Tears of joy. I entered here not knowing what a metric even was and now am learning how to derive the Schwartz child metric via differential geometry. Thanks Chris 👍
@@joesmith8288 Don't be jealous of him learning tensor calculus before you. EDIT: Oops. I just read that part about "tears of joy." I've got to retract my statement.
Chris, the clarity of your videos, and your don’t skip the hard stuff rigor, have not only made an understanding of the Einstein field equations accessible to a seriously curious high school senior and certainly to a similarly curious college undergraduate, but also you have shown them the mathematical tools needed to solve the equations rather than simply a magical formula at the end. The only comparison I can think of are the Feynman lectures on physics. If there were an equivalent Nobel or Fields prize for what you have done to demystify for young students (any age, really)the many hang points of just partial understanding so common in normal teaching, you deserve that prize.
I understand it will be a lot of work for you, but covering the kerr metric in the future would be extremely helpful. There are little to no derivations i can find for that metric online and most GR lectures skip over the derivation and jump right into using it.
I agree it's really needed & important but I dunno if it's possible to do it without e.g. NoahExplainsPhysics derivation of time translation symmetry = energy conservation: it's done in a very formula heavy way that means he only has 2.9k views. So PLEASE do the Kerr metric as no one else does it but please try to present it in a more palatable, mainstream way? I dunno how one would go about doing this.
the thing is that the derivation of the kerr metric is absolutely diabolic, it took literally decades to figure it out, and considering that as you say, it is important, that is pretty scary. So no, of course you won't find "mainstream" derivations, such thing does not exist, I would look the original papers if you really still want a derivation.
A lot of viewers are more interested in retracing Einstein's path to learning Special relativity, equivalence principle, tensor analysis, and using the weak-field approximation to derive Newtonian gravity. The complete GR solutions from Schwarzschild and Friedman are very useful, but it's well above the mathematical abilities of a person not enrolled in a physics graduate program. The more advanced gravitational models with frame dragging, measurable gravitational waves, and black hole rotations are done by computer programs. The first approximation used by Einstein to recover Newton's theory of gravity and confirm the precession of Mercury's orbit, the gravitational deflection of light, and a gravity law that propagates at the speed of light were enough for Einstein and Hilbert to publish their field equations in 1915.
This Mr. Schwarzschild figured out Einstein's method in a couple months? Sitting in a trench with bullets overhead?? What a man! As are you Chris, thanks
Hello Chris! I recently graduated high school, and your videos on general relativity played a crucial role. In Denmark, every high school student must complete a special study project (SRP) during their final year. The project has to be at least 15 to 20 pages long and research a self-selected topic. I chose to focus my project on "The Derivation of the Schwarzschild Metric and Black Holes" (my physics teacher thought it might be challenging but not impossible for me to tackle). Thanks to your videos, I was able to grasp complex concepts and present them effectively in my project. During the oral exam, where I had to defend my written work, both examiners were thoroughly impressed and commented that it might be the best project they had ever seen. Many thanks to you and your invaluable videos, Chris! And by the way, my physics teacher and I are now your biggest fans after watching your videos almost on loop the last few months :-)
There is a typo on timestamp 15:50 and corresponding slide #54 from the presentation. In the bottom line g22 is non-zero. Thank you for wonderful lecture!
Great video! For the section where you are finding the constant k, how can you make the equivilence that - 2GM/rc^2 = k / r when you changed coordinates for r in the derivation r = sqrt(C)r? Thanks
That honestly slipped my mind. I'll have to think about that. I think the difference in r-coordinates only becomes apparent close to the Schwarzschild radius. For the earth the Scharzschild radius is only 1cm or so, so there's no real difference. The sun's r_s is make 3km, so again no real difference. I'll talk about that in 108b.
BTW: About staticity of the metric - the only assumption you can go into the derivation with is spherical symmetry (and that einstein equations hold). a] The linear angular components of the metric dtheta, dphi will vanish as the spacetime separation must not be dependent on direction of angels you travel by. The g_tr and g_rt components must also be zero at least on the coordinate origin - as the radial geodesics begin there, perpendicular to time geodesics. Then you look at geodesic equation which tells you that Gamma^mu_rr is zero, which can be expressed in terms of metric components and due to some symmetries in the formula it follows that g_(mu)r,t =0, so the geodesics remain perpendicular everywhere. Recap: Angles cannot be represented linearly, g_tr component is also zero due to the fact that it is zero at the origin and this property can be stretched, so only remaining components are the diagonal ones. Metric is diagonal! (note: only vanishing of the linear dt terms in metric would be sufficient, we however have diagonality which is much stronger). b] Now one introduces the diagonal metric form into EFE. You go the usual route with general (now diagonal) metric: metric -> christoffel -> riemann -> ricci tensor (and ricci scalar = 0 we know that already) -> einstein tensor = 0 From the G^0_1 component of einstein tensor one learns immidiately that the g_rr,t = 0 and further on with some math juggling one learns from G^0_0 that it should be proportional to inverse of g_rr => it also does not depend on time (or rather the constant of proportionality can depend on time only -> can be some function f(t) which can later on be hidden into scaling time coordinate instead of the metric). So the metric does not depend on time from einstein equations and does not contain linearly dt (spacetime interval does not depend on the direction of time) as the metric is diagonal from geometry => it is static. I know this is a bit nitpicky and im not even trying to correct anyone, just saying it as an interesting caviat for some of the more advanced viewers. I usually visit this channel quite regularly as I get to recap some of my previous lectures and I always draw some inspiration and new insight here so thanks a bunch!
