What is Curved Spacetime? Curvature Visually Explained!

[@nuranichandra2177]

This video is simply mind bending. No one can explain such complex concepts in simple terms. I watched it twice to comprehend it better. Great job.

[@whyofpsi]

Wow, thank you for the kind words, I really appreciate that :)

[@smlanka4u]

The universe is flat. The video wrong. The curvare in areas of space is a result of gravity. But quantum gravity likely depends on neutrino oscillation (graviton force).

[@whyofpsi]

Indeed, the universe as a whole, smoothened out, appears to be flat.

[@everythingisalllies2141]

Except the video, although explaining Math and Geometry well, is only reinforcing to nonsense claim that there can be such a thing as "Spacetime" and that its curved, not flat. This is a nonsense idea from Einstein, and its is destroying real Physics. Math is not Physics. There is no such thing as "Spacetime", Time is not a Dimension to travel along or through, and there is nothing about reality that can be thought of a curved or flat. You make the error of assigning reality with the properties of abstract Math.

[@factChecker01]

@@everythingisalllies2141 , Einstein's theories have solved so many puzzles in experimental results and have been experimentally verified in so many ways, that denying them is like belonging to the Flat Earth Society.

[@vijaysahani3464]

The best explanation of space-time curvature through graphics.

[@whyofpsi]

Thank you very much! That's awesome to read :)

[@akm_io]

This was explained so well! Can’t wait for more videos from you 😄

[@whyofpsi]

Thank you for the kind words! Next video is coming :)

[@Ninnanannetristi]

Great! Despite my mathematics low knowledge, I finally understood what Riemann tensors are. Beautiful and clear graphics and exposition. Thanks!

[@whyofpsi]

Ah, that makes me happy. It's why I made this video! Thank you for taking the time to comment :)

[@-_Nuke_-]

That was excellent! I think we need to start using the phrase "aging towards" something. Because that's what falling really is, you just age towards the center of mass of the Earth, no forces applied.

[@whyofpsi]

That's a very interesting way of putting it. Thanks! :D

[@sanketparekar]

You cannot say "falling" since you said there are no forces applied...

[@whyofpsi]

I think you touch upon an interesting point. You connect the word "falling" with applied forces, by which I assume you mean the force of gravity. However, in general relativity, gravity is not considered a force, but the concept of falling still exists. Therefore, in this sense, falling can be used without any reference to a force, no?

[@sanketparekar]

@@whyofpsi thanks for your reply. When you say "Apple falls...", you actually accept that it falls due to gravitational force. If you want to use general relativity, you should say that the Apple collides into the earth. According to general relativity, Apple is actually moving forward into the fabric of space-time linearly, but it is the mass of earth which curves this space time and hence the Apple collides into the earth

[@whyofpsi]

Alright, we agree on the physics of the event :)

[@jackcau1845]

wtf why this have only 300 views?? great video!

[@whyofpsi]

I guess the algorithm still has to learn what the fitting audience is before it can start recommending :) Thanks for commenting, that also drives up engagement!

[@jackcau1845]

@@whyofpsi i think your channel will grow fast soon, this is really high quality stuff. glad to be one of the first subscribers!

[@whyofpsi]

@@jackcau1845 Thank you very much, I will remember you ;)

[@wolf5393]

Because this generation and in general people would not like to give their brain any sort of challenge. In past there were lots of challenge thus using brain was the way to reduce their pain and be in better condition but now people have gone so stupid they would go and watch tic tok shit and scroll reels whole day Or do the same steps as if they were sheeps who will follow stupid moves bit they would not do Or learn anything good. Your content was good bro keep it up world is like this u would not get attention likeof tictok trends but the attention you will get will be of the best quality and may be not in quantity.

[@whyofpsi]

I do tend to agree that the attention span of younger generations has reduced significantly. And the cause of this has been the rise of retention driven social media. However, let's hope when they grow older they will find something that interests them and draws their attention instead :)

[@adiblokhandwala1115]

This video is amazing, hope you'll continue teaching complex concepts by breaking it into simple chunks like this

[@whyofpsi]

Thank you for the kind words :) I'm working on new content!

[@TheJara123]

oh my God you are back again!!! a beautiful presentation superbly explained math...Please keep posting your fantastic videos.

[@whyofpsi]

Wow, that makes my day! Thank you!! :) I will, making one about determinism next! Share the love for science

[@TheJara123]

@@whyofpsi delighted to know. There are no videos on KZbin on The Majestic General Relativity Theory, or SRT, visualy on your level, Dialect channel and yours are superbly paced and very well explained complicated maths. I would, like many others, love to see your take on GRT equation and it's physics, math, gravitational waves, Mercury problem etc.. It would be a great treat....something to treasure and something to make life interesting and bearable... Thanks for the wonderfull efforts you put on these videos making..

[@whyofpsi]

@@TheJara123

[@icebeartwo]

This was really an amazing video, I am happy for the growth of this channel. Keep going!

[@whyofpsi]

HI ICEBEAR :D I love to see you here :)

[@cerealpuffsalmomd1625]

Good visuals, they really help! Looking forward to your next topic!

[@whyofpsi]

Thank you, I'm super happy I could help you! I am working on the next one indeed :)

[@MC-wg3fm]

I like how you basically started explaining the concept from a two dimensional perspective and made sure to say how important it is to have a 3rd dimension to fold a the paper in and how you showed the difference be the sphere and cylinder in three dimensions and how important it was. But then just quickly glossed over the whole 4th dimension and had a 5th dimension with an X through it as if it wasn’t possible.

[@whyofpsi]

The important thing here is that intrinsic curvature doesn't need an extra dimension to be curved into. Since space-time, if curved, is intrinsically curved, we would not need a fifth dimension.

[@phyx1s]

Seems like what could be meant here is that space is "compressed", like dots on a 2D surface getting closer or further away rather than equal distance what would be in a linear space. All dots still being in the same 2D space. As soon as the dots need to move our of the 2D surface another dimension is needed. If the 2D space would intrinsically curved without the need to have another dimension, it would be impossible to detect as all measurements would intrinsically include these features and always give the expected outcome, i.e. the 90 degree angle in the intrinsic curved space would have a different shape compared to space where 3D would be possible but space is not.

[@Black-mx3hc]

@@phyx1s elaborate

[@Black-mx3hc]

@@whyofpsi what goes with the idea that it's intrinsically curved?

[@whyofpsi]

@@Black-mx3hc One way to characterise intrinsic curvature is that it can be measured *without* needing to zoom out in an extra dimension. For instance, on the surface of a sphere you can draw a triangle, and its inner angles will always sum up to be more than 180 degrees.

