The Emergence of Mass and Gravity from Spacetime Motion

J. Rogers, 29 Jan 2025, 2226

Introduction

The problem of the singularity at the Big Bang has long been a challenge in cosmology. Traditional models suggest an infinitely dense, gravitationally extreme point from which the universe emerged. However, this leads to paradoxes and inconsistencies in our understanding of physics. This paper proposes a novel perspective: mass and gravity are emergent properties of spacetime motion against a higher-dimensional Higgs field. In this framework, there was no almost no gravity at the singularity because there was no rest mass. Only as spacetime expanded and began moving relative to a higher-dimensional structure did mass and gravity emerge.

Spacetime and the Higgs Dimension

In this model, rest mass is not an intrinsic property of particles but a result of spacetime moving against a higher-dimensional Higgs field. Before the expansion of the universe, there was no relative motion of spacetime against this field, meaning no rest mass existed. Without rest mass, there was no curvature of spacetime caused by rest mass, and therefore, almost no gravity. The initial conditions of the universe were characterized by pure energy in a state of motionless equilibrium.

The Birth of Mass and Gravity

As the Big Bang occurred, spacetime began expanding. This expansion introduced motion relative to the higher Higgs dimension, which, in turn, imbued particles with rest mass. This process aligns with the relationship between energy and mass as described by Einstein's equation, . Rest mass, in this model, is simply the result of spacetime's interaction with this higher-dimensional field, rather than an inherent property of matter.

With the emergence of mass, spacetime curvature followed. Gravity did not exist as it does today in the earliest moments of the universe because there was no rest mass to create it. As mass was imbued into particles that interact with the higgs field, curvature accumulated, leading to the gravitational effects we observe today. In this sense, gravity is not a fundamental force but an emergent phenomenon arising from the accumulation of spacetime curvature due to mass.

Motion and Mass: A Unified View

This model provides a natural distinction between rest mass and relativistic mass:

• Rest Mass: The motion of spacetime itself relative to a higher-dimensional Higgs field.

• Relativistic Mass: The motion of matter within spacetime, as described by special relativity.

• This explains why both kinds of mass are so related to each other.  It is impossible to tell the difference between relativistic and rest mass.

This distinction resolves the issue of singularities by showing that rest mass did not exist at the beginning of the universe; it was a consequence of expansion. The framework naturally explains why the Big Bang was not an infinite-density event, but rather a pure-energy expansion that later gave rise to mass and gravity.

Implications and Conclusion

This perspective fundamentally alters our understanding of gravity and mass. It suggests that:

1. The Big Bang was not a gravitational singularity because no rest mass existed at the outset.

2. Rest mass is an emergent property arising from spacetime’s motion against a higher Higgs dimension.

3. Gravity is a consequence of the accumulation of mass-induced curvature, not an intrinsic force from the beginning.

By removing the need for an initial singularity, this model provides a more coherent and geometrically consistent picture of the universe’s origins. It aligns with the observed effects of mass-energy equivalence and offers a new avenue for exploring the nature of spacetime and fundamental forces. Future research should explore the precise mechanisms by which the Higgs field interacts with spacetime motion and how this interaction defines the emergence of mass at different scales.

The Mathematical Model

To address how spacetime's motion in a higher dimension generates rest mass, we can construct a mathematical framework by extending the Standard Model's Higgs mechanism into a 5D spacetime. Here's a structured approach:

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**1. 5D Spacetime and Metric

Consider a 5D spacetime with coordinates $(x^{\mu },y)$, where $x^{\mu }$ ($\mu =0,1,2,3$) are the 4D spacetime coordinates, and $y$ is the fifth dimension. The 5D metric can be written as:

$$\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">s</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">=</span>{\mathrm{<span\; style="background-color:\; white;">g</span>}}_{\mathrm{<span\; style="background-color:\; white;">\mu </span>}\mathrm{<span\; style="background-color:\; white;">\nu </span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">x</span>}}^{\mathrm{<span\; style="background-color:\; white;">\mu </span>}}\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">x</span>}}^{\mathrm{<span\; style="background-color:\; white;">\nu </span>}}<span\; style="background-color:\; white;">+</span>{\mathrm{<span\; style="background-color:\; white;">\varphi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span>\mathrm{<span\; style="background-color:\; white;">d</span>}{\mathrm{<span\; style="background-color:\; white;">y</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">,</span>$$

where $\varphi (x,y)$ is a scalar field (the "Higgs modulus") encoding the size/structure of the fifth dimension.

