General relativity has no stage: the gravitational field and the geometry of space are the same object
Every field theory before general relativity was written on a page that was already there. Maxwell’s equations describe a field spread over a spacetime whose distances and durations were fixed before anyone wrote down a single equation. The field is the actor and the spacetime is the stage, and the stage never responds. That arrangement is so natural it is hard to notice you are making an assumption.
General relativity dissolves it. The metric is not the arena in which the gravitational field lives; the metric is the gravitational field. There is no second geometry underneath telling you how far apart two points are while the first one bends. Take away the metric and you are left with a bare manifold, which knows about smoothness and about topology and about nothing else: no lengths, no angles, no durations, no light cones, no way to say which of two events came first.
Einstein spent three years fighting this consequence and losing. The argument he constructed against himself, the hole argument of 1913, is the first demo below, and its resolution is the single most important input to everything that follows in this module. Then comes the distinction that gets confused more often than any other in this subject, between writing a theory in arbitrary coordinates and building one that supplies its own geometry. The lesson ends in 1986, with the change of variables that turned general relativity into a theory with the phase space of an SU(2) gauge theory, and handed the quantum programme something it could actually work with.
Take a solution, choose a region containing no matter, and drag every field inside it while leaving everything outside untouched. The result is another solution with identical initial data. Read the outcome two ways, in the two readings Einstein himself held three years apart, and watch which numbers change.
Read this way, the drag is a catastrophe. The two configurations are identical everywhere outside the hole, so they share the same initial data, and yet they differ inside: the field equations do not determine what happens. This is why Einstein abandoned general covariance between 1913 and 1915.
Einstein built the hole argument in 1913, while he and Marcel Grossmann were working on the Entwurf theory, and he built it as a proof that generally covariant field equations were impossible. The logic looks airtight. If the equations hold in every coordinate system, then dragging the fields inside a matter-free region by a smooth map that is the identity on the boundary sends solutions to solutions. The two solutions agree on the entire initial-data surface outside the hole and disagree inside it. Specify the past completely and the future is still not fixed. Determinism fails.
Einstein believed this for three years and produced field equations that were not generally covariant because of it. The Entwurf theory gave the wrong perihelion precession for Mercury. In November 1915 he abandoned the restriction, wrote down the covariant equations, got Mercury right in a week, and had to explain to himself what was wrong with the hole argument. The answer he sent Paul Ehrenfest in January 1916 is that all our experience consists of coincidences of events, and both configurations contain exactly the same coincidences. Two light rays either meet or they do not, and dragging the fields cannot change that.
Stated more sharply, and this is the form John Stachel revived in 1980 and John Earman and John Norton sharpened in 1987: a point of the bare manifold has no physical identity. Nothing distinguishes it from its neighbours until fields are laid on top, and if you drag the fields you have simply relabelled which point is which. The question “what is the metric at the point p” is not a question about the world. The question “what is the metric where those two particles collide” is. Localisation in general relativity is relational, and it is relational all the way down: things are located with respect to each other, never with respect to a manifold that exists independently of them.
The consequence for a quantum theory is immediate and severe. Every physical observable must be invariant under dragging the fields around, which rules out anything of the form “the field at coordinate x”. Whatever a quantum state of geometry turns out to be, it cannot be an amplitude assigned to field values at points. This is exactly why the states in the later lessons of this module are abstract graphs. A graph has no coordinates for a diffeomorphism to shuffle. All it has is combinatorics, and combinatorics is what survives.
Five theories, four structures each. For every one, ask which structures were fixed by hand before anything was solved and which came out of the equations. Then delete the fixed ones and see what is left standing. The trap is the third entry, which is fully covariant and completely background dependent.
still flat spacetime, given in advance
still the Minkowski metric, now with position-dependent components. Its Riemann tensor is identically zero and it obeys no equation of motion: it is an input, dressed up.
Christoffel symbols that are now nonzero, but computed from a metric that was known before any equation was solved
the only things that actually obey a field equation
| theory | generally covariant | background independent |
|---|---|---|
| Newton | no | no |
| Maxwell | no | no |
| SR, any coordinates | yes | no |
| graviton expansion | yes | no |
| general relativity | yes | yes |
Fully generally covariant. Every equation is a tensor equation, valid in rotating coordinates, accelerating coordinates, any coordinates at all. Covariance has been bought entirely with notation.
Erich Kretschmann pointed out in 1917 that general covariance, on its own, has no physical content whatsoever. Any theory can be written covariantly. Take special-relativistic electromagnetism, switch to rotating coordinates, and every equation grows Christoffel symbols and covariant derivatives and starts to look like general relativity. Nothing about the physics has changed. The Minkowski metric is still there, still handed over before any equation is solved, still solving nothing itself. Its Riemann tensor is identically zero no matter how baroque the coordinates get. Covariance was bought with notation.
Elie Cartan made the same point from the other direction in 1923 by rewriting Newtonian gravity as a generally covariant theory of a curved connection on a spacetime with absolute time. Newton-Cartan theory is covariant and has curvature and is still Newtonian physics, complete with a preferred slicing into instants and a flat spatial geometry nobody ever solved for. Two centuries of absolute space survived the translation into tensor notation unharmed.
The property that actually distinguishes general relativity is background independence: no geometric structure is fixed before the equations are solved. The metric appears in the theory only as an unknown. It carries energy, it radiates, it responds to matter and matter responds to it, and there is nothing behind it playing the role that Minkowski space plays for a photon. The manifold and its topology and differentiable structure are still assumed, which is a real assumption and one the spin foam models later in this module try to weaken, but the geometry is not.
