The Minkowski Geometry We Live In But Never See

---

The Geometry We Live In But Never See

Minkowski spacetime is unsettling in a specific way. Not so much because it is conceptually hard, but because it is categorically unlike the Euclidean geometry our instincts expect - and yet it largely stays out of sight.

We live in a world whose metric admits null vectors, whose orthogonality behaves oddly at the light cone, and whose causal structure is rigid in ways no Euclidean space can mimic. Still, daily life feels like three-dimensional space with time tacked on as a separate parameter. Where has the weirdness gone?

The usual explanation is that the speed of light is enormous, so relativistic effects are small. True, but shallow. The deeper explanation is geometric plus biological: Lorentzian structure is real, but our species only samples a thin, very timelike region of it, under strong thermodynamic and cognitive boundary conditions.

Begin with the geometry. Minkowski space is not Euclidean four-space with a sign flipped as a mere technicality. Minkowski mixed signature changes the rules: a nonzero vector can have zero norm; the orthogonal complement of a null (lightlike) direction fails to be transverse; at null surfaces (light cones), “normal” and “tangent” collapse into the same direction.

This is why you cannot “model” even 1+1 Minkowski space as a surface inside any Euclidean space to get an intuitive feel for it. A Euclidean embedding inherits a positive-definite metric; it simply has no place to put null vectors. Spacetime diagrams are therefore not models but coded projections: what your intuitions see on the page is not literally what is happening.

So why does such alien structure not intrude? Partly because everyday life is carried out deep inside the timelike cone. For ordinary speeds, worldlines cling close to the time axis, and the Minkowski interval looks Newtonian (space and time separate and different*). The geometry is not Euclidean, but we keep walking in a narrow region where the difference barely registers.

Yet one everyday fact is already a clue. Time and space present themselves to us as categorically different kinds of thing. In a straightforward four-dimensional Euclidean universe, by contrast, “time” would be just another axis - in principle rotatable into “space” - and that felt distinction would be hard to justify as anything other than an arbitrary psychological quirk. Minkowski spacetime, at least, builds in a deep and invariant difference between timelike and spacelike directions.

The most distinctive feature of Minkowski space is also the least inhabitable: the null directions. The light cone defines the boundary between possible and impossible causal influence. But no massive organism can live on a null worldline - our worldlines are timelike. There is no rest frame of light, no proper time along a null curve, no “lived experience” of that geometry from within. The sharp edge of the metric is precisely the edge we cannot stand on.

Then add the thermodynamic arrow. Lorentzian geometry by itself does not demand an irreversible time, but it cleanly separates timelike from spacelike and makes causal order frame-invariant. Our experienced asymmetry of time - memory, anticipation, decay, the sense that causes precede effects - is a dynamical fact about a low-entropy past. Yet it sits naturally inside a spacetime where “time” is not just another axis you can rotate into “space”. In Euclidean four-space, that experiential distinction would be an awkward add-on. In Minkowski space, it is at least compatible with the underlying geometry.

Relativity becomes visible mainly when different inertial slicings are forced into comparison: moving clocks, synchronisation disputes, high rapidities, long baselines. Absent those comparisons, Lorentzian structure is present but quiet - like the curvature of the Earth to a pedestrian.

---

* Newtonian space-time is not “Minkowski with a different sign” (all pluses?) but a different kind of geometric structure altogether, one far less elegant.

Minkowski space is a four-dimensional manifold equipped with a single non-degenerate Lorentzian metric of fixed signature, so one invariant object simultaneously defines intervals, orthogonality, proper time, and a light-cone causal structure.

Newtonian (Galilean/Newton-Cartan) space-time is typically formalised on a four-manifold too, but it has no non-degenerate spacetime metric: instead it carries an absolute time function (time is absolute, universal, and geometrically prior to space) that foliates the manifold into three-dimensional simultaneity slices, plus a Euclidean spatial metric that only measures distances within each slice. 

Because this “metric” structure is degenerate, there is no invariant spacetime interval between arbitrary events and no geometric mixing of space and time under boosts. So relativity’s unified causal geometry fractures into separate notions of absolute time and instantaneous Euclidean space in a mechanistic way.

