Your Daily Equation #27: Curvature and Parallel Motion

Episode 27 #YourDailyEquation: In his general theory of relativity, Einstein described gravity in terms of the curvature of space and time. Join Brian Greene for an introduction to the mathematics of curvature, which Einstein used to fashion his gravitational field equations.
Even if your math is a bit rusty, join Brian Greene for brief and breezy discussions of pivotal equations and exciting stories of nature and numbers that will allow you to see the universe in a new way.
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[@richardschultze9926]

Dr. Greene, thank you for the time and thought that you are daily devoting to this series. It has been important for me during this pandemic period. It has been a daily time of intellectual stimulation and learning new ideas. I really appreciate it.

[@semmering1]

I can not enough agree with that statements, Brian you helped also me over this difficult times with your daily brain training..

[@zanedexter990]

you prolly dont care but does someone know of a tool to get back into an Instagram account? I was stupid forgot the account password. I love any assistance you can offer me!

[@bartk07]

Professor Greene with "Your daily equation" and Professor Carroll with "The biggest ideas in the Universe" - Perfect duo! I could listen and learn from them all day long. We live in happy times indeed. Big thank you!

[@platonicdescartes]

I have likewise been consuming both of their content ravenously. Also let me give a recommendation to PBS SpaceTime and FermiLab, who produce great KZbin content, even in the time of COVID-19.

[@frede1905]

"We live in happy times indeed". Yeah, yeah, sure...😓

[@Dr10Jeeps]

I really look forward to these daily sessions with Brian Greene. His ability to explain and communicate concepts is second to none. Thank you, thank you.

[@kagannasuhbeyoglu]

Thanks a lot Dr. Greene. Wonderful series.

[@stefanadler]

Greetings from Dusseldorf (Germany) and THANK YOU for this interesting series. It's like a beacon of light and these are really excellent explanations! Thank you

[@siddharthchavan910]

🤩🤩 Brian you are a wonderful teacher, fantastic you simplify hard content to extend that it is graspable I wish I had teacher like you!! THANK YOU SO MUCH 😊

[@shubrajchuckowree5670]

Absolutely brilliant, thank you Professor Greene

[@woody7652]

Thank you, Brian!

[@jlo3349]

I studied GR and parallel transport in uni already....but I never felt like i understood the physical interpretations until I watched this short and concise episode of daily equation. Thanks again Prof. Greene, you are a truly amazing teacher - God among mortals!

[@romyroll]

aaaaah the summation convention! :)) Dr. Greene, it'd be great to have a session on intuition for tensor notations, covariant/contravariant coordinates, and their importance to this topic. Thanks a million for the daily equation

[@nickciggs3968]

You have just opened my mind to math again. As a semi-retired psychonaught, this lesson in particular has bridged for me some conceptual/expression gap. I feel closer to whole in my understanding of things. I want to know more. As much as you can share.

[@gedlangosz1127]

Another fantastic lecture. Long may these continue.

[@kansuerdem2799]

Dr Green , i love your teaching .. your room, your view through the windows .. :) ...

[@squarerootof2]

Yeah, It's never windy outside.

[@martijn130370]

Thanks a lot for another interesting lecture! I am looking up Riemann Curvature now - amazing how many things this man came up with in his short life. His Riemann Hypothesis still is the single most important unsolved problem in mathematics today, ever since it was posed in 1851 (or especially since Hilbert put it on his wishlist)

[@mahadlodhi]

I was musing over the same thing, i mean gauss and riemann are probably the two most influential mathematicians of modern era. The work that they did on non Euclidean space must certainly would back in their own time have been considered a purely mathmatical manipulation devoid of any physical significance, even by themselves as well. But turned out later on that they long ago had unknowingly had solved the problems to which einstein was led by the general postulate of relativity in the description of physical phenomena.

[@dominiqueschlogl8700]

Riemann hypothesis? Zeta-Function? Connection to quantum physics?

[@sakettiwari5603]

Professor Greene when are Maxwell's equation coming in a series of videos.

[@tomsmith4542]

nice visualization with the ball !

[@sarahhoward8455]

Thank you 📚💯

[@devikab.s.3613]

Great explanation Sir.. parallel transport was some alien concept when I studied it, but it is very much clear now.

[@tewariasim]

Hi Brian As is always the case of science, it answers few questions but raises a lot more questions Wish This was more of an interactive session. Truly outstanding nonetheless. Regards

[@neeteshmudgal1209]

thank you so much proff.,, great insight by mathematical tool.

[@globaldigitaldirectsubsidi4493]

Brian Greene rocks

[@adityajha9665]

Since I'm a high school student of science....so i love mathematical interpretation , even more than theoretical interpretation....

[@timjones1815]

Professor Greene, you are throwing so much illumination on the themes within Roger Penrose's 'Road to Reality', which I have been grappling with a limited degree of success...thank you!

