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00:00 - 16:0016:00 - 21:00

00:00
@Relativisticcucumber i hated it soooo much
00:16
Why don't we usually include complementary function in solutions to Poisson's equation?
I have seen in Jackson that they would just write the Coulomb solution .$$\phi(t,\vec{r})=\int\frac{\rho\left(\vec{r}',t\right)}{|\vec{r}-\vec{r'}|}dV'$$ with some boundary terms depending on whether it's a Dirichlet or a Neumann BVP.
Yeah, they don't claim it to be the full solution either but I was wondering whether there is some non trivial reason not to include the solution to Laplace equation in the solution to Poisson's equation.
01:09
@Mr.Feynman have u seen breaking bad
02:06
@Sanjana When you write down an integral equation as opposed to the differential equation, the boundary condition/initial value is also stated inside. As such, there would be no more complementary function because it is already part of the integral.
I get the first sentence but not the second one...how are complementary functions also part of the integral?
02:20
@Sanjana By putting the boundary/initial condition inside the integral, you would already fix the complementary part
@naturallyInconsistent Oh...so we are choosing the complementary function in such a way that the boundary condition is obeyed?
03:17
what is the natural metric on $S^n$, the unit sphere?
is it like the euclidean metric?
 
2 hours later…
05:11
It has a homogeneous metric which is the sphere metric
there's no Euclidian metric on the sphere by the Gauss Bonnet theorem
Well, in even dimension, anyway
06:11
Currently trying to figure out who invented spacetime, which is tricky because there are some curveballs
ie if ypu consider Wells' Time Machine
> “Clearly,” the Time Traveller proceeded, “any real body must have extension in four directions: it must have Length, Breadth, Thickness, and—Duration. But through a natural infirmity of the flesh, which I will explain to you in a moment, we incline to overlook this fact.
There are really four dimensions, three which we call the three planes of Space, and a fourth, Time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the begi
Which predates relativity by a bit
I've read the claim that Hinton made some spacetime-like ideas in the 1880's in his book about the 4th dimension, but it is only sort of true
He used time as a tool to help visualize the fourth dimension
07:02
Earliest case seems to have been "A friend of d'Alembert"
fqq
fqq
@Relativisticcucumber makes you wanna Kubo–Martin–Schwinger?
That's pretty advanced
 
2 hours later…
09:04
when deriving the energy stored in magnetic field (defined as the work done to set up a current against the back EMF produced), we consider $ \nabla \cdot J = 0$. Is this approximation valid?
That's the approximation of when the current is steady
Since the real equation of continuity for charge is $\partial_t \rho - \nabla \cdot J = 0$
when im building the current from 0, its not steady right. JDJ says, "we may imagine the building process ocurs at infinitesimal rate, so $\nabla \cdot J=0$"
but it doesnt occur at infinitesmal rate right?
I guess it depends on how fast it is going?
If $\partial_t \rho \ll 1$ then you could use this
in wires i suppose its true?
what about a soup of charges flying in all sorts of directions (plasma?)
I mean if it's at equilibrium, the charge density is likely to be constant
if you're doing non-equilibrium stuff it could vary yeah
I don't know that much about MHD though so idk
09:11
the process of building a current from 0 to some I is not a process happening at equilibrium though, right?
i guess the correct word would be "establishing" a current
I mean it could be in such a way that $\partial_t \rho = 0$
Since in a wire you have basically just a line of positive charges (the ions of the conductor forming the wire) and a line of negative charges (the free electrons) moving along it
The total charge is zero on average
you could presumably just have this type of process going on at various speeds during the establishing current
hmm, that makes sense
what is mhd
I'm not entirely sure how the establishing current works microscopically so I can't guarantee that it happens that way
But it is possible
MHD is magnetohydrodynamics
Study of electrical fluids like plasma
@Slereah I would assume its based on electron electron collisions, modelling it could to be a daunting thing idk.
