Relativity 107b: General Relativity Basics - Manifolds, Covariant Derivative, Geodesics
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- čas přidán 1. 06. 2024
- Full relativity playlist: • Relativity by eigenchris
Powerpoint slide files: github.com/eigenchris/MathNot...
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You are free to continue watching to the next video, but if you feel you are getting confused, here are some other videos on geodesics and the covariant derivative:
TC 15 - Geodesics + Christoffel Symbols on Sphere: • Tensor Calculus 15: Ge...
TC 16 - Geodesics on Sphere example: • Tensor Calculus 16: Ge...
TC 17 - Flat Space Definition: • Tensor Calculus 17: Th...
TC 17.5 - Component Definition: • Tensor Calculus 17.5: ...
TC 18 - Extrinsic Definition: • Tensor Calculus 18: Co...
TC 19 - Intrinsic Definition: • Tensor Calculus 19: Co...
TC 20 - Abstract Definition: • Tensor Calculus 20: Th...
0:00 Introduction
1:35 Equivalence Principle and Manifolds
6:15 Extrinsic vs Intrinsic views of Manifolds
10:29 Tangent Vectors on Manifolds
16:20 Covariant Derivative Notation
22:10 Levi Civita Connection
28:58 Geodesics
33:44 Summary
Now I'm doing my final General Relativity exam. This video is the signal that everything will be okay.
I too gave my final GR exam 2 weeks ago. This channel really helped me with basics.
haha gonna suck when General relativity is proven wrong by 2050
The best notification of my day
Agreed.
Double agreed
I agree too
Yes!
+1
I can admit when I'm not gonna understand something, and this is one of those times. Be back in 3 years after I learn differential geometry
You're probably one of the people who should check out the links in the description. This video is basically 3-4 hours of content squashed into 36 minutes. The videos I've linked cover it at a much slower pace.
You don´t need 3 years, learn linear algebra first, then check out Tensor Calculus from Eigenchris and be back in 3 months
@@eigenchris thanks so much
Check Tensor for beginners and Tensor Calculus, both from Eigenchris.
Chris explains everything you need to know awesomely, and you'll get ready also for Riemann and Ricci.
I am only an intermediate in maths and have never studied any maths in any university, yet I understand your great explanations. I have learned some notations including shorthands.
1:09 Einstein Summation Notation
1:38 Equivalence Principle
3:31 Manifolds
6:17 Extrinsic geometry vs intrinsic geometry
9:27 General Relativity (1915)
15:05 Tangent spaces - extrinsic view
15:49 Derivative operators = basic vectors
16:36 Covariant derivative in flat space
17:19 Christoffel symbols
22:11 Comparing vectors in flat space
22:26 Comparing vectors in curved space
25:45 Fact #1 - Metric compatibilty
26:42 Fact #2 - Torsion-free
27:03 Metric compatibility + torsion-free
If you know multivariable/vector calculus I can recommend the tensor calculus series (and the tensors for beginners series before it). It goes in more depth explaining these and more.
neopalm2050 I know just some scraps of multivarible/vector calculus and same with tensor calculus. I have been watching videos on those and looking at which books to buy. I am roughly 16 months learning maths from mostly CZcams and forums, but until recently, I was lacking books to study from.
Now I am ordering books. Just over the last few months, I've bought some books on analysis, linear algebra, calculus, classical mechanics, quantum mechanics, engineering mathematics, physics, discrete mathematics, abstract algebra, etc. I have roughly 15 books. I have yet to order books on number theory, ordinary differential equations, partial differential equations, etc.
This really explained some subtle points that I hadn't fully grasped before, thanks.
Glad to hear it. I'm curious: which parts did it help with?
@@eigenchris In particular the difference between intrinsic and extrinsic geometry, where we use derivatives as unit vectors. I have been doing discrete calculus work on graphs using the theory of differential form, and I think I understand better where some of what I have been using comes from.
Always a pleasure. I usually have to watch them quite a few times so by this point I have viewed like hundreds of hours of Eigenchris.
Me too to get all the various subtleties of the mathematics and to go through them in your head takes me quite a long time for me.
It's either reinforce this beautiful theory or watch some movie I've seen a hundred times.
I choose this
Damn, im really impressed how far you are talking this series. Keep up the good work
This is so good. I just have enough mathematical training to be able to understand these concepts, though I am not that far in my studies at university, it is very interesting to get an overview of the topic in a way that is accessible. The explanations of the notational gotchas are particularly helpful
Wow. You've done it again. Brilliant deeply useful summary. Right on the nail.
Best series of lectures. Much much better than any University lectures
In the intrinsic case , those three "like" vectors are indeed on a complex situation.
As you mentioned an 4-D manifold has its limiting appearance.
I really like this series. It made my day.
Covariant Derivative finally makes sense to me. Thank you very much.
One of the main channels that helped my mathematical intuition
nothing to say.just woww..one of the best video series in utube for understanding GTR.
Good work Chris! Waiting for the next ones!
The unique channel covering all graduate topics
I'm coming back to your videos after a long time, your voice has really changed!
SUPER MATERIAL!!! Thank you! I Understand Everything!
I was watching your without Tesnor Algebra and Tensor Calculus but i completed all the video expect Ricci Tensor i found those video are really useful here most of the concept make scence
Thanks chris ....was very eagerly waiting for your GTR series...
I love you.
love U too 😂😂
He’s mine.
thank you NSA, for this feed.
Any manifold can be embedded in a high dimensional space (like 2*N or 2N+1 D). So, extrinsic or intrinsic has little difference, at least in mathematics.
As someone who is in 7th grade. I don't know why i am here.
@eigenchris
I liked your explanation of the covariant derivative notation.
I'm currently working my way through "A First Course in General Relativity" by Bernard Shutz and he doesn't really explain it
as well as you did. So thanks.
