Global vs. Local Conservation Laws

"The total charge of the universe never changes." It sounds like the strongest possible statement of conservation — and it is in fact the weakest one physics entertains. Watch what it permits: a coulomb vanishes from this laboratory bench and, in the same instant, materializes on the far side of the Moon. Total before: $Q$ . Total after: $Q$ . The global ledger balances perfectly, and yet something absurd has happened — charge has teleported, no current flowed, nothing crossed the intervening space. Nothing in "the total is constant" forbids it. What forbids it is locality, and this article is about why locality is not a stylistic preference but a relativistic ultimatum. "Simultaneously vanishes here, appears there" is not even a frame-independent sentence: in a moving observer's frame the appearance precedes the disappearance, and for a measurable interval the universe holds the wrong total. Global conservation, implemented nonlocally, contradicts itself the moment you boost. The only version of conservation that survives relativity is the local one — the quantity in any region changes solely by transport through the region's boundary — and its mathematical form is exactly the continuity equation whose grammar the previous article codified. That is the doctrine of this article: local implies global (a two-line integration), global does not imply local (the teleportation counterexample), and relativity strikes the second option down. The doctrine has teeth. It is why energy spent by the Sun is honestly in transit for eight minutes; why "the fields carry energy and momentum" is forced rather than poetic; and why, when general relativity is reached, the discovery that gravitational energy has no local density will be the correct kind of scandal.

Key takeaways

Global conservation: one number, the total, is constant. Local conservation: every region's content changes only by flux through its boundary — the continuity equation.
Local $\Rightarrow$ global: integrate over all space; the boundary flux dies for localized systems. Global $\nRightarrow$ local: teleportation balances the total while violating every local ledger.
Relativity kills global-only conservation: simultaneity is frame-dependent, so "vanish here, appear there" gives some frames an interval with the wrong total — even the global law fails outside the original frame.
Conservation with a source term is not violation but transfer: field energy's $-\mathbf{J}\cdot\mathbf{E}$ reappears in matter's ledger; the tracked pair is conserved.
The same doctrine repeats across physics — fluid mass, QM probability, heat — and famously strains in general relativity, where gravitational energy has no local density.
Locality is why "energy in flight" is real: the Sun's output is stored in space for the eight minutes it travels, auditable sphere by sphere.
KEY IDEA: local always implies global; global alone permits acausal teleportation — and relativity therefore demands every conservation law be local, i.e. a continuity equation.

The core insight

Global conservation fixes a total; local conservation explains it through a density and a flux tied by a continuity equation, $\partial_t\rho + \nabla\cdot\mathbf{J}=0$ . Local always implies global; global alone would permit relativity-violating teleportation — so physical conservation laws are local.

Where this lives today

Locality-of-conservation is a working constraint in modern practice. Numerical relativity and computational fluid dynamics distinguish conservative schemes — which enforce the discrete flux ledger exactly, cell to cell — from schemes that merely approximate it, because leaked mass or energy compounds into qualitatively wrong shocks and orbits; the Yee grid's exact discrete charge conservation (E6 twin's observation) is the electromagnetic member of this family. Detector physics runs on the same doctrine: "missing transverse momentum" at the LHC is a local audit — momentum that left the interaction region without crossing any tracker or calorimeter surface is inferred as neutrinos or new physics, a teleportation-test performed billions of times per second. The doctrine's frontier lives in quantum information and gravity. The no-communication theorem is locality-of-conservation's quantum cousin: entanglement correlates distant outcomes but cannot transport charge, energy, or signal — measured "collapse" moves no conserved quantity, which is precisely why it coexists with relativity. And quasi-local energy in general relativity — how much gravitational energy is inside this surface — remains an active research literature (Brown–York, Wang–Yau), physics still negotiating the terms on which its deepest interaction will honor the ledger rule every other force signed in 1918.

The physical picture

Think of money in a closed economy. "Global conservation" says the total cash supply is constant — an auditor's statement about one number. "Local conservation" is the far stronger rule that every dollar leaving your wallet does so through a transaction at the wallet's boundary: it cannot evaporate from your pocket and rematerialize in a stranger's account across town. The continuity equation is that ledger rule written for a density: watching any box, every unit lost is accounted for by current crossing the walls, with no escape hatch. The teleportation scenario is worth taking seriously precisely because it is not obviously absurd — a theory could be built that way. Imagine charge $q$ deleted at point $A$ at noon and inserted at point $B$ , one metre away, also at noon. No trajectory, no current, total constant. In pre-relativistic physics this is merely ugly. Relativity upgrades ugly to inconsistent. "At noon here and at noon there" is a claim about simultaneity of separated events, and the relativity of simultaneity (special relativity, recalled by name from its own track) says moving observers disagree: for an observer sweeping past at half the speed of light, the insertion at $B$ happens almost two nanoseconds before the deletion at $A$ . For those two nanoseconds her universe contains $Q+q$ — the global total itself is wrong in her frame. Conservation implemented nonlocally is not conservation at all; it is a bookkeeping coincidence of one preferred frame.