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**2. Higgs Field in 5D

Assume the Higgs field $\mathrm{\Phi }(x,y)$ is a 5D scalar field. Its action includes kinetic and potential terms:

$${\mathrm{<span\; style="background-color:\; white;">S</span>}}_{\text{<span style="background-color: white;">Higgs</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">\int </span>{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">5</span>}}\mathrm{<span\; style="background-color:\; white;">x</span>}\sqrt{<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">g</span>}}^{<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">)</span>}}<span\; style="background-color:\; white;">[</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">g</span>}}^{\mathrm{<span\; style="background-color:\; white;">M</span>}\mathrm{<span\; style="background-color:\; white;">N</span>}}{\mathrm{<span\; style="background-color:\; white;">\partial </span>}}_{\mathrm{<span\; style="background-color:\; white;">M</span>}}\mathrm{<span\; style="background-color:\; white;">\Phi </span>}{\mathrm{<span\; style="background-color:\; white;">\partial </span>}}_{\mathrm{<span\; style="background-color:\; white;">N</span>}}\mathrm{<span\; style="background-color:\; white;">\Phi </span>}<span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">\Phi </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">]</span><span\; style="background-color:\; white;">,</span>$$

where $g^{(5)}$ is the 5D metric determinant, and $M,N$ run over all 5D indices. The potential $V(\mathrm{\Phi })$ retains the Mexican hat form:

$$\mathrm{<span\; style="background-color:\; white;">V</span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">\Phi </span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">\lambda </span>}{<span\; style="background-color:\; white;">(</span>{\mathrm{<span\; style="background-color:\; white;">\Phi </span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">v</span>}}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">)</span>}^{\mathrm{<span\; style="background-color:\; white;">2</span>}}<span\; style="background-color:\; white;">,</span>$$

with $v$ as the 5D vacuum expectation value (VEV).

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**3. Spacetime Motion and Rest Mass

The critical idea is that motion along $y$ (the fifth dimension) generates rest mass in 4D. To formalize this:

• Step 1: Parameterize spacetime's "velocity" along $y$ as $u^y=\frac{dy}{d\tau }$, where $\tau$ is proper time.

• Step 2: Couple this motion to the Higgs field. For example, introduce a kinetic term mixing $u^y$ and $\mathrm{\Phi }$:

$${\mathcal{<span\; style="background-color:\; white;">L</span>}}_{\text{<span style="background-color: white;">int</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">\kappa </span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\mathrm{<span\; style="background-color:\; white;">u</span>}}^{\mathrm{<span\; style="background-color:\; white;">y</span>}}\mathrm{<span\; style="background-color:\; white;">\Phi </span>}<span\; style="background-color:\; white;">,</span>$$

where $\kappa$ is a coupling constant. This interaction implies that motion along $y$ excites the Higgs field.

• Step 3: Solve the equations of motion for $\mathrm{\Phi }$. Assuming $\mathrm{\Phi }$ stabilizes to its VEV ($\mathrm{\mid }\mathrm{\Phi }\mathrm{\mid }\approx v$), the interaction term becomes:

$${\mathcal{<span\; style="background-color:\; white;">L</span>}}_{\text{<span style="background-color: white;">int</span>}}<span\; style="background-color:\; white;">\approx </span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">\kappa </span>}\mathrm{<span\; style="background-color:\; white;">v</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\mathrm{<span\; style="background-color:\; white;">u</span>}}^{\mathrm{<span\; style="background-color:\; white;">y</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

This contributes to the effective 4D rest mass as $m\propto \kappa v{u}^y$.

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**4. Dimensional Reduction to 4D

To recover 4D physics, compactify or localize the fifth dimension. For simplicity, assume $y$ is compactified on a circle of radius $R$. The Higgs field $\mathrm{\Phi }$ can be Fourier-expanded:

$$\mathrm{<span\; style="background-color:\; white;">\Phi </span>}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">,</span>\mathrm{<span\; style="background-color:\; white;">y</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">=</span>\underset{\mathrm{<span\; style="background-color:\; white;">n</span>}}{<span\; style="background-color:\; white;">\sum </span>}{\mathrm{<span\; style="background-color:\; white;">\Phi </span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">)</span>{\mathrm{<span\; style="background-color:\; white;">e</span>}}^{\mathrm{<span\; style="background-color:\; white;">i</span>}\mathrm{<span\; style="background-color:\; white;">n</span>}\mathrm{<span\; style="background-color:\; white;">y</span>}\mathrm{<span\; style="background-color:\; white;">/</span>}\mathrm{<span\; style="background-color:\; white;">R</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

