 We've seen that Newtonian theory can be interpreted as requiring a matter-filled universe to be dynamic, if all matter in the universe is ever at rest. Then in the past it had to have expanded from a big bang singularity, come to rest, and in the future it necessarily will collapse into a big crunch singularity. If we want to explore the cosmological predictions of relativity, we need to extend the theory we've developed. That theory applies to empty space, but we need equations that describe matter-filled space. To start, let's review the framework we've developed so far in this series of videos. As always, there's no need to understand the equations in detail. They can be glossed over. But if you want to dig in, the details are there in previous videos, especially in appendix videos. We are free to choose four spacetime coordinates, three of space and one of time, more or less arbitrarily, with the caveat that we can't know their precise geometric significance until we solve the following set of equations. At the heart of relativity is the metric tensor, Gij, a 4x4 array of quantities that allows us to calculate the proper time, ds, between events separated by coordinate changes, dx1, 2, 3, and 4. ds squared equals the sum over i and j of Gij, dxi, dxj. There are four spacetime velocities, ui, for i equals 1, 2, 3, and 4. ui is the change in the ife coordinate over the change in proper time. We also use the notation x.i. The equations of motion give us the four accelerations, u.i, in terms of the four velocities and the so-called Christoffel symbols, gamma ijk, which are derived from the metric tensor. These geodesic equations express the fact that of all possible ways for a clock to move between two spacetime events, the natural or free fall motion, is that which typically results in the largest possible clock reading. In relativity, gravity manifests as spacetime curvature. The Riemann tensor, Rijkl, derived from the Christoffel symbols, specifies all possible ways spacetime might be curved. If all its components are everywhere zero, then by simply falling freely, we obtain a global, flat, or inertial coordinate system, free of gravity. If any component of the Riemann tensor is nonzero, then all coordinate systems must be curved, that is, have gravity. Summing various components of the Riemann tensor, we obtain the Ricci tensor. Einstein's field equations for empty space are that the Ricci tensor vanishes everywhere. We say that empty space is Ricci-flat. The physical significance is that for a short period of time, a dust cloud having all particles initially at rest with respect to each other may deform in shape, but will maintain its volume. As we've seen throughout this series, this relativistic equation of gravity allows us to rigorously describe gravitational time dilation, non-elliptical orbits, and black holes. But it only applies to empty space. We need a modified version to describe the presence of matter. Einstein struggled to find this modification. When he finally did, his general theory of relativity was complete. In empty space, each of the 16 components of the Ricci tensor are zero. Empty space can be curved, but its curvature is constrained by this equation. In non-empty space, presumably the modified equation will relate spacetime curvature to the distribution of matter. By matter, we mean all non-gravitational forms of energy. Let's first see how this works in Newtonian theory. It's most convenient to work with gravitational potential instead of directly with the gravitational field. If we have mass Big M, then a unit mass at a point a distance r1 away will experience a gravitational field G1. Suppose we move the unit mass to another point, a distance r2 away, and we ask how much work is required to do this. We first move along a radial line. We need to push against the gravitational field, hence we have to do work. Then we can move along an arc perpendicular to the gravitational field, which requires no work. We associate a number U with each point, called the gravitational potential, such that the difference in U values is the energy needed to move a unit mass between the points. For Newton's law of gravity, this potential is minus M over r. The gravitational field can then be related to the variation of the potential through space. The behavior of the potential in empty space is governed by Laplace's equation. For a point mass M, the potential at distance r0 away is U0. Consider a sphere of radius A that does not enclose the mass. Some points on the sphere would be farther away, hence have higher potential, while other points will be closer, hence have lower potential. The average of the potential over the sphere turns out to equal the potential at the center. We can define the Laplacian of the potential as the limit as the sphere shrinks to 0 of 6 over A squared times the average potential on the sphere minus the potential at the center. We then have Laplace's equation. We can add any number of additional masses outside the sphere, and this will continue to be true. Therefore, it's a general law. The Laplacian of the potential vanishes everywhere in empty space. In general relativity, the metric tensor plays the role of gravitational potential, and the relativistic version of Laplace's equation is the Ricci-Flatnis condition. In matter-filled space, Laplace's equation is replaced by Poisson's equation. Suppose our sphere is now filled with uniformly dense matter. Starting at the center, we want to know how much work is required to move a unit mass to the surface. When we are at distance r from the center, our unit mass will feel a traction to the mass inside a sphere of that radius. The resulting gravitational field is minus m over r squared, where m is the volume four-thirds pi r cubed times the mass density mu. This gives a gravitational field that grows linearly with r. If we take the potential at the center to be zero, we find that the potential on the sphere is two-thirds pi mu a squared. Putting this into our definition of the Laplacian, we get Poisson's equation. The Laplacian of the potential equals four pi mu. If mu equals zero, this reduces the Laplace's equation. This is the equation we seek a relativistic version of. Before we proceed, let's first consider the implication of Poisson's equation for Newtonian cosmology. For nonzero mu, we cannot have a constant potential, because then Poisson's equation would read zero equals something nonzero. If the potential is not constant, then the universe cannot have zero gravitational field. A changing potential means work is required to move masses around. Hence they must be subject to gravitational forces. Before Hubble's discovery of the expanding universe, it was commonly assumed that the universe is static on large scales. To account for this, modifications of Poisson's equation were proposed. For example, as shown here, an additional term minus lambda u can be added to the left side. In fact, Einstein discussed this modification in his 1917 paper, Cosmological Considerations on the General Theory of Relativity, as a way to motivate a similar modification he made to the relativistic version. We'll consider that later. The important point is that this modified equation has a constant potential solution, u equals minus four pi over lambda mu. This describes a universe with no gravitational field. Of course, the price we pay is a modification of Newton's law of gravity. The resulting modified Newtonian potential has an additional exponential factor, e to the minus square root lambda r. Of course, the unmodified potential describes our solar system very well. But for a given r, if lambda is small enough, the modification is negligible. So maybe the lambda term exists, but its effects only become important for cosmologically large distances. With this modification, the ambiguities in Newtonian theory disappear. We can call lambda a cosmological constant. As we'll see, Einstein proposed a cosmological constant for general relativity, a constant that is now associated with so-called dark energy.