I'm glad that helped. For me, the notation was one of the worst parts of learning the covariant derivative.
this is freaking insanly AWESOME. Stringtheory here I come 😃
I love your videos man keep it up
thank you for this work.
this video has a really really short abstract about some topics of riemman geometry but, is a good one
Thanks a lot for these videos Chris ❤️. Do you know if it is possible to find some GR exercises online with solutions to them?
Very good, I could only wish I had videos like this for every other course in physics!
Amazingly good.
It's taken me many CZcams videos to understand why gen relativity never made any sense to me. But I finally realized I was trying to visualize a 4 dimensional phenomenon.
At 5:13, is the reference to the pseudo-metric we use to get squared length the Minkowski metric with the mostly minuses convention?
Yes. Or the mostly-plus convention. The point is that the squared length can be +, 0 or -.
I have a brief question, as I was covering the basics of differential forms I remembered about your tensor calculus videos on covariant derivative, and found the fact that there can be no constant vector fields on Manifolds. I did some more searching and found that you cannot integrate tensor fields in curved space-time (which also matched up with your definition of covariant derivative) but how ever in the generalized stokes theorem, we can see a differential form being integrated on a surface, so is theorem only for only flat surfaces? I would be glad to hear the answer.
I'm pretty sure you can integrate vector fields and tensor fields on a manifold. The issue is that the result of the integral can possibly depend on the path you take. I'm not familiar with all the details of the generalized stoke's theorem, but my understanding is that it gives certain conditions for when you can calculate an integral of a tensor field just by knowing the values of the tensor field on the boundary of some manifold, and therefore you can "forget" about the tensor fields inside the manifold. For example, in conservative vector fields, the result of an integral only depends on the endpoints of the path, not on the exact path you take. But not all vector fields are conservative, so this fact doesn't apply to all vector fields. I'm not 100% sure, but I think in the language of generalized stoke's theorem, you require differential forms you integrate over to be "exact", which is the equivalent of a conservative vector field. But not all differential forms are exact, so the theorem doesn't always apply.
@@eigenchris oh yes that actually makes sense sense as vectors are more likely path dependent but forms aren't. Thanks 👍:)
@@pythoncure6755 I think forms can be path-dependent. I'm a bit foggy on this, but I think "exact" forms are the ones which are not path-dependent. A form f is "exact" if you can write it as the exterior derivative of another form: f = dF. As an example, the form -ydx + xdy is not exact. It's basically like a counter-clockwise "whirlpirl" and if you draw a loop around the origin and integrate, you'll get a different answer than if you follow 2 loops around the origin (or 3 or 4, etc.). This is like a vector field that can't be written as the gradient of a function.
@@eigenchris I suffered the internet for a bit and yes, stokes theorem depends on exact forms and also the fact that exact forms are not path-dependent. Thanks Eigen Chris you cleared my doubt I really appreciate it. ☺️
Thank you for the video.
Glad you liked it!
@eigenchris if a black hole is a perforation in spacetime shouldn’t the manifold we model on have a perforation also?
I'm not sure how to answer that, as I haven't studied it. Sorry!
There are some inconsistencies with this. You can develop an understanding of this by reviewing Gauss's revelations on space:
Physics has to disregard the delusions of mathematicians and find a Physics Geometry that is consistent with the scientific method. The mathematics must be defined as Gauss attempted but found no one capable.
Here it is Gauss to Bessel Goettingen 9 April 1830 …
"The ease with which you delved into my views on geometry gives me real joy, given that so few have an open mind for such.
My innermost conviction is that the study of space is a priori completely different than the study of magnitudes; our knowledge of the former (space) is missing that complete conviction of necessity (thus of absolute truth)
that is characteristic of the latter;
we must in humility admit that if number is merely a product of our mind."
This can be resolved by creating the geometry by rotating an observer to establish the axis for north and south pole references. The perceived measurement is observed by humans using one of their perceiving senses which provide information to their brain. In sight it is the eye that yields the reality, in sound, it is the oscillation of the observers’ nerves in the ear or other parts of their body.
The Doppler effect establishes the change measured in time.
The Doppler effect is the derivative (the change of) the E-field in respect to time, not space.
Space is simply a mathematical delusion:?)
Space does not exist in physics, it exists as delusion in the mind of mathematicians who do not require observation.
The inconsistency in mathematics exists in the first assumption of Euclid. It is responsible for the generation of irrationality:?) Gauss may well have understood the issue. It is evident in Bessel's response to Gauss that Bessel did not understand what Gauss was talking about:?)
I hope this information allows you to sort out how the delusion of space came into being. I will answer any questions you have. I can be reached privately if you prefer at 713 922 3227. I am preparing a redefinition of geometry that clarifies the mistaken assumption proclaimed by Euclid and perpetuated by Newton as well as others including Einstien who followed Euclid's mistaken first assumption.
THANK YOU
I watched your suggested videos of tensor calculus series and found out that when a vector is parallel transported then the covariant derivative of that vector along that path will always be zero . But at 35:03 you said covariant derivative measures how much a vector deviates away from parallel transport . But I thought that vector field will not deviate when parallel transported . Is there something I missed ? I am confused on this one .
I maybe didn't explain it great... When we parallel transport a single vector, the covariant derivative is zero. But when we have a vector field, with a vector defined everywhere on the manifold, that vector field might be changing in different ways that are different from parallel transport. The covariant derivative determines how much a vector field deviates away from parallel transport.
@@eigenchris I got it , I was just confusing vector with vector field and btw you mentioned that it is vector field which deviates in tensor calculus video .
Hello @eigenchris, thanks for this great series of video on Relativity!
I'd like to make one correction though: on this video at 3:04 you said that "in small/local regions of spacetime, GR disappears" but this isn't true. The tidal forces still exist whatever you're looking in a small region of spacetime or not. If you imagine yourself in a floating box on the sea with an altimeter in your hand and no way to look outside, then, assuming that the box stands perfectly still on the water's surface, you will notice a change in the value read on the altimeter because of the tidal forces caused by GR, even in your small/local region.
