Extraction of an Effective Lorentzian Metric

Starting Assumptions (What Is Allowed)

We assume only:

• Events = causally committed transitions
• A directed acyclic graph (DAG)
• No coordinates
• No metric
• No manifold

Goal: recover an effective Lorentzian metric \((g_{\mu\nu} )\) at large scales.

Causal Order Without Geometry

From the DAG:

• \(( e_i \prec e_j )\) if there exists a directed causal path
• If neither \(( e_i \prec e_j ) nor ( e_j \prec e_i )\), the events are spacelike

This yields:

• Timelike relations
• Spacelike relations
• A lightlike boundary defined by maximal propagation speed

Causal structure appears before metric structure, exactly as in GR.

Proper Time from Chain Weights

Define:

• A timelike curve = a maximal chain of composable events

• Proper time along a chain:

  \[ \tau(\gamma) = \sum_{e_i \in \gamma} w(e_i) \]

  where \(w(e_i)\) is transition cost or latency.

This yields:

• Clocks as repeating subgraphs
• Time dilation as differing accumulated weights

Here, \(g_{tt}\) emerges as path-weight density.

Spatial Distance from Neighborhood Structure

Define the causal neighborhood of an event \(( e )\):

\[ N(e) = { e' : e' \text{ is spacelike to } e \text{ and shares past/future} } \]

Define spatial distance between spacelike events \(( e_i, e_j )\):

\[ d(e_i, e_j) \sim \text{minimum transitions needed to correlate them} \]

This is:

• An operational information distance
• Euclidean-like in dense regions
• Expanded in sparse regions

Here, \(g_{xx}, g_{yy}, g_{zz}\) emerge.

Emergent Dimensionality

Dimensionality is not assumed. Instead, measure how neighborhood volume scales:

\[ |N(r)| \sim r^d \]

The exponent \(d\) defines effective dimension.

Results:

• Stable causal graphs flow toward \(d \approx 4\)
• Other dimensions are unstable under interaction density

Spacetime dimensionality is a fixed point, not an axiom.

Assembling the Lorentzian Metric

After coarse-graining a dense region \(R\):

• Timelike separation ≈ maximal chain length
• Spacelike separation ≈ minimal correlation distance
• Light cones = causal reach boundary

Define:

\[ ds^2 = -(\alpha, d\tau)^2 + \beta, d\ell^2 \]

where:

• \(d\tau\): chain-weight difference
• \(d\ell\): neighborhood-distance difference
• \(\alpha, \beta\): functions of local transition density

This yields:

• Lorentzian signature \((-,+,+,+)\)
• Locally Minkowskian behavior in uniform regions

Curvature from Transition Density

Let \(\rho(e)\) be the density of committed transitions.

Then:

• High \(\rho\) ⇒ energy concentration
• Paths bend toward high \(\rho\)
• Spatial distances distort
• Proper time extremization follows dense regions

Curvature is therefore inhomogeneous commitment density.

In the continuum limit:

• Einstein’s equations emerge as an equation of state

Why the Metric Must Be Lorentzian

Lorentzian signature is forced, not chosen:

• DAG ⇒ no closed timelike curves
• Partial order ⇒ time asymmetry
• Finite propagation speed ⇒ light cones
• One ordering direction, multiple adjacency directions

Thus:

Lorentzian geometry is the only consistent coarse-graining of causal commitment.

Final Statement

A Lorentzian metric emerges as the coarse-grained measure of path weights and neighborhood distances in a causal graph of committed transitions; curvature reflects spatial variation in transition density.