Leonard makes this easy to understand. I have no background other than curiosity on this subject. And the concept suddenly clicks on my head. Im watching these videos all nights for a very long time. Usually replay them from time to time. Everytime I understand a little more. I love it
1:21:56 That's what Cooper saw in the Tesseract in Interstellar. He was actually looking at the Torus of Murph's room. And since it was explained that the tesseract was placed there by some "higher dimensional beings", it makes sense that "they" made it in form of cuboids or rectangles but those which were a quantized evolution of her room. "They" made it easier for Cooper to navigate. BLOWING OUR MINDS EVEN AFTER 8 YEARS. BRILLIANT!
They expressed all of their time as a spatial dimension attributed to the view of "bulk beings" - this is related to what theory describes the singularity inside a black hole: as space and time acting the same and which all world lines lead to the singularity. Plus the 'tesseract' seemed more akin to some kind of computer program. So he goes through the black hole, time and space become one in a computer program where he can access his world line. It's an awesome concept but I'm having trouble seeing where in that concept that anything is quantized? We need to know how to quantize gravity to understand the singularity but that idea in the movie was all relativity. Is this what kip wrote in his book? Or your take?
Honestly, I'm about to buy some textbooks. I'm not going to college anymore, but the pursuit of knowledge goes beyond that. Thank you for making these lectures public!
Regarding the discussion of geometries at @1:15:00; In fact Simon Donaldson showed in the 80's that there are many pairs of compact simply connected smooth four-manifolds which are homeomorphic but not diff eomorphic. That is, there are 4-manifolds which are topologically identical, but have different geometries (i.e. metrics). So the question of, "What is the geometry of our spacetime?" is much more complicated.
That what might confused a bit the lecture is when LS has put the observer at the center of a nested 3-sphere topology space. In this case the radius never comes back to π but instead is growing continuously. This is corrected when the observer is on the surface of a 3-sphere and locking down one direction, then the 3-spheres increase initially in radius and then decrease back to a point at π. Βut this does not happen when you locate your observer at the center of a nested 3-sphere topology.
There is a logical fallacy at around 36:20. Susskind claims that a galaxy at r=π would look big (i.e. big θ) but faint. But if we assume light is not absorbed on its way to us, that galaxy will also look very bright. (The law that apparent luminosity goes 1/r^2 is valid only for flat space. It should be replaced by 1/sin^2(r) in a spherical space.) In fact, if we live in a spherical space but (erroneously) measure r by luminosity using the 1/r^2 law, we would believe we are in a flat space, because the sin^2 in the estimation of r and in the metric cancel each other.
Always a trip. Talking about 'space' as a 'fabric' or a thing like a solid that can be shaped, curved. It isn't just the intervening distance between to material objects but it is a thing itself, it is something with substance of a sort. Flat Space, Curved Space talking about the stuff between things as if it were not simply a nothing/space but a material. If it isn't empty/nothingness between things then there is no such thing as nothing, the opposite of matter and things? a kind of matter?
if we lived on a torus then the universe could still be homogeneous but not isotropic since curvature in one direction would be different to curvature in the other, except for special cases
These are so great lectures, like stanford relativitiry series was. I wish there were more content like this on youtube. BTW, I wonder why space is expanding but the time is not, I'd love to have asked that if I were at the physical lecture!
Actually background independent theories of quantum gravity (like LQG) promote viewing space not as an entity or fabric, but as the gravitational field itself to put gravity on the same footing of the other fundamental interactions. In that view, space and time do not make sense anymore as concepts. Can we argue on the existence of nothingness? An entity always "exists" relative to another, even relative to the gravitational field; that's when the defining properties of the entity itself arise.
Dear Sir Leonardo, I had a question that nobody can explain. I thought this was the right moment to repeat it. If the universe is expanding, the galaxies are supposed to get away from each other, while dark matter, with its ripping action makes more expansion. Yet it is observed that almost all galaxies have either bumped into other galaxies a number of times or are on their way to hit another. Our Milkyway will hit Andromeda galaxy in 4 billion years, but in that time they are supposed to get further away from each other. Could you kindly explain what is the true celestial mechanics?
