The way to understand Einstein’s equations for general relativity is an opening towards Heaven.

Although Einstein is a legendary figure in science for a large number of reasons — E = mc², the photoelectric effect, and the notion that the speed of light is a constant for everyone — his most enduring discovery is also the least understood: his theory of gravitation, general relativity. Before Einstein, we thought of gravitation in Newtonian terms: that everything in the universe that has a mass instantaneously attracts every other mass, dependent on the value of their masses, the gravitational constant, and the square of the distance between them. But Einstein’s conception was entirely different, based on the idea that space and time were unified into a fabric, spacetime, and that the curvature of spacetime told not only matter but also energy how to move within it.

This fundamental idea — that matter and energy tells spacetime how to curve, and that curved spacetime, in turn, tells matter and energy how to move — represented a revolutionary new view of the universe. Put forth in 1915 by Einstein and validated four years later during a total solar eclipse — when the bending of starlight coming from light sources behind the sun agreed with Einstein’s predictions and not Newton’s — general relativity has passed every observational and experimental test we have ever concocted. Yet despite its success over more than 100 years, almost no one understands what the one equation that governs general relativity is actually about. Here, in plain English, is what it truly means.

Einstein’s original equation relates spacetime curvature to the stress-energy of a system (top). A cosmological constant term can be added (middle), or equivalently, it can be formulated as dark energy (bottom), another form of energy density contributing to the stress-energy tensor.Credit: © 2014 University of Tokyo; Kavli IPMU

This equation looks pretty simple, in that there are only a few symbols present. But it’s quite complex.

- The first one, G
_{μν}, is known as the Einstein tensor and represents the curvature of space. - The second one, Λ, is the cosmological constant: an amount of energy, positive or negative, that is inherent to the fabric of space itself.
- The third term, g
_{μν}, is known as the metric, which mathematically encodes the properties of every point within spacetime. - The fourth term, 8πG/c
^{4}, is just a product of constants and is known as Einstein’s gravitational constant, the counterpart of Newton’s gravitational constant (G) that most of us are more familiar with. - The fifth term, T
_{μν}, is known as the stress-energy tensor, and it describes the local (in the nearby vicinity) energy, momentum, and stress within that spacetime.

These five terms, all related to one another through what we call the Einstein field equations, are enough to relate the geometry of spacetime to all the matter and energy within it: the hallmark of general relativity.

A mural of the Einstein field equations, with an illustration of light bending around the eclipsed sun, the observations that first validated general relativity back in 1919. The Einstein tensor is shown decomposed, at left, into the Ricci tensor and Ricci scalar.Credit: Vysotsky / Wikimedia Commons

You might be wondering what is with all those subscripts — those weird “μν” combinations of Greek letters you see at the bottom of the Einstein tensor, the metric, and the stress-energy tensor. Most often, when we write down an equation, we are writing down a scalar equation, that is, an equation that only represents a single equality, where the sum of everything on the left-hand side equals everything on the right. But we can also write down systems of equations and represent them with a single simple formulation that encodes these relationships.

E = mc² is a scalar equation because energy (E), mass (m), and the speed of light (c) all have only single, unique values. But Newton’s **F** = m**a** is not a single equation but rather three separate equations: F_{x} = ma_{x} for the “x” direction, F_{y} = ma_{y} for the “y” direction, and F_{z} = ma_{z} for the “z” direction. In general relativity, the fact that we have four dimensions (three space and one time) as well as two subscripts, which physicists know as indices, means that there is not one equation, nor even three or four. Instead, we have each of the four dimensions (t, x, y, z) affecting each of the other four (t, x, y, z), for a total of 4 × 4, or 16, equations.

Instead of an empty, blank, three-dimensional grid, putting a mass down causes what would have been “straight” lines to instead become curved by a specific amount. In general relativity, space and time are continuous, with all forms of energy contributing to spacetime’s curvature.Credit: Christopher Vitale of Networkologies and The Pratt Institute

Why would we need so many equations just to describe gravitation, whereas Newton only needed one?

Because geometry is a complicated beast, because we are working in four dimensions, and because what happens in one dimension, or even in one location, can propagate outward and affect every location in the universe, if only you allow enough time to pass. Our universe, with three spatial dimensions and one time dimension, means the geometry of our universe can be mathematically treated as a four-dimensional manifold.

In Riemannian geometry, where manifolds are not required to be straight and rigid but can be arbitrarily curved, you can break that curvature up into two parts: parts that distort the volume of an object and parts that distort the shape of an object. The “Ricci” part is volume distorting, and that plays a role in the Einstein tensor, as the Einstein tensor is made up of the Ricci tensor and the Ricci scalar, with some constants and the metric thrown in. The “Weyl” part is shape distorting, and, counterintuitively enough, plays no role in the Einstein field equations.

