SpikeyFreak
12-27-2010, 09:06 AM
Interesting read from someone at reddit explaining expanding cosmology:
The whole "expanding universe" thing is, unfortunately, a bit misleading at first glance. Normally when we throw the word "expanding" around, we're talking about things getting bigger in some sense. The deficit is expanding, my waistline is expanding, something like that.
Not so, when the subject turns to modern cosmology.
See, the idea that lies at the core of what's generally called the "standard model of cosmology" — that is, the cosmological model of the universe that best explains all our observations — is one of metric expansion.
Metric expansion basically works like this: Given any two fixed points in space, the distance between them is not a constant. It increases with time. That does not mean the two points are moving away from each other. Those two points are fixed, pinned down as it were. They ain't moving. But the distance between them is increasing.
This is a surprisingly simple idea to express mathematically. You just write down the equation for calculating the distance between any two points — the one we use in this universe is similar to, but not the same as, the good ol' Pythagorean theorem that imaginary people living in an imaginary Euclidean universe would use — and toss in a coefficient that depends on time. We call that coefficient a(t), and give it the name "the scale factor." The distance between any two points in the universe is the coordinate distance — that is, the distance you get when you use that almost-Pythagorean equation I alluded to — times the scale factor, which in turn depends on the age of the universe.
If you know anything about basic geometry, this should give you a splitting headache. How can the distance between two unmoving points vary? The answer is that in Euclidean space — the space we talk about when we're studying basic geometry — it can't. The distance between points in Euclidean space is constant with respect to time … and indeed, with respect to everything else except the points' positions. But the geometry of our universe is not Euclidean geometry. On certain scales — the scale of your living room, for instance — it sure looks Euclidean. But on larger scales, or at high relative velocities, or in the presence of strong gravitation, it's very much not Euclidean. And one of the non-Euclidean properties of the geometry of our universe is that distances between fixed points can vary with time. It's permitted by the rules of geometry that govern our universe, and furthermore it appears to be fact.
Now, this might all sound like mathematical wankery and abstract folderol. But it really isn't. Take a minute to google up a recent experiment called Gravity Probe B. Gravity Probe B did something remarkable: it directly measured the geometry of spacetime around the Earth. And the way it did it was very, very clever.
Imagine a sheet of paper with an arrow drawn on it. The arrow starts somewhere, and points off in some arbitrary direction; doesn't matter which one. Now imagine moving the arrow around on the paper while keeping its direction constant. Think of it like a game of pin-the-tail-on-the-donkey. The arrow is the donkey's tail, and you can move the pin holding it down wherever you want, as long as you keep it pointed in the same direction.
Move the arrow around any path you like, ending back at the same place where it started. You can move it in a circle, or in a complicated curlicue, or whatever. When you get the arrow back to the same point where it started, you'll see that it points in exactly the same direction it did when we began. We moved the arrow around a closed path, and its direction did not change.
That's Euclidean geometry at work, right there. But as we talked about before, the geometry of our universe is not Euclidean. In our universe, if you do that same experiment — move an arrow around without changing its direction — it may not necessarily end up pointing where it pointed when you started.
That's what the Gravity Probe B experiment did. Except instead of an arrow, it used incredibly precise gyroscopes. A gyroscope, due to its angular momentum, resists any motion that would change the direction of its axis of rotation. If you get a gyroscope spinning in a sufficiently low-friction environment, it becomes a sort of compass, always oriented in the same direction. The Gravity Probe B experiment carried a gyroscope on a closed path around the Earth — aboard an orbiting satellite — and compared the direction it pointed when it was done to the direction it was pointing when they started … and found a difference.
Now, the reason for this has to do with gravitation. The Earth's mass induces a curvature in the structure of spacetime around our planet; that's how gravity works. But another result of this curvature is that the parallel transport of a vector — moving an arrow around without changing its direction — results in a deviation. This was long predicted by general relativity, but the Gravity Probe B experiment actually tested it directly. We went out there and directly measured the geometry of the universe. And I think that's pretty damn awesome.
