Hubble's law

Over many videos now, we've been talking about how every interstellar object is moving away from Earth. And we've also been talking about how the further something is away from Earth, the faster it's moving. What I want to do in this video is to put a little bit of numbers behind it, or even better conceptualize what we've been talking about. So one way to think about it is that, if at an early stage in the universe, I were to pick some points. So that's one point, another point, another point, another point. Let me just pick nine points so that I have a proper grid. So this at an early stage in the universe. If we fast forward a few billion years-- and I'm clearly not drawing it to scale-- all of these points have all moved away from each other. So this point is over here-- actually, let me draw another column, just to make it clear. So if we fast forward a few billion years, the universe has expanded. And so everything has moved away from everything. And let me color code it a little bit. Let me make this point magenta. So this point, the magenta point is now here. This green point has now moved away from the magenta point. And now this blue point has now moved away from the magenta point in that direction. And we could keep going. This yellow point is maybe over here now. I think you get the general idea. And I'll just draw the other yellow points. So they've all moved away from each other. So there's no center here. Everything is just expanding away from things next to it. And what you can see here is not only did this thing expand away from this, but this thing expanded away from this even further. Because it had this expansion, plus this expansion. Or another way to think about it is, the apparent velocity with which something is expanding is going to be proportional to how far it is. Because every point in between is also expanding away. And just to review a little bit of the visualization of this-- one way to think of this, if you think of the universe as an infinite flat sheet. You can imagine that we're just taking a sheet of, I don't know, some type of sheet of stretchy material and just stretching it out. We're just stretching it out. That's if we kind of imagine a more infinite universe that just goes off in every direction. We're just stretching that infinite sheet out. So it has no boundaries, but we're still stretching it out. Another way to visualize it-- and this what we did earlier on-- Is you can imagine that the universe is the three dimensional surface of a four dimensional sphere. Or the three dimensional surface of a hyper-sphere. So at an early stage in the universe, the sphere looked like this. And these points here-- that magenta point is right over here. The green point is right over there. Then we add the blue point up here. And then let me just draw the rest of the yellow points. And the yellow points are here. They're all on the surface of this sphere. Obviously I'm only dealing with two dimensions right now and it's nearly impossible, or maybe impossible, to imagine a three dimensional surface of a four dimensional sphere. But the analogy holds. If this is a surface the balloon, or the surface of a bubble, if the bubble were to expand over a few billion years-- and once again, not drawn to scale. So now we have a bigger bubble here. This part of the surface is all going to expand. So once again, you have your magenta. You have your blue dot. You have your green dot right over here. And then let me just draw the rest in yellow. So they will have all expanded away from each other on the surface of this sphere. And just to make it clear that this is a sphere, let me draw some contour lines. So this is a contour line. Just to make it clear that we are on the surface of an actual sphere. Now with that out of the way, let's think about what is the apparent velocity with which things are moving away? And remember, we're going have to say, not only how far things are moving away, but we're going to say how far they are moving away from-- if the observer is us-- depending on how far they already are. So what we're going to do-- we could say is-- let me write this down. All objects moving away from each other. And the apparent relative velocity is proportional to distance. And what I've just written down here-- and this is why I wrote it down-- this is a rephrasing of, essentially, Hubble's Law. And he came up with this by just observing that when he looks-- especially the further out he looks, the more redshifted objects are. And not only were they moving faster and faster away from Earth, but they seem to be moving faster and faster away from each other. So this is just a restating of Hubble's Law. Or another way to say it is, from any point, let's say from the earth, the velocity that something appears to be moving is going to be some constant times the distance that it is away from the observer. In this case, we are the observer. And we put this little zero-- so this H here is called Hubble's Constant. And it's a very non-constant constant. Because this constant will change depending on where we are in the evolution of the universe. So we put this little zero here, this little sub 0 right over here, to show that this is Hubble's Constant right now. And when we talk about distance, we're talking about the proper distance right now. And this has to be very important because that proper distance is constantly changing as the universe expands. So the now will actually change slightly from the beginning of this video to the end of this video. But we could roughly say in kind of our current period of time. And when we say proper distance, we're talking about if you actually had rulers. And if you were to just lay them down instantaneously-- obviously we can't do something like that. But we can imagine doing something like that. So that's what we're talking about it. So just to give a sense of, or do a little bit of math of how fast things are actually moving apart-- let me actually write it someplace where I have more space-- the current Hubble Constant is 70.6 plus or minus 3.1. So we have observed some variation here. There is some error to our actual measurements. Kilometers per second per megaparsec. And remember, a parsec is roughly 3.2, 3.3 light years. So another way to think about it is, if this is where we are in the universe right now. And if this object right over here-- if this distance right over here is one megaparsec, so 1 million parsecs. Or 3.26 million light years from Earth. So just so we have a sense, this is roughly 3.26 million light years from Earth. Then this object will appear to be moving away. Although it's not moving in space, just the space that it's in is stretching in such a way that it looks to be moving at, based on its redshift, 70.6 kilometers per second away from us. So this is a huge velocity. 70.6 kilometers per second. So this is a pretty fast velocity. But you have to remember, this is over one megaparsec. The Andromeda galaxy is not even a megaparsec away. It is about 2.5 million light years. So it's about 0.7 or 0.8 of a megaparsec. So if you look at a point in space a little bit further than the Andromeda galaxy, it will look to be, right now, receding at about 70.6 kilometers per second. But what if you were to go twice that distance? If you were to look at something that's almost 7 light years away? 2 megaparsecs away? So if you were to look at this object over here, how fast would that be receding? Well, if you just look at it over here, it's 2 megaparsecs away. So it's going to be twice this. You're just going to multiply its distance-- 2 megaparsecs times this. The megaparsecs cancel out. So 70.6 times 2 is-- it's going to look to be moving, It's not moving in space. Remember, space is just stretching. So its velocity, It's apparent velocity, will be 70.6 times 2. So that's 141.2 kilometers per second. And one question you say, well how did Hubble know-- you can observe the redshift of objects moving away from us. But how did he know that they were moving away from each other? Well, if you were look at the redshift of this object, and say, wow, that's moving away is 70.6 kilometers per second. And then you were to look at the redshift of this and say, wow, that's moving away from us at 141.2 kilometers per second. Then you also know that these two objects are moving away from each other at 70.6 kilometers per second. And we could keep doing this over different distances. But hopefully this gives you a little bit bigger sense of things. And just remember, even though I said this is a huge distance-- a megaparsec is further than it is to the Andromeda galaxy. The Andromeda galaxy is the nearest large galaxy to us. There are some smaller galaxies that are closer to us that are kind of satellite galaxies around the Milky Way. But the Andromeda is the nearest large galaxy to us. And we also know that we're talking about hundreds of billions of galaxies in just the observable universe. So very quickly, as you go near the edge of the observable universe, these velocities, the apparent distance at which things move are moving away from us, start to become pretty significant.