The Einstein field equations are not just one equation, then, but rather a suite of 16 different equations: one for each of the “4 × 4” combinations. As one component or aspect of the universe changes, such as the spatial curvature at any point or in any direction, every other component as well may change in response. This framework, in many ways, takes the concept of a differential equation to the next level.

A differential equation is any equation where you can do the following:

- you can provide the initial conditions of your system, such as what is present, where, and when it is, and how it is moving,
- then you can plug those conditions into your differential equation,
- and the equation will tell you how those things evolve in time, moving forward to the next instant,
- where you can plug that information back into the differential equation, where it will then tell you what happens subsequently, in the next instant.

It is a tremendously powerful framework and is the very reason why Newton needed to invent calculus in order for things like motion and gravitation to become understandable scientific fields.

When you put down even a single point mass in spacetime, you curve the fabric of spacetime everywhere as a result. The Einstein field equations allow you to relate spacetime curvature to matter and energy, in principle, for any distribution you choose.Credit: JohnsonMartin / Pixabay

Only, when we begin dealing with general relativity, it is not just one equation or even a series of independent equations that all propagate and evolve in their own dimension. Instead, because what happens in one direction or dimension affects all the others, we have 16 coupled, interdependent equations, and as objects move and accelerate through spacetime, the stress-energy changes and so does the spatial curvature.

However, these “16 equations” are not entirely unique! First off, the Einstein tensor is symmetric, which means that there is a relationship between every component that couples one direction to another. In particular, if your four coordinates for time and space are (t, x, y, z), then:

- the “tx” component will be equivalent to the “xt” component,
- the “ty” component will be equivalent to the “yt” component,
- the “tz” component will be equivalent to the “zt” component,
- the “yx” component will be equivalent to the “xy” component,
- the “zx” component will be equivalent to the “xz” component,
- and the “zy” component will be equivalent to the “yz” component.

All of a sudden, there aren’t 16 unique equations but only 10.

Additionally, there are four relationships that tie the curvature of these different dimensions together: the Bianchi Identities. Of the 10 unique equations remaining, only six are independent, as these four relationships bring the total number of independent variables down further. The power of this part allows us the freedom to choose whatever coordinate system we like, which is literally the power of relativity: every observer, regardless of their position or motion, sees the same laws of physics, such as the same rules for general relativity.

An illustration of gravitational lensing and the bending of starlight due to mass. The curvature of space can be so severe that light can follow multiple paths from one point to another.Credit: NASA / STScI

There are other properties of this set of equations that are tremendously important. In particular, if you take the divergence of the stress-energy tensor, you always, always get zero, not just overall, but for each individual component. That means that you have four symmetries: no divergence in the time dimension or any of the space dimensions, and every time you have a symmetry in physics, you also have a conserved quantity.

In general relativity, those conserved quantities translate into energy (for the time dimension), as well as momentum in the x, y, and z directions (for the spatial dimensions). Just like that, at least locally in your nearby vicinity, both energy and momentum are conserved for individual systems. Even though it is impossible to define things like “global energy” overall in general relativity, for any local system within general relativity, both energy and momentum remain conserved at all times; it is a requirement of the theory.

As masses move through spacetime relative to one another, they cause the emission of gravitational waves: ripples through the fabric of space itself. These ripples are mathematically encoded in the Metric Tensor.Credit: ESO / L. Calçada

Another property of general relativity that is different from most other physical theories is that general relativity, as a theory, is nonlinear. If you have a solution to your theory, such as “what spacetime is like when I put a single, point mass down,” you would be tempted to make a statement like, “If I put two point masses down, then I can combine the solution for mass #1 and mass #2 and get another solution: the solution for both masses combined.”

That is true, but only if you have a linear theory. Newtonian gravity is a linear theory: the gravitational field is the gravitational field of every object added together and superimposed atop one another. Maxwell’s electromagnetism is similar: the electromagnetic field of two charges, two currents, or a charge and a current can all be calculated individually and added together to give the net electromagnetic field. This is even true in quantum mechanics, as the Schrödinger equation is linear (in the wavefunction), too.

But Einstein’s equations are nonlinear, which means you cannot do that. If you know the spacetime curvature for a single point mass, and then you put down a second point mass and ask, “How is spacetime curved now?” we cannot write down an exact solution. In fact, even today, more than 100 years after general relativity was first put forth, there are still only about ~20 exact solutions known in relativity, and a spacetime with two point masses in it still is not one of them.

Yours,

Dr Churchill

PS:

- The Einstein field equations appear very simple, but they encode a tremendous amount of complexity.
- What looks like one compact equation is actually 16 complicated ones, relating the curvature of spacetime to the matter and energy in the universe.
- It showcases how gravity is fundamentally different from all the other forces, and yet in many ways, it is the only one we can wrap our heads around.