Figure

Teleportation passes the global audit in one frame and fails it outright in another — locality is the only frame-proof conservation

The remedy is continuous transport: charge that leaves $A$ and arrives at $B$ by flowing, at or below light speed, through every intermediate surface. Then every frame sees a causally ordered current, every box's books balance, and the global total is constant in all frames as a theorem rather than a fluke. This is what the continuity equation encodes, and the logic runs in one direction only — that asymmetry is the first derivation below. Once the doctrine is stated, electrodynamics is revealed to have been obeying it all along, in three escalating instances. Charge: the continuity equation, proven twice already (E4's counting, E6's Maxwell identity, both recalled by name — this article re-proves neither). Energy: the fields will be forced to carry a density $u$ and a flux $\mathbf{S}$ , because energy leaving a battery arrives in a resistor across intervening space, and locality demands it be somewhere in between; that forcing is executed over the next four articles. Momentum: same again with a tensor flux, resolving a genuine crisis (currents that violate Newton's third law) later in this chapter. Locality is not decoration on electrodynamics — it is the reason fields must be granted mechanical properties at all.

The doctrine, proved

Derivation ALocal implies global - and the converse fails by counterexample

Two claims: (i) any quantity obeying a sourceless continuity equation has a constant global total; (ii) constancy of the global total does not imply local conservation. Together they establish that the local law is strictly stronger — the content of this article's title. Strategy: (i) is an integration with one honesty clause about the boundary at infinity; (ii) is proved the only way a non-implication can be: by exhibiting a counterexample that satisfies one and violates the other.

Step 1. Let $\partial_t\rho+\nabla\cdot\mathbf{J}=0$ hold everywhere. Integrate over a ball $V_R$ of radius $R$ and apply the divergence theorem (E0, by name):

\frac{\mathrm{d}}{\mathrm{d}t}\int_{V_R}\rho\,\mathrm{d}\tau = -\oint_{S_R}\mathbf{J}\cdot\mathrm{d}\mathbf{a}.

Step 2. Let $R\to\infty$ . Honesty clause: the right side vanishes only if $|\mathbf{J}|$ falls off faster than $1/R^2$ — true for any localized system (charges confined to a finite region; radiation fields carry $\mathbf{J}=0$ ). Then $\mathrm{d}Q_{\rm total}/\mathrm{d}t=0$ : local conservation delivers the global law as a corollary, with the fine print exposed rather than hidden. (For genuinely infinite systems — a uniform current filling the universe — the global total is not even defined, and only the local law survives as a statement. The local law is primary in every sense.)

Step 3. Now the counterexample for (ii). Define, for points $A\neq B$ separated by displacement $\mathbf{L}$ :

\rho(\mathbf{r},t)=q\,\theta(t_0-t)\,\delta^3(\mathbf{r}-\mathbf{r}_A)+q\,\theta(t-t_0)\,\delta^3(\mathbf{r}-\mathbf{r}_B),\qquad \mathbf{J}=0.

The global total is $q$ at every instant — global conservation holds identically.

Step 4. Test locality: around $A$ , a small ball has $\mathrm{d}Q_A/\mathrm{d}t=-q\,\delta(t-t_0)$ with zero flux ( $\mathbf{J}=0$ ); the continuity equation fails by a point burst there, and by the mirror burst at $B$ . Could a cleverer $\mathbf{J}$ repair it? Only one that transports $q$ continuously from $A$ to $B$ — at which point the configuration is no longer teleportation but ordinary flow. The failure is intrinsic to the instantaneous redistribution, not to a poor choice of current. Global-but-not-local is thus a consistent mathematical specification — and physics must reject it on physical grounds, which is the next derivation's job.

Step 5. Note the asymmetry now established: the local law is strictly stronger, and the global law is exactly as much of conservation as survives when you throw away all spatial information. Every conservation statement in the rest of this chapter will be issued in local form first, with the global version as the trivial integral. Step 1–2 reproduce the standard localized-system argument (Zangwill §2.1); the counterexample's global constancy is checked by direct integration of the two delta terms, and its local failure matches the lone-creation obstruction of the previous article (the same point burst, staged twice).