[@AndrewPa]

As a physicist would like to thanks author for great video

[@whyofpsi]

That's awesome to hear from a fellow physicist! :)

[@davidmudry5622]

Brian Greene tells Alan Alda there is no gravity in free fall. NIST WTC FAQ 31 "the upper section came down essentially in free fall." NIST Shyam Sunder John Gross "gravity is a downward driving force." Weight is a Force, weightlessness is not a Force. F = ma Weight = Mass x Gravity or W = mg...In free fall weight = weightless = zero weight...So that means in free fall it's either mass or it's gravity that must = zero. But mass cannot = zero so it must be gravity that = zero in free fall. How can NIST WTC FAQ 31 say the upper section came down essentially in free fall, and that the weight from gravity was the driving force when gravity & weight = zero in free fall? Gravity is a phenomenon. According to Newton's law of Universal Gravitation, gravity is a force between masses3. However, according to general relativity, gravity is only a pseudo force. kzbin.info/www/bejne/e2WWXnacqtB4ndU kzbin.info/www/bejne/joPVYp6XjbB1qbc kzbin.info/www/bejne/g6moaYFrfqZ5jck kzbin.info/www/bejne/faavfZKrpapnsM0 kzbin.info/www/bejne/lXjNepqAhdNnbac

[@im4science77]

This is a great visual explanation. I have been learning GR on my own for the last many years and have seen a lot of explanatory videos on the topic. But this video is better than them all. Thank you and hope to see more such videos in the channel.

[@whyofpsi]

Awesome! I'm super happy to hear that :D

[@ian731]

this and the video from science click is wesome

[@whyofpsi]

Thanks :)

[@frankdimeglio8216]

​@@ian731 Consider what is the man (AND THE EYE ON BALANCE) who IS standing on what is THE EARTH/ground. Consider TIME AND time dilation ON BALANCE. What is E=MC2 is taken directly from F=ma, AS the stars AND PLANETS are POINTS in the night sky ON BALANCE. This CLEARLY explains and proves the fourth dimension. c squared CLEARLY represents a dimension of SPACE ON BALANCE !!!! Indeed, E=mc2 is taken directly from F=ma; AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). Accordingly, ON BALANCE, the rotation of WHAT IS THE MOON matches it's revolution. Notice the TRANSLUCENT blue sky ON BALANCE. Consider what is THE EYE ON BALANCE, AS c squared CLEARLY represents a dimension of SPACE ON BALANCE. Gravity/acceleration involves BALANCED inertia/INERTIAL RESISTANCE, AS TIME is NECESSARILY possible/potential AND actual ON/IN BALANCE; AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE); AS what is E=MC2 is taken directly from F=ma; AS GRAVITATIONAL force/ENERGY is proportional to (or BALANCED with/as) inertia/INERTIAL RESISTANCE. “Mass”/ENERGY involves BALANCED inertia/INERTIAL RESISTANCE consistent with/as what is BALANCED electromagnetic/gravitational force/ENERGY, AS ELECTROMAGNETISM/energy is CLEARLY AND NECESSARILY proven to be gravity (ON/IN BALANCE). Accordingly, ON BALANCE, what are OBJECTS may fall at the SAME RATE. By Frank Martin DiMeglio

[@robbatayaki5505]

I usually dont subscribe when new channel. But here I did. Finally a visual explanation of tensor !

[@whyofpsi]

Awesome to hear :) Thank you. More content coming in the future!

[@Philoreason]

To do a parallel transport in a curved space you have to first define what you mean by "parallel", right? What does it mean to be parallel on the surface of a sphere?

[@whyofpsi]

You are completely correct. I hesitated including this in the video, but then decided not to. Parallel transport is rigorously defined by the connection on that space, which is a combination of the corresponding metric and its derivatives.

[@sonyalindee8676]

Well this takes Star Trek to another level. So if we ever got a good handle on warp speed the mathematics alone for trajectory would be mind boggling.🖖🏼Chekov would been a genius navigator.🌌

[@whyofpsi]

Who knows :)

[@lowerbound4803]

Thank you for trying to make the concept accessible!!

[@whyofpsi]

You are very welcome! :))

[@gustavogarcia1528]

Great work, very simple to understand for someone like me that is an ignorant in maths.

[@whyofpsi]

Thank you very much! I hope it helped you see some beauty in it. Never give up on math struggles :)

[@faisalsheikh7846]

Incredible quality of content sir you are phenomenal keep it up❤

[@whyofpsi]

Thank you, I'm glad you like it and perhaps learned something :)

[@peterdegroot466]

Amazing !! I finally understand how curved space-time works ; many thanks for this excellent explanation !

[@whyofpsi]

You are very welcome! I'm super glad I was able to convey the message :))

[@terrycrooke1]

I've Subscribed... on the content of one video!! Now that's the first for me.... well done!!!

[@whyofpsi]

Cool, that's an honour! Thank you. More to come in the future :)

[@terrycrooke1]

I will wait with interest

[@spranav9878]

This is a masterpiece ❤️

[@whyofpsi]

Thank you very much! That means a lot to me :)

[@lj823]

Amazing! I could only-kinda follow the math when you got to tensors, but the visual of not meeting at the same point, I got. TY! It clicked.

[@whyofpsi]

I am super happy that it clicked for you, that's what it's all about for me :)

[@josephfredbill]

This is excellent. A point where I am struggling is why is it not that the cyclinder surface is curved in one dimension but the sphere surface is curved in two. Im aware that a flat plane cannot wrap a sphere - isnt this this the extrinsic/intrinsic difference? Id welcome more explanation on this point, which it seems to me is at the core of things.

[@whyofpsi]

I've never thought of it like that. Though you need to be able to generalise it I think. What you're suggesting is that intrinsically curved surfaces are curved in all of their dimensions (for a sphere 2), whereas extrinsically curved spaces in at most 1. That's interesting, I think you are right. Thanks!

[@josephfredbill]

@@whyofpsi well > 1 anyway. I suspect (intuitively) that the unwrappable property appears because the dimensions are orthogonal - you need at least 2 obviously. Question is, does orthogonality of dimensions always produce unwrappability - are there any shapes that are curved in 2 dimensions that are wrappable by a flat plane. I think not.

[@wknz48]

Really a great video with lovely explanations!

[@whyofpsi]

Awesome to hear! Thanks for the kind words :)

[@davidmudry5622]

@@whyofpsi Brian Greene tells Alan Alda there is no gravity in free fall. NIST WTC FAQ 31 "the upper section came down essentially in free fall." NIST Shyam Sunder John Gross "gravity is a downward driving force." Weight is a Force, weightlessness is not a Force. F = ma Weight = Mass x Gravity or W = mg...In free fall weight = weightless = zero weight...So that means in free fall it's either mass or it's gravity that must = zero. But mass cannot = zero so it must be gravity that = zero in free fall. How can NIST WTC FAQ 31 say the upper section came down essentially in free fall, and that the weight from gravity was the driving force when gravity & weight = zero in free fall? Gravity is a phenomenon. According to Newton's law of Universal Gravitation, gravity is a force between masses3. However, according to general relativity, gravity is only a pseudo force. kzbin.info/www/bejne/e2WWXnacqtB4ndU kzbin.info/www/bejne/joPVYp6XjbB1qbc kzbin.info/www/bejne/g6moaYFrfqZ5jck kzbin.info/www/bejne/faavfZKrpapnsM0 kzbin.info/www/bejne/lXjNepqAhdNnbac

[@MrMaitreyann]

Thank you, you cleared a long standing doubt....😊❤️

[@whyofpsi]

Cool, I'm very glad I could help :)

[@marcobenatar7638]

Brilliant video, thanks a lot for the explanations. I'm looking forward to following some other videos you posted.

