The Maths of General Relativity (7/8) - The Einstein equation
2021 ж. 4 Қаң.
123 361 Рет қаралды
In this series, we build together the theory of general relativity. This seventh video focuses on the Einstein equation, the key ingredient of the theory which allows us to relate our mathematical model to the physical world.
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Alessandro Roussel,
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This is literally the best series about general relatively on YT
Agree
This is a good one, too: kzhead.info
Yes. It really is. This type of high-quality educational content has been one of the few benefits of the pandemic.
This is the best use of the word literally that I've ever seen.
@@dritemolawzbks8574 it existed before the pandemic in french. This is the translated chanel.
Again: Who else got goosebumps from first moment of the background music?
Me
It's just so classic to set the mood 😎
yeah, the spacetime music always does its thing ...
Music sounds like from the Social Network OST by Trent Reznor and Ross Atticus
@@TNTsundar I'd make a wild guess that the music is by Alessandro Roussel himself, too. He has a few electronic tracks on his channel. This track is much more subdued, but this should have been expected if it had been created as a background music for a video.
This is just phenomenal! Alessandro Roussel's magnum opus is probably the clearest, simplest and most intutive presentation of this beautiful theory in existence✨. The glimpse at the Schwarzschild metric cleared up so much mystery about black holes! Eagerly waiting for the finale.....
I can already hear the word “indeed”
so satisfying :P
Yes...
3:17 ‚Solving such a intricate problem on paper is almost impossible to solve it‘ I bet Einstein knew this when he finished his theory.
He couldn't even find any solutions to his own equations. The first (non-trivial) solution, the Schwartzschild metric, was found by Karl Schwartzschild in 1916, the year after Einstein initially published his equations
You've just taught an elderly English major the basics of general relativity, a subject from which I have cowered these 70-someodd years. Pour yourself a beer, put your feet on the coffee table, and glory in a job well-done.
The music is so dramatic. It is deserved.
This is so well done, you even pronounce "Einstein" in correct German.
Understandable, have a nice day
It hasn’t even premiered yet...
Actual derivation of Einstein field equation is presented in drphysicsa channel in a video spanning over 1hr. The videos of scienceclickenglish channel are vital in getting a physical grasp of the various quantities like ricci tensor,christoffel symbol ,etc. Scienceclick english has done brilliant job by presenting simplified videos on a very hard topic. The equivalence principle has been dealt earlier in veritasium videos. But the beautiful ways in which scienceclickenglish has dealt the complex subject is very much commendable
Could not agree more, Rajarshi! To bring in a comparison of didactic approaches, there is also an excellent series of 5 lectures on the introductory diffgeo and GR, 1-1.5 hours each, read by Kip Thorne to the grad class, which covers it all (kzhead.info/channel/PL1XfECM855xnXjlacd6UkPzKo5IqRgYpP.html; there are more advanced ones at astro-gr.org). If asked in which order to watch, I would, honestly, stumble. I (used to?) know the subject well enough, but I'm rusty--I dropped out of academia 25 years ago. I'd certainly go to ScienceClic for a quick refresher rather than the classic Wheeler, Thorne et. al. _Gravitation_, at least to begin with. For a noob autodidact, on the other hand, these videos may be a bit too fast paced and too condensed, so it needs good thoughtful pauses and rewinds, but it will help create a good plan for the future deeper self-study. In all, the overall presentation of the material is very well thought through: I can literally _feel_ the hours of work behind every minute of the video. It's exactly as simple as it only could be to remain true, and not simpler.
yeah the DrPhysicsA videos are great, too. but this is def more accessible
That is not a derivation, the only way this can be derived is from the action principle, which not rly is a formal derivation.
this is so awesome, finally the final form of explanation of GR on KZhead, this is so valuable man !
thanks a lot !
"Black hole" narration lets me get goosebumps lol
Thank you so much for such a clear explanation! I especially learnt WHY the Einstein Equation is not final; because it is not derived nor solvable mathematically. It is however most effective in explaining so much of cosmology. It was the sheer genius of Einstein that this eq was discovered.
This is simply awesome! How can such a hard topíc be made so interesting and clear? The background music, the precise narration, the illustrations, the eureka moment. I just had an epiphany.