Go deeper · Derivation B · 2 minRelativity outlaws the global-only option

Derivation BRelativity outlaws the global-only option

The counterexample above is mathematically consistent, so its rejection needs physics. We show that nonlocal conservation is not Lorentz-consistent: in boosted frames even the global total fails, for a computable interval. Strategy: take the teleportation events, apply the relativity of simultaneity quantitatively, and measure the damage. The machinery of Lorentz transformations belongs to the relativity chapter (E10) and the mechanics track (both by name); here we need only its most famous first-order consequence.

Step 1. In frame $S$ : deletion of charge $q$ at event $\mathcal{E}_A=(t_0,\mathbf{r}_A)$ and insertion at $\mathcal{E}_B=(t_0,\mathbf{r}_B)$ , simultaneous, separated by $\mathbf{L}=\mathbf{r}_B-\mathbf{r}_A$ .

Step 2. View from frame $S'$ moving at velocity $\beta c$ along $\mathbf{L}$ . The Lorentz transformation of time, $t'=\gamma(t-\beta x/c)$ , gives the two events different times:

\Delta t' \;=\; t'_B-t'_A \;=\; -\,\gamma\,\beta\,\frac{L}{c},

where the minus sign says the insertion at $B$ occurs first (for $\beta>0$ ). The events are spacelike separated — no signal connects them — so their order is genuinely frame-dependent; that is the relativity of simultaneity doing exactly what it always does.

Step 3. Audit $S'$ 's ledger: for the interval $|\Delta t'|=\gamma\beta L/c$ , both the inserted charge at $B$ and the not-yet-deleted charge at $A$ exist. The universe's total in $S'$ is $Q+q$ — global conservation is violated, not by bookkeeping error but as a direct kinematic consequence of implementing conservation nonlocally. Numbers: $L=1$ m, $\beta=0.5$ ( $\gamma=1.155$ ): $|\Delta t'| = 1.155\times0.5\times1/(3.00\times10^8) = \mathbf{1.9\ ns}$ — an eternity by modern metrology; a frame-dependent violation a fast electronics lab could in principle clock.

Step 4. Close the loophole candidates. Could the excess be unobservable? No: charge sources fields — during $\Delta t'$ , Gauss's law over a large sphere must report $Q+q$ ; the violation is electromagnetically visible, which is the field-level inconsistency the E6 twin proved (Maxwell's equations cannot even be solved with a non-conserved source — recalled by name). Could we declare $S$ the "true" frame? Only by abandoning the relativity principle wholesale. The sole surviving option: the transfer must be a causal current, $\mathcal{E}_A$ and $\mathcal{E}_B$ connected by a timelike-or-null chain of flow through every intermediate surface — which is precisely the continuity equation.

Step 5. State the moral at full strength: in a relativistic universe, conserved and locally conserved are the same word. Whatever is conserved must have a current; whatever has no possible current cannot be conserved. This is why the discovery of each conservation law in this chapter is operationally the discovery of a flux — Poynting's vector, Maxwell's stress tensor — and why those objects are physics, not notation. the simultaneity shift $\Delta t'=\gamma\beta L/c$ is the standard textbook result for spacelike-separated simultaneous events; the numerical value recomputes as 1.93 ns; and the argument's structure (order-reversal for spacelike pairs) matches the causality discussions in the relativity references cited below.

Figure

Local implies global — nested audit surfaces pass the same flux until, at infinity, there is nothing left to cross

The template across physics

The doctrine is track-wide, and electrodynamics is its showcase rather than its owner. The same sentence — density rate plus flux divergence equals transfer — is conservation of mass in a fluid ( $\rho$ and $\rho\mathbf{v}$ : the mechanics track's continuity equation, by name), conservation of probability in quantum mechanics (the probability current of the QM track, by name — locality is why probability doesn't teleport between detector arms), and conservation of heat in a rod (Fourier's law feeding the diffusion equation — the same mathematics that gave the previous chapter its magnetic skin depth). In every case the physics enters through what the flux is made of; the law is the shared grammar. Within this chapter the template will be conjugated three more times, with one structural upgrade each: energy (scalar density $u$ , vector flux $\mathbf{S}$ , transfer $-\mathbf{J}\cdot\mathbf{E}$ — the next four articles), linear momentum (vector density $\mathbf{g}$ , tensor flux $-\mathsf{T}$ — the stress-tensor articles, by name), and angular momentum ( $\mathbf{r}\times\mathbf{g}$ and its flux — closing with the Feynman disk). The escalation in rank is forced: a vector's worth of conserved stuff needs a vector of currents, one per component. And one honest frontier: general relativity strains the doctrine in a famous way. Gravitational energy admits no local density — the equivalence principle lets you transform gravity away at any point, taking any candidate energy density with it; energy conservation survives only quasi-locally or for isolated systems (ADM/Bondi masses), and the 1915–1918 crisis over this — Einstein, Hilbert, Klein, and the analysis that drove Noether's theorems — is among the deepest episodes in theoretical physics. Within special-relativistic field theory, where this track lives, the doctrine is exact; the GR subtlety is flagged here so it is a known landmark rather than a future ambush (details belong to the relativity track, by name).

