Abstract
Relativity is built on spacetime intervals and transformations that preserve them. In Shunyaya Symbolic Mathematics, each coordinate and energy–momentum component carries an alignment channel a, embedding stability directly into the fabric of spacetime. Under collapse phi(m,a) = m, all constructions reduce to classical relativity.
Symbolic spacetime point (events with alignment)
Represent an event as
X = ( (t, a_t), (x, a_x), (y, a_y), (z, a_z) )
Notes
• Magnitudes (t, x, y, z) are classical coordinates (units: s, m, m, m).
• Alignments (a_t, a_x, a_y, a_z) encode stability of each coordinate.
• Constants such as c are treated as (c, +1) by default.
Interpretation: an event is not only “where and when” but also “how stable” those coordinates are relative to the centre.
Symbolic interval (Minkowski form with alignment)
Classical Minkowski interval (signature +,-,-,-):
s^2 = c^2 * t^2 - x^2 - y^2 - z^2
Symbolic interval (componentwise squares via M2, sums via oplus, differences via ominus):
S^2 = (c, +1)^2 otimes (t, a_t)^2
ominus [ (x, a_x)^2 oplus (y, a_y)^2 oplus (z, a_z)^2 ]
Square via repeated M2-multiplication:
(m, a)^2 = ( m^2 , tanh( 2 * atanh(a) ) )
Properties
• Collapsed magnitude phi(S^2) equals the classical s^2.
• The alignment channel of S^2 distinguishes stable vs fragile separations even when s^2 matches.
Symbolic Lorentz transformation (1D boost along x)
Classical (gamma = (1 - v^2/c^2)^(-1/2)):
t' = gamma * ( t - v * x / c^2 )
x' = gamma * ( x - v * t )
y' = y
z' = z
Symbolic (scalar products via otimes; sums/differences via oplus/ominus):
(t', a_t') = (gamma, +1) otimes (t, a_t)
ominus (gamma * v / c^2, +1) otimes (x, a_x)
(x', a_x') = (gamma, +1) otimes (x, a_x)
ominus (gamma * v, +1) otimes (t, a_t)
(y', a_y') = (y, a_y)
(z', a_z') = (z, a_z)
• Magnitude channel reproduces the classical Lorentz boost.
• Alignment channel follows the weighted rapidity-average of contributing terms, so misaligned space–time components can reduce net stability after a boost.
Collapse: if all alignments are +1, this reduces to the classical transformation.
Naming clarity (two rapidities).
• Kinematic rapidity (relativity): eta = atanh(v/c).
• Alignment rapidity (Shunyaya): u_a = atanh(a).
These are distinct variables.
Symbolic four-vectors and invariants
Four-position:
X^mu = ( (c*t, a_t), (x, a_x), (y, a_y), (z, a_z) )
Four-momentum:
P^mu = ( (E/c, a_E), (p_x, a_px), (p_y, a_py), (p_z, a_pz) )
Classical invariant:
E^2 = (p*c)^2 + (m*c^2)^2
Symbolic invariant (componentwise M2 products):
(E, a_E)^2 = [ (p, a_p) otimes (c, +1) ]^2
oplus [ (m, a_m) otimes (c^2, +1) ]^2
• Collapse recovers Einstein’s relation exactly.
• The alignment channel distinguishes strong-but-fragile from strong-and-stable energy configurations at equal classical energy.
Symbolic curvature (GR sketch)
Attach alignment to each metric component:
g_s[mu,nu] = ( g[mu,nu] , a[mu,nu] )
• Levi-Civita connection, Riemann tensor, and contractions are formed on the magnitude channel as usual; the alignment channel is transported by an auxiliary evolution (e.g., diffusion/relaxation in coordinate space; see PDE templates).
• Field equations gain an alignment companion that indicates “stability of geometry.”
Collapse: if all a[mu,nu] = +1, we recover the classical geometric content.
Boundary, synchronization, and constants
• Synchronization conventions (e.g., Einstein synchronization) carry default alignment +1 unless declared otherwise.
• Units and universal constants (c, G, hbar) are (constant, +1) unless a model explicitly studies their effective stability.
Worked mini-example (same s^2, different S^2)
Events
A = ( (t, +1), (x, +1), (0, +1), (0, +1) ), with t = 2, x = 1
B = ( (t, +1), (x, -0.6), (0, +1), (0, +1) ), with same magnitudes
Classically:
s^2 = c^2 * 4 - 1
Symbolically: both share the same collapsed s^2, but B’s spatial Nearo (a_x = -0.6) lowers the alignment of S^2, flagging a more fragile spacelike separation.
Numerical and modeling notes
• Work in alignment rapidity:
u_a = atanh( clamp(a, -1+eps, +1-eps) ), eps = 1e-6
Present a = tanh(u_a).
• For linear combinations (e.g., Lorentz sums), use the global addition rule (U,W accumulator) to maintain associativity and boundedness for alignment.
• Constants multiply as (constant, +1) otimes (variable, a_var).
• Always report both the collapsed invariant and the alignment channel.
Takeaway
Symbolic Relativity extends spacetime physics with alignment-aware coordinates, intervals, and transforms:
• Lorentz transformations preserve classical invariants under collapse yet expose drift-induced stability changes.
• Energy–momentum gains an alignment axis, separating strong-but-fragile from strong-and-stable states.
• Curvature inherits an alignment field, hinting at centre-aware companions to Einstein’s equations.
Thus relativity becomes not only the geometry of spacetime, but also the geometry of stability within spacetime.
Navigation
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Disclaimer
Observation only. Results reproduce mathematically; domain claims require independent peer review. Defaults: mult_mode = M2, clamp_eps = 1e-6, |a| < 1 with rapidity u = atanh(a). All formulas are presented in plain text. Collapse uses phi(m,a) = m.
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