2. Introduction to tensors.
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- Опубліковано 23 лип 2024
- MIT 8.962 General Relativity, Spring 2020
Instructor: Scott Hughes
View the complete course: ocw.mit.edu/8-962S20
UA-cam Playlist: • MIT 8.962 General Rela...
The notion of 'coordinate' bases. Several important 4-vectors for physics: 4-velocity, 4-momentum, 4-acceleration, and their properties. 1-forms, and tensors more generally. Using the metric and its inverse to raise and lower tensor indices. Introduction to tensor fields. The number flux 4-vector, and its use in defining a conservation law.
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A high school physics teacher, I want to follow up GR so many years after my graduation, and this open courses lecture is greatly helpful. Thanks a great lot!
Great initiative from the side of MIT to share such valuable and prestigious sessions to common people while some universities are not willing to share.Please do similar for econ classes.Hats off to you MIT.
Google topdoctor Yola
“common people”?? As in commoners, riff raff, peasantry? These binaries we’ve all soaked up need to be done away with but your main point is right on. All universities should release these types of videos
Regular guy would say I will drink water, the professor will hydrate himself.
A great offering by MIT. Making these courses available to everyone makes the world a better place. This open altitude is not found everywhere but it is at the heart of what makes a great place great.
1:19 Recap
4:20 Basis vectors
8:54 Transformation of basis vectors
24:01 Operations using four vectors.
59:44 conservation of 4momentum
When he said "That's the damn vector", I felt that.
Was waiting for this course!
Wonderfully explained. Thank you.
I’ve begun learning General Relativity using Bernard Schutz’s book. These lectures (so far) match the notation and topics of the textbook. This is truly a gold mine of information! Thanks many times!
MIT ocw is just the right way to learn in quarantine
True
I'm taking a math course in calculus on manifolds which covered tensors and it seems like the physicists use a pretty different language...
AT LAST , general relativity from MIT with videos. GREAT DAY FOR ME!
As with any of these advanced and technical courses, the views drop off exponentially with each successive lecture. Kudos to those who kept with it all the way to the end.
tbh I repeatedly re-watch the same parts of the same lecture to aid understanding before moving on.
It is because one needs exponentially more time to understand each lesson!😅😅😊😊
Finally after so many years.
The general relativity.
Good
Thanks MIT OCW. A lot of thanks ♥️♥️♥️
Thank you for sharing! 🙂
oh thanks MIT OCW now i understand tensor calculus
myyyy gooooooddddd i have never commented on any video in my 10+ years of youtube but i wanna say that
THANK YOU MIT i really needed this ive watched all three semesters of quantum by prof Zwiebach and im thankfull to the ocw team
ps a little QFT would be nice
Just starting this series (and watching Stanford's Leonard Susskind's series on GR concurrently). At the beginning of this video, Prof. Hughes is drinking water and mentions that his "daughter has some kind of virus." This was recorded in Jan. 2020 -- just as COVID was appearing on the radar. The first recorded case in the US was January 20, 2020 (and that was in Washington state, on the opposite side of the country from MIT). So I presume this was just an "ordinary bug" his daughter contracted. Still, knowing what was to come in the next few weeks and months; it was enough to give me pause. In fact, I will be interested to see if this becomes an issue in the course later on.
Correction at 47 min for spherical coordinates: the "dφ" is missing on the final term.
It should be something like this:
vec{dr} = dr vec{r} + r dΘ vec{Θ} + r sin Θ dφ vec{φ}
en.wikipedia.org/wiki/Spherical_coordinate_system
I noticed it too! It happens at 47:20
Thank you for sharing
How did e1_bar.e2_bar become a 4x4 vector, what matrices/vector are exactly e1_bar and e2_bar?
I like how this guy is moving relatively fast!
I can't thank you enough for sharing this
6 February 2020 : "My daughter has some kind of a virus"😅
Great lecture btw. Keep it up MIT!
Thanks, Finally some good classes about tensors
Tensor defined simply is the wave contribution to force vector. More is not needed.
