RCCM / Visual field guide / 01
The Asymmetric Metric-Tensor State
One place. One moment. Three readings: capacity, sliding, twisting.
Here is where each reading lives in the local Cartesian, comoving matrix.
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
Light = positive · dark = negative. This symbolic key assumes positive eᵢ and bᵢ; the examples below use each value’s actual sign.
Read the shorthand
q = αs² = Pstatic / Pc
Remaining pressure capacity. Read it as a percentage: 100% is the full budget; 80% means four-fifths remains.
ei = α v⊥,i / c
bi = α tp Ωi
e and b are shorthand for dimensionless matrix entries, not electric and magnetic fields in SI units. α is the transverse coupling ratio; tp is the structural relaxation time; c is the acoustic phase limit.
About this guide: These pictures explain RCCM on its own terms. They combine its proposed field descriptions with the stated rules for how objects respond. They are teaching examples, not a full simulation or proof of the model. Electric and magnetic numbers below show direction and relative size; they are not measurements in volts/metre or tesla.
Five comparisons / Same scene, one change
Make the numbers do something familiar
Start with something familiar: a watch, a compass, a charged bead, a falling rock, or hair standing up. See why it responds, then compare the two sets of numbers. Each pair changes one named input. Each matrix is a reading at one place.
The rotation test / keep the 3D structure
Turn a screw upside down. It keeps the same handedness.
A right-handed screw stays right-handed when you turn it over. A circle’s “clockwise” appearance, however, switches when you look at it from the other side. That is the distinction the earlier facing-flow picture lost. A charge sign cannot be the clockwise appearance of one loop.
Imagine a garden hose joined end-to-end into a doughnut. Water can travel along the hose, around the central hole, and also swirl around the hose’s circular cross-section. Follow one speck of dye doing both: it winds like a corkscrew inside a curved pipe. Turning the whole pipe preserves the relationship between those two motions.
Around the central holeThe long way around the doughnut. Called toroidal circulation.
Around the tubeA small loop around the hose’s cross-section. Called poloidal circulation.
Both togetherA winding path in three dimensions. Its handedness belongs to the relationship between the turns.
Hold the camera still. Turn the object.
This example closes after two trips around the hole and three around the tube, making a trefoil knot. The wire doughnut is a see-through guide; the thick line is the path to follow. The thin grey arrow marks the doughnut’s axis. This is a concrete example of handedness, rather than a specified RCCM particle blueprint.
Your object · change one thing
Turning changes the pose. Reversing one winding changes the relationship. Reversing both retraces the same knot. The controls compare configurations; changing the drawn knot instantly does not demonstrate a physical process that converts one charge into another.
Now put a test charge on every side.
The charge-only electric field of a positive point source points outward in every direction. A negative source gives the inward pattern. Turn the source and its net charge stays the same. A positive test charge follows the field; a negative one responds against it. That gives like-sign repulsion and opposite-sign attraction without choosing a “front” of the source.
Positive source
Negative source
These arrows show the electric field, not fluid being pumped out of or swallowed by the core. At each place, eₓ, eᵧ and ez record the arrow’s three components. Recovering the source charge needs the field around a whole enclosing surface. Follow those arrows into the tensor →
This is the prescribed point-charge electric field, also the leading distant field of a localized net charge. A finite object can additionally have electric dipoles, magnetic moments and other structure whose interactions depend on orientation. Electric field and charge response; Gauss’s law.
What RCCM connects—and what the circle did not establish
RCCM identifies charge with a defect’s winding or handedness, and identifies the surrounding electric sector with transverse slip. Its proposed pressure mechanism uses the combined motion of the surrounding fields: where velocities reinforce, dynamic pressure rises and static pressure falls; where they cancel, the reverse occurs.
The missing step is to construct that whole surrounding field from the charged core. The Coulomb section assigns reinforcing or opposing facing flows from charge sign, but does not work through arbitrary independent rotations of the two defects. Adding a second circulation makes a rotation-resistant handedness possible; it does not, by itself, prove the required electric field or force.
The earlier two-arrow picture treated one chosen facing arrangement as the full mechanism. The replacement keeps three things visible: the internal winding, the external field it must produce, and the source rule still needed to connect them.
Exact geometry, helicity, and the source boundary
The geometry uses u = 2s and v = ±3s on a torus. These produce mirror trefoils. The two turns are measured relative to the torus’s own cycles, so rotating the torus preserves them. Reversing all flow sends s to −s and reverses both signed windings; the un-oriented knot and its handedness are unchanged. Rolfsen, Knots and Links, chapter 3C, discusses mirror torus knots. The fixture does not assign electric charge to either trefoil, and these winding counts are not charge magnitudes.