@Slereah Thats cool. technically, based on the definition, shouldnt plasma physics be the most general study of EnM? its literally a soup of charged particles i dont think it can get any more general than that
Nah, the charges have to obey like fluid mechanics laws, too
hence the hydro
10:10
There was an infamous question about rolling motion of a circle. do you know what it was
i forgot what it was
it wasnt a physics question. it did not involve any forces. it was a math question. i think they told you to find the rolling speed and it was counter intuitive
it is a cursed problem. i think it may be ambiguous
from the PoV of the smaller circle, it does 3 rotations. But from the PoV of the outside observer, it does 4 rotations
so it is ambiguous in asking "how many rotations"
10:26
no it isn't
the second answer there does the correct analysis, this problem is not "ambiguous", it just requires you to think carefully about how the points actually move when the circle rolls
@ACuriousMind it is calculating the number of rotations about O1. The number of rotations about O is still 3. So, "how many times does the circle turn around itself" depends on "whose PoV". From the PoV of the smaller circle, it's 3 rotations. From the PoV of the outside observer, it's 4 rotations
when most people give the "wrong" answer to this question, they are just thinking in the PoV of the smaller circle
nope, there's no ambiguity here, the number of rotations is not relative to anything
it is a fact that the point P travels a total distance of 4 times the circumference of the small circle, i.e. the small circle has to turn 4 times
there's no "POV" from which there are only 3 rotations - that's just our misleading intuition, not something anyone would reasonably observe
@ACuriousMind imagine you are on a planet in isolation in space and you are only aware of the planet's spin. so, you would calculate your number of rotations relative to the planet's center. However, it could be that another far away planet is revolving around the our planet in an orbit. if we switch to the PoV of that planet, then we add an extra rotation to our motion
I think it's easier to think about this thinking about O: First, convince yourself that in order for O to move by a distance D (whether on the circle or on a flat plane), the small circle has to turn $D/(2\pi r)$ times
then, observe that for the small circle to travel around the big circle, O moves on a circle of radius $R+r$, i.e. it travels a distance of $2\pi(R+r)$
10:43
that makes sense. i will think about it
and there's nothing relative about how often the small circle has to turn: There is some apparatus that has to actually make the small circle turn, and that apparatus knows whether it put in the effort to make 3 or 4 revolutions.
@ACuriousMind Look at the second image in the first answer. it shows that the circle has completed a full revolution at the 90 degree quadrant of the bigger circle. imagine yourself rolling the smaller circle over that 90 degree quadrant. the smaller circle's boundary has been painted such that, as it rolls, the paint gets removed from it. Now, at the end of the 90 degree quadrant, would you say all the paint from the circle has been removed?
or would you say all the paint gets removed at the 120 degree mark from the first picture in the same answer? @ACuriousMind
@RyderRude no, but what does that have to do with anything? it just means that as we roll along the big circle, our point of contact with the circle changes in a manner different from what would happen along a flat plane
it means that you would have to re-paint the smaller circle only three times over in order to fully traverse the bigger circle and paint the bigger circle (assuming the paint gets transferred to the bigger circle). This is a measurement apparatus that shows that it's only three rotations @ACuriousMind
no
it means you haven't thought carefully enough about what the paint measures
10:56
it measures rotations from the PoV of the smaller circle
i am sure that the number of rotations is relative. i will try to come up with a mathematics of this relativenwss
what is a "rotation from the PoV of the smaller circle"
sure, if you define a rotation here as everything between "P is in contact with the big circle" and "P is in contact with the big circle again", then there's only 3 rotations
the point is that that's not what a "rotation of the small cricle" is
i will try to come up with a definition. i'm sure it's relative. remember that the distance traveles is itself a relative quantity. your argument depends on the distance traveled
when you switch to the pov of the bigger circle, you are adding an additional velocity to the smaller circle
this results is the extra distance traveled
from its own PoV, this extra distance isnt there as it is not moving relative to itself @ACuriousMind
what
none of my arguments have anything to do with velocity
i explained this in my "isolated planet" experiment. if you switch to the PoV of the orbiting planet, your own planet gains a rotational velocity (as your planet is orbiting wrt thta planet)
Again, It's just geometry - the distance traveled by the center of the small circle is the distance of 4 rotations. In the frame where the center of the small circle is stationary, it's the center of the big circle that travels that distance (but it's the same distance - $2\pi(R+r)$).
11:03
@ACuriousMind you havent mentioned velocity explicity. but the distance covered is dependent on the frame of reference. the extra distance is arising only because you are looking at it from the PoV of the bigger circle at rest
you seem to try to make this into some weird physics argument about different observers but really if you carefully define what "one rotation" means the problem either goes away or you realize you're using two incompatible definitions of "rotation"
@ACuriousMind the distance traveled is relative.
