This video is sponsored by brilliant In our everyday lives we experience and observe three spatial dimensions We can move left to right up and down and then forward and back If you wanted to tell me to meet you somewhere even outside The earth three numbers is all I need but this isn't quite complete because if I wanted to meet you for lunch Let's say I would need three numbers like a longitude a latitude and the floor to meet you on but I'd also need a time Because we need this fourth value to completely specify some event in the universe. We consider time to be the fourth dimension Now time is not a spatial dimension that extends beyond height length and width but rather it's called a temporal dimension It doesn't behave like the other dimensions and we definitely can't point in the direction of time Let's say since we're embedded in 3d space But in simplest terms since we need that fourth value of time to specify an event. We consider to be the fourth dimension Now we used to think space and time were totally independent, but then Einstein came along and found that's not the case They were in fact linked together and the mathematical model that fuses these two things became known as space-time What Einstein's general theory of relativity says is that massive objects curved space-time? This means that distances between points and also times between events are warped around massive objects and this has strange consequences like time ticking differently on different planets and massless photons being bent by massive objects because of that curvature The equations that describe all this can definitely look intimidating But really what you see is on one side of this equation is math regarding the geometry or curvature of space-time And on the other side we have an expression has to do with mass and energy The famous saying by physicist John Wheeler is that space-time tells matter how to move and matter tells space-time how to curve Now in order to understand more of the underlying mathematics behind how our universe really works We need take a closer look at the word curvature Because in general we tend to think that something is curved if it's simply not flat So for example, most people would tell you this is curved but this is not However, there's more than one way to define curvature and something we're going to investigate. More of is Gaussian curvature Which might not be exactly what you'd expect For example when it comes to Gaussian curvature this and this are the exact same or something I'm sure many of you guys know is that you cannot have a world map that? Perfectly represents a globe without some distortion and the reason behind this has to do with Gaussian curvature But moving to the actual math a piece of paper has zero Gaussian curvature, which is probably no surprise, but why is that? It's because at any point if I draw a little line segment in any direction, it's completely flat. So we say that has zero curvature Then if I draw another segment perpendicular we see the same thing zero curvature and when we multiply them we of course get 0 which is the Gaussian curvature at that point and that would apply to anywhere on the piece of paper, however things change when we look at a sphere because now at any point if I draw a little segment It's not easy to tell but that segment has curved outwards just a little bit and that outward curvature we say is positive Gaussian curvature And if I do the same thing? Perpendicular that segment has curved outwards as well So since we have a positive times a positive Then we say at that point there is positive Gaussian curvature And again the same would apply to any point on the sphere now for negative curvature I really couldn't find much around my apartment, but I think my electric shaver will do the trick right at this point not everywhere But right here If I draw a little segment here, it curved outwards gets that positive curvature But if we go this other way perpendicular to it It curves inwards so negative so that positive times negative curvature means at that point there's a negative Gaussian curvature overall a More general surface that has that negative Gaussian curvature would be something like a saddle shape or even the inside of a torus But now how do we figure this out for something like a cylinder? For example? Because that any given point there are several segments I could draw through it that all curve a little differently So what's the official way to figure this out? Well if I want to find the Gaussian curvature at that point I need to find the segment that goes through it with the most curvature and the moment the least and then multiply those together So in this case, this is the segment with the most curvature curse outward just a little so it's got that positive curvature Then this segment the one perpendicular to it has the least curvature In fact, this didn't curve at all just went straight down the side. So it has zero curvature That means we have positive times zero curvature which gives us zero Gaussian curvature at that point But the same thing could be said about any point on the lateral surface of the cylinder So yes, the cylinder is curved but it's Gaussian curvature is zero So now we're ready for a theorem I found really interesting and that's the theorem agree geom which was discovered by Gauss and what this theorem says Pretty much is that if you smoothly deform a surface its gaussian curvature at any point is not going to change so because this has zero Gaussian curvature The theorem says that this at every point has zero Gaussian curvature as well Which isn't too hard to see if we take any point Let's say right at the top any point there and find the gaussian curvature Well, yes one segment right here does curve has positive curvature But remember we have to find the segment with the least curvature and that would be this one here Which doesn't curve at all it's just a straight segment so that zero times