In the movie Interstellar, they play with the fact that tidal forces still exist when the crew has landed on a planet with a lot of water and near a supermassive blackhole.
I think both of the examples you give rely on the tidal forces acting on a relatively huge body, over relatively long periods of time. The point I was trying to make is that spacetime, being a manifold, is locally flat. (That is, any given point on the manifold can be approximated by a flat tangent plane in the nearby spacetime.) Basically, I'm was trying to say it's difficult to build a machine that will quickly tell you how much spacetime is curved/not flat that operates only at a single point. Unlike, say, a machine that can measure electric or magnetic fields at a point, which we can build.
@@eigenchris just a follow up question, if spacetime is locally flat, but they also say it is cosmologically flat too. is there any actual geometry to that statement, is it still intrinsic?
cheers
@@kunx5387 When people talk about "the universe being flat", they're talking about the large-scale structure of SPACE, not SPACETIME. A spatially flat universe that expands overtime (as ours does) is not flat in terms of SPACETIME... its Riemann Curvature Tensor for spacetime will be non-zero. I also say "large scale structure" because individual stars and black holes can bend space in ways that are relatively "local" (it's pretty much not noticeable if you go far enough out). I'm talking about this my next video (110b), which is in-progress.
It seems that torsion free is related to the generalized stokes of tensors on manifold which in turn relates to boundary conditions for Einstein's equation. If one would like to match solutions for one part of space to others such as Rindler frames or matching the inside to outside of a black, one requires how normal and tangential components of the metric tensor cross the boundaries. What is the status of boundary conditions for GR?
@ 5:10 I have stopped the video feed;
It appears that he May have filled a missing detail in my speculation;
I am working on the Neutral Kaon as a probe into generalized Spacetime features which distinguish fermionic from baryon with the idea that in a gravity localized reference frame time-by his squiggles here-become "time-like" ( Squiggles is the cat that escaped the Quantum Paradox ) when "Condensed" *[CDM]* by the "compressive" "force" of an as-yet unseen │ "undiscovered" … "thing" which does not exist: A *◘ NEUTRAL◘* such that the 3-Quark model "tensions" an "unbound" with slight "discrepancy" which is documented by persons with vastly greater training than me with the g-2 Muon thing:
I Propose the Russians are trying to corner the Helium Market or-alternately-that the RH spiral chirality is only local to a "Geodesic" which occurs in regions where this compressive results in tangible matter-hence would vary ± over vast regions larger than Galactic Scales ○ From this we can devolve novel notions like backward time which would be an anti-particle & other ideas which should be kept to our private thoughts · Since the instructor notes to skip ahead to ,,, since I do not care about notation; We see the changes in the directional vector of Squiggly the cat are noted as delta angle of a scattered herd of cats during a Compton scattering interaction with this _unseen_
*○ Note for others ○*
_Herding cats is one of the most difficult problems that many will ever encounter_
Since Λ is shown ▬ We take the idea that "application of right-hand rule" in electronics is very relevant to the tensor dynamics which causes the appearance of tangible matter at a rate of 5% · Since we are not "torsion free" ( obviously ) then something is causing the ± Bias orthogonal to the plane of the ecliptic which also shows on a larger scale ◘ Note I am not aware of what stokes of tensors or Rindler frames are - I have no formal training-I am a You Tube comment poster; My qualifications end there; I work in trance and note that all of this being published in 1915 long before the mechanics to show it exists propends discovering exactly whom or what is working for Schrödinger's Cat Coalition at which time we can publish: Physicists Have Finally Figured Out a Way to Save Squiggles from Schrödinger's Cat → Remember it is Publish or Perish!!!!!
Great video
Thank you for the video! Does d/dy acts like a derivative operator? Or is just another notation for basis vector?
It's both. You can apply the basis vector to a function to take its derivative.
Great presentation as usual Chris, but how do you ‘apply a basis vector’ and which ‘function’ should it be applied to?
Thank you very much.
It's 1 AM and I have a Calc II exam after tomorrow, what am I doing here :') Chris, you're like chocolate, you're bad for me but I still go on a rampage to watch your videos
chris is like vegetables for me, since I am vegan, he is all I get, and since I am weird, I really like it.
Good luck.
@@pinklady7184 thank you pink lady
@@korwi7373 niceee!!
@Julez O'Neil actually that would normally be the case. However, my university has decided to do it in dispair space, a less known topological space whose kernel maps my inability to deal with stokes theorem to my grades
At 22:00 @eigenchris explains that the other notations measure vector components... but how is that different from a vector? Vector components are one expression of that vector, right?
When a vector is decomposed into components, there are two "parts" you need to keep in mind: the basis vectors and the components measured along those basis vectors. The same vector can have different components in different bases. This is a very important concept in relativity because you are constantly looking at the same physical situations from different frames of reference (different bases) and so you'll measure different numbers (components) in each frame. I tried to emphasize this at the start in the 102 videos.
Thank you for the great videos! What book do you use as a reference?
It's not any one particular book. It's a mix of books, PDFs, wikipedia, video, and anything I can find online. I've used Sean Carroll's free online GR notes, the free online textbook "Exploring Black Holes" by E.F. Taylor, and the "Gravitation" textbook by Misner, Thorne, and Wheeler, but the latter is very heavy and not easy reading.
@@eigenchris thank you. Keep up the amazing work!
is it just me or are the lower indices on the christoffel symbols on lines 3 and 4 at 26:38 in the wrong order?
All your videos are really helpful and I like then a lot .This video was very clear but I have 2 questions.1 The mwtric compatibiluties physical meaning is the curved generilization of Newtons first law?
2.The Tortion free connection is chosen beacause due to the carvature of space time we do not use vectors anymore and we use derivative operators?
Thank you very much for your time ,continue the good work!