I'm not saying anything more about this
@ACuriousMind but i agree here. the problem works with one particular definition of "rotation".the problem just doesnt specify it
if the problem was like "as you watch the smaller circle go around the bigger circle, how many times does it turn about itself"?, the answer would unambiguously be 4
because it fixes the frame of reference to "us watching"
when people answer 3, they are really doing some version of my paint experiment
11:30
Apparently we have an ancient greek manual for the use of a stelle for astronomical purpose
"Patron set this up for Zeus Epopsios. Winter solstice. Should anyone wish to know: off 'the little pig' and the stele the sun turns."
Where is the little pig
No photo is included
"The little pig was a rock formation way out in the water."
11:47
the reason that the paint experiment measures the number of rotations about the center of the smaller circle, is that the paint can only get exhausted when the point has made a full rotation about the center, as this is when all the points on the circle have been in contact with the surface beneath
12:18
this answer also talks about the two types of revolutions : math.stackexchange.com/a/1356200
12:48
@SillyGoose I would have gone for the natural extension: Start with $\mathbb R^{n+1}$ and convert to polar coördinates, then restrict to the hypermanifold of $r=1$. It might be standard, rather than natural.
13:30
the mathematical version of my logic is that the rotation speed of a spinning object, and hence the distance traveled by a point on it in a particular time, is dependent on the frame of reference.
to see this, we have to look at the situation in the question in reverse
start with a frame of reference where there's a circle whose center is at rest and its rotating at speed $\omega$.
@SillyGoose Define "natural" :P
and now switch to a frame of reference where the center of the same circle is now rotating at speed $\omega _2$
in this frame of reference, the circle rotates at a speed higher than $\omega$ and any point on it travels more distance in the same time compared to the distance traveled in the previous frame
in the question, there are just two natural frames of references and it's ambiguous which one to use.
@RyderRude yes, they certainly do
@user726941 what philosophies have you studied?
the question expects you to use the frame of reference of the person looking at the question
ones around consciousness
13:43
oh. you mean illusionism, dualism, etc
I mean Bohm's contribution to it.
i'm not familiar
I have finally come around to read about green function method for BVPs after avoiding it for quite long. It's pretty cool
it's 1hr long...
@user726941 how did this make you change your view of consciousness
@user726941 thanks
14:13
after 2:36, Penrose talks about the importance of quantum interpretations
14:25
@Relativisticcucumber glad to see you got the reference
Why is $\Pi^{\mu\nu}(q)=A(q^2)g^{\mu\nu}+B(q^2)q^{\mu}q^{\nu}$ the QED boson self energy $\Pi^{\mu\nu}(q)$ most general form?
There's only so many tensors of that form you can build with the metric and the momentum
why only metric and momentum in the first place?
I agree that's the most general you can have with metric and momentum
quoting P&S $g^\mu\nu$ and $q^\mu q^\nu$ are the only tensors that can appear in $\Pi^{\mu\nu}$. That's what baffles me
I think it's mostly just that if you work out the rules of the QED diagrams you can show that it's basically the only objects to pop up in the theory?
@Mr.Feynman what other non-scalar quantity do you think plays a role here?
Mhh, I don't have any to propose but I didn't consider that as a good reason to accept it was the most general ansatz
So we're just saying that since in the calculation only the metric and $q$ are involved, they combine to form the polarization tensor
that's the constraint, isn't it?
14:36
sure
ok, that was the same as the fermion self energy being a combination of $\gamma^\mu p_\mu$ and $I$
dummy question :P
not the stupidest question, books tend to not really do such things in a lot of details
and you can do it with a lot of details
one can divide by zero in a number system called a "Riemann sphere"
in there lies the devil
why
what is the general abstract algebra statement of which "you cant divide by zero" is a special case of?
14:43
I meant the details Slereah mentioned
A lot of physics does the "These are all the things you can generate from those" without explaining how to prove it
which is of the purview of invariant theory, if you wish to know more on the topic
A wheel is a type of algebra (in the sense of universal algebra) where division is always defined. In particular, division by zero is meaningful. The real numbers can be extended to a wheel, as can any commutative ring. The term wheel is inspired by the topological picture ⊙ {\displaystyle \odot } of the real projective line together with an extra point ⊥ (bottom element) such as ⊥ = 0 / 0 {\displaystyle \bot =0/0} .A wheel can be regarded as the equivalent of...
