dying on impact when walking as anything within your field acts normally, so you'll be able to walk normally (gravity doesn't change either). Any interaction with the outside world where a portion of the object is outside of your field (say a person you punch) will experience the full effect of your time-accelerated extra newtons per second (besides that the time-acceleration field's boundries will likely wreak havoc on their blood supply and chemical balance).

As you start going faster you notice an air-pressure buildup in front of you, slowing your down. The time-field prevents the air in front of you from getting away easily into the non-accelerated air around it. It does try to flow around you but the normal flow is choked by the boundries of the time-field. This kind of pressure makes you think off a space-ship re-entering atmosphere and heating up and makes you afraid for a moment, but you'll never reach anywhere near enough speed to heat up (https://what-if.xkcd.com/28/ ;) ). In fact, you'll be cooling down as the air that previously insulated you is now flowing against you at accelerated speed. How fast you cool down depends on the temperature of the air, but it's there.

As the air in front of you builds up and pushes against you, you start feeling a "pull" behind you. Normally when you move the air around you fills up the area you left. Unfortunately most of the air is slowed down, and it'll be up to the high-pressure area in front of you to try and fill it. This accellerates the air around your body even more, causing potential choked flow (https://en.wikipedia.org/wiki/Choked_flow). Not pleasant in any way, you are going to get cold!

As you accelerate the air will cool you down and the higher pressure air will start to choke you (https://biology.stackexchange.com/questions/41639/breathing-becomes-harder-in-certain-wind-conditions). As you run faster you risk dying of hyPERoxia (https://en.wikipedia.org/wiki/Hyperoxia), rather than hyPOxia.

The cold is likely going to be the first problem, but it also has a laughably simple solution: Wear a coat. The worst problems happen at your skin due to the time difference, but a coat would cover and insulate the air at the point where the worst time difference is. You could simply wear a backpack with a few coats (and a raincoat, they work wonders against wind), and each time you want to accelerate time faster you stop for a moment, put on a thicker coat and then start running again. This should work wonders for protecting you against the cold and the choked flow.

You'll still die of hyperoxia if you go too fast though. Only with airtanks close to your body would this be solvable, just be sure that all the air will be within the maximum air acceleration zone or the pressure difference in the tank might cause problems for your breathing.

I don't know at what point the cold or the pressure will be a problem. But this is just a start to figure out at what speed you could go at the maximum. Considering some people have survived multi-MACH velocity aircraft breakups and survived (linked in the What-if, they survived through their pressure suits and immediately slowing down so only go this fast for a fraction of a second) you could probably be going pretty damn fast with the right preparation, just not for long.

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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.

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    $\begingroup$ The question specifies a gradual transition between normal time and quick time, so you don't get an event horizon, you just get a Doppler shift. (Even a sharp transition would likely generate a Doppler shift, not a trap.) $\endgroup$ – Mark Feb 2 at 0:09
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One other issue to consider, as the title asks about issues/limits generally (not just suffocation). Would pressure gradients be stable? Meaning, if your air has different physics (different distribution of velocities, and different power inflow/outflow "as seen from the other side", would you have a problem that air in your bubble is more or less energetic simply because of being in that part of.spacetime, and therefore an unstable situation arises in that air tends to constantly leave the space, leading to low pressure/vacuum, or constantly move into it? Could this lead to an equilibrium pressure (or lack of one) that was harmful to your hero?

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You look hot!

Consider: Your normal body temp is 300K has a peak wavelength of 9.7 micron. If you run at 3 times speed, your radiation's wavelength just shrank by a factor of 3, corresponding to 900K, a sullen red glow.

Go up to 5 time speed up and to an outsider, and you're going to be bright yellow.

But the other problem occurs too.

Light from outside will be red shifted. Even at a speedup of 2-3 visible light will be red-shifted into infrared -- you'd have to see by what the rest of us mortals call UV. During the day there is enough UV for you to see, although the world will be dim. It will also have seriously distorted colour. For an idea look at infrared colour film.

At night you would be blind at speedups of over 2. You might have to move like a cockroach, who apparently doesn't have enough brain to see and run at the same time. They sprint. Stop. Sprint. Stop.

Or you could carry a flashlight. It's photons would be shifted to UV going out, and converted by to your visible coming back. Note that this flashlight could do serious damage to people around you. Bright UV sources are not good for people's eyes.

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Let us begin with a time bubble with a given surface $S$, time factor $F$, and opacity $R$.

