{
  "artifact_id": "qft.artifact.holography-quantum-gravity.wormholes-gravitational-path-integrals-and-ensembles.eternal-ads-bridge-time-map",
  "owner_page_id": "qft.topic.holography-quantum-gravity.lorentzian-einstein-rosen-bridges-and-two-boundary-states",
  "title": "Eternal AdS bridge: causal diagram and boundary-time map",
  "kind": "causal-and-state-map",
  "schematic": true,
  "not_to_scale": true,
  "reader_question": "How do the two boundary time evolutions appear in the eternal black-hole causal diagram, and why can equal-forward evolution change an anchored bridge slice without making the bridge traversable?",
  "takeaway": "H_R-H_L generates the stationary boost that fixes the thermofield double, whereas H_L+H_R changes tau = t_L + t_R and the prescribed boundary-anchored maximal slice; neither spatial connectivity nor a two-sided correlator supplies a causal channel.",
  "scope": {
    "geometry": "nonextremal, uncharged, nonrotating eternal AdS black-hole saddle",
    "suppressed_directions": "all angular or transverse directions",
    "boundary_system": "two identical, uncoupled boundary theories with matched energy spectra",
    "regime": "controlled large-N semiclassical holographic description",
    "quantitative_status": "qualitative causal incidence and exact boundary-state identities; drawn lengths and curvatures are not quantitative"
  },
  "conventions": {
    "metric_signature": "+---",
    "left_boundary_time": "t_L increases toward the physical future on the left boundary",
    "right_boundary_time": "t_R increases toward the physical future on the right boundary",
    "stationary_killing_field": "xi = partial_(t_R) in the right exterior and xi = -partial_(t_L) in the left exterior",
    "time_sum": "tau = t_L + t_R",
    "boost_generator": "H_- = H_R - H_L",
    "equal_forward_generator": "H_+ = H_L + H_R",
    "left_basis": "the left energy ket is written |E_n^*>_L to make the antiunitary identification explicit"
  },
  "regions": [
    {
      "id": "left-exterior",
      "label": "I_L",
      "boundary": "timelike conformal boundary partial M_L"
    },
    {
      "id": "right-exterior",
      "label": "I_R",
      "boundary": "timelike conformal boundary partial M_R"
    },
    {
      "id": "future-interior",
      "label": "II",
      "endpoint": "future spacelike r = 0 singularity"
    },
    {
      "id": "past-interior",
      "label": "IV",
      "endpoint": "past spacelike r = 0 singularity"
    }
  ],
  "causal_structure": {
    "horizons": "four event-horizon branches meet at the bifurcation codimension-two surface B",
    "future_directed_curve": "a causal curve entering from the right boundary crosses a future horizon and terminates at the future singularity",
    "forbidden_inference": "the curve does not reach the left boundary, so the bridge is not a causal channel",
    "spatial_connectivity": "Sigma_0 and Sigma_max(tau) are spacelike boundary-to-boundary slices, not causal trajectories"
  },
  "slices": [
    {
      "id": "reflection-symmetric-slice",
      "notation": "Sigma_0",
      "anchors": {
        "t_L": "0",
        "t_R": "0"
      },
      "relation": "passes through the bifurcation surface B"
    },
    {
      "id": "equal-forward-maximal-slice",
      "notation": "Sigma_max(2T)",
      "anchors": {
        "t_L": "T",
        "t_R": "T"
      },
      "relation": "a schematic representative of the maximal spacelike slice for tau = 2T; the curve's drawn arclength is not the regulated volume"
    }
  ],
  "state_identity": {
    "time_evolved_state": "|TFD; t_L, t_R> = exp[-i(H_L t_L + H_R t_R)] |TFD> = Z^(-1/2) sum_n exp[-beta E_n/2 - i E_n(t_L+t_R)] |E_n^*>_L |E_n>_R",
    "thermal_weights": "p_n = exp(-beta E_n) / Z",
    "boost_invariance": "(H_R - H_L)|TFD> = 0",
    "marginals": "rho_L and rho_R remain thermal under either local time evolution"
  },
  "evolutions": [
    {
      "id": "stationary-boost",
      "anchors": "(t_L, t_R) = (-s, +s)",
      "generator": "H_R - H_L",
      "time_sum": "tau = 0",
      "state_effect": "the TFD is invariant",
      "bulk_interpretation": "the slice and anchors are related by the stationary Killing isometry",
      "licensed_conclusion": "no physical bridge-growth inference follows from this redundancy"
    },
    {
      "id": "equal-forward-evolution",
      "anchors": "(t_L, t_R) = (T, T)",
      "generator": "H_L + H_R",
      "time_sum": "tau = 2T",
      "state_effect": "each Schmidt coefficient acquires the phase exp(-2 i E_n T)",
      "bulk_interpretation": "the prescribed boundary-anchored maximal slice changes in the same stationary spacetime",
      "licensed_conclusion": "a regulated maximal volume may change only within a declared semiclassical prescription"
    }
  ],
  "claim_ceiling": {
    "traversability": "with uncoupled boundary theories, cross-boundary correlations do not transmit a signal",
    "volume": "V_max^ren requires boundary anchors, slice or homology class, extremization branch, regulator, and subtraction scheme",
    "observability": "the bridge volume is a nonlocal relational geometric diagnostic in a specified classical solution, not a local diffeomorphism-invariant observable or an automatically established exact boundary operator",
    "finite_N": "the diagram does not assert that an exact finite-N state is literally a single classical geometry"
  },
  "visual_encoding": {
    "solid_horizontal_curve": "reflection-symmetric spacelike slice Sigma_0",
    "long_dashed_curve": "later equal-forward boundary-anchored slice Sigma_max(2T)",
    "arrowed_solid_curve": "future-directed causal trajectory ending at the singularity",
    "solid_boundary_arrows": "physical future directions of t_L and t_R",
    "short_dashed_boundary_arrows": "orientation of the stationary Killing field xi",
    "pale_regions": "directly labeled Kruskal regions; color carries no independent meaning"
  },
  "caption": "Schematic radial Penrose diagram of a nonextremal, uncharged, nonrotating eternal AdS black hole, with angular directions suppressed. Both physical boundary times increase toward the future, while the stationary Killing field satisfies xi = partial_(t_R) = -partial_(t_L). Consequently H_R-H_L leaves the thermofield double invariant, whereas equal-forward evolution by H_L+H_R changes tau = t_L+t_R and the maximal spacelike slice anchored at those times. In the uncoupled geometry, a causal curve entering a future horizon ends at the future singularity rather than reaching the opposite boundary. The Einstein-Rosen bridge is slice-dependent spatial connectivity, not a timelike tube; the drawn curve length does not represent V_max, which requires a declared extremization and renormalization prescription. Not to scale.",
  "alt_text": "Compactified two-sided AdS black-hole diagram with timelike left and right boundaries, four Kruskal regions, crossed past and future horizons, spacelike singularities, and a central bifurcation surface. A zero-time spacelike slice crosses the bifurcation surface, while a second slice joins equal future boundary times through the interior. A future-directed causal path entering the black hole ends at the future singularity rather than reaching the other boundary. A side panel states that H_R-H_L fixes the thermofield double, while H_L+H_R changes t_L+t_R and the anchored slice."
}