## Miscellaneous Science Questions

For the discussion of the sciences. Physics problems, chemistry equations, biology weirdness, it all goes here.

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Pfhorrest
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### Re: Miscellaneous Science Questions

I wasn't sure if space-filling curves actually did what I thought they might do. Thank you for clarifying that they in fact do. (Although I'm still not 100% clear on how exactly they manage that).
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Tub
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### Re: Miscellaneous Science Questions

Hang on.. the discrete iterations of space-filling curves may be bijective, but the limits aren't. Space-filling curves are necessarily self-intersecting.

You can use them to prove that | R² | <= | R |, and it's easy to prove that | R | <= | R² |, but that's just an existence proof. It's not constructive.

If you want a bijection, it can't be continuous (i.e. not a curve).

Bitwise de-interleaving is not a bijection, either, unless you resolve the whole 0.999... = 1.0 ambiguity.

Eebster the Great
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### Re: Miscellaneous Science Questions

That's true, it suffices to prove there is a surjection, but it doesn't explicitly construct a bijection.

Flumble
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### Re: Miscellaneous Science Questions

Tub wrote:Bitwise de-interleaving is not a bijection, either, unless you resolve the whole 0.999... = 1.0 ambiguity.

Oh, bollocks.
Although, does that ambiguity exist since it holds for the components (0.999...,0.999...)=(1,1)=(1,0.999...)=(0.999...,1) too?

Tub
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### Re: Miscellaneous Science Questions

Although, does that ambiguity exist since it holds for the components (0.999...,0.999...)=(1,1)=(1,0.999...)=(0.999...,1) too?

Go ahead and interleave these pairs. If interleaving is a bijection, all four of them must yield the same number.

There is only a countably infinite amount of numbers with two representations, and the whole thing is fixable, but it needs to be fixed if you want your bijection. And after the fix, your bijection will be a lot less beautiful and a lot less simple to use for further proofs.

measure
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### Re: Miscellaneous Science Questions

Tub wrote:There is only a countably infinite amount of numbers with two representations, and the whole thing is fixable, but it needs to be fixed if you want your bijection. And after the fix, your bijection will be a lot less beautiful and a lot less simple to use for further proofs.

Can't you just start by replacing all the terminating decimal representations with the non-terminating equivalents? It shouldn't matter going the other way since both representations correspond to the same real number. All four of the (1,1) pairs would yield 0.9999..... -> 1.

Eebster the Great
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### Re: Miscellaneous Science Questions

Stack exchange has a great answer. Interleaving digits works in the following way:

First you need to exclude terminating representations, so ½ = 0.4999..., not ½ = 0.5000.... Next, instead of interleaving individual digits, you interleave chunks of digits. For every number in R, you separate it into chunks which start with at least zero zeroes and end with a nonzero digit. For instance, the number .1020030004... becomes 1 02 003 0004.... Since we are excluding numbers that end with infinite sequences of zeroes, this is always well-defined. Now, interleaving the chunks of two numbers gives a unique real number. For instance, (½,⅓) maps to 0.4393939... = 29/66, and (1/11,1/13) maps to .0907096099092093... = 907096099092093/9999999999999999. This can map (0,1]2 to (0,1] bijectively. Then you need bijections from (0,1] to R and (0,1]2 to R2, but these are trivial.

The post also mentions that Cantor instead used continued fraction representations to construct a bijection.

Xanthir
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### Re: Miscellaneous Science Questions

Oh yeah, space-filling gives you a bijection, but the nice thing about infinite sets is that you don't need a bijection, just two injections. Mapping the reals to the complex is done trivially via the function `f(x) = x` ^_^
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tomandlu
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### Re: Miscellaneous Science Questions

Yet another relativity question...

At the centre of the Earth, you would be in a state similar to freefall*, yet at the bottom of a gravity well. Does the freefall bit have any impact on relativity or just the gravity well?

* well, identical, provided you mean the sun.
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doogly
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### Re: Miscellaneous Science Questions

yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."
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### Re: Miscellaneous Science Questions

The relativity bit is that being in a box in place at the bottom of a gravity well feels just like being in a box rapidly accelerating towards the Earth's surface.
The thing about recursion problems is that they tend to contain other recursion problems.

tomandlu
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### Re: Miscellaneous Science Questions

doogly wrote:yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."

I think I'm being stupid. Thinking about it, and afaik, freefall has no impact on the effect of a gravitational body with regard to time. e.g. being in freefall towards a black hole isn't going to help you... I guess I had this vague notion that the gravity well might be more of a W shape, rather than V, iyswim.

