Applications to Quantum Physics

Mass of Relativistic Particles

Relativistic particles — such as electrons or protons — move at speeds close to the speed of light (about 300,000 km/s). At such velocities, their mass appears to increase.

For instance, a car with a rest mass of 800 kg would be perceived as weighing several thousand tons if it were accelerated to relativistic speed. This effect was established by Einstein in Special Relativity. This phenomenon is described in our book - see the footnote.

At relativistic velocities, spacetime becomes compressed, as illustrated in the figure below. For a car, this compression is loosely comparable to aerodynamic drag, expressed by the familiar kv² term that opposes motion. Overcoming this resistance requires additional energy, producing for an external observer the impression of increased mass.

A similar mechanism appears in quantum mechanics: particles are slowed by spacetime compression, producing the illusion of greater mass.

Note: One may object that this explanation relies on the spacetime of General Relativity (GR) rather than that of Special Relativity (SR). This is correct; however, GR extends SR, as shown in the Einstein Field Equations (see Appendix in our book). In GR, the Minkowski space of SR is generalized first to a Gaussian space and ultimately to a Riemannian manifold. At relativistic speeds, geodesics become more constrained. The relativistic relation presented here can also be derived from SR using length contraction effect. A concise introduction to SR is provided in our book — see footnote.

Contraction–Expansion of Spacetime

Einstein developed two theories of relativity:

• Special Relativity (SR) This theory describes the relationship between the four-dimensional spacetime coordinates (x, y, z, t) as experienced by an external static observer and those experienced internally by an object in motion.
• General Relativity (GR) GR explains the contraction and expansion of spacetime in the vicinity of a massive object — conceptualized here as a mass-volume. Although GR and SR share foundational principles, the phenomena they describe differ significantly.
  • Fig. A: Far from any mass-volume, spacetime remains flat. Geodesics are straight lines, and both space and time are uniform: t1=t2.
  • Fig. B: A mass-volume induces curvature in spacetime. As a result, geodesics — and thus time intervals and spatial distances — are altered. Time t1 becomes different from time t2. The Schwarzschild solution, a specific case of the Einstein Field Equations (EFEs), allows us to quantify this distortion for objects with spherical symmetry. See the appendix of our book.

Note: Appendix of the book presents a reformulated version of the Schwarzschild metric, condensed into two pages rather than the original 33-page derivation.

The Twin Paradox

The “twin paradox” is a classic thought experiment in relativity: one twin travels into space while the other remains on Earth.

This scenario reflects the phenomenon illustrated in Figure B above. The twin on Earth experiences a time interval t1, while the traveling twin experiences a different interval t2. Upon returning, the spacefaring twin will have aged differently from the sibling who stayed on Earth.

Note: In realistic situations, the difference in aging is extremely small — typically only a few nanoseconds.

Deflection of Light

In 1919, Arthur Eddington confirmed Einstein’s general relativity during a total solar eclipse: starlight passing near the Sun was deflected by about 2 arcseconds. Why does this occur?

As discussed earlier, any mass-volume naturally curves spacetime. Einstein proposed that spacetime has elastic properties, which can be illustrated through analogies. In the experiment below, spacetime is simulated using foam. Light follows geodesics — curved paths — represented by the contours drawn on the foam.

Note 1: The curvature is convex relative to the mass-volume, contrary to the concave diagrams often used in contemporary physics. Von Laue supported this view in 1927 (see Appendix of our book).

Note 2: In this experiment, the foam is curved by the cylinder’s volume — not its mass. This volume qualifies as a mass-volume because the foam cannot penetrate the cylinder’s interior.

Note 3: The inter-geodetic spacing can be measured directly on the chart above. The measured distances match — within experimental accuracy — the values predicted by the Schwarzschild metric. This is a real photograph, not a simulation, and therefore cannot be manipulated. It provides empirical support for the Spacetime Model, as the experiment relies on mass-volumes rather than mass. The material of the central cylinder — wood or lead — is irrelevant. Full details are provided in Appendix A6 of our book.

Note 4 (for physicists): The exact structure of spacetime remains unknown and is likely more complex than it appears. Some arguments support the idea of a “quantified continuum” (see Part 4 of our book), which falls within the domain of rheology. Spacetime is not a solid body, but it may exhibit rheological behavior. The magnitude of spacetime curvature is extremely small: Δr/R = 1.4166 × 10^-39 for a proton. This tiny value implies that, regardless of the function involved, the process operates in a linear regime, consistent with the Schwarzschild metric.

Understanding E = mc²

How does matter become energy?

Einstein’s equation E = mc² shows that mass can be converted into energy by multiplying it by the square of the speed of light (≈ 3 × 10⁸ m/s). This principle has been experimentally confirmed many times since 1905.

The underlying mechanism becomes clearer when “mass” is replaced by the concept of a mass-volume, as proposed in our formulation of E = mc².

Imagine immersing a balloon in water (Fig. A). If it bursts (Fig. B), a depression and surface turbulence appear. Similarly, when a mass-volume collapses or vanishes, it generates waves in spacetime.

This is the principle behind the atomic bomb: the excess mass of uranium-235 or plutonium-239 forms a mass-volume. Its sudden disappearance creates a spacetime “tsunami” of radiation — X-rays, gamma rays, UV, etc.

This phenomenon is intuitive when viewed through mass-volumes, but remains puzzling when framed in terms of mass. Einstein demonstrated the effect, but never fully explained its foundational cause.

