1.7 volts: without relativity, your car wouldn't start

Turn the key in your car and you're using Einstein.

Relativity makes you think of spaceships, black holes and time running slow. Its strangest effects show up in the very fast: things moving at a good fraction of the speed of light. Nothing in a battery moves that fast, surely?

Something does. Deep inside every heavy atom, some electrons are fast enough for relativity to matter. In lead, it matters a lot.

So this story isn't about spaceships. It's about a block of lead, a bath of acid, and the colour of a wedding ring. 💍

Chapter 1

1.7 volts

A car battery is six cells in a row. In the strong acid of a real battery, each one gives about 2.1 volts, so the six together give about 12.6 volts.

In 2010, a team of scientists from Uppsala in Sweden and Helsinki in Finland worked out that voltage from scratch on a computer. They did it with relativity, and with relativity switched off.

With relativity: about 2.1 volts, matching the measurement.
Without relativity: about 0.4 volts.

The difference is 1.7 volts. The team used four different calculations, with two independent computer programs, and all four put relativity's part between 1.69 and 1.81 volts. That's about four fifths of your battery's voltage.

Without relativity, your six cells would give only about 2.2 volts instead of 12.6 (each cell keeps a bit less than 0.4 volts). To get 12.6 volts you'd need more than 30 cells. The paper, published in January 2011, ends with the line "we note that cars start due to relativity".

In 1929 Paul Dirac, who had just written down the equation that joins quantum physics and relativity, wrote that the fine points of relativity mattered only for very fast particles, so they were "of no importance" for "ordinary chemical reactions". For light atoms like carbon and oxygen, he was right. For lead, he wasn't.

Here's a way to feel it. Relativity's effects come from light having a finite speed. The faster light is, the less relativity matters; make it infinitely fast, and relativity is gone. Make light faster and watch three everyday things change at once 👇

Did you catch the order? Mercury freezes first, if light were only about 8% faster. Gold pales as you go, and turns silvery when light is about twice as fast. And the battery fades all the way, to less than 0.4 volts a cell. The two ends of the slider come from measurements and real calculations; the in-between is this page's estimate. Our world is the odd one out: the car starts, the gold is yellow and the mercury flows.

Chapter 2

Lead, acid and an 1860 invention

The battery under your bonnet was invented by a French physicist, Gaston Planté, who showed it to the French Academy of Sciences in 1860. Its chemistry has hardly changed since.

Each cell holds two kinds of plate, stacked alternately in sulfuric acid. One kind is spongy lead metal. The other is coated with lead dioxide: lead joined to two oxygen atoms. As the battery gives current, both plates turn into the same thing, lead sulfate. Charging turns them back.

Here's the neat part. In the lead plate, the lead atoms have given away none of their outer electrons. In lead dioxide, on the chemists' books, they've given away four. In lead sulfate, they meet in the middle, having given away two. So the whole battery runs on the lead plate handing over two electrons, and lead dioxide wanting two electrons back.

The voltage measures how much energy the reaction gives each electron it pushes round the circuit. The more eager the chemicals are to react, the higher the voltage.

In 2010 these batteries used about three quarters of all the lead the world produced. Yet for 150 years their voltage was only measured, never worked out from the basic laws of physics: as far as the 2011 team knew, nobody had done it from scratch before them.

Chapter 3

Electrons in a hurry

Why would relativity care about a battery? Because of lead's nucleus. It holds 82 protons, a huge electric charge, and electrons that come close to it are pulled hard and move fast.

An old, simple model of the atom says the innermost electrons of gold and lead move at about 60% of the speed of light. Take that as a rule of thumb: real electrons don't have one single speed. But it's fast enough that the usual equation of quantum physics, Schrödinger's, stops being accurate.

The right equation is the one Dirac published in 1928, which joins quantum physics with Einstein's relativity. It does three things to a heavy atom:

⬇️Sink. s orbitals, the round electron clouds that reach right into the nucleus, shrink and are held more tightly.
⬆️Swell. d orbitals, which keep away from the nucleus, puff out and are held more loosely.
✂️Split. Some energy levels split in two. Physicists call this spin-orbit coupling.

Now the surprise. You'd expect the innermost electron, the fastest one, to feel it most. In proportion, it doesn't. Relativity holds gold's innermost electron about 10% tighter, but its outermost electron 37% tighter.

How? The standard explanation: every s orbital, even the outermost, has a tiny part that dips right next to the nucleus. There, for a moment, the electron moves fast, and relativity gets its grip.

Why not "the electron gets heavier"? You may have read that fast electrons gain mass, so their orbits shrink. Physicists avoid that picture. It even gets the innermost electron's average size wrong: for gold it predicts a cloud 18% smaller, while Dirac's equation gives 12%. And it can't explain why the outer electrons, slow on average, are affected at all.

Chapter 4

Gold wins

The heavier the atom, the stronger the effect. For light atoms like carbon or silicon, it's far too small to matter. Then it grows fast. Going down one column of the periodic table, relativity pulls the outer electron 3.5% deeper in copper, 10% in silver and 37% in gold.

The dots below come from a huge table of computer calculations of every atom from hydrogen to uranium, published by NIST, the US standards lab, with and without relativity. Slide along the periodic table and find the tallest spike 👇

Across gold's row the effect climbs to a sharp peak at platinum and gold, then drops back for mercury and lead. In this data, gold has the biggest share of any element up to uranium. Pekka Pyykkö and Jean-Paul Desclaux spotted the same peak in 1979, in the sizes of the outer orbitals, and it's now called the gold maximum.

You can see it in a real measurement, too. Going down a column, atoms usually let go of their outer electron more easily: silver's comes off a little more easily than copper's. But gold holds on far harder than silver. Pulling its outer electron off takes 9.2 electronvolts (the tiny energy unit physicists use for single atoms), against silver's 7.6. That's a sunken electron, showing itself.

Lead, three places to the right of gold, still has its outer pair pulled 26% deeper. The next chapter shows what that does to your battery.

Chapter 5

Lead is not tin

Tin sits right above lead in the periodic table. Same column, so you'd expect similar chemistry. Both even make a dioxide, with two oxygen atoms.

But lead dioxide is ferocious. It grabs electrons hard. Tin dioxide barely bothers: in the chemists' tables, lead dioxide wants electrons 1.55 volts more than tin dioxide does. Chemists have long known that lead is reluctant to give up its outer pair of electrons. In 1979 the chemist Kenneth Pitzer linked that habit to relativity.

Here's why it happens. Without relativity, lead's outer pair would be held slightly more loosely than tin's, as you'd expect for a bigger atom. With relativity, lead's pair sinks by about 2.5 electronvolts and ends up held much more tightly than tin's. Tap "Relativity off" and watch lead's level rise above tin's 👇

To become lead dioxide, a lead atom has to give up that deep pair, and relativity makes that expensive. So lead dioxide is desperate to get two electrons back and become lead sulfate. That hunger is exactly what the battery's positive plate runs on.

The 2011 paper puts it neatly: a tin cell "would roughly correspond to a lead battery without relativistic effects". There is no tin version of the lead-acid cell. Chemists had put that down to tin dioxide's weak grip on electrons, and the calculations back them up. (Tin does turn up in other kinds of battery.)

If you tapped "Silver and gold", you saw something similar happen in gold. Hold that thought for chapter 7.