This is exactly the property the perturbative programme of the previous lesson gives up in its first line. Splitting the metric into a fixed background plus a small ripple requires choosing the background first, and everything afterwards depends on that choice: the vacuum, the notion of a particle, the causal structure, the very propagator you expand in. Loop quantum gravity’s bet is that this is not a technical convenience to be corrected later but the source of the trouble, and that a quantum theory of gravity has to be constructed without ever writing down a background metric. That commitment is what makes the rest of the module look so unfamiliar: no Fock space, no particles, no fields at points, and states that are combinatorial objects rather than functions on a spacetime.
One point of space, described by a frame rather than a metric. Rotate the frame internally and every component of the triad and the connection changes while the geometry does not move at all. Then slide the Barbero-Immirzi parameter and watch the connection rescale while every classical quantity ignores it and the quantum area gap does not.
| 1 | 2 | 3 | |
|---|---|---|---|
| x | 0.605 | 1.372 | 0.126 |
| y | -0.704 | 1.042 | 0.000 |
| z | 0.078 | 0.099 | 1.009 |
| x | y | z | |
|---|---|---|---|
| 1 | 0.089 | -0.116 | -0.011 |
| 2 | 0.060 | 0.051 | -0.008 |
| 3 | -0.013 | 0.004 | 0.135 |
| x | y | z | |
|---|---|---|---|
| x | 1.026 | -0.643 | -0.278 |
| y | -0.643 | 1.412 | 0.128 |
| z | -0.278 | 0.128 | 1.616 |
Spin the frame and watch which numbers move. Every component of E and of A changes, because the internal rotation is a genuine transformation of the variables, and yet the ellipsoid does not shift by a pixel and the last digit of the metric, the volume element and the patch area stay put. That redundancy is what the Gauss constraint removes: nine numbers in the triad, six in the metric, three rotations in between. The gamma slider is the stranger case. It rescales the connection while leaving every classical quantity alone, which is why gamma cannot be measured by any classical experiment, and then it multiplies the area gap in the bottom panel, which is the one number in this demonstration that quantum geometry would let you observe.
To quantise a theory canonically you first split spacetime into space and time, which in a diffeomorphism-invariant theory is a choice of gauge rather than a fact. Arnowitt, Deser and Misner did this for general relativity in the early 1960s. The configuration variable is the spatial metric on a slice and its conjugate momentum is built from the extrinsic curvature, the rate at which the slice is bending into the future. The dynamics is entirely constraints: four of them per point, and no evolution equation at all. The trouble is the Hamiltonian constraint. Written in these variables it contains the Ricci scalar of the spatial metric and inverse powers of the determinant, so it is not polynomial in the canonical variables, and thirty years of attempts to define the corresponding quantum operator got nowhere.
In 1986 Abhay Ashtekar changed variables. Instead of the metric, use a frame: three vector fields Eai, the densitised triad, carrying an internal index i that runs over a three-dimensional internal space. The metric is recovered as a product of two triads, which is the crucial move, because a quantity quadratic in the new variable is only linear in the old one. Instead of the extrinsic curvature, use a connection built from the spin connection of the triad plus the extrinsic curvature. In these variables the constraints of general relativity become polynomial and the phase space is that of an SU(2) Yang-Mills theory, with the triad playing the role of the electric field. A century of gauge theory technique suddenly applies to gravity.
Two prices, both visible in the demonstration above. First, a frame carries nine numbers where a metric carries six, and the extra three are an internal rotation that changes every component of the triad without changing any geometry. This is the new SU(2) gauge symmetry, and the Gauss constraint is what removes it. It is the same constraint that will later force the nodes of a spin network to be invariant tensors. Second, the free parameter. Ashtekar’s original 1986 connection was complex, which is what made the Hamiltonian constraint genuinely polynomial, but it forced awkward reality conditions on the quantum theory. Fernando Barbero found the real version in 1995 at the cost of an extra term in the constraint, and the real family comes with a free parameter multiplying the extrinsic curvature. Giorgio Immirzi pointed out in 1997 what that parameter does.
What it does is genuinely strange, and the module will keep running into it. Different values of the Barbero-Immirzi parameter are related by a canonical transformation, so classically it is invisible: no measurement of a length, an area, a volume or a trajectory can reveal it. But the transformation relating two values is not implemented unitarily in the quantum theory built on these variables, so the resulting quantum theories are inequivalent, and the parameter multiplies every geometric eigenvalue. It sets the size of the smallest quantum of area. The usual way of fixing it is to demand that counting horizon states reproduce the Bekenstein-Hawking entropy, which gives a number near 0.2375; that calculation is the subject of a later lesson, and both the value and the method of arriving at it have been disputed. A theory that has to be told the size of its own quanta by an outside argument is not finished.
Even with the variables fixed, be clear about what has and has not been achieved. The Gauss and spatial diffeomorphism constraints are solved cleanly in the quantum theory, and the results that follow from them, the discrete area and volume spectra, are the solid core of the subject. The Hamiltonian constraint is not. Thomas Thiemann constructed a well-defined operator for it in 1996, and there is still no agreement that it has the right classical limit, nor a unique choice among the available versions. The problem of dynamics is the open problem of loop quantum gravity, and it has been open since the beginning.
There is one more thing to say about the connection, and it is what the next lesson begins with. A connection is not something you can evaluate at a point in a quantum theory without producing the usual infinities. What a connection is naturally built to do is transport a vector along a curve. Smear it along curves instead of at points, and the resulting variables are finite. That is where the loops come in.