As usual, what a beautiful video. Many thanks for taking the time to do them and share. Do you have any recommendations for someone who wants to learn more?
@@eigenchris There is a typo on timestamp 15:50 and corresponding slide #54 from the presentation. In the bottom line g22 is non-zero. Thank you for wonderful lecture!
The Einstein equation is said to PROVE that the mass/energy Tensor on the RHS of the equation leads to the bending of space time and creates the resultant paths of bodies as they traverse space time It does NO such thing. What it does is first to describe bodies moving through flat space and then develop the changes required , by very complex partial derivatives , with other Tensors using the same method , to conclude that the RHS of equation produced curved space time as indicated by the LHS ie it as essay in reasoning of Einstein using complex maths as the support. For example as a practical error ; when the bending of light by gravity was measured the results were completley different depending on where the measurement took place.
Thanks Chris. I was looking at Schwarzschild metric derivation yesterday and the only thing I could find was a 3-part series after watching which I was more confused. Chris to the save today! Thank you again and Google for keeping tabs of what I saw.
Can you do the same for Robertson-Walker-Metric and Cosmology??? And explain, how it's related with Friedmann-Equations and all the dynamics/evolution of the universe.
My plan is to do that in 108c (not for earth specifically, but any mass in a Schwarzschild spacetime). I'm pretty sure the solution is analytical. But the solution is very similar to the Newtoniam result unless you're very close to the mass. Basically the gravitational potential gets an extra 1/r^3 term that goes to zero very quickly as r gets big.
You can look these coefficients up on wikipedia or any GR textbook. en.m.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution#Calculating_the_Christoffel_symbols
Thank you for the great work you did. It is possible to make a video about extracting torsion from metrics (the line element) that describe the topological defect space-time. For example, the extracting torsion from the line element describes a kind of dislocation in space-time.
This is the first I've heard of this concept. I'm not familiar with it at all. I probably won't be making a video on it. Do you know where I can read more?
@@eigenchris The metric can have curvature or torsion or both of them. If you wish, I can email you a series of resources and articles in this context.
Hey, I dont understand this argument of "space is flat for infinite distance from the mass". This sounds like another axiom to me. Why are we allowed (or maybe even forced?) to use intuition in addition to the einstein field equations? Shouldnt we get this as a result, instead of having it as an assumption?
It's common when solving differential equations to specify "boundary conditions". Sometimes this means sepcifying the field at infinity. I think there would be similar requirements for, say, Maxwell's equations.
For EM field inverse square law was experimentally tested to a high accuracy. For gravity some people propose MOND, and while experiments are still inconclusive, Newtonian gravity better fits the data.
I am very excited for the new journey, I have always looked forward to it. I also have completed all of the Tensor series and as well as general relativity series... Now I will go beyond it. Thanks for uploading this video.
You are assuming a static metric at 8:20. Can you also tell in which frame? Because we know time is relative and mixes up with space when changing frames. So if you assume a static metric, I can come up with a frame where it is no more static. Kindly also specify the frame where it is static, because it creates confusion.
@@eigenchris When I did my exams: I was told of a method of computing the connection coefficients that didn’t involve using the formula: it involved writing out the Lagrangian and then computing the geodesic equation using the Euler Lagrange equations: the connection coefficients would then be deduced via the geodesic equations My teacher always preferred this method then using the general formula, they thought it was much less fidly
In theory, it's possible to have a geometry where R=0 but R_μν is not zero. But if we are constrained by the Einstein field equations, and are in a vacuum where T_μν=0 (and also take the cosmological constant to be zero), then R=0 and R_μν = 0 are equivalent conditions.
@@eigenchris Thanks! I see. There's one more step: after R vanishes from the EFE, all the terms go to zero except for the first term. Thus R_μν must also vanish.
