We have now reached the question that was waiting for us at the end of the previous essay:
Where does geometry come from?
At first, this may seem like an odd question.
Geometry appears to be everywhere.
We measure distances.
We draw coordinates.
We calculate angles.
We determine trajectories.
General relativity gives us a remarkably successful mathematical description of gravitational phenomena in terms of geometry.
Surely geometry is simply there.
But our investigation has gradually made that assumption less secure.
If spacetime is not fundamental, and if physical reality is fundamentally relational, then geometry cannot be the arena in which the fundamental processes occur.
It must itself be something that emerges.
The question is therefore not:
How is geometry distorted by matter?
but:
How does a physical world acquire geometry in the first place?
That is a much deeper question.
Geometry begins with relations
Consider two physical events.
There is some relation between them.
Perhaps one occurs before the other.
Perhaps one can causally influence the other.
Perhaps a physical process connects them.
At this stage, we need not imagine a pre-existing distance between them.
There is simply a physical relation.
Now consider many events.
Relations begin to form a structure.
Some events are connected.
Some are not.
Some sequences are possible.
Others are excluded.
Some processes recur.
Some relations remain stable.
Once this structure becomes sufficiently regular, we can begin to represent it geometrically.
This suggests a fundamental inversion:
Geometry does not create the relations. Relations create the possibility of geometry.
The geometric description is a way of representing an organised relational structure.
Space is not necessarily what allows things to be related.
Rather, a sufficiently stable organisation of relations may be what we experience and mathematically describe as space.
From interval to geometry
We can make the idea more concrete.
Suppose we have physical events A and B.
We establish a reproducible relation between them using physical processes that function as clocks and rulers.
We can assign an interval.
Repeat this with many pairs of events.
We obtain a system of intervals.
Now ask whether those intervals fit together consistently.
If they do, a geometric structure begins to emerge.
We can assign coordinates.
We can describe trajectories.
We can identify neighbourhoods.
We can speak about curvature.
But notice what happened.
We did not begin with geometry.
We began with physical relations that exhibited regularity.
Geometry was the mathematical structure that captured those regularities.
This does not mean geometry is merely invented.
The regularities are physically real.
The geometry is real because it is an extraordinarily successful expression of their relational organisation.
But its reality is the reality of a structure, not necessarily of an independent substance.
The map is not the territory—but neither is it arbitrary
There is an old temptation to say that geometry is merely a description.
That would go too far.
If geometry were arbitrary, general relativity could not make such extraordinarily accurate predictions.
The curvature encoded in the metric has physical consequences.
Clocks behave differently.
Light follows different paths.
Objects accelerate.
So geometry is not merely a convenient picture imposed upon reality.
It tracks something physically real.
The relational ontology therefore does not reduce geometry to a human convention.
It says something more subtle:
Geometry is a real description of stable physical relations, but its reality need not make it fundamental.
A map can be real as a representation without being the landscape.
Likewise, geometry can be physically real without being the deepest level of physical ontology.
Why smooth geometry?
Now we encounter a more difficult question.
If fundamental reality consists of individual actualisations and relations, why does the universe look smooth?
Why does space appear continuous?
Why can we describe the trajectory of a planet using a smooth curve?
Why can general relativity represent the gravitational field using differentiable geometry?
Nothing about the notion of an individual physical event guarantees smoothness.
Indeed, one might expect the opposite.
At a sufficiently fundamental level, reality may be discrete, irregular, probabilistic or otherwise unlike the smooth continuum of classical geometry.
So the existence of smooth spacetime becomes something to explain.
Perhaps smoothness is an emergent property.
The coastline analogy
Imagine looking at a coastline from an aeroplane.
It appears smooth.
Zoom in.
You see bays and promontories.
Zoom in further.
You see rocks.
Further still, grains of sand.
At each scale, a different description becomes appropriate.
The coastline has not become less real because its apparent smoothness disappears at smaller scales.
Smoothness was a large-scale property.
Geometry may be similar.
At macroscopic scales, the relational structure of physical events is sufficiently regular that a smooth manifold provides an excellent representation.
At smaller scales, that description may cease to be appropriate.
The underlying structure need not itself be geometric.