[@craigwall9536]

Even more than English Composition, apparently...

[@Wrotan]

Love the basketball example. Dr. Greene is always good for a laugh.

[@salimyaghi605]

Dr.Greene, thank you for amazing words. Is it scientifically correct to say that gravity is due to field property of space-time or its just a pure objective geometric property of space time ? thank you

[@bharatrajesh698]

Always a fantastic feeling to lean into our own ignorance sometimes 🤣 Hopefully I'll get to work on these tools one day!

[@paulc96]

Hi Prof. Greene, Greetings again from West Wales. I hope you are keeping well. Just a quick question please. Sorry if I sound a bit dim, but I don’t understand where the Lambda exponent comes from, (and what it represents), in the Riemann Curvature Tensor equation. You say that the other four indices (rho, sigma, meu & neu) are all a number from 1 through to n, but there is no mention of Lambda ? (And I did watch it more than once.) I would be really grateful if you could explain it, for the benefit of us “mathematically challenged” viewers. Thanks. Best wishes & Stay Safe. Paul C.

[@taseen707]

Sir,could you make a video on the boltzman's entropy equation and the blackhole entropy equation(hawking radiation)

[@HebaruSan]

I find that the introduction of the "Einstein summation convention" marks the point in any advanced physics presentation where the symbols stop meaning anything.

[@rv706]

Why?

[@vinniejudilla3921]

Hello from Jacksonville Florida

[@shishirjha7744]

Where is this house ? Background is so calm & serene. BTW, before any one starts with a text book on GTR, one must go through the two lectures on GTR by you. Absolute beauty.

[@mini30coupe]

Sir, I cannot find an answer to the curvature in space. In a display it always shows the Earth as though it is in a net with the moon orbiting caught in this "dip" in the net. My question is shouldn't the curvature be showing that "dip:" completely around the earth and not just the regularly shown earth in the net. Shouldn't that net actually curve entirely around the earth?

[@taseen707]

Sir,could you make a video on maxwell's equations

[@gpcrawford8353]

The Italians also added insights to reimanns differential geometry including tensor calculus his classmate Marcel Grossman taught him this .

[@andreranulfo-dev8607]

27:18 And that's why we learn FOR loop in algorithm.

[@dominiqueschlogl8700]

Riemann, Ricci and Christoffel seem to require their own series😊

[@gautomdeka581]

Professor greene please explain those math. I didnt get clearly i wanna know so deeply

[@geoffreyfaust3443]

FYI If you look at Thomas Reid's 1764 essay, The Geometry of Visibles, you can see that he anticipated Gauss, Lobachevsky, Bolyai and Riemann's non-Euclidean geometries by pointing out that the back of the eye is curved and yet our mind correctly interprets that a triangle in space has 360 degrees, i.e., that the real triangle is not one of the non-Euclidean varieties, even though the projection on the back of the eye is not a flat plane.

[@geoffreyfaust3443]

Oops, make that 180 degrees!!!!!!!!!

[@onderozenc4470]

Even the centrifugal acceleration manifests itself as the precession of the orbital velocity vector around the rotational axis.

[@aliim.s.p4151]

Hello from Alexandria Sir

[@nazishahmad1337]

In my opinion , Teaching a bit of Tensor calculus before getting into Reimann curvature tensor and it's derivation would be a better option..

[@erwinmarschall8879]

(a) Riemann, please (b) I think "teaching a bit of tensor calculus before" isn't a good idea for the intended audience. (c) Prof. Greene announced at 29:22 "I will pick that calculation up for the die-hard equation enthusiast sometime".

[@somathebluemoon]

About how space-time fabric is curved by the presence of an object, is density the major factor in determining the degree to which the fabric is warped? Aka, the intensity of the force of gravity

[@mikeclarke952]

The short answer is no, mass is the major factor. If the Sun collapsed into a BH the Earth would still remain in it's exact orbit. However as you guessed the Sun's now greater density would curve S-T so strongly no light can escape. This very strong curvature is also very close to the Sun's centre, only within a few 100 km.

[@telekinetic6472]

Given the centre of the mass (or density) is in the centre of the sphere. Would this make that typical animation that was shown - (the one with the moon orbiting the earth) actually incorrect - As the "curved mesh" of space should converge towards the centre of the earth ...not to the bottom of the earth?

[@taseen707]

Sir,could you explain the weird shape of the CY Manifold?And is there a equation behind the prediction of the number of dimensions of CY Manifolds(why is it six dimensional)

[@divyeshraj6306]

Well, the CY manifold can be of any number of dimensions!!!. The CY manifold when employed as a way to compactify the extra dimensions in string theory, has to be 6 dimensional as we have 10 dimensional space time, in which we can observe only 4,so the extra dimensions need to be compactified into a manifold with certain nice properties and it turns out that CY manifold is the right object!!