(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).
How can I think my way out of the problem when the problem is the way I think?

Pfhorrest
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### Re: Miscellaneous Science Questions

tomandlu wrote:(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).

Objects in orbit are in freefall, and objects in orbits at different heights experience time differently, and an object floating in the middle of a hollow at the center of the Earth is essentially just in the lowest possible orbit, and so will experience time differently than any other orbit, or the surface.
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p1t1o
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### Re: Miscellaneous Science Questions

tomandlu wrote:
doogly wrote:yeah, if instead of being in freefall, you were firing a jetpack, it'd have some additional energy contribution. Is that what you're thinking of? I'm not sure what you mean exactly by "impact on relativity."

I think I'm being stupid. Thinking about it, and afaik, freefall has no impact on the effect of a gravitational body with regard to time. e.g. being in freefall towards a black hole isn't going to help you... I guess I had this vague notion that the gravity well might be more of a W shape, rather than V, iyswim.

(edit) to put the question as simply as possible, does time at the centre of the Earth move at a faster or slower rate than time on the surface of the Earth? (I assume the latter).

I believe there is something called "gravitational time dilation":

https://en.wikipedia.org/wiki/Gravitati ... e_dilation

So yes, I think that merely by being in a gravity well, your time will flow differently, regardless of any thrust, acceleration or freefall.

This will be in-addition-to the difference in time dilation caused by relative velocity.

The stronger the gravitational potential, the slower time moves.

Eebster the Great
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### Re: Miscellaneous Science Questions

Wikipedia claims Earth's core is 2.5 years younger than its surface, which gives a time dilation of 5.5×10-10. So it's real, but it's not exactly significant. The difference between the surface of the Earth and the high orbit of GPS satellites (20,180 km) is about 5.2×10-10, or about 45 μs/day, but since the satellites are moving, there is also time dilation due to special relativity of about -7 μs/day, giving a net 38 μs/day correction required to account for special and general relativity.

The effects of gravitational time dilation really don't become significant until you get close to black holes or neutron stars. If you've seen the movie Interstellar, time dilation is what was going on on Miller's planet, and is why most of the cast aged more slowly than Romilly in a higher orbit. If you actually fall into a black hole, things get a lot more complicated, and you can't really even use the same coordinate system, because those coordinates have a singularity at the event horizon.

Sableagle
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### Re: Miscellaneous Science Questions

Is there an SI unit roughly equivalent to a snowball's chance in Hell?

I'm not sure how to answer without using a prefix like "pico" and I'm not sure how to answer with a prefix like "pico."
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cyanyoshi
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### Re: Miscellaneous Science Questions

Sableagle wrote:Is there an SI unit roughly equivalent to a snowball's chance in Hell?

FemtoSnowball.png

I'm not sure how to answer without using a prefix like "pico" and I'm not sure how to answer with a prefix like "pico."

You could always compare the probability to some of the examples here.

gmalivuk
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### Re: Miscellaneous Science Questions

You could always use negative (deci)bels, but then the absolute value of the number conveys its size rather than any prefix.
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### Re: Miscellaneous Science Questions

"It's likely in the same sense that a Minus Forty on the Richter Scale is powerful."
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p1t1o
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### Re: Miscellaneous Science Questions

The monkeys and typewriters one is an absolute classic

Eebster the Great
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### Re: Miscellaneous Science Questions

Unfortunately, monkeys don't produce particularly random characters on their first try at a typewriter, so the actual probability is likely far lower. (Source: six Sulawesi crested macaques' first try at a computer keyboard.)

Flumble
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### Re: Miscellaneous Science Questions

Can radiation break a torch/flashlight and, if so, how? (It happens in the Chernobyl miniseries and is based on accounts of those people according to the commentary.) Note that these are 80's battery-powered torches.

Pfhorrest
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### Re: Miscellaneous Science Questions

Are modern flashlights no longer battery-powered!?
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KittenKaboodle
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### Re: Miscellaneous Science Questions

I'd imagine a similar mechanism to that which "breaks" people, breaking apart molecules and changing certain chemical reactions etc.., possibly in extreme cases just direct heating which isn't good for batteries. I certainly wouldn't want to observe it myself, and considering the relative fragility of biological systems I'd be skeptical of any "first hand" reports unless made via a Ouija board. Though if you saw it on TV it must be true.