I've never heard of "metric compatibility" being compared to Newton's 1st Law. I think the GR equivalent of the 1st Law would be to saw that objects with no forces on them travel along geodesics, given by the geodesic equation. I don't think there are good physical reasons to assume metric compatibility + torsion free connection that we could guess ahead of time. I think they are the "easiest"/most straight-forward way to define a connection in curved spacetime, so they were the properties Einstein used, and in practice they seem to agree with experiment. I think there are some more "fringe" theories of gravity that allow for a connection with torsion, but I don't know much about those.
@@eigenchris I undetstand what you say and to be honest is one of the answers I usually take for this question.But I read somewhere that because in physical space tangent vectors to geodesics are velocity vectors amd metric compatibility preserves the angles and the length is like saying when we parallel transport a vector the velocity vector only twicks because of carvature and it does not change length ,so the only accelaration parallel transport gives is the twick of the vector due to carvature.Again I am not sure about that statement but I found it interesting ,agaian thank you for your time
@@eigenchris Indeed there is one - it's called Einstein-Cartan theory and was developed by Elle Cartan.
THANKS !
@eigenchris is the any good texts that go in to the nitty gritty math of derivatives as basis vectors?
Off the top of my head, I don't know. Is there a particular question you have about the concept?
@@eigenchris I dunno, I just find the correspondence between linear operators and derivatives to be fascinating on its own right.
@@eigenchris even if i feel like its alright, i would like to see a formal proof of why d/dx is equal to the versor e_x, and so why d/d lambda is equal to the tangent vector to the curve. really, al the magic in differential geometry follows from this "trick, and i would like to really grasp it. Can you help me?
Donuts is a better example of a manifold than a saddle, because donuts are tasty AND the beginning of very special kinds of manifolds in topology. :P
3:14, if GR is locally indistinguishable from SR, why is there a separate need to formulate quantum gravity? The Dirac Equation already accounts for special relativity. Or is it not "local" so much as "local and low density"?
By "local", I mean "the tangent space at a point in spacetime". The more mathematical way of saying "GR is locally indistinguishable from SR" is to say "spacetime is a manifold", where manifolds have flat tangent spaces that are tangent to each point. On those flat tangent spaces, the rules of special relativity apply, because SR applies to flat spacetime. You brought up the Dirac Equation... QM might have a different definition of "local" that the one I'm referring to. I'm not considering QM here.
26:29 just a question, how do you get those Christoffel symbols from those covariant derivatives? Like, what’s the derivation?
That's just the definition of the Christoffel symbols. The covariant derivative of one basis vector in the direction of another basis vector will be some new vector, which is written as a linear combination of basis vectors. The coefficients in this linear combination are the Christoffel symbols. You might want to look at the 105e video if you haven't seen it yet. It covers the covariant derivative in flat spacetime.
@@eigenchris I understand that (or think I do), but as you said, the covariant derivative gives us a new vector which is shows as a *combination* of basis vectors. However, at 26:29, the covariant derivative of e_μ (in the direction of e_σ) is a product of a Christoffel symbol and a new basis vector; not a combination. I’ve looked it up on the internet and found the actual identity, but can’t find a derivation. Is the derivation just plugging in a basis vector into the normal covariant derivative formula, or is it some other thing?
Tl;dr: I’m probably missing the point and being absolutely stupid
@@ieatbananaswiththepeel4782 Are you familiar with the Einstein summation notation? There are summations going on over the alpha index in the line at 26:29, but the summation symbols are not explicitly written to save space (this is the Einstein notation). So there is actually a linear combination of basis vectors with Christoffel symbols coefficients.
@@eigenchris OHHHHHHHHHH thanks so much. I knew I was forgetting something. I’m really new to Einstein notation so I always make dumb mistakes like this. Thanks for replying! You’re the best :)
Covariant derivative of matric tensor is zero in any direction, so how can we make christoffel symbol (at time 26:27) by covariant derivative of matric?
That is not the covariant derivative it is the ordinary derivative
The covariant derivative of the metric tensor is zero in any direction if the connection is metric-compatible. But at 26:27, this is not the covariant derivative of the metric tensor, this is the covariant derivative of the metric tensor *components*. The covariant derivative of the metric tensor involves derivatives of the basis as well (which are covector-covector tensor products). This creates additional terms, which end up cancelling out with the terms seen in this video, giving zero. You might want to watch my Tensor Calculus video 20 to get all the details on this (linked in description).
Hari vayapat sarvartra samana means we have equal potential at all places but consciousness make it unequal according to concentrate press
I love this
Sir is it possible to reach at the velocity grater than 3*10^8 m/s by constant acceleration?
It's not. You can look at my "105a" video to learn about constant acceleration in special relativity. In Newtonian physics, constant acceleration leads to a parabolic path in spacetime, with unlimited speed. But in special relativity, constant acceleration leads to a hyperbolic path in special relativity, with an asymptote at the speed of light, giving a limit on the maximum speed.
@@eigenchris Thanks a lot sir.
Your video "105a" clear my doubts.
thanks
this is gold
I'm still somewhat stuck on the basics. Is there a necessary relationship between TpM and the Rn of the local homeomorphism?
If I understand correctly TpM does not have any structure beyond being a vector space. So it has no topology or notion of distance(yet). But this will be where the tensors live once we define them.
I think I'm a bit confused how spacetime can be both, locally euclidian _and_ locally minkowski. Both are statements about the metric of a local space and seem contradictory.
Or am I misunderstanding "euclidian"? Does it not mean Rn with euclidian metric?
But these spaces are not the same or at least used for different purposes? Do we ever need to refer to the euclidian space again after using it to define the manifold?
But can't we just start with locally minkowski? Or do we get in trouble defining the open sets? It looks like the lightcones would be convenient.
But I supposes then we don't have a manifold anymore, by definition.
So I'm going to lay out some of the definitions from topology.