The geometry may be what emerges after enormous numbers of physical relations have been organised into stable patterns.
Geometry as coarse-graining
This suggests a useful concept: coarse-graining.
A microscopic description may contain enormous detail.
A macroscopic description suppresses most of that detail while preserving the structures relevant at the larger scale.
Temperature is a familiar example.
A gas contains an enormous number of molecular degrees of freedom.
We nevertheless describe it using a small number of variables such as pressure, density and temperature.
Those variables are not arbitrary.
They capture stable large-scale regularities.
Perhaps spacetime geometry works similarly.
The fundamental relational structure may contain enormous microscopic complexity.
At sufficiently large scales, a small number of geometric variables capture the persistent regularities.
The metric would then be a remarkably compressed description of relational behaviour.
Geometry would be real because it preserves what matters.
It would be emergent because much of the underlying detail has disappeared from the description.
The metric as relational grammar
This gives us another way of thinking about the metric.
A metric tells us how physical intervals relate.
It specifies which trajectories are possible.
It determines causal structure.
It constrains the behaviour of clocks and rulers.
It therefore functions almost like a grammar of physical relations.
It does not describe a material substance called spacetime.
It specifies the systematic relationships among possible physical events.
This is particularly important in general relativity.
The metric is not simply a map laid over a pre-existing universe.
It is part of the mathematical structure that determines how physical processes can relate to one another.
That makes it deeply relational.
Gravity without a curved substance
This brings our earlier interpretation of gravity back into view.
We have been treating gravity not as the bending of a physical substance called spacetime, but as the systematic variation of spatial and temporal intervals through the gravitational field.
On this interpretation, there is no need to imagine a four-dimensional material object bending.
Instead, physical processes exhibit a particular pattern of relational intervals.
Clocks nearer a mass behave differently.
Spatial intervals change.
Free-falling trajectories converge or diverge.
Light follows the corresponding null paths.
The geometry is the mathematical expression of these systematic relations.
Thus what we call gravitational curvature is not necessarily the curvature of a thing.
It is the structure of the relations.
That distinction becomes crucial once geometry is regarded as emergent.
The geodesic is what matters physically
This also clarifies our earlier claim that it is not spacetime itself that should be imagined as curved.
A geodesic is a relation between a physical trajectory and the geometry used to describe it.
A freely falling body follows a path that is locally inertial.
We describe that path as a geodesic.
The geometry tells us what counts as a geodesic.
But physically, what we observe is not “curvature”.
We observe the behaviour of clocks, rulers, particles and light.
Curvature is inferred from the systematic relationships among these physical processes.
Thus there is a hierarchy:
physical processes
↓
relations among processes
↓
regular relational structure
↓
geometric description
The geometry is not an extra physical ingredient added at the end.
It is the organised form of the relations.
Where does dimensionality come from?
Now we can ask an even more radical question.
Why three dimensions of space?
Why one dimension of time?
Why four-dimensional spacetime?
We normally take these as starting assumptions.
But if geometry is emergent, dimensionality becomes something to explain.
A fundamental relational structure need not begin with three spatial dimensions.
It might possess some entirely different organisation.
Three-dimensional space could emerge because a particular class of relational patterns supports stable structures that behave as though they inhabit three dimensions.
This would be analogous to other emergent properties.
The number of variables needed to describe a macroscopic system need not equal the number of microscopic degrees of freedom.
Dimensionality itself might therefore be a large-scale invariant of the relational organisation.
That would be a profound result.
Space would not merely have a shape.
Its dimensionality would itself be an achievement of physical organisation.
Why these dimensions?
The question becomes particularly interesting because the dimensionality of space is not arbitrary in physics.
Many familiar physical structures depend sensitively upon dimensionality.
The behaviour of fields changes.
The stability of orbits changes.
The possibility of complex structures changes.
The relationship between inverse-square laws and geometry changes.
If three spatial dimensions are an emergent relational property, then the persistence of those dimensions must reflect something about the structure of physical possibility.
Perhaps only certain relational organisations can support stable, complex patterns.
Perhaps dimensionality is selected by the dynamics.
Perhaps it is a consequence of the way potential actualisations can be related.
We do not yet know.
But the relational ontology turns an unexplained background fact into a legitimate physical question.