Hands down one of the best series on general relativity I’ve seen. Simplifies very complex concepts in a short period of time.
I never expected I could be so excited about a new video featuring so much math. Amazing teaching.
This is the best series on General Relativity on KZhead so far! Thank you so much for this effort!
Sir your dedication about physics charges us to do so .👏👏👏
I love how you put the physical concpets in a really simple way to understand. Helped me a lot. Can't wait for the last part of the series!
Man, I've just read your CV and seriously I'm asking to myself how you could have managed all that stuff in your head in such a small amount of time. Congrats. And by the way, this small series is outstandingly cool.
haha i just read it after seeing this comment... i feel like a useless human being now hahaha
Imagine Einstein thought about this kind of stuff in his head without no computer or no help from anyone.
excellent work gentlemen, I just love these videos. Cant wait to binge watch all 8 episodes in a row. These are a great compliment to the Lenard Susskind lectures
I just want to say thank you for the whole series! Thats education at the next level.
We have Eugene, but your explanation of GR is by far the most extensive and efficient - per visuals; build-up and time - which I have encountered on youtube. Thanks to people of you, I do not need books anymore!
Thank you for this. This explains a question I've been thinking about recently, namely why the Einstein Equation is usually presented in that form, instead of moving both the Ricci and the Energy Momentum Tensor on the same side. That would equate the metric to two energy terms. But equating the curvature to the content is of course also useful and maybe more so.
Just brilliant! Thank you very much for this series .. eagerly awaiting episode 8
Hello I am a bachelor and this was the first time I formally study general relativity. I can say that your work helped me a lot! It was brilliant! I believe this is the best material on the internet to explore the concepts behind this subject.
2:23 I have always found it interesting that newton's gravitational constant appears in einstein field equation, even though both theories describe gravity as a different concept; does that come naturally as a result or is it chosen by convinience? (like you stated, to recover newton's theory under particular conditions)
As you said, G is chosen so that we can recover the Newtonian equations when spacetime is static and almost flat. Let me just add that this may not be the last piece of the puzzle: there is a set of theories that aim to modify general relativity in order to get a better description of, say, inflation, dark matter and dark energy. In some of those theories, the gravitational constant G is not actually a constant, but described by an additional dynamical field
@@leastaction_224 i dont get it; is G chosen? what about merging quantum mechanics and general relativity
In Newtonian physics, we describe gravity as an attractive force acting between bodies with mass. Newton said that this force must be proportional to the product of their masses and inversely proportional to the square of their distance. He called G the constant of proportionality. The value of G was later measured many times and we are now pretty sure it is indeed a constant. Now, Albert Einstein enters the chat. He described gravity as the effect of the curvature of spacetime itself, making use of the Einstein field equation(s) described in this video. Now, the video correctly says that this equation cannot be proven: it is a postulate of the theory. Therefore, in a way, it is "chosen", as you said. If you are interested, let me elaborate a little bit more. The Einstein field equation is written keeping in mind that we want it to have a certain set of properties. In particular, we want to relate the curvature of spacetime (described by the metric tensor, the Christoffel symbols and the various curvature tensors) with the energy and momentum of the "stuff" in the universe. We want this relation to be valid in all possible frames of reference (this is a fundamental requirement of every physical law) and we want to make sure that it does not violate physical laws we already know, like conservation of energy. These requirements alone fix the form of the equation as you see them in 1:53, except for the fraction in the left-hand side (the one multiplying the energy-momentum tensor). That coefficient is chosen so that, when gravity is weak and static (i.e. when the metric tensor looks like the one of flat spacetime) we recover the newtonian description. This requires the presence of the factor G in the numerator. Now, as for merging quantum mechanics with general relativity... that's a really complicated topic. The main idea is that we cannot simply apply the rules of quantum mechanics to the gravitational field (as we know we can do with the electromagnetic one). The main problem is that quantities that should be finite number turn out to be infinite, and we have no way to "get rid" of these infinities. Various strategies have been proposed, none of them has ever been confirmed by an experiment.
