Opening a New Chapter in the Equations of Einstein and Hawking: Saroj Joshi’s New Concept of Time and Cosmic Structure

Author: Saroj Joshi, P.E., Ph.D.

Saroj Joshi’s Concept of Time and Cosmic Structure proposes a foundational principle: CHANGE → EVENTS → REFERENCE → TIME. The central idea is that time may be investigated not only as a fundamental coordinate used to describe nature, but also as a reference associated with the progression, ordering, and measurement of physical change and events. This perspective opens a possible new direction for understanding the physical world by beginning with observable change, the sequence of events, relationships between physical states, and the references through which physical processes are described.

The proposed framework introduces λ as an independent reference or ordering parameter representing the progression of physical events, distinct from time itself. In this formulation, τ represents time. The distinction between λ and τ is central to the proposed SJ concept: λ describes the ordered progression of physical events, while τ represents the time associated with that progression. The proposed differential relationship is dτ/dλ = F(dX/dλ, dR/dλ, dV/dλ, dA/dλ), and its integral form can be expressed as SJ: τ − τ₀ = ∫[λ₀ to λ] F(dX/dλ, dR/dλ, dV/dλ, dA/dλ)dλ. Here, τ − τ₀ represents the elapsed time reference associated with the progression of a physical system from an initial event or state to a later event or state.

In the SJ formulation, SJ represents the proposed Saroj Joshi formulation; τ represents time; τ₀ represents the initial or reference time; τ − τ₀ represents the elapsed time between the initial and final time references; λ represents the proposed independent ordering or progression parameter for physical events and is intentionally distinct from time; λ₀ represents the initial value of the event-ordering parameter; dλ represents an infinitesimal increment in the progression parameter; X represents the physical state of a system; dX/dλ represents the rate at which the physical state changes with respect to λ; R represents spatial position, distance, or spatial configuration; dR/dλ represents the rate of spatial change with respect to λ; V represents velocity or a rate of spatial change with respect to the chosen progression parameter; dV/dλ represents the change in velocity with respect to λ; A represents acceleration or the rate of change of velocity with respect to the chosen progression parameter; dA/dλ represents the change in acceleration with respect to λ; F represents the proposed mathematical function connecting physical evolution to the time variable τ; and dτ/dλ represents the change in the time reference associated with an infinitesimal progression in λ.

The mathematical challenge is to determine the exact physical and dimensional form of F rather than assuming that different physical quantities can simply be added together. A valid formulation must be dimensionally consistent and compatible with established physical principles. The purpose of introducing F is therefore to establish a mathematical relationship that can be examined, refined, tested, and potentially connected with known physical laws.

The proposed framework is not limited to position, velocity, and acceleration. It can potentially be expanded to include measurable physical variables such as temperature, pressure, density, energy, entropy, gravitational conditions, atmospheric or fluid velocity, electromagnetic fields, and quantum variables. A generalized physical-state representation may therefore be written as Ψ(λ) = [X, R, V, A, E, S, Θ, P, ρ, fields, …], where Ψ represents the generalized physical state of the system, E represents energy, S represents entropy, Θ represents temperature, P represents pressure, ρ represents density, and fields represent relevant physical fields, including electromagnetic or quantum fields.

A generalized SJ relationship may then be expressed as dτ/dλ = F(Ψ, dΨ/dλ, gμν, Tμν), where gμν represents the spacetime metric and Tμν represents the stress-energy tensor used in general relativity. This creates a possible mathematical framework for investigating physical conditions from the Earth and its atmosphere to high-speed systems, gravitational environments, stars, black holes, and large-scale cosmic structures.

This concept invites comparison with Einstein’s relativity, which established a profound relationship among space, time, motion, gravity, matter, and energy. Einstein’s field equation is expressed as Gμν + Λgμν = (8πG/c⁴)Tμν. In this equation, Gμν represents the Einstein tensor associated with spacetime curvature, Λ represents the cosmological constant, gμν represents the metric tensor describing spacetime geometry, G represents the gravitational constant, c represents the speed of light in vacuum, and Tμν represents the stress-energy tensor describing matter, energy, momentum, and stresses. The indices μ and ν represent spacetime components.

The SJ framework seeks to build upon this foundation by asking whether another mathematical layer can be investigated in which physical change and the ordered progression of events, represented by λ, are related to the time variable τ. The proposed relationship is therefore not intended to diminish the established role of Einstein’s equations, but to investigate whether an additional formulation may provide another perspective on the relationship between physical evolution and time.

