Quaternion-Hypersphere Theory of Spacetime

by Norbert Nopper

Definition of a Quaternion

A quaternion is defined as:

$$q = w + p\mathbf{i} + r\mathbf{j} + s\mathbf{k}$$

where $w, p, r, s \in \mathbb{R}$ and the basis elements satisfy:

$$\mathbf{i}^2 = \mathbf{j}^2 = \mathbf{k}^2 = \mathbf{i}\mathbf{j}\mathbf{k} = -1$$

Theory

A spacetime event is represented as a quaternion:

$$q = ct + x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$$

where the components map as $w \equiv ct$, $p \equiv x$, $r \equiv y$, $s \equiv z$. Here $t$ is time, $c$ is the speed of light, and $x, y, z$ are the spatial coordinates of the Cartesian coordinate system. Scaling time by $c$ ensures all four components share the same unit of length [m].

Any point $q$ lies on a hypersphere S³ of radius $R$, satisfying:

$$|q| = \sqrt{c^2t^2 + x^2 + y^2 + z^2} = R$$

This places time and space on equal footing with a Euclidean signature $(+,+,+,+)$. Note that this differs from the Minkowski signature $(-,+,+,+)$ of standard Special Relativity; in this framework, concepts such as Lorentz invariance and causal structure require separate treatment. In particular, causal ordering may emerge from the constraint that all events lie on S³ and from the monotonic growth of $R$, rather than from the metric signature itself.

All components $ct, x, y, z$ are expressions of the same underlying energetic reality — spacetime is energy. The quaternion norm $|q| = R$ is the Schwarzschild radius of the total mass $m$ of the universe, directly proportional to the mass-energy:

$$E = \frac{c^4}{2G} R$$

where $G$ is the gravitational constant.

Expansion of the Universe

When energy materializes into massive particles (e.g. pair production), the total mass $m$ increases and the radius $R$ of the hypersphere grows — the universe expands. Thus $R$ is not a constant but a function of the total mass formed:

$$R = \frac{2Gm}{c^2}$$

Here $E$ and $R$ track the mass-energy component only. At the singularity, no mass has yet formed ($m = 0$, $R = 0$), while energy exists in non-mass form (e.g. radiation). As radiation converts to mass, $m$ grows, $R$ grows, and total energy (mass-energy + radiation) is conserved throughout.

State	Description	Hypersphere
Singularity	No mass formed ($m = 0$)	$R = 0$
Big Bang	Energy → mass begins	$R$ starts growing
Present	Ongoing conversion	$R$ is large
Future	Conversion approaches completion	$R \to R_{\max} = \frac{2GE_{\text{total}}}{c^4}$

Cosmic expansion is driven by the conversion $E = mc^2$, not by a cosmological constant.

Novelty

Each ingredient of this theory has precedent in the literature:

• Quaternions for spacetime — explored since Hamilton, with contributions by Silberstein and others.
• S³ hypersphere cosmology — proposed in various forms by Suntola, Carroll, and Ramírez.
• Schwarzschild radius as cosmic scale — central to black hole cosmology models.
• Euclidean signature $(+,+,+,+)$ — used in quantum gravity via Wick rotation (Hawking–Hartle).
• Expansion without a cosmological constant — pursued by several alternative cosmologies.

What is novel is the specific synthesis: a single quaternion $q = ct + x\mathbf{i} + y\mathbf{j} + z\mathbf{k}$ with Euclidean norm constraining all events to S³, whose radius $R$ is identified as the Schwarzschild radius of the total mass of the universe via $E = \frac{c^4}{2G} R$, with cosmic expansion driven entirely by energy-to-mass conversion.

Observable Implications

Several features distinguish this framework from standard ΛCDM cosmology:

• No cosmological constant — Expansion is driven by energy-to-mass conversion rather than dark energy. The predicted expansion history differs from ΛCDM, particularly at late times when conversion slows and $R$ asymptotically approaches $R_{\max}$.
• Euclidean signature — The $(+,+,+,+)$ metric predicts no fundamental distinction between timelike and spacelike intervals. Any observed Lorentz-invariant phenomena must emerge as effective behavior, potentially testable through high-precision interferometry or cosmological observations at extreme scales.
• Finite maximum radius — The universe has a definite upper bound $R_{\max} = \frac{2GE_{\text{total}}}{c^4}$, implying a closed spatial geometry. This could leave imprints in the cosmic microwave background (CMB) as suppressed large-angle correlations or matched-circle signatures.
• Schwarzschild radius coincidence — The observable universe's radius should track the Schwarzschild radius of its total mass content. This relationship can be checked against current cosmological data for the observable mass-energy density and Hubble radius.

Outlook

Once $R$ reaches $R_{\max}$, the long-term fate of the universe remains an open question within this framework. Two scenarios are possible:

• Stable — If mass-to-energy reconversion processes (e.g. Hawking radiation, matter–antimatter annihilation) are negligible at cosmic scales, the universe remains at $R_{\max}$ as a static S³.
• Cyclic — If reconversion is significant, $m$ decreases, $R$ shrinks, and the universe may contract back toward $R = 0$, potentially triggering a new cycle of expansion.

Determining which scenario holds requires further investigation into the long-term balance between mass formation and reconversion, and may be constrained by future cosmological observations.

References

• Cartesian coordinate system
• Mass–energy equivalence
• N-sphere
• Quaternion
• Schwarzschild radius
• Spacetime
• Time