---

 

Does a Thrown Ball Reveal the Curvature of Spacetime?

---

Does a Thrown Ball Reveal the Curvature of Spacetime?

The question sounds innocent: you throw a ball, it traces a neat parabola, and you wonder whether you have just watched Einstein at work. The difficulty is that “curvature” is doing two very different jobs in this story. One sense concerns the visible curvature of a spatial trajectory drawn by an object as time passes. The other concerns the invariant curvature of four-dimensional spacetime itself. These are related, but they are not the same thing. The confusion arises when they are either collapsed into one another, or artificially torn apart. The clean way forward is to contrast three observer stances and be explicit about what each can and cannot infer.

1. The ground-based observer: “The ball falls.”

You stand on the Earth and watch the ball arc downward. In your coordinates, the path through space is approximately parabolic. This is not in dispute. The question is what that parabola represents.

In general relativistic terms, you are not an inertial observer. The Earth’s surface is prevented from free fall by internal stresses in matter; your accelerometer reads roughly 1g. Your frame is therefore non-inertial. When you describe the motion of freely falling objects from such a frame, inertial effects appear that look exactly like a gravitational force.

From this standpoint, the ball’s parabolic trajectory is the spatial trace of a geodesic described in a non-inertial coordinate system. This is a perfectly legitimate description, but it does not by itself settle the question of spacetime curvature. Curved spatial paths can arise either because spacetime is curved or because the observer’s frame is accelerating. From the ground alone, those possibilities are not disentangled.

2. The local free-fall observer: “The ball is (almost) straight.”

Now imagine you are launched alongside the ball, sharing its free-fall motion. In your immediate neighbourhood you are very close to an inertial frame. Over short distances and times, the ball does not exhibit any strong downward acceleration relative to you. Its motion is close to uniform; its worldline appears nearly straight in your local coordinates.

This is the equivalence principle in its proper domain: local, approximate, and powerful. It guarantees that along any freely falling worldline one can choose coordinates in which gravitational effects largely disappear. This is technically represented by the ability to set the connection coefficients (the Christoffel symbols) to zero at a specific point.

But this does not mean spacetime is flat. What vanishes locally are the connection coefficients, not the curvature. Curvature reveals itself only when you consider a region rather than a point or a single worldline. If you watch another freely falling object nearby, you will eventually observe relative acceleration between the two. Those tidal effects cannot be transformed away. They are the signature of the Riemann curvature tensor.

3. The deep-space inertial observer: “The ball follows a geodesic of curved spacetime.”

Now consider an inertial observer far from the Earth, equipped with a telescope, floating freely and not accelerating. This observer is the “smoking gun” witness. Because they know they themselves are inertial, any deviation they see in the ball’s path cannot be attributed to their own coordinate acceleration.

This observer does not need to appeal to accelerating coordinate systems to explain what they see. They observe a ball moving in the gravitational field sourced by the Earth, and in general relativity that means the ball follows a geodesic of the Earth’s spacetime geometry. When this observer asks, “What curve does this geodesic trace in three-dimensional space as a function of my time coordinate?”—referring to the coordinate time t of a distant clock—the answer in the weak-field, low-velocity regime is: approximately a parabola.

That near-parabolic shape is not an illusion, nor is it a coordinate trick. It is precisely how timelike geodesics in the Earth’s weak gravitational field project into ordinary space when described using a reasonable global time coordinate. In this limited but perfectly legitimate sense, the parabolic trajectory is explained by spacetime curvature. The Earth’s mass curves spacetime; free particles follow geodesics; those geodesics, when viewed spatially, look parabolic to an excellent approximation.

Why the parabola is still not “the curvature”

Here is the crucial distinction that must be made sharply. Although the near-parabolic trajectory is a consequence of spacetime curvature, it is not itself a direct measure of that curvature. A single geodesic, by itself, does not encode the Riemann tensor. Different curved spacetimes can support geodesics whose spatial projections look very similar over short distances.