The same truth about the universe that causes parallel vectors transported around closed paths to deviate also permits metric expansion. And metric expansion explains all that weird, bizarre stuff we see when we look up at the night sky. The universe isn't expanding into anything. It isn't really expanding at all, in the sense that people normally use the word. Rather, stuff that's at rest relative to other stuff is staying pretty much where it is … but all distances in the universe are gradually increasing with time.
Yeah, visualizing metric expansion is one of the hardest things one has to do when one studies physics, in my opinion.
Basically the way it works is this. Consider any two fixed points in the universe. (If you want to get technical, by "fixed" I mean they're at rest relative to each other, and they're both in reference frames in which the cosmic microwave background is isotropic.) There's some distance between them, call it X.
Now wait a little while.
The distance between those two fixed points is now X′, where X′ is definitely larger than X.
The two points have not moved. But the distance between them has increased.
This is possible because the distance between any two points is a function of the underlying manifold — that's the technical term for it. We normally think of the world around us as fundamentally being Euclidean, just like what we studied in high-school geometry class. This turns out not to be the case. It's tough to spot the difference, because it's only significant on scales that we don't normally interact with — galaxies and black holes and such — but the geometry of the universe is not Euclidean. It's different, and one of the ways in which it's different is that the metric — that is, the distance between any two given points — is a function of time. The older the universe, the farther apart any two points in the universe will be.
Now, how we got here is a bit of an interesting story. See, early in the 20th century it was observed the light from distant galaxies appears redder than it really ought to be. Around that same time, Einstein had just demonstrated that the universe makes a lot more sense if the speed of light is constant in all reference frames, and that raised the implication that the light from objects that are moving away from us should be red-shifted. So for a while, everybody thought distant galaxies were moving away from us. Which was fine, because that fit with what was then the widely accepted idea of the Big Bang: a colossal explosion in space, from which all matter has since radiated outward. These distant galaxies, it was believed, were just coasting on their residual primordial momentum.
But there are some problems with that, three of which are worth talking about here. First of all, wherever we look, we see galaxies moving away from us. It's clearly not the case that we ourselves are moving. Which means we, ourselves, lack that primordial momentum we see everywhere else. We appear, by all observations, to be the sole stationary point at the exact center of a universe full of Big Bang debris. Which is hard to swallow.
Second, there's the fact that not everything appeared to be moving away from us at the same speed. If we were at the center of the universe, at the point where the Big Bang explosion occurred, we'd expect to see everything radiating outward from us with a constant velocity. It isn't. And stuff isn't slowing down, either. In fact, it appeared to be speeding up! The further away a galaxy was, the faster it appeared to be going. Which made just no sense.
Finally, there was the problem of time. The same theory that tells us an object moving away from us at a significant speed will appear red-shifted when we look at it also tells us that it will appear to progress more slowly through time than we do. A clock on a fast-moving spaceship will be seen by us to run more slowly than our own clocks. Now, obviously there are no clocks in distant galaxies, but there are rigidly periodic astrophysical phenomena. Because these are distant galaxies, they appear red-shifted … but they do not appear to be time-dilated. That is, it does not appear to be the case, from our observations of these periodic phenomena, that their clocks are ticking more slowly than our own, as would be consistent with the high recessional speed the cosmological red-shift seemed to imply.
Long story short, we simply couldn't find a solution that explained what we saw in the sky. So people started thinking harder about the problem. Eventually some particularly smart people discovered — partly in cooperation, partly independently — that if you let go of the assumption that distances between fixed points are constant with respect to time, suddenly it all makes sense. It suddenly became clear that the cosmological red-shift — as it's called — is not a consequence of radial motion away from us at all, but rather the result of a completely unrelated phenomenon that just happens to look like a Doppler effect.
