Relativity 107f: General Relativity Basics - Einstein Field Equation Derivation (w/ sign convention)

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0:00 Overview of Derivation
6:42 Metric Compatibility + Cosmological Constant term
12:53 Contracted Bianchi Identity
20:54 Solving for Kappa (Einstein Constant)
28:08 Trace-Reversed Form
29:46 Sign Conventions
35:09 Summary

[@arshadali2312]

You're in a position to publish all your KZbin lectures in book form -- color pictures on glossy paper. In 40 years of looking at books on differential geometry, Riemannian geometry, and general relativity, I have never seen anything comparable.

[@hugoqueiroz7648]

Humanity will thank this

[@CallOFDutyMVP666]

Amazing work eigenchris. Truly Legendary. I've probably watched 30 hours+ of your videos, between the Tensor Calculus series and Relativity. I was just watching Tensor Calculus 14 today for review. All those hours of notes I have, truly worth it, your videos have changed my outlook on everything. Thank you for everything.

[@valiyahamza4526]

¹¹¹

[@eigenchris]

ADDITIONAL NOTES AND ERRORS: For anyone who wants additional justification for why the Einstein Field Equations look the way they do, you can look at en.wikipedia.org/wiki/Einstein_tensor#Uniqueness and en.wikipedia.org/wiki/Lovelock%27s_theorem. You can also google the Einstein-Hilbert Action if you want to derive the EFE using the principle of least action. Error at 23:18... I accidentally wrote the inner product of vectors instead of the outer product. Whoops.

[@mohammedkhan4990]

Hands down, One of the best GR videos on youtube!! Takes 8 years of GR insight and sums it up in 37 mins.

[@tea3er250]

I finished Tensor Calculus series and General Relativity series 107 videos. Thank you for your excellent works. I was a physics student decades ago, and had been working as software engineer after graduation. Last year, I retired, I returned to study physics, in particular General Relativity.

[@javiermk1055]

MSEE retired here, studying GR now for fun!

[@eigenchris]

Special Thanks to the Physics StackExchange user J. Murray who helped me when I was stumbling finding a derivation of Kappa: physics.stackexchange.com/questions/658968/why-does-this-derivation-of-the-einstein-field-equations-only-work-with-the-trac

[@Handelsbilanzdefizit]

I really appreciate, that you put arrow-signs over your vectors. This makes it much easier to follow. Many explainations lead to a mess of indices. Most tutors make it unnecessary complex.

[@AndreaPancia1]

It's amazing what you have been able to summarize and explain in a way many may attempt to understand, far better than what they have struggled to do up to now. Thanks a lot.

[@michaelzumpano7318]

I only watched the first 6 minutes tonight, but you gave an incredible intuitive insight into the construction of the field equations. I really love your videos.

[@michaelperrone3867]

Thanks for uploading all these very instructive videos about how general relativity works: it has been extremely instructive and helpful

[@PriyanshuAnand10]

Thank you Eigenchris. Your video helped me drastically for my research paper on special and general relativity. Truly grateful to you.

[@SpecialKtoday]

Fantastic, thank you so much!

[@eigenphysics6155]

Awesome video chris sir.

[@davidwagner6116]

Thanks Eigenchris, l think i finally get it! I'm 60 and have wanted to know the details for ages.

[@nandospm]

My most awaited you tube video

[@KrishnaSharma-le7ux]

Outstanding work...

[@boukharroubamediane119]

your videos are as beautiful as the identity of Euler "e[^(i*pi)]+1=0" 😊. They are nicely clear and well explained!! your efforts in preparing those videos are very grateful. Thanks very much to you and your coworker J. Murray Thus, I subscribe, like and share. Good lucks.

[@riadhalrabeh3783]

Brilliant.. very clear .. and thank you.

[@rupeshrajkamal6493]

Amazing work forever , sir..

[@cks5275]

Thank you very much!😃👍

[@leonardp8517]

Excellent work as always, I am hyped for the next videos!! Maybe you can answer therse questions I haven't found answers to online: Is there any intuitive geometric interpretation of the Einstein tensor, like how one can interpret the components of the Ricci tensor as geodesic deviation? Also, what kind of surfaces give a nonzero Einstein tensor? I have tried the sphere, saddle, and a few other random ones, but everything I tried produces an all-zero Einstein tensor.

[@eigenchris]

Unfortunately I don't know the answer to either of those questions. I'd be interested to know if you find out.

[@user-ox9fg8wd9j]

Thank you so much 🙏

[@narfwhals7843]

This was a mouthful. Thank you very much! The small perturbation of the metric linearizes the field equations, right? I assume you're going to go into that a bit in the video on gravitational waves. I'm really interested in the differences of linear and non-linear physics recently. I'm not really sure where I'm going with this so lets just say I'm commenting for the algorithm.

[@eigenchris]

Yes, that approach is called "linearized gravity". It is used in a number of places, including the derivation of gravitational waves. I am still studying it, myself.

[@imaginingPhysics]

A nice way to view GR in the weak field limit is: en.m.wikipedia.org/wiki/Gravitoelectromagnetism

[@asmaiqbal3821]

your videos are really informative and interesting. please make some videos on basic understanding of quantum mechanics too

[@eigenchris]

I don't plan on making videos on basic quantum. The channel Quantum Sense is currently making an intro-to-quantum playlist that's almost done. You can check them out.