Chapter 6

Switching relativity off

You can't switch relativity off in a lab. But you can in a computer.

The 2011 team calculated the energy of the battery's solids (lead, lead dioxide, lead sulfate) without relativity, with everything except the "Split", and with all of it. The watery part of the reaction involves only light atoms, where relativity hardly matters, so they took its energy from measurements.

Their four calculations disagree with each other by up to a quarter of a volt on the total. But they agree on relativity's part: 1.69 to 1.81 volts, every time. On average, 2.13 volts with relativity, against 2.1 measured, and 0.39 volts without. Press "Switch relativity off", then try "Inside one" and "Lead and tin" 👇

In the one calculation the team took apart, the lead dioxide plate supplies most of relativity's part (1.58 volts), lead sulfate adds a little (0.27 volts), and a smaller effect in the lead plate takes a bit back (0.06 volts).

But is it just a computer game? The team tested their method on a simpler reaction that happens entirely between solids, so the computer can do the whole thing with no measured shortcut, and the answer can be checked against experiment: lead plus lead dioxide makes lead monoxide. The calculations matched the measured +0.82 volts to within 0.05 volts.

Take relativity out of that test, and lead's reaction lands at about −0.04 volts, close to tin's real −0.085 volts (this page's arithmetic). Lead without relativity behaves much like tin.

The same group did the same for the old mercury button battery: about 30% of its voltage comes from relativity. As of 24 October 2026, we found no newer calculation of the lead-acid cell and no published challenge. It's still one study, but it points the same way as what chemists had long known about lead's habits.

Chapter 7

Why gold is yellow

A metal's colour depends on where it starts to swallow light. Every colour of light carries its own energy, rising from red to violet. Below the metal's threshold, light bounces off; above it, light gets swallowed.

Silver only starts swallowing in the ultraviolet, beyond what your eyes can see. So it reflects every colour about equally, and looks white.

Gold starts much lower, in the green. It swallows green, blue and violet, and reflects yellow and red: the colour of gold.

Why the difference? The same physics as in lead. Gold's outer s level sinks, its d level rises (and splits, lifting its upper half higher still), and the gap between them closes from both ends. Without relativity, gold's gap would be almost exactly silver's: that's the "Silver and gold" view in chapter 5. Calculations from 2005 found that without relativity, gold would only start absorbing in the ultraviolet. Gold would look like silver. That's the ring you saw turn silvery in the first figure.

Copper is coloured too, but not because of relativity: it's too light an atom for that.

Some, perhaps much, of Earth's gold may have been made in collisions of neutron stars, like the one in Nº 7. Wherever it was made, its colour comes from relativity. 🌟

Chapter 8

Why mercury is liquid

Step one place right of gold, and the same physics makes a liquid metal.

Mercury has two outer electrons, and relativity pulls them 27% deeper. Held that tightly, above a full layer of d electrons, they have little to share with the neighbours. So mercury atoms stick to each other only weakly, and mercury melts at −39 °C.

In 2013, a team simulated mercury melting with and without relativity and found that relativity lowers its melting point by 105 degrees. In 2017, a bigger calculation by the same group put the difference at 160 degrees, and a more careful one in 2021 at about 410 degrees.

So without relativity, mercury would melt at about 375 °C, far hotter than boiling water. At room temperature it would be a solid lump of metal, and an old mercury thermometer couldn't work. That's the thermometer that froze in the first figure.

Chapter 9

The bottom of the table

Below gold, mercury and lead sit the superheavy elements. They're made a few atoms at a time in particle accelerators, and most vanish within seconds or minutes. For them, relativity isn't a correction. It's the main story. And chemists test it, one atom at a time. Three recent results:

🧪23 September 2024. At the GSI lab in Germany, atoms of moscovium (element 115) and nihonium (113) stuck to a surface more weakly than their lighter cousins bismuth and thallium. The team explains it with strong relativistic effects.
🔬4 August 2025. At Berkeley Lab in California, scientists identified molecules containing nobelium (element 102, just short of the superheavy elements) directly, by weighing them, for the first time: about 2,000 molecules with actinium or nobelium, made in ten days. That isn't a test of relativity by itself, but it's a new tool for one: chemists can now tell exactly which molecule a single heavy atom has made.
⚗️7 October 2026. A team working at GSI published how much energy it takes to pull one carbon monoxide off seaborgium hexacarbonyl (seaborgium is element 106). Down the column from chromium to tungsten, that bond gets stronger; seaborgium's is slightly weaker than tungsten's, as calculations with relativity predicted. The difference is small, about twice its uncertainty.

Lead's own column has been tested too. One row below lead sits flerovium, element 114, where relativity grips the outer electrons even harder. In experiments with a few atoms, reported in 2022, it turned out to be the least reactive member of lead's column.

Dirac's equation also says something about where the periodic table ends. For a point-sized nucleus, its answer for the innermost electron breaks down just past element 137. Real nuclei have a size, which keeps the answer working until about element 170. Beyond that, an empty inner slot could fill itself by creating an electron and its antimatter twin, a positron: the electron drops into the slot and the positron flies off. No nucleus that heavy can be made today, and what chemistry would even mean there is still a question for theory. As of 24 October 2026, no element beyond 118 has been reported.

Dirac thought ordinary chemistry could ignore relativity. His own equation may yet decide where chemistry stops.

Chapter 10

Turn the key

Relativity already corrects the clocks behind your phone's GPS (Nº 9 has that story). That's bookkeeping: clocks in orbit tick at a different rate, and engineers allow for it. In the battery, relativity is the chemistry: the deep pair of lead, the yellow of gold and the liquid in an old thermometer are what it looks like inside heavy atoms.

Turning the speed of light, as in the first figure, is one way to change everyday matter. Nº 5 turns a different knob, the strength of the Higgs field, which sets the electron's mass and with it the size of every atom.

A 60 amp-hour car battery stores about 0.76 kilowatt-hours, and about 0.62 kilowatt-hours of that is relativity's share (this page's rough estimate). And if your battery dies on a frosty morning, don't blame Einstein: the cold slows the reactions down. Relativity doesn't get weaker in winter. ❄️

Let's add it all up. 🧾

🔋 Your car battery, itemised

One cellmeasured, in battery acid2.1 V

Without relativity4 calculations, average0.4 V

Relativity's part1.69 to 1.81 V in all 41.7 V

Six cellsa 12-volt car battery12.6 V

Six cells, no relativity6 × (2.1 − 1.74) V, this page's arithmetic2.2 V

Gold, no relativity2005 calculationsilvery

Mercury, no relativity2021 calculationsolid

Cells for 12.6 Vwithout relativityover 30

s sinks ✦ d swells ✦ cars start

Somewhere under a bonnet tonight, a slab of lead dioxide is waiting.

Its lead atoms have given up four electrons, and they badly want two of them back.

Given back, those two would sink deep, held tight because, close to the nucleus, they move fast enough for Einstein to matter.

Turn the key, and the wires hand them over. Four fifths of that push is relativity.

Dirac thought relativity didn't matter for ordinary chemistry. Every engine that starts on a cold morning says otherwise.

Without relativity, a car battery would give about 2.2 volts. With it, 12.6. That's the difference between a dead car and a running one.