@bolbteppa yes. i just watched a video about this. the T element eats everything it touches
Wheel theory is more commonly used in the computer context where the bottom element is the NaN
They do not like the bottom symbol in their code
Although I think NaN predates wheel theory
Should credit Konrad Zuse for the wheel theory
14:50
surprisingly, division by 0 is never required to get the roots of any polynomial while negatives and irrationals and complexes are required
whelp, infinity is not a number
it's like the field of real numbers is happy without division by zero
all properties hold nicely and all polynomials have roots
there should be general field theoretic version of this statement
because we can talk about polynomials on arbitrary fields
"Wheel theory" does not really exist outside of that Wikipedia article and its few references.
It doesn't have that many uses except for floating point arithmetic
Galois theory relates polynomials to groups
14:53
and really in that context the wheel's center is essentially a garbage bin
the problem started when people thought light traveled infinitely fast: c = infinity
c = 1/0
@user726941 wtf are you talking about
who ever thought light travelled "infinitely fast"
@RyderRude wtf are you talking about, too
Over the reals, not all polynomials have roots.
and that you don't need division by 0 is only "surprising" if you don't understand how either division or polynomial roots work
@ACuriousMind yes
5
Q: Infinite Speed of Light

Jonathan L.I recently watched a video that stated that Newtonian Mechanics assumed an infinite speed of light. That same video, "PBS Space Time: The Speed of Light is Not About Light", stated that if the speed of light had no bound, then space, time, and matter would essentially not even exist. Can someone ...

@RyderRude so what do you mean "it's like the field of real numbers is happy without division by zero all properties hold nicely and all polynomials have roots"???
15:00
i mean that, at most, we only care about polynomials over fields, so we are happy with irrationals, negativea and complexes
@ACuriousMind sorry
there is no polynomial which requires the solution to x•0=1
what
can we please at least try to say things here which a) make sense and b) are correct?
4
i mean you dont need to define this to get the roots of any polynomial
2
Q: Contradictions of explanations for the speed of light

Sovereign InquiryI've recently started reading the book Biocentrism, by Robert Lanza and Bob Berman, which in its mind-blowing chapters, discusses the perceptions of reality, space, and time. Lanza states that: If one could travel at lightspeed, one would find oneself everywhere in the universe at once. This ind...

there's an infinite number of nonsense equations which we don't need a solution for to determine polynomial roots
why do you think we cant divide by zero?
i'm just looking for a fundamental reason
beyond "let's look what happens when we define it"
15:07
why do you think h bar is the avenue to throw around falsified/unsubstantiated claims?
dividing by zero would mean multiplying by the reciprocal of zero
okay so the contradictions follow from the "ring axioms"
i got this comment: "If we assume our algebraic structure of interest satisfies as least the ring axioms, then, as Andrew shows, 0=1, so all elements in the set satisfy x=x.1=x.0=0, and the set consists only of the element 0. So the end result is a very uninteresting and simple set"
this does not assume reals, only rings
@user726941 this is constructive to the conversation in what way, sorry?
and the rings axioms are very natural
zero has no reciprocal or multiplicative inverse
15:12
@user726941 the discussion is "what happens when you define it to be something"
if your algebraic structure follows ring axioms, then division by zero causes contradictions
isn't that enough?
@naturallyInconsistent , you remember I was having a confusion over self energies being captured by those macroscopic electrostatic energy formulas? Now I have a feeling in some way or the other, a semblance of self energy is being accounted for. I think also, the jump from $\frac{1}{2}\sum q_{i} V_{i}$ to $\frac{1}{2}\iiint \rho V d\tau$ is the primary culprit of most of my problems
15:15
this isnt specific to reals or complex numbers. it assumes only rings
@nickbros123 yay?!?