positive curvature leaves us with zero Gaussian curvature overall at that point it didn't change and the same goes for any point on the surface So mathematically it now makes a little more sense why I can wrap a piece of paper Around a cylinder without there being any wrinkles or anything like that because we saw this is zero Gaussian curvature everywhere on the lateral surface and when I deform this it's zero Gaussian curvature is not going to change at any point and As it smoothly maps onto the cylinder with no wrinkles or distortion or anything like that However, we saw that the sphere has positive Gaussian curvature and Because this must maintain its zero curvature as I deform it Then I cannot wrap a piece of paper around the sphere and the masses if I were to try it's guaranteed to crumple and If this were I don't know a map and I tried to wrap it around a globe then work guaranteed to have Distortion and If I try to give a piece of paper negative curvature, you see the same thing happen It's basically I have to give it positive curvature here, but then we got to bend it up this way Which if you try it's just it's not gonna happen without it crumpling just a little bit Okay, now that we have some background with that it's time to talk about people living on these surfaces And by the way, we're assuming that these surfaces are all very large Now to a small creature living on these surfaces Whether it's a giant sphere a flat plane or a saddle the area around you will appear to be flat and two-dimensional So my question is even though this is the case. Can we determine we're on a curved surface without leaving it? Like could we determine that the earth is not flat? Without going into space or going all the way around The answer is yes, but especially for this first explanation you need very precise equipment But let's assume the earth is a perfect sphere to figure that. In fact is not flat One thing you could do is draw a circle on that surface of a known radius then from there You'd measure the distance around the circle or circumference and if the surface is curved You'd notice that the circumference does not have a length of 2 pi R as it should It actually be a little smaller if we're on a sphere Because remember in your eyes You just walked in a perfectly straight line to trace out or measure that radius, but in reality you curve just a little That means the circle you drew would lie on a plane Slightly inside the sphere and when those have a radius slightly smaller than the distance you walked Meaning the circumference will be 2 pi times a value slightly less than that red R which you thought was a flat line segment and from this you could conclude that you're on a surface with a positive curvature at least locally If on the other hand, the circumference was greater than 2 pi or you could conclude maybe you're on some saddle shape with negative curvature Now yes, this would be a difficult task on such a large non Perfect sphere like Earth, but there is more we can do without going around the entire thing or leaving it Because one of the biggest differences between curved surfaces is what you'll encounter when moving parallel to someone else So if you and your friends start walking parallel to each other on a flat surface, you'd never get any closer or further apart But if you're on a sphere and you and your friends start walking perfectly straight you think you're walking parallel paths at first But you'd slowly notice you and your friend getting closer together until your paths would eventually cross Even though you thought you're walking straight the entire time and this is something that happens when you have that positive curvature For another thing you'd see on a flat surface is that if you started walking perfectly straight then made a right turn Walk the same distance and repeated this it would take three turns before you got back to your original spot But if you're on a sphere it'd be possible for you to make two right angle turns before returning to your original position This wouldn't always be the case, of course, but it is true that on a positively curved surface The internal angles of triangles always add to more than 180 degrees So if we go to brilliant sight the secret to a problem like this Where you have to find the sum of all interior angles is that longitudinal lines are always perpendicular to the equator Which is why they appear parallel at least when they're very close then since they meet at the top you have a triangle whose angles will definitely be greater than 180 and that excess will depend on That top angle in this case Negatively curved surfaces on the other hand have internal angles that sum to less than 180 Which means if we want to see whether our universe is open closed or flat aka doesn't have negative positive or zero curvature we could make a big triangle and measure the internal angle since that directly relates to that intrinsic curvature of our universe and scientists have done this before by looking at the Cosmic Microwave Background and they found that the universe is mostly Flat as in parallel lines behave as expected if there's no curvature kind of the least exciting answer unfortunately But on top of the small margin of error the exact shape of our universe and its global topology are still up for debate So we found that geometry and spaces that aren't flat or non Euclidean geometry. It's really weird, but there's still more we haven't Like in flat space we know the shortest distance between two points is a straight line However on a curved surface, we usually don't have straight lines so we called the shortest path and point A to point B along a surface a geodesic as an example on a sphere the shortest path between two points or the geodesic is always part of a great circle a Great circle is basically the largest circle you can draw around a sphere So what you see here or the equator would be examples of that You can think of geodesics