I like that story in particular because it illustrates the point that when theory doesn't match observation, sometimes what you have to let go of is not just the theory that's giving you trouble, but also one of your fundamental assumptions about the universe. Much of 20th-century physics, from relativity to FLRW cosmology to quantum theory, was marked by this sort of letting go of some fact about nature that was intuitive and obvious and undeniable and wrong.
You've basically got it right. A ray of light has a wavelength, yeah? And wavelength is, as the name implies, a length. It's expressed in terms of units of length — meters, light-years, whatever.
Well, length is a function of time in our universe.
Picture a distant galaxy. Like really distant, ten billion light-years or whatever. In that galaxy is a star, and stars (duh) emit light. A particular ray of light comes out of that star and heads — purely by random coincidence — in our direction.
Now, let us further assume that that ray of light was generated by some known atomic process. A particular energy-state transition in a particular atom. Okay? I bring this up for a reason that'll become clear soon.
Now. The ray of light begins its journey. It has some fixed energy — because it was created out of a particular interaction — and for light, energy is proportional to wavelength. So this ray of light has some wavelength, λ.
Let us further assume that between that distant star and our telescope lies absolutely nothing. This is not the case, but we're just imagining this scenario, so let's go with that. Between here and there, there's pure vacuum.
The ray of light propagates through empty space, as rays of light are wont to do. It travels for ten billion years — because the star that emitted it is ten billion light-years from here.
Now let's freeze time in our minds when the ray of light is exactly half an inch from our telescope's detector. It's just sitting there, not yet having interacted with our detector but about to, having made the ten-billion-light-year journey from that distant star. It's been in transit for ten billion years — again, because that star is ten billion light years away.
If we examine that ray of light — in our minds; remember, this is all impossible and we're merely imagining it — we'll see that its energy is less than what it was when it was emitted. Its wavelength is longer than it was. It's now, let's call it, λ′. It's the same ray of light; it hasn't interacted with or been scattered by anything along the way. But it's changed.
Why? Because the scale factor of the universe has changed during those ten billion years. See, when the ray of light was emitted, its wavelength was actually λa(t₀), where the quantity a is the scale factor, which is a function of the age of the universe, and t₀ is the age of the universe at the time the ray of light was emitted. Now we're at t₁, ten billion years later, and a(t₁) is numerically larger than a(t₀). So the wavelength of the ray of light, λa(t₁), is now greater than it was when it was emitted.
This is the cosmological red-shift. The wavelength of a ray of light grows longer as it travels through empty space. How much longer it grows is directly proportional to how long it's in transit … which is why galaxies that are twice as far from us appear to be twice as red-shifted.
Wanna hear something neat? This phenomenon doesn't just affect light coming out of distant stars. You know about the cosmic microwave background, yeah? It's often popularly described as an "echo" of the Big Bang, but that's a bit wrong. It's actually the light that was emitted during a period in the universe's history when everything was much denser — because lengths were smaller — than it is today. At that time, matter and energy were interacting like crazy, and the universe was a sort of hot soup of, most likely, monatomic hydrogen plasma. This soup was so energetic — that is to say, its energy density, or energy per unit volume, was so high — that it radiated tons of electromagnetic radiation. Eventually, somewhere on the order of thirteen and a half billion years ago, the scale factor of the universe grew to the point where it was possible for electrons to stay bound to protons, and the hydrogen plasma condensed into hot hydrogen gas. At that time, the universe became transparent to visible light — literally. Before that, a ray of light wouldn't make it very far in the universe before it interacted with some free electron or hydrogen ion. After that, rays of light could propagate freely through space without interacting much.
But there were still all these energetic photons around. They didn't go away, and they weren't all absorbed by all the new matter laying around. They just hung out, radiating through space in all directions.
But over time, the scale factor continued to increase, and the wavelength of all this leftover radiation increased along with it. So gradually these energetic photons "dimmed," until today they're pretty much all in the microwave spectrum. We see this as a sort of nearly-uniform glow in the sky, apparently coming from everywhere. It's the light that was emitted by everything during that period in the universe's history when all distances were shorter, all volumes were smaller, all densities were larger and everything was so hot it glowed.
