Quantum Gravity Field Equation

Welcome to the Quantum Gravity Field Equation

The Theory of Quantum Gravity - Incorporating Gravity into the Quantum Framework

The aim of quantum gravity is to resolve the incompatibility between the physics of general relativity (GR) and quantum mechanics (QM) into one cohesive framework, explaining extreme environments like black holes, and the Big Bang, by describing all forces and matter under a single set of laws, including gravity within the quantum framework.

 

Key Reasons to Resolve the Incompatibility Between GR and QM

GR describes gravity as the curvature of spacetime caused by mass and energy, while the Quantum Field Theory (QFT) describes the electromagnetic, weak, and strong nuclear interactions (forces) as quantum fields, their mathematical descriptions fail to reconcile at extreme scales. The fundamental issue is that when attempting to combine them, the equations of GR produce infinite, non-sensical mathematics, rather than precise, discrete (quantized) quantum values.

 

Core Principles of Planck’s Quantum Theory

1. Quantized Energy (Quanta): Matter cannot emit or absorb energy continuously. Energy transfer happens in distinct, minimum units, which Planck named quanta.
2. Planck’s Constant: Planck introduced a new universal constant to define the relationship between the frequency of radiation and the size of the energy quantum.
3. Energy-Frequency Relationships: The energy (E) of one quantum is directly proportional to the frequency of the radiation.
4. Whole-Number Multiples: A body can only emit or absorb energy in integer multiples of a quantum. Energy changes cannot occur in fractions of a quantum.

 

A reason gravity is difficult to unify with quantum mechanics is in Einstein's field equations, in which the gravitational constant is very small, and the speed of causality is very large. This means that because of the rigid nature of the spacetime fabric it takes a significant amount of mass-energy to make a minute change in the geometry of spacetime.

Resolving the incompatibility and incorporating gravity in the quantum framework would reveal a mathematical description where all forces and matter are different manifestations of a single underlying principle, potentially explaining the universe’s fundamental structure.

 

Resolution Proposition

Spacetime is the geometry of mass-energy and gravity is its curvature, then is it possible that spacetime is the geometry of sub-particles and gravity is its curvature at a sub quantum scale?

This resolution proposition hits on the exact boundary of modern theoretical physics. The simple and direct answer, is Yes, it is possible, and it is a major focus of research, however, is not yet correct in the ways General Relativity is proven.

Here is a breakdown of the proposition compared to current scientific understanding:

1. Spacetime is the geometry of mass-energy and gravity is its curvature.

• This is Einstein's General Relativity.

2. A hypothesis (Quantum Gravity).

• Mass-energy (sub-particles) determines the geometry of spacetime, and gravity is its curvature at a quantum scale.
• Sub-particles (like electrons or quarks) cause quantum-scale curvature. In this view particles aren’t just in space, they are excitations of fields that are the geometry of spacetime itself.
• Gravity at a quantum scale: Gravity at that level is not just a smooth curve, but perhaps the collective effect on these tiny quantum fluctuations.

3. The Challenges.

We do not have a final theory yet for a few reasons:

• QM cannot determine precisely both position and momentum simultaneously (Heisenberg Uncertainty Principle).
• Renormalization Problems: We cannot currently calculate the effects of gravity at quantum scales without getting nonsensical infinity answers.
• Leading Contender: Theories trying to incorporate gravity into the quantum framework include Loop Quantum Gravity (which posits space itself is quantized, or made up of discrete loops), and String Theory (which treats particles as tiny vibrating strings in higher dimensional spacetime).

 

Summary

The proposal is essentially the quest for a Quantum Theory of Gravity. It is not yet established as ‘correct’ experimental evidence; however, it is a leading contender for how the universe works at its deepest level.

 

Think of it this way:

• GR: Gravity is the curvature of a smooth fabric.
• The idea: Gravity is the geometry of a woven fabric (where sub-particles are the threads).

 

In GR, the distinction between gravity as the curvature of a smooth fabric and gravity as the geometry of a woven fabric represents the transition from classical, macroscopic descriptions of spacetime to emerging, microscopic quantum perspectives.

• Smooth Fabric (Classical GR): Gravity is the warping of a continuous four-dimensional spacetime continuum.
• Woven Fabric (Quantum Gravity / Emergent Gravity): Gravity is the overall geometry formed by discrete, interconnected threads (such as sub-particles).

 

While smooth fabric describes the effects of gravity (how things move around masses), woven fabric attempts to describe cause of gravity at a fundamental level (how spacetime itself is built from quantum interactions). The woven fabric view suggests that if you zoom in close enough, space is not a smooth, elastic sheet, but a tight woven, discrete tapestry of relationships.

 

Approach

The aim of the quantum gravity field equation is to show that the interchangeable equivalence laws governing Einstein's energy equation and the conservation of energy equally apply to gravity and force.

Spacetime and force are different manifestations of the same mass-energy source. They work in parallel, but not as independent, separate systems. They are intertwined, partner manifestations of the same source, working together to dictate how energy behaves.