as the paths we perceive to be straight as we walk along a surface at least when that surface is Very large like if you went outside and started walking perfectly straight right now in any direction Assuming an ideal sphere you'd be walking along some great circle of the earth These are also what we see for flight paths planes are traveling along geodesics If you want to fly from New York to Madrid You would not fly along the latitude line as that's part of a small circle Which is not a geodesic and therefore not the shortest path You'd instead take a slightly different path, which is part of a great circle But on a cylinder for example a geodesic looks more like a spiral in fact I know this is a geodesic because if you were to unroll a cylinder back to a flat plane The geodesic should remain the shortest path aka a straight line But to relate this back to the universe you first need to know that light Naturally travels the path not of shortest distance but of shortest time When light travels from air to a new medium like glass we notice it will refract bending at a different angle The reason is because that is the path that will make light go from point A to point B the fastest Light doesn't travel as fast and glass compared to air. So although this may be the shortest path It doesn't minimize time because in this case the light spends more time in the glass where it's moving slower Now when light moves through the vacuum of space to minimize time it travels along geodesic curves, which are typically straight lines but around massive objects light bends and Because light follows geodesics we can say that mass bends those geodesic lines or really we can say that space is curved simply put if geodesics are not lines and there's some indication that the space is curved and This was a huge breakthrough improving that general ativy was in fact Correct. Seeing that it wasn't just things with mass that were affected by gravity Okay, so far we've gone over Gaussian curvature geodesics and some non Euclidean geometry But we need one more thing. We need a way to calculate the distance between nearby points on curved surfaces because when it comes to a flat plane This is really easy to do just give me any two points and I can use the Pythagorean theorem to find the distance between them But if we want to be able to do this on curved surfaces, we need something else and that brings us to the metric tensor The metric tensor was discovered by a mathematician named dream on and it's this Mathematical tool that allowed Einstein to complete his general theory of relativity Now if we have a flat plane and a vector going from the origin to 3 comma 4 which will dis label with this notation Finding the length of that vector isn't very hard We know that X component is 3 which I labeled Delta X and the y component is 4 which I'll label Delta Y Now I'm just going to deal with distances squared here. So I don't have to have any square roots, but this doesn't change anything Anyways, we all know the general solution to find that distance squared is Delta x squared times a constant plus another constant times Delta X Delta Y Plus another constant times Delta Y Delta X plus another constant times the change in Y squared Ok, I know that was probably frustrating since of course The distance squared is just Delta x squared plus Delta Y squared the good old Pythagorean theorem Which would tell us that the distance squared is 25 or the distance is 5 but the equation I put is still technically correct so long as we assign values of 1 0 0 and 1 to the constants respectively, but ok. Why did I add those terms in? It's because the Pythagorean theorem only works in a flat space when you define your basis vectors, very nicely For this example, these were our basis vectors Which I'll label a 1 and E 2 just the unit vector in the X and y direction so when I said 3 comma 4 was the vector that just meant we take 3 e ones combined with 4 e 2 s And find the distance from the tail of the first to the tip of the last But now what if instead we use these as our basis vectors to describe any point Because now that same coordinate would be considered 1/2 e 1 + 2 e 2 Now if you want to find the distance the Pythagorean theorem will not work 1/2 squared + 2 squared won't get us even close to 25 But if we use the general equation from before we can make it work The Delta X and Delta Y are really just II 1 and E 2 you see here, so I'll just plug those in Yeah, it's not very good notation. Since it's not really a Delta X or Delta Y, but I'm just going to keep those consistent ok, now I wasn't going to do this but for anyone who just needs to know where this almost Pythagorean theorem looking equation comes from Here's a quick explanation When we have a vector like 3 comma 4 just like we saw earlier Which I'll write with I hat and J hat components now To find the length squared we can take the dot product of the vector with itself to do this Most of you guys know we multiply the X or I hat terms together and then add that to the Y or J hat terms multiplied together yielding a distance squared of 25 just like before and When we have technically the same thing with those strange basis vectors now to find the distance squared we still do the dot product But it will look a little different at first Because technically to do the dot product we have to essentially foil Making it 1/2 e 1 dot 1/2 e 1 the first terms plus 1/2 e 1 dot 2 e 2 the outer terms then the same thing with the inner terms and then the last This is how we find the vector length squared and know that these are dot products not multiplication symbols In fact, this is what we did beforehand. The three times three came from 3i 3i And sixteen came from four J dot four J But we forgot the outer terms of 3i for J. And the inner terms of for J dot 3i It's just that since the I hat and J hat or X and y direction are at right angles to each other When we do a dot product those terms