Causal order may come first
There is another possibility.
Perhaps geometry does not emerge directly from arbitrary relations.
Perhaps causal order is more fundamental.
Suppose physical events can influence one another in certain ways.
Some events can causally precede others.
Some cannot.
This produces a partial ordering of events.
At sufficiently large scales, perhaps that causal structure can be represented geometrically.
Then spatial and temporal geometry would emerge from a deeper pattern of causal possibility.
This would be a particularly elegant possibility because causality is already central to both quantum theory and relativity.
The light-cone structure of relativity tells us which events can influence which others.
Perhaps the geometric structure of spacetime is, at a deeper level, the large-scale expression of an underlying causal organisation.
Geometry would then be secondary to causality.
From causal structure to metric structure
But causal order alone does not give us everything.
Knowing that A can influence B does not necessarily tell us the precise interval between them.
We need additional relational information.
How much temporal separation?
How much spatial separation?
What invariant structure do clocks and physical processes reveal?
Thus we might imagine a hierarchy:
causal possibilities
↓
relational intervals
↓
metric structure
↓
smooth geometry
This would provide a natural route from a fundamentally relational ontology to general relativity.
The precise order remains an open question.
But the conceptual architecture is suggestive.
Geometry need not be primitive.
It may be the large-scale synthesis of several kinds of stable physical relation.
The metric may be an emergent invariant
Perhaps the deepest insight is that the metric need not be identified with a microscopic object.
It may instead be an invariant pattern.
Imagine many different microscopic configurations producing essentially the same macroscopic relational behaviour.
At the microscopic level, the details differ.
At the macroscopic level, the same metric emerges.
This would explain why geometry can be so robust.
The underlying reality may fluctuate enormously while the large-scale relational structure remains stable.
Just as the temperature of a gas does not depend on the precise position and momentum of every molecule, the macroscopic metric might not depend upon every microscopic relational detail.
The geometry is therefore not fragile.
It is an attractor of relational organisation.
Geometry as an attractor
The word attractor is useful here, although we should not take it too literally.
A complex dynamical system can contain many microscopic states that nevertheless converge upon the same large-scale behaviour.
Different initial configurations can produce the same stable pattern.
Perhaps something similar occurs with physical geometry.
Many microscopic relational configurations could give rise to the same effective metric.
The classical spacetime description would then represent a stable region of relational possibility.
This would also explain why general relativity can be so insensitive to microscopic details.
It describes the attractor behaviour.
Not the microscopic machinery beneath it.
This would change the meaning of Einstein's equations
If geometry is emergent, Einstein's equations would acquire a different conceptual status.
They would not necessarily be the fundamental laws governing a spacetime substance.
They could instead be effective equations describing the stable large-scale organisation of physical relations.
This would not diminish them.
Quite the opposite.
A successful emergent equation is a powerful statement about what large-scale reality must look like, regardless of its microscopic constitution.
Thermodynamics did not become less important when statistical mechanics explained its foundations.
Its domain became clearer.
Likewise, general relativity could remain exactly the right theory of macroscopic gravitational phenomena while ceasing to be the fundamental ontology of the universe.
A striking reversal
This gives us a reversal of the conventional quantum-gravity problem.
The usual picture is approximately:
general relativity gives us fundamental spacetime
↓
quantum theory tells us matter is quantum
↓
we need to quantise spacetime
The relational picture is:
quantum potential produces actual events
↓
actual events establish relations
↓
relations organise into stable patterns
↓
stable patterns acquire effective geometry
↓
geometry obeys general-relativistic dynamics
If this is even approximately correct, then the apparent conflict between quantum theory and general relativity has been misidentified.
The theories may be describing different levels of organisation.
Quantum theory describes the structure of potential and actualisation.
General relativity describes the emergent relational geometry of large-scale actuality.
The problem is not necessarily to force them into one formalism.
It may be to understand the transition between the levels.
The geometry of actuality
There is a beautiful symmetry here.
Quantum theory begins with potential.
General relativity describes actual relational structure.
The first tells us what can happen.
The second tells us how actual physical processes are organised.
Perhaps geometry is therefore not the container of possibility.
It is the large-scale structure produced by actualised possibility.