 

A Non-linear Problem

While spacetime geometry and stress-energy (or stress-energy-coupled geometry) are intrinsically and intricately linked and operate within a unified, and symmetric system, current physics cannot make them work together mathematically in all situations. 

 

A Non-linear Solution

Modeling the co-evolution of stress-energy-coupled geometry. This is a creative and viable way to model spacetime and force that attempts to bypass the singularities found in standard general relativity by focusing on energy balances.

 

Energy Equation

E = mc². The rest mass (m) and energy (E) are equivalent, scaled by the square of fundamental speed of causality (c²). Mass and energy are interchangeable forms of the same thing. The speed of causality squared is the conversion factor, converting mass into energy and energy into mass, showing that mass is a concentrated form of energy. This constant value ensures the equation's unit's balance. The speed of causality squared works as a multiplier that quantifies the energy equivalent of mass. The equation shows that a small amount of mass can be converted into a large amount of energy, and a small amount of energy can be converted into a large amount mass due to the value of the speed of causality squared. In the equation, the increased mass of a body multiplied by the speed of causality squared is equal to the energy of that body.

m = E/c². The electromagnetic radiation (emr) energy released from an object is equal to the mass lost by that object multiplied by the speed of causality squared.

 

Fundamental Interactions

The electromagnetic (em) and strong (s) interactions propagate and mediate at the speed of causality in a vacuum. The em interaction is mediated by the exchange of photons, which are quantum particles of light that travels at the speed of causality. The electromagnetic interaction itself propagates through space at the speed of causality; c appears in Maxwell’s equations, connecting electric and magnetic fields, where c represents the speed of causality in a vacuum. While the interaction is mediated by c, the square of the speed (c²) appears frequently in em formulas that relate electric and magnetic fields (showing that magnetism is a relativistic consequence of electricity) or when converting energy to mass.

The strong interaction is mediated by gluons, which are massless and therefore propagate at c. The energy required to break bonds (binding energy) is often converted to mass using E = mc².

The weak (w) interaction plays a significant and specialized role in the behaviour of em fields and interactions.

• At low energies the em fields act independently, governed by Quantum Electrodynamics (QED).
• At high energies / nuclear scale, the weak interaction is crucial for transformation of particles that produce em fields.
• The photon and weak intermediate vector bosons (W, Z), that mediate the weak nuclear interaction, are intrinsically linked, being descendants of a single electroweak field.
• The weak interaction plays a role in decay and transformation. It does not cause em fields to decay; it is responsible for the decay of particles that generate em fields and transforming the underlying source of em change.
• The em interaction conserves particle identities, the weak interaction changes the flavor of quarks and leptons (e.g., transforming a neutron into a proton).

 

Nucleon Interconversion (via Weak Interaction)

The proton and the neutron are closely related, interchangeable forms of the nucleon and can be converted into each other. While they are not identical as neutrons are slightly heavier and have no charge, they are treated as different states of a nucleon.

 

Beta-Minus Decay

A neutron converts a proton, releasing an electron and an antineutrino.

 

Beta-Plus Decay

A proton converts to a neutron, releasing a position and a neutrino.

 

Interchangeability via Weak Interaction

• Protons and neutrons are two different forms of a nucleon, and within an atomic nucleus, they interact and exchange identities through the strong interaction (via pion exchange) and can be converted through the weak interaction (via beta decay). The weak interaction changes the flavor of quarks. In this process, a proton (uud) converts to a neutron (ddu) by changing an up quark to a down quark (beta-plus decay), and vice versa (beta-minus decay).
• Quark Level: This process occurs because the weak interaction can change a down quark in a neutron into an up quark in a proton, and vice versa.

In the framework of Quantum Field Theory (QFT), forces are not actions at a distance, but results of particles exchanging the virtual gauge bosons. The total of 11 bosons (3 for the weak interaction and 8 for the strong interaction), account for all mediators of the nuclear forces within the Standard Model of particle physics (excluding the photon for electromagnetism and the hypothetical graviton for gravity).

 

How Interactions Align with E = mc²

The weak, strong, and em interactions align with energy conversion, described by Einstein’s mass-energy equivalence, E = mc².

• The weak interaction of radioactive decay causes mass-energy conversion.
• The strong interaction of binding nucleons causes energy-mass conversion, which contributes to most of the visible matter in the universe.
• The em interaction of binding atoms and molecules causes energy-mass conversion. In addition, the em interaction cause rest mass to convert into useable energy in the form of light.  

The strong interaction accounts for approximately 99% of the mass of atomic nuclei, and the em field contributes approximately 0.1% to the mass of nucleons. The Higgs field, which provides inertia mass to elementary particles (electrons, quarks), accounts for approximately 1% of ordinary mass.

 

Unified Interaction Equation 

E (⟹ w) = m (⟹ s, em) x c². Both the em and the strong interactions are mediated by massless gauge bosons (photons for em and gluons for strong interaction), which means changes in these fields propagate at the speed of causality in a vacuum. The velocity of em waves, the speed of causality, and the propagation speed of changes in the color field for the strong interaction is c =1. The strong and weak interactions propagate through field interactions via the exchange of 11 distinct force-carrying gauge bosons. The factor c² is fundamentally used in context to both em and strong interactions, primarily as the conversion factor between mass and energy in determining the energy scale of force carriers.