Andromeda and the Milky Way are gravitationally bound galaxies so their gravitational influence on one another dominates over universal expansion. Hubbles law and universal expansion deals on very large scales of the universe where peculiar motion is negligible.
Naimul Haq Yes the accelerated expantion (dark energy), makes the galaxies go further away from us. Andromeda is the exeption, it is close enough to the Milky Way so they are trapped in each others gravity and they will collide, all other galaxies will escape. So galaxies which are close enough to eachother will collide, and the galaxies which are too far apart will flow outwards, so you will get a population of galaxies which have merged but they will be moving apart from eachother after that
+Naimul Haq Just so you know, you're asking about dark energy there: not dark matter. The two are very different things. As for your actual question, only the galaxies in our local galactic cluster may move towards us. Galaxies in the universe tend to exist in clusters; some galaxies in some distant, gravitationally bound group may move towars one another, but they will never move towards anything outside of their group. The expansion of the universe will always push apart any things galaxies/objects/specks of dust that are not close enough to be gravitationally boud together. So the distance between galaxies in a group bay get bigger and smaller, but the distance between galaxies in different and unbound groups will only ever get bigger.
Wikipedia: Observable_universe Summarized: Take a distant galaxy at a distance of 13 Gly (redshift 8,2) at the time it emitted the light. Light needed 13 Gy to reach us, in the meantime the universe expanded and the galaxy is NOW much further away (about 30 Gly, can be calculated e.g. by the Hubble-law). The part of the universe we have received signals is flat. But the bigger part ("observable universe") from which we have received no signals until NOW, could it be curved?
I noted two little mistakes: 1. the hyperboloid drawn in the lecture has a positive curvature. In order to obtain the negative curvature, one has to look as the surface that its approaching the cones from outside (saddle shape). 2. A torus embedded in 3D is not flat, one cannot roll it out on a table without strechting it
Some confusion comes from the difference between embedded tori and topological tori that are globally flat. I noticed that I had to keep track of the difference while he was talking. Sometimes when physicists talk of toroidal geometry they mean the flat geometry that's finite yet unbounded, and I think in the context of cosmology that's what Prof. Susskind means.
Slight correction: 13 Gly is NOT correct. "Proper distance" for redshift 8,2 is 30 Gly, so the proper distance at time of photon-emission is 30 Gly/(1+ 8,2) = 3,26 Gly (NOT 13 Gly)
?!! But the hyperboloid is not homogeneously curved, it is not even negatively curved. At the bottom it is positively curved, like a sphere, and at 'the edges' it is asymptotically flat like a cone! To make this surface homogenous, one has to draw it in the Minkowsky space...
***** Like the professor mentioned in his talk, as far as we can tell, the universe is flat. Though this can be a local anomaly where the universe is much larger that what we can see and is curve. This is similar to us on Earth where on the surface it looks flat.
Markus Garbiso The Friedmann eqations were used in twentieth century cosmology to try and determine the ultimate fate of our universe, with few people thinking that the flat scenario would be a likely finding - since a universe might be expected to only stay flat for a short period, before shifting to an open (or closed0) state because its expansion (or contraction) would alter the density of its contents. Matter density was assumed to be key to geometry - and estimatess of the matter density of our universe came to around 0.2 atoms per cubic metre, while the relevant part of the friedmann equations calculated that the critical density required to keep our universe flat would be 5 atoms per cubic metre. Since we could only find 4% of the required critical density, this suggested that we probably lived in an open universe but then we started coming up with ways to measure the universe’s geometry directly. There’s a You-Tube of Lawrence Krauss (of Physics of Star Trek fame) explaining how this is done with cosmic microwave background data (from WMAP and earlier experiments) - where the CMB mapped on the sky represents one side of a triangle with you at its opposite apex looking out along its two other sides. The angles of the triangle can then be measured, which will add up to 180 degrees in a flat (Euclidean) universe, more than 180 in a closed universe and less than 180 in an open universe.