Limiting cases

case	statement	reading
localized system, $R\to\infty$	$\mathrm{d}Q_{\rm tot}/\mathrm{d}t=0$	global law as corollary; needs $J$ falling faster than $1/R^2$
finite box	$\mathrm{d}Q_V/\mathrm{d}t=-\oint\mathbf{J}\cdot\mathrm{d}\mathbf{a}$	the auditable form: every loss at the walls
sourced ledger	$\partial_t\rho_X+\nabla\cdot\mathbf{J}_X=\sigma_X$	transfer to a named partner, not violation (previous article's taxonomy)
static	$\nabla\cdot\mathbf{J}=0$	flux in = flux out everywhere; steady transport
non-relativistic physics	locality optional in principle	but no known law uses the loophole — continuity everywhere
general relativity	no local gravitational $u$	conservation quasi-local only; the honest frontier

Common trapA constant total is not conservation — and a source term is not violation

Twin misreadings, opposite directions. First: "the total never changes" is the weak statement — it permits acausal teleportation and fails under boosts; only the continuity equation states what physicists mean by conservation. When a theorist says energy is conserved, they are asserting the existence of a flux. Second, inverted: seeing $\partial_t u+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ , students conclude field energy "isn't really conserved." It is transferred — the same term credits the mechanical ledger, and the tracked pair balances exactly. Violation means a burst with no partner and no flux; transfer means the partner is named. Physics contains many sourced ledgers and, as far as ever measured, zero violated ones.

Worked examples

1 · Pricing a teleportation: how wrong the boosted books go (confidence builder)

Suppose $q=1.0$ nC could be deleted at $A$ and simultaneously inserted at $B$ , $L=1.0$ m away (frame $S$ ). An observer moves along $AB$ at $\beta=0.60$ . Quantify her ledger violation.

Step 1. Simultaneity shift: $\gamma=1/\sqrt{1-0.36}=1.25$ , so $|\Delta t'|=\gamma\beta L/c = 1.25\times0.60\times1.0/(3.00\times10^8) = \mathbf{2.5\ ns}$ , with the insertion first.

Step 2. During those 2.5 ns her universe carries excess charge $+q$ . Is it detectable in principle? Gauss's law on any surface enclosing both sites: total enclosed $Q+q$ — an extra nanocoulomb sources an extra flux $q/\varepsilon_0=113$ V·m. Real instrumentation resolves nanocoulombs and nanoseconds comfortably; the violation is physical, not philosophical.

Step 3. Scale the horror: teleport the same charge across a synchrotron hall, $L=100$ m: $\Delta t'=250$ ns. Across the Earth–Moon distance: $\Delta t' = 1.25\times0.6\times3.84\times10^8/3\times10^8 = \mathbf{0.96\ s}$ — a full second of wrong-total universe. The violation grows linearly with distance, which is why nonlocal conservation is not a small-correction sin but a structural one.

Step 4. The repair: any causal transport — a wire, a beam, a wave — replaces the spacelike pair with a chain of timelike steps; every frame then sees deletion-before-arrival and a current in between, and both audits (global and every-box local) pass in all frames. Conservation and causality are not two constraints but one.

2 · The Sun's energy, audited in flight (exam level)

The Sun radiates $L_\odot=3.83\times10^{26}$ W. Use locality to audit the energy between Sun and Earth: check the flux at 1 AU against measurement, and weigh the energy in transit.