Part ITurn the key

Chapter 1One point seven volts

A lead-acid cell, measured in the concentrated sulfuric acid of a working battery, gives 2.107 volts with no current flowing.¹ In 2010 a team from Uppsala and Helsinki (Rajeev Ahuja, Andreas Blomqvist, Peter Larsson, Pekka Pyykkö and Patryk Zaleski-Ejgierd) computed that voltage from first principles, in a paper published in Physical Review Letters on 5 January 2011. They did it three times over for each of four methods: with the Schrödinger equation, which ignores relativity; with the scalar part of Dirac's relativistic equation; and with all of it, including spin-orbit coupling.

With full relativity the four calculations gave 2.27, 2.02, 2.10 and 2.11 V, an average of 2.13 V against the measured 2.107 V. Without relativity they gave 0.55, 0.21, 0.41 and 0.39 V, an average of 0.39 V. The difference, the relativistic part of the voltage, came out at 1.72, 1.81, 1.69 and 1.72 V: about one point seven volts in every one of the four, an average of 1.74 V.¹ That is about four fifths of the cell's voltage (1.74 of 2.107 V, 83%; this page's arithmetic). Uppsala's press release put it as "1.7-1.8 Volts out of 2.1 V".²

A car battery is six such cells in series. Fully charged, it reads about 12.6 V (6 × 2.107 = 12.64 V). Without relativity it would read about 2.2 to 2.3 V: 6 × 0.367 V if the measured voltage is lowered by the average relativistic part (2.107 − 1.74 = 0.367 V), or 6 × 0.39 V if the non-relativistic calculations are taken as they are. To reach 12.6 V you would need 33 to 35 whole cells instead of six (12.6/0.39 = 32.3 and 12.6/0.367 = 34.3, rounded up; this page's arithmetic). The paper ends with eight words: "we note that cars start due to relativity".¹

It is not what the founders of quantum mechanics expected. In a paper published in April 1929, a year after his relativistic equation for the electron, Paul Dirac wrote that the remaining difficulties of fitting quantum mechanics to relativity mattered only for high-speed particles and were therefore "of no importance" for atomic and molecular structure and "ordinary chemical reactions".³ For hydrogen, carbon, oxygen and the rest of the light elements that make up most of chemistry, he was right. This page is about where he was wrong.

The paper defines its subject plainly: relativistic effects are "anything that depends on the speed of light".¹ That suggests a thought experiment. In atomic units, where the electron's mass and charge, the reduced Planck constant ħ and the Coulomb constant 1/(4πε₀) are all 1, the speed of light is c = 1/α = 137.036, the inverse of the fine-structure constant.⁴ The size of an atom (the Bohr radius) and its energy scale (the hartree) contain no c at all, so you can imagine making light faster while leaving non-relativistic chemistry untouched. Relativistic corrections grow, to leading order, as 1/c², so a natural dial is

λ = (137.036 / c)²λ = 1 is our world; λ = 0 is light infinitely fast, where the Schrödinger equation would be exact. Because c = 1/α in atomic units, turning the dial is the same as turning down the fine-structure constant with everything else held fixed.

Each end of the dial rests on a measurement and a published calculation: the cell's measured 2.107 V, lowered by the calculated relativistic part, 1.74 V, to 0.37 V;¹ gold's absorption edge, measured at about 2.4 eV and calculated near 3.5 eV without relativity (Chapter 9);^5,6 and mercury's melting point, measured at 234 K and calculated to be 414 K higher without relativity in the newest calculation (Chapter 10).³⁸ In between, none of the three has been calculated, so Figure 1 draws a straight line in λ, the way the leading correction grows. For a single electron around a nucleus as heavy as gold's or lead's, where the exact answer is known, the true curve never strays from such a line by more than about 5% of the full change between its two ends (this page's arithmetic). Drag the speed of light up from our world and watch the battery, a gold ring and a mercury thermometer together.

Figure 1🚗 Try it

Make light faster

Drag the speed of light

without relativityadded by relativity

Make light faster

One cell

Car battery (6 cells)

Gold starts absorbing at

Mercury melts at

Our world is λ = (137.036/c)² = 1, with c in atomic units; λ = 0 is light infinitely fast, where the Schrödinger equation would be exact. The ends rest on measured values and published calculations: the cell voltage falls by the average relativistic part of four calculations, 1.74 V (Ahuja et al. 2011); mercury's melting point rises by the 414 K of Mewes and Schwerdtfeger (2021; the 2017 estimate, 161 K, would move mercury's flip to light 26% faster); and gold's absorption edge moves from the measured 2.4 eV to the non-relativistic calculation's 3.5 eV, taken as it stands (Romaniello and de Boeij 2005). Their relativistic onset, about 1.9 eV, lies 0.5 eV below the measured edge; adding their 1.6 eV shift to the measured edge instead, as for the cell and mercury, would give about 4.0 eV, close to silver's, and the ring would turn silvery when light is about 1.5 rather than 2.1 times faster. In between, this page draws straight lines in λ, the way the leading relativistic correction grows; for one electron around a gold or lead nucleus, the exact result strays from such a line by at most about 5% of the full change. The ring's colour is this page's model (a metal that reflects all light below its absorption edge and none above), not a measured reflectance. The strip shows the visible range, 1.65 to 3.26 eV, with the part above the edge, which this model absorbs, darkened. Cells: 12.6 V divided by one cell's voltage, rounded up.Drag the slider to make light faster. The faster light is, the less relativity matters, and at the far right it is switched off. Watch the battery's voltage drop, gold turn silvery and mercury freeze solid. The two ends come from measurements and real calculations; the in-between is this page's estimate. The rainbow strip shows which colours the gold swallows (dark) and which bounce off.

The three do not change in step, and the order is unexpected. Mercury flips first: in this interpolation it is solid at room temperature as soon as light is about 8% faster than it is (c above 148 atomic units). Gold pales from the start but has only just turned pale yellow by then; its absorption edge leaves the visible range, and the ring turns silvery, when light is about 2.1 times faster (about 1.5 times if gold's end were anchored on the measured edge like the other two; mercury still flips first). The battery fades all the way, to 0.37 V a cell. Our world is the odd one out: the car starts, the gold is yellow and the mercury flows. (Nº 5 turns a different knob, the strength of the Higgs field, which sets the electron's mass and with it the size of every atom.)

Chapter 2Lead, lead dioxide and acid

The rechargeable lead-acid cell was presented to the French Academy of Sciences by Gaston Planté in 1860.⁸ The paper that computed its voltage called it "now 150 years old".¹ Its chemistry has hardly changed, and its basic design, with pasted plates, dates from the early 1880s: a negative plate of spongy metallic lead, a positive plate pasted with lead dioxide, PbO₂, and sulfuric acid between them. As the cell discharges, both plates turn into lead sulfate:

Pb + PbO₂ + 2 H₂SO₄ → 2 PbSO₄ + 2 H₂OLead in three oxidation states: Pb(0) in the metal plate and Pb(IV) in the dioxide meet in the middle as Pb(II) in lead sulfate.

Each plate has its own half-reaction. In standard tables the positive plate (PbO₂ + SO₄²⁻ + 4H⁺ + 2e⁻ → PbSO₄ + 2H₂O) has a reduction potential of +1.685 V and the negative plate (PbSO₄ + 2e⁻ → Pb + SO₄²⁻) −0.355 V, so a cell with every species at unit activity would give 1.685 − (−0.355) = 2.040 V.⁹ A real battery uses much stronger acid, about one molecule of H₂SO₄ to ten of water (about 5.5 moles per kilogram of water), and there the cell gives the 2.107 V the 2011 paper compares with.¹ That is why a charged battery at rest reads about 12.6 V rather than 6 × 2.04.