When people say a manifold is "locally Euclidean", they aren't talking about the tangent space TpM. They're talking about something in topology called an "open set" being homeomorphic to R^n. An open set is a generalization of an open interval... loosely speaking, it's a set of points without boundary points (that's not a formal definition, just a metaphor). And R^n in this case NOT being treated as a vector space with a Euclidean metric... R^n is just a set of n-tuples of real numbers. So saying "open sets are homeomorphic to R^n" just means that you can take a "patch" or "chunk" of the manifold and match the points up with R^n (so these local patches will not have "holes" inside, since R^n has no holes).
Now, moving on to the idea of a metric... not every manifold has a metric. It's something "extra" you add onto the manifold. As a counter example, symplectic manifolds from the phase space of Hamiltonian mechanics don't have a metric on them (these spaces have "position" and "momentum" directions, and measuring "distance" between position and momentum isn't very sensible). A manifold with a metric g (positive-definite, never outputs negative values) is called a "Riemannian Manifold". A manifold with a PSEUDOmetric g (can output negative numbers) is called a PSEUDO-Riemannian Manifold". A metric takes 2 vectors from a tangent space TpM and outputs a number. In relativity, we use a specific kind of pseudometric called a Lorentzian metric, which is defined as a metric with signature (+ - - ... -) or (- - ... - +). I kind of use "Minkowski" and "Lorentzian" interchangeably in my videos, which might be a mistake.
I'm not sure if that clears it up. If you want relativity videos with a more formal build-up of the topology concepts, you can check out "The Winter School" lecture series on CZcams by Frederick Schuller. I personally found all this topology talk a noisy distraction when I was first learning relativity, but obviously not everyone feels that way, or they might want to learn it after getting the basics.
@@eigenchris Thank you for always putting so much effort into your replies!
I certainly see why you'd want to gloss over the topology stuff, but my dad kept asking me things like "but how do we even know what "closeby" means on the manifold?" and I had no answers...
I'll check out those lectures.
My issue is that the local homeomorphism maps open sets from the manifold to open sets in Rn. These open sets in Rn are generally defined to be the epsilon balls, which require a distance function to define the radius. So we take the euclidian distance function r²=x²+y². If I understand correctly that is what euclidian _means_ . So there is a certain arithmetic defined on Rn and it has a distance function. Is that not a metrizised vector space?
And it looks like we must use at least some of this once we define the differential structure to get a differentiable manifold, which we need to get to the tangent spaces.
Now these _are_ vector spaces and _not_ manifolds(or boring ones) so we can give them a single metric tensor that is constant or "flat". If this has lorentzian signature we call it Minkowski Metric and our manifold is "locally minkowski".
So the statement "locally euclidian" and "locally minkowski" are basically unrelated because they refer to different spaces.
The metric tensor field is then a function that assigns a metric tensor to each TpM. It tells you "given your local metric, this is what the metric looks like over there in relation to that"? So if my local metric tensor is -1,1,1,1(which I can always transform into if I want) I can say these numbers change by this much based on the function at different points and I can use that to do the integral to calculate distances along curves.
How do I define "the distance" from a pseudo metric? Do I say it's the maximum if it's timelike and the minimum if it's spacelike?
I'm not sure if Topology on its own can answer the question about what "closeby" means. Homeomorphisms let us treat topological spaces like rubber so we can pull points apart or squash them together as we like. You're right that open balls on R^n are defined using a euclidean distance function of sorts, but it seems to me that distance function is pretty divorced from the metric used to measure distances on a manifold. You could take a chunk of the unit circle (say the open interval between 0 and 90 degrees) and map it to the entire positive half of the real line using x = tan(θ), which sends the points all the way out to infinity. I think the purpose of open balls isn't to give you a notion of distance on the manifold; it's more to just tell you how the various patches of the manifold are linked together. The true 1x1 metric on the unit circle would be [1] in θ coordinates and [1/(1+x^2)^2] in x coordinates on the real line (you get this from squaring the derivative of arctan, standard 2-covariant metric behaviour). You can see that this metric has basically nothing to common with the euclidean metric on the real line. The x-coordinate is worth "less and less distance" as x goes to infinity, because the tan(θ) function spreads the points on the circle near 90-degrees very far out apart. Similarly, the Lorentzian metric has basically nothing to do with the metric used to define open balls in R^n. And yes, R^n is technically a vector space, but I don't think that fact is particularly useful, other than just telling us what the dimension of the manifold is and telling us how many numbers we need to identify a point. I'm not sure I understand your last question about defining the distance from a pseudo-metric. I would say "just integral the lengths of the tangent vectors along a curve", but I'm guessing that's not what you meant.
@@eigenchris So the euclidian metric on Rn is just used as a tool to create the structures. Locally euclidian and locally minkowski just work on separate levels.
I'm still interested if we can use the minkowski metric to define open sets, but I won't pester you with that. I found a paper I'm going to read on the induced topology of the light cones.
My last question was about how we chose one of the infinitely many curves that connect two events to represent the spacetime interval between these events. It can't just be the shortest or longest, since the curves can go infinite in either direction in a pseudo metric(or s can even be complex). And the geodesic equation isn't necessarily unique.
@@narfwhals7843 Creating open sets with the minkowski metric would be a bit weird since, unlike "open balls" from the euclidean metric (which are basically circles/spheres), open sets defined using the Minkowski metric would involve hyperbolas that go off to infinity. Maybe you can get away with doing that somehow, but to me that just seems like a headache.
There isn't a unique way to draw a unique curve between two points. A spacetime interval is something we associate with a curve, not something we associate with 2 endpoints. In special relativity with cartesian coordinates we sometimes think of the spacetime interval as being between two points because we can easily draw a straight line between them and use that as our curve. But generally speaking we can draw any curve we like between 2 points (although timelike curves must always have a timelike tangent vector, and spacelike curves must always have a spacelike tangent vector). "Timelike geodesics" between two points will locally maximize proper time between the points, and "Spacelike geodesics" will locally minimize proper distance between two points, so you can use geodesics to pick out a curve between two points, but geodesics are not the only option.
Great!
