Video Transcript: The Curvature of Space

Gravity was first understood mathematically by Isaac Newton in the 1500s and he had a very simple mathematical relationship for the gravity between two objects.  And that is such a powerful, simple mathematical relationship, we still teach it in schools and university.  General relativity is a more sophisticated model that actually predicts how things will behave even more accurately.  And that model is based on the idea of spacetime, we hinted at a little bit before, this idea that space and time are woven together into this quantity we call spacetime, but in particular, that gravity arises through the warping of this spacetime.  

So here's kind of the idea.  The idea is that we're used to Newton's idea of gravity, which is, you know, I have some mass, you have some mass, and things that have mass, well, they pull towards each other.  And that's why there's gravity between the Earth and the Moon, for example.  In general relativity, though, the model is different.  It says, anything that has mass causes spacetime to be warped, to be bent, to be curved. 

Now, the way to picture, it's difficult to imagine because space is three dimensional, and what does it mean for a three dimensional space to be curved, but we'll talk about that in a minute.  The picture you have in mind at first is kind of like this first picture in the gallery.  You can think of it almost like a rubber membrane like a trampoline maybe, or a sheet of rubber.  And if you put something heavy in it, well then, that sheet gets warped.  Now, it's a two dimensional sheet, but it gets kind of curved and bent.  And other objects then are going to try to move in straight lines through that space.  But because the space is curved, then a straight line now becomes what seems like a curved path.  

So the model now says the Moon orbits around this Earth, not necessarily because there's this magical force pulling towards each other but because the space itself that the Moon is traveling through is warped and so because of that warp, it moves in this orbit.  

It's very bizarre model, it seems strange, like why would that be the case?  Like it doesn't make much sense.  But it makes predictions.  And those predictions are weird.  And those predictions are true.  And that is what makes us think wow, this is the way the universe actually works; much more bizarre than we would have expected.  

Now, what this, let me just illustrate one of these predictions, okay, they illustrate how weird this really is.  One of these predictions that was meant to be a test of general relativity was to say, okay, if you’ve got something really, really massive like the Sun, then it should warp spacetime so much that even light, which has no mass, so it shouldn't have any gravity, according to Newton, only things with mass have gravity, but light doesn't have mass.  So Newton's prediction for gravity here would be to say, well, there's no gravity here, the light doesn't move at all, but with general relativity, it's like, well, if space is warped, then even light can be bent by gravity.  And so the thought was, we need something really massive, like the Sun.  So if the Sun is here, and there's light trying to go past the Sun, maybe that light could be bent.  

Now a perfect way to test this is with a solar eclipse.  Because when a solar eclipse happens, the Moon blocks the Sun, and during just a minute or two, the sky becomes completely black, and you can see stars in the middle of the day.  Now, if you were to look at a star’s location, right next to the eclipsed Sun, well, that starlight, you can see in the picture, the second picture in the gallery, that we know where those stars are, very accurate positions, and what you can do is you can look at that starlight and if you see that that star looks like it's in a slightly different position that means the light has been bent ever so slightly.  

So this was tested.  This was tested.  This was an observation made in like the early 1920s, or 1919, somewhere around there, I think.  Maybe it was the 30s; I don't know.  But Sir Arthur Eddington went out South America, South Africa, somewhere like that, and he made this observation at the total eclipse.  And he in fact confirmed that the star’s position was deviated; that the light was bent by gravity.  This was seen as a profound, you know, this was in the newspapers, like this is confirmation, Einstein's a genius.  This is the way the universe works.  This is amazing.

Now, Einstein was already convinced that this was true.  To him, the mathematics of general relativity were so beautiful, that it absolutely had to be true.  So it came as no surprise to him.  But to everyone it was a prediction, a bizarre prediction, which was confirmed by observation that really cements everyone's belief to say, yes, this must be true.

But here's the crazy thing.  This implies lots of bizarre things.  Now, clearly, we talked about some of the implications like black holes and other things like that.  One of the other things, though, it implies is that what we have here is a warping of space.  Now, when you think about a rubber membrane, like a part, you know, the flat part of a trampoline, that's two dimensional, you could have one direction two dimensions.  But if you are a little bug, you live on that rubber membrane, you don't jump up or down, it's just flat.  

But when you put a bowling ball on there, it gets warped.  When you warp something, you have to warp it into a new direction.   You're warping it into a higher dimension.  So if we live in a three dimensional space, and somehow that three dimensional space is warped, or curved, or bent, then you'd ask the question, well, where is it bent into? I mean, I know there's up and down, forward and back, and left and right.  Now, which direction is it being bent? Like, where's the fourth dimension that it's being warped into.

And this is amazing, like our minds, literally cannot comprehend a fourth dimension, it's just outside of our existence.  What's curious is that in mathematics, we can easily write equations that can describe four dimensions, five dimensions, a hundred dimensions.  The math is easy to write out.  I can easily calculate the volume of a five dimensional sphere.  That's easy to do.  But it's impossible to understand.  It's impossible to visualize what does a five dimensional sphere even look like?  This is a really cool part of cosmology to me as we start thinking about higher dimensions.  

There's a beautiful story that was written called Flatland.  And this was written to kind of try to illustrate, by analogy, how difficult it is for us to understand higher dimensions.  And so Flatland is the story of a place which is completely flat, a universe that exists in only two dimensions.  Imagine that, I mean, imagine what a room would look like in two dimensions.  And I'm sitting in a room right now it has a ceiling, a floor, it has walls on all sides, and it has a door.  It's a box.  But in two dimensions, if you're a little circle living on a piece of paper, a room would just the square around you.   That would trap you because you can't go up and over the line, you're stuck in this two dimensional world.  And a door would just be a line that maybe opens and closes.  

