Hamilton's principle/Explainers/Gravity from an action

Technical version: The Einstein-Hilbert action

Hamilton's principle: map
Reading the volume How to read this volume Notation Glossary Equation index Figure gallery
Reading by level Start here: Comparing paths The mathematics: Functionals and the space of paths Stationary is not least: The Jacobi condition and conjugate points The applications: The brachistochrone The history: Hamilton 1834 and 1835
Sources Sources Chronology
Related Noether's theorems On Physical Lines of Force

This page explains how gravity comes out of the same method. It uses no mathematics. The technical page it goes with is named above.

The demand

Two stationary light paths around a mass.

Electromagnetism from an action built electromagnetism from one requirement: a certain adjustment must have no physical effect, at every point independently.

Gravity comes from a similar demand, applied to something more basic.

Nothing physical may depend on your choice of coordinates.

Coordinates are labels. Where you put the origin, which way the axes point, what units you use — all of that is your bookkeeping, not the world's. And you should be free to choose differently in different places.

Insist on that, write the simplest cost rule that respects it, and general relativity comes out.

What is being varied

Free fall as a straight path in a curved geometry.

There is one thing to decide first: what the theory is about.

For electromagnetism it was a field spread through space. For gravity, the thing that varies is the shape of space and time itself — specifically, the rule for measuring distances and durations from place to place.

That rule can differ from place to place. Where it does, the geometry is curved.

So gravity is not a force acting in space. It is a statement about the shape of space, and the cost rule is a cost rule for shapes.

Free fall is going straight

The boundary term in one dimension and in a field.

The consequence for motion follows without extra assumptions.

An object with nothing pushing it travels as straight as the geometry allows. That is the shortest-path idea from Where the minimum fails, now applied to space and time together.

Near a massive body, the geometry is curved. Straight in that geometry looks bent from a distance. That bending is what we call gravity.

A falling apple is not being pulled. It is going straight through a curved geometry. Nothing distinguishes free fall from doing nothing at all — which is why astronauts float, and why you feel nothing during a fall until you land.

The edge of the problem

Something happens here that did not happen anywhere else in this volume, and it is worth flagging.

Every calculation so far quietly discarded a leftover term at the boundary of the region considered. The usual justification is that nothing is being nudged at the edge, so the term vanishes.

For gravity it does not vanish. The cost rule involves shapes in a way that leaves something behind at the boundary even when the edge is held fixed.

The fix is to add a matching term by hand that cancels it.

This is not tidying. That boundary term turns out to carry the energy of the gravitational field, and treatments that discard it get the equations right and lose everything that depends on the value of the total. The warning issued much earlier in this volume — that a leftover term must be argued away rather than assumed away — is paid for here.

Where energy goes wrong

Now the deepest consequence, and it is the one Noether's second theorem was written for.

For ordinary theories, energy conservation is a real dynamical statement with a definite local meaning. You can point at a region and say how much energy is in it.

For gravity you cannot.

Every attempt to define the energy stored in the gravitational field at a point gives an answer that depends on your choice of coordinates. Change coordinates and the energy density changes, though nothing physical has.

For a long time this looked like a failure of ingenuity. It is not. Noether proved that any theory with this kind of local symmetry has exactly this property: its energy conservation becomes an identity, true by construction, rather than a dynamical law.

There is no local gravitational energy to find. The searching failed because the thing does not exist.

That is a rare kind of result — a proof that a research programme cannot succeed, with the reason given.

Energy has not stopped being conserved. For a system that settles down at large distances, a perfectly good total energy can be defined, and it lives in exactly the boundary term of the previous section. What has gone is the ability to say where the energy is.

A prediction you can photograph

The volume's central point turns up in the sky.

A massive galaxy bends light passing near it. Light from a distant object behind it can therefore reach us by more than one route, curving round different sides.

We see several images of the same object. Astronomers photograph them regularly.

Each route is a legitimate straightest path. Only one is the quickest. The others are perfectly good paths that are not shortest — which is exactly what this volume has been saying since its first page, now in a photograph.

The routes have different lengths, so the light on them left at different times. When the distant object flickers, the images flicker one after another, sometimes weeks apart.

What the method achieved

Stand back and look at what was assumed.

Not that gravity attracts. Not an inverse square law. Not that mass bends light. Only that the laws must not depend on your choice of labels, and that the cost rule should be the simplest one respecting that.

Out came a theory that predicts the bending of light, the precession of Mercury, the slowing of clocks in a gravitational field, and gravitational waves — all confirmed, some of them decades later.

That is the strongest case for the whole approach in this volume. Write down what a theory must ignore, and the theory writes itself.

Where to go next

The volume's final argument, explaining why any of this works at all, is at Sum over histories.