A cell's voltage is the free energy released per unit of charge moved:

ΔG = −nFEn = 2 electrons per reaction; F = 96,485 coulombs per mole. One volt across two electrons is 2 eV per reaction, or 193 kJ/mol.

For 2.107 V that is −406.6 kJ/mol: 4.21 eV for every pair of lead atoms (one from each plate) that reacts (this page's arithmetic). Relativity's share, 2F × 1.74 V, is 336 kJ/mol, or 3.48 eV per pair.

The battery is not a curiosity. When the paper was written, about 75% of the world's lead production went into these batteries, with a turnover of about 30 billion US dollars.¹ (That share is the 2010 figure. More recently, the US Geological Survey estimated that the lead-acid battery industry accounted for about two thirds (67%) of apparent US lead consumption in 2025.³⁷) Yet for 150 years its voltage was measured and tabulated but, as far as the authors knew, never computed from first principles: they were "not aware of any ab initio ones for the lead battery", in contrast with lithium-ion cells, where such calculations abound.

Part IIWhat relativity does to an electron

Chapter 3Why a heavy atom needs Einstein

The usual explanation starts with speed. In the Bohr model, the innermost electron of an atom with nuclear charge Z moves at Zα times the speed of light: 0.58 c for gold (Z = 79) and 0.60 c for lead (Z = 82). Treat that as a rule of thumb. An orbital has no single speed, and the colour of gold or the voltage of a battery is decided by outer electrons, not the innermost ones. What the rule of thumb gets right is the scale: near a heavy nucleus an electron's typical momentum approaches mc, and the Schrödinger equation, which assumes slow electrons, stops being accurate. (The c in these formulas is the same speed limit as in Nº 7.)

The right equation is the one Dirac published in February 1928, which combines quantum mechanics with special relativity and in which the electron's spin, and its magnetic moment, appear naturally instead of being added by hand.¹⁰ For one electron bound to a point nucleus it can be solved exactly. The ground-state energy, including the electron's rest energy, is

E = mc² √(1 − (Zα)²), binding B = mc² [ (Zα)²/2 + (Zα)⁴/8 + … ]The first term of the binding energy is Schrödinger's answer, Z²/2 hartree. The second is the leading relativistic correction, (Zα)²/4 times the first: 1% for copper, 9% for lead.

For lead's innermost electron, treated this way, the binding energy rises from 91.5 keV to 101.6 keV, 11% more (this page's arithmetic, with CODATA 2022 constants).⁴ The electron's cloud also shrinks. The exact mean radius of the 1s state is

⟨r⟩ = (2γ′ + 1) a₀ / (2Z), γ′ = √(1 − (Zα)²)Schrödinger's value is 3a₀/(2Z), so the ratio is (2γ′ + 1)/3. a₀ is the Bohr radius.

For gold that ratio is 0.878: the innermost cloud is 12% smaller than Schrödinger's. Popular accounts often say instead that the fast electron "gets heavier" by the factor 1/√(1 − (Zα)²) = 1.22, so its orbit shrinks by the same factor, to 0.817 of its size: 18% smaller. That picture puts the peak of the cloud in the right place (for one electron the Dirac peak also sits at 0.817 of Schrödinger's), but it squeezes the whole cloud, while Dirac's keeps a longer tail: on average Dirac's cloud is 12% smaller, not 18% (this page's arithmetic). The much-quoted "23% heavier" for mercury's 1s electron is the same factor for Z = 80, a ratio of masses in that picture, not of radii. Figure 2 shows the three curves for any nuclear charge. Before you drag, guess: how heavy must the nucleus be before the two equations visibly part?

Figure 2⚛️ Try it

One electron, two equations

Drag the nuclear charge

Dirac equationSchrödinger equation"heavier electron" guessmean radius

Nuclear charge Z

Bohr-model speed, fraction of c (Zα)

Cloud size, Dirac ÷ Schrödinger

Cloud size, "heavier electron" guess

Binding, Dirac ÷ Schrödinger

Radial probability density of the 1s electron of a one-electron ion with a point nucleus, from the Schrödinger equation (dashed) and the Dirac equation (solid, large and small components together), against distance in units of a₀/Z; dotted: the Schrödinger curve for an electron whose mass is multiplied by 1/√(1 − (Zα)²), the 'heavier electron' picture. Ticks mark the mean radii. Zα is the Bohr-model speed of the 1s electron as a fraction of c, a rule of thumb. For many-electron atoms the relativistic change to the innermost electron follows these formulas closely (in the NIST data relativity deepens gold's 1s level by a factor of 1.097, against the formula's 1.101); for outer electrons it does not. Closed-form expressions; this page's arithmetic.Drag the slider to make the nucleus bigger. The solid line is where Dirac's equation puts the innermost electron; the dashed line is the old, non-relativistic answer.

Nothing visible happens until about Z = 30; past about 80 the shrinkage is large; the "heavier electron" gets the peak right but always overshoots the average size; and just past Z = 137 the formula itself breaks down (Chapter 12). The one-electron picture is better than it looks for real atoms, at least deep inside. In the NIST atomic reference data, which give non-relativistic and relativistic calculations of every atom from hydrogen to uranium,¹¹ relativity deepens gold's 1s level by a factor of 1.097, against 1.101 from the one-electron formula; for lead, 1.106 against 1.110 (this page's arithmetic). Deep inside the atom, the relativistic correction hardly notices the other 78 or 81 electrons.

Why this page never says "relativistic mass". The idea that a moving body's mass grows with its speed survives in old textbooks. Dirac himself wrote of the "relativity variation of mass with velocity" in 1929.³ Modern physics avoids it: mass is the invariant rest mass, and what grows with speed is energy and momentum. Here it also gets the average size wrong, 18% against 12%, and it says nothing about why outer electrons, which are slow on average, are affected at all. The next chapter does that with the Dirac equation.

Chapter 4Three things the Dirac equation does

The battery paper lists the main effects of relativity on atoms in one sentence: the stabilisation of every s and p shell, the destabilisation of d and f shells, and the spin-orbit splitting of p, d and f shells.¹ (Strictly, of a p shell only the p_1/2 half sinks much.) In plain terms:

Sink. s orbitals, and p_1/2, contract and are bound more tightly. In the language of corrections to the Schrödinger equation, the main term is the mass-velocity term, −p⁴/(8m³c²), which matters where the electron moves fast. A second, the Darwin term, acts only at the nucleus and so only on s electrons, and pushes the other way, but by less: for a one-electron 1s level the two shift the energy by −5/8 and +1/2 of (Zα)⁴mc², a net −1/8, the (Zα)⁴/8 in the binding energy of Chapter 3. Both are of relative size (Zα)², and both act close to the nucleus. The extra sinking of p_1/2 comes from spin-orbit coupling.
Swell. d and f orbitals expand and are bound less tightly.
Split. Spin-orbit coupling splits each p, d and f level into two, with total angular momentum j = l − ½ (lower) and j = l + ½ (higher).