@@leastaction_224 I see; thanks for clearing up my doubt
This is an excellent question, which I've asked myself a long time ago. The answer is: Einstein's theory doesn't need 8piG/c^4 but just a constant to connect curvature and energy-momentum in the field equation (this constant has a name: the Einstein constant). However, in order for Newtonian gravity to be reproduced in the low-gravity, low-speed approximation, that constant needs to be set this way. Fascinatingly, it turns out that, once you set this constant this way, the equation also works in the high-gravity, high-speed scenarios. Why? Nobody knows. But the equation describes the data. It's part of the theory.
Best explanation of general relativity with visuals! Congratulation!
Yes
These are some of the best videos I've ever seen in my entire life. Please do more!
I would like to give a million like to this series. It is the best explaination how all of math build blocks work together for the Einstiein equation. Thank you so much.
I waited for this part! Thank you!
You are the best man thanks for this interesting serie you made it simple and consistent at the same time bravoooooo
Yet another wonderful class. Thanks.
You guys have no idea what beauty you are producing.
Please please do a similar series to explain Quantum Physics, you are the best physics teacher on KZhead!!!!
1:25 how did Einstein came up with his equation ??? What example did he took ... Simply brilliant
If you could do a video explaining electromagnetism that'd be awesome - especially how the fundamental force produces macroscopic electromagnetic fields, i don't get that
This was absolutely brilliant
Just mind blowing.. Congratilations and many thanks for these 8 videos..
I'm really loving it
Normal people: name their babies with normal names Parents of physicists: what about "Schwarzschild"
For the man who discovered the first black hole solution, Schwarzschild ("black shield" in german) is quite appropriate :p
This is complicated and mostly, nearly entirely, beyond my ability. But I love it!
Love the programming ! Good job 👍 !
This is Amazing!
DAMMN I barely understand half of it and I realize now how GENIUS this all is.
ScienceClic is Einstein's grandson
Excellent work, action music. Thank you. Go on like this. Love for deeper understanding is equivalent with an revolutionary ability of truth contagion (Plato's banquet). Mathematics on the left member of field equations, physics on the right of field equations. The easily perceived nature (even with "magnifying" scientific material instruments), that is aroud us and presented inside the our unknown conscience, seems to be the "outer part" of a "mental infinite world" by something as "Plato's Mathematical ideas", maybe the "Multiverse" information tank of all infinite probable events constracting alternate realities. In the invariant quantity dΩ = |g(t,r,θ,φ)|^ (1/2) . dV of the General Theory of Relativity, with the elementary (complex?) volume dV of spacetime, we observe that the square root of the determinant |g| of the metric tensor seems asprobability density of unknown events, because this quantity corresponds to the probability density f = f(x1, x2, ... , xn) of events (!) in the also invariant relation dP = f .dV of the probability theory, with random variables x1, x2, ..., xn the any n magnitudes. What does this mean? It may be helpful that in quantum physics the square of the measure of the wave function ψ [ that is | ψ | ^ 2 = f(t,r,θ,φ) ] is the probability density f with random variables again the n = 4 coordinates t, r, θ, φ of spacetime: dP = | ψ | ^ 2 . dV.
very helpful...... keep making such videos
Great video! I was lost in the end due to the math. Try CIG Theory which offers an equivalency between both sides of the field equation.
Amazing thank you so much for this effort!
Wonderful videos
Excellent video. At 2:54, I'd always seen these equations as partial derivatives whereas the video used total derivatives. Is this in error?
@ScienceClic English: Could you do a follow up series about calculating certain examples? For instance how to calculate the bending of light in the gravitational field of Earth? And what is the difference to free fall? Or the precission of mercury? How come the origin of gravity is not the beding of spacetime but the difference in time dilation? Thanks for this great series. Best explanation of the math of GR!
The 8th / last episode will be about 3 examples of this type ;) The precession of Mercury is a bit more tricky though so I keep this idea in mind for a later video maybe. Glad you like the series ! Also, the origin of gravity is the bending of spacetime. It's only as a first approximation that it also corresponds to the gradient of time dilation
@@ScienceClicEN perfect! Thanks :)
ScienceClic English Fun Fact: Einstein used the precession of Mercury in order to prove his theory in 1915. However then there was no Schwartzschild metric or any other ‚tool‘ at hand.
Our universe is so weird that the mathematics to describe it, is so complex that it is not needed anywhere else outside the General Relativity (GR). Have you ever heard of Ricci tensor, Ricci scalar, the metric tensor or even the likely more common energy-momentum tensor in any context (in mathematics, physics or anywhere else) completely unrelated to GR ?