[@whyofpsi]

Cool, thank you! More content is on the way :D

[@bbk1984]

You seem a very learned mathematician/physicist.. Im a layman.. But we both don't know for sure how this universe began.. So you are a genius to me in relative terms.. Other intelligent life forms in some other dimensions will be genius to you in relative terms.. May be we will never know

[@whyofpsi]

I am by no means a genius. I just put in the time and effort to go through the rigorous mathematics and education.

[@JEETxNinja]

Wow 👍 lots of love from INDIA 🇮🇳😀

[@whyofpsi]

Thanks! :)

[@threefiverealty]

That was the best way I’ve seen it explained. Can you do a video on how a photon appears as a particle and a wave?

[@whyofpsi]

Thanks for the kind words. I will definitely do a video about that in the future, not sure when since they take a while to make. The next one will be on the arrow of time :)

[@Ryannnnnnnnnnnnnnnn]

Not sure the thought experiment at the end works for me. That we don’t end up in the same place seems more a Cartesian phenomenon. On the cartesian plane the rockets are going to be moving apart by virtue of launching at perpendicular from a sphere’s surface. So they get farther apart as they rise, so obviously the outer ring is going to be longer than the inner. The whole thing can be understood without thinking about spacetime curvature at all.

[@whyofpsi]

That's a very good remark. I should have specified in the video, that this experiment and its result does not rest on the fact that the earth's surface is curved.

[@marcelob.5300]

This is great, thanks.

[@whyofpsi]

I'm super glad you liked it! :) Thanks for commenting :D

[@zapazap]

A promising chamnel sir. You have my subscription.

[@whyofpsi]

Cool, much appreciated! :)

[@maarirs12894]

This channel will soon b as famous as veritasium

[@whyofpsi]

We'll see about that ;) I'm just happy this video got to reach as many people as it did.

[@sesam703]

Great video. I need more, much more, of this 👍

[@whyofpsi]

Thanks! I'm in the process of making a new one, about the arrow of time :)

[@sairamakrishnasrivathsavpa9255]

Thank you for this excellent video!! Subscribed this channel

[@whyofpsi]

Thank you for commenting, I appreciate that :) And you're very welcome, I'm glad you liked it.

[@Aegis4521]

Ur name 😭

[@quantisedspace7047]

Thanks. This is the best explanation of curvature that I've ever seen. The whole 'space is expanding into what?' question is becoming clearer to me, as a consequence of this.

[@whyofpsi]

Thank you for the kind words, it makes my day! I'm happy I could help you navigate this wonderful topic (: PS: you are my first super thanks ever, many thanks ^^

[@JorgeRomero-jt2ne]

Amazing video! Thank you!

[@whyofpsi]

Thank you, I really appreciate that :)

[@douche8980]

What I gather from this video is that spacetime is curved so that you wl eventually wind up no matter which direction you are traveling through space so long as you are traveling in a straight 1D line. Therefore if other universes do exist and there is a series of multiverse that does not effect our universe in any way since our universe is all there is in this realm.

[@whyofpsi]

Technically, the geometry of a space tells you about geometrical facts, such as the sum of inside angles of a triangle. It's the topology that tell you whether you can get at the same point again by going endlessly in one direction.

[@edfriedman6284]

There's no question that the mathematics part of the video is very well done. My concern is that there is too little emphasis on the idea that the universe as a whole is FLAT. This result comes from precise measurements of the cosmic microwave background. The video is correct in saying that spacetime is curved near a mass, but don't forget that the universe is mostly empty and its overall geometry is the same as what we learned in the 8th grade; triangles contain 180 degrees, parallel lines don't cross, etc. These are the definitions of a flat geometry. I always make the point that there is a difference between geometry (say a flat spacetime) and shape (such as the cylinder that the author correctly states is 'flat').

[@whyofpsi]

You are completely correct in saying that space as a whole, i.e. smoothened out is flat. The geometry of a shape is what can be defined as the intrinsic properties of that shape.

[@kayhan_qfpn]

But how do we really know space is flat? What about the theory that universe beyond what we see can be so slightly curved and if big enough, loop back around on itself?

[@anandbavkar8572]

simply I have no words for how amazing it is. Keep up the great work.. Looking forward to your video on the Math's of Curved spaces... May your channel, grow and grow.. I'm wondering how you acquire this physical intuition about "Metric Tensor"? I will appreciate any recommendation about books or some other sources for self study on Tensors or SP or GR emphasizing physical intuition... Thank you.

[@whyofpsi]

Thank you for the kind words :) The video on the in-depth mathematics will be on my other channel: "Pen and Paper Science", it has a better audience there. I've studied GR from Sean Carroll's book on the topic. The intuition about the metric tensor and Riemann tensor came from taking the time to think deeply about them :)

[@anandbavkar8572]

@@whyofpsi Thank You very much!! Immediately after looking at the content of your channel you have won a subscriber...

[@davyerni2136]

Curved spacetime is the lense effect around black holes because of the immense gravity near the black hole .

[@whyofpsi]

Exactly, the curvature of space-time can be observed by gravitational lensing of light :)

[@Jasmin-lg3gf]

Yes, mass curves 4D spacetime. But what does the curvature look like in a massless space? There are also experiments for this, and so far they have shown that the universe is flat. These experiments are completely missing here in the video.

[@whyofpsi]

It follows from the theory of General Relativity, that if there is no mass (or energy since both are equal), that space-time is indeed flat. You are completely right there. The experiments you refer to are those that show that on very large scales (thus smoothing out the distribution of mass), the universe is indeed flat.

[@Pugetwitch]

Than there's no curvature because mass is what creates it.

[@Jasmin-lg3gf]

@@Pugetwitch If I am correctly informed, then this statement is not true. (if by "it" you mean the curvature) For example, if the universe were a 3D spherical surface, then massless space should also have curvature. Pretty much every video about the shape of the universe shows this shape, although as yet we have no proof of it.