The emr of photons is emitted from atoms by electron energy transitioning from a higher energy level to a lower level, and travel freely at c in a vacuum over long distances, as massless neutral bosons. Gluons, due to their color charge, are restricted within hadrons (including protons and neutrons) by the strong interaction, meaning that even though gluon fields propagate at c in a vacuum, they can only travel freely over short distances. This is a fundamental distinction between QED - photons and Quantum Chromodynamics (QCD - gluons).

 

Interaction Summary

The energy stored in em, weak, and strong interactions forms mass. The binding energy of these fundamental interactions contributes to the total mass in E = mc². The nature of the contribution is different to the strong interaction (which binds quarks to create most of the mass of nucleons), whereas the Higgs field gives fundamental particles their intrinsic mass. The weak interaction contributes to mass primarily through the Higgs mechanism, which makes its force carriers (W and Z bosons) massive. Because the weak interaction is mediated by massive bosons, it contributes directly to the total mass-energy of the system, however, its contribution to the mass of matter (nucleons) is tiny compared to the strong interaction.  

 

Gravity

Gravity = curvature of spacetime, where objects follow the straightest possible paths (geodesics) through this geometric curvature. The stress-energy tensor (density, pressure, momentum) is the source that dictates the curvature of the four-dimensional spacetime continuum. Spacetime can have curvature even in regions where the stress-energy tensor is zero (a vacuum), such as around blackholes or in the presence of gravitational waves, as the curvature is carried from distant sources.

Spacetime perturbations (gravitational waves) propagate at the speed of gravity, which equals the speed of causality (c) in a vacuum; speed of gravity = speed of gravitational waves = c.

 

Gravity Equation 

St = g. Gravity is equal to the geometry of spacetime, and this geometric curvature is caused by the stress-energy tensor. The speed of gravity, specifically the propagation of gravitational waves in a vacuum, is equal to the speed of causality (c). Experimental and theoretical physics confirm that gravitational waves and light (electromagnetic radiation) propagate at the same speed in a vacuum.

According to the equivalence principle, mass and energy are identical sources of gravity. In parallel to the energy equation, the gravity equation expresses the gravity equivalence to the curvature of spacetime, meaning, spacetime and energy are not separate, but interwoven components of the same entity. The square of the speed of causality (c²) in the Einstein's energy equation acts as the conversion factor that quantifies the equivalence between mass-energy and the curvature of spacetime. 

All energy curves spacetime, therefore a high energy, massless photon will cause a tiny amount spacetime curvature, causing a minuscule gravitational wake (perturbation in spacetime metric). While a photon in a vacuum does not create a “gravitational wave (which requires an accelerating mass), it does create a distortion in the spacetime metric, which acts like a gravitational field accompanying the photon at the speed of causality. Showing that it is theoretically possible to reconcile the mathematical descriptions of quantum gravity at extreme scales.

The gravity equation demonstrates the Block Universe model, which defines the simultaneity of time. All moments in time - past, present, and future are equally relative and exist within a four-dimensional spacetime model. This view is consistent with relativistic physics, where space and time are combined into a single, unified, and fixed entity.

 

Quantum Gravity Field Equation 

g (⟹ St) ⟹ E (⟹ w) = m (⟹ s, em) x c². Gravity is caused by the geometric curvature of spacetime, which itself is caused by energy. Energy equals mass, where the weak interaction of radioactive decay causes mass-energy conversion; the strong interaction of binding nucleons, mediated by gluons, causes energy-mass conversion; the em interaction of binding atoms and molecules causes energy-mass conversion, and rest mass to convert into light, resulting in a net loss of system mass, multiplied by the square of the speed of causality, establishing the scale of conversion.

Simply, the weak, strong, and em interactions cause mass-energy and energy-mass conversion. Stress-energy generates spacetime geometry, and this geometric curvature is the origin and source of gravity; c² represents the square of the speed of causality which acts as the fundamental conversion factor that quantifies the equivalence between mass-energy and the curvature of spacetime.

All mass-energy, including the energy stored in weak, strong, and electromagnetic fields, are equivalent to mass and cause the curvature of spacetime. Objects and particles follow paths guided by this curvature. Mass-energy density and spacetime geometry are intrinsically co-dependent and occur simultaneously.

The interchangeable equivalence of mass-energy, the geometric curvature of spacetime (gravity), and quantum field interactions are all different representations of a single, underlying symmetry principle.

 

Quantum Gravity Field Framework

St (g) = E (mc²) ⟹ w (s, em). The geometry of spacetime, and its curvature of gravity equals the stress-energy within it, caused by the energy associated with the weak, strong, and em interactions.

The geometry of spacetime is proportional to and sourced by energy-momentum, and gravity is coupled by the speed of causality. The curvature of spacetime is directly proportional to the stress-energy tensor, which includes mass density, momentum density, energy flux, stress (pressure), em fields, weak, and strong interactions within matter. The gravitational field carries energy (via non-linear mathematics) and contributes to the overall curvature.