Expansion of the Universe

How can we explain the expansion of the universe discovered by Riess, Perlmutter, and Schmidt (Nobel Prize 2011), especially when the expansion rate is not uniform?

The figure below illustrates a distribution of galaxies. The blue galaxy (A), located near the center of the universe, experiences equipotential spacetime pressure from all directions (white arrows). In contrast, the red galaxy (B), positioned near the periphery, is subjected to strong inward pressure from the central region (large black arrow), while the outward pressure is minimal due to the absence of surrounding galaxies (small black arrow).

Consequently, central galaxies remain essentially stationary, whereas peripheral galaxies are pushed outward. This imbalance in spacetime pressure produces an expansion of the universe — one that is non-uniform and continuously evolving.

Note: This behavior was predicted by the Spacetime Model in 2006. As of 2025, researchers in Christchurch, New Zealand, have published results confirming this prediction (see the PDF in Appendix of our book).

Filaments of Galaxies

Why do galaxies cluster along filaments?

The red galaxy is subject to two distinct forms of pressure:

• Axial pressure: exerted along the filament’s axis. Galaxies located within the axis (shown in beige) absorb part of the global spacetime pressure of the universe.
• Radial pressure: stronger and originating from all surrounding galaxies across the universe.

If a galaxy attempts to move away from the filament, the external radial pressure naturally redirects it back toward the filament.

Note: This is a proposed interpretation — not a definitive explanation.

Mass Excess – Nuclear Fission

Mass excess is the difference between an atom’s actual mass and its mass number A, which corresponds to the total number of nucleons (protons + neutrons) in the nucleus.

For example, consider an element X with a mass number of 19, composed of 10 protons and 9 neutrons. If its actual atomic mass is 19.3 atomic mass units (amu), the mass excess is: 19.3 − 19 = 0.3 amu.

The figure below illustrates how individual nucleons, when grouped into a nucleus, can result in a measurable mass excess.

• Fig. A: The nucleons are isolated. Their total mass is obtained by multiplying the mass of a single nucleon by 19.
• Fig. B: The 19 nucleons are grouped into a nucleus. A portion of spacetime — shown in orange — becomes trapped inside the nucleus.
• Fig. C: From the outside, the nucleus appears compact and sealed. The orange region is confined and no longer visible. The nucleus behaves as a hermetic volume, and its mass effect is greater than in Fig. A due to the imprisoned spacetime.

This excess mass corresponds to the trapped orange region. If the nucleus is broken apart, this confined spacetime is released and converted into energy.

Notes:

1. The nucleus shown here is imaginary and not the K-19.
2. Significant mass excess occurs primarily in heavy nuclei.
3. For further details, see the Bethe–Weizsäcker formula (presented elsewhere on this site).
4. Certain halo nuclei, such as lithium-11 (Li-11), may present anomalies, but these light nuclei are not relevant to nuclear fission.
5. Mass excess is typical of heavy nuclei such as uranium-235.
6. This 2D diagram should be interpreted in four dimensions for full conceptual accuracy.

Mass Defect – Nuclear Fusion

Nuclear fusion is the process of merging light atomic nuclei to form a heavier one. It is the principle behind experimental reactors such as ITER and NIF.

Fusion requires extremely high temperatures, achieved either through magnetic confinement (TOKAMAK) or, more recently, high‑power lasers.

During fusion, protons and neutrons rearrange, modifying both the volume and the surface of the resulting nucleus. As shown in the simplified figure, the fused nucleus has a smaller surface area than the two initial nuclei. This reduction increases the mass effect, and the difference — linked to the surface of mass-volumes — is released as energy.

Apparent Mass Increase of a Particle in a Crystal

Why does a particle’s mass appear to increase when it moves through a crystal? The explanation is straightforward.

In crystals, atoms form a regular lattice that creates tunnel-like paths. As a particle travels through one of these tunnels, the mass-volumes of the atoms (nucleons and electrons) curve the surrounding spacetime. This curvature makes spacetime denser at the center of the tunnel than at its ends.

To illustrate: imagine a high-speed train entering a tunnel. Air compresses inside, increasing resistance and slowing the train. From a distance, it may seem as if the train’s mass has increased — though it has not. The denser medium simply resists motion more strongly.

Similarly, in a crystal, the particle’s rest mass remains unchanged, but the denser spacetime slows its motion, creating the illusion of increased mass.

Enigma of Black Hole Mass — Radius

We conclude this section with an example that continues to puzzle the astrophysics community. No physicist fully understands the underlying mechanism. Yet the solution is surprisingly simple.

This example once again confirms that our explanation of mass and gravitation is the correct path forward.

Black holes remain one of astrophysics’ great mysteries. In everyday experience, mass is proportional to volume — double the mass, double the size, assuming constant density. Black holes defy this intuition: their mass scales not with volume, but with radius, which appears counterintuitive.

Explanation:

• The mass effect is proportional to the volume V,
• …because greater volume increases spacetime curvature and pressure.
• It is also inversely proportional to the surface area S,
• …because spacetime pressure acts on the surface — smaller surface means stronger pressure (as with a stiletto heel).
• The mass effect is therefore proportional to V/S.
• The volume V scales as R³, where R is the radius,
• …and the surface S scales as R².
• Thus, V/S = R³ / R² = R.

Therefore, the mass of a black hole scales with its radius, not with its volume.

This result supports the Spacetime Model, offering a new perspective on mass and gravitation in astrophysics.