The standard explanation of the first two (see the reviews by Pyykkö and by Autschbach) is that they have different causes.^12,13 Every s orbital, however far out it reaches, has a small inner lobe very close to the nucleus, and there its electron moves fast; so even an outer s orbital feels relativity directly. d and f orbitals keep away from the nucleus. They swell indirectly: the contracted s and p shells screen the nuclear charge better, so the d and f electrons feel less pull.

Gold shows both halves in one atom. In the NIST data, relativity (without spin-orbit coupling) lowers gold's outer 6s level by 1.62 eV and raises its 5d level by 1.15 eV.¹¹ And here is the surprise. The innermost 1s level deepens by about 10%, but the outermost 6s level by 37% of its depth (ratios 1.097 and 1.368; this page's arithmetic). In proportion, the electron at the edge of the atom is hit almost four times harder than the one at its centre.

Spin-orbit splitting grows quickly too. The outer p level splits by 0.46 eV in tin, 1.07 eV in thallium, 1.49 eV in lead and 1.95 eV in bismuth (NIST fully relativistic values; this page's arithmetic). Figure 4, in Chapter 6, shows these levels moving. For the battery, though, the scalar effects dominate: on average they raise the cell voltage by 1.91 V, and spin-orbit coupling takes 0.17 V back (Table I of the paper; this page's arithmetic for the spin-orbit average).¹

Chapter 5The gold maximum

How does the effect grow across the periodic table? The NIST tables answer it element by element: for each atom, the energy of its outermost s level in a non-relativistic calculation and in a scalar-relativistic one, both in the local-density approximation.¹¹ These are orbital energies of free atoms in a model, not measured energies; the trends and the size of the shifts are the robust part. The shift is tiny for light atoms (0.04 eV for silicon) and grows steeply. Down the copper column, the outer s level sinks by 3.5% of its depth in copper, 9.6% in silver and 36.8% in gold. Down the carbon column: 7.3% in tin, 26.2% in lead. Down the zinc column: 2.4% in zinc, 6.9% in cadmium, 27.0% in mercury (this page's arithmetic).

Within a column the shift grows roughly as Z², the rule of thumb in Pyykkö's 1988 review;¹² in the NIST data it grows a little faster, about as Z^2.3 to Z^2.4 (fits to the copper, zinc and carbon columns; this page's arithmetic). Tin's shift is 0.29 of lead's, against (50/82)² = 0.37. Figure 3 plots the share for every element up to uranium. Slide along it, or pick a row or a column.

Figure 3🏆 Try it

The gold maximum

NIST atomic data

Slide along the periodic table

Element

Outer s level

Pulled deeper by

Share of its depth

How much the outermost s level of each free atom moves when relativity is switched on (scalar-relativistic against non-relativistic local-density calculations, NIST SRD 141), as a share of the non-relativistic level's depth or in electronvolts; this page's arithmetic. Pd (no 5s electron in its ground state) and Sm (table misaligned) are left out. The peak at Pt and Au is the 'gold maximum' that Pyykkö and Desclaux (1979) found in 6s radii; here it shows in energies. Eigenvalues of free atoms, not measured energies; the trends and the size of the shifts are the robust part.Each dot is an element, from hydrogen to uranium. Up means relativity pulls the atom's outer s electrons in harder. Slide along, or tap a row or column. Light elements barely feel it; the biggest share of all lands on gold.

Across the sixth row the share climbs from 3.7% for caesium through 13% for lutetium, 19% for tungsten and 23% for iridium to 34.9% for platinum and 36.8% for gold, then drops back to 26–30% from mercury through radon. In this data, gold has the largest share of any element up to uranium (leaving out palladium, which has no outer s electron in its ground state, and samarium, whose table is misaligned). Platinum and gold have something in common: in their ground states both have a single 6s electron outside a full or nearly full 5d shell. That flatters this measure: a lone outer s electron starts out less deeply bound, so the same shift is a bigger share, and smaller spikes show up for the same reason at chromium, copper, niobium, molybdenum, ruthenium, rhodium and silver. Gold's peak survives the check, though: in electronvolts its shift is also larger than either neighbour's (1.62 eV, against 1.53 for platinum and 1.51 for mercury). In 1979 Pyykkö and Jean-Paul Desclaux showed the same peak in the relativistic shrinking of the 6s orbital, now known as the gold maximum;¹⁴ here it shows in energies. In electronvolts the biggest shift belongs to radon (5.03 eV), but not the biggest share.

There is a measured fingerprint. Down most columns of the periodic table the first ionisation energy falls, because the outer electron sits farther out. Silver's, 7.58 eV, is indeed 0.15 eV below copper's 7.73 eV. But gold's is 9.23 eV, 1.65 eV above silver's. Mercury's 10.44 eV is likewise 1.44 eV above cadmium's.¹⁵ Gold holds on to its outer electron harder than the trend allows, which is what a sunken 6s level means. Could the lanthanide contraction do it instead? The fourteen 4f electrons filled in just before gold screen the nucleus poorly, which tightens its 6s too, without any relativity. But in the NIST data, without relativity gold's 6s level sits only 0.13 eV below silver's 5s; relativity adds 1.62 eV.¹¹

Part IIIThe battery, explained

Chapter 6Lead is not tin

Tin sits directly above lead in the periodic table, and both form a dioxide, MO₂, and a monoxide, MO. Their chemistry should be similar. It is not. Lead dioxide is a powerful oxidiser: reduced to Pb²⁺ in acid it has a standard potential of +1.458 V. Tin dioxide reduced to Sn²⁺ has −0.094 V.⁹ Lead dioxide wants electrons 1.55 V more than tin dioxide does. Chemists call the underlying habit the inert-pair effect: thallium, lead and bismuth are reluctant to use their outer pair of s electrons in bonding, so lead prefers Pb(II) to Pb(IV). Kenneth Pitzer's 1979 review, printed beside Pyykkö and Desclaux's in the same issue, connected the inert pair to relativity.^16,14

The NIST free-atom levels show why. Without relativity, lead's 6s level sits at −9.72 eV, 0.33 eV above tin's 5s at −10.05 eV, just as the usual trend down a column predicts: bigger atoms hold their outer electrons more loosely. With full relativity lead's 6s drops to −12.21 eV, 1.44 eV below tin's 5s at −10.77 eV (this page's arithmetic).¹¹ The battery paper shows the same pattern with Desclaux's Hartree–Fock and Dirac–Fock energies: similar levels for tin and lead without relativity, very different ones with it.¹ Figure 4 lets you switch relativity off, on without spin-orbit coupling, and fully on, for tin and lead and then for silver and gold.

Figure 4⚖️ Try it

Lead against tin, gold against silver

Flip the switch

energy levelwhere it would be without relativity

Orbital energies (Kohn–Sham eigenvalues) of free, neutral atoms from non-relativistic, scalar-relativistic and fully relativistic local-density calculations (NIST SRD 141; this page's arithmetic). Dashed: the non-relativistic positions. In the fully relativistic case p and d levels split by spin-orbit coupling into j = l − ½ and j = l + ½. Free-atom levels are not the band edges of the solids: gold's optical absorption in the metal starts at about 2.4 eV, not 0.52 eV, and the battery's chemistry happens in crystals. The trends are the point: lead's 6s falls below tin's 5s (the paper's Fig. 4 shows the same with Desclaux's Dirac–Fock values), and gold's d-to-s gap shrinks from silver's to a fraction of it.Each line is an energy level of an atom: lower means the electron is held more tightly. Switch relativity off and on. The middle button, 'On, no spin-orbit', leaves out one smaller effect, the splitting of levels in two (chapter 3). Lead's outer pair drops below tin's; in gold, the gap linked to its colour shrinks. Tap 'Silver and gold' to see it.