This would explain why geometry constrains future events.
Once a relational pattern has stabilised into geometry, it becomes part of the conditions determining what can happen next.
The relationship is therefore circular, but not viciously so:
potential produces actuality
↓
actuality produces relational structure
↓
relational structure produces geometry
↓
geometry constrains future potential
↓
future potential produces new actuality
The universe is not merely sitting inside a geometry.
It is continually producing and reproducing the conditions under which geometry remains meaningful.
Geometry and affordance
This brings us back to the idea of affordance.
A geometric structure does not merely describe where things are.
It determines what trajectories are available.
A light cone determines which events can causally influence one another.
A gravitational field alters the trajectories available to freely falling bodies.
A spatial interval constrains the possible relationships among physical processes.
Geometry therefore describes a structured field of physical affordances.
This makes the emergence of geometry particularly interesting.
If geometry is an emergent system of affordances, then the universe is not first given a geometric stage and then populated with possibilities.
Rather:
stable physical possibilities generate a structure of affordances that we describe geometrically.
That is a much more natural fit with the ontology developed throughout this series.
Why geometry feels fundamental
If geometry is emergent, why does it seem so fundamental to us?
Because we live inside the emergent regime.
Our bodies, instruments and measurements are all constructed from stable macroscopic patterns.
We never ordinarily encounter the underlying relational dynamics directly.
We encounter their large-scale regularities.
Space therefore feels like the most obvious feature of reality because the physical structures from which we ourselves are constituted already participate in the same emergent geometry.
We inhabit the attractor.
Our conceptual intuitions have been formed inside it.
No wonder it feels fundamental.
The possibility of geometry without spacetime
At this point we can make a stronger claim.
If geometry emerges from relational structure, then the fundamental level may not possess anything recognisable as spacetime.
There may be no:
- points;
- coordinates;
- distances;
- dimensions;
- smooth curves;
- metric tensor.
These may all belong to the emergent description.
This sounds radical only because we have become accustomed to treating the mathematical language of spacetime as though it were the vocabulary of fundamental existence.
But physics has repeatedly taught us that successful descriptions can cease to be fundamental without becoming false.
Classical trajectories are not fundamental.
Temperature is not fundamental.
Rigid bodies are not fundamental.
Yet all are real.
Perhaps spacetime belongs to this family.
But then what is “here”?
The question becomes almost philosophical.
If there is no fundamental space, what does it mean for something to be somewhere?
Perhaps “somewhere” is itself an emergent relation.
An event is not first assigned a location and then related to other events.
Its location is constituted by its relations to other events.
To say that A is here and B is there is shorthand for a structured set of physical relations between A, B and the surrounding system.
Space is therefore not a container.
It is the organisation of relative placement.
This may be why the relational ontology feels so different.
It does not abolish location.
It explains it.
And what is “when”?
The same reasoning applies to time.
An event does not occur “at” a pre-existing temporal coordinate.
Its temporal location is expressed through relations among physical processes.
Clocks establish reproducible sequences.
Causal relations impose order.
Stable processes provide standards of duration.
From these relations, temporal geometry emerges.
Time is therefore not necessarily a substance flowing beneath events.
It is the stable organisation of temporal relations among events.
Space and time become two aspects of a deeper relational structure.
That is precisely what general relativity already hints at.
Our proposal is simply to push the relational insight one level deeper.
The geometry we observe may be a limit
We can now formulate a particularly important possibility.
Perhaps classical spacetime is a limit description.
At sufficiently large scales, the relational dynamics settle into a regime in which:
- intervals become well-defined;
- causal order becomes stable;
- dimensionality becomes effectively fixed;
- geometry becomes smooth;
- and gravitational behaviour is captured by general relativity.
At smaller scales, those properties may lose their determinate classical meaning.
This would mean that asking what the metric is doing at the smallest possible scale may be like asking what temperature a single molecule possesses.
The question may not be wrong because the answer is unknown.
It may be wrong because the concept no longer applies in the same form.
The end of the road for geometry
This gives us a possible explanation for why attempts to combine quantum theory with general relativity have repeatedly encountered conceptual difficulty.
We have been trying to carry the geometric description beyond the regime in which it is fundamental.
