 Welcome to Relativity 13. In this video, we explore the cosmological implications of general relativity. For more than half a century, Big Bang cosmology has been widely accepted as our best model for the evolution of the universe. From an initial state of extreme density and temperature, the universe has been expanding for almost 14 billion years. More recently, we've discovered that the expansion is now accelerating instead of slowing down as one would expect if the process was dominated by gravity. As much as our common sense might revolt against this bizarre picture, the Big Bang model is supported by a vast body of evidence, one of the most striking examples being the cosmic microwave background. In this video, we will see how physics predicts the universe has no choice but to be in a dynamic state of either expansion or contraction, and how observational data confirm this incredible prediction. A quantitative discussion of cosmology requires us to adopt some mathematical model of the universe. This model has to be simple enough to be solvable and, at the same time, accurate enough to reasonably represent the real universe. In most ancient models, Earth was assumed to have a privileged position in the cosmos. The most successful of these models was due to Ptolemy. In the Ptolemaic model, Earth rests at the center of the universe. The sun, moon, planets, and stars follow daily orbits about the Earth. The peculiar motions of the planets relative to the stars is accounted for by additional smaller orbits, called epicycles. The modern point of view is that Earth is not special, the so-called Copernican principle. In the model of Copernicus, the observed motions of the sun, moon, and planets through the sky are explained by assuming Earth is simply another planet orbiting the sun and spinning once per day on its axis. Copernicus assumed all orbits are circular. Later, Kepler demonstrated that orbits are actually elliptical. Later still, Newton developed his laws of motion and gravitation, which explained elliptical orbits from fundamental universal principles. This made it possible, in theory, to describe the motion of all objects in the universe, and therefore to develop a rigorous theory of cosmology. A central question is the size of the universe. One possibility is that the universe is finite, in which case it would have a well-defined center. At the turn of the 20th century, a common opinion among astronomers was that the universe is finite and corresponds to what we now call the Milky Way Galaxy. Another possibility is that the universe is infinite and has no center, although there might be an infinite number of local centers of mass concentration. This was Newton's surprisingly modern view. He wrote, But if the matter was evenly disposed throughout an infinite space, it could never convene into one mass, but some of it would convene into one mass and some into another, so as to make an infinite number of great masses scattered at great distances from one another throughout all that infinite space. We will provisionally assume an infinite universe with a uniform distribution of matter as formalized in the cosmological principle, which can be stated as The spatial distribution of matter in the universe is homogeneous and isotropic when viewed on a large enough scale. Homogeneous means that all properties are the same at all points in space. Isotropic means that all properties are the same in all directions in space. Following this principle, our model for the universe is a uniform cloud of dust with the same physical properties at all points and in all directions in space. This includes energy matter density, pressure, temperature, and so on. Here's a visualization of our model for, let's say, matter energy density. At a given time, there is a single value for this quantity for the entire universe. As we move into the future, suppose this value decreases, we can visualize this by the screen getting darker. Moving into the past, the value might increase, which we visualize by the screen getting brighter. So although its value may change with time, at a given time any physical property has a single value throughout the entire universe. Now, this might seem like a ridiculously simplistic model, and clearly wrong, because physical properties certainly do vary across space. Density can be almost zero in the near absolute vacuum of deep space, or incomprehensibly large in the core of a neutron star. But the key is the phrase, when viewed on a large enough scale. Every dot in this image from the Sloan Digital Sky Survey is a galaxy with color encoding distance from Earth. The image width corresponds to about 6 billion light years. There's a lot of structure to be seen. But as we examine coarser pixelations of the image, the image looks ever more uniform. Different lines of evidence show that in the observable universe, the cosmological principle is a reasonable assumption on large enough scales. So we can expect that a model based on this principle should give us a good representation of the overall evolution of the universe, while obviously not accounting for the formation of stars, galaxies, and other, quote, small scale structures. An analogy is that Earth is not a homogeneous sphere. Its inhomogeneous, anisotropic surface has a huge effect on weather, the distribution of living organisms, and all aspects of our environment. However, all of that is negligible if we want to calculate the force of attraction between Earth and Moon. At that large of a scale, Earth can accurately be treated as a homogeneous sphere. Before the 20th century, there wasn't enough observational data to support a rigorous theory, and cosmology was little more than conjecture. However, it was known that a rigorous theory of an infinite universe based on Newtonian physics would confront a serious problem. We can see this by asking the question, in an infinite homogeneous and isotropic universe, what is the gravitational field at a given point? To give our eyes something to focus on, we'll visualize our model as an infinite array of point masses, even though the actual model is perfectly uniform. Let's arbitrarily choose one of these masses and try to figure out the gravitational field it experiences. That field is the sum of the forces of attraction due to each of the other masses in the universe. For any given mass, there will be another mass at the same distance, but in exactly the opposite direction. The forces due to these two masses will cancel. Extending this pairing off to all masses in the universe, we come to the conclusion that the gravitational field in an infinite, homogeneous and isotropic universe is zero everywhere. This seems rigorous enough. However, there's another way to analyze the problem. Let's choose an arbitrary point as, quote, the center of the universe. Centered on this point, the infinite universe is a sphere of radius big R, where big R goes to infinity. Now we choose any particle in the universe. We'll color it orange and assume small r is its