What curvature controls, invariantly and unavoidably, is the relative motion of nearby geodesics. If the deep-space observer watches two balls thrown with slightly different initial conditions, they will see their separation evolve in a way that cannot be mimicked in flat spacetime by any choice of coordinates. That relative acceleration—tidal deviation—is the coordinate-independent manifestation of spacetime curvature.

“Mostly timelike rather than spacelike curvature”

This phrase gestures at a genuine feature of the weak-field regime. Near Earth, the dominant deviation of the metric from flatness lies in the time-time component (g₀₀). This encodes gravitational time dilation and reproduces the Newtonian potential in the appropriate limit. For slow-moving objects (where velocity v is much less than the speed of light c), this time-warping largely governs their motion.

Spatial curvature is also present, but its effects on slow projectiles are subleading. If the ball were thrown at relativistic speeds, or if we were observing the path of light, the spatial components of the curvature would become just as prominent as the temporal ones. It is one unified spacetime geometry, but the ball’s low velocity makes it primarily sensitive to the temporal "stretch" of the metric.

The answer, stated precisely

A thrown ball near Earth follows a timelike geodesic of the Earth’s weakly curved spacetime. A ground-based observer describes that geodesic as a parabola because they are in a non-inertial frame. A local free-fall observer sees the motion as nearly straight because curvature is negligible over small regions. A deep-space inertial observer sees neither illusion nor fiction: they see a genuine geodesic of a curved spacetime whose spatial projection is approximately parabolic.

---

Addendum: Why do different observers see the same shape?

It may seem counter-intuitive that the earthbound observer (in a non-inertial frame) and the deep-space observer (in an inertial frame) both conclude the ball follows an approximate parabola. One might expect such different perspectives to yield different geometries. However, they converge because the Earth’s gravitational potential and the ground’s physical acceleration are numerically and geometrically coupled.

For the ground-based observer, the parabola is an inertial effect. Because the ground is constantly pushing you upward, you are in an accelerating frame. In such a frame, a free particle appears to accelerate in the opposite direction. Your description follows the Newtonian kinematic: z(t) = z₀ + v_z0t - ½gt². This is a "fictitious" force result, but the mathematical plot is a literal parabola.

For the deep-space observer, the interpretation is inverted. They see the ball following a worldline that is as "straight" as the curved geometry allows (a geodesic). Because they are observing a slow-moving object in a weak field, the metric component governing time dilation (g₀₀ ≈ 1 + 2Φ/c²) dominates the math. When they project this 4D worldline onto their 3D spatial grid, the resulting equation for the trajectory yields the exact same ½gt² relationship.

Ultimately, they must converge because of the Equivalence Principle. If the "fictitious" parabola seen on the ground did not match the "geometric" parabola seen from space, an observer could distinguish between gravity and acceleration simply by throwing a ball. The fact that they see the same shape is not a coincidence; it is a requirement of the symmetry between acceleration and gravity that lies at the heart of General Relativity.

---

Why is our universe four dimensional and Lorentzian?

---

Why Four-Dimensional Lorentzian Spacetime?

Our universe has three large dimensions of space and one of time, and its large-scale geometry is Lorentzian (one negative sign in the metric). A good part of the reason lies in the mathematics of partial differential equations: which kinds of equations support genuine time evolution with finite signal speed, and which do not.

This essay develops the case in undergraduate-friendly prose, explaining every symbol used. Even if parts of this post are too hard, consider it as a pointer to what you need to understand to properly address the issue. Implicit is the idea that there imay be some underlying physical process capable of generating 'universe manifolds' with a distribution of dimensionalities and metric signatures.

This post was put together with GPT5.2 and Gemini.

---

1. Static fields: Laplace’s equation

In n spatial dimensions Laplace’s equation (a special case of Poisson’s equation) is

∇²φ = Σ_i=1ⁿ ∂²φ/∂x_i² = 0.

Here φ(x) is a scalar field (electric potential, temperature, etc.). The Laplacian ∇² (“del squared”) sums second spatial derivatives, measuring the total local curvature of the field.

Domain quantification for Laplace’s equation

Let φ: ℝⁿ → ℝ be twice differentiable and let Ω ⊆ ℝⁿ be the region of interest. Writing “∇²φ = 0” means

∀ x ∈ Ω,   ∇²φ(x) = 0.