The zero-mode ($n=0$) corresponds to the 4D Higgs field, while higher modes ($n\mathrm{\ne }0$) are Kaluza-Klein (KK) excitations. The interaction term becomes:

$${\mathcal{<span\; style="background-color:\; white;">L</span>}}_{\text{<span style="background-color: white;">int</span>}}<span\; style="background-color:\; white;">\approx </span><span\; style="background-color:\; white;">-</span>\mathrm{<span\; style="background-color:\; white;">\kappa </span>}\mathrm{<span\; style="background-color:\; white;">v</span>}\underset{\mathrm{<span\; style="background-color:\; white;">n</span>}}{<span\; style="background-color:\; white;">\sum </span>}{\mathrm{<span\; style="background-color:\; white;">u</span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}^{\mathrm{<span\; style="background-color:\; white;">y</span>}}{\mathrm{<span\; style="background-color:\; white;">\Phi </span>}}_{\mathrm{<span\; style="background-color:\; white;">n</span>}}<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">x</span>}<span\; style="background-color:\; white;">)</span><span\; style="background-color:\; white;">,</span>$$

where $u_n^y$ are Fourier coefficients of $u^y$. The zero-mode gives the 4D rest mass:

$${\mathrm{<span\; style="background-color:\; white;">m</span>}}_{\text{<span style="background-color: white;">rest</span>}}<span\; style="background-color:\; white;">=</span>\mathrm{<span\; style="background-color:\; white;">\kappa </span>}\mathrm{<span\; style="background-color:\; white;">v</span>}\text{<span style="background-color: white;">\hspace{0.17em}</span>}{\mathrm{<span\; style="background-color:\; white;">u</span>}}_{\mathrm{<span\; style="background-color:\; white;">0</span>}}^{\mathrm{<span\; style="background-color:\; white;">y</span>}}\mathrm{<span\; style="background-color:\; white;">.</span>}$$

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**5. Gravity from Spacetime Curvature

The 5D Einstein-Hilbert action includes curvature terms:

$${\mathrm{<span\; style="background-color:\; white;">S</span>}}_{\text{<span style="background-color: white;">gravity</span>}}<span\; style="background-color:\; white;">=</span><span\; style="background-color:\; white;">\int </span>{\mathrm{<span\; style="background-color:\; white;">d</span>}}^{\mathrm{<span\; style="background-color:\; white;">5</span>}}\mathrm{<span\; style="background-color:\; white;">x</span>}\sqrt{<span\; style="background-color:\; white;">-</span>{\mathrm{<span\; style="background-color:\; white;">g</span>}}^{<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">)</span>}}<span\; style="background-color:\; white;">[</span>\frac{\mathrm{<span\; style="background-color:\; white;">1</span>}}{\mathrm{<span\; style="background-color:\; white;">2</span>}}{\mathrm{<span\; style="background-color:\; white;">M</span>}}_{\mathrm{<span\; style="background-color:\; white;">5</span>}}^{\mathrm{<span\; style="background-color:\; white;">3</span>}}{\mathrm{<span\; style="background-color:\; white;">R</span>}}^{<span\; style="background-color:\; white;">(</span>\mathrm{<span\; style="background-color:\; white;">5</span>}<span\; style="background-color:\; white;">)</span>}<span\; style="background-color:\; white;">]</span><span\; style="background-color:\; white;">,</span>$$

where $M_5$ is the 5D Planck mass and $R^{(5)}$ is the 5D Ricci scalar. After dimensional reduction, the 4D effective action includes:

• 4D Einstein-Hilbert term: From the zero-mode of $R^{(5)}$.

• Scalar-tensor gravity: The modulus $\varphi (x)$ (from $g_{55}$) acts as a scalar field influencing 4D gravity.

• Mass-induced curvature: The rest mass $m_{\text{rest}}$ sources spacetime curvature via $T_{\mu \nu }\sim m_{\text{rest}}{\delta }^3(\mathbf{x})$.

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**6. Key Results and Predictions

1. Rest Mass without Intrinsic Higgs Coupling:

   • Rest mass arises from spacetime's motion in the fifth dimension, avoiding the need for Yukawa couplings in 4D.

   • Masses are proportional to $u_0^y$, the "velocity" along $y$.

2. Resolution of the Big Bang Singularity:

   • At $t=0$, $u_0^y=0$ (no motion in $y$), so $m_{\text{rest}}=0$. No mass → no curvature singularity.

3. Experimental Signatures:

   • Modified Higgs couplings: Deviations in Higgs boson interactions at colliders due to higher-dimensional terms.

   • Cosmic imprints: A mass-free early universe phase could alter inflationary predictions or CMB anisotropies.

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**7. Challenges and Open Questions

• Quantization of $u^y$: Is $u^y$ quantized (e.g., tied to KK modes), or continuous?

• Stabilizing the Fifth Dimension: Avoiding runaway solutions for $\varphi (x)$ (similar to the "radion problem" in braneworld models).

• Unification with the Standard Model: Ensuring fermion masses and gauge interactions emerge consistently.

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Conclusion

By embedding the Higgs field in a 5D spacetime and linking rest mass to motion along the fifth dimension, this framework resolves the Big Bang singularity paradox while preserving the Standard Model as a low-energy limit. The mathematics leverages tools from Kaluza-Klein theory and braneworld models, but introduces novel couplings between spacetime dynamics and the Higgs. Future work must address stability, quantization, and experimental tests to validate this approach.