[@lourencoentrudo]

Great video as ever!! Question: When generalizing the poisson equation, how do we know that the Einstein tensor is the right one? It seemed to me that the criteria that its divergence should be zero should acommodate other tensors constructed with the Ricci tensor. Or is the Einstein tensor the only tensor constructed with the Ricci tensor and whose divergence is zero?

[@eigenchris]

This might help answer your questions: en.wikipedia.org/wiki/Einstein_tensor#Uniqueness

[@MBTTRADES]

nice vid!

[@sebastiandierks7919]

Maybe you shouldn't call it a "derivation", more a justification/plausibility check on theoretical grounds. In the end, experiments had to show that it's the "correct" theory of gravity (within its range of applicability). You cant really derive the most fundamental equations of physics like the standard model Lagrangian etc. Anyway, that's just semantics, amazing video with a huge information density.

[@changarturo]

Great videos! Thank you! Could you please make a video with an example of General Relativity soon?

[@eigenchris]

I'm taking a break now. But I should be back in mid-October with black holes. I'll eventually cover gravitational waves and cosmology as well.

[@William880915]

This series of video and the tensor calculus one are truly marvellous and are explaining things thoroughly in a nice way with sufficient context to understand things clearly. I am just commenting that in this video it would be better to at least mention the Lovelock’s theorem because that theorem tells us the Einstein tensor is really the unique choice (barring some constant). I myself learned about the Lovelock’s theorem when I found myself unsatisfied with the explanation that since the covariant derivative needs to be zero, we must add the -Rg/2 term, and I was wondering whether there are other tensors that would work as well, and then I found the Lovelock’s theorem, which kinds of rules other possibilities out. I believe I’m not the only one that would have this kind of question and mentioning Lovelock’s theorem would make this already marvellous video even more perfect. Thank you again for all your great efforts in creating these wonderful video series.

[@biblebot3947]

23:10 Neat outer product there

[@eigenchris]

:'(

[@ToddDesiato]

That was really well done. Thank you. Did I understand correctly? That, the only justification for equating kappa*T^uv with G^uv, was because their covariant derivatives were both zero. There had to be more to it than that. Einstein could've simply put the cosmological constant term alone on the LHS and it would've satisfied that condition. I prefer the interpretation where, T^uv tells us all the forces acting on matter at that location, and G^uv is simply what we measure using deformed rulers and clocks.

[@eigenchris]

I tried to justify the EFE as a generalization of Poisson's Equation. the G^uv is like a relativistic generalization of the laplacian of the newtonian potential, and T^uv is like a generalization of the mass density. I agree the use of forcing the divergence of R^uv to be zero, and getting G^uv is a bit weird, but I don't know a more intuitive way of doing it. If you were to just put the cosmological constant term without the G^uv, the equations would not reduce to Newtonian gravity in the non-relativistic limit, so that is why G^uv is needed. You can read these wikipedia articles for more justification on why we use G^uv: en.wikipedia.org/wiki/Einstein_tensor#Uniqueness and en.wikipedia.org/wiki/Lovelock%27s_theorem

[@canyadigit6274]

Thanks

[@ridnap]

Hey Crhis, I love your videos. I was wondering if there is/are special book/s on which your Tensor Calculus series is based on? I want to learn more about the mathematical theories behind all of this physics and want some good book recomendations :D Thank you in advance!

[@eigenchris]

I can't say there's a particular textbook that matches my Tensor Calc series. Part of the reason I made it is because I didn't feel satisfied with a lot of the "intro to tensors" stuff I found online. I felt a lot of it was very formula-heavy and not based on intuition and geometry enough. Some Wikipedia articles are decent. It uses the "stack" picture for covectors that I like: en.wikipedia.org/wiki/Linear_form#Visualization. For relativity, The "Gravitation" textbook by Misner, Thorne and Wheeler has some good parts on tensors used for General Relativity but it's a huge book and not not a light introduction. For additional online GR notes, you can look at Sean Carroll's notes: www.preposterousuniverse.com/grnotes/ or watch Alex Flournoy's videos: kzbin.info/aero/PLDlWMHnDwyljkfy3EBSMlM5D5KQiUSpsB .

[@ridnap]

@@eigenchris Great! Thanks very much!

[@walter--]

Your voice is so monotone and still I keep watching, listening. There is still hope for me...

[@MessedUpSystem]

Wasn't there a way to derive the Einstein Field Equations through extremal action? But I guess it would be significantly harder anyway

[@eigenchris]

Yes, that's the Einstein-Hilbert Action, which has a Wikipedia article you can read. I lack experience with Actions/Lagrangians so I won't cover it right now. In this video I follow the justification of generalizing Poisson's Equation to be relativistic, and also enforcing local conservation of energy-momentum (which means the EFE must have zero divergence). I'm not yet sure how to justify why the Einstein-Hilbert action is the "correct" action for GR. I know in SR, the action is just the proper time, because inertial worldlines (geodesics) maximize proper time. I'm assuming the E-H action generalizes this idea to curved spacetime using the Ricci scalar somehow, which gives us geodesics in curved spacetime that also maximize proper time.