I just stumbled into the video. I’m far from being competent in GRT but have some experience in mathematical modelling. The suprising fact is that obviously the mass/energy of our spherical star/planet has nothing to do with tensor T. How is that even possible? Why is there the „Newtonian“ constant G in front of T when Newtonian gravitation has to be imposed a posteriori onto the metric? This really seems a bit disappointing. So the field equations as they stand do not include the Newton case. What then is the sense of G in front of T when when gravity is not a force but the consequence of the bending of spacetime? Why? Why is our star not part of the energy-stress-tensor T (T11)? And what does that mean „low velocity and weak gravity“, who says that our star will cause „weak“ gravity and whose velocity is low? Which criterion makes gravity „weak“ a p r i o r i? Do we have to conclude that a little bit of energy, pressure and stress in T really is the cause of „strong“ gravity? Is that an empirical fact? How do you handle galaxies? And for velocity? Is the observers velocity at infinity (which in itself is nonsense) low? No, thats zero. So what velocity is low? Btw, in which sense is the star non-rotating, not rotating around his own axis?
The T=0 used in this video is only for the vacuum *surrounding* the earth, as I point out at 3:15. Since T is the generalization of the mass density, you can think of this as seeing mass density to zero outside the earth. Newtonian gravity is brought in at 24:47, when I force the potential from Newtonian gravity to match up with the results predicted in GR. If you're up for watching more, you can check out my "Relativity 107d" video on Newton-Cartan theory. This is a theory of Newtonian gravity that uses curved spacetime, and it makes the jump to GR much easier. The Einstein Equations are a generalization of Poisson's Equation, where the metric g replaces the gravitational potential, and the energy momentum tensor T replaces the mass density. If you want a solution that takes the interior of a star into account, you can google the "interior schwarzschild solution". I don't know if "weak" has a formal definition. It's basically a limit where relativistic effects are negligeable. So there is no time dilation either from large velocities, and we don't need to take non-Newtonian GR effects into account from large masses. Rotations are considered objective in relativity, because rotating bodies have centripetal acceleration (velocities are relativity but an rotating object fundamentally has non-inertial motion).
4:10 how do you go from saying that the trace of the Ricci Tensor is 0, to the claiming that each entry must be zero? Is that derivable from the symmetries of the Ricci tensor? I tried watching your Tensor Calculus videos to find an explanation, but I couldn’t find one.
08:00: It's actually _Killing vector_ (with capital 'K') because the vectors don't take any lifes but are named after the German mathematician Wilhelm Killing (1847-1923).
Can anyone help me out because I'm grapling with a thought experiment of mine: an object falls freely towards a pointlike large mass in a (hence curved) 1+1D spacetime, its worldline will look like an osillation - lets say it falls through a tunel because just hey, let's suppose. That oscillation is a geodesic of the curved n+1D (i.e. 3D) shape, isn't it? We should be able to imagine that very shape, should we not? So, if existing at all, what does it look like and if not, where am I getting things so terribly wrong?
Cool explanation! Thank you! Could you tell in which literature the introduction of the function C(r) is described to satisfy spherical symmetry (time 10:10, slide number 28 in PP presentation)?
at 21:47 you have got :( (drA)^2/ 2A^2 ) - ((drA)^2/4A^2).... so I make this to be equal to ( - (drA)^2/ 2A^2 ). I hope that I am not wasting your time!
Question sir. In the case of rotating black hole, time isnt orthogonal to spatial components, r, theta, phi so that g_mu*v can change under timd reversal t --> -t? g_ti --> -g_ti. If time is orthogonal to spatial components, then g_ti = 0 and hence g_mu*v wont change?
Thank you for your video. I don't understand what "spherical symmetry" means for metric tensor. Like, normally if we want to say if sth is symmetric, we compare its values in different positions. But since covariant derivative of metric is zero when it is torsion-free, does that mean it's symmetric for all coordinates?
If you use the mass energy equivalence for a single static charge,you know have a hole in hole in hole in hole with 360° twist,charge without charge,mass with out mass. A static charge is its own CP inverse .Until "The Particle Problem in the General Theory of Relativity" can be answered G.R. will always be incomplete.
If I wish to get A replaced in B and vice versa, in the same line element of schwarzschild , what are the turns and modifications I have to do to obtain so?
Thanks. It might be out by the end of October. Most of the slides are done, but there are some concepts I still don't understand, so I'll need to spend more time learning and fact-checking.
24:20 when I find the solution to the equation I get k/r - 1, I dont know if I did something wrong but I inputed the equation into different calculators and got the same thing. If you read this can you please show me your solution for the equation
The curvature of spacetime only depends on the derivatives of the metric, not the actual metric value. So I don't think small additive numbers like this change the physics.
@@nenhard I should maybe be a bit more careful... the metric itself does appear in the Einstein Field Equations, but as long as we ignore the cosmological constant, and the Ricci scalar is zero, we don't need to worry about the constant.