00:51 When he said virus, only corona came in my mind.
RISHABH KAUSHIK this was from Spring 2020 so...
that's crazy, he probably had covid
And he said drinking water constantly would be preventive. It souds wierd...
@@brukewossenseged4241 He is probably immune now. XD
@@user-pk5rc4or2w drinking a lot of fluids help our immune system battle virus though
tensors are love
This thing saved my grades.
"My daughter has some kind of a virus". Famous last words lol 😄 Great set of lectures though. Loving them 👍
Thank you MIT
Thanks so much for sharing this
Just perfect
Are there any plans to post introductory QFT lectures as well?
I have no idea what MIT's plans are, but Alex Flournoy has a great series on that topic (well, more specifically, particle physics) that I highly recommend.
@@conoroneill8067 Thank you. I just discovered his lectures. It's a treasure.
28:12 man of culture
This is what I want for years, my university won't allow selecting courses from other majors, I'm in computer engineering btw:-)
Finally I can try to understand GR
me too from CSE :)
can anyone give me link of SR
One doubt: Why u.u implies a negative sign in -γ²+vγ²= -1? Because ds = (-cdt,dx) and ds/dt = (-1,0)?
does anyone know the meaning of symbol 2 an 3
Are all these concepts of special relativity,(any quick reference to learn those if so)
because I am having a bit of trouble with calculations
Same
Yeah the Lorentz transformation portion is from special relativity, mit ocw has some videos for the same as well with relatively shorter lectures.
Are there any tensors in this introduction to tensors? It should have been called "short recap of special relativity"
what does that vector notation (v ~) mean ?spacelike component ?
any letter with a tilde under it denotes the spatial part of that 4-vector
It represents the usual 3 vector.
11:10 amazing singing
are there any solutions to the problem sets given? Thank you for the knowledge
There are no solutions available for the problem sets but there are hints available for some of them (PS 1, 3, 4, 7). See the course materials on MIT OpenCourseWare for more info at: ocw.mit.edu/8-962S20. Best wishes on your studies!
The negative sign in front of time comes from the fact that a pulse of light expands like a growing sphere. dx+dy+dz=cdt. Thus, -cdt+dx+dy+dz=0 for light it equals something else for everything else, call it ds. Hence ds=-cdt+…
Aren't inner products supposed to obey positive definiteness? How can we define an inner product that can be negative?
just redefine it for Minkowski space, simple as that
Recorded early february 2020? Cant wait to see the covid fallout as the series progresses.
So for the Kronecker delta at 6:40, is there a reason we use the mixed indices over, say, a twice covariant or twice contravariant Kronecker? I've never been good at distinguishing the usage as it seems we just take the tensor product between 2 co / contravariant vectors, or the tensor product between a covariant and contravariant vectors.
What
@@MasterCivilEngineering The kronecker delta shown at 6:40, has 1 index high and the other low. Doesn't this insinuate that our kronecker delta is made out of the tensor product of 1 covariant and 1 contravariant vector?
For instance, if we have the kronecker with double lower indices, that would imply that it was constructed out of the tensor product of 2 covariant vectors, right?
I could be wrong, sorry.
Delta is not covariant or contravariant.
@@user-pk5rc4or2w yes but it can be made up from the tensor product of covariant or contravariant vectors, right?
This is why it's so infuriating that he does not explain this. In the same time it took him to say "look it up on Wikipedia" he could have explained the notation. Now if you actually follow his advice, the Wikipedia page will send you down a deep, deep rabbit hole that has nothing to do with what he was trying to do here: A more compact notation to write *the exact same thing* he wrote out before explicitly in the definitions of e1 to e4.
Is any indian here studying in MIT, I just wanna ask that how much percentage is needed in 12th to get into MIT as in field of experimental physicist. I am currently 14, and have studied all of these things which aint of my level and i understand it. I want to study in MIT after my 12th, thats why i want to ask.
"Depending upon what kind of muscle relaxants you enjoy taking over the weekend..."
What a legend.
I just wrote a better google brain using this course. Thanks Professor.