  • $S$ describes the shape of the surface itself (in which will be required to do integrations). Evidently, $S$ is a closed continuous surface (otherwise nothing makes sense).

  • The time factor $F$ indicates the factor in which time speeds up inside $S$.

  • $R$ is the reflectivity, a dimensionless number between 0 and 1, which indicates the probability of a random molecule hits $S$ and gets reflected back.


Velocity distribution

We'll start with Maxwell-Boltzmann distribution, and compare it inside and outside of $S$. The probability density function is: $$ f(\mathbf v) = \left(\frac{m}{2\pi kT}\right)^{3/2}\exp\left(-\frac{m\mathbf v^2}{2kT}\right) $$

Where $k$ is the Boltzmann constant, $T$ is the temperature of the environment. This gives the probability density of finding a particle with velocity $\mathbf v$. This probability will shift due to the time-field.

For sake of simplicity, let's move to inside the field, and analyse the outside (we could do the opposite, and, we should get the same results). Outside, the time is slowed down by factor $F$.

Meaning, the velocity of everything is slower by a factor $F$. Thus, the probability function outside shifts:

$$ f_F(\mathbf v) = \left(\frac{mF^2}{2\pi kT}\right)^{3/2}\exp\left(-\frac{m\mathbf v^2 F^2}{2kT}\right) $$

To see that's the actual probability, one can calculate the speed mean $\langle v\rangle$ and see that it got slowed down by $F$. Other possibility is $v_{rms}$.

The added $F^2$ inside the exponential near the velocity is explained, but the added $F^2$ near temperature outside the exponential, is solely due to normalization -- we are, after all, talking about a probability density function.

The mean keeps being zero even with $F$, but now the standard deviation of the velocity distribution is different. Notice, it makes no sense the temperature to change, because every single physics phenomenon has slowed down: velocities did so, but also heat transfer, and everything else involving actual movement.

Equilibrium is reached when the number of molecules per unit time leaving the bubble equals the number of molecules per unit time entering the bubble. For $F=1$, this is trivial. For $F\neq 1$, if we're inside the bubble, particles outside will be moving much slower (the shift in the velocity distribution). Initially, less will enter the bubble, and more will leave, causing the density of particles in the bubble to decrease, until equilibrium is reached: until flow is equal. So, we expect the air inside the bubble to become much more thin, then in the outside (for $F>1$). In reverse, if $F<1$, atmosphere inside will have greater pressure than outside. Our objective, is to calculate the molecular density ratio (or the pressure ratio) with respect to $R$ and $F$. :).

We need to calculate the exact number of particles/molecules per unit time which are going from outside to inside (crossing the barrier). Call it, $s_{io}$ That number, is: $$ s_{io} = \frac{dN_{io}}{dt} = \int_S \int_{\mathbf v\cdot\mathbf n \le 0} f(\mathbf v)\mathbf v\cdot \mathbf n dS d\mathbf v $$

Good luck with that integral! You're going to need it.


Atmosphere inside the time bubble

Actually, we already have plenty of luck: the gas is spreaded isotropically, there's no external potential, and all sorts of nice approximations we can make, because it is true.

Let's simplify. Let $n_i$ be the density inside the bubble, and let $n_o$ be the density outside. Same for the pressures $p_i$ and $p_o$. We'll assume the time bubble has a 'cubic' form, which will allow considerable simplification of the surface integral. But, due to isotropy, the result remains valid for complicated shapes. Furthermore, let us be inside the bubble. Let $s_{io}$ the flow from inside to outside (the number of particles per unit time per unit area leaving the bubble). And let $s_{oi}$ be the flow from outside to inside (the number of particles per unit time per unit area entering the bubble). We'll calculate these quantities for $yz$ plane, but, any other face calculations are analogous.

$$ s_{io} = \frac{1}{A}\frac{dN_{io}}{dt} = n_i (1-R) \int_{v_x \ge 0} v_x f(v_x, v_y, v_z) dv_x dv_y dv_z $$

Expanding: $$ s_{io} = n_i (1-R) \left(\frac{m}{2\pi kT}\right)^{3/2} \int_{0}^\infty v_x \exp\left(\frac{mv_x^2}{2kT}\right) dv_x \int_{-\infty}^\infty\exp\left(\frac{mv_y^2}{2kT}\right) dv_y \int_{-\infty}^\infty\exp\left(\frac{mv_z^2}{2kT}\right) dv_z $$

