When physicists talk about Einstein's equation they don't usually mean the famous E=mc2, but another formula, which encapsulates the celebrated general theory of relativity. Einstein published that theory a hundred years ago, in 1915. To celebrate its centenary we asked physicist David Tong of the University of Cambridge to explain what general relativity is and how Einstein's equation expresses it. You can watch his explanation in the video below, or read on.
Start with Newton
The general theory of relativity describes the force of gravity. Einstein wasn't the first to come up with such a theory — back in 1686 Isaac Newton formulated his famous inverse square law of gravitation. Newton's law works perfectly well on small-ish scales: we can use it to calculate how fast an object dropped off a tall building will hurtle to the ground and even to send people to the Moon. But when distances and speeds are very large, or very massive objects are involved, Newton's law becomes inaccurate. It's a good place to start though, as it's easier to describe than Einstein's theory.
Suppose you have two objects, say the Sun and the Earth, with masses and respectively. Write for the distance between the two objects. Then Newton’s law says that the gravitational force between them is
where is a fixed number, known as Newton's constant.
The formula makes intuitive sense: it tells us that gravity gets weaker over long distances (the larger the smaller ) and that the gravitational force is stronger between more massive objects (the larger either of and the larger ).
Different force, same formula
There is another formula which looks very similar, but describes a different force. In 1785 the French physicist Charles-Augustin de Coulomb came up with an equation to capture the electrostatic force that acts between two charged particles with charges and :
Here stands for the distance between the two particles and is a constant which determines the strength of electromagnetism. (It has the fancy name permittivity of free space.)
The problem with Newton
Newton's and Coulomb's formulas are nice and neat, but there is a problem. Going back to Newton's law, suppose you took the Earth and the Sun and very quickly moved them further apart. This would make the force acting between them weaker, but, according to the formula, the weakening of the force would happen straight away, the instant you move the two bodies apart. The same goes for Coulomb's law: moving the charged particles apart very quickly would result in an immediate weakening of the electrostatic force between them.
But this can't be true. Einstein's special theory of relativity, proposed ten years before the general theory in 1905, says that nothing in the Universe can travel faster than light — not even the "signal" that communicates that two objects have moved apart and the force should become weaker.
Why we need fields
This is one reason why the classical idea of a force needs replacing in modern physics. Instead, we need to think in terms of something — new objects — that transmit the force between one object and another. This was the great contribution of the British scientist Michael Faraday to theoretical physics. Faraday realised that spread throughout the Universe there are objects we today call fields, which are involved in transmitting a force. Examples are the electric and magnetic fields you are probably familiar with from school.
Albert Einstein (1879-1955) in 1921.
A charged particle gives rise to an electric field, which is "felt" by another particle (which has its own electric field). One particle will move in response to the other's electric field — that's what we call a force. When one particle is quickly moved away from the other, then this causes ripples in the first particle's electric field. The ripples travel through space, at the speed of light, and eventually affect the other particle. In fact, the particle that is moved also generates a magnetic field and emits electromagnetic radiation. The end result is a complex interaction of rippling fields — but the point is that the force is really one particle being affected by ripples propagating through the field of the other.
It took scientists a long time to fully develop this field picture of electromagnetism. The main credit goes to the Scottish scientist James Clerk Maxwell, who not only realised that the electric and magnetic forces were two aspects of a unified force of electromagnetism, but also replaced Coulomb's simple law of electrostatics with four equations that describe how electric and magnetic fields respond to moving charged particles. Maxwell's four formulas are some of the most amazing equations in physics because they capture all there is to know about electricity and magnetism.
Gravity and spacetime
So what about gravity? Just as with electromagnetism there needs to be a field giving rise to what we perceive as the gravitational force acting between two bodies. Einstein's great insight was that this field is made of something we already know about: space and time. Imagine a heavy body, like the Sun, sitting in space. Einstein realised that space isn't just a passive by-stander, but responds to the heavy object by bending. Another body, like the Earth, moving into the dent created by the heavier object will be diverted by that dent. Rather than carrying on moving along a straight line, it will start orbiting the heavier object. Or, if it is sufficiently slow, will crash into it. (It took Einstein many years of struggle to arrive at his theory — see this article to find out more.)