Hoorray, I understand now the black hole and the „meaning“ of the Schwarzschild Radius 😊 - but the equations.. 😬🤔 Thanks for this wonderful series! 👍
I was looking For such a Channel Thanku Soo Much I'm Active Subscriber Of Your Channel From now
Well done!
A superb explanation.
This is a brilliant series of GR videos!. This is the only place I've found any info about the alternative formulation of the equation and I can't find any more info about it!. What would the T scalar component be/correspond to in this alternative form where the stress tensor is on the left. Anyone know?
Great series. thank you. Also, i was hoping that Einstein’s equation would give an idea on "Why" energy flux/momentum has any affect on space time. but unfortunately, looks like in the end, it was written by empirical fitting. we still don't know "why " energy affect curvature... the fundamental interaction between energy and space-time
Incredible!
Any chance you guys could cover the singularity theorems that won roger penrose this years Nobel prize?
Wow really nice and easy to understand explanation of the connection of energy content and curvature. I like the series very much. But I must ask, how is this equation derived, can you make a video about that? Anyway thanks for this series, it made everything clearer about general relativity.
Glad you liked it ! The Einstein equation is not really derived, it is a postulate of the theory, the only equation that we take as an "axiom" in the theory. However you might wonder why Einstein supposed this equation and not something else ? There is quite a simple way to find this equation. The idea is that all theories in physics can be expressed as "principles of least action". Basically, any theory can be reduced to the postulate that : "the universe minimizes a certain quantity, S, called the action". This quantity must be a number. In our model, the only number that we have derived was the Ricci scalar R. Therefore we can simply suppose that S=R (or rather the integral of R). This corresponds to postulating that "the universe minimizes the average curvature of spacetime". Adding matter to this, and using the equations that describe the principle of least action (Euler-Lagrange equations), this yields Einstein's equation. So basically the Einstein equation comes from the postulate that the universe minimizes the Ricci scalar.
@@ScienceClicEN Although we mathematically take it as an “axiom”, it surely is not a mere lucky guess, that the action (or I suppose one of the terms in GR Lagrangian) depends on the Ricci scalar. Is there any physical intuition that could lead to this idea?
I think it's because the Ricci scalar is virtually the only scalar quantity that one can construct from curvature tensors. My personal physical intepretation would be that spacetime "wants" to be flat, just like a fabric with tension that would try to minimize its curvature
@@ScienceClicEN and maximize the time between?
OK, great that makes sense, it's like the Fermats principle which states that light always takes the path which requires least time or lagrangian mechanics. In principle it's the same equation just instead of the lagrangian you take the Ricci scalar.
AWESOME!!!! Thank you!!! But what about lambda - cosmological constant?
you have explained in the best way how mathematicans figure out that there could be a black hole in the universe, now i understand it at some level.
Excellent ! ! Thrilling ! ! Climatic ! ! I've never seen the Einstein equation written in terms to Tuv and T. From where did that come?.
Brilliant!
Thank you sir!!
this is so good
Light is an ordered vibration of gravitational quanta. This is determined experimentally using a hybrid fiber optic gyroscope (based on Michelson's experiment 1881-2015). Using a hybrid fiber optic gyroscope, the straight-line speed of vehicles can be measured.
All the examples are crucial for my brain to understand
you are great sir
The best of the best 🎉
THIS IS AMAZING
Sir thank you for this
Are you going to discuss the metrics for the case of electric charge or a rotating body?
What's strange, is, that there are similarities to quantummechanics. In GR, the tangentvector to the worldline of a particle has always lenght "1": 1 = g_xx dx²/ds² + g_xy dxdy/ds² + g_yx dydx/ds² + g_yy dy²/ds² In quantum mechanics, the sum over all particle probabilities has value "1": 1 = |Ψ_x|² + (Ψ_xΨ*_y=0) + (Ψ_yΨ*_x=0) + |Ψ_y|² That leads to: |Ψ_x| = dx/ds --> So I assume, the metric tensor can't be elimentary. It should be 16x16 and built up by other matrices (pauli matrices, clifford algebra or whatever)
It is enough to define the concepts to become subversive. E.g. Define the concept of "freedom" in the question "Which man will "freely" choose his eternal evil to go freely to hell?" before it is answered. In mathematics that communicate by physical reality the main axiom is the definition of the meenings and the terms that are under a logical connection before the answer in question or the solve of the problem.