If we measure the distance of a galaxy by measuring the apparent luminosity, wouldn't distant galaxies on a sphere look brighter than in flat space? Because the angle in which the light of the galaxy can reach the opening of our telescope grows in the same way as the angle we can see the galaxy. If so, we would underestimate the distance of the galaxy presumably by the same ratio that we overestimate the diameter of the galaxy, so we couldn't tell the difference between flat space and sphere.
How can he argue that homogeneity cannot be achieved with certain shapes if the observer is bound by the laws of nature? Even if the universe was 'U' shaped and we stood anywhere within it then the observer would still be bounded by space and any observations would be bent along with it. Thoughts?
argh, this continously turns my head in; if space, the universe, is curved, does it curve into 4D space, and if so how can measurements of angles in huge triangles, made in 3D space tell us anything. Please ELI5!
Are you asking is the 3D universe embedded in a 4D space we can't access? Like you can draw a 2D sphere embedded in xyz 3D space. He mentions when he draws that picture (don't remember when) that it doesn't necessarily mean that must be the case for us, I imagine he goes into it later but I'm not there yet lol
The area in which energy would be arriving at an observer would be visible from a larger surface area. It seems to me that you are correct if you are saying that the total energy arriving at that observer over the area received is the same, but if you are saying the brightness at each point is the same that would violate energy conservation.
Hello, ive bee trying to compare the equation at Cosmology Lecture 3 with the equation at Cosmology Lecture 3 and i cant figure out what happened to the 'sinr' and 'dr' terms in the latter. can anyone explain why are they not there?. i.e. why is it ds^2 = -dt^2 + a(t)^2 (domega2^2) instead of ds^2 = -dt^2 + a(t)^2 (dr^2 + sin^2 r * omega2^2) . Ive been trying to follow along since lecture 1 but i am struggling with this one. please help!
Sorry to reply 5 years later but you don't need to include the dr and sin(r) terms because they are embedded in the dOmega term, ie, dOmega2^2=dr^2 + sin^2(r)dOmega1^2.
He's right, curves are one dimensional objects and surfaces are 2d objects, they might live on higher dimensional spaces tho, but that's not how you define dimension. You have to look at the dimension of the manifold (your object) once you flatten it down, like a curve, it can be flattend into the 1d real line, a surface can (locally) be flattened into the 2d plane and so on
@@alessandromanta9405 I'm not saying he's not right, I'm just asking why, you know, I'm ignorant, I'm trying to understand why he calls a circle 1 dimensional. In my limited understanding is two-dimensionale. Your explanation isn't very clear, because you don't say why your statement is true. But at least you suggested that I should look at the definition of dimension, probably if I understand the definition of dimension I will understand why a circle is one-dimensional. Thank you for trying to inform me, but you failed, sorry. Maybe is just my fault that I'm ignorant
@@sacredgeometrymusic3290 i'm sorry i didn't mean to say you implied he was wrong😂. What i told you is the idea behind the mathematical definition of the dimension of a manifold, you can look it up if you want to
This is purely terminology. A 2 sphere refers to the two-dimensional surface of a 3 dimensional ball. I believe that is the language that mathematicians use. A 3D ball includes all the points x^2 + y^2+z^2
@@Eldooodarino thanks buddy for the effort ❤️ still im not sure if it s clear what u mean. But i will just take it as matematician terminology without overthinking it
To be more precise: you have to think to a circle or a sphere of unitary radius. In this case, the lenght on the circonference (or on the circonference of maximum radius, i.e. meridians) are equal to the angle of co-latitude
Interesting. There is no center to the radially expanding universe, he tells us at 38:45. Yet it was all created by an explosion of nothing at a single point, called the big bang, and it therefore must be expanding around that point on any scale you can conjure mathematical anecdotes to define. That point would be the center of the expansion...would it not?
The person who films these lectures zooms the camera unnecessarily too much. Instead of filming large parts of the board, he fixies on Leonard's head and pursuits him everywhere.