[@ritemolawbks8012]

​@@Jasmin-lg3gf Are you sure you're not confusing the 4D local curvature by mass/energy that manifest as gravity, with the 3D spatial shape of the observable universe that determine the history and ultimate outcome? In cosmology, they use General Relativity and even Newtonian gravity to determine flatness to determine whether the universe is infinite and whether there will be collapse due to gravity. Are you wondering how the observable universe could be both spatially spherical and perfectly flat?

[@Jasmin-lg3gf]

@@ritemolawbks8012 I don't confuse the two and my first question was related to the 3D spatial shape of the observable universe, because the video was only referring to the local curvature. That is why the space should be massless. "Are you wondering how the observable universe could be both spatially spherical and perfectly flat?" I would actually be interested in the answer to this question. But I think you've already given it, because it's the time dimension that's curved, while 3D space is flat in both cases. But I could also be wrong.

[@jeromeblanchet3827]

Thumbnail explanation: The manifold on the cylinder is not curved space-time because all squares are the same as cartesian space-time. To be considered as curved space-time, squares have to be distorted with elongated or narrowed sides and angles.

[@whyofpsi]

That is indeed the case :) The surface of a cylinder is not intrinsically curved! :D

[@yonishafrir]

Awesome Video!

[@whyofpsi]

Thanks! :)

[@johneonas6628]

Thank you for the video.

[@whyofpsi]

You're very welcome, I'm glad you like it. And thank you for taking the time to comment :)

[@enalaxable]

You have a helpful way of exposing quite a complex subject. Can’t wait for the connections and R curvature video. I have a keen interest in hyperbolic spaces in data analysis. What is a good book for beginners you think? ( with all mathematical depth but with intuitive explanations as you very well use)

[@whyofpsi]

The in depth video about the mathematics will be posted on my second channel: Pen and Paper Science. I think it's very interesting to have hyperbolic spaces in data analysis (being a data scientist by day myself). I don't know any books, but I can say that I learned GR from Sean Carroll's book on the topic :)

[@enalaxable]

@@whyofpsi thank you I will follow P&P Sc too. Nice to see a fellow analyst following same topic. Some say Hyp. Geom. is more spacious than Euclidean, so can pack more information(?) maybe, may natural processes seem to follow this geometry. GR is a bit too deep for me now. But as my application is with 3D data I am looking into 3-sphere manifold in quaternion (4dim) hyp. geom. as a start. I hope it is good enough.

[@whyofpsi]

@@enalaxable awesome :)

[@GimbertLane]

Wonderful. Thank you!

[@whyofpsi]

Thank *you* for taking the time to express your liking, I appreciate that a lot :)

[@kavalkid1]

Fascinating! Time dilation causes the curvature of Space Time! Of course! (temporaly speaking) Excellent video! It is so important for math naturals like yourself to find the way to communicate with those of us who are math challenged. For the first time I can sense the "beauty" of math that is so often spoken of.

[@whyofpsi]

Whoa, thanks for the super kind words! I'm super happy you were able to get a glimpse of how beautiful science and mathematics can be :) I really appreciate the comment!

[@timjohnson3913]

Dialect recently came out with a video debunking this misconception that time dilation causes spacetime curvature and thus gravity. I believe a main part of the misconception argued by Dialect is that a KZbinr many years ago mistook Minkowski space to be actual spacetime (vs a useful tool/description of spacetime), and since then, many physics KZbinrs have used the same argument/misconception.

[@whyofpsi]

Time dilation is a consequence of the absence of absolute time. It has to do with reference frames. Thus only in this roundabout way through reference frames does it link with curvature. But you are correct that time dilation does not cause curvature, it's more like the other way around.

[@karenas1701]

Great video! please notice a small typo in 17:30 for the above right equarion says "... R^2 sin^2(theta^2) dtheta" ... the dtheta should have the ^2 instead of the theta.

[@whyofpsi]

Oof, that is very well spotted! Thank you! :D

[@whyofpsi]

If you wish, you can support me on Patreon. 👉 www.patreon.com/TimoKerr A 👍also helps out tremendously (;

[@emelradjo7788]

My uni professor linked here. Nice!

[@whyofpsi]

Whoa! I'm super curious to know from what university! It's the first time I hear this :) Was the video useful for the course material?

[@jensphiliphohmann1876]

12:00 f: "Δs² = Δr² + r²Δθ²" with _finite_ coordinate differences Δr and Δθ won't work since polar the r=const. coordinate lines are curvilinear. It has to be replaced by the infinitesimal formulation "ds² = dr² + r²dθ²".

[@whyofpsi]

That is correct :) Nice catch

[@whyjaywonders]

Just amazing. Superb video. Kudos. Subscribed :)

[@whyofpsi]

Thank you very much! :)

[@vts747]

Great job!

[@whyofpsi]

Awesome to hear, thanks! :)

[@SuperGameingGuy]

Very solid video. You did however mix some things up with extrinsic and intrinsic geometry. They are merely different ways to view a surface. Extrinsic geometry is when you are considering a closed surface that is within a space. So looking at a globe or a cylinder in full is considering the extrinsic geometry. This is the easy view because we can do visual analysis to determine it’s curvature. I can determine that the sphere has positive curvature because the of 90 degree triangles, the principle curvatures, or the fact that I can’t place a piece of paper on it without crinkles. The cylinder must be flat because of the principle curvatures (if you can draw a straight line on a surface, it must be flat) or the fact you can properly deform a flat piece of paper into a cylinder Intrinsic geometry is like viewing the shape on its surface. So like the way we view the earth. This is the more abstract way of viewing things so it isn’t as easy to detect curvature. This is why parallel transport is so necessary. Parallel transport doesn’t require us to look at the shape globally; local data is all we need! Eigenchris has an amazing, more rigorous series on this subject for supplemental research. His series “tensor calculus”

[@whyofpsi]

I'll check it out. Thanks :)

[@Black-mx3hc]

So would they still embedd in another dimension

[@joshuawhitworth6456]

Just divide one by each number and plot it on a graph to see the curvature of the universe. It's that simple.

[@whyofpsi]

That would give you the infinite harmonic series, which is one dimensional, so it couldn't account for different curvature in different directions, right? And it is fixed irregardless of the matter in the universe. How would you incorporate that?

[@joshuawhitworth6456]

@@whyofpsi I made some pretty trippy art with it. Plus I also can up with a new set of numbers no one seems to know called the holographic series which lines up with the harmonic series....1, 1/2, 1/2.5, 1/3, 1/3.333, 1/3.667, 1/4, 1/4.25, 1/4.5, 1/4.75, and so on forever. I call it holographic wave progression.

[@iwara]

Great! Well done!

[@whyofpsi]

Thank you! :D

[@deletefacebook8419]

There’s a phenomenon that manifest on the surface of spheres in the universe that points towards an increased complexity. It appears to be that the notion of reactions that must occur for this shape to form results in an increasingly complex dynamic that allows for the propagation of waves. Thus, the mathematical method you just described actually falls apart at the quantum level because in order to have one sphere you must have many.