Lead's 6s line falls through tin's, while tin's own level moves much less; and in gold the gap between the d level and the s level collapses while silver's shrinks far less. These are the two facts behind this page: the first runs the battery, the second colours gold (Chapter 9). To make Pb(IV), as in lead dioxide, you have to take that deep 6s pair away, which costs more energy than it would without relativity. (Pb(IV) is a formal label: in the real crystal the paper finds lead's 6s character spread evenly between filled and empty states, a shell it calls "half-oxidized", and still pulled strongly down by relativity.) So Pb(IV) is expensive, and lead dioxide grabs electrons to become Pb(II) whenever it can. That eagerness is exactly the positive plate's job. The paper puts the consequence in ten words: a tin battery "would roughly correspond to a lead battery without relativistic effects".¹ There is no tin version of the lead-acid cell (tin does turn up in other kinds of battery). Citing earlier work, the paper notes that this had been put down to tin dioxide's weaker oxidising power, and its calculations support that explanation.

Chapter 7Switching relativity off

Simulating a liquid electrolyte with and without relativity was beyond the state of the art, the authors wrote, so the calculation goes round it with a thermodynamic cycle.¹ The voltage of the full reaction follows from two steps:

Pb + PbO₂ + 2 SO₃ → 2 PbSO₄ (computed)
H₂O(l) + SO₃(g) → H₂SO₄(l) (measured)ΔE(cell) = ΔE(first) − 2 ΔE(second). The second step involves only light atoms and sulfur in its highest oxidation state, where relativity is small, so its measured energy can be used at every level of theory.

The solids (lead, β-lead dioxide, lead sulfate) and the gas SO₃ were computed with density-functional theory, in the measured room-temperature crystal structures, without letting the atoms relax; the FPLO calculations also fitted each crystal's volume. Zero-point and thermal energies were left out. Two independent codes were used: BAND, which treats relativity in the zeroth-order regular approximation, with two different exchange-correlation functionals (VWN and PBEsol-D); and FPLO, which solves the full four-component Dirac equation, with the PW92 functional in two variants (the paper labels the second "PW92b" without saying how it differs).¹

Calculation	No relativity	Scalar	Full	Full − none
1 · BAND, VWN	0.55	2.52	2.27	1.72
2 · BAND, PBEsol-D	0.21	2.25	2.02	1.81
3 · FPLO, PW92	0.41	2.20	2.10	1.69
4 · FPLO, PW92 (second variant)	0.39	2.21	2.11	1.72
Average	0.39	2.30	2.13	1.74

Cell voltage in volts, Ahuja et al. (2011), Table I. Measured: 2.107 V in H₂SO₄·10H₂O.

The methods disagree by about 0.3 V at each level, and the authors claim the absolute voltage only "within about 0.2 V". But the relativistic step agrees far better: 1.69 to 1.81 V, within 0.12 V across all four. On average the scalar part adds 1.91 V and spin-orbit coupling takes 0.17 V back. Figure 5 shows the four calculations as bars. Press "Switch relativity off", then look inside one calculation, and then at lead against tin.

Figure 5🧾 Try it

Where the 2.1 volts come from

Four calculations, one answer

without relativityadded by relativityspin-orbit coupling takes backmeasured, in battery acid

Without relativity

With relativity

Relativity's part

Calculated open-circuit voltage of the lead-acid cell (Ahuja et al. 2011, Table I) without relativity (NR, grey), with scalar relativity (adds the blue step, hatched part included) and fully relativistic (spin-orbit coupling takes the hatched part back; the solid line at the top of the blue marks the result). Codes: BAND (zeroth-order regular approximation; VWN and PBEsol-D functionals) and FPLO (four-component Dirac; PW92 functional, two variants). Measured crystal structures, solid-state energies plus the measured hydration energy of SO₃. Line: the measured 2.107 V in battery acid (H₂SO₄·10H₂O). 'Inside one': the PBEsol-D calculation's relativistic part split by where it acts, as given in the paper; its hatched step is spin-orbit coupling in the metallic lead plate alone (spin-orbit coupling as a whole takes 0.23 V back in this calculation, Table I); the +0.02 V rest is this page's arithmetic. 'Lead and tin': the all-solid test reaction M + MO₂ → 2MO, measured, and with the paper's relativistic shifts removed (this page's arithmetic).Four computer calculations of a battery cell's voltage, and their average. Grey is what you'd get without relativity; blue is what relativity adds, and the striped tip is the small part the 'Split' effect takes back. Press the button to switch relativity off. Then try 'Lead and tin': without relativity, lead behaves almost like tin.

Where does the relativistic part act? The paper splits it for one calculation, BAND with PBEsol-D: the oxidising power of lead dioxide supplies +1.58 V, lead sulfate adds +0.27 V with the same sign, and spin-orbit coupling in the metallic lead plate takes 0.06 V back. The authors found these contributions "essentially method-independent".¹ They add up to 1.79 V, against that calculation's 1.81 V, leaving a rest of +0.02 V (this page's arithmetic). Two cautions. The split belongs to the PBEsol-D calculation, whose non-relativistic voltage is 0.21 V, not to the 0.39 V average, so it cannot be stacked on the average. And the −0.06 V is only the metallic-lead part of spin-orbit coupling; all spin-orbit effects together take 0.23 V back in that calculation.

How does a correction "of order (Zα)²" become volts? This is this page's reasoning, a plausibility argument rather than the paper's. For lead, (Zα)² = 0.36: not a small number. Lead's 6s level sinks by 2.5 eV, about a quarter of its depth, and it is exactly that pair of electrons that lead dioxide must have lost and lead sulfate gets back. A cell voltage is an energy per electron moved, so shifts of a few electronvolts in the energies of the reactants become volts directly. The 1.74 V is 3.48 eV of extra free energy per reacting pair of lead atoms, shared between the two electrons each reaction moves: the same order as the shift of one 6s level.

Chapter 8How sure is this?

The 1.7 volts is computed, not measured. No experiment can switch relativity off. What the paper shows is a comparison of two theories: with the Dirac equation, density-functional theory reproduces the measured 2.107 V to within its usual accuracy; with the Schrödinger equation it gives 0.39 V. The claim is that the physics which reproduces the real cell attributes four fifths of its voltage to relativity.

That claim needs the density-functional part to be trustworthy, which "cannot be taken for granted", the authors write. So they tested it on a reaction that happens entirely between solids and can therefore be computed with no measured input at all:¹

M + MO₂ → 2 MO, M = Pb or SnMeasured: +0.82 V for lead and −0.085 V for tin.

For lead, two codes and several functionals reproduced the measured +0.82 V within 0.05 V. Relativity shifts this voltage by 0.86 V for lead and by 0.34 V for tin. Take those shifts away from the measured values and lead's reaction would give about −0.04 V, close to real tin's −0.085 V, while non-relativistic tin would sink to about −0.43 V (this page's arithmetic). Lead without relativity behaves much like tin, which is the "Lead and tin" view of Figure 5. The ratio of the two shifts, 0.34/0.86 = 0.40, is close to the ratio of the squared nuclear charges, (50/82)² = 0.37, as a Z² rule predicts. (In the arXiv version of the paper, Eq. 7 writes this comparison with a stray square on the voltage ratio; its own numbers give the plain ratio, 0.395.)