I plan to make them eventually, but I've covered that content in my previous "tensors for beginners" videos so I've been less motivated to make them. Do you have any particular questions you want answered?
we are saying g_teta,phi = 0 so that g be similar to spherical coordinate metric. do you only try a g with g_teta,phi = 0 to see if it works or is there more mathematical background when we say g_teta,phi = 0.
The "mathematical reason" is that, if we take a constant time, and constant radius, we want the resulting shell surface to have the same geometry as an ordinary 2D spherical surface, by spherical symmetry. I don't think I have a better reason.
I was a bit "loose" with the indices here. To be technically correct, I would need a kronecker delta on one of the terms to raise/lower the index appropriately, so that everything matches. The kronecker delta doesn't actually change any of the values in the formula, it just puts the indices in the right place, so I left it out for simplicity. Sorry if that's confusing.
I didn't get into the details much... it might become more clear in later videos. A rotating black hole causes additional gravitational effects in the direction of rotation (you can google "ergosphere" to learn more). If we reverse the direction of time, these gravitational effects reverse. So if we force spacetime to be the same in both time directions, it guarantees no rotation.
Tears of joy. I entered here not knowing what a metric even was and now am learning how to derive the Schwartz child metric via differential geometry. Thanks Chris 👍
cringe
@@joesmith8288 No u
The joy of learning
@@joesmith8288 Don't be jealous of him learning tensor calculus before you.
EDIT: Oops. I just read that part about "tears of joy." I've got to retract my statement.
@@dritemolawzbks8574 Ok sissy🤣🙋🙏
This channel is simply one of the most valuable things I've ever found not just on RUclips, but on the whole holly internet,
And I literally mean it.
Cringe
Amen : )
Chris, the clarity of your videos, and your don’t skip the hard stuff rigor, have not only made an understanding of the Einstein field equations accessible to a seriously curious high school senior and certainly to a similarly curious college undergraduate, but also you have shown them the mathematical tools needed to solve the equations rather than simply a magical formula at the end.
The only comparison I can think of are the Feynman lectures on physics.
If there were an equivalent Nobel or Fields prize for what you have done to demystify for young students (any age, really)the many hang points of just partial understanding so common in normal teaching, you deserve that prize.
I understand it will be a lot of work for you, but covering the kerr metric in the future would be extremely helpful. There are little to no derivations i can find for that metric online and most GR lectures skip over the derivation and jump right into using it.
100% agree for how important the kerr solution is for astrophysics it's poorly described in popular media
I agree it's really needed & important but I dunno if it's possible to do it without e.g. NoahExplainsPhysics derivation of time translation symmetry = energy conservation: it's done in a very formula heavy way that means he only has 2.9k views. So PLEASE do the Kerr metric as no one else does it but please try to present it in a more palatable, mainstream way? I dunno how one would go about doing this.
the thing is that the derivation of the kerr metric is absolutely diabolic, it took literally decades to figure it out, and considering that as you say, it is important, that is pretty scary.
So no, of course you won't find "mainstream" derivations, such thing does not exist, I would look the original papers if you really still want a derivation.
A lot of viewers are more interested in retracing Einstein's path to learning Special relativity, equivalence principle, tensor analysis, and using the weak-field approximation to derive Newtonian gravity.
The complete GR solutions from Schwarzschild and Friedman are very useful, but it's well above the mathematical abilities of a person not enrolled in a physics graduate program.
The more advanced gravitational models with frame dragging, measurable gravitational waves, and black hole rotations are done by computer programs.
The first approximation used by Einstein to recover Newton's theory of gravity and confirm the precession of Mercury's orbit, the gravitational deflection of light, and a gravity law that propagates at the speed of light were enough for Einstein and Hilbert to publish their field equations in 1915.
@@alwaysdisputin9930 if you "don't know" then how the hell can you be advising him you dodo??! Keep your mouth shut🙋🙏
This Mr. Schwarzschild figured out Einstein's method in a couple months? Sitting in a trench with bullets overhead?? What a man! As are you Chris, thanks
Hello Chris! I recently graduated high school, and your videos on general relativity played a crucial role. In Denmark, every high school student must complete a special study project (SRP) during their final year. The project has to be at least 15 to 20 pages long and research a self-selected topic. I chose to focus my project on "The Derivation of the Schwarzschild Metric and Black Holes" (my physics teacher thought it might be challenging but not impossible for me to tackle).
Thanks to your videos, I was able to grasp complex concepts and present them effectively in my project. During the oral exam, where I had to defend my written work, both examiners were thoroughly impressed and commented that it might be the best project they had ever seen. Many thanks to you and your invaluable videos, Chris!
And by the way, my physics teacher and I are now your biggest fans after watching your videos almost on loop the last few months :-)
There is a typo on timestamp 15:50 and corresponding slide #54 from the presentation. In the bottom line g22 is non-zero. Thank you for wonderful lecture!
its amazing how we can freely access this level of information
You've obviously put a heck-of-lot of thought, time, and energy into this fine video... Enjoyed it...
