Einstein’s Field Equations describe how mass and momentum create a curvature in space-time, thereby creating the apparent force of gravity. Einstein’s Field equations also describe phenomena such as gravitational waves, black holes, and the accelerating expansion of the Universe. This represents two of the four dimensions. Just as the Earth appears flat to us when we are looking at only a tiny portion of it, this curved space-time will also appear flat if we zoom in and look at only a tiny section. If we approximate this region with the mathematics of flat space-time, we can define an axis for one of the three spatial dimensions and an axis for time. Different observers in this region moving relative to each other will have different orientations for these axes, in accordance to the rules of Special Relativity. Suppose we move forward along the time axis seen by the observer moving in the direction of the red arrow. Our approximation for flat space-time is now no longer valid. We correct for this by rotating our red arrow along the line 90 degrees to the surface. We can now create a new flat space-time approximation for this new region. We can use this new approximation to again move forward in the direction of the red arrow. If we repeat this over and over again, using infinitesimally small time steps, we get the path that the object will follow through curved space-time. The path appears curved to us, thereby creating the illusion of a gravitational force. To describe four dimensional space-time, we need four coordinates, which we will refer to as X0, X1, X2, and X3. The numbers zero through three are index values, not exponents. Einstein’s Field equations remain valid no matter what coordinate system we choose. None of the coordinates necessarily need to correspond to the time coordinate of any observer. Curved coordinates can also be used to describe flat space-time. In order to better understand the critical concepts, let us first focus on a flat two dimensional space. The following arrows show the direction in which our position will change if we increment one of the coordinates at various points in space by a small amount. These arrows are what we refer to as the basis vectors. Let’s move one of the basis vectors by exactly one unit as shown. The rate at which the basis vector changes is described by the white vector. The white vector can be represented as a linear combination of the two basis vectors at this point. Let’s replace the numbers with the following symbols, each of which has three index values. The index value at the top indicates which basis vector it is multiplying. The two index values at the bottom indicate which basis vector is moving, and in what direction. These symbols indicate the rate at which the basis vectors are changing at each point in space. The basis vectors could be changing because we are using a curved coordinate system in a flat space-time. Or the basis vectors could be changing because we have a curved space-time, even though we have coordinates as straight as possible. Suppose we have a curved four dimensional space-time with coordinates as straight as possible, where X0 represents time, and X1 through X3 represent the three spatial dimensions. In situations where Newtonian physics gives accurate approximations, the acceleration due to gravity is approximately equal to the following symbols related to the curvature of space-time. In these cases, the gravitational acceleration is approximately equal to the rate at which the basis vector for time changes as it moves forward in time. This change is due to curvature. In order to know how a vector changes due to curvature, we need to know how to move it from one point to another. Suppose we move a vector by an infinitesimally small amount without changing the direction it is pointing. Every time we do this, we need to take the curvature into account by rotating the vector along the line 90 degrees to the surface. If we repeat this over and over again, we can move a vector over long distances, which we refer to as parallel transport. If we parallel transport a vector in flat space-time, when we return the vector to its starting position, it will always still be pointing in the same direction as before. The same is not necessarily true in curved space-time. We can define the curvature of space-time in terms of how much the angle of a vector changes when it returns to its original position. But, this angle also depends on the size of the area that the vector travels around. When the vector travels around this area, it is rotated by 90 degrees. If we double the area, we also double the amount by which the vector is rotated. We can therefore define the curvature in terms of the amount by which the vector is rotated divided by the size of the area around which the vector traveled. Note that when the vector travelled around the area in the counter clockwise direction, the vector rotated counter clockwise. And when the vector travelled around the area in the clockwise direction, the vector rotated clockwise. If the vector rotates in the same direction in which it is travelling, we will refer to this as positive curvature. If the vector rotates in the opposite direction in which it is travelling, we will refer to it as negative curvature. An example of negative curvature is a saddle point, such as the region colored in green. The sphere had the same curvature at every point. On the other hand, this surface has a different curvature at every point. We can refine our definition of curvature by defining the curvature at each point in terms of the amount by which the vector is rotated, divided by the size of the area, as the size of the area approaches zero. The curvature at each point is signified by the scalar variable “R”, which is one of the terms of Einstein’s Field Equation. Though, this should not be confused with the tensor variable that has an R with two index values. This tensor variable is also related to curvature, but it represents sixteen different numbers at each point in space-time. The other two tensor variables in Einstein’s Field equations also each represent sixteen different numbers at each point in space time. At each point in space time, the relationships between all of these different tensors must be as is specified by the equation, regardless of what coordinate system we use. But, the individual values of each of the tensor components