Imaginary Time: Stephen Hawking's Favorite Physics Concept Relativity by Parth G

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When asked about the one thing Hawking wished everyone knew about his work, he replied "Imaginary Time". #imaginary time #stephenhawking #physics
Hi everyone, in this video I wanted to discuss the basics of the concepts behind Imaginary Time. As some of you will have guessed, it has nothing to do with imagination, and everything to do with imaginary numbers.
In this video, we briefly discuss the definition of the imaginary number i, sometimes called j by engineers. It represents the square root of -1, and imaginary numbers can be represented on an axis perpendicular to the real number line. Together, the real and imaginary axes form what is known as an Argand diagram.
Imaginary time consists of a simple substitution where any mention of time in our mathematics is replaced with the imaginary number i, multiplied by a new quantity being defined, which we will now call imaginary time (tau).
Firstly, it's worth mentioning that this kind of substitution, where a real quantity is replaced with an imaginary one, is known as a Wick rotation after Gian Carlo Wick. The reason it's known as a rotation is because in going from a real value to an imaginary one or vice versa, we rotate from one axis to another on the argand diagram.
Wick rotations, especially imaginary time, are useful in relativity as a mathematical tool. In this video we see how using imaginary time can take us from a complex Minkowski spacetime (time has the opposite sign to spatial components) and bring us to more familiar mathematics where all components have the same sign. This makes many problems easier to solve mathematically, after which we can revert the imaginary time substitution.
But Stephen Hawking took the idea of imaginary time further than just a mathematical convenience. He hypothesized that in the early universe, time could take both real and imaginary values, and that at some point the imaginary time dimension was lost (which is why we only experience time running in one direction now). The reason he hypothesized this was because having an extra time dimension would allow us to deal with infinities in our mathematics much more easily due to the extra degree of freedom.
Specifically, most of our current theories suggest that the universe was initially densely packed into an infinitely small volume, meaning its density was infinite at the Big Bang. This is problematic for mathematics, and using imaginary time solved this issue.
Unfortunately, this was not testable according to Hawking, since we have no way of knowing if an extra time dimension existed. Additionally, other scientists have found it difficult to align the idea of imaginary time with other observations we can make of our universe today.
Here are some resources you may like:
Stephen Hawking Interview - • Stephen Hawking Extend...
My Relativity Playlist: • Relativity by Parth G
Relativistic Interval: en.wikipedia.org/wiki/Special...
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[@ParthGChannel]

Hey everyone, thanks so much for watching! I'd like to take a moment to thank Skillshare for sponsoring this video - the first 1000 people to use the link will get a free trial of Skillshare Premium Membership: skl.sh/parthg05211

[@baklolvlogger8413]

Thanks❤️❤️❤️❤️

[@sudhanshuranjan6928]

Thankyou sir for posting this amazing video 😊huge fan sir

[@ManpreetKaur-ry3vz]

Your videos and explanation very helpful and intuitive. Please make a video on thermal radiation in general and black body radiation in detail. How the atoms inside a material behave like while emitting thermal radiation and what is the thing on quantum level that makes black bodies so different ?

[@scoreprinceton]

Nice explanation of a very interesting topic. Is this leading to Minkowski near miss in discovering the theory of relativity before Einstein’s?

[@afriedrich1452]

The Minkowski metric is NOT: (ds)^2 = (icdt)^2 + (dx)^2 + (dy)^2 + (dz)^2 The Minkowski metric is: (ds)^2 = (cdt)^2 + (idx)^2 + (jdy)^2 + (kdz)^2 SEE: Wikipedia article on Minkowski space and search for first paragraph in section "Minkowski metric." It is not time that is imaginary. It is space that is imaginary.

[@DrDeuteron]

I met Hawking in an elevator. Hans Bethe was also in it. I felt pretty much like the dumbest person in the room.

[@dhanashrikulkarni5878]

You atleast had a chance! You're the luckiest one!❤❤

[@birendrachhotaray2263]

You nailed it!

[@fariesz6786]

@@dhanashrikulkarni5878 but that would just reinforce the notion that luck is inversely proportional to smarts

[@dhanashrikulkarni5878]

@@fariesz6786 LOL yeah😂😂

[@davidbrisbane7206]

So what? If you were Jennifer Lopez do you think you'd even care?

[@wallstreetoneil]

Hadn't thought about imaginary Time, but from my math background, the idea of time going from 'real' space that we usually experience, to vectoring off into complex space orthogonally at an angle according to your speed, still existing, but not recording at the same rate as you travel at faster speeds, until it reaches a completely imaginary length of i on the unit circle at speed c is actually very satisfying to my mathematical brain.

[@ChaseNoStraighter]

Maybe just say time is the three vector imaginary component of spacial velocity giving a complex 6D vector of fixed length C?

[@isreb]

I would go even further. Imaginary times become space values. So velocity transforms time to space. So at the speed of light there is no time as it becomes pure space (no real part which is time, but full imaginary part which is space). And velocity is the angle between them. Speed of light is an angle of 90 degrees or pi/2. This also explains why c has a limit. It is sin(pi/2) and is therefore 1 and v = sin(phi) which is at max c.

[@isreb]

I based a theory on that which can ease Einstein's field equation and also explains superposition of quantum mechanics and lots of phenomenon in a very intuitive way. Was just to busy and lazy until now to refine and publish it. :) But it basically all comes down to complex geometry.

[@positionthepositron]

Yeah, we are approaching zero or 1 at all intervals. Tangents and such

[@serenityindeed]

@@isreb Did you end up publishing this? Link to the paper?