[@deletefacebook8419]

It’s that the Mandelbrot maps out the complexities of such a system. It just seems that the Mandelbrot is the description of a very hyperbolic and real spherical plane.

[@whyofpsi]

The reason why General Relativity does not adequately describe gravity at the Quantum scale is well known. However, because the theory breaks down in one regime of scale does not mean it is wrong on the scale it proved to be right. General relativity to this day is still the best description of gravity at very large scales.

[@whyofpsi]

@@deletefacebook8419 What is the Mandelbrot?

[@deletefacebook8419]

@@whyofpsi the Mandelbrot is a mathematical function of what is essentially a self replicating system that utilizes chaos in order to produce a bifurcation. Which is what you see in the galactic clusters, our veins, the trees, the sun, it’s basically extremely visible for every complex structure in the universe basically.

[@deletefacebook8419]

@@whyofpsi the bifurcation diagram also allows us to predict population collapse.

[@jameshylen8606]

Very nice video! At 12 minutes, you seem to have glossed over the fact that there are two different r, and use only one in your formula; yes they go to the same when you go to differentials, but it might confuse some viewers if they actually tried to calculate something macroscopic with it.

[@whyofpsi]

Oh, thanks for pointing that out! Well spotted!

[@mike814031]

6:50 wait but why exactly doesn't the universe need something to expand into?

[@whyofpsi]

What I try to get across there, is that just because the universe would be curved, doesn't mean there is an extra dimension that it is curved into. In the same sense that the curvature of a sphere does not require a third dimension to be curved into. It is intrinsically curved. Does this make sense?

[@Pugetwitch]

Because there is nothing yet there and it is creating what it will expand to on its own accord. The universe is literally creating space.

[@markthom7965]

Matter tells light how fast it can go as well. The more massive an object, the slower light is able to go away from it.

[@whyofpsi]

I think the wording is chosen poorly. Matter tells space how to curve, and curved space tells how light will travel. It doesn't quite influence the velocity of the light, since that is always constant.

[@markthom7965]

The speed of light is not always constant. If it It was then there would not be any black holes.

[@mathieupowell8828]

Very good!

[@whyofpsi]

Thank you! :)

[@evergreen408]

Thanks for outstanding content. One question : Intrinsic curvature means the curvature can be verified without having to zoom out. But , whether it needs a higher dimension to curve into , isn’t it a different problem ? Why should one leads to another.

[@whyofpsi]

Thank you for the kind words! :) If you can determine the curvature without having to zoom out, you can be certain the surface is curved without having to curve into another dimension. Since if it didn't, you would have to zoom out to observe it. Is this clear?

[@evergreen408]

@@whyofpsi thanks for answering my question. I am still not clear on the issue. Perhaps due to my limited knowledge. Again , Why not having to zoom out is the necessary condition for not existing an additional dimension. Why the two must be coupled. For example We can imagine flatlanders living on the surface of a sphere. They can determine the curved two dimensional surface of the sphere without leaving the surface. Then they conclude the sphere is not bent in three dimension.

[@Black-mx3hc]

@@evergreen408 technically a sphere has extrinsic curvature as there are an infinite amount of 2d curved circles that make up that sphere

[@WildEngineering]

great video, channels like yours (and mine xD) deserve more recognition.

[@whyofpsi]

Agreed!

[@PrasannaSimhaM]

Very nicely explained. I am a cardiac surgeon (I am starting mathematics from the basics once again starting with vectors and scalars) trying to learn general relativity from the basics and this really helps. Being primarily an "intuitive learner" this has helped a lot though I do realize as one progresses it becomes a sort of "This is what the maths predict" and intuition does break down but this does help understand where one is heading even if it is "warped" You had alluded to another video giving further detailed explanations but I cannot find it. Please post the link.

[@whyofpsi]

Thank you for the kind words :) The more in depth, math related videos will be on my second channel; Pen and Paper Science. It might also be helpful for you since you start from the basics again :) Thank again and good luck on your science endeavours :D

[@dougoconnor]

Amazing video

[@whyofpsi]

Thanks for taking to time to comment, I really appreciate that :)

[@hinata8511]

Your computer graphics for plotting the curved surfaces are really nice! What software do u use to generate them?

[@whyofpsi]

Thank you! I use Manim, which is a Python Library. It has a very nice and helpful community.

[@hinata8511]

@@whyofpsi Thanks for the reply! I'll check it out

[@glcpit7797]

are vector's ( scalar ) components dependings from polar coordinates ?

[@whyofpsi]

The components of a vector are indeed coordinate system dependent. If that's what you're asking? :)

[@glcpit7797]

@@whyofpsi how to espress sum of two vectors, u and v, directly by polar versor when u and v do not have the same origin ?

[@whyofpsi]

@@glcpit7797 First, you transform the vectors from cartesian coordinates to polar coordinates. This is as simple as doing matrix multiplication with the well known transformation matrices. Then, you would want to shift both vectors to the origin, which is vector addition. Then, you add them together by adding the corresponding components together. This is technical, and something I cover on my other channel: "Pen and Paper Science"

[@mrk24107]

Excellent !!

[@whyofpsi]

Thank you!! :)

[@bloodyorphan]

The reason I call spacetime curved is for one simple reason , The mass of an orbiting object is calculated with Orbital Mass = M+(M*V) which implies the orbiting mass is in "Stable Equilibrium" when changing its' direction. Instead of Collision Mass = M*V^2 which is an unstable energy reaction far more volatile than an orbital mass equation. **Einstein** PS: This was proven by JPL and NASA back in the mid to late 1980's

[@whyofpsi]

I have never heard of those types of masses. What is the normal mass called then? The collision mass as you put it, is the kinetic energy.

[@bloodyorphan]

@@whyofpsi What ? , you said you understood General Relativity ? What would an Orbital mass equation actually be trying to achieve ?? What is Inertial mass ?? You are trying to figure out how fast the ISS needs to go to stay above the Earth... Or MOND, using the radius to define spacial density at any given point in a larger mass grouping of a Galaxy. Time_dilation_calc = (Temp in degrees Celsius/5)^2 , now convert the galaxies mass at given radius and caculate its' time-dilation ratio, now multiply its' velocity by that number to understand dark matter ... **EINSTEIN 1981~present**

[@dirk_t_wachter]

Fantastic video, thank you. Is there a slight error though in that the trajectories shown on the sphere (e.g. thought experiment) are not parallel? I mean the second red leg is not drawn along a latitude i.e. its angle from the first red leg is not 90°, while the blue turn is clearly a right angle on the surface of the sphere. It does not alter the argument, but I did find it confusing the way it is drawn... The lines will still not end up in the same point, but they will intersect rather than miss each other.

[@whyofpsi]

That's interesting. Would you be able to say that the spacial curvature of the earth can be neglected in the entire thought experiment, since it's locally flat. This does not change the experiment.