Integrating: $$ s_{io} = n_i (1-R) \left(\frac{m}{2\pi kT}\right)^{3/2} \left[\frac{1}{2}\frac{2 kT}{m}\right] \left[\frac{2\pi kT}{m}\right] = n_i (1-R)\left(\frac{m}{2\pi kT}\right)^{1/2}\frac{kT}{m} $$

Thus: $$ s_{io} = n_i (1 - R)\sqrt{\frac{kT}{2\pi m}} $$

Analogously, we can calculate the other factor the same way: $$ s_{oi} = \frac{1}{A}\frac{dN_{oi}}{dt} = n_o (1-R) \int_{v_x \le 0} v_x f_F(v_x, v_y, v_z) dv_x dv_y dv_z $$

And, we'll arrive at: $$ s_{oi} = n_o (1 - R)\frac{1}{F}\sqrt{\frac{kT}{2\pi m}} $$

Demanding equilibrium, we finally arrive: $$ \frac{s_{io}}{s_{oi}} = \frac{n_i}{n_o} = \frac{1}{F} $$

Using ideal gas law, $P = nkT$, we also get the pressure ratio between outside and inside: $$ \frac{p_i}{p_o} = \frac{1}{F} $$

The pressure inside falls as $1/F$. If pressure outside is 1atm, if $F=10$ times faster in the bubble, then, the pressure inside will be 0.1 atm. Not a good pressure to breath while running..... Notice that the pressure ratio does not depend on the opacity of the time bubble $R$, as the $R$ cancel out.


Time to reach equilibrium

Now, because these are equilibrium equations, there's no clue in how much time it will take to reach equilibrium. That's what we are going to investigate here. But, that is not hard to calculate. For that, we define $N_i$ the amount of molecules inside, and $N_o$ the amount of molecules outside. Due to conservation of particles, if 1 particle left the barrier, then, there exists one less particle inside. We apply it: $$ \frac{dN_i}{dt} = \frac{dN_{oi}}{dt} - \frac{dN_{io}}{dt} $$

Therefore: $$ \frac{V}{A}\frac{dn_i}{dt} = \frac{1}{A}\frac{dN_{oi}}{dt} - \frac{1}{A}\frac{dN_{io}}{dt} = s_{oi} - s_{io} $$

Thus, we are left with the following first order differential equation: $$ \frac{V}{A}\frac{dn_i}{dt} = \frac{n_o}{F}\alpha -n_i\alpha ,\quad\quad \alpha = (1 - R)\sqrt{\frac{kT}{2\pi m}} $$

Fortunately, that is easy to solve. The solution is simply: $$ n_i(t) = n_o \left(1 - \frac{1}{F}\right)\exp\left(-\alpha\frac{A}{V} t\right) + \frac{n_o}{F} $$

We can put the solution in the form: $$ n_i(t) = n_o \left(1 - \frac{1}{F}\right)\exp\left(-\frac{t}{t_c}\right) + \frac{n_0}{F} $$

Where $t_c$ is call the time constant. That time gives us a clue, an intuition, of how long the system takes to relax to equilibrium. That time is: $$ t_c = \frac{V}{A}\frac{1}{\alpha} = \frac{V}{A}\frac{1}{1-R}\sqrt{\frac{2\pi m}{kT}} $$

Notice something important: apparently, the time constant does not depend on the time factor $F$ (hopefully I made no mistakes in the calculations -- feel free to check) (though now that I thought about it, it makes a bit of sense: if two environments were at different pressures, and one opened a wall for equalization, the time constant wouldn't depend on the pressure difference itself, rather, the geometry of everything. In here, equilibrium causes a pressure difference, and, again, time constant only depends on geometry and temperature, not on the pressure difference, and thus, the factor $F$).

The greater the volume of the bubble, the greater the time. The greater the area, the smaller the time it takes. The greater the opacity $R$, the greater the time it takes. The greater the temperature of the environment, the less time it takes. And so on.


A numeric example

Let's plug some numbers! The volume of the human body is approximately $V\approx 0.07 m^3$. The area of the human body is roughly $A\approx 1.7 m^2$. Assume temperature $T=300K$, with atmosphere composition of nitrogen, $m = 2.3258671 \cdot 10^{-26} Kg$. Boltzmann constant $k = 1.38\cdot 10^{-23} J/K$. Let's put no opacity, $R=0$. This gives us a time constant $t_c$ of.... [adds suspense]........ $t_c = 0.00024s$. That is, 0.2 milliseconds. Um... I guess I suggest that your hero never activates the time bubble for too greater $F$, otherwise, in less than a millisecond later, equilibrium will be reached. If $F$ is too high, that could be compared to explosive decompression.

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