--Sharer Spikey
The whole "expanding universe" thing is, unfortunately, a bit misleading at first glance. Normally when we throw the word "expanding" around, we're talking about things getting bigger in some sense. The deficit is expanding, my waistline is expanding, something like that.
Not so, when the subject turns to modern cosmology.
See, the idea that lies at the core of what's generally called the "standard model of cosmology" — that is, the cosmological model of the universe that best explains all our observations — is one of metric expansion.
Metric expansion basically works like this: Given any two fixed points in space, the distance between them is not a constant. It increases with time. That does not mean the two points are moving away from each other. Those two points are fixed, pinned down as it were. They ain't moving. But the distance between them is increasing.
This is a surprisingly simple idea to express mathematically. You just write down the equation for calculating the distance between any two points — the one we use in this universe is similar to, but not the same as, the good ol' Pythagorean theorem that imaginary people living in an imaginary Euclidean universe would use — and toss in a coefficient that depends on time. We call that coefficient a(t), and give it the name "the scale factor." The distance between any two points in the universe is the coordinate distance — that is, the distance you get when you use that almost-Pythagorean equation I alluded to — times the scale factor, which in turn depends on the age of the universe.
If you know anything about basic geometry, this should give you a splitting headache. How can the distance between two unmoving points vary? The answer is that in Euclidean space — the space we talk about when we're studying basic geometry — it can't. The distance between points in Euclidean space is constant with respect to time … and indeed, with respect to everything else except the points' positions. But the geometry of our universe is not Euclidean geometry. On certain scales — the scale of your living room, for instance — it sure looks Euclidean. But on larger scales, or at high relative velocities, or in the presence of strong gravitation, it's very much not Euclidean. And one of the non-Euclidean properties of the geometry of our universe is that distances between fixed points can vary with time. It's permitted by the rules of geometry that govern our universe, and furthermore it appears to be fact.
Now, this might all sound like mathematical wankery and abstract folderol. But it really isn't. Take a minute to google up a recent experiment called Gravity Probe B. Gravity Probe B did something remarkable: it directly measured the geometry of spacetime around the Earth. And the way it did it was very, very clever.
Imagine a sheet of paper with an arrow drawn on it. The arrow starts somewhere, and points off in some arbitrary direction; doesn't matter which one. Now imagine moving the arrow around on the paper while keeping its direction constant. Think of it like a game of pin-the-tail-on-the-donkey. The arrow is the donkey's tail, and you can move the pin holding it down wherever you want, as long as you keep it pointed in the same direction.
Move the arrow around any path you like, ending back at the same place where it started. You can move it in a circle, or in a complicated curlicue, or whatever. When you get the arrow back to the same point where it started, you'll see that it points in exactly the same direction it did when we began. We moved the arrow around a closed path, and its direction did not change.
That's Euclidean geometry at work, right there. But as we talked about before, the geometry of our universe is not Euclidean. In our universe, if you do that same experiment — move an arrow around without changing its direction — it may not necessarily end up pointing where it pointed when you started.
That's what the Gravity Probe B experiment did. Except instead of an arrow, it used incredibly precise gyroscopes. A gyroscope, due to its angular momentum, resists any motion that would change the direction of its axis of rotation. If you get a gyroscope spinning in a sufficiently low-friction environment, it becomes a sort of compass, always oriented in the same direction. The Gravity Probe B experiment carried a gyroscope on a closed path around the Earth — aboard an orbiting satellite — and compared the direction it pointed when it was done to the direction it was pointing when they started … and found a difference.
Now, the reason for this has to do with gravitation. The Earth's mass induces a curvature in the structure of spacetime around our planet; that's how gravity works. But another result of this curvature is that the parallel transport of a vector — moving an arrow around without changing its direction — results in a deviation. This was long predicted by general relativity, but the Gravity Probe B experiment actually tested it directly. We went out there and directly measured the geometry of the universe. And I think that's pretty damn awesome.