[@wulerhaufung9468]

The second red line will not follow a latitude but instead a big circle whose center coincides with the center of the sphere. The only lines that will follow a latitude are those on the equator.

[@dirk_t_wachter]

@@whyofpsi Lol, of course not 😀 Maybe I didn't make clear what I meant by this, or I am misinterpreting the animations (?). Unfortunately I can't paste images or screen shots here, but... The red vs the blue trajectories, as first shown in 2D (flat plane & cylinder) and then in the 3D space (sphere). The turns are all supposed to be 90° for simplicity, or at least the right turn (blue) and the left turn (red) should be the same angle regardless of dimension?

[@dirk_t_wachter]

@@wulerhaufung9468 Yes, that makes sense. From the perspective of the blue vs the red traveller though, and to compare the two equal-length paths, the second red leg could have been chosen parallel (i.e. latitudes are parallel, big circles are not) to the first blue leg. Both travellers start out perpendicular to each other, travel the same distance, then turn the same angle (90°), then travel another identical distance, but unlike in flat space, they end up in a different place. That's what is being illustrated

[@wulerhaufung9468]

@@dirk_t_wachter The point that this thought experiment wants to make is about the role that 'turning 90 degree' part plays in these actions. Following the latitude after the first section of red's trajectory is not turning 90 degrees and then going straight. And that's why the author went through all the troubles talking about vectors and tensors.

[@poetmaggie1]

I understand right everything you’re telling us is provable with mathematics, but there’s no way to prove it physically.

[@whyofpsi]

There have been experiments where the curvature of space-time was implicitly measured. Also, the GPS system gets its accuracy from taking curvature into account :)

[@danieljuneau229]

This explains well how coordinates work with traveling a curved line instead of a straight line. This doesn’t prove to me that “space is curved”. If you travel from earth on a spaceship in a “straight line”, you aren’t actually traveling in a straight line because you are already traveling in another direction that the earth already set you in motion. Therefore you will be traveling in at least two directions causing a curved path. Other objects in space are also never traveling in just a straight line.

[@whyofpsi]

You raise an important point. It is true that when you leave the earth perpendicular to its surface, you will travel on a curved line, compared to someone not on earth. However, the experiment at the end is more of a thought experiment, where you can imagine the earth not spinning, and not round. These are approximations that are valid in specific regimes, for example when you consider small distances. As for the rotation of the earth, we can keep it out of the picture, since it does not change the curvature of space-time in the presence of its mass. Does this resolve questions you had?

[@danieljuneau229]

@@whyofpsi even taking out the spinning direction, you will be traveling in what ever direction the earth is traveling. If a baseball is traveling left from your point of view and then it gave itself a force upwards, like a rocket, it would go in a curved path up and left. Nothing in space can travel in a straight line because all objects are initially traveling in multiple directions causing a curved path. All objects are also traveling at different velocities of each Cartesian coordinate. This all explains the orbits and timing of the planets aka… “space-time”. Frame of reference/relativity play a role in this also. Saying that space can curve, which is a non physical object doesn’t make sense to me. This video does explain well where an object would end up following a certain curved path.

[@whyofpsi]

@@danieljuneau229 Ah, I see what you mean. The fact is that when talking about curvature of space-time, special-relativity is still valid. This means that you do not need to distinguish between inertial reference frames, these are observers traveling at constant speed compared to each other. You are correct that there is indeed no preferred speed or direction in the universe. This, however, does not mean you cannot distinguish a curved from a straight path through space-time. If a path is curved for a specific inertial reference frame, it will be curved for ALL inertial reference frames. This is a very important point: Curvature, as defined by the Riemann tensor, is NOT dependent on the coordinate system. A triangle on a sphere will NEVER have a sum of angles of 180 degrees, whatever coordinate system you choose.

[@danieljuneau229]

@@whyofpsi you cannot distinguish between an actual curved or straight line object’s path. That’s absolutely based on frame of reference. In the scenario with the ball, let’s say that is earth. A rocket shoots straight up from the ball/earth. If you are a person on earth/ball, you will see a straight path. Remember the ball/earth is flying left. Let’s say another ball/planet is flying right. If the planet flying right can see the rocket, then from their frame of reference the rocket is on a curved path. As far as the triangle on a sphere, it can’t actually be defined as a triangle because a triangle requires straight lines. A “triangle” on a sphere has more than three angles, it’s just the focus is on the three more sharp abrupt angles.

[@Fkd-About-Found-Out]

in 4D spacetime you need 1 line of causality going out from the sun to represent the start and stop point to an orbit, if you mark that arbitrary straight line out from the sun it has one line of continuity, in a 3d world flattened out to a cone you need 2, in 2D space on a map you will see you need 4, if you look at a polar orbit each time the line changes direction is where causality bumps it from this perspective, it looks like it goes up, then left, then down then right, or vice versa, but its always doing that 360 degree journey in spacetime in 1D space you need 8 lines of causality, so it does suggest in 5D you need 0.5 lines or maybe none... but the 3D world around us is conical, not spherical. it works so nicely, flattening out those straight line orbits in space you see how it all works, your resting point when you arent moving is trying to pull you into your orbit as it should be in spacetime, so you feel gravity, if you was at the centre of the earth any mass you tossed would fly up, not down as your terminal velocity at that point in curved spacetime its much shorter of an orbit.. the cone is gravitational potential map of local spacetime, you could do this for any large body, just using its radius and circumference, even the observable universe

[@eigentensor]

Is the background made with Chaotica?

[@whyofpsi]

The space background was taken from another video, which had no copy-right attached to it ;)

[@eigentensor]

@@whyofpsi ah ok, thanks for letting me know. I'm the author so always trying to find it in the wild ;) Cool channel!

[@Jaggerbush]

New channel? I recognize your voice from your other channel.

[@whyofpsi]

What a coincidence! Do you mean the Pen And Paper Science channel, because that's me :)

[@Jaggerbush]

Awesome!!! I love longer content - this is great- keep ‘em coming.

[@whyofpsi]

@@Jaggerbush Thank you, I will! There is indeed an audience for long-format videos :)

[@bbk1984]

Definitely brings me sleep.. Thanks

[@whyofpsi]

Haha, you're welcome :)

[@hooked4215]

13:49 We can consider infinitesimal distances only in the case that space is not quantized.

[@whyofpsi]

It is still an ongoing and extraordinary interesting problem whether space(time) itself is quantised or not! :D

[@whyofpsi]

Any feedback is welcome! Expect more videos in the future! :) ⏱ Timestamps: 0:00 Intro 1:26 Part I: The Intuitive Explanation 7:10 Parallel Transport 9:43 Part II: The Mathematical Explanation 23:02 The Riemann Tensor 26:20 Part III: Example of Curved Spacetime. 🚀 Hidden away: If you want, you can always support me via Patreon: www.patreon.com/TimoKerr

[@user-lb8qx8yl8k]

It's a great video!!