would just come out to 0 So in summary the 1/2 and 1/2 or 1/2 and 2 and so on correspond to the Delta X and Delta Y whereas the actual dot product of e 1 and e or e1 and e2 correspond to the constants I'll put more detailed links down below, but just to get to the point These are the constants that make everything work for this choice in basis vectors You can go ahead and do the math, but it will come out to 25 Those constants of 16 2 2 and 17 force I'll organize a little better in a matrix because this here is our metric tensor and What we saw before with the Pythagorean theorem where the constants were 1 0 is 0 1. Well, that's also a metric tensor Now this is kind of a boring example because both metric tensors dealt with flat space but it showed the general idea and that is when the Pythagorean theorem fails whether it be for weird basis vectors or More importantly those right triangles being on curved surfaces The metric tensor makes the necessary Corrections allowing us to find those distances and fully describe the curvature of that surface You'll find the metric tensor is typically written with the letter G and some subscripts in order to describe higher dimensional surfaces we just expand the matrix for three-dimensional space the metric tensor would look like this, but we can keep going as For four-dimensional space-time, it looks like this 16 numbers which is really just 10 because of the symmetry these 1/2 is all we need at each point in four-dimensional space to completely describe how that space is curved so Again this identity matrix represents flat space where there is no curvature as this is where the Pythagorean theorem holds but this extends to higher Dimensions where this 3 by 3 identity matrix just leads to the distance formula in three dimensional flat space And although it's not really a topic for this video. We can see something similar in the Minkowski metric It's not exactly the identity but it is used for flat Euclidean space time in four dimensions And it encompasses the postulates of special relativity like the speed of light being the same in all reference frames But when it comes to general relativity space time is now curved leading to a more complex metric Although that is of course beyond a video like this in Einstein's equations We saw earlier on the side that represents the curvature of space-time. We see the metric tensor that help make that possible Now once general relativity was accepted in the scientific community. It completely changed the way we understood gravity You may have heard before that gravity is in fact, not a force and this was a result of general relativity We used to think of gravity is some magical force between two objects that happens over a distance But now we say there is no force. It's just the curvature of space-time So like when you throw a ball through the air, we like to think it's accelerating downward due to the force of gravity But Einstein said this is the wrong way to look at it. In fact, since there's no force We can say the object is moving along a straight line in space-time Well, this may sound crazy This comes from the fact that flat space is being warped by the mass of the earth Curving space-time around it. The parabolic path. We see is just an illusion because we can't perceive that underlying curvature So hopefully you're seeing that the mathematics or our universe runs pretty deep and of course we haven't even scratched the surface But what about those mind-bending theories regarding our universe and what goes beyond it? Like what about the math? That only works in 26 dimensions or the shapes that theoretical physicists think could describe higher dimensions of space-time Well, all that stuff is coming in the next video But if you made it this far you probably enjoy this kind of content and if you're looking to gain a deeper understanding of what? We covered here. I highly recommend checking out brilliant org who I'd like to thank for sponsoring this video Brilliant is an educational platform that hosts over 50 interactive math and science courses They not only teach you the technical information. You need to know for the various subjects, but they also challenge you with interactive practice problems Programming and just in general more hands-on approach. So you have a real fundamental understanding of what you're learning If you're like me and really enjoyed the unique applications of mathematics, then you won't be disappointed by what they have to offer If you like the topics in this video, for example, then their astronomy course would definitely be something you'd enjoy It starts off by going over the essential technical information to get you started with astronomy and astrophysics topics But then it goes more into what you saw earlier such as geodesics Services with different curvature and the mathematics of non Euclidean geometry as it pertains to the shape of the universe Then they go much further covering dark matter dark energy black holes and more if you're looking to understand more of the strange mystery It's about our universe. I highly recommend checking this out and they're adding new courses all the time like recently they came out with a puzzle science course where anyone can develop foundational physics knowledge while playing around with fun interactive puzzles and For the more advanced people they have things like differential equations courses that go beyond what I even saw my college curriculum So there's something for everyone to learn So if you want to get started right now and support the channel You can click the link below or go to brilliant org slash major prep to get 20% off your annual premium Subscription and with that I'm gonna end that video there If you guys enjoyed be sure to LIKE and subscribe Don't forget to follow me on Twitter and join the major Facebook group for updates on everything hit the bell if you're not being notified and I'll see you all in the next video