[@LawrencRJUTube]

I was so impressed with your "solving" (not really the right word) of the "mirror round trip 'paradox' " that I thought I would check into your attempts to "clarify" the general theory -- and The first episode of the two I saw (where you had your back to a window with a view of snowy slope rather than a messy desk) was meaningful but, you lost me on final math regarding the quantification of the difference in the tensors that have been moved through two different paths with parallel transport - that showed no change on the flat surface but did show a change on the basket ball. However, in the lecture earlier where you pointed out that simulated gravity (where the platform being pulled by an acceleration equal to G, or the "centrifugal" force produces weight) is EXACTLY the same as "non-simulated" gravity. The question I have with regards this EXACTNESS is this: Since we know that there's an extremely slight time dilation with "non-simulated" (real) gravity, such that time passes more slowly at the bottom of the Eiffel Tower than at the top, there should be an equal time dilation with simulated gravity. That would mean that there is a slight time dilation that is commensurate whatever weight is being felt whether it is a "simulated weight" or a "non-simulated" weight. So in the "paradox" I presented to you, when A and B got back together with an "unmentioned observer" C, both their clocks will be the same as you pointed out -- but C who never left his inertial frame of reference will have a clock that is "fast" compared to the clock's of A and B. The question I have is how much of the difference is due to the dilation the C observed in both A and B and how much of the difference will be due to the two 1G burns in the first and second part of the journey? Some of it, if not all of it, should be due to the acceleration. Probably only a negligible amount due to acceleration because it takes a very long time to show any difference between two clocks that are separated by even a mile in elevation. But if we are splitting hairs I would guess that the amount by which Cs clock is fast would be due to both!? One other question: If space is slightly curved on cosmic scale (rather than flat as it seems to be) wouldn't any two observers in space be in different frames of reference even if there is no relative velocity between them. Such that each observer would regard the time on the other as slower than his own? (and be foreshortened in the direction of their separation and have clocks on the closer side of the "other" asynchronous with clocks on the further side -- the whole ball of wax of special relativistic effects.) At one point you said "now we're cooking with gas". I had just started teaching in a rural school after teaching in a city school and a farm boy raised his hand and said "Mr. Lawrence, round here we don't say "cooking with gas" we say "diggin where there's tayters." :-) That boy was the laziest student in my class. But he became the most successful of any other student I had in that school. He recently retired from being head of Blue Seal Feeds for Western New York State. What he lacked in ambition he made up for by being very personable.

[@MyWissam]

Is a connection unique for a space, or can one have varied connections defining different structures (?) on the space?

[@briangreene6975]

The connection is generally NOT unique. You CHOOSE the connection to define what you mean by parallel. (Some connections are singled out by particular considerations--e.g. if there is a metric (a distance function), then there is a unique (symmetric) connection which the metric yields.) But generally speaking, the connection is for you to choose.

[@gpcrawford8353]

Brian Greene I’m a maths dumbo but have been reading about GR as well as your clear daily equations am sure that in the initial draft Einstein helped by Marcel Grossman got muddled up with the connections and produced a wrong result . Two astronomers one German one American on learning that a total eclipse could test it set of for the Crimea to observe said eclipse in August 1914 by the time they arrived the 1st world war had started and were arrested by the Russians as spies and the special telescope the the American had devised was confiscated. Hasten to add as America was neutral at time he got sent home the German got interned I believe. Apparently Einstein had not realised that time was warped as well so had to go back to the drawing board in fact from what I can make it this time warping is the major contributor to gravity.

[@indysbike3014]

So it can curve but does it mean it can also be compressed?

[@mikeclarke952]

Well spacetime is expanding since the big bang, so it must be able to contract. I don't know if contraction is the same as compression though.

[@aliim.s.p4151]

tho the possibility to see my question is (1/the number of comments) I won't be depressed, what are spacetime crystals sir and what breaks the time or TTS sir ???

[@mireillecourteau48]

A thing that I don't understand with the concept of gravity related to curvature is how this curvature influence "local" phenomenon like ocean tides on earth (which are related to the moon and sun)???

[@aluaki16]

Please prepare your clases before you start and your cámara in correct place, thank you!

[@adilsonsf]

I've learned to demonstrate Maxwell's equations, but this is more advanced. I think I have to study more math. I'm an engineer not a scientist . But, roughly I get the idea. 😏

[@fernweh3726]

What happened to episode n. 24?

[@HebaruSan]

There was a two and a half hour Q&A uploaded between 23 and 25

[@rv706]

I'll blow your mind: the Christoffel sumbol Γ, despite "having three indices", is *not a tensor!* Now I'm responsible for your sleepless night.