In words: φ is harmonic at every point of Ω. In electrostatics, for example, with charge density ρ, the potential satisfies Poisson’s equation

∇²V(x) = −ρ(x)/ε₀   in ℝ³,

and therefore Laplace’s equation holds only in charge-free subregions (where ρ = 0).

Physical meaning

Think of φ as temperature. The sign of ∇²φ compares φ at a point to the average of φ over a small surrounding sphere:

• ∇²φ > 0 means φ is locally below its neighbourhood average (so, under diffusion dynamics, it would tend to increase with time).
• ∇²φ < 0 means φ is locally above its neighbourhood average (so it would tend to decrease with time).
• ∇²φ = 0 means perfect local balance: the value equals the neighbourhood average (the mean value property).

It is important to stress: ∇²φ = 0 does not mean φ is uniform. Harmonic functions can vary smoothly. For example, φ(x,y) = x in two dimensions has Laplacian zero but is clearly not constant. What Laplace’s equation really guarantees is that there are no interior maxima or minima: extrema occur only on the boundary.

Thus Laplace’s equation describes equilibrium in source-free regions. Mathematically it is an elliptic PDE: it is controlled by boundary data and does not describe time evolution.

---

2. Smoothing in time: the diffusion equation

The diffusion (heat) equation is

∂u/∂t = D ∇²u,

where u(x,t) is temperature or density, t is time, and D is the diffusion coefficient (units length²/time). Diffusion smooths irregularities. Why do we say it has “infinite propagation speed”?

Point disturbance. Start with u(x,0) = δ(x) (a spike at the origin). The solution for t > 0 is a Gaussian (the heat kernel)

u(x,t) = (4πDt)^−n/2 exp(−|x|²/(4Dt)).

The standard deviation in each coordinate is √(2Dt) (so the typical radius grows like √(2nDt)). For any t > 0 the Gaussian is nonzero for every x (though tiny far away), so the disturbance has instantaneous support everywhere. Diffusion is therefore parabolic: it smooths, but it does not impose a finite signal speed. (In real materials diffusion is an effective, coarse-grained description; microscopic physics remains causal.)

---

3. Signals and causality: the wave equation

The one dimensional wave equation is

∂²u/∂t² − c² ∂²u/∂x² = 0,

with wave speed c. Solutions are travelling waves u(x,t) = f(x − ct) + g(x + ct). Disturbances propagate at finite speed c: if the initial data are localised, the solution vanishes outside the region that the wave has had time to reach.

The minus sign matters because it makes the equation hyperbolic. Hyperbolic PDEs have real characteristic curves (here x ± ct = constant), which define a finite domain of dependence: the value at a point depends only on data in its past light-cone (in 1D, its past interval). Elliptic equations (like Laplace’s) have no such real characteristics; they are controlled globally by boundary conditions instead of evolving locally in time.

---

4. The box operator and why only one time dimension

Relativistic wave equations use the d’Alembertian (“box”)

□ = g^μν∂_μ∂_ν.

In flat spacetime (or locally in a freely falling frame), with Lorentzian signature (−,+,+,+), this becomes

□ = −(1/c²)∂²/∂t² + ∇²,

and equations like □φ = 0 are hyperbolic: they support finite-speed waves (electromagnetism, gravitational waves) and a well-posed initial value problem.

With all plus signs (Euclidean signature), the operator becomes elliptic: there are no light cones and no genuine wave propagation.

With two or more minus signs (multiple time dimensions), the problem is deeper than “the energy goes negative”. One typically loses a natural notion of a Cauchy surface (a spacelike “snapshot” carrying enough data to determine evolution), and the initial value problem is generically ill-posed. In field theory language, it becomes difficult (often impossible) to maintain both stability and unitarity: there is no clean, positive-definite conserved energy that plays the role of a stable Hamiltonian generating time evolution.

So: no time gives no dynamics; multiple times tend to destroy well-posed evolution and stability; exactly one time gives hyperbolic dynamics, finite signal speed, and a coherent causal structure.