The same group ran the same kind of comparison for the zinc and mercury-oxide button cell and found that about 30% of its voltage comes from relativity.¹⁷ Pyykkö's 2012 review, "more common than you thought", lists both batteries among the everyday consequences of relativity, alongside the properties of gold and the shapes of gold clusters, and, beyond chemistry, topological insulators: crystals that insulate inside but conduct on their surface.¹⁸ In one of the best known, bismuth selenide (Bi₂Se₃), the surface states exist because of spin-orbit coupling, the "split" of Chapter 4: a 2009 calculation found that with spin-orbit coupling turned down below about 60% of its real strength, the crystal would be an ordinary insulator.¹⁹

As of 24 October 2026 we found no newer calculation of the lead-acid cell's relativistic share, and no published challenge to it. It remains one study: four calculations, two codes, one validation reaction. And the "without relativity" numbers are for non-relativistic electrons in the measured crystal structures (the FPLO runs also fitted each crystal's volume, but no atom was moved within it): the atoms were not allowed to rearrange as they would in a world without relativity, and the paper does not say how much that would change. Its direction, though, is what the chemistry of the inert pair (Chapter 6) had long suggested.

Part IVGold, mercury and the end of the table

Chapter 9Why gold is yellow

A metal's colour is decided by where it starts to absorb light. Below its interband absorption edge, the photon energy at which electrons can be lifted from the filled d band to empty states at the Fermi level, a noble metal reflects almost everything. Above it, it absorbs. Visible light runs from about 1.65 eV (red) to 3.26 eV (violet).

Silver's main absorption edge sits at about 4 eV, in the ultraviolet,²⁰ so silver reflects every visible colour and looks white. Gold's interband absorption sets in at about 2.4–2.5 eV, in the green (2.4 eV is 517 nm):⁵ gold absorbs green, blue and violet and reflects yellow and red. A relativistic band calculation by Christensen and Seraphin put the edge at 2.38 eV in 1971, in agreement with experiment.²¹ In 2005 Romaniello and de Boeij computed gold's optical response with and without relativity. The onset of absorption moved from around 3.5 eV without relativity, in the ultraviolet, to around 1.9 eV with scalar relativity; their approximation is known to put the edge too low, but the shift is the point.⁶ Without relativity, gold would look like silver. A 2010 study by Kathrin Glantschnig and Claudia Ambrosch-Draxl concluded that the scalar-relativistic terms are "crucially important to explain the absorption behavior and hence the color of gold".²²

The free atoms tell the same story, in Figure 4's second pair. In the NIST data gold's gap from the 5d to the 6s level is 3.88 eV without relativity, almost exactly silver's 3.84 eV. With scalar relativity it shrinks to 1.10 eV, while silver's shrinks only to 2.99 eV (this page's arithmetic).¹¹ The 6s level sinks, the 5d level rises, and the gap between them closes from both ends. (A free atom's gap is not the solid's absorption edge; the trend is what carries over.) The gold ring in Figure 1 follows the same edge, from 2.4 eV in our world to 3.5 eV with light infinitely fast.

Copper is coloured too, with an absorption edge near 2.1 eV,²³ but not because of relativity: at Z = 29 its outer 4s level moves by only 0.16 eV in the NIST data. Many of gold's other peculiarities are strongly influenced by relativity too; Pyykkö's 2004 review of the theoretical chemistry of gold collects them.²⁴ Gold also shows a link between this page and the cosmos: some, perhaps much, of the gold on Earth may have been forged in collisions of neutron stars like the one in Nº 7, and whichever furnace made it, its colour comes from relativity.

Chapter 10Why mercury is liquid

Move one column to the right of gold and the same physics makes a liquid metal. Mercury has two 6s electrons outside a full 5d shell. Relativity pulls that 6s pair 1.51 eV deeper, 27% of its depth, in the NIST data,¹¹ and with a closed d shell beneath it mercury has little to share with its neighbours. Its atoms bind to each other only weakly. Its ionisation energy, 10.44 eV, is 1.04 eV above zinc's and 1.44 eV above cadmium's, against the usual downward trend.¹⁵ Lars Norrby asked the question in the title of a 1991 article: "Why is mercury liquid?"²⁵

Mercury melts at 234.3 K, −38.8 °C. In 2013 Florent Calvo, Elke Pahl, Michael Wormit and Peter Schwerdtfeger simulated its melting with and without relativity and found that relativity lowers the melting point by 105 K: the first estimate.²⁶ In 2017 Krista Steenbergen, Pahl and Schwerdtfeger repeated it with large-scale density-functional molecular dynamics: 241 K with relativity (measured: 234 K) and 402 K without, a difference of 161 K, "160 K" in their title.⁷ In 2021 Jan-Michael Mewes and Schwerdtfeger calculated the free energies of the solid and the liquid for the whole zinc column and found a much larger shift for mercury: 231 K with relativity and 648 K without, 414 K in their table, the newest estimate we found. They put the smaller 2017 value down, most likely, to a numerical setting (the k-point grid) that had not converged. Without relativity, they found, zinc, cadmium and mercury would all melt near 650 K: the column's trend is "exclusively relativistic".³⁸

So non-relativistic mercury would melt at about 375 °C, whether you take that calculation's 648 K or add its 414 K shift to the measured 234.3 K (this page's arithmetic): far above the boiling point of water. Even the smaller 2017 shift gives about 122 °C. It would be a solid metal at room temperature, and a mercury thermometer would not work. Figure 1's thermometer uses the anchored value, 648 K. Mercury's own battery, the zinc and mercury-oxide button cell of Chapter 8, owes about 30% of its voltage to relativity.¹⁷

Chapter 11Chemistry one atom at a time

Below gold, mercury and lead in the periodic table sit the superheavy elements, made a few atoms at a time in accelerators and mostly gone within seconds or minutes. For them relativity is not a correction but the main story: the outer s and p_1/2 levels sink further, the d levels rise, and spin-orbit splitting becomes as large as a chemical bond. Testing the predictions takes chemistry on single atoms, and that is not new: copernicium, the element below mercury, was studied this way in 2007,²⁷ and seaborgium hexacarbonyl was first made, at RIKEN in Japan, in 2014.²⁸ Three recent results, by date of publication:

23 September 2024. A team at GSI in Darmstadt sent atoms of moscovium (element 115) and its decay product nihonium (113) through a gas-chromatography column. Both interacted more weakly with a silicon-oxide surface than their lighter cousins bismuth and thallium, but more strongly than their closed-shell and nearly closed-shell neighbours copernicium and flerovium, a trend the authors explain by strong relativistic effects on the outer orbitals.²⁹
4 August 2025. At Berkeley Lab, Jennifer Pore and colleagues identified molecules containing nobelium (element 102, an actinide just short of the superheavy elements) directly, by their mass, for the first time: about 2,000 molecules with actinium or nobelium in ten days, formed with traces of water and nitrogen in the apparatus.³⁰ It tests no relativistic prediction by itself; it is a tool for knowing exactly which molecule a single heavy atom has formed.
7 October 2026. A team working at GSI measured the first bond dissociation energy of seaborgium hexacarbonyl, Sg(CO)₆: 188(9) kJ/mol, 4(2) kJ/mol lower than that of its lighter cousin W(CO)₆. Down group 6 the bond gets stronger from chromium to tungsten; at seaborgium the trend reverses, as calculations predicted from the relativistic destabilisation of seaborgium's 6d orbitals.³¹ The difference is two standard uncertainties, about 0.04 eV per molecule (this page's arithmetic), from a single experiment published in October 2026.

