stopping time

When she woke up on the couch I tried to absentmindedly look the other way. She was doing that slow stretch thing that made me think of cats in windowsills.

Hey, how long have I been sleeping? Sleepy smile, slightly embarrassed. Looking around for a clock or some hint from the sunlight.

"I'm not sure" -and I wasn't. I had not checked for a while and had been sitting on the other side of the room, half-awake, watching her sleep. I was thinking about where her dreams were taking her. Her breathing was slow and her eyelids moved in a soft motion
up,down

No, really, what time is it? , standing up now, trying to get her bearings. I forgot my watch so I kinda need to know.
"Really, I don't really know, I turned all the clocks off, the phone too."

Are the doors locked? Trying to look half serious, half kidding.
"Oh of course, I am just trying to make the day last longer. If you have to leave, that's all right."
She walked over slowly, sly smile and wraps arms around me and puts her head on my shoulder, yawns and blinks.
She is whispering in my ear:
Are you trying to make our time last longer?
"Of course" I say to her back, "Of course"
Well that's sweet, but I have to go.

I am not the first person to try this with her, I'm sure. The difference is that it worked for me. Time did stand still.

All you have to do is accelerate yourself to relativistic speeds. As you approach c, time in other reference frames will slow and eventually stop.

Oddly enough, to other people, your time will be the one that slows down. Go figure.

(probability theory):
Given a sequence of random variables X₁, X₂, ... (the generalisation to a sequence of sigma fields is obvious if you care about that sort of thing, and irrelevant if you don't know or care), a stopping time is a random variable T for which the events {T=t} (for any fixed t) depend only on the values of X₁, ..., X_t-1.

A stopping time is just that: think that you participate in a sequence of trials, the result of trial #i being X_i. You may stop the sequence at any time. Then obviously your decision to stop at time t depends only on the first t values.

Examples:

1. Stop a sequence of coin tosses at the second head.
2. Play a slot machine until either you lose all $100 you started off with, or you win $1000000.
3. Stop a sequence of die rolls as soon as more than half the rolls came up 6.
4. Employ a monkey to bang on a typewriter until it produces the complete text of Hamlet 2: The Prince is BACK.

Nonexamples:

1. Repeatedly bet double or quits on a fair coin toss stopping just before you lose.
2. Employ a monkey to bang on a typewriter until 30 minutes before it produces the complete text of Hamlet, then invite all your neighbours to see your educated monkey (bet on it producing Hamlet within 30 minutes).

As you can see, a stopping time is feasible, whereas other rules might not be. And, in fact, various theorems about betting are true for a fixed number of trials or for a stopping time (a fixed number is a stopping time!), but not for just any rule. Wald's theorem is probably the most important elementary result for stopping times.

Thanks to the wonderful node by StopTheViolins on the electromagnetic spectrum, a thought came to my mind about those various plot/scifi cliches where the whole plot revolves around a device's ability to stop time.

One would initially think that to stop time, you would stop time for the entire universe. Not so. One could generate a pocket in which all time would cease. But what is the ceasing? Shows represent it in a cessation of all movement. But at what level does the movement cease? Macro? Micro? Nano? Subatomic? In most cases, this does not appear to affect light, but this is where they are (most likely) wrong.

Now the electromagnetic spectrum comes into play. In all the shows with the bad plots, when time freezes, the people with the device are immune to the effects. They can see everything. One would assume that they are in a micropocket of unaffected space. This is all well and good, but now, assuming this does, in fact, affect the electromagnetic spectrum, light itself would be stopped before reaching the pocket. What would one see when looking out of the pocket upon the world?

Nothing.

At least, nothing in the visible sense. Assuming you stopped time totally, light would not be able to reach your eyes until you passed through the space in your pocket, at which point it would pass through, bounce off you, and exit, immediately being frozen again (you probably would leave condensed trails of light as you moved). For things to remain visible, you would have to slow time relative to the EM spectrum. You could slow down time enough for light, at its peak visible frequency, to about 810 THz. Everything would appear as either black, or dark red. As you moved, things behind you would appear even darker, while what is in front of you would be lighter. The doppler effect, first hand, as your living room does a red-shift in front of you.

Freaky.

Now assuming the genius with the device was actually SMART, he would bring with him (or her) a pair of infrared goggles. This allows you to slow time down enough that the lower edge of the infrared range is visible. Sans goggles, everything would be black, because light, now being down to 300GHz, would no longer register in the eye. But with your goggles, which converts the now slow light back to visible light, allows you to see.

Below this wonderful range would take special equipment, after a while eventually requiring a TV (modified, of course), to maybe navigating with a radio, as the frequency of light gets lower and lower, as time gets slower and slower relative to you, to the point that you stop time. At which point light stops moving, and you are stuck in the dark.