So this Flatland, he describes this Flatland, and one day, this Flatland is visited by a mysterious object that can jump through walls, that can appear and disappear at any moment, that changes sizes.  And this mysterious object takes the circle out of this Flatland so he can see it from above.  

This mysterious object is a sphere.  And imagine if you have a piece of paper and you live on that piece of paper, imagine what a sphere that can just move up and down, how that would change, and it can move side to side, like it lives in those two dimensions but it has one additional dimension.  In addition to moving at any location, it can go up and down.  Now imagine what that would be like.  

Well, if you encounter one of the walls on this room, in this two dimensional world, well you can just go up and over; you would just appear on the other side of that wall.   If you were to pass into that, like you touch that surface, well, that sphere would start all of a sudden like a little circle, and then that circle will get bigger, like it's cross section of that sphere.  So it would appear all of a sudden out of nowhere as a circle, and then it would get bigger, smaller, change, and disappear.

This is a very interesting analogy.  The point is it's basically impossible for us to comprehend

what a higher dimension object would interact with us.  But some of the things we know is that the limitations we experience as three dimensional beings would not be limitations to a four dimensional being.  Imagine, in this world, somehow there was a four dimensional being that visited; somebody who was able to live in these three dimensions, but also just had one additional dimension.  Well, then they wouldn't be stopped by a wall at all.  And we would be amazed that someone would appear through a wall.  But they would have just stepped over it in the additional dimension.  We would be amazed that they seemingly disappear and reappear somewhere else.  But to them, it's just kind of hopping into that extra dimension.  It's nothing.

I can't help but think about the stories in the Bible about angels appearing, about Christ appearing to the disciples in the upper room, even though the doors were locked.  What if there is just one extra dimension, that's a spiritual dimension, that the angels have access to or live in?  Or, maybe there's more than one dimension, but even just one additional dimension allows for an entire universe that is invisible to us.  We'll revisit that a little later, but think about that.  Having one more dimension enables another entire universe that's invisible to us.  

We have three dimensions.  With three dimensions, you add one more dimension to that and that means all of those three dimensions could exist… just moved up one inch in that dimension.  Now this is maybe a little confusing, but think about it in terms of papers?  I'm getting ahead of myself.  It's so cool.  We'll revisit this in our last lesson.  

But what general relativity suggesting to us is that there may be something happening with higher dimensions than what we're accustomed to experiencing, because spacetime, it seems, is warped and bent into these other dimensions, in some sort of way.  Okay.

Now, can we measure this warping of spacetime?  Now at one level, space is warped by things that have mass.  But what about the universe as a whole?  Maybe there's a general curvature that this fabric of space has a curve overall.  

I mean, it could be completely flat.  Like this picture, this is the fourth picture in the gallery.  But the picture on the far right, a universe which is completely flat.  Now, again, this is three dimensional, plus time, but it's not curved overall into any pattern.  Or maybe it's positively curved, the whole universe is actually like the surface of a sphere.  And all these analogies are empty because even the surface of the Earth is really two dimensional.  It's not a full 3d thing that's curved into some fourth dimension.  But still, we could have this positive curvature so that if you kept going in one direction, out into deep space, you would eventually come all the way back to your starting point.  That'd be like a positive curvature.  And then negative curvature; it could be that our whole universe is shaped like a saddle where things are curved upward.  These are the kinds of the three possible curvatures that our universe could have.  

Well, it’s kind of like, well, who cares which curvature we have?  Well, it plays into an interesting question, which is, if our universe is positively curved, then our universe could be finite in size.  It might not be infinitely large.   Like the earth is really, really big but you could measure the size of the Earth.  I mean, it's finite, it doesn't go on forever.  But these negative and flat curvatures, they could also be finite in principle, but there's no edge to them it seems.  They could go on forever; they could be infinite.  So if our universe is negative or flat curvature, it's entirely possible that our universe is a size of infinitely large.  We don't know, we just don't know.  But the curvature can be a clue to help tell us, to help us understand what is our universe like on the biggest scale, the universe as a whole.  

Now, there are ways to make these observations and you'll read about them.  Astronomers have come up with ways to do this.  You can look at triangles, things like that.  One way to try to measure the curvature is by looking at the number of galaxies you see as you look out into space.  It was a little weird, but the idea is, if you're in a flat universe, as you look out into space, you would expect to see a certain number of galaxies as you look further and further back.  The further away you're looking, the more space you're covering, and you'd expect to see more stars, or galaxies.  If you're in a positively curved universe, as you look further and further away, eventually you reach a point where you would expect to see less and less galaxies.   And, if you're in a negatively curved universe, you would see way, way more galaxies as you look further and further away.  So there's relatively simple ways of testing this.  And there are more sophisticated ways of measuring the curvature of the universe.  

These methods lead to more or less the same conclusion, which is that our universe appears to be pretty close to flat.  Now flat, not to say we're a piece of paper, but to say that the overall curvature of the whole universe is generally flat.  Is it infinite? Is it finite? We don't really know.  There's a limit to what we can see.  But we know that the curvature on the whole is flat.

Okay, so lots of cool stuff.  Extra dimensions that are possible, we don't really understand them.  We're going to revisit them a little bit later, and the curvature of the universe as a whole.  You'll see how now in astronomy, we're asking these really big questions.  We're not just talking about galaxies and where are they located, but we're talking about the universe as an object itself, which is really cool.  

All right, we'll see you next time.