I read they had problems with the robots examining the Fukushima reactors, and while a robot and video cameras are probably much more delicate than a simple battery and incandescent lamp, if the radiation is intense enough I imagine it could damage anything.

Thesh
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### Re: Miscellaneous Science Questions

My first guess is the flashlights didn't get regular battery changes, and so dead batteries were a common occurrence that people didn't notice as much when there wasn't a disaster. I can't find anything on alkaline batteries, but I see an 8.4% loss in capacity for lithium ion after 2.7 million rads (which is fatal for most humans).

https://www.sciencedirect.com/science/a ... 3X1401088X
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Eebster the Great
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### Re: Miscellaneous Science Questions

2.7 Mrad is thousands of times a typical fatal dose of X-radiation. I also have a hard time believing the X- and gamma rays released by the Chernobyl meltdown could have damaged batteries significantly.

A post on Reddit claims the batteries were discharged (not damaged) by the ionizing radiation. I also don't see how that's possible.

p1t1o
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### Re: Miscellaneous Science Questions

Do you guys remember batteries in the 80's? A common-or-garden torch would last about 3 minutes before the light started to go dim, and no recharging, you had to crack open another fresh pack. Most likely explanation is that they were just run down and with the confusion and adrenaline etc.

Its hard to see any mechanism by which nuclear radiation at the levels in question could significantly disrupt a battery powered torch.

I would assume that the lethal dose of radiation for something as simple as a chemical battery and filament lamp, is way waaaaaaay higher than that required to disrupt the delicate and complex biochemistry of a human. If the humans are still standing, so are the torches.

Eebster the Great
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### Re: Miscellaneous Science Questions

I don't remember batteries in the 80s, though the basic technology was the same. On the one hand, it sort of figures that they would have improved over the past 30-40 years, but on the other hand, I don't know what actually changed to increase capacity (or longevity in storage). It's not like lithium ion batteries, where many obvious improvements in density have been made.

ijuin
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### Re: Miscellaneous Science Questions

The greater change in torches/flashlights over the past couple of decades has been the switch from incandescent bulbs to LED lights, resulting in a decrease in power requirement of more than 80%.

That said, as several people in this thread have observed, any radiation that would damage such a simple and robust device as any halfway-decent battery-powered light would be more than enough to kill humans who are not protected with shielding better than any protective suits yet invented.

p1t1o
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### Re: Miscellaneous Science Questions

I think there were numerous incremental developments in battery design over the years, batteries today, even non-Li ones, are much better than they used to be.

It is possible that it is due to reductions in manufacturing cost meaning that batteries may use the same tech, but are higher quality for a lower price (you ever had the misfortune of buying cheap, shitty batteries that last for about 30 seconds?) But I would wager that there have been improvements in design too.

cyanyoshi
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### Re: Miscellaneous Science Questions

Since gravitational waves have energy, does this mean that they exert their own gravity too? I have seen their amplitudes measured by strain (the relative change in lengths), but... is that really all there is to them? I'm not quite sure how to wrap my head around the relationship between the description of gravitational waves as ripples in space-time, and the "force" of gravity coming from the geometry of said space-time.

Also, could you in theory make a black hole by hurling a bunch of gravitational waves at one point in space?

Eebster the Great
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### Re: Miscellaneous Science Questions

I think the answer to all of those questions is "yes." In particular, g waves will follow null geodesics, so they will appear to fall towards a gravitational well just like light or anything else. So Newton's third law would appear to require that they also exert their own gravitational "force." There's probably a better way to express this without invoking Newton's third law though, since it doesn't strictly speaking apply here. I'm not completely certain.

PM 2Ring
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### Re: Miscellaneous Science Questions

Yes, the energy of gravitational waves creates gravity. This makes them non-linear, so calculations involving extremely intense gravitational waves are hard. But fortunately, most gravitational waves don't carry much energy, so we only need to worry about this for things like colliding black holes, or the final stages of binary neutron stars spiraling into each other.

On Physics Stack Exchange, Professor Michael Seifert calculates that the power of the gravitational waves emitted by the Earth revolving around the Sun is roughly 196 watts. At that power level it takes around 14,500 years to emit energy equivalent to 1 gram of matter, and 1 gram doesn't have much of a gravitational field.