What's a google bain?
@@davidhand9721 An automata owned by google.
I just performed "The Wave" an 1980's Breakdance move with both arms.........I've displaced Space/Time
If your hands get really cold and maybe you have some minor heavy metal toxicity in your joints, your get frost bite that looks like a ring , a black ring . It disappears when you warm your hands by the fire . Thats the ring . That's pain
It is crisp clear he is like talking in a real classroom. Can you help me please tell me which brand of microphone he is using?
He's most likely using a Sennheiser or a Rode wireless lavalier microphone. Pretty much any good quality lavalier microphone should work well.
@@mitocw I really appreciate it. I was looking for Sennheiser and Rode online and now you just confirmed to me they work great for my classes and talks.
Genius
Just small heads up to channel admin, love the labelling of each part of the video, but some small spelling mistakes may confuse viewers (as a four-vector is intended to relate to the number 4, so "for-vector" isn't correct and doesn't help conveying that). In particular, the 2nd, 12th and 13th sections need that small fix!
Awesome lecture, watching this instead of what my university offers for GR because this gives me a thorough, wider mathematical context and amazing physical intuition. Brilliant, thank you very much for making this available!
Thanks for your note! The chapters you see are actually auto-generated ones made by UA-cam... using it's auto-generated captions. We've double-checked the captions we made and they are correct. We'll see about manually adding chapters to this video to avoid the auto-generated confusion. 😜
Sir which book you have preferred for this
Carroll
@@iaexo thank you dear
That’s just Lorentz’s perspective. ;-)
Which level is this lecture for, Bachelors or Masters, because I am in 3rd year of my bachelors and I can not understand more than 30% of this lecture in first 30 minutes.
4th year Tensor analysis
MIT website says its a graduate level course but you can understand it if you know linear algebra, had some introduction to tensors and had introduction to special relativity and electromagnetism
@@seanpaul2562 actually I didn't have any knowledge about tensors and just came here to understand basics, but then realized that it is a bit advance level.
@@umerfarooq638 i seconded Dirk Knight's advice. You have to read the textbooks on this subject before these can sink in.
I first learned about tensor when I was in 2nd year undergrad (physics) from video lectures. But I can only confidently say i'm quite good at it after reading caroll's and schutz's texts and watching some lectures.
Some exercises will help you along the way. You can find them in caroll's first chapter.
When "Time Duration Timing" grew up and matured, it became Relativistic Tensors, and more specific to purposes in the languages used.
Eg in the "common usage" language of Radio Circuit Astronomy, Navigation is via a floating Quiescent zero-centrality differentiates of Spectroscopy, a measure reference-framing that has equivalent meaning to Reciproction-recirculation Singularity positioning integration in Newtonian Fluxion-Integral QM-TIME quantization identification by Math-Phys-Chem and Geometry, pure-math e-Pi-i emitter-receiver, spectral phenomena.
It seems like the captionist mistakes "Kronecker Delta" for "chronic or delta".
Thanks for spotting that! The caption will be updated.
At 47:21 it should be rsinθ dΦ instead of just rsinθ
I really liked how he just slipped in the contravariant nature of the basis vectors as opposed to the covariant nature of the representation coefficients.