The same truth about the universe that causes parallel vectors transported around closed paths to deviate also permits metric expansion. And metric expansion explains all that weird, bizarre stuff we see when we look up at the night sky. The universe isn't expanding into anything. It isn't really expanding at all, in the sense that people normally use the word. Rather, stuff that's at rest relative to other stuff is staying pretty much where it is … but all distances in the universe are gradually increasing with time.
Yeah, visualizing metric expansion is one of the hardest things one has to do when one studies physics, in my opinion.
Basically the way it works is this. Consider any two fixed points in the universe. (If you want to get technical, by "fixed" I mean they're at rest relative to each other, and they're both in reference frames in which the cosmic microwave background is isotropic.) There's some distance between them, call it X.
Now wait a little while.
The distance between those two fixed points is now X′, where X′ is definitely larger than X.
The two points have not moved. But the distance between them has increased.
This is possible because the distance between any two points is a function of the underlying manifold — that's the technical term for it. We normally think of the world around us as fundamentally being Euclidean, just like what we studied in high-school geometry class. This turns out not to be the case. It's tough to spot the difference, because it's only significant on scales that we don't normally interact with — galaxies and black holes and such — but the geometry of the universe is not Euclidean. It's different, and one of the ways in which it's different is that the metric — that is, the distance between any two given points — is a function of time. The older the universe, the farther apart any two points in the universe will be.
Now, how we got here is a bit of an interesting story. See, early in the 20th century it was observed the light from distant galaxies appears redder than it really ought to be. Around that same time, Einstein had just demonstrated that the universe makes a lot more sense if the speed of light is constant in all reference frames, and that raised the implication that the light from objects that are moving away from us should be red-shifted. So for a while, everybody thought distant galaxies were moving away from us. Which was fine, because that fit with what was then the widely accepted idea of the Big Bang: a colossal explosion in space, from which all matter has since radiated outward. These distant galaxies, it was believed, were just coasting on their residual primordial momentum.
But there are some problems with that, three of which are worth talking about here. First of all, wherever we look, we see galaxies moving away from us. It's clearly not the case that we ourselves are moving. Which means we, ourselves, lack that primordial momentum we see everywhere else. We appear, by all observations, to be the sole stationary point at the exact center of a universe full of Big Bang debris. Which is hard to swallow.
Second, there's the fact that not everything appeared to be moving away from us at the same speed. If we were at the center of the universe, at the point where the Big Bang explosion occurred, we'd expect to see everything radiating outward from us with a constant velocity. It isn't. And stuff isn't slowing down, either. In fact, it appeared to be speeding up! The further away a galaxy was, the faster it appeared to be going. Which made just no sense.
Finally, there was the problem of time. The same theory that tells us an object moving away from us at a significant speed will appear red-shifted when we look at it also tells us that it will appear to progress more slowly through time than we do. A clock on a fast-moving spaceship will be seen by us to run more slowly than our own clocks. Now, obviously there are no clocks in distant galaxies, but there are rigidly periodic astrophysical phenomena. Because these are distant galaxies, they appear red-shifted … but they do not appear to be time-dilated. That is, it does not appear to be the case, from our observations of these periodic phenomena, that their clocks are ticking more slowly than our own, as would be consistent with the high recessional speed the cosmological red-shift seemed to imply.
Long story short, we simply couldn't find a solution that explained what we saw in the sky. So people started thinking harder about the problem. Eventually some particularly smart people discovered — partly in cooperation, partly independently — that if you let go of the assumption that distances between fixed points are constant with respect to time, suddenly it all makes sense. It suddenly became clear that the cosmological red-shift — as it's called — is not a consequence of radial motion away from us at all, but rather the result of a completely unrelated phenomenon that just happens to look like a Doppler effect.
I like that story in particular because it illustrates the point that when theory doesn't match observation, sometimes what you have to let go of is not just the theory that's giving you trouble, but also one of your fundamental assumptions about the universe. Much of 20th-century physics, from relativity to FLRW cosmology to quantum theory, was marked by this sort of letting go of some fact about nature that was intuitive and obvious and undeniable and wrong.