[@whyofpsi]

Thanks! :) I'm glad you liked it. Something specific that was unclear?

[@MusicEngineeer]

Very well done content. I sometimes find the background music mildly distracting though, especially, when it contains a lot of blings and blongs (i.e. attention grabbing events). I'm Looking forward to the video with the "daunting" math! I'm sure, you'll make it accessible!

[@whyofpsi]

Thanks for the feedback! The choice of background music, or whether to have any at all, was a hard one.

[@MarkusBHZ]

I'd cut the lofi soundtrack. Personally I find it quite mundane and annoying to be honest. Your video is amazing and perhaps needs no soundtrack at all.

[@tedsheridan8725]

I have one question that's been bugging me. It's clear that intrinsic and extrinsic curvature have different properties. But both the cylinder and the sphere require a 3rd dimension to 'curve into', regardless of any test of the type of curvature (parallel transport, sums of triangle angles, etc.) So why is it that you say that the curvature of space-time (4D), which is intrinsic, does not require a 5th dimension to curve into? Wouldn't that be like saying you could have something like a sphere - which has intrinsic curvature - existing purely in 2D space, i.e. not curved into a 3rd dimension?

[@whyofpsi]

The tricky part of intrinsic curvature is that the curvature is defined without ever having to reference an extra dimension. If something can exist and exhibit properties without needing the reference of an extra dimension, then that dimension is strictly not needed. It's hard to think about, because we always operate in the 3D world, but a 2D sphere is curved even if no third dimension exists.

[@tedsheridan8725]

@@whyofpsi So you're saying that the 2-sphere does not require a 3rd dimension to exist. It's just a collection of curved lines in the plane, correct? Does it still have a front and back, like the two sides of the plane? And also, couldn't you say the same thing about the cylinder - that it doesn't require a 3rd dimension to exist?

[@whyofpsi]

@@tedsheridan8725 I am saying that you can define, and thus measure, the curvature of the surface of a sphere without there having to exist a third dimension. On the surface of a cylinder, you will never be able to measure its curvature, unless you zoom out, which DOES require a third dimension :D

[@tedsheridan8725]

@@whyofpsi Ok - That makes more sense, thanks. So in the case of curved spacetime, we can measure curvature without REQUIREING a 5th dimension. But there is nothing to suggest that there ISN'T a 5th dimension right? Just as for a sphere it's quite - I'd say more - natural to think of it as embedded in 3 dimensions, shouldn't it be an open question as to whether spacetime is embedded in a 5th dimension or not? Or is it just a philosophical point that can't be answered?

[@whyofpsi]

@@tedsheridan8725 That is more of a pragmatic idea: adding extra dimensions where they are not explicitly needed to explain behaviour is just not something you'd want to do. It's unnecessary complexity. :) In a sense, the way you want to go about adding an extra dimension is to look at an unexplained observation in the literature, and try to explain it using a fifth dimension. Of course, it all has to be very rigorous :)

[@iABabyCuh]

Thumbnails always get me😂

[@whyofpsi]

Do you feel it was misleading? I'm still learning the thumbnail game. Also, did you nonetheless learn something from the video? :D

[@iABabyCuh]

@@whyofpsi No, I actually think your thumbnails are both awesome and funny. And also I didn’t watch the entire thing yet because I usually listen to these things on the long drive home from work. I’ll make sure to comment on it later.

[@TheJara123]

No more new videos? Please post more videos about GRT, Gravitational waves etc...

[@whyofpsi]

I have been super busy with other projects (and a full time job as a data scientist). One of these days my course on statistics launches, then I will have more time again and might start back into making these videos :) Thanks :D

[@TheJara123]

@@whyofpsi I understand, course on statistics? Great!! But I am more eager to see your take on GRT, GWave!!

[@kordta]

It is like overtaking on wide long turn. The inner lane vehicle moves slower but looks like you have to drive really fast to catch up with it. It has nothing to do with time but with distance itself. The rocket that first travelled further away needs more speed to catch up with the one near the surface. Then the lower orbit rocket needs exactly the same time to climb to the above trajectory as the other one. So the one above will have more distance to travel in order to reach the crossing point. Since the speed of both rockets is the same... I don't really get it what it has to do with time dilation.

[@whyofpsi]

I should have put this more clearly in the video. For the rockets, we don't consider the curvature of the earth's surface. It can be done on a flat surface just as well, with the same result. If we do take into account the curvature of the earth's surface, we would get a difference that is larger than what we would expect when only considering the curvature of the ground.

[@WildGamez]

Brilliant

[@whyofpsi]

Thanks, I'm really glad you enjoyed it :)

[@nicolascalandruccio]

If the radius of the cylinder is 1, it's exactly a piece of paper. Is it normal?

[@whyofpsi]

If the radius is indeed 1, then it would be a piece of paper with one dimension being pi. The other dimension being the height of the cylinder. Is this what you mean?

[@nicolascalandruccio]

@@whyofpsi Hi, I meant if the radius is 1, the metric tensor is the Kronecker delta. Therefore, it seems flat while it is not. So, there is something wrong or I don't understand.

[@whyofpsi]

@@nicolascalandruccio You are completely correct. For a cylinder you can indeed find a Kronecker delta metric, which indicates that it is indeed flat; it's extrinsically curved! It's curved only because it is folded into an extra dimension, but if you were tiny and living on its surface, you wouldn't be able to tell :)

[@putrid.p]

I don't understand how intrinsically curved 3D space doesn't need a 4th spatial dimension to exist, whilst an intrinsically curved 2D surface (such as a sphere) needs the 3rd dimension.

[@danielvolinski8319]

9:00 You are moving the vector in the sphere but it is not parallel to itself! How can you call this "parallel transport" when the vector does not remain parallel?

[@whyofpsi]

This is indeed a very delicate point. I think the confusion comes from the fact that here, we are dealing with "parallel" on a 2D surface, not in 3D space. What's important is that at each step along the trajectory, you construct a tangent space to the sphere's surface, and then compare the direction of the vector *within this tangent space* to the previous one. I'm not sure it this is clear now. The difficulty arises from the fact that parallel transport on a 2D spherical surface requires 3D to visualise. Though it's important to remember that theoretically, there need be no mention of a third space in which the sphere is embedded. :)

[@danielvolinski8319]

@@whyofpsi It's not clear at all. Why do you need a tangent space if the original vector is not necessarily on this tangent space?. Let me ask you this, once you have the first vector at the origin of the trajectory, how do you obtain the second vector on a nearby point in the trajectory; what is the procedure?