---

5. A brief note on “why three space dimensions?”

The argument above already explains why a Lorentzian signature with one time dimension is the minimal structure needed for causal propagation. The “why three space dimensions?” question is partly separate, but there is at least one clean geometric fact worth stating. In n spatial dimensions, the field of a point source spreads over the surface area of an (n−1)-sphere, so long-range forces scale as

F(r) ∝ 1/rⁿ⁻¹   (for n > 2),

giving the inverse-square law only when n = 3. This does not “prove” that only three spatial dimensions can support complexity, but it does show that the familiar hierarchy “stable atoms - stable chemistry - long-lived planetary systems” is not generic as n varies: change the dimensionality and you change the basic fall-off of the forces on which bound structures depend; the inverse-square law in three spatial dimensions uniquely supports robust bound orbits, while in other dimensionalities bound systems are typically far more fragile or non-generic (i.e. they require precise tuning, and a small perturbation typically causes escape or collapse).

---

6. Why ‘one time dimension’ and a Lorentzian metric

We can now answer the title question directly.

If there were no time dimension at all, spacetime would reduce to a purely spatial (positive-definite) geometry - a metric with signature (+,+,...,+). The natural second-order field operators would then be elliptic (Laplace/Poisson-type): they would impose global constraints fixed by boundary data, rather than generating time evolution and finite-speed propagation.Those equations describe static balance: solutions are fixed by boundary conditions and source distributions. There would be no evolution, no wave propagation, and no causal unfolding of events - a universe “frozen” into equilibrium.

If there were two or more time dimensions, the basic wave operator would carry two or more minus signs. Then the usual framework of physics strains or breaks: there is no privileged notion of “time evolution from initial data”, because one generally lacks a natural Cauchy surface (a 'snapshot' of the universe at one instant of time that contains enough information to predict the future) and the associated well-posed initial value problem. In quantum field terms one typically encounters negative-norm states or energy functionals not bounded below, undermining stability.

With exactly one time dimension, the box operator is hyperbolic. That hyperbolicity yields light cones: the set of events that can influence (or be influenced by) a given event at speed ≤ c. Light cones divide spacetime into past, future, and “elsewhere”, providing the mathematical backbone of finite signal speed and causality. Given suitable initial data on a spacelike slice, the future evolution is (in the relevant sense) determined.

So the Lorentzian metric with exactly one negative sign is not an arbitrary convention. It is the minimal signature that supports stable, causal dynamics rather than static constraints or ill-posed evolution.

---

7. In summary

Elliptic equations (Laplace/Poisson) describe static constraints. Parabolic equations (diffusion) smooth disturbances but have instantaneous support. Hyperbolic equations (waves) give finite signal speed and well-posed evolution from initial data.

A Lorentzian signature supplies exactly the sign structure needed for a hyperbolic box operator and therefore light cones: finite-speed propagation and a coherent causal order. Remove time and you remove dynamics; add extra time directions and you generically lose well-posed evolution and stability. In that sense, 3+1 Lorentzian spacetime sits in a narrow “window” where predictable causal physics is possible, and with it cosmological structure and life.

---

Addendum: Greg Egan’s Orthogonal as a stress test

Greg Egan’s Orthogonal trilogy is a useful imaginative stress test. He explores a universe with a purely Euclidean metric (+,+,+,+). It is mathematically consistent, but it becomes physically alien: the usual hyperbolic wave structure, invariant signal speed, and light-cone causality are not available without substantial compensating assumptions. Read as fiction with working mathematics, it is a good way to feel how tightly “Lorentzian” is entangled with “signals, dynamics, and causality”.

Egan escapes stasis in 'Orthogonal' by decoupling “time” from the metric: although the geometry is positive-definite, he introduces a preferred non-metric evolution parameter along which fields evolve. This produces dynamics and propagation, but only by abandoning geometric causality and invariant signal speed - showing that, without a Lorentzian signature, time and causality must be imposed artificially rather than arising naturally from the geometry.

---

.

Foliation in SR and GR - a ChatGPT briefing

From ResearchGate

Introduction

Foliations formalize the concept of the present moment in special and general relativity. Unlike the naive intuition that 'now' is a unique and universal moment in the history of the universe, relativity theory demonstrates that foliations depend on the observer's state of motion and local spacetime curvature.

