RCCM / Illustrated cheat sheet / 0229 September 2026
A field at a point. A charge inside a surface.
The asymmetric tensor gives a local reading of the continuous fluid. Electric charge belongs to the surrounding field pattern and the region it encloses.
How many samples determine charge? One can suffice with known spherical symmetry. Four can check a suggestive pattern. No fixed number of isolated samples guarantees the answer for an unrestricted field. In general, measure the electric flux over a complete closed surface.
Scope: this guide uses the newer RCCM-GfX-2.tex for the tensor definitions. The charge pictures are prescribed electrical examples. The missing derivation from fluid motion to a charged cavity is identified in §7.
01 / Place the sampler in the fluid
The field continues between the cavities.
A cutaway of a three-dimensional fluid volume. The dots mark selected readings; the field also exists between them. The pale cavities are excluded interiors, not parcels of fluid.
Imagine hovering beside a cavity with a tiny sampler. Move it into the fluid above, below, behind, or between cavities. At each place and moment it returns a tensor:
Û(x, t)
x names the location; t names the instant. The matrix records the model’s local capacity, slip and internal vorticity there.
A cavity occupies a region. Its boundary and the surrounding flow extend over many locations. One tensor sample is not an object record containing that cavity’s charge, orientation and history.
A numerical grid stores a finite set of these field readings. Its boxes divide the calculation; they do not divide the medium into little physical objects.
Point → local state. Neighbourhood → spatial variation. Closed surrounding surface → enclosed charge.
02 / Read one event
Four rows are four coordinate directions.
Ûμν(x, t) = Sμν + Aμν
t
x
y
z
t
−q
−ex
−ey
−ez
x
ex
1/q
−bz
by
y
ey
bz
1/q
−bx
z
ez
−by
bx
1/q
This is GfX §3’s local Cartesian, comoving form. Its 16 entries use seven independent shorthand values: q, three e components and three b components. The four rows are not four spatial samples.
Capacity / symmetric q = αs² = Pstatic/Pc The remaining pressure budget sets the model’s temporal and spatial interval weights.
Slip / time–space ei = α v⊥,i/c A normalized rate of transverse displacement. It carries motion information at this instant.
Twist / space–space bi = α tp Ωi Normalized internal Clebsch vorticity. Ω is defined from that rotational phase, not automatically from the entire velocity field.
Unit check: q, e and b are dimensionless. α is a coupling ratio, c a speed and tp a relaxation time. Uppercase E below is a calibrated electric field in volts/metre. A conversion from e to E must be supplied before calculating charge in coulombs.
CSS comparison: a shear transform maps positions to new positions. This matrix records local field quantities. Opposing contributions can cancel when added at the same location; opposite readings at different locations remain two parts of a spatial pattern.
03 / Your four-sample experiment
Opposite arrows can surround the same charge.
In a radial exterior, the left and right sides of one positive cavity point in opposite directions. The meaningful comparison is each arrow’s direction relative to the cavity.
Exact radial patterns for two separate isolated fixtures, viewed in the same coordinate orientation. For two cavities sharing a medium, these arrows describe the nearby readings when each cavity’s contribution dominates. Other fields add to those readings.
Compare corresponding sides. The positive fixture’s left arrow reverses in the negative fixture. So does its right arrow. Your four samples detect this reversal.
Keep the symmetry assumption. Left and right readings do not tell you what happens above, below, in front or behind. They establish the full pattern only when a justified model supplies the missing directions.
04 / Surround a region
Count the signed field through a closed skin.
Place an imaginary sphere entirely in the fluid, enclosing the cavity. At each tiny patch, feel which way is outward. Measure the electric field’s component along that direction, multiply by patch area, then add over the whole sphere. Surface-flux definition.
More outward → Q > 0
A positive radial fixture. Every patch contributes positively.
More inward → Q < 0
A negative radial fixture. Every patch contributes negatively.
Balanced → Q = 0
A uniform field crosses an empty region. Entry and exit cancel.