As for concentrating a bunch of gravitational waves so as to make a black hole, although it's theoretically possible, I don't know what physical arrangement could do that. Maybe you could do it by putting a large number of pairs of rotating black holes into a small enough region.

doogly
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### Re: Miscellaneous Science Questions

PM 2Ring wrote:Yes, the energy of gravitational waves creates gravity. This makes them non-linear,

Oh huh, I would phrase it the opposite -- gravitational waves are linear perturbations of a metric, and full GR is nonlinear, but if you're in a regime where you need the higher order terms you are no longer working with a gravitational wave.
Definitely carry energy though, which is cool.
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p1t1o
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### Re: Miscellaneous Science Questions

I have some gravity wave questions:

Are gravity waves expected to show wave/particle duality?
Is there an associated particle with gravity waves?
Is the graviton obsolete or just unverified or...?
Is a gravity wave obstructed, absorbed, reflected or refracted by anything?
Which of these question are non-sensical?

And over-the-top XKCD question:

Is a gravity wave "laser" (Gwaser?) plausible?

Probably not by using "stimulation by emitted radiation" but any kind of intense, coherent, highly directional, low-divergent beam?

PS: If "Gwaser" takes off as a scientific term, you heard it here first.
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### Re: Miscellaneous Science Questions

p1t1o wrote:Is there an associated particle with gravity waves?
Yes, the graviton
Is a gravity wave obstructed, absorbed, reflected or refracted by anything?
They are absorbed by everything, but bight things don't absorb them much. Black holes are black, things close to black holes are dark. I don't know about reflection.
Are gravity waves expected to show wave/particle duality?
Yes, but they're very hard to detect and create; so basically we never expect to see it directly.

If we wanted to recreate the experiments that showed duality with photons and electrons we'd have two problems.

1) We can't create or reposition significant sources of gravitation waves. We have to work with the randomly placed sources.

2) We can create 1 mm thick objects to detect/ absorb/reflect every photon or electron (within certain energy ranges). The vast majority of gravitation don't have a strong interaction with any device we build. Even if we build a device the size of Jupiter, we would still we only catching a tiny potion.

When we combine those, we basically can't do controlled experiments with gravitons.
Is the graviton obsolete or just unverified or...?
We're pretty confident something exists worth calling a "graviton". The specifics depend on quantum gravity; which is to say, we don't know the specifics yet. AFAIK, all major theories allow for gravitons.
Is a gravity wave "laser" (Gwaser?) plausible?
The energy beam is perfectly fine. The device to create the energy beam is not and woul probably negative masses or some weird thing.
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PM 2Ring
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### Re: Miscellaneous Science Questions

doogly wrote:
PM 2Ring wrote:Yes, the energy of gravitational waves creates gravity. This makes them non-linear,

Oh huh, I would phrase it the opposite -- gravitational waves are linear perturbations of a metric, and full GR is nonlinear, but if you're in a regime where you need the higher order terms you are no longer working with a gravitational wave.
Definitely carry energy though, which is cool.

I'm more than happy to defer to your expertise in this topic.

Eebster the Great
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### Re: Miscellaneous Science Questions

It's gravitational waves. Gravity waves are just waves driven by gravity, like surface waves in the ocean.

Is a gravity wave obstructed, absorbed, reflected or refracted by anything?
They are absorbed by everything, but bight things don't absorb them much. Black holes are black, things close to black holes are dark. I don't know about reflection.

Gravitational waves can be reflected to an extent in the sense that a gravitational wave that is absorbed by a massive object will cause it to move back and forth, thus generating its own gravitational waves, some of them travelling in the opposite direction.

I don't know what luminosity has to do with anything.

doogly wrote:
PM 2Ring wrote:Yes, the energy of gravitational waves creates gravity. This makes them non-linear,

Oh huh, I would phrase it the opposite -- gravitational waves are linear perturbations of a metric, and full GR is nonlinear, but if you're in a regime where you need the higher order terms you are no longer working with a gravitational wave.
Definitely carry energy though, which is cool.

It's not really the same thing though. By comparison with EM for instance, Maxwell's equations actually are linear, so electromagnetic waves are fundamentally linear, as is every other electric and magnetic interaction. With gravity that's not the case, and especially when you get colliding gravitational waves, there is no way to model it linearly.

Pfhorrest
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### Re: Miscellaneous Science Questions

Eebster the Great wrote:
Is a gravity wave obstructed, absorbed, reflected or refracted by anything?
They are absorbed by everything, but bight things don't absorb them much. Black holes are black, things close to black holes are dark. I don't know about reflection.

I don't know what luminosity has to do with anything.

I took the luminosity to be metaphorical. Being "black to gravitational waves" means absorbing them completely, like normal blackness means absorbing light completely. Being "dark" means absorbing most of it, but not all. Etc.
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