Why one would believe that a balloon can expand forever? Why can't one think that, everything we see as expanding might be revolving /orbiting around something that we can't see?
Leonard makes this easy to understand. I have no background other than curiosity on this subject. And the concept suddenly clicks on my head. Im watching these videos all nights for a very long time. Usually replay them from time to time. Everytime I understand a little more. I love it
Even to a layman; Professor Susskind is profoundly enlightening. Thank you for this. What a time to be alive.
Dr Susskind is very good lecturer and teacher.
1:21:56
That's what Cooper saw in the Tesseract in Interstellar. He was actually looking at the Torus of Murph's room. And since it was explained that the tesseract was placed there by some "higher dimensional beings", it makes sense that "they" made it in form of cuboids or rectangles but those which were a quantized evolution of her room. "They" made it easier for Cooper to navigate.
BLOWING OUR MINDS EVEN AFTER 8 YEARS.
BRILLIANT!
They expressed all of their time as a spatial dimension attributed to the view of "bulk beings" - this is related to what theory describes the singularity inside a black hole: as space and time acting the same and which all world lines lead to the singularity. Plus the 'tesseract' seemed more akin to some kind of computer program.
So he goes through the black hole, time and space become one in a computer program where he can access his world line. It's an awesome concept but I'm having trouble seeing where in that concept that anything is quantized? We need to know how to quantize gravity to understand the singularity but that idea in the movie was all relativity. Is this what kip wrote in his book? Or your take?
As brilliant as Dr Susskind is, he is also a very good lecturer and teacher. Many people at his level have trouble teaching us mortals.
I'm addicted.....love these lecs .....they are priceless
indeed
No doubt
@@professord8888 Better late than never 🤣 But it still holds true 👍
Honestly, I'm about to buy some textbooks. I'm not going to college anymore, but the pursuit of knowledge goes beyond that. Thank you for making these lectures public!
There excellent scientifics who are not good teachers but Leonard is excellent in both!!
priceless content ....pure mind candy
+realcygnus you are very aDICKted. good for u
Thank you for your time professor
Regarding the discussion of geometries at @1:15:00; In fact Simon Donaldson showed in the 80's that there are many pairs of compact simply connected smooth
four-manifolds which are homeomorphic but not diff eomorphic. That is, there are 4-manifolds which are topologically identical, but have different geometries (i.e. metrics). So the question of, "What is the geometry of our spacetime?" is much more complicated.
That what might confused a bit the lecture is when LS has put the observer at the center of a nested 3-sphere topology space. In this case the radius never comes back to π but instead is growing continuously. This is corrected when the observer is on the surface of a 3-sphere and locking down one direction, then the 3-spheres increase initially in radius and then decrease back to a point at π. Βut this does not happen when you locate your observer at the center of a nested 3-sphere topology.
I love the RUclips service
❤Thank you very much Professor and class
There is a logical fallacy at around 36:20. Susskind claims that a galaxy at r=π would look big (i.e. big θ) but faint. But if we assume light is not absorbed on its way to us, that galaxy will also look very bright. (The law that apparent luminosity goes 1/r^2 is valid only for flat space. It should be replaced by 1/sin^2(r) in a spherical space.) In fact, if we live in a spherical space but (erroneously) measure r by luminosity using the 1/r^2 law, we would believe we are in a flat space, because the sin^2 in the estimation of r and in the metric cancel each other.
Always a trip. Talking about 'space' as a 'fabric' or a thing like a solid that can be shaped, curved. It isn't just the intervening distance between to material objects but it is a thing itself, it is something with substance of a sort. Flat Space, Curved Space talking about the stuff between things as if it were not simply a nothing/space but a material. If it isn't empty/nothingness between things then there is no such thing as nothing, the opposite of matter and things? a kind of matter?
if we lived on a torus then the universe could still be homogeneous but not isotropic since curvature in one direction would be different to curvature in the other, except for special cases
mindblowing lesson this one!