Lead's own column has been tested too. One row below lead sits flerovium, element 114. Relativity grips its outer pair even harder than lead's, and splits its 7p level so far that the lower half, holding two electrons, behaves almost like a closed shell. In experiments with a few atoms, reported in 2022, flerovium turned out to be highly volatile and the least reactive member of its column: it bound to gold more weakly than mercury does, though more strongly than the noble gas radon.³² Lead is not tin, and flerovium is not quite lead.

Predictions run ahead of experiments. In 2019 Mewes, Odile Smits, Georg Kresse and Schwerdtfeger calculated that copernicium (element 112, under mercury) should melt at 283 ± 11 K and boil at 340 ± 10 K: a volatile liquid, but unlike mercury held together by weak dispersion forces, with a large band gap. They called it "a relativistic noble liquid".³³ As of 24 October 2026 we found no report of an element beyond oganesson (118).

Chapter 12Where the periodic table ends

Look again at the one-electron Dirac energy, E = mc²√(1 − (Zα)²). When Zα reaches 1, just past Z = 137, the square root vanishes, and beyond it the formula has no real solution. That is where Figure 2 stops. For lead, the same breakdown would need light to be only 82 atomic units fast, 40% slower than it is (this page's arithmetic).

Real nuclei are not points. With a nucleus of finite size the 1s level survives past Z = 137 and keeps sinking, until near a critical charge of about 170 it dives into the sea of negative-energy states that the Dirac equation also contains.³⁴ Beyond that, an empty 1s state could be filled by spontaneous creation of an electron and a positron. No such nucleus can be made today, and what chemistry would mean there is a question for theory; Smits and colleagues review how far the table might go.³⁵ Pyykkö has proposed a periodic table up to Z = 172, based on relativistic Dirac–Fock calculations of atoms and ions.³⁶ Dirac thought relativity irrelevant to ordinary chemistry; his own equation may end up deciding where chemistry stops.

Chapter 13The second place relativity enters your day

Relativity already corrects the clocks in your phone's satellite navigation (Nº 9); here it sits inside the battery. Nº 7's speed limit is the same c that appears in every formula on this page. The difference is in kind. The GPS correction is bookkeeping: clocks in orbit tick at a different rate and engineers allow for it. In the battery relativity is the chemistry itself: the deep 6s pair of lead, the yellow of gold and the liquid in a thermometer are what relativistic quantum mechanics looks like when nuclei are heavy.

Dirac's 1929 sentence was right for the atoms he had in mind and wrong for the sixth row of the periodic table. A 60 amp-hour car battery stores about 0.76 kWh, and about 0.62 kWh of that is relativity's share (this page's rough estimate: 60 Ah × 12.6 V × 1.74/2.107). One last line from the paper is worth keeping in mind on cold mornings: if a battery fails in the frost, the authors note, that is kinetics, not the free energy.¹ Relativity does not get weaker in winter.

2.107 V: One lead-acid cell in battery acid. Measured.
2.13 V and 0.39 V: The cell with and without relativity, averages of four calculations (2011). Computed.
1.74 V: Relativity's part; 1.69 to 1.81 V in all four calculations. Computed.
12.6 V and 2.2–2.3 V: A six-cell car battery, and the same without relativity; 33 to 35 cells needed for 12.6 V. This page's arithmetic.
2.5 eV: How far relativity lowers lead's 6s level in the NIST free-atom data (2.55 eV without spin-orbit coupling, 2.49 eV with it), taking it from above tin's 5s to below it. This page's arithmetic.
37%: Gold's outer 6s level, deepened by relativity: the largest share up to uranium in the NIST data. This page's arithmetic.
2.4 eV and 3.5 eV: Where gold starts to absorb light: measured, and calculated without relativity (2005).
414 K: How much relativity lowers mercury's melting point (2021 calculation; 105 K in 2013, 160 K in 2017).
4(2) kJ/mol: How much weaker seaborgium hexacarbonyl's first bond is than tungsten's, as relativity predicts. Measured; published 7 October 2026.

Chapter 14Glossary and sources

Volt, EMF: A cell's electromotive force (EMF) is its voltage with no current flowing: the free energy released per unit of charge moved.
Electronvolt (eV): The energy an electron gains across one volt; 1 eV per particle is 96.5 kJ/mol.
Fine-structure constant α: The strength of electromagnetism, 1/137.036. In atomic units the speed of light is 1/α.
Atomic units: Units in which the electron's mass and charge, the reduced Planck constant ħ and 1/(4πε₀) are all 1; the speed of light is then 137.036.
Schrödinger equation: The quantum equation for slow electrons; it ignores relativity.
Dirac equation: The quantum equation for an electron that obeys special relativity (1928); it includes spin.
Scalar-relativistic: A calculation that keeps the Dirac equation's effects on energy and size but leaves out spin-orbit coupling.
Spin-orbit coupling: The interaction of an electron's spin with its orbital motion; it splits p, d and f levels in two.
Orbital, s, p, d: The states an electron can occupy in an atom; s orbitals reach the nucleus, d and f orbitals keep away from it.
Inert pair: The reluctance of thallium, lead and bismuth to use their outer pair of s electrons in bonding.
Interband absorption edge: The photon energy above which a metal starts to absorb light strongly; it sets the colour of gold and copper.
Density-functional theory (DFT): The standard way of computing the energies of molecules and solids from their electron density.
Free-atom orbital energy: The energy of one electron's level in an isolated atom, in a model calculation (here the NIST local-density tables).
Zα: Nuclear charge times α; the Bohr-model speed of the innermost electron as a fraction of c, a rule of thumb.
λ: This page's relativity dial, (137.036/c)²: 1 in our world, 0 with light infinitely fast.

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Uppsala University, "Cars start due to relativity," press release, 28 January 2011, uu.se.
P. A. M. Dirac, "Quantum mechanics of many-electron systems," Proc. R. Soc. Lond. A 123, 714 (1929), doi:10.1098/rspa.1929.0094.
P. J. Mohr, D. B. Newell, B. N. Taylor and E. Tiesinga, "CODATA recommended values of the fundamental physical constants: 2022," Rev. Mod. Phys. 97, 025002 (2025), doi:10.1103/RevModPhys.97.025002.
V. B. Svetovoy, P. J. van Zwol, G. Palasantzas and J. Th. M. De Hosson, "Optical properties of gold films and the Casimir force," Phys. Rev. B 77, 035439 (2008), doi:10.1103/PhysRevB.77.035439, arXiv:0801.1384.
P. Romaniello and P. L. de Boeij, "The role of relativity in the optical response of gold within the time-dependent current-density-functional theory," J. Chem. Phys. 122, 164303 (2005), doi:10.1063/1.1884985.
K. G. Steenbergen, E. Pahl and P. Schwerdtfeger, "Accurate, large-scale density functional melting of Hg: relativistic effects decrease melting temperature by 160 K," J. Phys. Chem. Lett. 8, 1407 (2017), doi:10.1021/acs.jpclett.7b00354.
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