[@davidharris7810]

Hi Chris. Another great video. One part has flummoxed me. The Einstein Field Equations have lower indices and in a number of your slides you have them in the upper position. If these are fields then should they all be lower indices with covector basis (1-forms)? Or, have I really missed something here?

[@eigenchris]

The EFE can be written with upper, lower, or "mixed" indices. You can just raise or lower the indices using the metric or inverse metric as needed. When lower indices, they are written using a basis of covector-pairs (comibined using the tensor product). When upper indices they are written using a basis of vector-pairs (combined using the tensor product).

[@luudest]

I am not sure wether you mentioned this in an earlier videos: What units do the Tensors have? Do they all have the same units?

[@eigenchris]

From a mathematical point of view, I don't have a great way of answering that. The important thing is that every upper (or lower) index in a tensor component must be summed with a lower (or upper) index on a basis vector/covector. The components of 4-velocity vector would be unitless, but the units of the 4-momentum vector would be mass, and the units of 4-force would be mass/second. So the units depend on the physical context.

[@soroushzare6991]

First of all, I want to thank you for your wonderful training. What book do you suggest in this regard so that we can see most of these details there?

[@eigenchris]

I used "Gravitation" by Misner, Thorne and Wheeler to learn some of this, but its derivation of Kappa is pretty quick and hard to interpret. In one of my comments above I link to a Physics StachExchange question I asked that helped me understand better.

[@soroushzare6991]

@@eigenchris thank you for your efforts

[@edmundwoolliams1240]

I can't take his voice seriously since I watched his Topology video, and momentum one XD

[@oldfire3107]

Just really curious. Would you mind telling me what software did you use while creating these series? Great videos, btw!

[@eigenchris]

It's all Microsoft Powerpoint. There's an equation editor in the "insert" tab (also obtains from Alt+= in a textbox), and animations and slide transitions in the respective tabs for those as well.

[@oldfire3107]

@@eigenchris who looks pretty neat. Nice work.

[@BiswajitBhattacharjee-up8vv]

A constant, multiple to energy momentum tensor is the stiffness to the fabric or field of U.. Is it a viscous flux or fluxes are independent entity, density of which is Linked to Einstein tensor. Not clear in this superb video.

[@jfkt9467]

Can you analyse it in 1 dimension, ie x and t??

[@Kelvin-ed6ce]

Is there a copy of these notes/slides somewhere? thanks

[@canyadigit6274]

Why is spacetime sometimes curved even if the stress energy tensor is 0 (ex.kerr vacuum solution)? What’s said to be causing the curvature if there’s nothing in the space?

[@eigenchris]

The curvature of spacetime is measured by the Riemann Tensor, which isn't in the Einstein Field Equations. Only the Ricci Tensor is in the EFE. The Ricci Tensor only gives 10 of the 20 independent components of the Riemann Tensor (related to volume changes along geodesics). The other 10 components of the Riemann Tensor are related to changes in shape along geodesics (tidal forces). (I think these are given in the "Weyl tensor", not sure.) This is good, though. We wouldn't expect spacetime to be flat outside the earth just because it's a vacuum. We still expect to observe gravity in vacuum regions outside planets and stars because we see objects fall, and moons/planets orbit. We can have a source of spacetime curvature like a planet, but the spacetime curvature will extent beyond the planet's mass region, into the surrounding vacuum.

[@imaginingPhysics]

It is the same thing in poisson equation formulation of newtonian mechanics or electrostatics: the force field exists outside the source as dictated by Laplace's equation.

[@peterkrauspe9217]

@@eigenchris Hi Chris, I understand your argument partially, but I stumbled upon the same point as @CanyaDigit. It's a bit weird that the mass in the center doesn't need to appear somehow in the EMT (e.g. in it's region of spacetime) . It seems that curvature may appear even though there is no mass at all. The mass seems to come mysteriously from the "backdor" into the metric tensor, by solving for theese arbitrary constants which are used to create a mostly general spherical symmetrical solution using the behaviour in the newtonian limit. I've read the same explanation in several books, but for my understanding this is still unsatisfactory.

[@eigenchris]

@@peterkrauspe9217 It's not much different than newtonian physics where mass creates a gravitational field that extends to infinity everywhere around it. The empty vacuum space around a planet still has a gravitational field due to the planet. And in GR, spacetime is still curved around the planet in the vacuum surrounding it out to infinity.

[@peterkrauspe9217]

@@eigenchris yes this is not the point I have trouble with curvature in the vacuum when there is a mass somewhere, but in the newtonian eqns you have the mass included in the equation and not set to zero as T in the Einstein eqns. I just expected that T contains everything which is part of the universe , which should include the mass in the center ( even though this is a model universe) . It seems similiar to the sin/cos solution of the homogenious maxwell‘s eqns, where a wave may exist even though there are no charge or magnetic flux densitys. But I never expected maxwells eqns to describe the whole universe…

[@artsciencesartsciences7957]

thank you for your efforts your videos are excellent they have helped me many times can you (if you know) make videos that explain general relativity from spinors i.e. A spinor approach to general relativity

[@eigenchris]

I'm not at the point where I understand spinors enough to do that. I know there's a book by Roger Penrose called "Spinors & Space-Time: Volume 1, Two-Spinor Calculus & Relativistic Fields", however I found it pretty difficult to follow. Can I ask what your motivation for learning this formalism is?