[@NateROCKS112]

Timestamps: 0:00 Intro 0:52 What Are Complex Numbers? 3:28 Wick Rotation 4:30 Skillshare Sponsorhip 5:29 Special Relativity 7:08 Reality of Complex Time (Stephen Hawking) 8:28 Conclusion

[@eigenchris]

This idea of relating statistical mechanics and quantum mechanics using a Wick rotation is new to me. Wikipedia says Schrodinger's equation and the heat equation are related by a Wick rotation. I'll have to look into that more. I've never come to terms with understanding why a factor of i is in the Schrodinger equation.

[@athelstanrex]

EIGENCHRIS YOU ARE THE BEST

[@Abhisruta]

Wow, you are here.

[@arnold-pdev]

I *think* the "i" is there to produce oscillatory solutions. Consider a x'+x=0 (exponential soln) vs x"+x=0 (oscillatory soln). However, you can also get an oscillatory soln out of the first order eq x'+ix=0. Physicists probably wanted a first order evolution eq because there is more theory out there for first order eqs, eg spectral theory and dynamical systems. Also, you only need one IC, which might be a math necessity in the funky uncertain world of QM

[@brooksbryant2478]

I’d noticed that the schrodinger equation was i times the heat equation before, I never knew that was called something though

[@matheuseliaspereira9566]

The imaginary units come to make sure the equation works for systems that conserve or dont conserve energy, according to Schrödinger himself in his second paper from 1926

[@sashwat5489]

The only Wick rotation I know of involves two men in a bar and a pencil✏

[@dystopian3342]

** 3 men in a bar**

[@maxwellsequation4887]

What? Is that another commoner joke?

[@JabranImran]

@@maxwellsequation4887 The baba yaga

[@goodplacetostart9099]

@@JabranImran The boogeyman

[@styckykeys2200]

You have any idea who you just pissed off? That was Gian Wick

[@davefalls1027]

at 1:22 - extremely very well stated "what happens if we define the square root of a negative number to be something"

[@andor888]

I think the use of "complex" here does correlate to everyday use. Numbers with one dimension are simple, numbers with more than one dimension are complex. That part is actually very intuitive.

[@chixenlegjo]

“Imaginary Time” sounds like a weird segment of a kids show.

[@CesarDainezi]

The square root of 9 is only positive 3, otherwise the square root wouldn't be a function. That's why every time we have a x^2 = a, then x = +- sqrt(a) and we have to put that +- sign before the square root, to enable the answer to be also the negative root.

[@tomkerruish2982]

The reason engineers (especially electrical engineers) use "j" rather than "i" is because electrical current is already "i" (or "I").

[@mosemusica]

I would argue that imaginary time actually has a lot to do with our imagination, it takes a lot of it to understand the implications here!

[@Ghost-hy1tl]

Tell me why and how

[@alphalunamare]

If they had defined SQRT(-1) as a 'shit' number would you assume that the implications were taking a dump in the morning? 'Imaginary' is just a very unfortunate adjective to have used. Maybe that's why Engineers use J for 'Jack Off' ?

[@ai_marsbar4325]

I agree and disagree/ propose the following constructively speaking that is: " what if we replaced "time/'pending on/regardless of how long it will take to understand (you wrote: it takes a lot to understand the implication/z) like in Pi, could we at least/ possibly be getting closer to whatever subject/identity of it?

[@Ivan.Wright]

@@alphalunamare We'll if the human imagination exists primarily as electromagnetic fields then it is correct to say. When energy changes phase from electron (material/"real") to em perturbation (imaginary) it does so at 90° like the number line.

[@Nat-oj2uc]

@@alphalunamare I like it. 1+ 3 of some shit makes sense

[@mahadlodhi]

Loved the video. As einstein remarked in his book emphasizing the importance of minkowski's realization: "But the discovery of Minkowski, which was of importance for the formal development of the theory of relativity, does not lie here. It is to be found rather in the fact of his recognition that the four-dimensional space-time continuum of the theory of relativity, in its most essential formal properties, shows a pronounced relationship to the three-dimensional continuum of Euclidean geometrical space. In order to give due prominence to this relationship, however, we must replace the usual time co-ordinate by an imaginary magnitude ict proportional to it. Under these conditions, the natural laws satisfying the demands of the (special) theory of relativity assume mathematical forms, in which the time co-ordinate plays exactly the same role as the three space co-ordinates. Formally, these four co-ordinates correspond exactly to the three space co-ordinates in Euclidean geometry. It must be clear even to the non-mathematician that, as a conse quence of this purely formal addition to our knowledge, the theory perforce gained clearness in no mean measure.....Without it the general theory of relativity, of which the fundamental ideas are developed in the follow ing pages, would perhaps have got no farther than its long clothes."

[@profdc9501]

Another interesting aspect of imaginary time is the relationship between the diffusion equation and the Schrodinger equation, which is one major motivation to make the Wick rotation. Feynman and Kac while at Cornell proved a theorem (the Feynman-Kac formula) which shows that the solution to the diffusion equation can be found using the expectation value of all paths of the random process, an important result in the path integral formulation of quantum mechanics.

[@td5786]

nitpick: @2:37 sqrt(a*b) = sqrt(a)*sqrt(b) doesn't hold for complex numbers, otherwise i^2 = sqrt(-1)^2 = sqrt(-1)*sqrt(-1) = sqrt(-1*-1) = sqrt(1) = 1, so i^2 = 1

[@anirban7493]

I liked how Hawking used the idea of imaginary time to describe the no boundary condition. He said universe has a begining in real time and it may also end with a big crunch in real time but in an imaginary time it has no beginning no end like the surface of the earth, big bang is at the North Pole and big crunch is at the South but there are no boundaries. A particle travelling through complex time coordinates can travel forward and backward just like we can move around anywhere on earth. Note: the expansion of the universe is accelerating, none of us know why, may be because of dark energy & we still don't know what it is, but the fact is , it will not end in a big crunch if the universe is accelerating.

