By Kevin Borland
(Former MIT Physics Major, DNA Researcher, and Enthusiastic Amateur Theorist Returning to Old Questions)
Introduction
Recently I watched a video by Sabine Hossenfelder about Penrose rotation and the geometric underpinnings of general relativity. Even after studying physics years ago at MIT, I realized how many conceptual corners I hadn’t explored back then.
The video triggered a line of thinking that I have not seen addressed explicitly in textbooks or online discussions — and I want to present it not as a definitive claim, but as an invitation for feedback from working physicists and fellow enthusiasts.
This post is a conceptual proposal / thought experiment, grounded in familiar physics but attempting to push a bit beyond the usual framing.
The seed of the idea:
If gravity is geometry, what about the other forces?
In general relativity:
- Gravity is encoded in a Lorentzian 4-dimensional geometry with signature (– + + +).
- The structure generates phenomena like time dilation, geodesics, and Penrose spinor rotations.
By contrast, the Standard Model gauge interactions (U(1), SU(2), SU(3)) live in:
- internal “gauge spaces”
- with positive-definite (Riemannian) metrics,
- and dimensions 1, 3, and 8 respectively.
These are usually treated not as literal dimensions of spacetime,
but as internal manifolds attached to each spacetime point via fiber bundles.
That’s mathematically elegant, but conceptually quite different from the simple “geometry = physics” picture of general relativity.
The discomfort: Why only one Lorentzian geometry?
It has always struck me as a deep asymmetry that:
- Spacetime has a time-like dimension and Lorentzian signature.
- All internal gauge spaces have purely Euclidean signatures.
- Only gravity is modeled on a geometry with causal structure (light cones, boosts, etc.).
If geometry is supposed to unify physical law, this seems oddly selective.
That led me to a question:
**Is it possible to classify all “geometric signatures”
(–, +, –+, ++, –++, +++, etc.)
in the same way chemistry classifies atoms?**
In other words:
Could there exist a “periodic table of gauge geometries” —
a finite or structured set of stable geometric building blocks,
each corresponding to a possible force or interaction?
And might different universes “choose” different geometric sets the way chemistry chooses different elements?
The proposal (tentative and exploratory)
1. Each gauge interaction corresponds to a stable geometric signature.
For example:
- Gravity: –+++ (Lorentzian 4-space)
- Electromagnetism: + (1D Abelian U(1))
- Weak force: +++ (3D SU(2))
- Strong force: ++++++++ (8D SU(3))
This is the usual classification, but seen through the lens of geometry rather than group theory.
2. Why these signatures, and only these?
In chemistry the elements arise from:
- discrete nuclear charge,
- quantum stability,
- periodic shell structure.
Analogously, perhaps the allowed gauge geometries emerge from:
- stability under quantization,
- anomaly cancellation,
- positivity or negativity of the metric,
- or deeper geometric consistency conditions we do not yet understand.
3. Why no forces corresponding to “–+”, “++”, “––+”, etc.?
We observe only a few gauge geometries. But mathematically, many more signatures are conceivable.
Why isn’t there a force with an internal space of signature “–+”?
Or a 2-dimensional Abelian force with signature “++”?
Or a non-Abelian force with a time-like internal dimension?
The Standard Model offers no geometric explanation — it simply lists the symmetries that happen to work.
4. Could “shared dimensions” determine which interactions couple?
Gravity interacts with all forms of energy because it shares the time-like – dimension.
U(1) interactions couple to particles with electric charge because they share the U(1) internal coordinate.
In a broader classification, perhaps interactions arise when two geometric signatures overlap in at least one dimension (time-like or space-like).
This could naturally explain:
- why certain forces couple to each other,
- why others remain hidden,
- why gauge sectors in some theories appear “dark,”
- and why only specific combinations of symmetries survive at low energies.
This is speculative, but geometrically appealing.
5. Other universes might choose different geometric sets.
If multiple gauge geometries exist in principle, then a universe could be characterized by which geometric “elements” are present.
Different universes could have:
- different signature patterns,
- different dimension overlaps,
- different stable gauge geometries.
This is reminiscent of the “string landscape,” but framed purely in terms of geometric signatures rather than compactification details.
Why bring this up?
I am not claiming I’ve discovered a theory of everything,
or that this approach is correct or complete.
Rather, I’m suggesting:
- the Standard Model gauge groups seem to have a “periodicity” or selection pattern,
- general relativity has a unique geometric signature,
- and the asymmetry between Lorentzian spacetime and Euclidean gauge spaces may not be fundamental.
The idea that there might exist a higher-level geometric classification — a “periodic table of gauge geometries” — seems like an interesting line of inquiry that I have not seen explored explicitly.
If any professional physicists (including possibly Sabine Hossenfelder, whose video inspired this) have thoughts on whether this is:
- obviously ruled out,
- already well-known under a different name,
- related to the topology of Lie groups,
- or perhaps even worth pursuing formally,
I would greatly appreciate the feedback.
Conclusion
This is a conceptual proposal from someone trying to reconnect with his physics roots and think about geometry, forces, and symmetry in a fresh way.
My central question is simple:
Why do only a few geometric signatures appear in nature,
and is there a deeper organizing principle behind them?
If a “periodic table of gauge geometries” is possible, it might offer:
- a more unified view of interactions,
- a geometric perspective on why the Standard Model looks the way it does,
- and a way to understand how different universes might be built from different geometric primitives.
I’m sharing this idea openly in the hope that those with more expertise may find it interesting, critique it, or point me toward existing work in this area.
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