The best courses of GR I have ever seen !! Thank you !!
Btw if you want to make your matrices look better you can just add in blank rows to fix the non aligning issue
Great video! For the section where you are finding the constant k, how can you make the equivilence that - 2GM/rc^2 = k / r when you changed coordinates for r in the derivation r = sqrt(C)r? Thanks
That honestly slipped my mind. I'll have to think about that. I think the difference in r-coordinates only becomes apparent close to the Schwarzschild radius. For the earth the Scharzschild radius is only 1cm or so, so there's no real difference. The sun's r_s is make 3km, so again no real difference. I'll talk about that in 108b.
@@eigenchris Thank you! I never really got a satisfactory explanation for that.
I just want to point out that, at 25:05, you spelled Newoton's instead of Newton's(and you are the best physics youtuber)
BTW: About staticity of the metric - the only assumption you can go into the derivation with is spherical symmetry (and that einstein equations hold).
a] The linear angular components of the metric dtheta, dphi will vanish as the spacetime separation must not be dependent on direction of angels you travel by. The g_tr and g_rt components must also be zero at least on the coordinate origin - as the radial geodesics begin there, perpendicular to time geodesics.
Then you look at geodesic equation which tells you that Gamma^mu_rr is zero, which can be expressed in terms of metric components and due to some symmetries in the formula it follows that g_(mu)r,t =0, so the geodesics remain perpendicular everywhere.
Recap: Angles cannot be represented linearly, g_tr component is also zero due to the fact that it is zero at the origin and this property can be stretched, so only remaining components are the diagonal ones.
Metric is diagonal!
(note: only vanishing of the linear dt terms in metric would be sufficient, we however have diagonality which is much stronger).
b] Now one introduces the diagonal metric form into EFE.
You go the usual route with general (now diagonal) metric: metric -> christoffel -> riemann -> ricci tensor (and ricci scalar = 0 we know that already) -> einstein tensor = 0
From the G^0_1 component of einstein tensor one learns immidiately that the g_rr,t = 0 and further on with some math juggling one learns from G^0_0 that it should be proportional to inverse of g_rr => it also does not depend on time (or rather the constant of proportionality can depend on time only -> can be some function f(t) which can later on be hidden into scaling time coordinate instead of the metric).
So the metric does not depend on time from einstein equations and does not contain linearly dt (spacetime interval does not depend on the direction of time) as the metric is diagonal from geometry => it is static.
I know this is a bit nitpicky and im not even trying to correct anyone, just saying it as an interesting caviat for some of the more advanced viewers. I usually visit this channel quite regularly as I get to recap some of my previous lectures and I always draw some inspiration and new insight here so thanks a bunch!
8:50 so static metric implies the time axis is orthogonal to the space axes.
Cool. Good to keep in mind.
Excellent video. Extremely clear and didactic. Thank you very much.
By the way,@24'32 A(r) shouldn't be with the form A(r) ) = 1 - k/(r + constant ) ?
The thing that confuses every time I see this derivation is what we mean when we say spherical symmetry ,I didnt really get that .Also amazing video!
It basically means we can rotate the space part of spacetime around the center of mass and see no change.
@@eigenchris Thank you for your quick answer!
As usual, what a beautiful video. Many thanks for taking the time to do them and share.
Do you have any recommendations for someone who wants to learn more?
You could take a look at Sean Carroll's notes on General Relativity. www.preposterousuniverse.com/grnotes/
“The Final Theory: Rethinking Our Scientific Legacy “,Mark McCutcheon.
@@eigenchris There is a typo on timestamp 15:50 and corresponding slide #54 from the presentation. In the bottom line g22 is non-zero. Thank you for wonderful lecture!
The Einstein equation is said to PROVE that the mass/energy Tensor on the RHS of the equation leads to the bending of space time and creates the resultant paths of bodies as they traverse space time
It does NO such thing.
What it does is first to describe bodies moving through flat space and then develop the changes required , by very complex partial derivatives , with other Tensors using the same method , to conclude that the RHS of equation produced curved space time as indicated by the LHS
ie it as essay in reasoning of Einstein using complex maths as the support.
For example as a practical error ; when the bending of light by gravity was measured the results were completley different depending on where the measurement took place.
Thanks Chris. I was looking at Schwarzschild metric derivation yesterday and the only thing I could find was a 3-part series after watching which I was more confused. Chris to the save today! Thank you again and Google for keeping tabs of what I saw.
Can you do the same for Robertson-Walker-Metric and Cosmology??? And explain, how it's related with Friedmann-Equations and all the dynamics/evolution of the universe.
That will be in the Relativity 110 videos that I hint at near the beginning.
Could we potential derive the orbits of the earth around the sun using general relativity? Or does no analytic solution exist?