Step 1. Local conservation in steady state ( $\partial_t u = 0$ on average, no absorption in vacuum) demands the same power cross every concentric sphere: $\oint\mathbf{S}\cdot\mathrm{d}\mathbf{a}=L_\odot$ for all $r$ . At $r=1\,\mathrm{AU}=1.496\times10^{11}$ m:

\langle S\rangle=\frac{L_\odot}{4\pi r^2}=\frac{3.83\times10^{26}}{4\pi(1.496\times10^{11})^2}=\mathbf{1361\ W/m^2},

which is the measured solar constant — the number every energy article of this chapter uses as an anchor. The inverse-square law of illumination is revealed as the continuity equation's steady-state clause: nothing created, nothing lost, so the same total spreads over growing spheres.

Step 2. Weigh the light in flight inside Earth's orbit. Energy takes $r/c = 1.496\times10^{11}/3.00\times10^8 = 499$ s to cross, so the vacuum inside 1 AU holds $E_{\rm transit} = L_\odot\times(r/c) = 3.83\times10^{26}\times499 = \mathbf{1.9\times10^{29}\ J}$ of solar energy currently between Sun and Earth — about the Sun's entire output for eight minutes, stored in empty space as field energy density $u$ . Global-only conservation would let this line item be zero (energy "arrives when it arrives"); locality makes it a real inventory, and Energy Density of the Electromagnetic Field (two articles ahead, by name) prices it per cubic metre.

Step 3. Cross-check the inventory independently: $u=\langle S\rangle/c$ at every radius (radiation in transit at $c$ — the transport identity the Poynting articles will derive), integrated: $\int_0^{r} (L_\odot/4\pi r'^2 c)\,4\pi r'^2\,\mathrm{d}r' = L_\odot r/c$ ✓ — same $1.9\times10^{29}$ J by the density route. Two ledgers, one answer: the hallmark of a genuinely local conservation law.

Step 4. The relativity coda: mass-energy leaves the Sun at $L_\odot/c^2=4.3\times10^{9}$ kg/s — four million tonnes per second, every gram of which is accounted for on every intervening sphere en route. The Sun does not "lose mass globally"; it exports it, through surfaces, on schedule.

Origins · 1 minAlbert Einstein and how this was discovered

Albert Einstein

1879–1955 · Bern, Berlin, Princeton — who made locality the law of laws

Einstein appears in this article twice over. The 1905 special relativity paper demolished absolute simultaneity, and with it the innocence of "vanishes here, appears there at the same moment" — after 1905, every conservation law had to carry a current or carry contradiction. But the sharper story is 1915–1918: with general relativity complete, Einstein and Hilbert discovered that gravitational energy refused to be localized — candidate energy densities could be transformed away at any point, and the status of energy conservation in GR became a genuine crisis, debated in print by Einstein, Hilbert, Klein, and Levi-Civita. Hilbert handed the puzzle to Emmy Noether, whose 1918 theorems — conservation laws from symmetries, and the special "improper" character of conservation in generally covariant theories — resolved it with mathematics that now underwrites every conservation statement in physics. Einstein called her intervention "penetrating mathematical thinking"; this article's doctrine, that conservation means continuity equations and continuity equations mean currents, is the settled form of the argument they were having.

History

1747

— Franklin's charge conservation: the first global ledger of electrical science

1847

— Helmholtz's energy-conservation memoir: one currency across mechanics, heat, electricity

1884

— Poynting and Heaviside localize energy flow: conservation acquires a flux

1905

— special relativity abolishes absolute simultaneity: nonlocal conservation becomes self-contradictory

1915

— general relativity triggers the energy-localization crisis

1918

— Noether's theorems: symmetries generate conservation laws; the GR case is structurally special

1962

— ADM formalism gives isolated-system energy in GR its rigorous quasi-local meaning

Going further

References. Griffiths, Introduction to Electrodynamics, 4th ed., §8.1–8.2 (Charge and Energy; Momentum) · Jackson, Classical Electrodynamics, 3rd ed., §6.7–6.8 (Poynting's theorem; conservation laws) · Purcell & Morin, Electricity and Magnetism, 3rd ed., §4.2 (charge conservation) · Zangwill, Modern Electrodynamics, §2.1 and §15.1 (continuity, conservation laws). Key formulas used in this article: Poynting's theorem $\partial_t u+\nabla\cdot\mathbf{S}=-\mathbf{J}\cdot\mathbf{E}$ with energy density $u=\tfrac12(\varepsilon_0E^2+B^2/\mu_0)$ and Poynting vector $\mathbf{S}=\tfrac1{\mu_0}\mathbf{E}\times\mathbf{B}$ ; momentum density $g_i=\varepsilon_0(\mathbf{E}\times\mathbf{B})_i$ with the Maxwell stress tensor as its flux. No numerical constants beyond $\varepsilon_0$ and $\mu_0$ enter the local-implies-global / global-not-implies-local logic. The quantitative simultaneity results are: $\Delta t'=\gamma\beta L/c = 1.9$ ns at $\beta=0.5$ , $L=1$ m; $2.5$ ns at $\beta=0.6$ ; $0.96$ s at Earth–Moon separation. Solar audit: $L_\odot/4\pi(1\,\text{AU})^2=1361$ W/m²; transit inventory $L_\odot r/c=1.9\times10^{29}$ J; solar mass-loss rate $4.3\times10^9$ kg/s.