It's the other way around
How does thermal energy singularity frequencies rebounding in magnetic fields of forced cycling circulation patterns that are planets and stars, collapse in on itself? What force constricts it? Thermal energy expands mass? It is a propellant. What is the mechanics of the attracting force and where does it originate? Contained thermal energy singularity frequencies in resistance in mass is outward force. What reverses this force to fall inward. As mass is reduced, its forward maximum momentum velocity increases towards its original state of thermal energy singularity frequencies out of entanglement of magnetic fields as thermal energy singularity frequencies in resistance to cold space. Space itself is maximum momentum velocity in resistance at maximum distance of forward maximum momentum velocity. You cannot exceed maximum momentum velocity of resistance that is the fabric of space itself. Mass amplification of the field of space is in entanglement of it, and distance of forward momentum is redirected into magnetic fields cycling circulation patterns of maximum momentum velocity in resistance as mass. The greater the entanglement amplification, the greater the redirection of forward momentum of amplification, the greater the distance of forward momentum slows as redirected trajectories of maximum momentum velocity in resistance in mass. Mass is equalization to resistance. It neutralizes resistance as occupied space of mass. Mass is the weakest point of resistance which mass trajectories are repelled to by resistance in vibrations of space surrounding mass. Mass is space accumulated in entanglement of thermal energy singularity frequencies rebounding in magnetic fields of forced cycling circulation patterns of mass. Thermal energy singularity frequencies and cold coexist throughout space as dark energy. This existing energy is space. It is vibrating at maximum momentum velocity of thermal energy singularity frequencies outside of entanglement. Entanglement is when these frequencies that are space, harmonize in entanglement and reduce in distance by redirected maximum momentum velocity, into magnetic fields of forced cycling circulation patterns of mass, in resistance. How does this become a dense black hole of thermal energy singularity frequencies? How does gravity pull? What force has the machnics to bend or curve the space surrounding it and how can it sustain its magnetic fields of forced motion holding mass together. Expansion is the strong force of thermal energy singularity frequencies. Thermodynamics is outward expansion to resistance. Ice is formed by lose of thermal energy. Space is comprised by thermal energy singularity frequencies that mass amplifies in forward momentum maximum velocity in entanglement as forward momentum. Explain how thermal energy that expands elements collapses as gravity? Or a black hole?
does he mean drinking water ?
Volume is low can hardly hear it
i'm a little sick of math, let's do some physics - prof Scott hughes 2020 what a classic 🤣🤣
What does hydrating have to do with preventing catching a virus?
It keeps the mucus membrane in your nose hydrated and at the right consistency to prevent infection through the nose.
And promotes general body function to boost immunity
This thread explains why the US is number in coronavirus...
Step by step video solution of civil engineering questions
Please film this again so that the stuff on the blackboard can actually be seen
You can get the lecture notes from the website just google Scott Hughes lecture notes and you can get all his 25 lecture notes
Tensor makes me cry :(
They really aren't too terrible. Virtually every mathematical object you've worked with *is* a tensor. Scalars, vectors, matrices and so on are all tensors. It gets difficult when you speak abstractly on them and derive their general transformation rules but if you approach them thinking "Hey, I've worked with these before", it isn't all that bad. :)
Edit: I'd also recommend a course on differential geometry / differential topology to really get a good understanding on manifolds, tensors, differential forms and the like. DG was my saving grace as we approached it , thinking "Okay, we're talking about surfaces (planes, spheres, etc)" and then we're told, "well hey, these objects are all examples of manifolds." So when you speak about a manifold, you may often times just refer to it as a surface. Then you go on and derive things like the first fundamental form (the metric), the jacobian (transformation matrix), etc. Very very easy, and much less time is spent on "How can I apply this general concept" and rather you focus on actual applications! (Side note, coordinates are yucky so differential forms are our solution when we want to be productively lazy).
X Is there a book that shall treat Differential Geometry with a Physics prejudice. Possibly a book written for people studying Physics (by a physicist)? Thanks!
UA-camr eigenchris has the BEST tutorial on Tensor in UA-cam: ua-cam.com/video/8ptMTLzV4-I/v-deo.html Please give it a try. He actually has tutorials on Tensor algebra, Tensor Calculus, and Relativity.
All hail tensors
omg learning general relativty for free
This video is based on a true story.
15:25
A lot of the he materia is not too clear because of poor camera work. The professor should use darker chalk too.
REGIÖNÄL WEÄTHER MÖDELING ÜITH PERFECT EVENTS
The title is very misleading...
Doing special relativity with "tensors" as this instructor is trying to do is a bit obfuscating. Lorentz transformations are simply actions of SO(3,1) on R^4 which give the coordinates in moving reference frames since SO(3,1) preserves the quadratic form t^2 - x^2 - y^2 - z^2 which is the invariant interval (happens to be the hyperbolic measure). No tensors required, only group actions. Essentially the same situation holds for electromagnetism since Maxwell's equations are Lorentz invariant. A "tensor" does appear in the form of Maxwell's tensor, but giving it this new name makes it seem much fancier than it really is, it's simply an SO(3,1) invariant bilinear map.