If I can add something, the idea that “the boundary of a boundary is zero” that comes from the second Bianchi identity is ensured by the divergence of a quantity, known as the Einstein tensor (equal to the left hand side of the Einstein field equations) being 0. Since we know that the stress energy momentum tensor is what we want to use as the source in the field equations, and it is conserved (therefore having zero divergence), we seek a curvature tensor that also has 0 divergence to relate it to, and the Einstein tensor appears to offer this to us. Therefore, the Einstein field equations are also a theory of the conservation of stress energy momentum: energy momentum is conserved *because* the boundary of a boundary is zero. That these two ideas are related is itself the mathematical foundation of general relativity.
From the previous lecture, you mentioned measuring a metric space beyond a localized object without viscosity (ideally, the sun and a satellite within its radius; wrt that example, g(mu,nu) was not 0, regardless of idealized phenomena). How does the Energy Density Metric behave wrt its Ricci Tensor counterpart when the object in question has a magnetic field, and there is viscosity therein?
6:19 Does the Schwarzschild metric tell us if space and time are curved by the same amount, or does it say that the curvature of one is greater than the other?
U are the best of the best
Fun fact: Shwartzchild came up with his solution as a soldier in the trenches in WW1.
Spelling 8/10 Gravity 10/10
@@colfrancis9725 Haha
What is the background music name?
It's a creation for the channel, you can find it on my SoundCloud : soundcloud.com/aroussel
Very worth full vedio
Dude I´m just taking General Relativity in University and I´m really not that far with my Bachelor. Thx to this series and some other great channels I already learned what we will work on in 3 months :O KZhead is getting wild these days.
I have a suggestion for an improvement. Show a specific grid Reyes the page being split in half on the right hand side have the metric cancer and the Christoffel symbols as live updates from change is in the face is factors with respect to changes in the coordinate system.. So the first thing that would happen is that the coordinate system would shift from a to B then the base inspectors would shift as well. Then metric Tesora would change that’s changing the Christoffel symbol. All one animation so the cause-and-effect is clear.
What is a metric cancer?
Point of clarification: it's interesting to note that the metric does not depend on the shape of the gravitating body or how the mass is distributed on the inside, but it doesn't work properly if any of its bits are at a higher altitude than the observer. The best you can do in that case is break it down into smaller chunks and solve the equations for each one separately. Another interesting tidbit: the known solutions to these equations do not allow the gravitating body to move. Seems like an oversight because we're pretty sure we've seen black holes bashing into one another for example.
Hmm not really, the metric does depend on the shape of the body, and GR can very well resolve the metric for moving objects. Maybe you were refering to the Birkhoff theorem ?
@@ScienceClicEN I was thinking of the Schwarzschild metric, but Birkhoff theorem seems to be what I was talking about in my 1st point and it has a more general scope. As for my 2nd point, the field equations may not preclude the case where the gravitating body is in motion, but I'm not aware of any metric solutions for that case. I think you have to break the timeline down into a series of snapshots and solve each one individually. The case of black holes in motion is particularly puzzling though because you can't push them around with contact forces. It's tantamount to moving the goalposts.
A single object in motion can be treated by a change of coordinates. For Black holes in orbit it's more complex but it can be done analytically in the weak field approximation, with the linearized Einstein equation. Usually the linearized equations are good for predicting kinetic effects in GR like frame-dragging.
@@ScienceClicEN Good point
So as an external observer, everything that reaches the event horizon will be frozen in time?
what software do you produce these videos with? in particular, the parts where equations move around and change color.
I do all my animations with After Effects, for the equations it was a bit painful but I found a few tricks to make the process faster. Basically I slice the equations into different layers for each term that I want to move, and then animate everything by hand.
I’m no mathematician but for the alternative version of the Einstein field equation wouldn’t the stress energy scalar term 1/2T guv cancel to 1/2Tuv Duv where Duv is the kronecker delta, to my knowledge this kind of tensor multiplication does associate so if you substitute in the definition of the stress energy scalar T into the field equation wouldn’t the metric and inverse metric tensors cancel to the delta leaving the stress energy tensor?