Builds on: Charge Conservation and the Continuity Equation · Divergence and the Divergence Theorem · Maxwell's Equations in Vacuum.

Leads to: Poynting's Theorem · Conservation of Linear Momentum in Electrodynamics · The Maxwell Stress Tensor.

Isn't the teleportation counterexample too silly to need refuting — who would propose it?

History did, repeatedly, in respectable dress. Newtonian gravity was an action-at-a-distance theory — energy bookkeeping across instantaneous forces was genuinely nonlocal, and it worked fine because Galilean physics has absolute simultaneity. Weber's electrodynamics (Maxwell's chief rival) had instantaneous potentials; early quantum mechanics flirted with nonlocal energy transfer in the BKS theory (1924), which proposed energy conserved only statistically — killed within a year by Bothe–Geiger coincidence experiments. The counterexample is the distilled form of real proposals, and relativity is the reason they all failed the same way.

Where exactly is the energy between the battery and the bulb — is that question even meaningful?

Locality makes it meaningful and answers it: in the fields filling the space around the circuit, with density u and flux S — that is the entire content of the Poynting articles ahead. The startling part (previewed honestly here, delivered in Poynting Vector in Static Field Configurations by name) is that the answer is not "inside the wire": the flux flows through the field outside the conductors, entering the bulb's filament sideways. Global conservation would never force such a question; local conservation forces both the question and its uncomfortable answer.

Does quantum entanglement violate local conservation — the collapse seems instantaneous?

No conserved quantity moves. Entangled measurements produce correlated outcomes at spacelike separation, but the no-communication theorem guarantees the local density matrix — hence every local expectation of charge, energy, momentum — is unchanged by the distant measurement. Each conserved quantity was locally present all along; what "collapses" is knowledge and correlation, neither of which carries a current. Nature appears to enforce a hierarchy: correlations may be nonlocal, but transport never is. Bell-test physics (QM track, by name) lives exactly on this line.

What is the precise relationship between Noether's theorem and this article's doctrine?

They are the two halves of "conservation laws are never accidents." Noether's first theorem: every continuous symmetry of the action yields a current satisfying a continuity equation — time translation gives energy, space translation momentum, gauge phase gives charge (previous article's FAQ, by name). So symmetry explains why conserved quantities exist and hands you their fluxes constructively. This article supplies the converse pressure: relativity demands that anything conserved must have such a current. Together: symmetry ⇒ local conservation law, and no other kind of conservation law is physically admissible. Noether's second theorem — the identity-type conservation of gauge theories — was the E6 twin's closing theme, and its GR version resolved the 1918 crisis in the physicist band above.

If gravitational energy has no local density, is energy conservation actually false in our universe?

It is subtler, not false, and the subtlety is well charted. For isolated systems (a merging black-hole binary in asymptotically flat spacetime) total energy is rigorously defined and exactly conserved — LIGO's observed waveforms balance the ledger: ~3 solar masses of energy radiated as the horizons' mass deficit. What fails is only the assignment of a pointwise gravitational energy density, because the equivalence principle lets any candidate be transformed away locally. Cosmology adds a genuine open ledger: photon redshift in an expanding universe has no global time-translation symmetry to protect a total. The honest summary: within special-relativistic physics, conservation is local and exact; in GR it is quasi-local and boundary-anchored; and knowing which regime you are in is part of literacy.

Why does the momentum version need a tensor flux when charge got away with a vector?

Counting components. Charge is one number; its flow through a surface needs one number per surface orientation — a vector J does it. Momentum is itself a vector: each of its three components is separately conserved and separately transported, so the flux must answer "how much x-momentum crosses a surface with THIS orientation" — nine numbers, a rank-two tensor. That object is the Maxwell stress tensor, built later in this chapter (by name), and its diagonal/off-diagonal entries are pressures and shears: conserved-quantity bookkeeping and continuum mechanics turn out to be the same subject.