[@stanvol]

Yes, that diagram was interesting, but I am wondering, would that particle be traveling through IT be going backwards or forwards in imaginary time, or through both space and time in imaginary time. Sorry for the question, I'm a begginer and quite young for this so I don't understand all of it.

[@anirban7493]

@@stanvol The is no real distinction between space and time in imaginary time domain. So moving backward and forward in imaginary time means the particle would move through both imaginary time and space simultaneously.

[@stanvol]

@@anirban7493 OK thank you for the explanation. I thought it was like that just wanted to make sure, thanks again.

[@Czarzhan]

I had always assumed the imaginary time component was a solution to the "many worlds" interpretation of quantum mechanics: Instead of waiting for an observation to collapse the superposition of a particle, going either one state or another, both outcomes occurred but in parallel realities, the distance between the two being measured in a direction at right-angles to the flow of time, or imaginary time.

[@Patatmetmayo]

This is what I've been thinking as well. The branching off that happens in many worlds could be described as creating an extra time dimension, namely the collection of all parallel realities. By creating an imaginary component of time it could be seen as the mathematical equivalent of adding an extra dimension to time that is orthogonal to real time. I believe that's what Stephen Hawking was aiming at, but for some reason his version is limited to the very early universe and separate from Everett's many worlds interpretation. I wonder why this concept has never been used by theoretical physicists working on many worlds? Maybe time to ask Sean Carroll about this?

[@reubenoakley5887]

​@@PatatmetmayoSeems likely that they have tried already, and either couldn't come up with anything, or are still thinking about it

[@informationparadox387]

Video's title : _Hawking's favourite concept_ : *_Ti-t_* Me to my brain : No please dont think that way!💀

[@masicbemester]

( ͡° ͜ʖ ͡°)

[@inamdarsaquib9528]

|-

[@haneen3731]

I love how you explain the basic foundations (especially math) needed to understand advanced concepts. As always thanks for sharing your knowledge!

[@abhiramvartak4149]

I am glad I found out this video and this channel after hours of searching relevant and understandable videos of this topic!

[@englishinenglish3473]

My first emotion : What a strange concept , is it real ? My brain : yep it's imaginary

[@Ewr42]

complex, eh? bit of both, actually

[@JohnVKaravitis]

3:29 then 5:29 -- I didn't have any real time to view this vid, so I just imagined that I saw it.

[@bamitsmanas]

Awesome!! Love the simplicity behind these concepts

[@darrinkinney2268]

I have always been surprised that nothing exists related to "imaginary frequency". So much of physics is based on frequency, and to envision there is another dimension to frequency is along similar concepts. We have a little trouble thinking about negative frequency, let alone an imaginary dimension. Although, from a math point of view, math doesn't care.

[@Daniel01101101]

Actually in the standard wave equations there is a meaning to imaginary frequency (even complex frequency). Usually when you have imaginary frequency it means the equation decays (or grows) exponentialy with the imaginary part of the frequency. For example: say a wave oscilates at a spatial frequency omega + i gamma along the x axis, that means the wave oscilates spatially at omega frequency and decays with gamma rate(every meter the amplitude decreases by e^gamma factor). This depends on the wave equation but in general such a concept exists (and is quite interesting in general). Quantum mechanics is usually restricted to real frequencies because its imaginary ones mean that energy is lost or gained externally, but there are cases where it is used in QFT.

[@trueriver1950]

@@Daniel01101101 The most obvious place that occurs is in quantum tunnelling. In the normal case when a particle travels through a region where it has positive energy the wave function is constant magnitude like a sin or cos If it tunnels through a barrier that needs more energy than the particle has, the energy effectively becomes negative and the wave function decays exponentially. This is not only a quantum effect. In classical electrodynamics a light wave can "tunnel" through a thin dialectric layer where it "should" bounce off. The equations are identical with those of QM, which is one reason we refer to that weird Schrodinger/Dirac thingy as a "wave function". Even more classically: related to that is that if you take the Laplace transform of a function whose values are imaginary everywhere you get something that looks like the Fourier transform of the modulus of that function (Can't remember the exact details my excuse is that I studied that some 46years ago)

[@ananintesarbinfaiz863]

Just a small confusion that I had, when writing sqareroot in that format at 1:10, shouldn't it just give us the positive value since that notation is for principal squareroot which has only one positive solution?

[@jorex6816]

Yes exactly. Sqrt(x^2) = |x| and not plus/minus x.

[@vinaykaushik812]

Or well, it could be the complex branch of the square root, he didn't mention which branch he was working with

[@ulfhaller6818]

Yes, you are right. I also got confused, but physicists are not always stringent with mathematics :). Apart from that, good video as always.

[@supptk]

I too was confused on seeing this because I have learnt that the square root of a number is the modulus, so it always has to be positive and not negative.

[@Twitchi]

Every paid for education I have done has said any squareroot of a positive number will give positive and negative solution (just like plotting x^2), can I ask where you guys are getting the "defined as" thing please?

[@areein4735]

Concise and succinct coverage..very handy..thumbs uped

[@abhishekkp7121]

Your videos are really informative. Keep on making these videos🔥

[@rakeshkumarkavalagi4653]

so underrated!! and thanks for the very informative video. hope you get more subs

[@Infinium]

Really interesting video, thank you for sharing!

[@robertschlesinger1342]

Excellent video, as always. Very interesting, informative and worthwhile video. Parth, you are changing the world for the better by raising people's understanding of science.