[@artsciencesartsciences7957]

@@eigenchris Thanks anyway, I'm trying to explain physics, especially general relativity in another way, maybe I find a theory that explains everything that's my motivation

[@souvikmandal1989]

I have some questions in context of Conservation of energy-momenta locally at 3:32. When we drop a ball with mass m from a certain height on earth surface (small height) h, its kinetic energy increases with the approach towards the ground. First of all, do you call this local or non-local incident? Secondly, as in general relativity gravity is not a real force , it can not transfer energy to the ball. Then how the kinetic energy of the ball increases? Also the conservation of energy-momenta is valid or not in this case?

[@eigenchris]

"Locally" means looking at a single spacetime point and anything "tangent" to that point (e.g. 4-velocity, 4-momentum, 4-acceleration). "Non-local" means looking at two different spacetime points. So when the EM tensor handles "local conservation of energy-momentum", it's only looking at a single point in spacetime and the objects that live in that point's tangent space. In Newtonian mechanics, when we're talking about "kinetic energy" and "potential energy", these are useful concepts because the "total energy" (K.E. + P.E) is a "constant of motion" throughout the falling object's path. The reason "energy" as a concept exists is because our system has "time translation symmetry". In general relativity, we can't always take for granted that the "energy" from Newtonian mechanics" is a constant of motion. But there may still be an "energy-like" quantity that is a constant of motion if our spacetime has time-translation symmetry. You'll see in the upcoming 108c video I'll use 2 constants of motion in Schwarzschild spacetime to calculate the Schwarzschild orbits. One of the constants of motion is kind of like an "energy" term, but it's not quite the same thing as Newtonian energy. When I eventually cover the FLRW-metric for the expanding universe, we're going to see there's no constant of motion for time translations, so there's no energy-type "thing" that's conserved. I know this is probably a lot to swallow, but I'll be covering all of this as the videos go on.

[@signorellil]

This is..... well, how can I spell it Chris? This is right. Which is more than it can be said of most of "science communication" videos around today

[@verticalgaming5434]

I use laplacians a lot in gr but I’m studying st, any langraungian vids (also Ik u can use them in gr sr and l. Mech)

[@johnpcourter4038]

This work still depends on an issue between real numbers and natural numbers Einstein avoids by fabricating a theory of spacetime. There is not zero included in any measurement of time, ever, as proof, this though throw work is more of a test on improper calculations not a form of mathematical proof. To correct Einstein’s General relativity is first started by addressing a diameter as a circles chord. This is explained way more complicated the it has to be.

[@tomkerruish2982]

I'm really looking forward to finishing this video later. I'm certain it's high quality like your others. Now for something completely different. General question: do those comments with ".fyi" addresses always lead to porn? I'm sick of them. It's okay to report them sight unseen, right?

[@eigenchris]

I know they're bot comments but I don't actually know what the link is. Apparently I can add "blocked words" in my channel settings, so I'll block "var[dot]fyi".

[@tomkerruish2982]

@@eigenchris Thank you. I know I've seen them show up in others' videos' comments, maybe Andrew's, I'm not sure.

[@Lennon959]

This comment is so late sir. I hope you may notice this. I have a question about Cosmological Constant. Is it constant to all observer or is it constant everywhere? Is it constant anytime? Like for example, an observer "escapes" the gravitational field of Earth (hence he is in outer space and experiences no curvature of spacetime). By using Einstein field equation, the Ricci tensor term, and the Ricci scalar term will be zero cause no curvature of spacetime (if I'm not mistaken). And when that observer observe the Earth, the energy tensor would not be zero, T_00 = rho*c^2. So to satisfy the Einstein field equation, we must use the Cosmological constant term with a factor of metric tensor. Since it is flat spacetime we can use the minkowski metric? (If I'm not mistaken). So if minkowski metric is gonna be used, what would happen to the other components of T_mu*v (i.e. T_xx , T_yy , T_zz)? Will they be both -rho*c^2 or they have different Cosmological constant? Thanks in advance sir.

[@davidharris7810]

Hi Chris. Further to my earlier comment should the derivation of the contracted Bianchi Identity use the metric tensors and not the inverse metric tensors to derive it with lower indices? Again, I may have completely misunderstood but hopefully you can clarify.

[@eigenchris]

I'm not sure there's an easy way to write the contracted bianchi identity with lower indices, because the index on the covariant derivative always has a lower index, and so the tensor components it is summed with must have an upper index. I guess you could write the tensor components with a lower index if you include an inverse metric with upper indices to balance things out, but that would just make the formula more complicated.

[@himanshuchaudhary5796]

Sir plzz provide the PPT of video for notes purpose

[@greenguo1424]

Hi Chris, I always wonder why we can just incorporate additional tensors into EFE if they are divergence-less?