@naturallyInconsistent explaining it in simple terms is important
@user726941 the standard maths, Peano, ZF, way to explain it is trivial
All that stuff about circles reminded me of the trajectory of Venus relative to Earth. I wrote an answer about that a few years ago, which inexplicably got a couple of upvotes a few weeks ago. So I added another diagram or two to the answer. :)
Here's a Sage script that plots the simplified trajectory of Venus relative to Earth, assuming that both orbits are circular, with the period ratio exactly 8/13 sagecell.sagemath.org/…PM 2Ring Aug 16 at 5:42
2
@naturallyInconsistent I am for now ignoring all the issues that both the mathematicians and material physicists (and I) may have with the assertion that $\langle \rho \rangle \langle \vec E \rangle \approx \langle rho \vec E \rangle $. If I can assert this, I am atleast at ease with the entire fiasco with energy. There are still some issues I hav with polarisation but one step at a time
15:21
no, you can only ever assert that for constant fields. i.e. it is never going to be true in practice.
Can I get away with saying slow varying fields?
but there is some linearity involved, and that might force the energy accounting to become true, maybe in some statistical limit
@ACuriousMind are those enough examples?
@nickbros123 oh, even things fast enough to matter to our eyes are usually taken to be a slowly varying field in some circumstances. So yeah, I think there will be quite a lot of leeway for saying slowly varying.
@PM2Ring it is very pretty
Wheel theory is breaking the ring axioms by having the bottom element
15:23
@user726941 A few crackpots don't really explain what you meant by "the problem started" or how this relates to the rest of the conversation
@user726941 those are people who are extremely far out of the scientific community, even in their own times.
I never said it did.
@RyderRude You can edit the code to make patterns for other ratios.
this is how wheel theory is able to allow division by zero. it is not a ring
@naturallyInconsistent I think atleast some materials my assertion may be true, no?
15:24
@user726941 then how are you thinking you are moving the conversation forward?
@nickbros123 Yes; but I think it might also involve a heavy amount of redefining, say, of both terms.
There is a common misconception I'm addressing
@user726941 what common?
@PM2Ring yes. i just tried. it becomes closer to a circle when you increase rvenus
@naturallyInconsistent both the terms are subject to change anyways, right? Don't we have like a million and odd averaging procedures for a million and odd materials?
it becomes more spiral-ish upon increasing Tvenus
15:27
32 mins ago, by user 726941
c = 1/0
idk how you decided on the number 13 :P
@user726941 This is not a common misconception, literally no one believes that except the few very obvious crackpots you linked.
@nickbros123 no, it cannot be changing for every material because then Macroscopic Maxwell's theory would have zero predictability. We at most can have different models for a few types of materials, but within a certain type, we must have some scheme that necessarily works.
we should note that the ring axioms are not symmetric wrt addition and mulitplication. this is why 0 does not have a multiplicative inverse
@user726941 and your assertion that it is common is [citation needed]
15:29
e.g. multiplication is axiomated to be distributive over addition, but not the other way around
@ACuriousMind did you also mean the ring axioms here
@RyderRude Please do not treat h bar as your personal echo chamber. Things that are this level of trivially obvious to everybody who have looked at university mathematics should not be something that needs to be particularly noted.
:64290595 When you have a scholastic debate, you made this assertion, you are the one with the burden of proof. If you want us to consider that it is common, you at least do a survey first and publish that data somewhere.
Here's a better version with an input GUI. I had to strip the other one down so it would fit into a comment.
It is not our job to be doing your debate prep for you
@naturallyInconsistent well, perhaps I was confusing this with the different scales of averaging that exists. Anyways, what did you mean by redefining the terms
@user726941 why do you think infinity has to do with the speed of light
15:35
@nickbros123 Remember that the microscopic E and $\rho$ fields are periodically spiky functions over the length scale of a unit cell, whereas the macroscopic versions are smooth over appreciable regions of the entire macroscopic sample? That shit.
@RyderRude you are not even reading your fellow bleep well enough! He did not say that he believed so.
@PM2Ring it's very cool:)
@user726941 sorry.i think you are saying that a few people in the past believed c=1/0
That's one reason I like to write code. It can let you simulate stuff related to physics & astronomy. And it's a hell of a lot easier than doing orbit calculations by hand, like people did for millennia.
@PM2Ring yes. and it's a very short code too for something this complicates
lol
that's because all the actual functionality is hidden in the parametric_plot, not because this would be simple to implement
@user726941 a few people in the past may have suspected that c was infinite, but no one credible would use division by 0 to describe it :)
15:41
@naturallyInconsistent if $\langle \rangle$ denotes an averaging procedure we define $ \chi_{macro}=\langle \chi_{micro} \rangle $ right? Are you talking about redefining this itself?