[@ChaineYTXF]

really nice. Thank you!

[@musicalbot9172]

I would like to point out a correction in the mathematics at 2:02 . The square root of a positive number is only positive. The square root of 36 is only +6 and not -6. Also, at 2:37

[@MillionaireMindset369]

Just like always your explanation is very good

[@desabathinareshma2537]

Amazing videos!

[@johnchristian5027]

Great Video man!

[@TheHumanHades]

Please make a video on quasicrystals. I really love your way of explanation

[@meccamiles7816]

Why has KZbin only now recommended your channel to me?! I LOVE your content!

[@frictyfranq321]

Don't judge a video by it's Title. And I did! And it was as expected. Really good!

[@djalexander968]

really helpful thanks, had to like and comment out of obligation due to skipping through the video so much XD, but the video is really good

[@imtiazshakil]

It’s really marvelous for me! I wanna to know more about it..

[@theartofmusic009]

Νice explaining video Parth. Have a good day from Cyprus

[@dogood8750]

Thanks, Parth I really like your videos I've always been interested in science and going to start a physics bachelor's next fall. I always liked science but wasn't able to understand the advanced math, but always able to get the quantitative principle. Now I know I have to pay my dues and learn the math to truly know physics which I am doing but I really like your videos because you really bridge the gap between the start and end of the finish line with many complicated math-intensive physics concepts which most courses don't do.

[@sujalsalgarkar360]

He brings so much quality content

[@SylwesterKogowski]

Actually, recently polish physics theorists from Warsaw and Oxford (A.Dragan and A. Ekert) came up with a theory, that time is actually made up of 3 dimensions (for superluminal objects) and also, that some particles travel at speeds faster than light, but they appear wave-like only because of how special relativity works with superluminal speeds, it actually forces them to be wave-like from our subluminal perspective and thus connects special relativity with quantum waves. They say that although the speed of light is cannot be achieved by anything with a mass, it goes quite opposite with hypothetical particles that are already at superluminal speeds from the very beginning. These cannot slow down to the speed of light because it would require infinite energy, they get faster the smaller energy they have.

[@wayneyadams]

The key word in your comment is HYPOTHETICAL, there are a lot of hypothetical results from equations but "that don't make em true!"

[@arthurdurham]

Thank you! I don't have any background in physics and every article explains it with the assumption you have a basis. This makes understandable for the non intellectuals, like myself

[@jn3917]

Sir Ur content and explanation is the best..feynman would be proud of you

[@superdrag65]

Three minutes in and it’s the best explanation of imaginary numbers I’ve seen yet. So glad I stumbled onto this channel.

[@BiswarupRay]

Isn't this explanation taught in schools?

[@gazsibb]

Parth you are a genius teacher. I loved the stepping stone way you led newbies from sqrt36 to sqrt4 X sqrt9 to sqrt of a number X sqrt-1 Brilliant 👏🏽👏🏽👏🏽

[@bozoclown2010]

thank you. good refresher for me on i.

[@prashantlale4976]

As usual this was fun!

[@ranakundu4898]

Just studying the chapter Complex numbers and got to watch this!!!😄😄😄 A perfect application in Physics. Really enjoyed the video........👍

[@positionthepositron]

Maybe complex time was not lost early in the universe, but exists as the smaller wrapped dimensions described in string theory. These additional time dimensions would not be interactive with the electromagnetic spectrum, but the gravity would be emergent, as in dark matter.

[@opbhuge1921]

Light of conciseness🙏🏻🙏🏻

[@Aledharris]

Very friendly man. Explaining things is what appears to be a simple to understand way. I’m just too thick.

[@michaeledwardharris]

Mind blown, again.

[@avinashthakur80]

That equation of interval doesn't look dimensionally valid. How are (length)² terms being added to (time)² ?

[@philipstuckey4922]

Good catch, there was a brief note explaining that c=1, but he didn't mention that t is actually tc (or maybe x, y, and z are actually x/c, y/c, and z/c respectively). This is a fairly common convention in some areas of physics, but it still should have been explained.

[@avinashthakur80]

@@philipstuckey4922 Yes, googled that and found the real equation to have (ct)²

[@tomkerruish2982]

@@avinashthakur80 When combined with imaginary time, you get (cit)², and thus Caltech is the fourth dimension.

[@junaidulislam1212]

*Thank* *you*

[@rikthecuber]

Hey Parth! Actually, root(9)=3 and not +-3. It is defined as the modulus of the result. It becomes +-3 only when it is in an equation or inequality and we have to take modulus on both end. Then we have, by definition of modulus, the plus-minus root(x).

[@chevasit]

Great! 😊

[@johnh7411]

One thing I don’t understand - ‘in the past’ refers to time on the real number line. So, how could that have any meaning for the imaginary part?

[@johnh7411]

What I mean is, in the video it says that imaginary time nay have been a bigger factor in the early universe. But, the imaginary time axis is perpendicular to the real time axis. How could a difference of past or present in real time have any bearing on what applies in imaginary time?

[@Ray-gr7yc]

Idk how I ended up in here, but i like it Very interesting channel.

[@TheyCallMeHacked]

The imaginary unit is not the square root of -1. The square root function is not defined for negative numbers (otherwise, you could prove things like -1=1). i is one solution of the equation x²+1=0

[@Draginx]

You're my favorite youtuber! I've watched about every single one of your videos and am subscribed! I love physics, science, and math and you're a great teacher! You should create a physics course on udemy or skillshare or something! I'd for sure buy it in a second~~!!!!!!