[@eigenchris]

I think of the cosmological constant term as like an integration constant. Similar to how we can always add a "+c" to indefinite integrals, we're allowed to add a divergenceless term to the EFE.

[@greenguo1424]

@@eigenchris Got it! So we can "make our own" FE as long as we find some divergenceless tensor, but those we end up finding for 4D happen to be the EFE terms as proven in Lovelock theorem right?

[@temp8420]

Question for Chris - if you can swap indices with metric tensor on both vectors and forms is there any real difference as the two metric swaps will cancel. You've probably covered it somewhere but I can't find at the moment. Many thanks.

[@eigenchris]

Sorry, I'm not sure I fully understand your question. Being able to swap the indices of the metric tensor means the order of the 2 input vectors doesn't matter. it basically means v·w = w·v. Does that answer your question?

[@temp8420]

@@eigenchris sorry Chris I mean in a form if we raise the index of the components to give contravariant components and drop the indexes of the basis vectors to give the basis forms we end up with a vector both done with a metric and it's inverse which two multiplied in a matix representing give 1 (are inverse) many thanks (again)

[@temp8420]

@@eigenchris thanks again for your help - I was trying to work out if there was any real difference between a vector and a form as you can pull one index down and the other index up on the vector or form with the metric tensor and it's inverse and swap vector into a form and vice versa. Apologies if this isn't clear or a silly question.

[@cinemaclips4497]

At 15:27, why does the basis of the Riemann Tensor have basis covectors in it? I don't remember anywhere in the derivation where we use covectors? Is it a typo?

[@eigenchris]

The rule is every index in the components of R needs an opposite index in the basis, since they are summed together. So the lower (covariant) indices of R must be matched with upper (cotravariant) indices on the basis covectors.

[@cinemaclips4497]

@@eigenchris okay I'm kinda understanding. Is it that basis covectors sum with lowered indices in the Riemann Tensor components and the basis vectors sum with the upper indices? Also is it that tensors with lowered indices have a Tensor product of basis covectors which makes its basis and tensors with upper indices have a tensor product of basis vectors which makes its basis?

[@domenicobianchi8]

put it like this, it seems the equation was simply guessed and kept because it worked. There was a formal derivation in the following years?

[@rv706]

Wait but.. You can't prove a tensor (components) identity involving derivatives in a coordinate system that has a special form at a point (e.g. normal coordinates) but not in a neighbourhood of the point: derivatives don't depend only on the values at a point but on a neighbourhood. So... is that proof of the second Bianchi identity correct?

[@avneetmalhi5723]

Eignchris when will you publish your new vedios on spinors

[@eigenchris]

I plan on starting before the end of 2022. For now I'm still taking a break/assembling notes together.

[@somshekharrakhe6224]

If we consider the Contracted Bianchi Identity, the covariant derivative of metric tensor is zero.. so shouldn't it give us that the covariant derivative of Ricci Tensor must also be zero?

[@eigenchris]

The covariant derivative of the metric itself is zero, but the partial derivatives of the metric componemts are non-zero. So we get non-zero terms with the covariant derivative of the Ricci tensor.

[@somshekharrakhe6224]

@@eigenchrisYes, but then what does the Metric Compatibility at 20:30 imply? Because there, it is mentioned that the derivative of the components is zero...?

[@doctorerty2061]

This might be a stupid question, but how exactly does ∇μTμν​=0 imply that the tensor on the left-hand side of the EFE (Einstein tensor) must follow ∇μGμν​=0? I know that ∇μTμν​=0 is just conservation of energy-momentum, and that makes intuitive sense to me. However, I just can't seem to figure out why the Einstein tensor must have a vanishing divergence because of that. Is there some kind of physical reason behind this, like that energy is not conserved if the divergence is non-zero, or is it just a mathematical one?

[@eigenchris]

If you take the divergence of both sides of the EFE, the results must be the same (zero).

[@doctorerty2061]

​@@eigenchris Hi eigenchris. Thank you so much for responding! I still don't think I quite understand. Could you please elaborate on why we must take the divergence of both sides and equate them? More precisely, what is the motivation behind doing such a thing? In the video, you said that the divergence of the energy-momentum tensor is zero and the divergence of the Ricci tensor is non-zero, and that therefore it cannot be correct to simply equate the two tensors. Is there some kind of property of tensors such that if you equate them, then their divergence must be the same? I am kind of new to tensor calculus, so I am sorry if the answer is obvious and that I am wasting your time.

[@sergiofranczak7595]

👏👏👏 👍👍👍👍

[@nellvincervantes6233]

Question sir. Why use T_00 = rho*c^2 instead of T_00 = (1/2)rho*v^2 + PE/V when it is Low Velocity Limit or the Classical Limit, v

[@eigenchris]

The energy we're talking about from T_00 doesn't come from velocity or position (potential energy)... it just comes from the object's mass. Mass on its own is a form of energy that causes spacetime curvature. If you watch the 107d video, I show how mass curves spacetime even for Newtonian gravity, without Einstein's relativity involved at all. The (1/2)rho*v^2 would be kinetic energy, which would affect other components of the EM tensor if it was large enough, but they go to zero in the low-velocity limit since the T_00 component will dominate. As for potential energy, I'm not sure if the concept of PE makes sense in curved spacetime. The concept of curved spacetime essentially "replaces" the concept of potential energy... since curved spacetime determines how an object will move due to gravity. You might try googling "potential energy general relativity" to learn more because I haven't looked into this.