My plan is to do that in 108c (not for earth specifically, but any mass in a Schwarzschild spacetime). I'm pretty sure the solution is analytical. But the solution is very similar to the Newtoniam result unless you're very close to the mass. Basically the gravitational potential gets an extra 1/r^3 term that goes to zero very quickly as r gets big.
"calculating is pretty boring but necessary... " 🧐 My question is, who is capable of checking all of your calculations?
You can look these coefficients up on wikipedia or any GR textbook. en.m.wikipedia.org/wiki/Derivation_of_the_Schwarzschild_solution#Calculating_the_Christoffel_symbols
Hi there, at 15:50, I think you crossed out the wrong term! I looked like you kept -d2g12 but you actually kept d1g22, otherwise fantastic video!
Would you consider making a video on the Tolman-Oppenheimer-Volkoff solution?
I've never heard of that one, so I'm not going to be much help. I'm focusing on my current series on spinors for now.
Happy New Year 2023 Chris. All the Best for you and your family. Thanks a lot for all the videos you make to cover tensoralgebra and cosmology...👍👍🙏
Thank you for the great work you did. It is possible to make a video about extracting torsion from metrics (the line element) that describe the topological defect space-time. For example, the extracting torsion from the line element describes a kind of dislocation in space-time.
This is the first I've heard of this concept. I'm not familiar with it at all. I probably won't be making a video on it. Do you know where I can read more?
@@eigenchris The metric can have curvature or torsion or both of them. If you wish, I can email you a series of resources and articles in this context.
Hey, I dont understand this argument of "space is flat for infinite distance from the mass". This sounds like another axiom to me. Why are we allowed (or maybe even forced?) to use intuition in addition to the einstein field equations? Shouldnt we get this as a result, instead of having it as an assumption?
It's common when solving differential equations to specify "boundary conditions". Sometimes this means sepcifying the field at infinity. I think there would be similar requirements for, say, Maxwell's equations.
For EM field inverse square law was experimentally tested to a high accuracy. For gravity some people propose MOND, and while experiments are still inconclusive, Newtonian gravity better fits the data.
I was looking for this specific topic form you 😁😁😁 thanks I really have learned a lot from your Channel 😊
I am very excited for the new journey, I have always looked forward to it. I also have completed all of the Tensor series and as well as general relativity series... Now I will go beyond it. Thanks for uploading this video.
You are assuming a static metric at 8:20. Can you also tell in which frame? Because we know time is relative and mixes up with space when changing frames. So if you assume a static metric, I can come up with a frame where it is no more static. Kindly also specify the frame where it is static, because it creates confusion.
Isn’t it easier to compute the Chrisoffel symbols by deriving the geodesic equation rather than computing the formula with the metric
Sorry, I don't follow. I would have figured you'd need to already know the christoffel symbols to use the geodesic equation in any meaningful way.
@@eigenchris When I did my exams: I was told of a method of computing the connection coefficients that didn’t involve using the formula: it involved writing out the Lagrangian and then computing the geodesic equation using the Euler Lagrange equations: the connection coefficients would then be deduced via the geodesic equations
My teacher always preferred this method then using the general formula, they thought it was much less fidly
@@cameronspalding9792 I'm not so familiar with Lagrangian methods. If you have a link to a page that explains this method, I could take a look.
people.uncw.edu/hermanr/GRcosmo/euler-equation-geodesics.pdf
How do you get 1-c/r when you solve that differential equation? I'm getting 1+c/r.
you are more than wonderful
Thank You so very much for sharing and posting.
who would ever dislike one of these videos?
Thank you, I have been waiting for your video
Waiting attentively for your 108d and e 😇🌹
You are the best .
Brown Edward Johnson Eric Gonzalez Barbara
Gravity inside the black hole follows Newton's shell theorem. Gravity in the center of a black hole is zero.
Question: How does the Ricci Scalar being zero imply the Ricci tensor being zero? (4:10) Isn't Ricci Scalar a sum of R^i_i?
In theory, it's possible to have a geometry where R=0 but R_μν is not zero. But if we are constrained by the Einstein field equations, and are in a vacuum where T_μν=0 (and also take the cosmological constant to be zero), then R=0 and R_μν = 0 are equivalent conditions.