These are so great lectures, like stanford relativitiry series was. I wish there were more content like this on youtube.
BTW, I wonder why space is expanding but the time is not, I'd love to have asked that if I were at the physical lecture!
Where can I get all the full details/derivation of metric of 2d/3d sphere, 2d/3d hyperpbola? Do you know any books?
@@youtubeshortsviral1361 nope
@@youtubeshortsviral1361
Space time & Geometry by : Sean & Carrol
Actually background independent theories of quantum gravity (like LQG) promote viewing space not as an entity or fabric, but as the gravitational field itself to put gravity on the same footing of the other fundamental interactions. In that view, space and time do not make sense anymore as concepts. Can we argue on the existence of nothingness? An entity always "exists" relative to another, even relative to the gravitational field; that's when the defining properties of the entity itself arise.
Dear Sir Leonardo, I had a question that nobody can explain. I thought this was the right moment to repeat it. If the universe is expanding, the galaxies are supposed to get away from each other, while dark matter, with its ripping action makes more expansion. Yet it is observed that almost all galaxies have either bumped into other galaxies a number of times or are on their way to hit another. Our Milkyway will hit Andromeda galaxy in 4 billion years, but in that time they are supposed to get further away from each other. Could you kindly explain what is the true celestial mechanics?
Andromeda and the Milky Way are gravitationally bound galaxies so their gravitational influence on one another dominates over universal expansion. Hubbles law and universal expansion deals on very large scales of the universe where peculiar motion is negligible.
Naimul Haq Yes the accelerated expantion (dark energy), makes the galaxies go further away from us. Andromeda is the exeption, it is close enough to the Milky Way so they are trapped in each others gravity and they will collide, all other galaxies will escape. So galaxies which are close enough to eachother will collide, and the galaxies which are too far apart will flow outwards, so you will get a population of galaxies which have merged but they will be moving apart from eachother after that
+Naimul Haq Just so you know, you're asking about dark energy there: not dark matter. The two are very different things. As for your actual question, only the galaxies in our local galactic cluster may move towards us. Galaxies in the universe tend to exist in clusters; some galaxies in some distant, gravitationally bound group may move towars one another, but they will never move towards anything outside of their group. The expansion of the universe will always push apart any things galaxies/objects/specks of dust that are not close enough to be gravitationally boud together. So the distance between galaxies in a group bay get bigger and smaller, but the distance between galaxies in different and unbound groups will only ever get bigger.
Wikipedia: Observable_universe
Summarized: Take a distant galaxy at a distance of 13 Gly (redshift 8,2) at the time it emitted the light. Light needed 13 Gy to reach us, in the meantime the universe expanded and the galaxy is NOW much further away (about 30 Gly, can be calculated e.g. by the Hubble-law).
The part of the universe we have received signals is flat. But the bigger part ("observable universe") from which we have received no signals until NOW, could it be curved?
I noted two little mistakes: 1. the hyperboloid drawn in the lecture has a positive curvature. In order to obtain the negative curvature, one has to look as the surface that its approaching the cones from outside (saddle shape). 2. A torus embedded in 3D is not flat, one cannot roll it out on a table without strechting it
Some confusion comes from the difference between embedded tori and topological tori that are globally flat. I noticed that I had to keep track of the difference while he was talking. Sometimes when physicists talk of toroidal geometry they mean the flat geometry that's finite yet unbounded, and I think in the context of cosmology that's what Prof. Susskind means.
Thanks profesor. That is my THEBEST lectures. Very good for listening.❤️🤗🤗 Amaizing subjectes. Amaizing Lessons.Thanks.
Slight correction: 13 Gly is NOT correct. "Proper distance" for redshift 8,2 is 30 Gly, so the proper distance at time of photon-emission is 30 Gly/(1+ 8,2) = 3,26 Gly (NOT 13 Gly)
King el-kyk, u can still draw hyperbolic outa Taurus.. its finite dynamic cyclical toroid cosmo-universe🔁♾️🔃
What a potential!
very Nice
?!! But the hyperboloid is not homogeneously curved, it is not even negatively curved. At the bottom it is positively curved, like a sphere, and at 'the edges' it is asymptotically flat like a cone! To make this surface homogenous, one has to draw it in the Minkowsky space...