@ 15:27 So by what is the S-vector replaced in curved spacetime?
It doesn't really get a replacement. We can still draw parameterized curves on manifolds (worldlines would be parameterized curves), but vectors can only live on tangent spaces at a point.
@@eigenchris So the path through the point determines the direction of the vector? Thanks for your reply!
When you refer to partial derivatives as “operators,” do you mean they are observed values, as is the implication in physics?
By "operator", I mean that they take a function and output a new function. I know in quantum mechanics, observable quantities are associated with Hermitian operators, but that's not what I'm talking about here.
I also prefer to call the classic metric the “positive definite norm” with actual “generalized metric” dropping the requirement that the norm squared has to be positive.
I thought the Lie derivative also helps "connect" tangent spaces, how is it conceptually different from the connection here?
The connection is an "additional structure" on the manifold, and comes in the form of the connection coefficients. There are many different connections we can make, because there are many different choices for the connection coefficients. In GR we use the Levi-Civita connection, which is sort of the "standard" connection on a Riemannian manifold. I think the Lie derivative can be defined on any two vector fields without any additional structure (that is, there's no connection coefficients we need to define).
Warning: I had not heard of the lie derivative until I saw this comment so keep that in mind while reading.
The lie derivative, from what I've seen, does not give a torsion free connection. L_x (y) = - L_y (x) meaning when you swap the lower two indices in the connection coefficients you actually have to put in a minus sign. Also, it doesn't even consider a metric so when you add one there's a possibility that "parallel transport" will change lengths and angles. "Parallel transport" along a path seems to depend not only on the path taken by the vector, but also the velocity vector field outside the path. Because of this, I believe that the lie derivative is not fit to handle the concept of geodesics.
19:51 how can we ignore summation with lambda derivatives and keep the equation unchanged? 😀
If you like, you can take lambda to be one of the x^u coordinates, and all the other terms in the summation will go to zero.
A Primer is always a good idea, but this one would be around the 3rd in series, after AM-FM time-timing Communication In-form-ation, and putting Quantum-fields resonant Circuitry in Perspective Principle maybe, because the development of equivalents and structures that follow are all derived from Reciproction-recirculation Singularity positioning integration, (other words for Circuitry), the superimposed features of QM-TIME pure-math relative-timing ratio-rates empirical laws of logarithmic coordination in eternal-constant, vertically integrated metastability. Ie we may declare the presentation "not even wrong", because it is 100% correct in a more appropriate fractal conic-cyclonic scaling reference-framing context, an orientation excluded from the description of the pure-math e-Pi-i resonances mechanism.
Definitely a case of "thinking for yourself", everyone is unique and starting from the floating point attention to local experience. This is why Primer N⁰1 is the recognition of Quiescence, the Eternity-now Interval Conception here-now-forever where everyone feels lost and alone without an example to emulate. There's no chance of a "cure" for this in the universe of probabilistic correlations, of Observable Actuality and uncertain time-timing. (Measurement Problem)
Too many drugs
I understand metric compatibility since without it 4-velocities would change their lengths unprompted, but what strange consequences lead to wanting the connection of spacetime to be torsion free?
I admit I don't fully understand why we choose the "torsion-free" property to be true. I think if your connection is not torsion-free, then basis vectors will "spin around" unnecessarily as you parallel transport them around. It might be a case of "we chose it because it's simple, and it ends up matching experiment", but I really don't know. Maybe google can help you out more?
Wait no, I've figured it out. Torsion corresponds to: if one follows the flow of "increase coordinate x" then follows the flow of "increase coordinate y" then that would end up in a different place than if one follows the flow of "increase coordinate y" then follows the flow of "increase coordinate x" even when the amounts of increasing each coordinate is the same. That'd have to mean that following the flow of increasing a coordinate does something other than just increase only the coordinate in question at a constant rate.
You may be on the right track, but this isn't quite torsion. The operation you're describing of "following one flow, and then the other, and then doing it in reverse and comparing the results" is called the "Lie Bracket". My Tensor Calculus 21 video covers the Lie Bracket (and also torsion) so it might be of interest to you. Basically, any sensible coordinate system will have a zero Lie Bracket, because the coordinate curves need to form closed boxes. The Lie Bracket isn't actually related to the connection, because you can calculate it without using the connection coefficients. Torsion, on the other hand, is directly related to the connection because when something is torsion-free, it means we can swap the lower indices of the connection coefficients. It turns out this is the same thing as saying "the torsion tensor is zero" (I also cover the torsion tensor in Tensor Calculus 21). All that being said, I don't know *why* our universe has a torsion-free connection. My best explanation right now is that "it's a simple property we tried to use, and it turns out to match experiment". But that's all I've got right now.
@@eigenchris Hold on. Isn't it called "curvature free" when you impose that parallel transport can't spin vectors?
I guess I have to backtrack my statement because it's true that when you parallel transport a vector around a loop in curved space, it can end up facing a different direction. But in ordinary flat space (say, 3D euclidean space), if we use a connection that isn't torsion-free and start moving a set of 3 basis vectors around using the connection, they can start "spinning" in an unnecessary way. Now, what does "unnecessary spinning" mean in curved space? I guess I don't really know. I was trying to use an analogy from flat space.
Is the derivative of a tensor again a tensor? The answer is no, and we can see this in the simple example of
a contravariant vector. One consequence is that the Christoffel symbols of the first kind
are not tensors, because they are sums of derivatives of the metric tensor.
But, surprise! Actually, geodesics are covariantly defined even that the nontensorial
Christoffel symbols appear in the geodesic differential equations. This is a theorem from differential geometry.
So, this equation for geodesics stays the same in every reference system.
We can think of the expression for geodesics as ordinary second derivatives modified by
"correction terms" that restore correction terms tensoriality.
After that , we define covariant derivative for a covariant vector, and covariant derivative of a tensor follows.
Parallel transport is related to covariant derivative along a curve, which is different from ordinary covariant derivative.