Then in general relativity, tensor fields are necessary since spacetime is a 4-manifold M with tangent spaces having the above structure. Curvature is thus measured by the connection on this pseudo-Riemannian manifold, and because it's a commutator of connections (covariant derivatives) this is a genuine section of 4-linear maps, a "(3,1)-tensor" since 3 arguments are dual vectors and only 1 is a tangent vector. The only reason higher rank tensors appear is because the metrics need to "play well" with one another upon parallel transport while remaining locally SO(3,1)-invariant.
He says at one point (I can't identify where) that he will give a "very precise definition" of what he's talking about. Not a single precise definition appears in the whole lecture!
The metric of SR which is used in the construction of the interval is a tensor so when you say no tensors are needed in SR that’s simply not true.
Also the idea that tensors are somehow more obscure or fancy than are Lie groups is a pretty subjective view. And an obvious reason to work with tensors in the SR setting in a course on GR should be pretty obvious: it allows for the laws to be put in a generally covariant form in a simpler setting than is the case for GR and you want students to use tensor notation in a familiar setting like SR. General covariance is of course one of the most appealing features of GR, that the laws take the same form for everyone is a profound innovation which falls right out the moment tensors are used.
@@muttleycrew Well, you don't really need the language of Lie groups, it just makes the point more concise. Basically SR can be summed up with "space-time interval is lorentz invariant" and classical physics is ported in by throwing in gamma as a "fudge factor". The simple idea of group actions isn't really obscure, it's the most straightforward way to make this intuition rigorous (in my opinion), and one can then mathematically encode events, measurement, physical quantities, and so forth (such as charge in E&M).
The language of general covariance can be safely ignored in SR where the situation is highly specialized. Of course, we have to introduce it to students of GR, but if physicists wouldn't faint when faced with an ounce of rigor they shouldn't be saying nonsense like "a tensor is a thing with indices that transforms like a tensor." This is not tongue-in-cheek, that's the actual "definition" used by most university physics courses on GR.
The Flagged Dragon you appear to have confused me with someone in need of a physics lesson I didn’t ask for and didn’t need
@Dirk Knight But you can get explicit numerical representations, they're matrices! My issue with this lecture is not a lack of abstraction per say. He talks tensors, tensors, tensors, but there's not a single explicit definition. Experimentalists actually do this quite well as they need to represent their quantities explicitly on computers, so they do come up with very precise definitions.
In the last lecture he explained his reasoning for this. Since the students have already taken Special Relativity in this class, he wanted to introduce some concepts that will be relevant in General Relativity, but in a more familiar subject first. He said that these concepts aren’t necessary for Special Relativity but would serve as a warm up for General Relativity. So I understand his reasoning for starting with examples from Special Relativity to teach tensors since the students are presumably more comfortable with the subject. At least that’s how I interpreted his explanation of this in the beginning of the first lecture.
Thnx from IIT Bombay
COVID-19?!
yeah what other virus
Leon TicklePickle why are distances not respected and why no one wears masks?
@@Giovanni2862 It's a spring 2020 course so this was recorded before people took it seriously. By the halfway point of the series, he's doing lectures to an empty room.
Respected MIT university professor in India country's MHRD minister ramesh pokhariyal told in corona situation INDIA's biggest exam JEE MAIN and NEET is starting 1st September 2020.but India's majority students says that please our exam conduct after2 months because india's sop is to weak and people not obey social distance and many state in heavy rain. therefore those states in very highly spread poor.and majority peoples exam centre very far like 100km to 300km how can they journey in lockdown state.I request to MIT UNIVERSITY HEAD PROFESSOR please write to email our mhrd minister to exam continue ofter 2 months because corona situation we are very afraid and how can we journey in corona viruses situation this situation very bad and headache situation. Please support domestic India's majority students oppose exame in current situation. Thank you for reading.please as soon possible as reply.