[@qualquan]

clear as mud

[@PlanetFrosty]

So, thinking about “repulsive” force of possibly dark matter how do we account for that?

[@TheRarest1]

Just wondwring if when consciousness changes the outcome of the double. Has anyone ever tried using animals? And does it ever make a difference If the person is aware of what it is they are even looking at?

[@TheRarest1]

Or unaware and just in the room

[@johnpinckney7269]

The definition of parallel has been played with. Obviously sliding the vector along the equator rotates the vector.

[@kyleklassen3660]

17:00 Two Words MATRIX LABORATORY AKA: MATLAB Signed yours truly, Kyle E.I.T.

[@aayushsoni9160]

But sir , you have moved the pen on the basketball by curving it's path in 3d

[@starroger]

Seems like an obvious way of proving whether the Earth’s surface is curved or flat. Also, if curved, how much so. The concept shown here just needs to be scaled up.

[@Valdagast]

Someone needs to send the good professor a longer extension cord.

[@trnod]

47 notices from app store... come on man... you got to check them out...

[@Sutha-ho1os]

Dear Brian, there is another way to do in a simple way. If you add two equal and opposite forces in the green point then it automatically moving the force with coupling. Your video great and I enjoyed. Thanks for your great teaching. But you are not telling some important points 😂😂😂...

[@ccarson]

Dad, why is there tape on my ball?

[@mikeclarke952]

Christoffel made me do it.

[@xaviermachiavelli5236]

Right none of that should of even happens ...happy endings🎉🎉🎉

[@kellymantei7465]

All matter has electron orbits making it repulsive to other matter, so isn't the matter repulsive nature of spacetime itself that forces apples to fall to the ground. Could I look at "gravity" as spacetime pressure where less dense objects are less repulsive to the pressure of spacetime.

[@heeraksharma1224]

What is the space that Curves? I mean dont the earth curves by suns gravity?

[@beaconterraoneonline]

“cookin’ with gas’ ... hydrogen presumably

[@Epoch11]

*Do you think it is cheating and perhaps counterproductive to use gravity to explain gravity........because on the sheet of rubber example, you need gravity to make the depression in the rubber. Is there perhaps a better way to explain these things?*

[@AndrewBarfield]

He addressed this issue in the last episode.

[@demetrismanikas3047]

The truth is that the sheet of rubber example is an oversimplification so that we have something to look at . Space is curved all around a massive object and there is no way to visualize that.

[@craigwall9536]

It would have to start with an explanation of why you need gravity to deform rubber. You don't. YOU need to learn to think abstractly. After that we can talk, but until then it's waste of time. Right now you're like the girl who was asked: "If it's 90 miles from Tulsa to Stillwater and you drive 90 miles per hour, how long does the trip take?" whereupon she burst into tears and cried "But I've never driven from Tulsa to Stillwater! How would I know???" _Abstraction._ It's a concept. You should get some.

[@frede1905]

@@demetrismanikas3047 Actually there is. Search up "hyperbolic geometry standupmaths". There, a guy running the YT channel "standupmaths" explores 3-D hyperbolic space in VR. Hyperbolic geometry is a kind of non-Euclidean geometry or curved space. And you don't need the space to be embedded in a higher-dimensional space to see how it works, like how a 2-D rubber sheet is embedded in 3-D space in the usual analogy.

[@demetrismanikas3047]

@@frede1905 I know what hyperbolic geometry is as Escher is my favorite painter! I will certainly see the video! Thanks

[@israelquito3072]

😵😵😵😵👍👍👍👍

[@meslaphysics]

Hello

[@vvgman]

why doesn't the momentum of the earth around sun slow down over time

[@mikeclarke952]

Gravity is still a conservative force, so in the vacuum of space there is very little friction (gas molecules dragging on Earth) so the velocity does not reduce and thus momentum does not reduce (p=mv).

[@vvgman]

@@mikeclarke952 what about the solar wind, doesn't that cause friction, what about the moons gravity, doesn't that slow the earth, the earth is slowly pushing the moon away.

[@dandelion6692]

🍏

[@trnod]

He takes his time to educate us for free, and someone has the audacity to give it a thumbs down.... has to be the president...

[@craigwall9536]

TDS much? The thumbs down probably came from someone fed up with the time-wasting _disorganization._

[@BeckBeckGo]

Get your battery replaced, that's your problem.

[@numankaraaslan]

24:05 hahahaha :D:D:D christ awful :)

[@Shenron557]

I reminded me of my undergrad in electrical engineering. We used to call Fourier transforms as "four-year" transforms because even if we study it for four years, we don't have a clue of what is going on 😂

[@paulc96]

@@Shenron557 Ha ha ha ! I hadn't heard that one before. Brilliant !!!

[@Goldslate73]

DAMN!

[@rktiwa]

You call it easy?