strongly depends on what coordinate system we select. In order to simplify the following explanation, let us assume that we are not dealing with four dimensional space-time, but just with a two dimensional space. Suppose we zoom in close enough for us to be able to approximate this region of space as flat. Keep in mind that the “1” and “2” are index values, not exponents. We want to know the length of the white arrow, but the yellow and red arrows are not 90 degrees to each other, so we can’t use the Pythagorean Theorem. Instead, the length squared will be given by the following equation. We can extend this concept from two dimensions to higher dimensions. When dealing with four dimensional space-time, this does not calculate spatial length, but what we call the “space-time interval.” In Special Relativity, the space time interval was defined with the following equation, which is the quantity all observers can agree on. In General Relativity, this equation is no longer necessarily true due to the curvature of space-time. Instead, we calculate the space-time intervals inside each infinitesimally small region using the tensor shown, which we call the “metric tensor.” Let’s now focus on the tensor with the “R.” We will first show what this tensor means in three dimensions, before moving on to four dimensions. We have a coordinate system and the basis vectors are not necessarily all 90 degrees to one another. As always, keep in mind that the following numbers are index values, not exponents. We are going to parallel transport the white vector around this infinitesimally small parallelogram. If we are in a flat three dimensional space, the vector will still be pointing in its original direction when it returns to its original position. But, if we are in a curved three dimensional space, then this is not necessarily true. We want to know how much the vector has changed, which means that we want to know “dV1”, “dV2”, and “dV3.” These quantities will be proportional to the product of the lengths of the two sides of the parallelogram, multiplied by some other terms. These other terms will be a linear combination of the components of the white vector. For the missing terms in the following equations, let us invent a new variable with the letter R, but it will be different from the variables with the letter R that we saw before. This new variable will have four index values. The top index indicates which component of “dV” it is calculating, the first bottom index indicates which component of “V” it is multiplying, and the last two bottom indexes indicate the two dimensions for the lines of the parallelogram. This is shown for only three dimensions, but let us suppose that we have extended this concept to four dimensions. This term in Einstein’s Field Equations can then be computed as follows. There is only one term remaining on the left hand side of the equation that we have not yet talked about. This is the cosmological constant governing the rate at which the expansion of the Universe is accelerating. This is the phenomena we associate with what we call dark energy. Let us now focus on the right side of the equation. Capital “G” is the gravitational constant, and it is the same gravitational constant that appears in Newton’s equation for gravity. This term with the “C” is the speed of light raised to the power of four. This last parameter with the “T” describes how much energy and momentum we have at each point. In classical physics, the momentum is mass multiplied by velocity. If the word “mass” always refers to the “rest mass”, which does not change with velocity, then this equation needs to be modified by adding the following variable, which increases to infinity as the velocity of the object approaches the speed of light. The vector for velocity has a component in each of the spatial directions. We can define the velocity vector as follows. We can also say that the object is moving through time at the speed of light, and extend the velocity vector to four dimensions as shown. Using the four dimensional vector for velocity, we can also create a four dimensional vector for momentum, which we will symbolize with the variable “p.” The variable “m” refers to rest mass. The term for momentum in the time dimension can be rearranged as shown. We can extend this definition of a four dimensional momentum vector to phenomena such as electromagnetic radiation, which has momentum and energy, but no mass. Suppose we multiply one of the components of the momentum vector with one of the components of the velocity vector, and we create a matrix showing all the different ways in which we can do this. If a volume of space-time contains no matter or energy, then all of these terms are zero for that volume. At each point we can calculate the values of each of these terms per unit volume, and signify the result with the following symbol. Provided that none of the objects have velocities close to the speed of light, then the term in the upper left hand corner will be far larger than any of the other terms, and all the other terms will be negligible by comparison. This is because the term in the upper left hand corner has the square of the speed of light in its equation, which is an extremely large number. This is the equation for the energy density. These are the values we have defined, but they are not the same as the values in Einstein’s Field Equations, which are symbolized as follows. These two sets of values can be calculated from each other by using the metric tensor. We now have the final component for Einstein’s Field Equations. We can call this the stress – energy – momentum tensor, and it tells us the density of energy and momentum at each point in space time. It includes the energy and momentum of substances such as matter and electromagnetic radiation, but it does not include the energy of curved space-time. For example, all the components of the stress energy momentum tensor could be zero while there are gravitational waves passing through the region. Throughout this video, we have visualized curved space-time as a two dimensional sheet embedded in a three dimensional space that obeys Euclidean Geometry. However, General Relativity does not require four dimensional space-time to be embedded inside a space of more than four dimensions in order for the mathematics to work. It could just be that the Universe is simply not governed by Euclidean Geometry.