It emphasizes how genius Einstein and his colleagues were, especially at their times.
VERY GOOD. CONCRETE AND CONCISE
Would the relationship between Content and Curvature be kinda similar to the relationship between an moving Charged Quantum Object and its Magnetic Field? Its movement induced the magnetic fields and the magnetic fields guides its movement?
It's quite similar yes. Energy is the equivalent of the electric charge but for the curvature of spacetime. And much like the electric / magnetic fields, you can say that static energy is the source for gravity, whereas moving energy is the source for "gravitomagnetism", most usually known as frame-dragging : en.wikipedia.org/wiki/Gravitoelectromagnetism
A question: is there any mathematical model without space time curvature concept with the same practical result?
I think graviton theories claim to achieve this, no?
Yes, alternatives to General Relativity (GR) exist. MND - Modified Newtonian Dynamics is one example that I'm aware of. This focuses on explaining unusual motion in systems of planets and stars. This may not replace all of GR but it is a large chunk of what GR is used to explain. However GR (and the curvature of spacetime) seems to be the most consistent, most tested and verified theory from the list of competitors so far. Nothing much lasts forever in Science, what we call curvature of spacetime today may be re-described as an Abtahi Quantum Phenomena in twenty years time.
No.
General relativity can be expressed as an effect on measuring rods and clocks without invoking spacetime geometry, though of course this description will be exactly equivalent to spacetime curvature.
@@colfrancis9725 thank you very much. I Know that MND is established to explain dark matter in galaxies ,however I am not sure that it can explain all gravity related phenomena .
I would like more info on the energy momentum tensor, like, how to calculate it if it even makes sense; maybe a practical example such as how einstein correctly predicted the precession of mercury orbit and a example of calculation inside a massive body (so that "Tuv" doesn't vanish); as I side note: yeah, I'm aware that it is where the rubber meets the road, hence why I'm curious : ) also, what if a force is acting on the object? the field equations make no direct reference to forces, so I'm assuming they are implied within the Tuv tensor, am I right? and if so, is there such a concept of conservative and non-conservative forces in einstein model, or does it treat all forces as conservatives? (e.g. a force which the line integral doesn't depend on the path, thus having a energy function associated to the initial and final points)
I'm confused about something: if we need an energy-momentum distribution (T tensor) to determine the spacetime metric, that being the distances, angles, curvature between things, etc. -- how is that distribution of energy-momentum content calculated in the first place? Wouldn't that require knowing the metric beforehand?
Partial answer / suggestion: Have you checked out episode 6/8 in this series about Energy Fluxes? Rough idea - in the simplest case you model everything as a "fluid", you only need to know a few things about that fluid. For example, the density, pressure and momentum (although ideally that would be at every point in space). The distribution of matter is given with the co-ordinates you are using and it doesn't matter how you measure lengths (including time) or angles in those co-ordinates, so the metric isn't required at that stage. For example, a density could be described as 1Kg of the fluid in every volume element. The volume element can be a wonky-shaped region of space given by 1 x-direction unit by 1 y-direction unit by 1 z-direction unit. It doesn't matter if the x,y, z directions were skewed, sheared, stretched or curvi-linear. Whatever wonky co-ordinates you specify the parameters with (density, pressure etc.) you will get the metric expressed in those same wonky co-ordinates. Hope this helps.
Isn't it beautifull when everything just works out?! :D
Does the Schwarzschild metric describe a Minkowski metric (empty space-time) even when "r" tends to infinite , still having mass?
In a way yes, we say that this metric is "asymptotically flat at infinity"
What about a uniform gravitational field throughout the universe? That sounds like it ought to be a simple scenario to model, yes? It would make a reasonable approximation to the surface of the Earth, wouldn't it?
That is essentially how the real universe behaves, since the distribution of matter, energy, and pressure throughout the universe is (on large scales) the same everywhere. So if we assume a uniform density and pressure everywhere, then Einstein's equation leads to a simple and very important metric that we use in cosmology, called the Friedmann-Lemaître-Robertson-Walker metric, or FLRW for short, after its discoverers. This metric describes a geometry which can either be flat (no curvature), or have constant positive or negative curvature, and a universe with each of these geometries will evolve in different ways.