[@nellvincervantes6233]

Ok sir thank you so much!

[@applealvin9167]

In This derivation, some approximations and assumptions are used, does that mean the Einstein field equation is an approximation?

[@eigenchris]

Which approximation are you talking about? Some approximations are used to get the weak gravity limit and derive the constant for the Einstein Field Equations, but that makes sense since Newtonian gravity is an approximation of General Relativity. Is there another approximation you're concerned about?

[@applealvin9167]

@@eigenchris yeah those are the approximations i was refering to, it seems a bit weird to me that you can derive the "exact" value of the constant if the derivation involves some approximations

[@Richard.Holmquist]

Man are your students lucky.

[@salfadelay2157]

One more peak conquered!

[@active285]

Just a small remark: Although I know that physicists seem to prefer using local coords, it's not always the simplest (and easiest) idea. Actually as mathematician I try to avoid them as best as possible. The Bianchi identities are almost trivial in the coordinate free version. Just write down the definition R(X,Y)Z = ∇_X ∇_Y Z - ∇_Y ∇_X Z for all permutations of the vector fields X, Y and Z. Then sum the three equations. Similarly, the method works for the second Bianchi identity, using the definition of the covariant derivative of a tensor, i. e. ∇_X R(Y,Z)V = ∇_X (∇_Y ∇_Z V - ∇_Z ∇_Y V) - R(∇_X Y,Z)V - R(Y, ∇_X Z)V - ∇_Y ∇_Z ∇_X V + ∇_Z ∇_Y ∇_X V for some other vector field V. Permute and sum the three equations. Thanks for teaching a stochastic differential geometer GR in such a nice way!

[@eigenchris]

Yeah, I have a more math-oriented tensor calculus playlist where I show that version, but I felt my GR videos were long enough as it was. There's apparently an ever shorter version that uses the exterior derivative "d", but I haven't put in the work to understand that version. I'm curious what stochastic DG is used for.

[@active285]

The deep connection between Brownian motion and the Laplacian being its generator (of a strongly continuous one-parameter semigroup in the sense of Hille-Yosida, namely the heat semigroup) seamlessly opens ways to study the local and global geometry of manifolds, or more generally martingales in vector bundles, by virtue of the paths of this stochastic process in an intrinsic way. Many questions related to the geometry of Laplace operators have a direct probabilistic counterpart. For example, Brownian motion explores the whole manifold for arbitrary small t > 0! Probabilistic methods often extend naturally to problems, like singular spaces, infinite dimensional spaces and sub-Riemannian spaces, where standard tools of global analysis or PDE methods may fail. Some technical assumptions, e.g. on the injectivity radius, may be dropped even on non-compact manifolds. Here is an answer of mine that might be helpful as well: math.stackexchange.com/a/1843820/238307 Notably, so called Bismut(-Elworthy-Li)-type formulae provide probabilistic derivative formulae for diffusion semigroups on possibly non-compact manifolds (by Bismut, Elworthy, Li, Hsu, Wang, Thalmaier; in the most abstract setting by my supervisor Thalmaier with Wang). The remarkable fact is that in this stochastic derivative formulae Ricci curvature only enters locally around a point and no derivative of the heat equation appears on the righthand side of the equation - “just” an expectation of a stochastic integral involving the function, local geometry and a flat real-valued Brownian motion. This potentially opens ways to study e.g. gradient estimates. Here are some nicely written notes: math.uni.lu/thalmaier/Stoch_Anal_2023/notes.pdf Of course there is also the book of Elton Hsu "Stochastic Analysis On Manifolds". Applications also extend to physics, e.g. scattering theory: Using those formulae the existence and completeness of the wave operators corresponding to the (Hodge) Laplacians induced by two quasi-isometric Riemannian metrics on a complete open smooth manifold can be proved under a mild local integral criterion but no assumptions on the injectivity radius. Other applications include estimates for the (covariant) derivative of the heat semigroup on differential forms, and covariant Riesz transforms, Hessians estimates, Harnack inequalities, Calderón-Zygmund inequalities and much more. A lot of work has been done in recent years in a more general setting, namely: Metric measure spaces (by Sturm, Lott-Villani, Savaré, Ambrosio, Gigli, Naber and many more). Of course on non-smooth spaces one has to give meaning to Ricci curvature. Lower Ricci bounds can be replaced so called curvature-dimension conditions (being equivalent to Ricci bounds on manifolds, serving as a definition in the non-smooth setting). Very recently people became interested in extending also GR to MMS, e.g. timelike curvature-dimension conditions, timelike lower Ricci curvature bounds on Finsler spacetimes, optimal transport in Lorentzian synthetic spaces, synthetic timelike Ricci curvature lower bounds, Rényi's entropy on Lorentzian spaces etc. I am trying to understand GR to dive in deeper :).