[@trbz_8745]

That last bit about the big bang just sounds like 2-dimensional time more than "imaginary time". Like, without the three spatial dimensions to move in matter would instead be moving through 2 or more time dimensions, which then collapses into a point and explodes "perpendicularly" into spatial dimensions, like a twisted hourglass.

[@arjyadebsengupta8159]

Thank you Sir

[@barrywilliams991]

Well explained. I had a professor who was a mathematical tool. Well, just a tool.

[@sudhanshurastogi1502]

I Don't know if u will read this or comment.....still... What i understand is the imaginary time is a tool to used to define the hypothetical frame work needed to justify the concentration of energy and mass at a given point. What i mean after the big bang every thing follows a certain rules (theory of everything/ unification). But one observes an anomaly before the big bang where the universe seems to deviate from these rules e.g - If every thing (energy and mass) is concentrated with in a point in a space ...that space will be at the highest temperature possible in the universe (i call it absolute maxima) but the space right next to will be close to absolute zero, thus the energy difference between the point and the space would be maximum possible. This simply put, knowing the nature of the universe is not possible. To simplify what i am saying is we need to have a concept of energy movement which would justify the the concentration of so much energy in a single point, thus the imaginary time... In my hypothesis of the big bang .....it starts like this 1. Universe expands and cools down to almost absolute zero 2. Particles reach a state of bose einstein condensate 3. Creating a uniform crystal like structure which is almost identical and symmetrical every where. 4. Energy can flow easily though out the system with no resistance. 5. A small defect some where in the fabric of space 6. Releasing a tremendous amount of energy via the defect 7. Channeling all that free flowing energy out though that same small defect 8. Thus creating the big bang. (for us it would look like that all energy was concentrated in a single point but it would be like a hole through with every thing expanded) The concept of imaginary time would help justify the realise of this energy trapped in the bose einstein condensate as it would be like a super conductor of energy (with no resistance) thus making it almost impossible to calculate the time due to the weird nature of the condensate

[@trueriver1950]

There is a fallacy in that -- a very common one but a fallacy even so. At the big bang there is no space just next to the point where all the matter is. There is only time going forward and no other spacetime dimension exists. The analogy is the North pole. At the pole there is no east or west, not even North. The only way is south.

[@sudhanshurastogi1502]

@@trueriver1950 what you are saying is true and I do know that.... But it was just to express what my thoughts are and what my findings suggests..... As to the proper explanation.... You and I need to talk.... As expressing in words and on KZbin is a bit hard

[@ktvx.94]

Man, these rotations are wicked

[@j.lo.5784]

Assuming Time has a real and an imaginary part - the imaginary part would "cause" things to happen - the real part of time would present "how fast" something happens compared to a "change in space"? That would give us a hint towards dark energy. And would assume that space itself should have some time like properties of change, the dark matter.

[@amshumansharma6967]

love this guy man

[@ChadWilson]

Another series here on KZbin talked about every number basically being able to have two components, the real and imaginary, ie, (2,2i) or (5,0i). What if time is three-dimensional like space, such that every particle has [x(r,i), y(r,i), z(r,i)]?

[@krumuvecis]

Why particularly three? Why not, for example, thirty?

[@dhanashrikulkarni5878]

Finally one video on my Idol!! ❤ Loads of repect for him and words of appreciation for your efforts!!😎

[@roysproule669]

Thanks!

[@ParthGChannel]

Thank you so much!

[@ramchandradey4059]

Thanks , i think this concept has inspired prof Penrose to conceptualize frequency having negative values like momentum . This may initiate many more move in understanding time at various level of cosmic evolution

[@devashishshah9021]

Please make a video on the cosmological constant/ expansion of the universe..

[@Elite7555]

To be precise, _i_ is *not* the square root of -1, but rather some number that squared with itself is equal to -1. Claiming _i_ to be square root of -1 leads to paradoxes.

[@ericterry4544]

That's the definition of square root.

[@countduku959]

Maths and physics are so cool!!!

[@Dyslexic-Artist-Theory-on-Time]

Time can be explained as a process of energy exchange formed by photon electron interactions. We have photon ∆E=hf electron couplings continuously transforming potential energy into the kinetic Eₖ=½mv² energy of matter, in the form of electrons as an uncertainty ∆×∆pᵪ≥h/4π probabilistic future comes into existence. All it takes for this to be logical is for the spontaneous absorption and emission of light waves to precedes absolutely everything that happens in our three-dimensional world.

[@jebarijihed]

I see always an eror happening when we define "i"! The square root function hence the symbol is ONLY defined for POSITF REAL numbers. The best way to define "i" is to say that i²=-1 not saying that i=is the square root of -1.

[@szxcrab7782]

The square root function (which ill call sqrt from now) is indeed only defined for positive numbers, _i_ is defined as the sqrt of -1 *because* it cannot be defined. Another thing, i² = -1 can be rewritten as i = sqrt(-1) through simple algebra (specifically applying sqrt to both sides). Note: didn't mean this rudely, just explaining.

[@jebarijihed]

@@szxcrab7782 The thing is the symbol of sqrt √ is not allowed but for positif real numbers. Also a more complete definition of i is : i is the root of 1+x^2=0. Still may be i'm wrong :)

[@d.l.7416]

In maths when something isn't defined for some values, we can just extend the definition. Like we had this problem where we couldn't do 3-5 because five is bigger, so there is less than nothing which doesnt make sense. So then we invented negative numbers. In the same we, we couldn't square root negative numbers, so we invented complex numbers.

[@d.l.7416]

Basically we can just say that sqrt does work for negative numbers. There's no reason to limit it to positive numbers so we extend the definition.

