Same Maths, Different Names

For the general reader

Physicists describe the world with equations that say how something changes from place to place and moment to moment. There are fewer of these equations than there are subjects. The same one can turn up in the study of sound, of light, of water, and of the shape of space, and because the people working in those subjects rarely read each other, each group tends to give it its own name and its own founding paper.

When two subjects share an equation, they may be able to share results. A measurement that is easy in a water tank might answer a question that is impossible to test near a black hole. But that only works if the match is real, and “real” has degrees. Sometimes the two equations are identical. Sometimes they agree only when something is small or slow, and stop agreeing when it is not. Sometimes they merely look alike. Popular accounts tend to blur these, and the blur is where overclaiming starts.

An example. Waves running up a river against the current slow down and stop at the point where the current flows as fast as they can travel. Ocean scientists knew this in the 1940s. Light trying to leave a black hole is stopped at the horizon, and under the right conditions the mathematics of the two situations coincides; physicists have since built “horizons” in water tanks and measured classical effects near them. The first audit in this series asks how far that coincidence actually goes, who noticed it and when, and what it does and does not tell us about gravity. The answer is more interesting, and more limited, than the headline version.

Each audit is a ledger, not an essay. It lists every field that has the structure, what each one calls it, what each one assumes, and a verdict on how exact the match is, with the evidence for each entry flagged so a reader can see what was actually checked.

Start here

Why I am doing this

My own interest is not neutral. I work on a hydrodynamic picture of gravity, and the audits are partly a way of finding out, honestly, what such a picture can borrow from established physics and what it still owes. Where that interest shows on a page, it is stated there.

How each audit is built

Every audit follows the same section order so that the ledgers can be compared across pages.

1. In plain terms. A short account for the general reader of what the structure is, what has been built or measured with it, and what it does not show.
2. What is being audited. The structure stated once, in one field’s notation, with the assumptions under which it holds.
3. The ledger. One row per correspondence: the name used and the earliest source located; an explicit dictionary of what plays the role of what; the direction of implication (does a medium produce a geometry, or is a geometry being rewritten to look like a medium); the assumptions each field adds or drops and where the correspondence breaks; and a verdict.
4. Independence. Which appearances were independent discoveries and which were deliberate transplants, marked uncertain where the record does not settle it. Several fields using a structure is not several discoveries.
5. Notes by row, then a dictionary of terms that name the same object across rows.
6. Verdict. The claims the ledger supports, separated from the claims people usually attach to them.
7. What this page does not claim.
8. Extensions. Relaxations of the assumptions, each searched before being written. Where a draft called something unexplored and it was not, that is recorded.
9. References, each flagged for how far it was consulted; a change log; and a method note.

The three verdicts

A verdict is given to a particular correspondence, the one named in the row’s dictionary column, not to a field as a whole.

exact

An exact correspondence between the stated mathematical models. The equations are the same equation.

limited

A correspondence obtained only after a specified approximation or limiting procedure, which is named in the row.

analogy

Selected features correspond, such as rays, a conserved quantity or a phenomenon, but equivalence of the governing equations is not established.

A shared equation is not a shared problem. Each audit is required to say whether the correspondence extends to the complete problem, meaning boundary conditions, admissible solutions and the observables each field actually measures, or stops at the equation. For anything touching gravity, an effective spacetime that governs how waves propagate is distinguished throughout from a derivation of gravitational dynamics; the analogue-gravity literature itself insists on this distinction and the audits follow it.

Two genres are kept apart. Most audits concern a shared structure. A smaller group concerns a proposed route to quantum mechanics or gravity, and asks whether a derivation goes through rather than whether two fields share an equation. Their verdicts separate assumptions, derived results and physical interpretation, and they are listed last in the catalogue.

The full catalogue

Thirty-six candidates, grouped by kind. The first series of eleven is marked in the order-of-work note below the table; the rest is a backlog, not a promise.

Change log