[@TheNewPhysics]

Am I correct to say that the introduction of Einstein's tensor was ad hoc, in the sense, that Einstein's tensor was created just to have a zero divergence, such that the erroneous (also ad hoc) initial einstein's equations to become "more" correct.? In other words, what you are providing is not a derivation. It is just the justification of an ad hoc equation.

[@eigenchris]

I think that's how Einstein originally came upon it. He was mainly motivated by the idea what "spacetime curvature = energy/momentum" and the most straightforward equation to do this was "R_uv = k*T_uv". He originally published this and then corrected himself later, with the full Einstein Tensor. If you're familiar with Lagrangian approaches to physics, you could look at the derivation of the EFE from the Einstein-Hilbert action, but you might run into the problem of saying the Einstein-Hilbert action is also ad hoc and not derivable (maybe you can derive it from something more fundamental, I'm not sure). You can also google "Lovelock's Theorem", which is a theorem that came over 60 years later, justifying why the EFE looks the way it does: en.wikipedia.org/wiki/Lovelock%27s_theorem

[@TheNewPhysics]

@@eigenchris Dear Eigenchris, I am a fan of your videos. You are a great teacher. I would like to ask you questions on similar derivations to your great "Curved spacetime for Newtonian Gravitation). I derived from first principles this equation of Gravity: If you follow the link, you will realize that the law of Gravitation was derived within a 4D spatial manifold (lightspeed expanding hyperspherical Universe topology) and has an epoch-dependent G. The epoch-dependent G was then propagated to derive the SN1a Absolute Luminosity G-dependence to be G^{-3.33}. Once the SN1a distances were corrected to accommodate an epoch-dependent G, the SN1a distances were parameterless predicted using d(z)=R_0*z/(1+z). Where R_0=14 billion light-years and Hubble Constant =69.69 km/(s.Mpc). So LEHU is supported by the laws I derived (which pass all GR and SR tests) and by the SN1a distance prediction. Since LEHU is not a solution to Einstein's equations, these observations debunk Einstein's equations. The argument is here: LEHU also eliminates the need for Dark Matter, shown here : I would like to talk to you about the geodesics equations for my potential which is like Newtonian Gravitation plus velocity-dependent components. My problem is that just recognizing the local homology between LEHU and Minkowski spacetime is not sufficient for some critics. I would like to go from my law of Gravitation to the Schwarzschild metric. From your presentation, one can derive the Schwarzschild metric just considering spherical symmetry and a null Energy-Momentum Tensor (the space outside the massive body). Once one has the Schwarzschild Metric one can use it to replicate both Gravitational Lensing and the Mercury Perihelion precession. This raises the question: If one can go from Newton's Law of Gravitation to Schwarzschild Metric, then one can go from Classical Mechanics to the correct expectation of Gravitational Lensing and Mercury Precession Perihelion Precession. Am I wrong? What am I missing?

[@TheNewPhysics]

@@eigenchris You might notice that my message didn't have any one of the required links (from quora). So, I will have to count on your intelligence to make sense of what I wrote. You can always ask any questions about what is missing.

[@mrslave41]

The expansion of the universe is not predicted by General relativity 0:14

[@it6647]

It seems as if the rule of connection coefficient being 0 at point p is a bit inconsistently applied Sure, the ones with the product of two connection coefficient term was a result of a derivative being taken somewhere in the derivation, but then later on, when the deivative of that product is taken, you set all the resultant term to zero which makes me think that that product was always zero Could you please clarify this?

[@eigenchris]

Do you mind giving me some timestamps in the video for the parts you're concerned about? It's been a while since I uploaded this and my memory is a bit foggy.

[@it6647]

@@eigenchris 16:22-40 16:49

[@eigenchris]

​@@it6647 The rule I'm trying to follow is: connection coefficients on their own can be sent to zero, but DERIVATIVES of connection coefficients may not be zero. You can imagine a parabola whose vertex is at the origin. It's value at the origin is zero, but its derivate is non-zero. At 16:22 I cross out connection coefficients that are on their own, without derivatives. The terms in the lower box have their derivative taken (see the 1st term in the top box... there is a derivative operator there). After doing product rule at 16:50, we end up with connection coefficients on their own without derivatives, so I set them to zero.

[@it6647]

@@eigenchris Alright so basically you ensured that the derivative will always be zero, used some foresight to make sure you don't prematurely turn the connection coefficients to 0 if they are going to be differentiated in the further proof So even though Riemann coeffs' 3rd and 4th term are zero at that instant, it by no means implies that their derivative is zero, which is why you chose to keep them around until the later half of the proof?

[@eigenchris]

@@it6647 I think you've explained it correctly. I make no assumptions about what the derivative of the connection coefficients are... I just keep them around, and when no further differentiation needs to happen, I set all the (undifferentiated) gammas to zero, and this greatly simplifies the formula.

[@fazilnajeeb]

I am right on time.

[@MrYellowm4n]

We want a french video like alexandre roussel did

[@eigenchris]

Are you talking about french narration or french subtitles?

[@Channel-zb1fi]

English narration but with a french accent.@@eigenchris

[@skypickle29]

How come we always talk of the 'expansion of the universe'? Could we not just as easily speak of the 'shrinking of us'? And does 'shrinking of us' really just suggest a changing metric? I really respect your mathematical clarity and hope you forgive my naive sounding question that has more to do with physics than math. I suppose I could 'mathematicize' my intuition by asking if kappa is changing instead of adding lambda (the cosmo constant) to the equation. A variable kappa might also acccount for the phenomena we label dark matter and dark energy, no?