In the flat spacetime of special relativity, foliations are distinct for inertial observers moving at different relative velocities. In the curved spacetimes of general relativity, particularly near regions of extreme curvature such as the event horizons of rotating black holes where spacetime becomes highly distorted leading to effects such as 'frame-dragging', foliations may not exist globally.

In these extreme scenarios, time is no longer orthogonal to space. This implies that even a relativized notion of a coherent 'now' can break down in such extreme environments, leaving no consistent way to define a universal 'now,' even for a single observer.

This sets up tomorrow's post: "There Is No Such Thing as ‘Now’".

---

Foliation in Special and General Relativity

Foliation is a geometric concept used in spacetime theories to represent the decomposition of a four-dimensional manifold into a family of three-dimensional hypersurfaces. This allows for a clearer understanding of spacetime dynamics, observers' perspectives, and the structure of the universe. Below, the concept is outlined in the contexts of both special relativity and general relativity.

1. Foliation in Special Relativity

Spacetime Structure

• In special relativity, spacetime is modeled as a flat, four-dimensional Minkowski manifold.
• It has a global structure, where spacetime can be divided into three-dimensional spacelike hypersurfaces labeled by a time parameter t.

Slices of Spacetime

• Simultaneity Hypersurfaces: A foliation in special relativity corresponds to slicing spacetime into surfaces of constant time t as perceived by an inertial observer. These slices represent "events happening at the same time" for that observer.
• Coordinate System: Using an inertial frame of reference, the Minkowski metric ensures a natural foliation where time and space are clearly separated. Note that spacetime foliations representing "now" for inertial observers in relative motion generally do not coincide as 3D hyperplanes in spacetime. This is due to the relativity of simultaneity in special relativity, which states that different observers in relative motion will disagree on what events are simultaneous.

Importance

This foliation is consistent across all inertial observers due to the uniformity of the Minkowski metric. It simplifies calculations in special relativity, particularly for problems involving dynamics or causality.

2. Foliation in General Relativity

Spacetime Structure

• In general relativity, spacetime is curved and described by a four-dimensional pseudo-Riemannian manifold with the Einstein field equations governing its geometry.
• There is no inherent global structure, and foliations depend on the spacetime geometry and the choice of observers or coordinates.

ADM Formalism

• In numerical relativity, the Arnowitt-Deser-Misner (ADM) formalism employs foliation to describe spacetime evolution:
  • Spacetime is foliated into a family of spacelike hypersurfaces Σ_t, each labeled by a time parameter t.
  • The metric on the spacetime is decomposed into:
    • The induced metric on the hypersurface Σ_t.
    • The lapse function, governing the rate of time flow between successive slices.
    • The shift vector, describing the relative motion of spatial coordinates between slices.

Dynamical Description

Foliation allows for the study of spacetime dynamics through "3 + 1 decomposition," where the Einstein field equations are split into:

• Constraint equations, governing each hypersurface's internal geometry.
• Evolution equations, determining how the geometry changes between hypersurfaces.

Applications

• Black Hole Spacetimes: Foliation helps describe event horizons and singularities by selecting hypersurfaces that adapt to physical features of the spacetime.
• Cosmology: In the Friedmann-Lemaître-Robertson-Walker (FLRW) metric, spacetime is foliated into constant-time slices corresponding to the universe's homogeneous and isotropic expansion.

Observer-Dependent Foliations

Observers in curved spacetime can define their own foliations, often dependent on their motion or gravitational effects. Examples include:

• Null Foliation: Hypersurfaces of constant null coordinates, used in the study of light cones.
• Constant Proper Time Foliation: Used for timelike observers, where slices correspond to the observer's proper time.