@@eigenchris Thanks! I see. There's one more step: after R vanishes from the EFE, all the terms go to zero except for the first term. Thus R_μν must also vanish.
amazing!
similarly,in 22:35 ,should it not be in the 3rd line, - rdrB/B^2
YEAH!YEAH!YEAH! Thank you so much
I just stumbled into the video. I’m far from being competent in GRT but have some experience in mathematical modelling. The suprising fact is that obviously the mass/energy of our spherical star/planet has nothing to do with tensor T. How is that even possible? Why is there the „Newtonian“ constant G in front of T when Newtonian gravitation has to be imposed a posteriori onto the metric? This really seems a bit disappointing. So the field equations as they stand do not include the Newton case. What then is the sense of G in front of T when when gravity is not a force but the consequence of the bending of spacetime? Why? Why is our star not part of the energy-stress-tensor T (T11)? And what does that mean „low velocity and weak gravity“, who says that our star will cause „weak“ gravity and whose velocity is low? Which criterion makes gravity „weak“ a p r i o r i? Do we have to conclude that a little bit of energy, pressure and stress in T really is the cause of „strong“ gravity? Is that an empirical fact? How do you handle galaxies? And for velocity? Is the observers velocity at infinity (which in itself is nonsense) low? No, thats zero. So what velocity is low? Btw, in which sense is the star non-rotating, not rotating around his own axis?
The T=0 used in this video is only for the vacuum *surrounding* the earth, as I point out at 3:15. Since T is the generalization of the mass density, you can think of this as seeing mass density to zero outside the earth. Newtonian gravity is brought in at 24:47, when I force the potential from Newtonian gravity to match up with the results predicted in GR.
If you're up for watching more, you can check out my "Relativity 107d" video on Newton-Cartan theory. This is a theory of Newtonian gravity that uses curved spacetime, and it makes the jump to GR much easier. The Einstein Equations are a generalization of Poisson's Equation, where the metric g replaces the gravitational potential, and the energy momentum tensor T replaces the mass density.
If you want a solution that takes the interior of a star into account, you can google the "interior schwarzschild solution".
I don't know if "weak" has a formal definition. It's basically a limit where relativistic effects are negligeable. So there is no time dilation either from large velocities, and we don't need to take non-Newtonian GR effects into account from large masses.
Rotations are considered objective in relativity, because rotating bodies have centripetal acceleration (velocities are relativity but an rotating object fundamentally has non-inertial motion).
4:10 how do you go from saying that the trace of the Ricci Tensor is 0, to the claiming that each entry must be zero? Is that derivable from the symmetries of the Ricci tensor? I tried watching your Tensor Calculus videos to find an explanation, but I couldn’t find one.
If you look back at the Einstein Field Equations at the top of the slide, and plug in R=0, Lambda=0, Tuv=0, you get Ruv=0.
This is beautiful. Thank you
08:00: It's actually _Killing vector_ (with capital 'K') because the vectors don't take any lifes but are named after the German mathematician Wilhelm Killing (1847-1923).
Can anyone help me out because I'm grapling with a thought experiment of mine: an object falls freely towards a pointlike large mass in a (hence curved) 1+1D spacetime, its worldline will look like an osillation - lets say it falls through a tunel because just hey, let's suppose. That oscillation is a geodesic of the curved n+1D (i.e. 3D) shape, isn't it? We should be able to imagine that very shape, should we not? So, if existing at all, what does it look like and if not, where am I getting things so terribly wrong?
Cool explanation! Thank you! Could you tell in which literature the introduction of the function C(r) is described to satisfy spherical symmetry (time 10:10, slide number 28 in PP presentation)?
at 21:47 you have got :( (drA)^2/ 2A^2 ) - ((drA)^2/4A^2).... so I make this to be equal to ( - (drA)^2/ 2A^2 ). I hope that I am not wasting your time!
I have a test in 30 minutes lol😂
Schwarzschild was born on 9 October 1873, this video was posted on 10 October 2021
It would be a nice video. Thank you
17:39 ,Great video! But I have a question, How are we justifying that the first 3 diagonal Ricci tensor components become zero?
Question sir. In the case of rotating black hole, time isnt orthogonal to spatial components, r, theta, phi so that g_mu*v can change under timd reversal t --> -t? g_ti --> -g_ti. If time is orthogonal to spatial components, then g_ti = 0 and hence g_mu*v wont change?
Thank you
Thank you for your video. I don't understand what "spherical symmetry" means for metric tensor. Like, normally if we want to say if sth is symmetric, we compare its values in different positions. But since covariant derivative of metric is zero when it is torsion-free, does that mean it's symmetric for all coordinates?
❤Thank you very much
If you use the mass energy equivalence for a single static charge,you know have a hole in hole in hole in hole with 360° twist,charge without charge,mass with out mass. A static charge is its own CP inverse .Until "The Particle Problem in the General Theory of Relativity" can be answered G.R. will always be incomplete.
Excellent presentations and explanations. Really appreciate it. Thank you.
When are releasing a one on the derivation of the Reissner-Nordstrom metric, and the Kerr metric
Never, unfortunately. I looked into it and gave up because ot was pretty complicated. There are other channels that cover it.
This stuff is interesting.
1:10-This timeline sucks
If I wish to get A replaced in B and vice versa, in the same line element of schwarzschild , what are the turns and modifications I have to do to obtain so?