The effect of parallel transport is to add correction terms involving the Christoffel symbols. If we replace the vector field
by a tensor field of any type, we simply need to add the appropriate correction term for each index.
Note that the covariant derivative along a curve leaves the type of a tensor unchanged, while the ordinary covariant derivative increases the covariant index by 1.
Thus we obtain a generally covariant definition of parallelism for vectors and tensors (in spacetime, for example).
Hey Chris, I have some feedback that might be useful. I’ve been watching your SR and GR playlist and it seems that you’ve been “reteaching” topics from your tensor calculus series. For example, in this video you went over the covariant derivative but you’ve already made a video that goes in depth about it. I think you shouldn’t explain topics from tensor calculus in the GR series and just start talking about them from the get go. It will save more time on your side and it will also make these videos better in my opinion. Thanks!
I understand where you're coming from. I think people who are familiar with my tensor calc series don't need to see this. However, I want the relativity videos to be as self-contained as possible. I'm going to do one more 20-30 minute video reviewing the curvature tensors and then I will move on to new stuff that involves actual physics.
@@eigenchris I like your approach to teaching--you review some stuff you talked about in tensor calculus and it obviates the need to go back and review the old videos. This is something that many professors do and I find it very helpful. I suppose if I were Einstein (not hardly), I would have completely learned and remembered everything you have said in your previous videos. It really helps me when you review some of the topics you have covered in the past because it makes learning the new topics easier and quicker...and you typically update the previous material in a slightly way which enhances your previous explanation.
Hard disagree. A video like this is actually extremely useful as an overview of the conceptual structure of GR. Both novices and experienced benefit from such an overview. In this area good summaries are hard to find, making this video especially valuable. Not surprised you used Sean Carroll's notes, they're fantastic.
Is equivalence principle enough to prove that spacetime is curved? I mean you only used the analogy that curved surfaces are locally flat..
I don't know how I missed this 🎯
At 27:36, how are we going to substitute the alpha into the equation with numbers?
Sorry, I don't understand the question. Alpha appears as an upper index and as a lower index, so it's a summation index. It involves indices 0,1,2,3 for spacetime. Does that answer your question?
@@eigenchris Do alpha substitute it with four digits 0 1 2 3, but we add
@@redbel2624 Yes. Just like I did with mu at 1:25, but in reverse.
I get that when we are thinking of a general manifold we can choose any connection we want, but how do we know that the levi civita connection works in space time?
I don't think I have a good answer for that, unfortunately. I don't think Einstein was that familiar with Riemannian geometry, and he needed advice from his friend Grossman to understand it. He may just have picked it because it was the simplest/most "sensible" connection to choose, and later we found that it agreed with experiment. You might have to turn to google to answer that question.
Here’s a partial answer: I know the following to be true of Riemannian manifolds, and I believe it’s also true for pseudo riemannian manifolds.
For every connection, there exists a unique torsion free connection with the same geodesics.
(I think) Einstein’s theory only cares about geodesics because paths in spacetime always have unit tangent vector, so the use of Levi-Civita boils down to metric compatibility.
@@eigenchris I mean, Minkowski, who was the one to build the geometric formalism of special relativity, seemed quite versed in it. Perhaps general relativity inherits the connection from that formalism?
@@theor4343 Oh, thanks for the insight! I wonder if there are some modified theories of gravity out there that use different types of connection though
@@lourencoentrudo that’s a central feature of yang mills theory. A Gauge potential is the physicist’s word for connection.
What an amazing summary - very glad to have this as a refresher before moving on. Can I make one annoying point? You’ve said “we can only detect the existence of gravity in large regions of space time”. This may be a little misleading. Is it more true to say “only in large regions of space time can we measure tidal forces and so be certain of being in a true gravitational field, as opposed to just being in an accelerating frame of reference” as we can of course detect the effects of gravity in large region of space time. Can’t wait for the next ones to drop!
At 32:30 isn't the first d /d-lambda MULTIPLYING d-x-sigma/d-lambda rather that acting on it? In which case it would not be a second-order derivative?
It's OK, I figured it out: the partial/partial-x-mu DOES act on d-x-sigma/d-lambda. So it's OK to multiply partial/partial-x-mu by d /d-lambda. 😎
CZcams: Recommends this video
Me: Thank the stars.
You can show to us this metrical magical way to take space and time vectors from the SR theory and Minkowski geometry when in flat space you couldn't have a spacetime fabric but when you paste that in a vector Riemann manifold this is possible? Because De Sitter didn't succeed to do that but please you could try it. Also, please, tall us which is the proper size of space where space and time are curved by the existence of mass, not gravity you are uses gravity, when you are talking gravity doesn't exist, and this because even the mass of the all Universe isn't enough to bend, even a bit of space.
I’m an absolute layman. Just curious - how are “large” and “small” quantified as it pertains to spacetime?
They aren't especially rigorous terms. From a mathematical point of view, if you imagine a ball, and pick a specific point to be the "north pole", you could imagine putting a sheet of cardboard on that point so that it sits on the ball. This is basically a "tangent plane" at that point on the ball. You could think of this tangent plane as being an "approximation" of the ball near that point. Although, the father away you move from the point, the worse the the tangent plane works as an approximation. This is basically the relationship between special and general relativity. Any given point in curved spacetime (General Relativity) can be approximated by a flat spacetime (Special Relativity), but the farther you move away from that point, the worse the approximation gets. So the division between "small" and "large" isn't well defined. It's just that farther you go from the point, the worse Special Relativity will be at giving the right answers,
@@eigenchris ok thanks
Is torsion-free equivalent to saying that the Lie derivative of the basis vectors is always 0?
Not exactly, but the Lie derivative/Lie bracket is closely related. In my "Tensor Calculus 21" video, I talk about the torsion tensor, which is T(W,V) = ∇_W V - ∇_V W - [W,V]. "Torsion-free" means this tensor is zero, which means ∇_W V - ∇_V W = [W,V]. You can check the wikipedia article on the "Torsion Tensor" or my Tensor Calc 21 video for more info.