3. Key Differences Between Special and General Relativity

Aspect	Special Relativity	General Relativity
Spacetime Geometry	Flat Minkowski spacetime	Curved spacetime with a dynamic metric
Global Structure	Global foliation exists universally	Foliation depends on local geometry
Coordinate Systems	Inertial frames define natural foliation	Observer-dependent or ADM formalism required
Applications	Simple dynamics and causality problems	Black holes, cosmology, and numerical studies

Conclusion

Foliation is a versatile tool in both special and general relativity, facilitating the analysis of spacetime's structure and evolution. While in special relativity it is straightforward due to the flat geometry, in general relativity, it becomes a sophisticated mathematical technique tailored to the curvature and dynamics of spacetime.

"The end of spacetime" - Nima Arkani-Hamed

Recently via Lubos Motl's blog.

"Nima Arkani-Hamed talks about the demise of spacetime, simplification in QFT, amplituhedrons which turn scattering amplitudes into high school geometry volumes, and other things."

Ideal if you have a spare ninety minutes 😏 .. . It's a public lecture, so not too technical.

"I have already been absent, non-existent"

Jenni Diski - writer

I thought this Jenni Diski (1947-2016) article worth noting. Here's an excerpt.

" I am appalled at the thought, suddenly, that someone at some point is going to tell me I am on a journey.

"But much as I hate it, the journey – that deeply unsatisfactory, often deceitful metaphor – keeps popping into my head. Like my thoughts about infinity, my thoughts about my cancer are always champing at the bit, dragging me towards a starting line.

From ignorance of my condition to diagnosis; the initiation into chemotherapy and then the radiotherapy; from the slap of being told that it’s incurable to a sort of acceptance of the upcoming end. From not knowing, to "knowing", to "really" knowing; from being alive and making the human assumption that I will be around "in the future", to coming to terms with a more imminent death. ...

"The end of the 'journey' doesn’t come until you either die cancer-free of something else, or die of the effects of a regeneration of the cancer cells. Good and bad; from here to eternity, and from eternity to here.

"But I have been not here before, remember that. By which I mean that I have been here; I have already been at the destination towards which I’m now heading. I have already been absent, non-existent.

"Beckett and Nabokov know:

I too shall cease and be as when I was not yet, only all over instead of in store.

From an Abandoned Work

The cradle rocks above an abyss, and common sense tells us that our existence is but a brief crack of light between two eternities of darkness.

Speak, Memory

"This thought, this fact, is a genuine comfort, the only one that works, to calm me down when the panic comes. It brings me real solace in the terror of the infinite desert. It doesn’t resolve the question (though, as an atheist I don’t really have one), but it offers me familiarity with:

“The undiscovered country from whose bourn/ No traveller returns.”

"I’ve been there. I’ve done that. And it soothes. When I find myself trembling at the prospect of extinction, I can steady myself by thinking of the abyss that I have already experienced. Sometimes I can almost take a kindly, unhurried interest in my own extinction. The not-being that I have already been."

---

Jenni Diski's insight here is real, but for those who know some physics a deeper consolation (perhaps) is that our lives persist in spacetime, a consequence of Einstein's great discovery which I wrote about in my sciencefiction.com article "Sub Specie Aeternitatis".

"Where is heaven, Father?"

Some seven or eight years ago I attended a Catholic event with Clare at the church in Andover. I think it was a lecture on some theological issue or other, or maybe a talk on the Missions. In any event it was a dark night and the church was packed.

At some stage in the evening the priest, an elderly, kindly man with a poor public speaking style, took questions from the audience: a kind of 'ask me anything'.

A quavering voice - evidently an elderly Irish woman - piped up from behind us: "Father, where is heaven?"

My jaw dropped: in this day and age?

The priest was, however, up to the job. He explained that previous orthodoxy had held that heaven was beyond the sky. However, NASA had sent a great many rockets and heaven was nowhere to be seen up there. However, modern physics was very strange with quantum effects between the atoms which no-one understood. Possibly it was here that heaven was located.

There was no follow-up question.

---

It made me think though. If heaven is nowhere to be found in this spacetime universe, could it really be found in Hilbert space? Perhaps in the primordial substance before spacetime geometry had ever congealed?

Let's ask Carlo Trugenberger: "Emergent 4D Quantum Geometry from Critical Space-Time Graphs".

