The Equations, Explained

How to Read This Page

An equation is a sentence. The equals sign is the verb, and each symbol is a noun standing for something you could, in principle, measure. Every section below does the same three things: says what the sentence means, names each symbol, and then puts real numbers in.

The equations run from the gentlest to the hardest. The last one, the field equations of general relativity, is among the most compact and most difficult statements in physics, and it is worth the climb.

1. The Photoelectric Law

In words: the energy of the fastest electron knocked out of a metal equals the energy of one particle of light, minus the price of escaping the metal.

• $K_{\max }$ is the kinetic energy of the fastest electron that comes out.
• $h$ is Planck's constant, a fixed number of nature: about $6.6\times {10}^{-34}$ joule-seconds.
• $\nu$ (the Greek letter nu) is the frequency of the light. Blue light has a higher frequency than red.
• $\varphi$ (phi) is the work function, the minimum energy needed to pull an electron out of that particular metal.

What to notice: brightness appears nowhere. A brighter lamp sends more particles of light, so more electrons come out, but each electron still gets the energy of just one particle. If $h\nu$ is smaller than $\varphi$, the answer is negative, which means no electrons come out at all, however bright the light. See The Photoelectric Effect.

2. The Random Walk

In words: the average of the square of the distance a jiggling particle has wandered grows in step with time.

• $x$ is how far the particle has moved from where it started, along one direction.
• The angle brackets mean "average over many particles, or many tries."
• $D$ is the diffusion coefficient, which measures how quickly the particle spreads. It is large for small particles in thin, warm liquids.
• $t$ is the time elapsed.

What to notice: it is the square of the distance that grows steadily. So the distance itself grows only as the square root of time. To wander twice as far, a particle needs four times as long; ten times as far takes a hundred times as long. That is why a drop of ink spreads quickly at first and then seems to stall.

Einstein's second step was to connect $D$ to things you can measure:

Here $T$ is the temperature, $\eta$ (eta) the thickness of the liquid, $r$ the radius of the particle, and $R$ a known constant from the study of gases. The only unknown left is $N_A$, the number of molecules in a standard amount of substance. Measure the wandering, and you have counted molecules. See Brownian Motion.

3. The Stretch Factor of Relativity

In words: this single number says how much time stretches and length shrinks for something moving at speed $v$.

• $\gamma$ (gamma) is the factor itself. It is never less than 1.
• $v$ is the speed of the moving object.
• $c$ is the speed of light, about 300,000 kilometers per second.

How it works: everything depends on the fraction $v^2/c^2$, the speed compared with light, squared. For anything in daily life that fraction is almost zero, so $\gamma$ is almost exactly 1 and nothing unusual happens. As $v$ creeps toward $c$, the fraction approaches 1, the quantity under the square root approaches zero, and dividing by a number near zero gives something huge.

• At a tenth of the speed of light, $\gamma$ is 1.005. Clocks run slow by half a percent.
• At half the speed of light, $\gamma$ is 1.15.
• At 87 percent, $\gamma$ is 2. A moving clock ticks at half speed.
• At 99 percent, $\gamma$ is about 7.
• At 99.99 percent, $\gamma$ is about 71.

At the speed of light itself the formula asks you to divide by zero. That is the mathematics saying that nothing with mass can get there. See Special Relativity.

4. Energy, Mass, and Momentum

In words: the total energy of anything is built from two parts, one from its momentum and one from its mass, combined like the sides of a right triangle.

• $E$ is the total energy.
• $p$ is the momentum, the amount of motion.
• $m$ is the mass.
• $c$ is the speed of light.

The triangle: picture a right-angled triangle. One short side is $m{c}^2$, the energy an object has just by existing. The other short side is $pc$, the momentum multiplied by the speed of light. The long side is the total energy $E$. This is Pythagoras's theorem with physical quantities for sides.

Two special cases:

• Something standing still has no momentum, so $p=0$ and the triangle collapses to a single side. What remains is $E=m{c}^2$. The famous equation is the special case of an object at rest.
• Light has no mass, so $m=0$ and the other side vanishes. What remains is $E=pc$: light carries momentum even though it has no mass, which is why sunlight can push a solar sail.

With numbers: because $c^2$ is about 90,000,000,000,000,000 in ordinary units, one gram of anything corresponds to about 90 trillion joules. See Mass–Energy Equivalence.

5. The Rule for Measuring Distance in Spacetime

This is where the notation becomes unfamiliar, so take it slowly.

Start with Pythagoras. On a flat sheet of paper, the distance $ds$ between two nearby points is given by $d{s}^2=d{x}^2+d{y}^2$, where $dx$ and $dy$ are the small steps across and up.

Now bend the paper. On a curved surface, such as a globe, that simple rule fails. A step of one degree of longitude covers 111 kilometers at the equator and almost nothing near the pole. You need a set of correction factors that say, at each location, how much real distance each kind of step is worth. That set of correction factors is the metric, written $g$.

• $ds$ is the true separation between two nearby events.
• $d{x}^{\mu }$ is a small step in one of the four directions of spacetime. The index $\mu$ (mu) is a label, not a power: it runs over 0, 1, 2, 3, standing for time and the three directions of space.
• $g_{\mu \nu }$ is the metric: the table of correction factors, one for each pair of directions.

The hidden instruction: whenever a label appears twice, once up and once down, you are meant to add up over all four of its values. So this short expression is shorthand for a sum of sixteen terms. Einstein introduced this convention himself to save writing. Because the order of a pair does not matter, only ten of the sixteen entries of $g$ are independent.

Why it matters: in general relativity, those ten numbers at each point are the gravitational field. Know the metric everywhere, and you know how every clock ticks and how every object falls.