Green meshes are measurement surfaces, not physical shells. Amber arrows show electric field, not material trajectories. A zero result can also enclose equal positive and negative charges.
The ordinary electrical readout · Gauss’s law
Qinside = ε0 ∮S E · n dA
S: the complete closed surface. n: the outward unit normal. E · n: the signed outward component. dA: a tiny patch area. ε0: the SI conversion factor.
The sum uses the total field, including contributions from outside the surface. Exterior charges can change individual arrows while contributing zero net enclosed flux. Gauss’s law reference.
To ask about one cavity: choose a surface enclosing only it. To ask about a whole system, enclose all of it. The surface can change shape while keeping the same charges inside.
To ask about local charge density: measure spatial variation: ρcharge = ε0 ∇ · E. Divergence compares neighbouring readings. A single nonzero E value does not establish local charge. Local charge equation.
Units: ε0 [C/(V·m)] × E [V/m] × area [m²] = Q [C]. The symbols q (capacity) and Q (charge) name different quantities.
05 / How many readings?
The missing information sets the sample count.
Samples
What they can establish
1, with known spherical symmetry
At known radius r, one signed radial reading Er represents the entire sphere: Q = 4π ε0 r² Er. The symmetry and calibration must already be justified.
2 per cavity; 4 across two cavities
Checks the left/right pattern in §3. Inferring charge still depends on radial symmetry and control of other field contributions.
6, or any other fixed finite number
No exact guarantee for an unrestricted field. A field can agree at every sampled point and differ between them. Three spatial dimensions do not imply six sufficient readings.
A complete closed surface
The mathematical flux specifies enclosed charge. In computation, approximate it with enough samples to resolve the surface pattern to a stated accuracy.
For a numerical estimate
QN ≈ ε0 Σa=1…N (Ea · na) ΔAa
Each sample needs its location, field, outward normal and represented area. ΔA is the patch’s area; uneven sampling requires appropriate weights. Compare readings at the same time in one specified frame. Transform samples taken in different local comoving frames into that shared frame first.
Cover the whole surface, including its far side.
Refine where the field varies rapidly; account for measurement noise.
Estimate numerical error using stated smoothness or resolution assumptions.
Report a sign only when the error interval excludes zero. For example, +0.2 ± 0.5 does not resolve polarity.
Repeated estimates becoming similar are useful evidence. Without bounds on unsampled variation, apparent convergence alone is not a guarantee.
Same charge. Harder sampling.
Move one known unit charge from the centre to 90% of a sampling sphere’s radius. The enclosed charge stays +1. The nearby surface receives a concentrated flux that coarse sampling can miss.
The same displaced source, sampled at 32 and 128 locations. Faded dots lie on the far side. These are numerical point-charge benchmarks, not derived RCCM cavities.
Samples
Centred
Displaced
32
1.000
0.738
128
1.000
0.851
512
1.000
0.985
2,048
1.000
1.000
Estimated Q / true Q, rounded to three decimals. There is no universal “2,048 is enough” rule; this result belongs to this fixture and sampling scheme.
Reproducible method: unit sphere, unit source at a = (0.9/√3)(1,1,1); normalized E(p) = (p − a)/|p − a|³. For i = 0…N−1, zi = 1 − 2(i + ½)/N, φi = iπ(3 − √5), pi = (√(1 − zi²) cos φi, √(1 − zi²) sin φi, zi). The equal-weight estimate is QN = (1/N) Σ E(pi) · pi; the centred case uses a = 0.
06 / Keep the mental pictures connected
Time, transverse motion and handedness.
A paused state can retain motion.
Your vibrating ring can occupy the same shape while moving in opposite directions. A geometry-only export loses that distinction; a state containing velocities preserves it.
GfX’s e entries retain a slip rate. A constant slip rate is ongoing relative motion, not automatically acceleration. In a fixed local inertial chart, a straight tilted trajectory through your snapshots records constant velocity; a bending trajectory records changing velocity. A steady circulating field can still accelerate its parcels.
Time is a physical tensor index here. CSS matrix3d() uses a homogeneous coordinate for point transforms. It does not make its fourth coordinate physical time.