8:20 Omega literally means "great O"
Debunking flat earth before it was a trend.
Flat: - Zero K - K=0 Spherical - Positive K - K=1/R2 > 0 Hyperbolic - Negative K
- K= -1/R2 < 0. What is K of the universe?
***** Like the professor mentioned in his talk, as far as we can tell, the universe is flat. Though this can be a local anomaly where the universe is much larger that what we can see and is curve. This is similar to us on Earth where on the surface it looks flat.
Markus Garbiso The Friedmann eqations were used in twentieth century cosmology to try and determine the ultimate fate of our universe, with few people thinking that the flat scenario would be a likely finding - since a universe might be expected to only stay flat for a short period, before shifting to an open (or closed0) state because its expansion (or contraction) would alter the density of its contents.
Matter density was assumed to be key to geometry - and estimatess of the matter density of our universe came to around 0.2 atoms per cubic metre, while the relevant part of the friedmann equations calculated that the critical density required to keep our universe flat would be 5 atoms per cubic metre. Since we could only find 4% of the required critical density, this suggested that we probably lived in an open universe but then we started coming up with ways to measure the universe’s geometry directly.
There’s a You-Tube of Lawrence Krauss (of Physics of Star Trek fame) explaining how this is done with cosmic microwave background data (from WMAP and earlier experiments) - where the CMB mapped on the sky represents one side of a triangle with you at its opposite apex looking out along its two other sides. The angles of the triangle can then be measured, which will add up to 180 degrees in a flat (Euclidean) universe, more than 180 in a closed universe and less than 180 in an open universe.
***** I'll have to check the video out. Though, I was trying to use what the professor said. Thanks for the info!
If we measure the distance of a galaxy by measuring the apparent luminosity, wouldn't distant galaxies on a sphere look brighter than in flat space? Because the angle in which the light of the galaxy can reach the opening of our telescope grows in the same way as the angle we can see the galaxy. If so, we would underestimate the distance of the galaxy presumably by the same ratio that we overestimate the diameter of the galaxy, so we couldn't tell the difference between flat space and sphere.
"Your mind does not want to visualize the 3-sphere" - Susskind
How can he argue that homogeneity cannot be achieved with certain shapes if the observer is bound by the laws of nature? Even if the universe was 'U' shaped and we stood anywhere within it then the observer would still be bounded by space and any observations would be bent along with it. Thoughts?
I can't tell if he's wearing his turtleneck or not
Great lecture
argh, this continously turns my head in; if space, the universe, is curved, does it curve into 4D space, and if so how can measurements of angles in huge triangles, made in 3D space tell us anything. Please ELI5!
there is no such thing as 4D space.
Are you asking is the 3D universe embedded in a 4D space we can't access? Like you can draw a 2D sphere embedded in xyz 3D space. He mentions when he draws that picture (don't remember when) that it doesn't necessarily mean that must be the case for us, I imagine he goes into it later but I'm not there yet lol
What is Mylar glasses?
btw at 37:13 lenny says something wrong, he say the galaxy far away on the 2 sphere would be fainter, it wouldnt.
apart from redshift if the 2 sphere is exspanding
Hello sir Where can I get all the full details/derivation of metric of 2d/3d sphere, 2d/3d hyperpbola? Do you know any books?
The area in which energy would be arriving at an observer would be visible from a larger surface area. It seems to me that you are correct if you are saying that the total energy arriving at that observer over the area received is the same, but if you are saying the brightness at each point is the same that would violate energy conservation.
milk is homogeneous
is this part 3 to external inflation
Hello, ive bee trying to compare the equation at Cosmology Lecture 3 with the equation at Cosmology Lecture 3 and i cant figure out what happened to the 'sinr' and 'dr' terms in the latter. can anyone explain why are they not there?. i.e. why is it ds^2 = -dt^2 + a(t)^2 (domega2^2) instead of ds^2 = -dt^2 + a(t)^2 (dr^2 + sin^2 r * omega2^2) . Ive been trying to follow along since lecture 1 but i am struggling with this one. please help!