We take:
geometry
and ask:
“What is its quantum state?”
But if geometry is an emergent large-scale structure, the question may be analogous to asking:
“What is the quantum state of temperature?”
There may be meaningful mathematical constructions corresponding to such a question.
But they will not necessarily reveal the fundamental ontology.
The deeper question is:
What microscopic relational structures produce the macroscopic geometric variable?
That is a very different research programme.
We still need a theory
Again, we should be careful.
Nothing we have said demonstrates that geometry is emergent.
We have not derived three spatial dimensions.
We have not derived the Lorentzian metric.
We have not derived Einstein's equations.
We have not identified the fundamental relational dynamics.
We have established a conceptual possibility.
And that possibility changes the questions that deserve to be asked.
If geometry is fundamental, we need to explain why.
If geometry is emergent, we need to explain how.
The relational ontology makes the second question particularly natural.
What would have to emerge?
We can now list the things a successful deeper theory would need to recover.
It would need to explain, at least in principle:
- causal order;
- spatial relations;
- temporal relations;
- dimensionality;
- metric intervals;
- smoothness;
- geodesic behaviour;
- gravitational effects;
- and ultimately the effective dynamics described by general relativity.
That is an extraordinary demand.
But notice what we are no longer asking it to do.
We are not asking it to manufacture a quantum version of a pre-existing spacetime.
We are asking it to explain why a world that is fundamentally relational and quantum should, at large scales, look like spacetime at all.
That may be the more profound problem.
The central reversal
We can now state the reversal at the heart of this series.
The conventional picture asks:
How does matter curve spacetime?
The relational picture asks:
How do physical relations become organised in the pattern that we describe as geometry?
The conventional picture asks:
How do we quantise spacetime?
The relational picture asks:
How does quantum relational potential generate a stable spacetime description?
The conventional picture begins with geometry.
The relational picture seeks to explain geometry.
That is not a small adjustment.
It changes the direction of explanation.
Perhaps gravity is not fundamental either
And this leads to a particularly provocative possibility.
If spacetime geometry is emergent, then gravity may be emergent with it.
Gravity would not be a fundamental force acting through spacetime.
It would be the large-scale behaviour of physical processes within the emergent relational geometry.
The familiar distinction between “gravity” and “spacetime” would then become somewhat artificial.
Both might be aspects of the same emergent structure.
General relativity would describe the dynamics of that structure.
Quantum theory would describe the potential actualisations from which the structure emerges.
The supposed incompatibility between them might therefore arise because we have mistaken two levels of description for two competing fundamental ontologies.
The furrow is beginning to show
There is a pleasing pattern emerging.
We began by questioning the ontology assumed in the black-hole information paradox.
We discovered that information is relational.
We discovered that the wavefunction is potential rather than an actual physical thing.
We questioned whether spacetime is fundamental.
We treated relations as primary.
We examined actualisation.
And now geometry itself has begun to look like an emergent pattern.
The direction has been remarkably consistent.
At each stage, an apparent object dissolves into a structure of relations.
Perhaps this is not because we are deliberately forcing the universe into a relational mould.
Perhaps the relational ontology is simply making visible assumptions that had previously gone unnoticed.
The furrow may indeed be guiding the plough.
The next question
We can now answer the question with which we began, at least provisionally.
Where does geometry come from?
Perhaps geometry comes from the stabilisation of physical relations.
Potential actualises.
Actual events enter into relations.
Relations accumulate into persistent patterns.
Persistent patterns acquire regularity.
Regularity becomes measurable.
And at sufficiently large scales, the resulting structure can be represented as a smooth geometry.
So the provisional sequence is:
potential
↓
actualisation
↓
relations
↓
stable relational patterns
↓
geometry
↓
gravitational dynamics
If this is right, then geometry is not the beginning of the physical story.
It is one of its achievements.
But that leaves us with a remarkable question.
If geometry emerges from relational organisation, and if the geometry we observe is precisely what general relativity describes, then perhaps we can finally ask the question that has been lurking behind this entire investigation:
What becomes of gravity when spacetime is no longer fundamental?
Perhaps gravity does not need to be quantised.
Perhaps it needs to be explained.
And that will be the task of the next essay.
🍷🙂
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