Amazing lecture! It's way better to start from a geometrical viewpoint including differential form algebra than to just portray GR as an extension of special relativity. Unfortunately in today's lecture it's mostly being taught the latter way. Once you've completely understood the mathematical formalism behind it you gain a much better understanding of the underlying properties of the theory. I was 12 when I began to study general relativity but in the book I was using it all seemed like only a more complicated extension of special relativity. The first time I was exposed to differential forms was when I began to study Polchinski's string theory books at the age of 14 and it revolutionized my understanding of GR and QFT as well. Now at the age of 16 I'm making contributions to actual theoretical research in particle physics and relativistic astrophysics. Thank you MIT for providing all these ocw lectures. Without them it would've certainly been much harder to develop all this knowledge.
sure..
handwritten blackboard makes life harder. That whole thing needs to be replaced with a giant screen that reflects whats on a tablet in his hand or something.
I was thinking how efficiently he handled it. He's doing a great job.
@@Zamicol how you got verified with only 4 subs
As long as you dont believe YOU'RE the time traveler giving yourself information. You can escape the looper trap . It just is what it is . Aliens , spooky shit , ghosts and stuff . Just not the ego , never the ego . And you're free too .
A _physicist_ explaining tensors?! Oh my fucking god NO!
Surendra Kumar chemistry
Some MIT clown should work on those blackboards in the middle of the night so in the morning they slide sideways as well as up and down...
Hi MIT Anna let's get acquainted the bot has got into the comments section let's report that freaking bot so UA-cam blocks the spam channels
ееее
Because retooling every engine on earth would be a ton of work . And diabetes rates and heat stroke sensitivity did not arise from political oppression
He should talk a bit slow.
Weak, he is missing the why. Too much physical idle motion. Really not a good professor. Grandiose for nothing.
I find it very strange that at one of the best universities on planet earth, i.e. MIT, the instructor teaching Einstein's GR does not know why the square of the spacetime displacement's component in time is negative. This is simply because time has to be described in a coordinate system that is orthogonal to the spacial coordinates, which just happens to be a multiple of the imaginary number, i.e. _i_ . But why imaginary? Because imaginary numbers, like time, are by definition cyclical, i.e. z = exp(i.theta). Not equaling the speed of light (C) to 1 would require the time-coordinate to be multiplied by iC, and the square of iCt is just -(Ct)^2 which is the first term in the spacetime displacement. If the time-component was not negative, then the spacetime displacement of 2 objects in *flat spacetime* separated by spacial coordinates (deltaX, deltaY, deltaZ) would not be equal to Zero, but it has to be Zero because the time it takes light to travel between these 2 objects is *exactly* equal to deltaD/C, where deltaD^2 = deltaX^2 + deltaY^2 + deltaZ^2, and indeed it is the *time* it takes for light to travel between points that ultimately defines the curvature of spacetime between the points, not the arbitrary coordinates in a coordinate system.
You are not explaining why there is a minus sign, you are just using an imaginary number to describe it, which is also derived from the minus sign fact. Using imaginary numbers is just an ease of writing down the displacement, and apparently, they don't use that notation any more. This is why the prof didn't mention it because it is not important.
@@johng7602 The problem is not that he didn't mention it. The problem is that he said he does not know. In fact, if you look at the textbooks, imaginary time is well mentioned in the Minkowski's metric. I'm not the first to mention it. Here I'm just giving the reason why it is so. However, I think it is *very* important, because it describes the 4 dimensional spacetime coordinates we live in differently than if it was 3 coordinates of space and 1 _real_ coordinate of time, because it shows us that as time increases or passes, the magnitude of the spacetime coordinates shrinks.
Plot Twist: His daughter was patient 0 for the C-19
what a useless class for tensor
his way of teaching sucks...too busy copying stuff from paper to board...compared to lectures from Schuller and Susskind...this is almost next to useless
I find it extremely useful
Surendra Kumar chemistry