Another lesson of Einstein's theory is that space and time can warp into each other — they are inextricable linked and time, too, can be distorted by massive objects. This is why we talk, not just about the curvature of space, but about the curvature of spacetime.
The general theory of relativity is captured by a deceptively simple-looking equation:
Essentially the equation tells us how a given amount of mass and energy warps spacetime. The left-hand side of the equation,
describes the curvature of spacetime whose effect we perceive as the gravitational force. It’s the analogue of the term on the left-hand side of Newton’s equation.
Massive objects bend spacetime. Image courtesy NASA.
The term on right-hand side of the equation describes everything there is to know about the way mass, energy, momentum and pressure are distributed throughout the Universe. It is what became of the term in Newton’s equation, but it is much more complicated. All of these things are needed to figure out how space and time bend. goes by the technical term energy-momentum tensor. The constant that appears on the right-hand side of the equation is again Newton’s constant and is the speed of light.
What about the Greek letters and that appear as subscripts? To understand what they mean, first notice that spacetime has four dimensions. There are three dimensions of space (corresponding to the three directions left-right, up-down and forwards-backwards of space) and one dimension of time (which only has one direction). If you want to understand how a moving bit of mass affects spacetime, you need to understand how it affects each of those four dimensions and their various combinations.
(As an analogy, think of the way you’d describe an object moving at constant speed along a straight line in Newton’s classical physics. You need two pieces of information: the direction and the speed of the motion. The direction is given by three numbers, each telling you by how much the object moves in each of the three directions of space. Therefore, the motion is described by a total of four numbers, three relating to space and one giving the speed. Since speed is distance covered per unit time, we need three bits of information relating to space and one to time, in order to describe the motion.)
Not just one equation
In Einstein’s equation the Greek letters and are labels, which can each take on the values 0, 1, 2 or 3. So really, the equation above conceals a whole collection of equations corresponding to the possible combinations of values the and can take:
and so on.
The value of 0 corresponds to time and the values 1,2 and 3 to the three dimensions of space.
therefore relates to time and the 1-direction of space. The term on the right-hand side describes the momentum (speed and mass) of matter moving in the 1-direction of space. The motion causes time and the 1-direction of space to mix and warp into each other — that effect is described by the left-hand side of the equation. (The analogue goes for an equation with and equal to 2 or 3.)
If the equation only has 1s, 2s or 3s, for example
then it relates only to space. The term on the right-hand side now measures the pressure that matter causes in the corresponding direction of space. The left-hand side tells you how that matter causes space in that direction to stretch.
If and both take the value 0, then the equation
only relates to time. The term now stands for energy, which causes time to speed up or slow down. The left-hand side of the equation describes that change in the flow of time.
Since each of and can take on four values, this gives a total of 4 x 4 =16 equations. However, it turns out that the equation with and (for and each equal to one of 0, 1, 2 or 3) is the same as the equation with and This reduces the total number of equations to ten.
In theory Einstein's equations allow you to work out exactly how massive objects, such as planets, stars, galaxies, or even black holes affect the spacetime they sit in. In practice though, things aren't quite as straight-forward. Einstein's equations are incredibly difficult to solve — supercomputers are needed to find solutions and coming up with new solutions is an active field of theoretical physics. One of the big current challenges is to figure out what happens to space-time when two very heavy objects, like black holes, collide.
How do we know that Einstein's theory is correct? In the hundred years since its publication, the theory has passed every test it has been subjected to. Despite its slightly esoteric nature, it's crucial in things most of us rely on every day, such as the GPS features in our smartphones and the Satnav devices in our cars. The theory does open up some new questions, which is why some physicists think it needs to be modified (see this article). But whether or not this really turns out to be necessary, there's no doubt that general relativity is one of the most amazing achievements in the history of science.
About this article
David Tong is a theoretical physicist at the Universiy of Cambridge. He works on quantum theory and general relativity.