Going Further 10.7: The Einstein Equation

We have just been introduced to two terms: the four-dimensional spacetime curvature, Gμν, and the four-dimensional mass–energy distribution, Tμν. How can we visualize these terms to try to gain a deeper understanding of what they represent?

We introduced the concept of a vector when we discussed velocity, and went into more detail when we discussed forces. A vector is a physical quantity that has both strength and direction. We can specify its components in the three spatial dimensions: x, y, and z. By specifying the sizes of these components, we also specify the direction that the force vector is pointing.

For example:

These force vectors are illustrated in Figure B.8.7.

Figure B.8.7: (a) A force vector with magnitude 10 N in the x-direction; (b) a force vector with magnitude 14.14 N at an angle 45 degrees between the x- and y-axes; and (c) a force vector with magnitude 17.3 N at an angle 45 degrees between the x- and y-axes and also 45 degrees between the x-y plane and the z-axis. Credit: NASA/Sonoma State/A. Simonnet

Vectors are useful mathematical tools to express forces that point in one particular direction. However, when we use vectors to represent (for example) the gravitational force, we are also assuming that the objects that are being affected can be treated as points. That is to say, we assume they are very, very small, so that the force can act on the objects as though all their mass was concentrated at one tiny point, which we can describe by its coordinates in three-dimensional space.

But what if we have really large objects? For example, Earth is rather large compared to satellites that are in orbit around it. So approximating the satellites as tiny points compared to Earth works pretty well. (However, as we have seen in our discussion of GPS, there are errors that arise that need to be corrected, which are about 1 part in one billion.) But Earth is not that much larger than the Moon, and both have bulges and craters. We might therefore expect that neither one of these objects looks like a tiny point to the other. Instead, we need to consider the entire Earth–Moon system when we do an accurate calculation of how they affect one another.

In these types of situations, we cannot describe the gravitational force as acting in a specific, linear three-dimensional direction. Instead, we need to specify the components of the gravitational acceleration acting in different directions in four-dimensional spacetime, so that we account for the curvature correctly. The four-dimensional spacetime curvature G, which has components, Gμν, is the mathematical way that we express this more complicated situation. Rather than being a series of three numbers (like a vector), it is represented mathematically by a special type of 4 × 4 matrix called a tensor. This is how the components of Gμν are defined:

\begin{equation} G_{\mu\nu}= \begin{pmatrix} g_{11}& g_{12}& g_{13}& g_{14} \\ g_{21}& g_{22}& g_{23}& g_{24} \\ g_{31}& g_{32}& g_{33}& g_{34} \\ g_{41}& g_{42}& g_{43}& g_{44} \end{pmatrix} \end{equation}

In this case, the diagonal components of the matrix appear in the expression for curvature that we defined previously:

\begin{equation} s^2= g_{11}\left(\Delta x\right)^2 + g_{22}\left(\Delta y\right)^2 + g_{33}\left(\Delta z\right)^2 - g_{44}\left(c\Delta t\right)^2 \end{equation}

For special relativity in flat space, the components (g11, g22, g33, g44) = (1,1,1,–1) yielding the familiar expression for the spacetime interval:

\begin{equation} s^2=\left(\Delta x\right)^2+\left(\Delta y\right)^2 + \left(\Delta z\right)^2-\left(c\Delta t\right)^2 \end{equation}

The matter-energy tensor, T, which has components Tμν, is a similar 4 × 4 matrix that describes the geometry of the field associated with the distribution of the matter and energy in spacetime.

Putting both of these tensors together into Einstein’s equation leads to a series of 10 independent equations that must be solved simultaneously in order to correctly calculate the mutual gravitational attraction between two (or more) large, complicated objects. In a few simple cases, these equations can be solved with pencil and paper; however, in all other cases, the solution requires numerical calculations by supercomputers.