QUANTUM GRAVITY · AN INTERACTIVE SIMULATION
Euclidean wormholesWhat you would see, and what the path integral computes.
Look through a ray-traced spherical throat into another sky, fly across it, or rotate to Euclidean time to watch why instanton wormholes pinch off into baby universes and statistical ensembles instead.
Regime 00 integrates null geodesics in a smooth Lorentzian wormhole metric (of the Morris–Thorne / DNEG family) so you can experience spherical gravitational lensing directly, before comparing it with the lecture's Euclidean saddles, AdS₃ black-hole boxes and ensemble statistics. Euclidean saddles themselves carry no Lorentzian travel route.
RELATIVISTIC OBSERVER SIMULATION · GPU RK4 GEODESICS
What an observer sees and experiences
In curved spacetime, a spherical wormhole mouth acts like a crystal lens onto another sky. Below, the first-person ray-traced sky is paired side-by-side with a synced 3D geometric embedding and camera rig: drag to look around, fly across the throat from Side A to Side B, and watch the camera frustum and center ray steer across the curved surface in real time.
Initialising relativistic RK4 geodesic shader…
Building geometric embedding & camera frustum…
What you are experiencing: real-time GPU 4th-order Runge–Kutta null geodesic integration through the smooth Morris–Thorne / DNEG throat metric. Rays with impact parameter b < ρ pass from Universe A (cyan) through the throat into Universe B (amber); grazing rays orbit near b ≈ ρ to form the Einstein ring.
How the observer ray-tracer relates to the lecture
The traversable view integrates null geodesics d²ℓ/dλ² = b² r'(ℓ)/r(ℓ)³ in the smooth DNEG / Morris–Thorne metric (James et al. 2015, [7]) so you can experience spherical lensing and throat transit directly. The lecture's Euclidean wormholes live in imaginary time τ = it and do not carry spacecraft across Lorentzian light cones; switching to regime 02 illustrates the Wick-rotated WKB pinch-off and baby-universe emission, while regime 03 integrates BTZ null rays inside the reflective AdS₃ box.
EUCLIDEAN GEOMETRY
Connecting boundaries in a calculation
Preparing 3D; the diagram below is already readable.
Use the buttons above to change connectivity
One surface connects both boundaries, illustrating a connected topology that a path integral may include. Blue and cyan distinguish the boundaries only; no particles travel through the surface and there is no physical timeline.
What exactly does this geometry show?
The 3D figure revolves r(x) = 0.58 + 0.25x² (−2.4 ≤ x ≤ 2.4) around an axis, making a two-dimensional surface with two circular boundaries embedded in three-dimensional drawing space. Disconnected mode shows two separate discs. Coordinates are drawing units: this is neither a complete spacetime nor a numerical solution of Einstein's equations.
BABY UNIVERSES
What if some degrees of freedom lie outside our description?
Baby universes are a theoretical construction in quantum gravity: closed spatial branches can enter the description as additional degrees of freedom. Unlike Schwinger pair production of charged particles inside a fixed spacetime, gravitational tunnelling changes the spatial topology itself.
In some models, treating these degrees of freedom can be reformulated as averaging over effective coupling parameters. α sectors distinguish fixing one theory from averaging many theories; parameters within a given sector do not jump randomly over time.[1]
One coin, or a bag of different coins
Repeatedly tossing one coin gives random outcomes within a fixed model. Drawing from a bag of coins with different biases and then averaging introduces differences between models. This is an analogy for two kinds of averaging, not a physical model of baby universes.
Switch between fixed sector and ensemble average in section 04 ↓AdS₃ / CFT₂
One entropy, two ways of counting
Holographic duality lets gravity and a boundary quantum theory test each other. For a BTZ black hole in three-dimensional anti-de Sitter space (AdS₃), high-energy state counting in the boundary two-dimensional conformal field theory reproduces the leading black-hole entropy. This illustration uses the non-rotating case; it is not an exact formula for every black hole.[2]
The curves coincide because of the adopted holographic parameter relations, not because they are independent observations. Higher energy permits more microstates; entropy measures the logarithmic growth of the state count.
ħ = kB = 1; E = Mℓ = Δ − c/12, cL = cR = c, J = 0. Adjusting c and E/c updates the black-hole shadow in observer regime 03.
Assumptions, units and range of validity
We use c = 3ℓ/(2G₃) and M = r₊²/(8G₃ℓ²), where ℓ is the AdS curvature radius and G₃ the three-dimensional Newton constant. For a two-dimensional CFT with large central charge and a sufficiently sparse low-energy spectrum, the leading Cardy entropy is universal over an extended high-energy regime. The slider uses E/c ≥ 1, on the high-energy side of that discussion. Finite-c values are illustrative; omitted corrections have not been estimated. The negative cosmological constant and boundary conditions of AdS cannot simply be transferred to a de Sitter description of our universe.[3]
ENSEMBLES & FACTORISATION
Multiply first, or average first?
Two non-interacting boundary systems should factorise within a fixed theory. If a gravitational calculation includes wormholes connecting the boundaries, their compatibility with this expectation needs explaining. Ensemble averaging offers one interpretation: dependence on a shared parameter can make the average of a product differ from the product of averages.[4]
Both sides share α, producing non-zero covariance after averaging. No signal passes between the sides here.
Try it · Reduce the spread to zero
The model has only two energy levels: 0 and 1 + α. Choose α ∈ {−s, 0, +s}, each with weight 1/3; βA = 1. Energies use an arbitrary reference energy ε₀, β uses ε₀⁻¹ and kB = 1.
The three-member average is calculated exactly, without random sampling noise. s is not derived from a wormhole action; these levels, distributions and values are neither CFT data nor quantum-gravity predictions.
From state counting to interactions
OPE coefficients describe how operators combine in a CFT. Appropriate averages of heavy operators admit universal asymptotic formulae in particular limits. Universality constrains statistics; it does not determine every individual coefficient.[5]
Further research constructs consistency-constrained ensembles of CFT data, matching some semiclassical three-dimensional gravitational calculations to ensemble averages, including multi-boundary Euclidean wormholes. The three-member model above explains averaging only; it does not implement these OPE formulae.[6]
What exactly are we averaging?
For general gravitational theories, the ensemble's members and measure, factorisation within a fixed theory, and extension to de Sitter each require further arguments. Success in particular models does not establish that all gravity must be an ensemble average.
WHAT THIS HELPS US ASK
What might wormholes help us understand?
In this research, Euclidean wormholes help test theories: they organise connected contributions to the gravitational path integral, probe black-hole microstate statistics and raise questions about compatibility between gravity and boundary quantum theories. Their potential value lies in understanding quantum gravity, rather than a demonstrated engineering application.