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