[@timjohnson3913]

Keeping the vector parallel means that you are not rotating the vector (in comparison to its surface) as you move it along the surface. It seems you are suggesting one should keep the vector parallel to the original vector. What you suggest is simply not parallel transport as it would require you to rotate the moving vector as it travels along its curved surface. Something that might be helpful: cut two cardboard arrows out and tape one to a wall. Now orient the 2nd arrow on top of the first and move it around at right angles, eventually returning it back on top of the original arrow. The arrows will always end up oriented in the same direction. Now do the same thing, but on a basketball.

[@pseudolullus]

@@danielvolinski8319 It is tangential because it needs to remain on the 2D surface. To me it is a bit easier to understand if you imagine at the same time the movement on a flat 2D surface. It is as if the sphere skipped a whole side, as the parallel vector on the third side of the curved triangle is pointing as it would in on the third side of the flat square (the same thing happens with the 2nd side).

[@danielvolinski8319]

Thanks to Science Intuition, Tim Johnson and potsdamcsc for your input, I really appreciate it. However, I understand Mathematics in terms of Computer Algebra System (CAS) so I am trying to mimic the animation of Parallel Transport on the sphere expressed on spherical coordinates (for Physicists). S=[R*sin(θ)*cos(φ),R*sin(θ)*sin(φ),R*cos(θ)] with 0≤θ≤π, 0≤φ≤2*π and R=2. The curve is C=[R,½*π*t,0] with 0≤t≤1 from point [R,0,0] (North pole) to point [R,½*π,0] (equator). The vector to be Parallel Transported starts at [R,0,0] and ends at [R+1,1,1]. Now I want to calculate the next vector which start at [R,π/100,0], how do I calculate the end of this vector?

[@DougShanahanMusic]

The intrinsic/extrinsic explanation was a bit strange. The sphere is intrinsically curved and so we do not need to zoom out to a 3rd dimension to see it, but that doesn't mean the 2d surface isn't curved into a 3rd dimension. We need 3 dimensions for a sphere, but you then say that because our universe is intrinsically curved, it doesn't require another dimension to be curved into. That doesn't make sense. It's one thing to say that being intrinsically curved doesn't require us to zoom out to a higher dimension to recognize the curvature, and it's another to say that curvature doesn't require a higher dimension in order to be curved.

[@whyofpsi]

I'd disagree, it is *really* the case that the surface of a sphere does not need a third dimension to exist. Though I could be wrong, but thanks for making me think about it :)

[@thesarariman]

Does the plane-sheet of paper really apply? The plane is infinite, it makes no sense to say that you put the edges toghether....

[@whyofpsi]

That's a good point. However, the analogy between a sheet of paper and a cylinder is the fact that they both have the same geometrical features. They do have different topological features (you can go straight and retrace your steps on a cylinder but not on a plane). Sorry for the confusion :)

[@MultiChrisjb]

So intriguing... I've never even heard of the word "inependent" before at 14:41. Do you have a reference or definition for it? I can't find it anywhere else.

[@whyofpsi]

It meant to spell "Independent", thanks for pointing it out :)

[@SaeedZ91]

Is time actually the density of space net of whatever it's made of?!

[@whyofpsi]

In science, generally you'd want to be precise and rigorous. What do you mean with the density of space? And what do you mean with "What space is made of? :)

[@SaeedZ91]

@@whyofpsi Curvature can be imagined in 2D space as you illustrated in your video. But in 3D space, can we say that curvature means that in some places space is denser? or we should say that space curves into the 4th dimension of space (not time) which we can't imagine it? And about the space's building blocks I meant that it should be made of something. So where it's denser (near the heavy objects) you move through more space (which is pressed into smaller length) and therefore you think your time is slower compared to others in less pressed space. And if it's deeper than that, we really need to smoke some pot before discussing it :)

[@Black-mx3hc]

This is a great video, tho I'm still a little confuse... But let me get this Technically this 2d surface is on a 3d object So like could we assume that our 3d universe as the 2d surface of the sphere and the the 4d spacetime as the 3d sphere

[@whyofpsi]

Thanks I'm glad you liked it :) The important thing is to grasp that a space can be curved without having to be "embedded" in a larger dimension. The 2d surface can be curved without ever speaking about a 3rd dimension. Our 4d space-time is curved, but that doesn't mean there is a fifth dimension in which it is curved. It's important also, to understand that you can only really separate 4d space-time into 3d space and 1d time for observers that are stationary with respect to each other. But yes, if we would imagine 1d space (a line) and 1d time, and say that this 2d space-time is curved, that would be analogous to the 2d sphere.

[@paeon21]

26:20 Doesn't this thought experiment only show that the two people are traveling relative to a spherical surface (i.e. the earth), but not necessarily that spacetime is curved? In other words, I assume that the curvature of spacetime should not be dependent on whether you're traveling on a sphere, like the earth, or a cube, hypothetically speaking. However, if we alter your thought experiment to two people traveling around one face of an imaginary cube-shaped planet, then the results of the thought experiment would seem to "show" that spacetime is flat just because the relative travel paths are relative to a flat surface of a 3D object in 4D spacetime? Or am I making too much of thought experiments?

[@whyofpsi]

That's the amazing thing, in this experiment you don't need to take into account that the earth's surface is curved. Like you say, they will arrive at different times even if the experiment is done over a flat surface. :)

[@carlosmana3806]

Great tutorial, error in Curvature;connection symbol; num 2.

[@whyofpsi]

Thanks for pointing that out! :)

[@fsysfreddy]

You have chosen a curved path by fling parallel to the earth, by doing so you didn’t have to calculate for curved space! Try to fly in strait lines in space and see if it is curved!

[@innertubez]

Wait, isn’t the surface of the earth intrinsically curved? Doesn’t that allow us to disprove the flat earth theory simple by testing the two 90 degree turns trip and checking whether we have arrived at our starting point?

[@whyofpsi]

Exactly :)

[@peterdamen2161]

You can also disprove the flat earth theory by e.g. showing that according to the flat earth theory the distance between Cape Town (South Africa) to Buenos Aires (Argentina) is much larger than it really is.......

[@tomusic8887]

Yes but what curves? After the video i say its time not space....but how does the apple/planet etc knows that when it falls?...still a bit of a mystery

[@jnsimhan]

Great video, arguably the best, explaining space-time curvature with such clarity and in such a simple way. Thank you. Subscribed immediately after seeing the video. Question: in the thought experiment at the end you explain why when moving closer to the earth the rocket ship would travel further horizontally. However, isn’t the fact that the earth’s gravity slows time predicated upon space-time being curved? Is this a circular argument, or am I missing something? Circular because time slowing down close to a heavy mass is based upon space-time curvature, and showing that space-time is curved (in this thought experiment) requires that time be slowed by the heavy mass.

[@whyofpsi]

Thank you for the kind words. I like your thinking, and to be honest, you are right :) Thanks for that :))

[@istvansipos9940]

great

[@whyofpsi]

Thanks :D