distance from the center. Let's color blue the region inside the sphere of radius small r. The matter in the blue region will exert a gravitational force on our particle directed toward the center. Let's color the rest of the universe green. We can think of this as a spherical shell of inner radius small r and outer radius big R, with big R going to infinity. Newtonian theory predicts that there is no gravitational field inside a spherical shell. So for any finite value of big R, the spherical shell exerts no gravitational force on our particle. Presumably we can let big R grow arbitrarily large and this will still be true. So we have a seemingly rigorous argument that the gravitational field at any point in the universe can be non-zero, with its magnitude and direction depending on what other point we arbitrarily call the center of the universe. The problem with both of these arguments becomes apparent if you write out the steps mathematically. In both cases you end up adding together an infinite number of non-vanishing contributions to the gravitational field. Mathematically this is problematic. For example, what is the sum of 1 minus 1 plus 1 minus 1 plus 1 minus 1 and so on without end? One argument is that you can group each plus 1 with a minus 1. And since 1 minus 1 is 0, you then have an infinite number of zeros added together which gives 0, so the sum is 0. But another argument is that you can isolate the first 1 and then group each minus 1 with a plus 1. Now you have 1 plus an infinite number of zeros, so the sum is 1. So which is it? In fact, this infinite series has no well-defined value. Mathematically we say it does not converge. Likewise, Newtonian theory does not converge on a unique answer for the gravitational field at a point in the infinite homogeneous isotropic universe. In the past this led some physicists to suggest that Newton's law of gravitation must not be valid at very large distances. We'll see that general relativity does not have this convergence problem and leads to unambiguous cosmological predictions. But first, despite its problems, let's look into the Newtonian model a bit more. Particularly our second line of analysis. There are parallels with the predictions of relativity that we can get with much less work. This will give us a useful point of reference as we work through the rigorous theory later on. Let's go back to the picture of an arbitrary point selected as the quote center of the universe and a particle a distance smaller away. Assume this particle has mass small m. And the mass in the blue sphere is big m. The blue mass attracts the particle towards its center. Using Newton's equations of motion and gravity, we write little m times acceleration equals minus big m times little m over r squared. We're using units in which the speed of light and the gravitational constant are one. If we assume no gravitational fields produced by the masses outside the blue sphere, then this is our particle's equation of motion. The double dots denote acceleration. If you plot r versus time, then r single dot, the velocity, is the plot slope, and r double dot, the acceleration, is essentially the plot's curvature. There's a link in the description to a video that discusses this in more detail. The mass big m is the volume four thirds pi r cubed times the mass density mu. r and mu may change over time, and we denote their values at time zero by r zero and mu zero. Substituting this expression for big m in the first equation and canceling a common factor of small m on both sides, we end up with a formula for the gravitational acceleration of our particle. This formula seems ridiculous because the acceleration depends on our arbitrary choice of a quote center of the universe. But let's express the distance r as a dimensionless scale factor a times r zero, distance at time zero. Then r double dot equals a double dot times r zero. Substituting these expressions into the previous equation, we find that the r zero terms cancel, leaving us with an equation for the scale factor alone, and containing a constant with units of time, big t, that depends only on the mass density of the universe at time zero. We call a the cosmic scale factor. Regardless of what point we choose to call the center of the universe, we will get this same equation. It doesn't refer to the distance between two particular points, but rather tells us how any distance in the universe changes with time. If a is initially one, and later is one half, then the distance between any two points in the universe will end up half of what it was initially. Let's highlight three masses in our model universe with green, blue, and yellow spheres, and let's take the blue sphere to be the quote center. Suppose as time goes on the scale factor decreases. This means that the entire universe shrinks towards the blue sphere. If instead we take the green sphere to be the center, then the universe shrinks towards the green sphere, and likewise for the yellow sphere. In each case, the relative distance, relative velocity, and relative acceleration between any two points in the universe is the same. The choice of a center has no effect on the evolution of the universe. It's simply for mathematical convenience. The cosmic scale factor equation says that a double dot is negative. This means that the curvature of a plot of the scale factor through time will curve downward. Let's assume at time zero, a is one, and a dot, the rate at which a is changing, is zero. This describes a universe in which all particles are at rest relative to one another. Solving the equation for future times, we get a graph that starts out flat and curves downward. The scale factor decreases ever more rapidly and reaches zero at time big T. We can also solve the equation for past times. The scale factor's past behavior is the mirror image of its future behavior. As we go into the past, the scale factor decreases ever more rapidly and reaches zero at time minus big T. As a approaches zero at T equals plus or minus big T, our equation tells us that a double dot approaches minus infinity. These are singularities where the equation is no longer valid. We'll black out these points on the graph to indicate that while our model describes the universe arbitrarily close to them, it fails at these singularities and consequently for all times beyond two. According to our model, in the past the universe was extremely dense. Any two particles were arbitrarily close at some given time. As time went on, the universe expanded and eventually came to rest. In the future, it will collapse and once again approach arbitrarily large densities. We might call the first singularity the big bang and the second the big crunch. Taken at face value, this curve tells us that a static universe at rest is not possible for more than an instant. If the universe is ever at rest, then it must have been expanding from an extreme state for a finite time in the past and after coming to rest will collapse to an extreme state a finite time in the future.