 The equivalence principle led Einstein to the position that the presence of matter curved spacetime, and that a body free from all forces, travels geodesics in this curved space. With Riemannian geometry and the Ricci tensor, he had what he needed to develop the curvature side of the equation. First, we fix the generalized n-dimensional coordinates to our four spacetime coordinates. The convention is for numbers to run from zero to three, with zero being the index for the time coordinate. Then we find the volume of the three spatial dimensions for a given time dimension. This is the Einstein tensor. Physically, this tensor determines how the volume of a small group of particles in free fall along geodesics will change. Our little geodesic experiment helps illustrate this important fact. Here we see the change in the volume as the particles diverge and converge along various geodesics. This changes the shape of the volume from a sphere to an ellipsoid, but the total volume remained unchanged. The Einstein tensor is zero, even as the curvature of the space is not zero. By the way, this sphere to ellipsoid phenomenon is called a tidal effect, because it is how the moon creates tides on the earth. In Newtonian physics, the force of gravity was created by mass, or more precisely mass density, the amount of mass per unit volume. In general relativity, we need to change this from mass density to energy density, because of the equivalence between mass and energy, and to take into account the motion or kinetic energy of the masses. So in addition to calculating the mass energy density of a volume of space, we need to account for the flow of energy through each surface of the volume. This information can be packed into a 4x4 matrix known as the energy momentum tensor, or stress energy tensor. Each element represents the flow of momentum across a surface. The first component represents classical energy density at a constant time. This was the only component used in Newton's equations. Similarly, the rest of the top row and left column is the energy flow across each surface. The rest are momentum flows across surfaces. For example, T12 keeps track of the flow in the x direction of momentum in the y direction. These are caused by pressure and stress at each surface. The final step for the gravitational field equations is to determine the constant of proportionality between the Einstein tensor that encapsulates curvature volume and the energy momentum tensor that encapsulates the total energy density. We use the boundary condition that the equations must produce Newton's equations for spaces with very little curvature. You can see that this is a very small number. What this means is that it takes an enormous amount of matter and energy to produce even a tiny amount of curvature in the space around it. This looks simple enough, but because they're tensors, it represents 40 equations with 40 unknowns. These are the Einstein field equations for general relativity. We'll go over what they predict for gravitational phenomenon near the earth, near the sun, and around a black hole in our next segment.