Sorry to reply 5 years later but you don't need to include the dr and sin(r) terms because they are embedded in the dOmega term, ie, dOmega2^2=dr^2 + sin^2(r)dOmega1^2.
but why does the universe even have a geometry?
It should have a shape? :)
6:09 Flat Earthers still don't believe it.....
why is he calling a circle 1 dimensional and a sphere 2 dimensional? shouldn't they be 2 and 3 dimensional?
He's right, curves are one dimensional objects and surfaces are 2d objects, they might live on higher dimensional spaces tho, but that's not how you define dimension. You have to look at the dimension of the manifold (your object) once you flatten it down, like a curve, it can be flattend into the 1d real line, a surface can (locally) be flattened into the 2d plane and so on
@@alessandromanta9405 I'm not saying he's not right, I'm just asking why, you know, I'm ignorant, I'm trying to understand why he calls a circle 1 dimensional. In my limited understanding is two-dimensionale. Your explanation isn't very clear, because you don't say why your statement is true. But at least you suggested that I should look at the definition of dimension, probably if I understand the definition of dimension I will understand why a circle is one-dimensional. Thank you for trying to inform me, but you failed, sorry. Maybe is just my fault that I'm ignorant
@@sacredgeometrymusic3290 i'm sorry i didn't mean to say you implied he was wrong😂. What i told you is the idea behind the mathematical definition of the dimension of a manifold, you can look it up if you want to
This is purely terminology. A 2 sphere refers to the two-dimensional surface of a 3 dimensional ball. I believe that is the language that mathematicians use. A 3D ball includes all the points x^2 + y^2+z^2
@@Eldooodarino thanks buddy for the effort ❤️ still im not sure if it s clear what u mean. But i will just take it as matematician terminology without overthinking it
How can you compare r and sin r ? They are on different orders of magnitude
No! r , despite the simbol, is an angol, and runs from 0 to Pie
To be more precise: you have to think to a circle or a sphere of unitary radius. In this case, the lenght on the circonference (or on the circonference of maximum radius, i.e. meridians) are equal to the angle of co-latitude
@@massimoandretta3529 thanks i understand it now
@@shyamvijay8985 Fine. Very glad to have clarified a question. 👍 Have a good study of cosmology at Stanford 😉
@@massimoandretta3529 Where can I get all the full details/derivation of metric of 2d/3d sphere, 2d/3d hyperpbola? Do you know any books?
Interesting. There is no center to the radially expanding universe, he tells us at 38:45. Yet it was all created by an explosion of nothing at a single point, called the big bang, and it therefore must be expanding around that point on any scale you can conjure mathematical anecdotes to define. That point would be the center of the expansion...would it not?
any reference books suggestions if any one knows ....gud book
Lecture notes are available online also his book on cosmic landscape is excellent
So in case of 3-sphere, is the 4th dimension time or another large hyper-spatial dimension?
12 flat earthers
9:57 (look at the image)😂
Order chage 80 Rest I
[S]
Yup, no such thing as nothing.
The person who films these lectures zooms the camera unnecessarily too much. Instead of filming large parts of the board, he fixies on Leonard's head and pursuits him everywhere.
Is this grandpa crazy? :))
X. Log
Log W plane
So order stop log z
D ± SnS
Chess point
🛣️.
..... 🤨
Black Energie in there ≈ m²
Log W u d
Ouderdom is geen excuus om je te misdragen, in mijn heelal.
Why one would believe that a balloon can expand forever? Why can't one think that, everything we see as expanding might be revolving /orbiting around something that we can't see?
One apple a day keeps the viewers away (1:12:00)
Sir please upload short lectures.
69
Log plane 📻
Log [S]
Log W s
Log [S] secand 📻
Chess.
🙄
Lost. I DI