[@eigenchris]

I'm not sure how to comment on the idea that Newton's "G" or the speed of light "c" are changing... I haven't looked into it. When I derive the Friedmann equations, you'll see how G, c, and Λ affect the expansion of the universe. You can look at the equations on wikipedia to see how they affect the expansion rate (a-dot) and the acceleration rate (a-double-dot): en.wikipedia.org/wiki/Friedmann_equations#Equations

[@HighWycombe]

Did we really derive Einstein's Field Equations in this 36 minute video? It feels like a bit of an anti-climax after spending many months studying all of eigenchris's Relativity videos from 101 upwards, making scrupulous hand-written notes, stopping off on the way to do the same with Tensor Calculus and Tensor Algebra. (I studied every video, spending about a week on each filling 4 ring binders.) Every video has been thorough and comprehensive all leading up to this one which I was expecting to be some sort of series climax. This "series finale" should have been an hour long and come complete with fireworks.) As it turned out, I was just starting to get a bit lost with 107f. I think maybe it didn't have the same rigour as earlier videos? There seemed to be jumps in the logic flow, equations referenced from "earlier" didn't seem to have been mentioned "earlier". What was a lower index earlier in the video reappears as an upper index. Also, the constant term. This is really fundamental, it's what makes the left-hand-side of the equation(s) equal the right-hand-side. We only looked at the simplified case of weak, gravity, low-velocity, and non-relativistic time when deriving the constant - the Field Equations are supposed to work in strong gravity, high velocity and relativistic time aren't they? (Most likely all this is just me, I did say I was starting to loose the thread.) I have learnt a great deal from eigenchris over these videos. (Thank-you eigenchris.) Eigenchris himself encourages us to look at other sources as well, so I think maybe I need to do that in order to get a wider perspective.

[@eigenchris]

Unfortunately, the truth is that this video and the many of the ones after it are going to be less elegant than the ones that came before. The equations in general relativity are much more complicated than the ones in special relativity, and we're often going to need to resort to assumptions and simplifications in order to get by. However, the Einstein Field Equations do work in strong gravity as well as weak gravity. We just need to make sure they reduce to Newtonian gravity in the limit of weak gravity. The "weak gravity" assumption is only used to calculate the constant the constant 8*pi*G/c^4, which is exactly what's needed to reduce the equations to Newtonian gravity in the limiting case of weak gravity. That said, if I failed to mention an equation "before", can you point it out? I've made so many videos that I may have neglected to teach a couple points.

[@HighWycombe]

@@eigenchris Thanks eigenchris. And apologies where I mentioned "jumps in the logic flow". That's just me failing to keep up. Hoping to understand GR is pretty ambitious for me personally.

[@flaparoundfpv8632]

*Strokes beard* ...yes, yes...

[@nimrasheikh636]

What is the significance of finding solutions of Einstein Maxwell field equations????

[@muhammad.748]

If deriving Einstein's equation was so simple, why did it take 15 years?

[@mahapeyuw5946]

Because beginnings are usually harder and a successful one takes out the years of mistakes leaving you with the fruit of the 15 years.

[@davidrandell2224]

If gravity is the earth approaching the released object instead of the released object approaching the earth- Galilean relativity of motion- what ‘use’ is Einstein relativity?

[@eigenchris]

I'm not sure I fully understand the question. Galilean relativity allows for the speed of light to change in different reference frames, whereas special/general relativity keep the local speed of light in a vacuum constant at a speed of "c".

[@davidrandell2224]

@@eigenchris The earth is the space elevator in the equivalence principle expanding at constant acceleration of 16 feet per second , I.e. gravity. Einstein missed this somehow when he had his greatest prompting concerning gravity: the man ‘falling ‘ off the roof would be not moving. All motion is relative. Something in the released object scenario- dropping or falling as it is erroneously called- is moving; which one(or both?) is it?

[@davidrandell2224]

@Imjust Observing “The Final Theory: Rethinking Our Scientific Legacy “,Mark McCutcheon for a rethink.

[@davidrandell2224]

The universe is expanding faster than “c” or spacetime is stretching, but Not even possible that the earth is expanding 16 feet per second constant acceleration; laugh. All motion is relative. The universe is not expanding.

[@davidrandell2224]

@Imjust Observing All atomic matter expands at 1/777,000th its SIZE: do the math for earth =16 feet per second constant acceleration. Grow a brain/backbone.

[@givemeabreak6374]

Einstein did not predict black holes. He thought they did not exist.

[@johnstfleur3987]

I AM JESUS CHRIST; THE PRESENT PROBABILITY STUDENT OF COLUMBIA UNIVERSITY.

[@SuperGrizzlybears]

I am here just to pretend that I can understand this 😂

[@jacobvandijk6525]

My gut-feeling says that you keep repeating yourself. Leave the milking to farmers, Chris! Try something new, mate. What about Quantum Mechanics and Quantum Field Theory ;-)