Fluid helicity measures how velocity and vorticity fit together over a volume. It is unchanged by ordinary rotations and changes sign under a spatial reflection. Reversing the entire velocity field reverses both velocity and its curl, leaving their dot product unchanged. Actual helicity depends on the full field and boundary conditions; two drawn circulation arrows do not calculate it. Bannikova, Kontorovich and Poslavsky, Helicity of the toroidal vortex with swirl. No identity Q = helicity is assumed here.
GfX §§1–3 separates transverse slip from internal vorticity; §10.6 identifies a winding sign with charge and proposes a parity/charge-conjugation connection. Condensed, “The Kinematic Origin of Charge” and “Coulomb’s Law,” supplies the torus, leakage and pressure claims. These passages do not specify this trefoil as an electron or demonstrate the sign of the interaction for every independent core orientation.
There is a further distinction: in ordinary electromagnetism, mirroring space reverses the electric-field vector and the outward surface normal together, preserving the enclosed charge. Charge conjugation instead changes the charge sign. GfX’s proposed link between those operations requires an additional physical identification; it does not follow just from the vector components changing sign. Read the focused source audit and rotation counterexample.
Try “Around the hole,” then turn the object 180°. The apparent spin changes. Switch to “Both together”: what survives the turn that the single circle could not show?
One charge · a field around it · three directions
Charge belongs to the object. The amber cells describe the field here.
Why turning a charge cannot reverse it · the 3D winding comparison ↑
A hair tip has an amount and sign of charge, called Q. At a nearby place, an electric field tells you which way a tiny positive charge would be pushed. The tensor stores that local electric direction as eₓ, eᵧ, ez. These are the three parts of one arrow.
Q · chargePositive or negative charge on the object.
e · electric field readingA direction and size at the sample location.
q · pressure capacityThe percentage of the pressure budget left there.
A positive source can give a negative eₓ reading
Stand on the left of a positive charge: it pushes a positive test charge left. Walk around to its right: now the push is right. The source charge stayed positive; your field reading changed direction.
A negative source reverses the arrows. A negative test charge responds opposite to whichever arrow it encounters.
Turn the field arrow. Watch its cells move.
In this drawing, x is right, y is up, z comes toward you. The earlier examples used only eₓ because their electric arrows lay along x. Upward arrows need eᵧ. Arrows pointing out toward your face need ez. A diagonal arrow can use all three.
The pair −eₓ / +eₓ records one component twice with opposite signs. It does not mean there is a negative charge in one cell and a positive charge in the other. The y and z pairs follow the same rule.
So where do you find the source charge?
Look at the field around a region. Arrows spreading outward indicate positive net charge inside; arrows converging inward indicate negative net charge. A single arrow does not tell you how much charge made it, or where that charge is. Even a place with no charge can sit in a strong electric field.
In ordinary electrostatics, Gauss’s law makes this precise: add the outward field through a closed surface to find the enclosed net charge. This is an additional field law, not an extra matrix cell. Source: OpenStax, Gauss’s law.
The proposed fluid underneath the electric reading
“Sliding” means sideways motion in the medium.
Put one hand flat on top of the other, then slide it sideways. That gives you a picture of shear: neighbouring parts moving past each other. RCCM gives the proposed space-filling fluid a sideways-motion component called transverse slip.
The motion belongs to the medium. A charged object has its own motion and response. A positive object responds along the electric arrow; a negative object responds against it.
For a travelling wave, “transverse” means across its direction of travel. It does not mean “always along x.” The motion can have x, y and z components, depending on the scene and your viewing frame.
eₓ = α vslip,x / c · eᵧ = α vslip,y / c · ez = α vslip,z / c
Read that as: take the local slip speed in each direction, divide by the model’s wave speed, and multiply by its coupling factor. Those three dimensionless numbers go in the amber pairs. The matrix records a local motion state; seeing how neighbouring layers differ needs neighbouring readings.
Where the source makes the connection · and what still needs a bridge
RCCM-GfX-2.tex §§1–3 defines the slip field and identifies its scaled components with the electric sector. The name and normalization establish the proposed mapping. A full derivation also needs a source/boundary rule connecting a charged defect’s winding to the surrounding field and its force on another defect.
Section 13 develops source-free transverse waves with divergence-free slip. Ordinary charge density instead appears in the sourced electric-field equation, or at excluded defect boundaries. A field may be divergence-free outside a charge while carrying nonzero flux around it. The source-free wave calculation alone does not supply that charged-boundary bridge. Condensed, “Coulomb’s Law,” proposes the handedness and interaction-pressure account illustrated above; the point-charge response examples here are declared teaching models.