You've basically got it right. A ray of light has a wavelength, yeah? And wavelength is, as the name implies, a length. It's expressed in terms of units of length — meters, light-years, whatever.
Well, length is a function of time in our universe.
Picture a distant galaxy. Like really distant, ten billion light-years or whatever. In that galaxy is a star, and stars (duh) emit light. A particular ray of light comes out of that star and heads — purely by random coincidence — in our direction.
Now, let us further assume that that ray of light was generated by some known atomic process. A particular energy-state transition in a particular atom. Okay? I bring this up for a reason that'll become clear soon.
Now. The ray of light begins its journey. It has some fixed energy — because it was created out of a particular interaction — and for light, energy is proportional to wavelength. So this ray of light has some wavelength, λ.
Let us further assume that between that distant star and our telescope lies absolutely nothing. This is not the case, but we're just imagining this scenario, so let's go with that. Between here and there, there's pure vacuum.
The ray of light propagates through empty space, as rays of light are wont to do. It travels for ten billion years — because the star that emitted it is ten billion light-years from here.
Now let's freeze time in our minds when the ray of light is exactly half an inch from our telescope's detector. It's just sitting there, not yet having interacted with our detector but about to, having made the ten-billion-light-year journey from that distant star. It's been in transit for ten billion years — again, because that star is ten billion light years away.
If we examine that ray of light — in our minds; remember, this is all impossible and we're merely imagining it — we'll see that its energy is less than what it was when it was emitted. Its wavelength is longer than it was. It's now, let's call it, λ′. It's the same ray of light; it hasn't interacted with or been scattered by anything along the way. But it's changed.
Why? Because the scale factor of the universe has changed during those ten billion years. See, when the ray of light was emitted, its wavelength was actually λa(t₀), where the quantity a is the scale factor, which is a function of the age of the universe, and t₀ is the age of the universe at the time the ray of light was emitted. Now we're at t₁, ten billion years later, and a(t₁) is numerically larger than a(t₀). So the wavelength of the ray of light, λa(t₁), is now greater than it was when it was emitted.
This is the cosmological red-shift. The wavelength of a ray of light grows longer as it travels through empty space. How much longer it grows is directly proportional to how long it's in transit … which is why galaxies that are twice as far from us appear to be twice as red-shifted.
Wanna hear something neat? This phenomenon doesn't just affect light coming out of distant stars. You know about the cosmic microwave background, yeah? It's often popularly described as an "echo" of the Big Bang, but that's a bit wrong. It's actually the light that was emitted during a period in the universe's history when everything was much denser — because lengths were smaller — than it is today. At that time, matter and energy were interacting like crazy, and the universe was a sort of hot soup of, most likely, monatomic hydrogen plasma. This soup was so energetic — that is to say, its energy density, or energy per unit volume, was so high — that it radiated tons of electromagnetic radiation. Eventually, somewhere on the order of thirteen and a half billion years ago, the scale factor of the universe grew to the point where it was possible for electrons to stay bound to protons, and the hydrogen plasma condensed into hot hydrogen gas. At that time, the universe became transparent to visible light — literally. Before that, a ray of light wouldn't make it very far in the universe before it interacted with some free electron or hydrogen ion. After that, rays of light could propagate freely through space without interacting much.
But there were still all these energetic photons around. They didn't go away, and they weren't all absorbed by all the new matter laying around. They just hung out, radiating through space in all directions.
But over time, the scale factor continued to increase, and the wavelength of all this leftover radiation increased along with it. So gradually these energetic photons "dimmed," until today they're pretty much all in the microwave spectrum. We see this as a sort of nearly-uniform glow in the sky, apparently coming from everywhere. It's the light that was emitted by everything during that period in the universe's history when all distances were shorter, all volumes were smaller, all densities were larger and everything was so hot it glowed.
--Sharer Spikey