A universe without gravitational waves

Imagine being told that you can make quantum gravity easier by removing one direction of space. That sounds like a reasonable bargain. Fewer coordinates, fewer components, fewer things to go wrong.

The surprise is that you lose more than a direction. In three spacetime dimensions, vacuum Einstein gravity has no local propagating gravitational waves. There is no little graviton wave packet you can launch across an otherwise empty universe. The usual local dynamical protagonist has left the stage.

It is tempting to close the book here. A theory of gravity with nothing waving: surely this is a very elaborate way of describing nothing?

The rest of this series is about why that conclusion fails.

A small room tells you very little

In three dimensions, the Riemann curvature tensor is completely determined by the Ricci tensor. Once the vacuum Einstein equations fix the latter, they fix the local curvature. With a negative cosmological constant, every sufficiently small vacuum patch looks like anti-de Sitter space. With zero cosmological constant, it looks flat.

This is a local statement. The word matters. A flat sheet and a cylinder can look identical to an observer who only examines a tiny neighborhood. Walking around the cylinder reveals information that the tiny neighborhood could never supply. Spacetime has more complicated versions of this distinction: identifications, noncontractible loops, and boundaries.

The early work of Deser, Jackiw, and ’t Hooft made the physical importance of global geometry especially concrete through particles and conical defects. A point source can change how a locally flat region fits together around it.^[1] The particle example includes matter, of course; it is not a vacuum counterexample. Its purpose is to train the eye. Curvature in the immediate neighborhood is not a complete inventory of the world.

What survives the subtraction?

Suppose you wanted to tell a friend exactly which universe you had built. “Every small patch has this curvature” would not be enough. You would also need to say how the patches join, which loops can shrink, and what happens at the edge. Even in the absence of matter, those questions survive.

And once there is an edge, another distinction appears. A change that is merely a relabeling in the interior can alter physically meaningful data at the boundary. Removing bulk waves does not automatically remove boundary excitations. We will meet this distinction in the next article, where it becomes an unexpectedly large symmetry algebra.

Witten’s 1988 treatment turned the special structure of three-dimensional gravity into a concrete quantization program.^[2] But “a tractable formulation” and “a uniquely specified quantum theory answering every gravitational question” are different achievements. Boundaries and the allowed global configurations still need to be chosen.

Why this is a good place to get confused

Three-dimensional gravity is useful because several questions that are tangled together in higher dimensions can be pulled apart. Which difficulty came from local waves? Which came from topology? Which came from summing over geometries? Which came from deciding what a boundary observer is allowed to measure?

That is also why calling it a toy model can be misleading. A toy is useful when removing details makes the mechanism visible. Here the mechanism that becomes visible is how much of gravity was never about local waves in the first place.

One qualification will stay with us: throughout this route, “no local gravitons” refers to ordinary vacuum Einstein gravity. Add suitable higher-derivative terms or matter and the story changes. We have chosen a particular laboratory, not established a theorem about everything called gravity in three dimensions.

For now, keep the room small and the questions large. The room looks simple. What happens at its boundary does not.