[@HectaSpyrit]

@@d.l.7416 Actually, there are problems when we try to use the square root function on negative numbers, in other words it cannot be properly extended to negative numbers. The reason for this has to do with the fact that, for any non-zero number z (complex or real), there is exactly *two* numbers whose square is z, which are oposites of each other. When dealing with positive real numbers, we define sqrt(x) as the *positive* real number whose square is x, the si called "principal" square root. Now for complex numbers the notion of sign doesn't make sense anymore so it would become difficult to say exactly which of the two roots to choose as the principal one, but even if we assume we can satisfiably choose one, we still encounter situations where defining *i* as sqrt(-1) leads to contradictions: -1 = i * i = sqrt(-1) * sqrt(-1) = sqrt((-1) * (-1)) = sqrt(1) = 1 The second to last equality is due to the fact that the square root of a positive number is defined as being positive too. So either *i* cannot be defined as the square root of -1, or it can but then the equality sqrt(-1) * sqrt(-1) = sqrt((-1) * (-1)) must be false, which means that sqrt follows different rules with positive reals than with negative ones, which doesn't seem like a satisfying extension of sqrt or definition of complex numbers to me, it even seems (to me) like it's defining a different square root function for negative numbers than the one for positive ones. A definition of i which is much more satisfying because it avoids these complications alltogether, is to say that sqrt remains untouched, and that the set of complex numbers is a set of things that include real numbers, on which + and * are defined, and that *also* contains something called *i* which, when squared, gives -1. It's a more algebraic approach and, for reasons I am not sure I can totally explain, I find it a lot more elegant. That square root thing always seemed to me like a bodge, an attempt to cram complex numbers with real ones in a way that is very forced and unnatural, like trying to cram a square peg in a round hole. Idk

[@IceMetalPunk]

This is really cool, but I think my biggest question is, "what is the physical interpretation of imaginary time?" Like, normal (real) time we can say is the sequence of events as things change (I guess?), or perhaps a progression towards entropy (maybe?). At the very least, even if I can't verbalize it, we all know what the experience of time is. So if time were allowed to have complex values... what would that even mean? How would an object changing over complex time differ from the same object changing over real time? What would the perception of complex time be like? These are the questions that hurt my brain.

[@jackzugna5830]

Time is not real but it is our perception, we have always understood it as an event that repeats itself in the same space, for example the rotation of the sun or the resonance of a quartz, are all events that are repeated in space (relative to the reference system used) after a certain "time" perceived by us.

[@IceMetalPunk]

@@jackzugna5830 So... the units by which we measure time are arbitrary (though based on universal constants), and the perception of time is subjective... but time itself is real. Without time, change would not be possible. Not to mention relativity shows that time passes at different rates depending on speed and gravity, and this is true even for things without any "perception" (i.e. particles that would normally decay before reaching Earth's surface can make it to the surface anyway, since they're moving fast enough that their decay is slowed by time dilation).

[@antronixful]

Thanks for the video, but I might say that √a=b, where b is defined as a positive (or zero) number, not ±b. That contradiction is because if we're solving some sort of equation where x²-a=0, because √() must be a positive number, we take the absolute value of x... |x|=√a, i.e. ±x=√a... For complex numbers, i²=-1 has two values (but that's another definition)

[@jorex6816]

Why should i^2 have two values?

[@antronixful]

@@jorex6816 because i and -i are not algebraically different. If you replace all demonstrations and definitions that that are using i with -i, they'll be totally equivalent. But if you replace all the definitions that have 1 with -1, it'll make a huge contradiction.

[@jorex6816]

@@antronixful Sure, that sounds like solving the equation x^2 = -1. There you have two values (+ and - i), as every quadratic has. But you could also use this definition for i: i = sqrt(-1), so if you would now replace i by negative i this obviously wouldn't be equivalent anymore. So doesn't it just make sense for the first equation I mentioned? Because for me it sounds like you're saying that i = -i

[@antronixful]

@@jorex6816 but that's not the definition of i... the real definition is the one that says that a complex number is a tuple of the form (a,b) where a and b are real numbers, that follows these rules: I) (a,b)+(c,d)=(a+b,c+d) II) (a,b)•(c,d)=(ac-bd, ad+bc) here, the tuple (x,0)=x and the tuple (0,1)=i if we use that rule: i•i=(0,1)(0,1)=(-1,0)=-1 But going back to the reason why i is algebraic equivalent to -i, is because there is not a notion of order in the complex numbers... You can't say that i>0 or -i>0, because the order is a tool to categorize real numbers, but that can't be assigned to a tuple. That's why when we treat complex functions we have to take a branch. Also, that's the reason of why taking the clockwise and anticlockwise path around a singularity in a complex plane is the same (if you recall from advanced calc, taking the negative path isn't the same as taking the positive path around a singularity), i.e. using i or -i for algebraic things is equivalent, but i≠-i.

[@walterufsc]

You are right. In fact, the square root symbol implies only the principal (or positive) root.

[@flpamr8850]

I really wished to see something like; in 4d we have a complete geometric object described by 4 vectors which forms: 1 scalar; 4 vectors; 6 bivectors; 4 trivectors; 1 quadvector. What does each dimention represents?

[@SomeMrMindism]

These are just the completely skew-symmetric tensors, representing (oriented) scalars, vectors, surfaces, volumes and hypervolumes. But there are infinite ranks of tensor, so you don't have a "complete geometric object" description. The Riemann curvature tensor, for example, in 4D space is a 256-dimensional geometric object

[@flpamr8850]

@@SomeMrMindism that makes sense. I guess I was thinking about some sort of basis object, which would make a bit more sense (I think). Because all you have is a set number of vectors and another set number of vectors which are the ones your transformation goes to; and metric, ricci or other tensors like that are written using those. So this called complete object would be what happens when you sum all possible geometric combination of this basis... Does that makes a little more sense?