@@eigenchris hmmm I see, I thought it would be equivalent since the geometric interpretation of the Lie derivative is a measure of how much two vectors fail to form a closed paralelogram
Yeah, it's a bit confusing, because both concepts involve a "gap" being closed. But importantly, "torsion-free" is a property of the connection, which means it is a property of the Christoffel symbols (we can swap their lower indices, as I say). The lie derivative doesn't involve the Christoffel symbols, so torsion-free isn't related to it (at least not on its own).
I'm guessing the Universe is curved, but is so large, our instruments aren't sensitive enough to measure it other than flat -- a manifold, but when we can, we will measure spherical size of Universe?
Please make a video on stress energy momentum tensor. You are amazing.
That will be the 107e video.
@@eigenchris thanks sir
And what about schwarzschild solutions of einstein field equations. Will you make any video on this topic?
And yeah you are doing a very good job. I can't afford book on these topics. So just watch your videos. And I understand all the stuff like I have a book
@@punitsolanki5744 That will be covered in the 108 videos.
@@eigenchris thank you so much again
legend
Can I share all your vedios in the Chinese version of CZcams, Bilibili? They are very good.
Sure. Thanks.
Spacetime in Special Relativity is uniformly curved, but the space component is flat unlike in General Relativity.
Sorry, I don't follow. As far as I know "flat spacetime" is the definition of Special Relativity.
@@eigenchris Space time is curved in the time dimension which is why the metric tensor isn't the identity matrix (as it would be in a 4d Euclidian space.) The curvature is uniform which is different from General Relativity but it is still not zero.
02:21 This, which is talking you that, is your inner ear. Because its mechanism is inertia mechanism and is for feeling increasing or decreasing acceleration, and not the stability. If you could feel the Earth's motions, can you imagine how you could live a such a life?
Why curved space time is torsion free?
I'm not sure. I think Einstein just went with the simplest possibility--which is assuming spacetime is torsion-free--and so far it's worked out pretty well.
Great video. But we live on a hyper Klein bottle. 3 dimensions of space but compactified time. The interior of event horizon is surface of entire universe. Conservation holds
Sin(cos(u/2)cos(v/2),cos(u/2)sin(v/2),sin(u)/2) 0
A Riemannian manifold has an inner product that tells "angle". This is what physicists call the metrix g. It is more then just a notion of length.
So we know how much surface we have on whole universe ???simple it's one PLANK length into total PLANK length square area
So we can develop a simple analogies of folding the plain paper
close to a black hole or even a neutron star it is not true that gravity can be detected only in large regions of spacetime, you will feel tidal forces in a short distance
Large is relative, what it means here is that locally the space is still flat, aka if you zoom infinitely it will look flat. What it means is a generalisation of a function being differentiable, but I'm not qualified enough to give you the details of how it can be expressed formally.
6:15 Bottom line should read space-like 😀
The space of SR, is not flat space, but is the curved geometry of Minkowski. This curved and open space. In a way this, in a more flat version, could be the space of the reality. How we are measuring the geometry? It's very simple. We are going a triangular between galaxies at the same far distance from Earth, and then we are measuring the sum of the three angles if the sum is less than 180 degrees then the shape of the Universe is a closed sphere like in Riemann's geometry. If the sum is greater than 180 degrees, then the shape is an open curved space like in the geometry of Minkowski. And if the sum, as we actually have found it, is everywhere =180 degrees, then the space is flat like the space of the Euclidean geometry. So, if such a huge mass like the mass of our visible Universe do not bend the space, then where, the heck, is doing that? The only reason to keep alive this GR theory, it's not the mathematical description of reality, but only to keep alive the myth of big bang and inflation, specially inflation which needs the energy of the empty space to exist. This "empty" energy is given by the spacetime fabric, which could exist empty, but full of energy in the absence of material and mass. This is the wrong philosophic interpretation of the world, given to Einstein by his professor teacher Mach, who believed that everything is only energy. He takes a function of the material, which is the energy, and makes it the creator of the material. Einstein, went few steps further in this philosophy and puts out of material existence, and the space and the time, and instead of being material function, they become a nonmaterial fabric, which, the field equations full sometime for the needs of inflation with "empty" energy. This mathematical cooking is very toxic for our logic, but idealism was always unscientific and toxic. Einstein believes in the god when he said to Bohr that god do not play dice, this is Einstein's right to believe in what he likes to, but this couldn't force reality to change in a curved and not existence. It's time humanity to move on, and to realize that all the world, the Universe is material, without limits no creation and will be there forever, and will partially, on the spots, it will always change from birth to death and from death to birth, with this two conditions existence the one inside the other and everywhere and any and the same time. No gods or humans with their mind or only with their will could change it. You could change the Universe, but this needs somehow physical reactions, not abracadabra. This has to be our main concept, and with this base we have to try to explain physical phenomena like redshift of the light from galaxies in distance, etc. Everything is material and on unstopped motion. This motion is the reason of the material's functions like interactions, energy, mass, field, charge, poles, time, space, information, shape, volume etc. If we found something new, and we do not know what to say about, first, we have to treated it like material, or material function, and then the description has to be done by the laws of motion. We could find these laws because they could be only the production of the main dialectical laws of motion. This product, the one moment will be there but the next no, because under the same laws, it has to change.
How out the Cavendish experiment.
I'll have to take a closer look and how this experiment relates to general relativity. I think this is an example of a "local gravitational experiment", which doesn't count as an external gravitational field. I think an experiment like this is what differentiates the Einstein Equivalence Principle from the Strong Equivalence Principle.
6:07 must be ||V||^2 < 0 (SPACE-like), not time-like.
whoops. missed that one.
👍
where is 101c tho?
I had one ready to upload, but I realized it contained some major factual errors because my understanding of GR wasn't good enough. I'll make 101c after I feel I understand GR properly.