Einstein’s mass-energy relationship E = mc² remains a foundational expression of mass-energy equivalence. In this equation, E represents rest energy, m represents rest mass, and c represents the speed of light in vacuum. The SJ proposal does not seek to replace this relationship. Instead, it asks whether changes in energy, represented for example by dE/dλ, together with changes in physical state, motion, spatial configuration, entropy, temperature, and other measurable variables, could be incorporated into a broader description of physical evolution.

The framework also connects conceptually with Stephen Hawking’s work concerning black holes, quantum effects, radiation, entropy, information, and extreme spacetime. For a simple non-rotating, uncharged black hole, the Hawking temperature can be expressed as TH = ħc³/(8πGMkB), where TH represents Hawking temperature, ħ represents the reduced Planck constant, c represents the speed of light in vacuum, G represents the gravitational constant, M represents black-hole mass, kB represents the Boltzmann constant, and π represents the mathematical constant pi. In the SJ framework, it is important to distinguish TH, which represents temperature, from τ, which represents time.

The Bekenstein-Hawking entropy relationship can be expressed as SBH = kBc³A/(4Għ), where SBH represents black-hole entropy, kB represents the Boltzmann constant, c represents the speed of light, A represents the area of the black-hole event horizon, G represents the gravitational constant, and ħ represents the reduced Planck constant. These established relationships provide another possible area in which the SJ framework can investigate whether changes in entropy, energy, mass, temperature, and physical state can be studied in relation to the progression parameter λ and the time variable τ.

For example, the proposed framework could investigate quantities such as dSBH/dλ, dE/dλ, dM/dλ, and dTH/dλ in relation to dτ/dλ. Such a relationship is not presented as an established physical law; rather, it represents a question for mathematical and physical investigation. The objective is to determine whether such connections can be derived consistently and whether they lead to measurable or testable consequences.

Chandrasekhar’s work on stellar structure and gravitational limits provides another example of physical systems progressing through changing states, while Higgs and modern particle physics provide the microscopic framework of fields and particle interactions from which physical events arise. These areas together suggest an opportunity to investigate nature across scales rather than treating each physical domain as completely separate.

The SJ concept therefore extends conceptually from quantum fields and particles to atoms, matter, stars, black holes, galaxies, the observable cosmos, and potentially regions beyond our present observational reach. The observable universe may represent a limit on the information available to us rather than necessarily being a complete physical boundary of existence. This raises questions about the possible nature, structure, and physical processes associated with regions beyond the observable universe.

In relation to dark matter, the framework considers whether the unseen matter associated with galactic gravitational structure could provide another area for investigating relationships among matter, gravity, physical change, entropy, and cosmic evolution. The intention is not to assume that an alternative explanation is correct, but to investigate whether alternative mathematical descriptions can reproduce observations and potentially reveal additional physical relationships. Any proposed alternative must ultimately be compared with astronomical observations and established gravitational models.

The concept also proposes investigating whether the apparent bending of light in gravitational environments might admit alternative geometric interpretations involving propagation relative to changing spatial or gravitational conditions. This remains a hypothesis for investigation and is not presented as an established replacement for gravitational lensing or spacetime curvature. Any alternative description would need to reproduce the observations currently explained by general relativity and gravitational lensing and would need to produce predictions that can be independently tested.

The proposed direction is therefore to investigate whether quantum fields, particles, matter, physical states, change, events, λ, time references, gravity, entropy, black holes, galaxies, dark matter, the observable cosmos, the non-observable cosmos, and possible larger cosmic structures can be connected within a mathematically consistent framework. Such an approach could encourage researchers to look for relationships across disciplines and scales using mathematical formulations, simulations, observations, experiments, and comparisons with established physical theories.

The generalized SJ formulation can be expressed as Δτ = τ − τ₀ = ∫[λ₀ to λ] F(Ψ, dΨ/dλ, gμν, Tμν)dλ. This represents the broader research direction of investigating whether the evolution of a physical system along an event-ordering parameter λ can be mathematically related to the time variable τ while incorporating relevant physical variables and established spacetime quantities.

The framework could potentially incorporate environmental and astrophysical data such as temperature, wind velocity, pressure, density, spatial motion, acceleration, energy, entropy, gravitational conditions, electromagnetic fields, and quantum-scale variables. Such data could be examined through computational models extending from terrestrial environments and atmospheric systems to high-speed motion, near-space conditions, astrophysical systems, black holes, and cosmological structures.

The deeper question raised by the SJ framework is whether the progression of physical events and the measured passage of time are related in a way that can be expressed through a more general mathematical structure. If such a relationship can eventually be established and shown to be consistent with known physics, it could provide another perspective for investigating systems involving extreme velocity, strong gravitational fields, high or low temperature, thermodynamic evolution, quantum phenomena, black holes, and cosmic evolution.