“Transverse” needs a reference.
For a plane shear wave travelling left-to-right, the local displacement can be up-and-down. Transverse compares local wave motion with wave propagation. It does not mean sideways relative to that parcel’s own motion.
In GfX §3 the slip is spatial relative to the continuum’s timelike four-velocity. In §13.1, its divergence-free plane-wave mode is transverse to the wavevector. A general cavity flow need not have one propagation direction.
Following a parcel removes its local bulk translation; it still leaves you free to turn your spatial axes.
Two circulation senses can encode handedness.
Your donut picture combines circulation along the large ring (toroidal) and around the tube (poloidal). In a helical flow, reversing either sense reverses handedness; reversing both retraces the same helix.
Rotating the whole pattern preserves its handedness. This supplies a candidate structure to investigate. Dynamical stability, protected topology and electric charge still need their own connecting laws.
The drawing shows flow directions around a toroidal boundary. A local e arrow, a local e · b value, or a knot’s appearance is not a supplied charge formula in GfX.
Turn your axes; keep the same charge.
Both vectors use unit magnitude here to compare directions; the field is normalized for this illustration.
On a fixed surface patch, E and the outward normal n change their coordinate components together. Their dot product stays the same:
(R E) · (R n) = E · n
A change of inertial velocity also mixes electric and magnetic readings. Charge density can change, while the total charge of the same isolated system remains invariant. Positive and negative names follow electrical convention, not the observer’s chosen orientation.
The simple matrix in §2 is a comoving form; a boost need not preserve its displayed diagonal pattern. Charge and relativity reference.
Time–space entries are not a picture of a warped trajectory. The tensor is a local state; an evolution law determines what happens next. In the quadratic interval, the antisymmetric part cancels because it is paired with the same displacement twice. GfX’s symmetric q weights carry that interval’s clock/length effect; slip can also affect q through the pressure budget.
07 / The remaining RCCM construction
Connect the electrical readout to the cavity flow.
GfX §3 supplies local slip/vorticity entries. Section 13.1 supplies source-free propagation. Section 10.6 assigns polarity to boundary winding. The next construction must connect a maintained cavity and its surrounding fluid to a measurable, nonzero Q.
Specify the full flow and boundary. Include the fluid outside the cavity and its continuing motion, with material conservation and stated boundary conditions.
Define the electrical readout. State how dimensionless tensor entries become E. Reconcile §3’s bare slip with §7.5’s dressed slip before using either.
Calculate enclosing flux. Produce an explicit charged example whose calibrated exterior gives Q ≠ 0 while the material balance remains satisfied.
Connect winding to polarity. Define the winding quantity and show why its two signs produce opposite Q that survives spatial reorientation.
Electric flux is not a count of fluid volume leaving the cavity. The conserved material current belongs to the total fluid motion. Equating it with electric flux requires an extra physical identification. Otherwise the original “where is the fluid created or destroyed?” problem has simply been put into an equation.
A useful boundary check: a radial field C r̂/r² has zero divergence away from its excluded core yet flux 4πC around it. The exterior alone does not explain what maintains that flux at the inner boundary. Conversely, a smooth divergence-free field throughout a filled volume has zero enclosing flux. This is why the cavity boundary matters.
Keep a positive source where it is. Slide your imaginary sampling sphere sideways until the source lies outside. Electric arrows still cross the sphere. What happens to the signed sum over its complete surface?
Check after predicting: its enclosed charge is now zero. The entering and exiting contributions cancel; individual field readings can remain large.
Reading route
Definitions, fixtures and the open connection.
RCCM-GfX-2.tex: §§1–3 define the field and matrix; §7.5 changes the slip definition; §10 treats covariance and winding; §13.1 supplies the source-free wave model; §14 gives the radial inverse-square exterior. This guide does not fill its missing charge construction from the older condensed manuscript.
OpenStax · Electric flux and Gauss’s law: the ordinary electrical measurement used for the prescribed charge fixtures. The finite-sampling experiment is calculated explicitly in §5.