Part 9 of 13. A symmetry whose parameter varies from point to point gives an identity among the equations of motion, not a new conservation law. That identity is gauge freedom, and it is why energy in general relativity is hard to pin down.
Noether's 1918 paper has two theorems. The first (Parts 5 to 8) takes a global symmetry, with a constant parameter, and gives a conserved current. The second takes a local symmetry, whose parameter is an arbitrary function, and gives an identity that the equations of motion satisfy whether or not they are solved.
Put a phase on each lattice site and an angle on each link. The hopping energy is , and the sum of link angles around a square is the lattice field strength . Both survive a local phase change if the links shift by .
Figure 1. A 7 by 5 lattice with seeded random phases, shown after one local shift. Needles are site phases, link colour is link angle, square shading is . A local shift turns each needle by its own angle and recolours the links; the squares and both gauge-invariant energies stay put. The naive hopping term , with no link, survives only the global shift.
The tree-gauge button spends the freedom: one shift per site but the first zeroes the 34 links of a spanning tree, and the 24 links left are fixed by the 24 square sums. Everything in the links beyond the field strength was gauge.
In the continuum this is Maxwell's theory, with , unchanged by . Its Euler-Lagrange equations satisfy
Figure 2. A seeded random potential on a 32 by 32 grid, differentiated by finite differences. Left: the size of , the amount by which Maxwell's equations fail. Right: its divergence , on the same colour scale. For the right panel is blank to rounding error. Switch to the symmetric , which has no gauge symmetry behind it, and the divergence is larger than itself.
With a source, , the identity forces : the field equations have no solution for a source that fails to conserve charge.
The general theorem shows up in a small mechanical model. Two masses on a line, plus a variable in the role of :
Shifting both positions by any function while leaves unchanged. With the Euler-Lagrange expressions, the change in the action is ; integrating by parts, for every ,
Figure 3. Three seeded random trial paths , not solutions, with . Thin lines: , and , each computed from its own finite differences of the paths. Thick line: their sum, flat at zero. The term () is the model's version of a photon mass: it is still symmetric under a constant shift, but not a time-dependent one, and the sum turns into .
Three equations with one identity are only two equations. The -equation, , has no second derivatives, so it constrains the initial data, as Gauss's law does. Nothing fixes .
Figure 4. Left: worldlines of the two masses (time upward) for several seeded choices of , all with and the same positions and velocities at . Each is an honest RK4 solution of the Euler-Lagrange equations. Right: the separation for every choice. The worldlines fan out, but the separation, which the gauge shift cannot touch, is the same curve every time.
Maxwell's theory is the same with for : one identity, one free function , Gauss's law as the constraint, and a determined future only for gauge-invariant quantities like .
The Einstein-Hilbert action is unchanged by any smooth change of coordinates, a local symmetry whose parameter is a vector field . The second theorem gives the contracted Bianchi identity
true for every metric. Of the ten Einstein equations , effectively only six are independent: the identity ties them together with four differential relations, and the four equations act as constraints on the initial data, as Gauss's law does. Four relations match the four free functions in .
The identity also forces , a covariant divergence, not a partial one that integrates to a conserved total. A conserved scalar energy needs extra structure: a timelike Killing vector, or an asymptotically flat spacetime, where the ADM and Bondi energies are defined at infinity. An expanding universe has neither.
Figure 5. A flat expanding universe filled with one fluid of pressure . The Friedmann equation and , here , integrated together by RK4. Plotted on log axes: the energy in a box expanding with the universe, against the scale factor . Dust (dashed) keeps its energy; radiation loses it as . The loss equals the work , but there is no outside system to receive it.
Hilbert and Klein brought Noether to Göttingen in the spring of 1915 for her invariant theory, while Hilbert worked toward the variational field equations he presented that November. Energy in the new theory stayed puzzling; Klein published his letters with Hilbert on it in 1918. Noether, refused a formal appointment because she was a woman and lecturing under Hilbert's name, answered it in Invariante Variationsprobleme (Nachr. Ges. Wiss. Göttingen, 1918): a local symmetry gives an identity among the field equations, not a conserved energy.
Charge conservation still comes from the first theorem, through the global part of the phase symmetry (Part 8); the local part makes it a consistency condition on sources. Act III turns to the algebra Noether and her students built in the 1920s, starting with the failure of unique factorisation in .