@RyderRude if this is your takeaway, that explains why you are this uneducated despite all the readings you have done
@PM2Ring we are very lucky to have these resources today ;)
@naturallyInconsistent do not refer to me as a "beep"
thank you
@RyderRude Yep. It still seems like science fiction to me that I can write & run powerful computer programs on my phone. :)
@nickbros123 I think that we are somewhat defining the macro field (and using the same symbols for the averaged fields), in such a way that the $\left<\rho\right>,\ \left<\vec E\right>,\ \left<\rho\vec E\right>$ are simultaneously kept constant.
15:45
@PM2Ring science fiction is just future science after all :)
or some people would say that about magic
gnnnnn no it isn't and the point of most science fiction stories is not the technology
i didnt mean it as 100% accurate ofc
it's literally like 1% accurate at best
the technology is not the point and the technology in lots of sci-fi is impossible or implausible while the actual technology we have is rarely predicted
@ACuriousMind Yes. matplotlib is ultimately doing most of the work there. I'm also doing the polar stuff with a symbolic vector function. It's only slightly more compact than using a "traditional" Python function, but it seems a bit more elegant. :)
look at 70s and 80s cyberpunk stories where the matrix is entirely cable-bound because no one imagined WiFi, or the bulky screens in any scifi before we actually invented flat screen
or flying cars
15:48
flying cars are 50% accurate. they exist now. they just arent used
video calls were predicted too
@naturallyInconsistent but in a paper I read, whose calculations I was not able to understand (if I'm able to find it again, I will link it) , they mention 2 other terms that come up, like $\langle \rho E \rangle = \langle \rho \rangle \langle E \rangle + a few extra terms. This is I think a calculation done using ensemble averaging (let me get back to you after I confirm it though, either Lorenz or ensemble). I ve not seen a method of averaging that fulfils what you are saying
this misunderstanding - that science fiction stories are in any meaningful way about the technology rather than about the implications of the imagined technology - is common in certain techy circles
@nickbros123 Because it might be a fool's errand. I think that no such good averaging technique exists, and there might well not ever be.
when you watch the Matrix and you come away with "Boy, wouldn't it be cool if he Matrix existed" or you read Neuromancer and go "Man, AI would be cool" you've missed the point
Scifi has certainly inspired a lot of people working in science & technology. But if you read old scifi that talks about stuff we have today, it tends to be very different. Even the most futuristic writers are still products of the era they grew up in and lived in.
15:52
@ACuriousMind I wonder why you continue to talk to people whose main composition is misunderstandings, especially those who demonstrates an inability to learn to correct those misunderstandings.
@naturallyInconsistent I don't want people to come here and see a take like this without anyone dissenting
@PM2Ring Asimov's take on Orwell is one of the most insightful. It is simultaneously too optimistic, and too pessimistic, but in ways that are wise.
I emjoyed reading Asimov back in the 70s. I still enjoy reading him today, occasionally. But some of the tech things in his stories can seem hilarious today. Eg, he has humanoid robots that can walk & talk like a human, but to do serious number crunching requires a device the size of a city block.
@ACuriousMind Well, there is a point where admin action to take care of trolls is the appropriate action, rather than dissent. I love to dissent against them too, but this is hurting the overall conversation
@naturallyInconsistent there is also another paper I read, and, my physics professor was also suggesting the same: take these energy formulae as axioms and enjoy life xD
15:55
4 messages moved to Trashcan
what did I do wrong now?
@RyderRude Apparently I need to be clear about something again: This was not an isolated warning. You're falling again into this pattern today, and "warp drives" is now like the third topic in a few hours you obviously don't know anything in-depth about
@nickbros123 Yes, and that is I think what everybody who have seen this particular abyss agrees is the correct way to retain sanity.
the disrespect you're earning from users like naturallyInconsistent is a consequence of your continued obstinacy
which, if it came out of the blue would certainly be inappropriate, but it doesn't come out of the blue
it certainly came out of the blue for @user726941
you should prevent this behavior, thats all
and i mentioned warp drives in a discussion about sci-fi being possible real. this is like the most comfortable thing anyone can mention in this discussion
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