"After a brief introduction to the problem of quantum gravity and the main solution approaches on the market I will focus on my new proposal of a quantum gravity model in which the fundamental degrees of freedom are information bits for both discrete space-time points and links connecting them.

"The Hamiltonian is a very simple network model consisting of a ferromagnetic Ising model for space-time vertices and an antiferromagnetic Ising model for the links. As a result of the frustration between these two terms, the ground state self-organizes as a new type of low-clustering graph.

"I will provide ample evidence that this simple network model has two critical points, an ultraviolet fixed point corresponding to fluctuating information bits and an infrared fixed point corresponding to an emergent geometric phase with space-time dimension 4.

"The model predicts that, at small scales, the space-time dimension decreases until space-time itself completely dissolves into a disordered soup of information bits.  The large-scale dimension 4 of the universe is related to the upper critical dimension 4 of the Ising model and to illustrate the dimension decoupling mechanism I will solve a toy version of the model in the mean field approximation.

"At finite temperatures the universe graph emerges without big bang and without singularities from a ferromagnetic phase transition in which space-time itself forms out of a hot soup of information bits."

So heaven might be 'a hot soup of information bits' (but perhaps that's the other place). Perhaps we could link this idea with Eternal Inflation to get a contemporaneous theological ontology.

---

How do we know that our current universe has three spatial dimensions? Because Clare needed a minimum of three strings to deploy her self-made bird feeder (two pie dishes from Poundland).

Three strings = three spatial dimensions

The designer shows her grasp of string theory

As I carefully explained to her, the argument works best in polar coordinates.

A tourist map of physics

This interesting cube-diagram is from Jess Riedels' blog.

The idea of this diagram is that the fundamental theories of physics can be placed in three dimensions as to how they treat gravity, the speed of light and Planck's constant h-bar, ℏ.

Gravity

G is Newton's gravitational constant as in his law F = Gm₁m₂/r². When we dial G to zero we're in domains where we ignore the effects of gravity. Most of quantum theory is here, as is special relativity and classical mechanics.

The speed of light

Strictly speaking the important thing about our universe is not the weirdness of the speed of light, it's that in the large the universe's structure is Minkowski spacetime, not our intuitive 3D Euclidean space + time. This is a difference in metric, as well-explained here.

If you dial the speed of light to infinity (so 1/c goes to zero) then the space + time metric tends to that of common sense.

Planck's Constant

This is the weird one. Quantum phenomena such as superposition and entanglement emerge mathematically from the non-zero value of ℏ, most clearly seen in the uncertainty principle.

However, the emergence of classical physics from quantum mechanics as you reduce ℏ to zero isn't too clear. The state space of quantum mechanics is an abstract structure, a complex high-dimensional vector space called Hilbert space, which is not directly mappable to spacetime.

---

You might say that seven out of the eight nodes on this cube are well-established - the central mystery of contemporary fundamental physics is the eighth, quantum gravity.

A long read

Clare and Nigel in Greece: 2007

A pointless holiday snap: we were travelling with the Andante archaeological travel company visiting Athens, Delphi, Corinth, Sparta and the site of the ancient Olympics at Olympia.

Terry Pratchett's humour doesn't really agree with me, although I respect his evident intelligence, wisdom and all-round national treasure status blah, blah, blah. I had therefore avoided "The Long X" sequence, X ∈ {Earth, War, Mars, ..., ...} on the grounds of anticipated boredom. Co-authorship with Stephen Baxter, big science and poor characterisation, did nothing to mitigate my fears.

A visit to the library and I weakened, bringing home {Earth, Mars}. The critics are right: nothing much happens for hundreds of pages. I quite like the idea of a countable (possibly countably-infinite) number of parallel universes accessible by 'stepping' -  a Euclidean fifth large dimension. That seems to have been where their joint imagination ran out.

But could the universe be even stranger? It would be soooo cool if the spacetime of our common-sense reality were an emergent property of some high-dimensional Hilbert space. Now wouldn't that be something!*

---

* I don't pretend to understand this paper, but I like the authors' aim:

" ... a new approach to the problem of unification of quantum theory with general relativity theory. Its key idea is to “general relativise quantum theory” instead of “quantising general relativity” ...".