Can we define the distortion of space time for a non rotating object of a different shape?
Thank you sir!
i cant fathom how fucking tedious it must be to type out all of these connection coefficient indices
Yeah, it's not the most fun. But once I write it out once, I can refer to it any time in the future easily from this video.
Excellent video! Very clarifying, thanks you very much
muito bom fera sexta 3h37 da madruga vendo vc derivar a metrica de Schwarschild chapadão por puro prazer é nois um abraço do brasil !!!!
Thank you so much for explaining how the coefficient C becomes 1 :)
Amazing
is there a name for the h metric in the weak gravity limit
Some people call it the "metric perturbation". Sorry for the delay. I guess I missed the notification for your comment.
Daaaaaamn, nice presentation, dude. I gotta watch this a few more times.
It is a fantastic work congratulations you are the best teacher thank you for your hard work
Sir I am requesting u Plzz provide this Ppt
Muhteşem.
How much days later you will uplode 108b video?
Because I'm waiting for extraordinary video.
I'm your very big fan Chris.
Thanks. It might be out by the end of October. Most of the slides are done, but there are some concepts I still don't understand, so I'll need to spend more time learning and fact-checking.
What a great vídeo, I loved it, thank You so much
1:07 🎶It’s better to burn out than to fade away!🎶
24:20 when I find the solution to the equation I get k/r - 1, I dont know if I did something wrong but I inputed the equation into different calculators and got the same thing. If you read this can you please show me your solution for the equation
dA/dr = (-A+1)/r
dA/dr/(-A+1) = 1/r
∫ = dA/dr/(-A+1) dr = ∫ 1/r dr
Evalute the integrals we get :
-log(-A(r) +1) = log(r) + C_1
=> A(r) = -e^(-C_1)/r + 1
A(r) = -k/r +1 = 1 - k/r
I have just gone over my working and I know where I have gone wrong. It is MY brain that is frazzled. Sorry!
That's okay. I appreciate people pointing out errors if they exist. Though I am glad this was not a real error. :)
I like your funny words magic man
This is excellent!! Really useful
Really really really thank for your all video.
Yo is that nietzsche
It's Karl Schwarzschild.
29:20 What if value φ at infinity is not zero but some very small number ε(M)?
The curvature of spacetime only depends on the derivatives of the metric, not the actual metric value. So I don't think small additive numbers like this change the physics.
@@eigenchris we can eliminate this constant by selecting a different scale. So no new physics, like you said.
@@nenhard I should maybe be a bit more careful... the metric itself does appear in the Einstein Field Equations, but as long as we ignore the cosmological constant, and the Ricci scalar is zero, we don't need to worry about the constant.
Equations are nonlinear and contain product of metric and its derivates.
Excellent!!!👍👍👍👏👏👏
Man, you are a stud
Nice videos
sorry to be a pest!
What happened to 106b and 106c? Those were planned, weren't they?
I plan to make them eventually, but I've covered that content in my previous "tensors for beginners" videos so I've been less motivated to make them. Do you have any particular questions you want answered?
@@eigenchris Maybe not, but I just remember looking forward to the next one after having seen 106a. But I can endure for a bit longer! :-)
we are saying g_teta,phi = 0 so that g be similar to spherical coordinate metric. do you only try a g with g_teta,phi = 0 to see if it works or is there more mathematical background when we say g_teta,phi = 0.
The "mathematical reason" is that, if we take a constant time, and constant radius, we want the resulting shell surface to have the same geometry as an ordinary 2D spherical surface, by spherical symmetry. I don't think I have a better reason.
@@eigenchris you are the best
Hello Chris. I don't understand at 25:33 , fifth line equation, you have a contravariant left side and a covariant right side. How is that possible?
I was a bit "loose" with the indices here. To be technically correct, I would need a kronecker delta on one of the terms to raise/lower the index appropriately, so that everything matches. The kronecker delta doesn't actually change any of the values in the formula, it just puts the indices in the right place, so I left it out for simplicity. Sorry if that's confusing.
Thank you for your answer Chris. Clear as usual. I'm getting great marks in GR thanks to you.
Best GR course ever
I have Turkish exam tomorrow yet I am still watching it... Someone help me
Excellent.
is there an algebra mistake at 10:41 ? I get : - ( drA)^2 / 2A^2
Is 10:41 the correct timestamp? I'm having a hard time seeing what you mean.
Great video as always. I cannot understand why if trade t por -t means that the body is non rotating. And why multiply some terms by C.
I didn't get into the details much... it might become more clear in later videos. A rotating black hole causes additional gravitational effects in the direction of rotation (you can google "ergosphere" to learn more). If we reverse the direction of time, these gravitational effects reverse. So if we force spacetime to be the same in both time directions, it guarantees no rotation.
Also, we multiply time by c to give it units of distance, to put it on the same level as x,y and z.