6. The Bending of Starlight

In words: the angle by which a ray of light is bent as it skims past a massive body.

• $\delta$ (delta) is the bending angle.
• $G$ is Newton's gravitational constant.
• $M$ is the mass of the body.
• $R$ is how close the ray passes to its center.
• $c$ is the speed of light.

What to notice: more mass bends more, and passing closer bends more. The $c^2$ on the bottom is enormous, which is why the effect is so small for anything less than a star.

With numbers: put in the mass and radius of the Sun and the answer is 0.0000085 radians. Converted to the units astronomers use, that is 1.75 arcseconds, about the width of a coin seen from two kilometers away. That was the number the 1919 eclipse expeditions set out to measure. See Testing General Relativity.

7. The Size of a Gravitational Wave

In words: the strength of a gravitational wave is the fraction by which it stretches a length.

• $h$ is the strain, a pure number with no units.
• $L$ is some length, such as the arm of a detector.
• $\mathrm{\Delta }L$ (delta L) is the change in that length as the wave passes.

With numbers: this is the simplest equation on the page and the most astonishing. For the first wave ever detected, $h$ was about ${10}^{-21}$: one part in a thousand billion billion. Over the four-kilometer arm of the LIGO detector, that is a change in length of about $4\times {10}^{-18}$ meters, a few thousandths of the width of a proton. See Gravitational Waves.

8. Why Lasers Are Hard at High Frequencies

In words: this compares two ways an excited atom can give up its light: by itself at a random moment, or prompted by passing light.

• $A_{21}$ measures how readily an atom emits spontaneously, with no prompting.
• $B_{21}$ measures how readily it emits when stimulated by light already present.
• $\nu$ is the frequency of the light, $h$ is Planck's constant, and $c$ is the speed of light.
• The numbers 2 and 1 label the upper and lower energy levels of the atom.

What to notice: the frequency appears cubed. Double the frequency and spontaneous emission becomes eight times more dominant compared with stimulated emission. Go up by a factor of ten and it is a thousand times.

A laser works only when stimulated emission wins. This equation says that gets steeply harder as the frequency rises, which matches the history: the first device of this kind worked with microwaves, visible light followed, and X-ray lasers came last and need enormous machines. See Stimulated Emission.

9. How Crowds of Identical Particles Behave

In words: the average number of particles occupying a state with a given energy, for the family of particles that like to share.

• $\bar{n}$ is the average number of particles in the state.
• $\epsilon$ (epsilon) is the energy of that state.
• $\mu$ (mu) is the chemical potential, which acts as a reference level set by how many particles there are in total.
• $T$ is the temperature, and $k_B$ is Boltzmann's constant, which converts temperature into energy.
• $e$ is the number 2.718…, the base of natural growth.

The whole story is the "minus one." Without it, this would be the ordinary formula for a classical gas. With it, something new can happen. If the energy $\epsilon$ of a state is far above $\mu$, the exponential is large, the minus one barely matters, and few particles occupy the state. But as $\epsilon$ comes close to $\mu$, the exponential approaches 1, the bottom of the fraction approaches zero, and the number of particles in that state grows without limit.

Cool the gas far enough and exactly that happens to the lowest state: a large fraction of all the atoms pile into it at once. That pile-up is the Bose–Einstein condensate. See Bose–Einstein Statistics.

10. The Field Equations of General Relativity

In words: the shape of spacetime, on the left, is determined by the matter and energy in it, on the right.

The Left Side Is Geometry

• $G_{\mu \nu }$ is the Einstein tensor. It is a measure of how spacetime is curved, built from the metric of section 5 and from how the metric changes from place to place.
• $g_{\mu \nu }$ is the metric itself.
• $\mathrm{\Lambda }$ (capital lambda) is the cosmological constant, a fixed number giving empty space a slight built-in tendency to expand. See Cosmology.

The Einstein tensor is itself shorthand:

$R_{\mu \nu }$ is the Ricci tensor, which describes how a small ball of freely falling particles, initially at rest relative to one another, begins to change its volume, and $R$ is a single-number summary of the curvature at a point.

The Right Side Is Matter

• $T_{\mu \nu }$ is the stress–energy tensor: a table recording, at each point, how much energy is there, how much momentum is flowing and in which direction, and how much pressure and stress.
• $8\pi G/c^4$ is the exchange rate between the two sides. $G$ is Newton's constant and $c$ the speed of light.

Ten Equations in One Line

The labels $\mu$ and $\nu$ each run over the four directions of spacetime, so the line stands for sixteen equations. As with the metric, the order of the pair does not matter, which leaves ten distinct equations, all tangled together, to be solved at once at every point in space and time. (Four built-in identities link them, so six are truly independent.)

Why Gravity Is Weak

Work out the exchange rate $8\pi G/c^4$ and it is about $2\times {10}^{-43}$ in standard units: a decimal point followed by forty-two zeros. It takes an enormous amount of matter and energy to produce a small amount of curvature. Spacetime is extraordinarily stiff. That is why you need an entire planet beneath your feet to feel a gentle pull, and why gravitational waves are so faint.

Why They Are So Hard to Solve

In most of physics, the stage is fixed and the actors move on it. Here the stage is one of the actors. Matter curves spacetime, the curved spacetime redirects the matter, the moved matter changes the curvature, and so on. The equations feed back into themselves.

Einstein himself at first found only approximate solutions. Exact ones are known for a handful of simple situations: a single round mass, a uniform expanding universe, a rotating black hole. For anything messier, such as two black holes colliding, the equations are solved step by step on supercomputers.

The One-Sentence Version

Matter tells spacetime how to curve, and curved spacetime tells matter how to move. Everything else on this page, from the bending of starlight to the ripples that LIGO detects, is that sentence worked out in a particular case. See General Relativity.