[@ASHISH_339]

Very Interesting and well explained 👍

[@Hevander75]

Parth you need a podcast

[@DeclanMBrennan]

Interesting concept. Once you do the imaginary substitution, can you use quaternions for the resulting 4D space? Many years ago, I was enjoying an ale by the side of the river Cam on a lovely summer day; when Stephen Hawking came speeding along the river path in his electric wheelchair with his nurse jogging beside him leaving a little cameo in my memories.

[@TheNagurentala]

I (with a Ph.D. in Geophysics) used to toy with an interesting extension of the same idea - treat mass (m) as the spatial projection of matter and charge (q) as the temporal counterpart. With normalized dimensions, the property of matter can be expressed as, say M+iQ where i is the imaginary unit. Interaction between two particles gets rid of the pesky negative sign for gravitational attraction (I used to cringe every time I told my students that this is just a matter of 'convention') but generates 'imaginary' forces - any takers? :)

[@profdc9501]

It sounds to me like what you are thinking about could be similar to analytic continuation, with the Lagrangian as a function of mass extended into the complex plane. The resulting Lagrangian should still have a stationary solution that is the same as the real-valued mass, but might offer some interesting alternatives when treating many body systems as their "imaginary masses" could be related together which could be used to find different stationary solutions.

[@TheNagurentala]

@@profdc9501 Since I left Geophysics a long time ago and Physics even longer than that, lmk if it is ok to communicate - [e-mail]

[@wayneyadams]

I realize you said "geophysics", but i have to agree with Sheldon Cooper when he says geology, and by extension any science(?) with the prefix geo is not a real science. LOL

[@extripl]

Actually part from 1:10 to 1:15 in not exactly true. Square root of any positive real number x is defined to be non negative (sqrt(x) >= 0). If we had have quadratic equation like x^2 = 2, then we would have have two solutions, positive and negative sqrt(2). Except for this, greate job!

[@walterufsc]

You are correct. In fact, the square root symbol implies only the principal (or positive) root.

[@aryanbaghel7881]

No we don’t have two solutions sometimes.......

[@Ewr42]

wow, mathemathicians trying to complicate stuff as always.. what even is a positive number and how do you choose which one is positive and negative? sqrt(x²) defines |x| not sqrt. it's like an integral, you may even say sqrt of 1 isn't i² or -1 but, sqrt(sqrt(1)) is still i, -i, 1, -1 and thus, 0+i, 0-i, 1+0i, -1+0i. Square roots ARE like anti-integrals and defined by polynomial expansions, specially newton's binomials just like derivatives. Thing is, newton and leibniz didn1t know about realtivity and quantum logicism, or anything about electricity or computation, really. when you get those tools in the future, or our present, you'll be able to see how 1=0 was never a contradiction, but a sideways answer. it's right in it's own way, which might be exactly wrong or exactly right, so you see, this idea is fractally self-referring and explains itself, 0, 1, and i bet with fewer than 100 pages i could prove that pascal's triangle exists in the platonic field of information(quantum energy or anything one might want to call it) and thus the universe can repeat itself not only in time, but locally. also in imaginary time and its imaginary surroundings. at least that's how i see it, i haven't watched the video yet nor have i any proof besides my art, poetry, equations, sculptures, inventions, experiments, drawings, paintings, and most of all my mind itself, but i think like hawking i can understand all this pretty easily. i feel so sorry for him not being able to draw pictures, diagrams and such, but also i feel the complexity of how that improved his cognition and made his mind so much more resilient to als

[@Ewr42]

besides, k=(-2)² =(-2)*(-2) =(2+i²)*(2+i²)= (2+i²)² = 2² + 2(2i²) + (i²)² (i²)(i²)2² + i²(i²2(2i²)) + i²i²i²i² (i²)²2² + i²(i²+i²(i²+i²)) + ((i²)²)² -1²*4 + -1(-1-1(-1-1)) + (-1)²)² /-1(*(1/-i)) 1. (-1)-1*4=> +1*4=>4 2. (-1)(-1-1(-1-1))=>(1+1(1+1))=>(2(2)) 1.=> -1²=+1 +1²=1, proof: 1*1=1. 4-4+1 4+0+(-1(4*-1)) 4-(4*i²) and thus, k= (4-4i²) 4(1-i²) = 4( 1+(-1)*i² ) k=2(2-2i²) prop: k²=16 k=2(2+2i*i)² (4+4i²)² 4²+2(4*4i²)+(4i²)² 16+(16i²)+(-4)² 16-16+16 = +-16. k²=+-16 =(16 * +- i²) sqrt(k²)= k = 4(1+i) sqrt(k)= (2+-2i) =(2*+-i²) thus, 2* +(-1)=-2 AND 2*-(-1)=+2 so, why is the first square root defined to be positive and again, what even is a positive number? 2 is an integer, natural, rational, i get it, but the signal, what does it mean? i claim that *+-(i²) is the same as the *1 and +0 transformations. not just in the way it transforms, but in needing two steps, we should account for the number of steps (or define negatives and positives as quantum entangled logically. being opposite, but none being default by definition) making it exactly like calculus, specially diferential calculus that leibniz messed up by saying one couldn't divide by zero, a problem of misunderstanding of HIS, not newton's and certainly not everyone else's. the idea works, besides that, it explains everything from plato to hawking and maxwell to einstein going through planck and poisson, deep into the everettian flatlands and outwards to the multiverse of quantum possibilities, states, universes, presents or anything you wanna call your experience of reality. you get entangled with the information and it sticks with you. you rotate it, applying i², making it 1=0, or perpendicular, seeing it in another way, you can express it through another i² of acting by the idea and transmiting it to other humans in the future, near or far. memes not only define our culture, they ARE our culture, and even deeper, they represent a deep shared-counciousness that if we managed to entangle the world's most brilliant minds into a single Hyper-Computer, we could maybe even measure the flow of information, energy and entropy with a bunch of EEGS, fMRIs, chips, hololens, virtual workplaces for 4d arts, een 5d with colors, 6d with sounds.. and so on and so forth because information comes to us from many different angles, we might as well act on it on many different angles too, we just need to win the fight on our ego and our cognitive dissonance saying this idea is too crazy, newton and einstein couldnt go deep into this imaginary black hole i'm inside it, crazy as cuckoo but the view is amazing.

