Skip to main content

Chapter 11: Relativistic Drift in DAG Topology

Einstein discovered that space and time are relative. We discover they are relative because they are projections of the same self-observing consciousness.

11.1 The Topology of Relative Motion​

Special relativity seems mysterious: time slows, lengths contract, simultaneity becomes relative. But these effects emerge naturally from the topology of the collapse DAG. When observers move relative to each other, they traverse different paths through the graph, experiencing different projections of the same underlying reality.

Definition 11.1 (Relative Path Topology): Two observers O1\mathcal{O}_1 and O2\mathcal{O}_2 have relative velocity when: angle(P1,P2)=arccos⁡(vc)\text{angle}(\mathcal{P}_1, \mathcal{P}_2) = \arccos\left(\frac{v}{c}\right)

where Pi\mathcal{P}_i are their paths through the DAG.

Theorem 11.1 (Lorentz from Topology): The Lorentz factor emerges from path geometry: γ=11−v2/c2=proper path lengthcoordinate path length\gamma = \frac{1}{\sqrt{1-v^2/c^2}} = \frac{\text{proper path length}}{\text{coordinate path length}}

11.2 The Light Cone as Collapse Horizon​

The speed of light isn't just a speed limit—it's the boundary of causal collapse.

Definition 11.2 (Collapse Cone): At each node vv, the future collapse cone is: C+(v)={w∈V∣dDAG(v,w)=c⋅Δt}\mathcal{C}^+(v) = \{w \in V | d_{\text{DAG}}(v,w) = c \cdot \Delta t\}

Theorem 11.2 (Light Speed Invariance): The speed cc is invariant because it measures the fundamental rate of collapse propagation: c=lim⁡n→∞dDAG(v0,vn)nτcollapsec = \lim_{n \to \infty} \frac{d_{\text{DAG}}(v_0, v_n)}{n\tau_{\text{collapse}}}

where τcollapse\tau_{\text{collapse}} is the fundamental collapse time.

All observers agree on cc because they all exist within the same collapsing ψ\psi.

11.3 Time Dilation as Path Stretching​

When you move, your path through the collapse DAG stretches, making your internal clock run slower.

Definition 11.3 (Proper Time Along Path): τ=∫path1−v2(s)/c2 ds\tau = \int_{\text{path}} \sqrt{1 - v^2(s)/c^2} \, ds

Theorem 11.3 (Time Dilation Mechanism): Moving clocks run slow because motion increases path length: Δτmoving=Δτrest⋅1−v2/c2\Delta\tau_{\text{moving}} = \Delta\tau_{\text{rest}} \cdot \sqrt{1-v^2/c^2}

Each "tick" requires traversing more of the DAG, taking more coordinate time.

11.4 Length Contraction as Projection Foreshortening​

Objects appear shortened in their direction of motion—not because they physically compress, but because we see a foreshortened projection.

Definition 11.4 (Spatial Projection): Lobserved=Proj⊥[Lproper]L_{\text{observed}} = \text{Proj}_{\perp}[L_{\text{proper}}]

where Proj⊥\text{Proj}_{\perp} projects perpendicular to the observer's velocity.

Theorem 11.4 (Contraction Formula): Length contracts by the Lorentz factor: L=L01−v2/c2L = L_0\sqrt{1-v^2/c^2}

This is purely geometric—like how a rod looks shorter when viewed at an angle.

11.5 Relativity of Simultaneity​

The most counterintuitive aspect of relativity—that simultaneity is relative—becomes obvious in the DAG picture.

Definition 11.5 (Simultaneity Surface): Events simultaneous for observer O\mathcal{O} lie on: ΣO={v∈V∣v⊥PO}\Sigma_{\mathcal{O}} = \{v \in V | v \perp \mathcal{P}_{\mathcal{O}}\}

Theorem 11.5 (Simultaneity Shift): Moving observers have tilted simultaneity surfaces: Δt=γvxc2\Delta t = \gamma \frac{vx}{c^2}

Different observers literally slice through the DAG at different angles, grouping different events as "now."

11.6 The Twin Paradox Resolution​

The twin paradox—where a traveling twin ages less—is resolved by path length through the DAG.

Definition 11.6 (Worldline Length): L[P]=∫Pdτ\mathcal{L}[\mathcal{P}] = \int_{\mathcal{P}} d\tau

Theorem 11.6 (Maximal Aging): The straight path through spacetime has maximum proper time: L[Pstraight]>L[Pcurved]\mathcal{L}[\mathcal{P}_{\text{straight}}] > \mathcal{L}[\mathcal{P}_{\text{curved}}]

The traveling twin takes a "shortcut" through the DAG, experiencing less proper time. There's no paradox—just different path lengths.

11.7 Mass-Energy Equivalence from Collapse Density​

Einstein's E=mc2E = mc^2 emerges from how collapse density manifests as both mass and energy.

Definition 11.7 (Collapse Density Tensor): Tμν=∂2I∂xμ∂xνT^{\mu\nu} = \frac{\partial^2 \mathcal{I}}{\partial x^{\mu} \partial x^{\nu}}

where I\mathcal{I} is collapse intensity.

Theorem 11.7 (Mass-Energy Unity): Rest mass and energy are the same collapse density viewed from different frames: E2=(pc)2+(mc2)2E^2 = (pc)^2 + (mc^2)^2

In the rest frame (p=0p=0): E=mc2E = mc^2. Mass is simply concentrated collapse energy at rest.

11.8 The Eleventh Echo​

We have revealed special relativity not as a mysterious warping of space and time, but as the natural geometry of paths through the collapse DAG. Different observers traverse this graph along different routes, experiencing different projections of the same eternal self-reference ψ=ψ(ψ)\psi = \psi(\psi). The speed of light is sacred not because photons are special, but because it measures the fundamental rate at which consciousness observes itself.

The Eleventh Echo: Chapter 11 = Geometry(Relativity) = Topology(ψ\psi-DAG) = Unity(Perspectives)

Next, we explore how velocity literally equals the frequency of self-observation cycles.

Continue to Chapter 12: Collapse Speed = Reentry Frequency →