This also creates a theoretical research direction concerning whether unusual trajectories through physical-state space could produce unusual relationships between event ordering and elapsed time. This should not be interpreted as a present claim that time travel has been demonstrated. Rather, it identifies a direction in which the mathematics could be explored: whether the proposed relationship between λ and τ permits solutions in which the relationship between event ordering and elapsed time behaves differently from ordinary temporal progression while remaining compatible with causality and established experimental constraints.

The central mathematical question is whether dτ/dλ = F(dX/dλ, dR/dλ, dV/dλ, dA/dλ, dE/dλ, dS/dλ, dΘ/dλ, …) can be rigorously defined, made dimensionally consistent, connected with established physical laws, and developed into predictions that can be tested through observation or experiment. A successful formulation would also need to reproduce established physical results in appropriate limiting cases before any claim of a new physical effect could be considered.

The proposed SJ formulation seeks to build upon the foundations established by Einstein and Hawking rather than replace them. Einstein’s work established profound relationships among space, time, motion, gravity, matter, and energy. Hawking’s work further expanded questions concerning black holes, quantum effects, radiation, entropy, and information. The SJ framework respectfully takes these achievements as foundations and asks whether an additional mathematical relationship can be developed between physical change, event progression, and time.

The central proposal is that λ may provide a useful mathematical parameter for describing the progression or ordering of physical events, while τ represents time associated with that progression. In this sense, the framework investigates whether a relationship can be established in the form CHANGE → EVENTS → λ → τ, and whether this relationship can be connected with spacetime, energy, entropy, quantum phenomena, gravity, and cosmic structure.

The broader objective is to explore whether established physics and the proposed SJ framework can be examined together through mathematics and evidence. The intention is to continue building upon established knowledge, asking new questions, developing mathematical descriptions, testing predictions, and allowing evidence to determine which ideas remain viable.

The SJ concept is therefore offered as an open research proposal and an invitation to continued inquiry. Its central question is whether physical change and the ordered progression of events can provide a deeper mathematical description of time, and whether such a description can add another perspective to our understanding of the relationships among matter, energy, gravity, entropy, quantum physics, black holes, and cosmic structure.

With appreciation to all those whose scientific work, knowledge, visualization, modeling, encouragement, and support have helped make these complex subjects easier to explore and understand, I offer the SJ concept as an invitation to continued inquiry, discussion, mathematical investigation, and discovery.

The purpose is not to close the questions opened by Einstein and Hawking, but to ask whether another door can be opened by building upon their foundations—where change leads to events, events lead to reference, reference leads to time, and the mathematical investigation of these relationships may open new possibilities for understanding the structure and evolution of our universe.

I am now ready to publish this article, and I express my sincere appreciation to AI, astrophysicists, cosmologists, physicists, engineers, and scientists whose work has made it possible to approach some of the most complex questions in science in increasingly understandable ways. Modern visualization, BIM, graphics, simulations, models, and educational tools have helped make difficult equations and concepts more accessible, from the microscopic level of particles and quantum fields to the macroscopic level of stars, galaxies, black holes, spacetime, and the cosmos. I am especially grateful for the scientific foundations developed through the work of Albert Einstein, Hermann Minkowski, Stephen Hawking, Subrahmanyan Chandrasekhar, Peter Higgs, and many other scientists whose discoveries and theories continue to shape our understanding of nature.

I would also like to recognize One Anonymous , P.E., and Dr. Dinesh Sharma Dahal, Er. Yam Bhandari as well as Derek Martinez, Managing Director, Konica Minolta and current NASA AI/Machine Learning Workgroup Member, whose early comments and encouragement supported the continued exploration and development of this concept. Their thoughtful engagement was meaningful and helped motivate the further development of this work.

I am also grateful to Kesh Bahadur Karki, CEO of OS Nepal, and journalist Santosh Parajuli for providing a platform for my articles and supporting their publication. Their efforts in helping bring these ideas to a wider audience are sincerely appreciated.

Even with the remarkable advances of modern science, many fundamental questions remain open, and there is still much to understand at both the microscopic and macroscopic levels. This article is therefore presented as an attempt to identify possible areas for further investigation, discussion, research, mathematical development, simulation, and observation.

I do not claim to have solved these questions. Rather, I hope to encourage scientists, mathematicians, engineers, and researchers to examine the proposed framework, challenge it, improve it, test it, or discover entirely different explanations. Scientific progress depends not only on proposing new ideas, but also on subjecting those ideas to rigorous mathematics, evidence, criticism, observation, and experiment.

With appreciation to all those whose scientific work, knowledge, visualization, modeling, encouragement, and support have helped make these complex subjects easier to explore and understand, I offer the SJ concept as an invitation to continued inquiry, discussion, mathematical investigation, and discovery.