[@Ewr42]

not the first one inside, for sure, i mean, craig two comments down and the first i saw after my second comment was exactky on how everything is made up of 2+2i and 5+0i reference: Chad Wilson 2 days ago Another series here on KZbin talked about every number basically being able to have two components, the real and imaginary, ie, (2,2i) or (5,0i). What if time is three-dimensional like space, such that every particle has [x(r,i), y(r,i), z(r,i)]?

[@mr_rede_de_stone916]

Very nice video, I love imaginary time. Just a quick thought though, I find it a little weird to teach complex numbers (plus, in such a rapid way) before suddently expecting the audience to know about Minkowski space-time. I guess it could be argued that in such a way everyone can find interest in the video no matter their level, but I think if you don't already know some stuff about the subject of the video you kind of miss the point watching it - so why introduce complex numbers if a background is needed anyways? Well, anyway keep up the good work.

[@Tuli0]

0:43 just a detale: the correct form of representing i is i^2: -1. Nice video by the way

[@IceMetalPunk]

What's the difference?

[@Tuli0]

@@IceMetalPunk mainly because it’s a definition, and operations like this: i^2: sqrt(-1)*sqrt(-1):sqrt(-1*-1):sqrt(1):1 doesn’t make any sense, so we avoid this problem not using a negative number directly on a square root

[@IceMetalPunk]

@@Tuli0 I don't quite understand still. You say that operation doesn't make sense, but it does, because sqrt(1) doesn't just equal 1. It equals +/-1, and in this case taking the negative solution remains consistent with the definition, no?

[@Tuli0]

@@IceMetalPunk but If you do so, you are supposing that sqrt(-1):-1 still noon sense

[@IceMetalPunk]

@@Tuli0 No, you're assuming that sqrt(1) is +/- 1, which it is. You're not assuming sqrt(-1) is -1 unless you choose the positive root of sqrt(1), which of course in this case is not a valid solution. It's like how when you have multiple solutions to a quadratic equation, sometimes one of the two solutions works out in the symbol manipulation but isn't valid and you have to discard it. In this case, sqrt(1) = +1 is invalid for this equation.

[@Marinerko]

Hey parth, could you explain quantum loop gravity? I really just can not grasp it really well

[@jliu214]

I still remembered when I first learnt the special theory of relativity as an undergrad, the expression for invariant interval (dx)^2+(dy)^2+(dz)^2-(cdt)^2 really makes me think that time looks like an imaginary axis while space on the real axis. May be time is imaginary anyhow. And I always wonder why we can easily feel 3 dimensions in space but only one dimension in time--in principle, physics does not forbid two dimensional or even multidimensional time if space and time are correlated. Pretty sure we are still only scratching the surface in understanding the universe.

[@winterrain870]

Interested in the relationships between QM and Statistical Mechanics.

[@smolboi9659]

Wick rotation sounds like the Hilbert transform.

[@aryanbaghel7881]

Can you make separate video on “Minkowski Metric”.

[@Ryoksos]

Hey, I hope this doesn’t sound pedantic, but when you’re explaining square roots, you say that the answer to the square root of a number is two answers, both the positive and negative, but just the square root symbol actually always implies only the positive answer. The answer only becomes positive and negative when you, say, have x^2 equals a number, in which case x equals plus or minus that number’s square root. If you already knew this and just omitted it for simplicity, I totally understand. Anyway, I love your videos and this one was very interesting so keep up the great work!

[@walterufsc]

You are right. In fact, the square root symbol implies only the principal (or positive) root.

[@exponentmantissa5598]

Its funny you mentioned engineers using j in place of i. I am an engineer and that is true. The piece that has always puzzled me is that actually engineers use both i and j. Typically I see j used in signal analysis such as e**jwt but I gets used elsewhere. If you ask an engineer what is the square root of -1 they will say i.

[@trueriver1950]

I disagree. The use of i removes the anomalous sign, but not the scale factor of c. You still have c^2.t^2 in the metric. I therefore prefer a different imaginary time where we set T = tic This has two enormous advantages. Not only do you end up with a "Euclidean" metric where there are no added constants or minus signs, but also we are already used to time passing tic by tic. Which of those reasons is more important depends on whether i have my prof hat on or my silly one at the, ummm, time

[@lorigulfnoldor2162]

Could imaginary time account for parallel universes, perhaps? Everett branches? As if at the time t2>t1 we can have some amount of different branch Universes splitting from one branch Universe at t1, but they are differentiated exactly by "how much of this complex t2 is imaginary and hom much of t2 is real"? Could it work like that, I wonder?

[@HeitorvitorC]

I love your videos, Parth. But these ads in the middle of a complex topic makes me get lost everytime. Maybe I'm just dumb

[@walterufsc]

It is possible that you have done just to simplify, but the dimensional inconsistency in the equation is striking. Obviously, the time must be multiplied by "c" (speed of light) in the space-time interval equation.