Use your CSS intuition · keep the meanings attached
Same 4 × 4 shape. Different jobs.
A CSS transform matrix takes a point and places it somewhere else. This RCCM tensor describes the physical field at a place and moment. Both use rows and columns to connect directions, but the labels and operations matter.
CSS: place the point
point → transform matrix → transformed point
Its coordinate list is [x, y, z, w]. An ordinary point enters with w = 1. That extra coordinate lets one matrix handle translation and perspective along with rotation and scale.
RCCM: describe the conditions
field + object + response law → motion
Its coordinate labels are [t, x, y, z]. Time is a physical direction, not CSS’s extra w. For a charged bead, the electric field and the bead’s charge give the force; its mass and existing velocity then help determine its motion.
This matrix really does transform the square
The grid is written in rows. CSS’s matrix3d() arguments read down each column in turn. CSS points +y down the page and +z toward you; the physics drawings above choose +y up. Source: W3C CSS Transforms.
What transfers from your CSS knowledge?
Keep the habits of labelling axes, tracking units, and asking what goes in and what comes out. Changing your viewing frame mixes components. A 4 × 4 grid is a useful representation; its shape alone does not specify its meaning.
How would a simulation use both?
Read the physical field, calculate how the object responds, advance its position and orientation, then build a CSS transform to draw that pose. The CSS matrix is the drawing step. Pasting Û into matrix3d() would give the browser unrelated transform instructions.
The closer mathematical connection: transforms can change how a metric looks
A spatial CSS scale of 2 doubles a length along that axis. In a metric, which measures squared lengths, the corresponding weight is 4. For a linear spatial transform M, the Euclidean metric expressed in the original coordinates is MᵀM. This connects transforms to measurement weights without equating them.
For RCCM’s two-lower-index tensor, a change of basis acts on both indices. If old coordinate differentials equal J times new ones, then Û′ = JᵀÛJ, as in GfX §10.1. A point transform instead applies one matrix to a point. The symmetric part of Û defines the proposed interval; the antisymmetric part cancels when paired with the same displacement on both sides. Its electromagnetic meaning needs the separate physical response law.
Keep a rightward electric field fixed. Replace a positive bead with a negative one. Which way should the physics move it—and which way should the CSS renderer then draw that motion?
The original cell-by-cell symbol guide
1 / Four diagonal cells / Symmetric
Capacity → clock & rulers
Picture a remaining-capacity reservoir. Its level sets both the local clock weight and spatial metric weights. Lower q means a smaller time coefficient and larger spatial coefficients.
time: −q · space: 1/q
The reservoir reads a pressure ratio, not a force. The clock/ruler factors are square roots of metric magnitudes, relative to the stated coordinates. The minus sign marks the time signature.
2 / Six time–space cells / Antisymmetric
Sliding → electric sector
Picture silk sliding along your skin. The arrow measures transverse slip of the medium. RCCM maps this sector to the electric field; a charged probe supplies the familiar force picture.
Û0i = −ei · Ûi0 = +ei
*The probe illustrates positive-charge response to an electric field. A cell is neither charge nor force by itself. Negative charge reverses that electric response; field direction and probe motion are separate readings.
3 / Six space–space cells / Antisymmetric
Twisting → magnetic sector
Picture a paddle wheel, then a compass. Vorticity has an axis through the wheel. RCCM maps this sector to the magnetic field; a misaligned magnetic moment gives the familiar turning couple.
Ûxy = −bz · Ûyx = +bz
The xy pair records rotation about z; yz about x; zx about y. Keep the field axis, the object’s magnetic moment, and its orientation distinct. These cells are not torque values.
Read the structure
Four directions at one event. Rows and columns are t, x, y, z—not four sensor locations. Swapping an off-diagonal pair reverses its sign.
Seven local inputs, sixteen slots. One q, three slip components, three vorticity components, with the frame and constants fixed. In the regular capacity regime, 0 < q ≤ 1; q = 0 makes 1/q singular.
From a reading to “what happens next?”
A reading is local. Neighboring readings reveal pressure differences and field variation. Object velocity, charge, orientation and material response help turn those fields into motion.
Uniform squeeze ≠ directional shove. Net force adds surface pushes; torque also tracks where they act. A regional average can hide opposite motions and internal structure.
Try one change: halve q. Which cells change?
The time entry halves in magnitude; all three spatial entries double. Holding slip and vorticity fixed leaves the off-diagonals unchanged algebraically. Whether that change is dynamically possible needs the model’s evolution rules.