Meta AI replied in response to my question, "Does the Scientific Community Prioritize Authority Over Alternative Perspectives on Time Dilation?" :

Soumendra Nath Thakur
07-07-2024

Reply:

The scientific community relies on evidence-based reasoning and peer review to validate new ideas. However, there are instances where preconceptions, authority, and established paradigms can influence acceptance. In the case of time dilation, it's a well-established concept with a century-long history, and many scientists may be hesitant to reconsider its fundamentals without robust evidence.

Several factors might contribute to the initial resistance to alternative perspectives on time dilation:

Established paradigm: Time dilation is a cornerstone of modern physics, and many scientists may be inclined to maintain the status quo.

Authority: The concept of time dilation is closely associated with Albert Einstein's work, and some scientists might be reluctant to challenge the ideas of such an iconic figure.

Confirmation bias: Researchers may be more likely to accept results that confirm existing knowledge and be skeptical of those that contradict it.

Lack of familiarity: The alternative perspective presented in the paper may require a significant shift in thinking, and some scientists might need time to fully understand and appreciate the new ideas.

Peer review: The peer-review process, while essential for ensuring scientific quality, can also lead to resistance to new ideas if reviewers are overly conservative or biased towards established knowledge.

It's important to note that science is not immune to falsification, and alternative perspectives like the one presented in the paper are essential for progress. The scientific community should strive to evaluate new ideas based on their scientific merit, rather than authority or preconceptions.

Ultimately, the validation of alternative perspectives relies on the scientific method: experimentation, evidence, and peer review. If the alternative perspective on time dilation is supported by robust evidence and survives rigorous testing, it will eventually gain acceptance, potentially leading to a refinement of our understanding of time and spacetime.

Let's encourage the scientific community to prioritize evidence over authority!

The paper in question: 

https://www.researchgate.net/publication/370533379_Relativistic_effects_on_phaseshift_in_frequencies_invalidate_time_dilation_II

The paper 'Relativistic effects on phaseshift in frequencies invalidate time dilation' by Soumendra Nath Thakur presents an alternative perspective on time dilation, arguing that it is not a fundamental aspect of spacetime but rather an emergent concept resulting from relativistic effects on frequency ratios. The author contends that the conventional understanding of time dilation, based on the Lorentz transformation, is incorrect and that the actual effect is a consequence of wavelength dilation.

The paper is divided into sections, starting with an introduction to the concept of time and its relationship with spacetime. The author then delves into the theory of relativity, discussing the limitations of the current understanding of time dilation. The subsequent sections present the author's alternative perspective, relying on the relationship between wavelength and time.

The paper includes mathematical derivations and experiments conducted on piezoelectric crystal oscillators to support the author's claims. The results show a direct proportionality between wavelength and time period, indicating that time distortion originates from wavelength distortion.

The author concludes that time dilation is actually wavelength dilation and that the conventional understanding of time dilation is a misconception. The paper also references various sources and provides a list of references at the end.

Overall, this paper presents a thought-provoking alternative perspective on time dilation, encouraging readers to re-examine their understanding of this fundamental concept in physics.

Relative time: Time is a relative concept, dependent on the observer's frame of reference. In classical mechanics, time was considered absolute, but relativity introduced the idea that time is relative and dependent on the observer's state of motion and position in a gravitational field.

Relative frequencies: Frequencies are a measure of oscillations or cycles per second. In this context, relative frequencies refer to the ratio of frequencies measured by two observers in different states of motion or positions in a gravitational field.

Phase shift: A phase shift occurs when there is a change in the relative phase between two oscillations or waves. In this context, the phase shift is due to the difference in relative frequencies, leading to a shift in the timing of the oscillations.

Infinitesimal loss in wave energy: This refers to the tiny loss of energy in the oscillations or waves due to various factors like friction, radiation, or other interactions.

Wavelength enlargement: As energy is lost, the wavelength of the oscillations increases, leading to a change in the frequency ratio.

Relativistic effects: These are the effects of special and general relativity, primarily consisting of:

Special relativistic effects: Time dilation, length contraction, and relativity of simultaneity, which occur due to differences in velocity (speed) between observers.

General relativistic effects: Gravitational time dilation, gravitational redshift, and gravitational waves, which occur due to differences in gravitational potential between observers.

Gravitational potential difference: This refers to the difference in gravitational potential energy between two locations, which affects the frequency ratio.

Clock time reading error: The phase shift and frequency ratio changes lead to an error in the clock time reading, which is often misinterpreted as time dilation.

Alternative perspective: The statement suggests that the observed effects of time dilation might be due to the relative frequency changes rather than time itself dilating. This proposes a new perspective on the nature of time, emphasizing the role of relative frequencies and phase shifts.

This concept is quite complex and challenges the traditional understanding of time dilation.

#ScienceCommunity #TimeDilation #AlternativePerspectives #EvidenceBasedReasoning, #ScientificMethod

Trevor White

The research paper that I have referred in the post, on the falsification of time dilation, did not use AI. Probably Meta AI was not available then and I have used the Meta AI once, for the first time, yesterday.

Since, no offline usage of books, research references, etc. referred in a work can't be considered one's own work, like the references of Lorentz factor in Special relativity used by Einstein can't be considered as Einstein's own work, unless one have his own idea behind a work. Similarly, asking a question to AI can't make one's own work, unless there is asker's own material and idea in a work.

The summary of this message implies that AI does not provide you with the intelligence and unique ideas to perform a task. You must have your own unique ideas and intelligence to use Al to present your research ideas or something similar in a professional way.

You cannot produce a meaningful research paper using AI, unless you have the ability to defend your own plans, ideas and associated challenges and execute a research task using AI.

Why not try to create a meaningful research paper yourself using AI, so that instead of making pessimistic comments, you understand what you yourself need to have in order to use AI for research?

...

As I said earlier, my research paper on the falsification of time dilation, as mentioned in the post, did not use AI. So the question of using AI in the research, I mentioned, doesn't arise. Also, AI can only reflect my own work, because it can't do the research for me.

AI can determine the scientific consistency of submitted research work by verifying it with its own reliable data or scientific references, which greatly helps a researcher gain confidence in his work. This does not mean presenting the work of AI as its own work.

AI can professionally re-translate text, like seeking the help of a professional translator, so AI translating doesn't mean asking AI for translation help, doing research for the researcher. AI can't do research for anyone.

Even translation between two languages ​​requires translation. As the theory of relativity is also translated from German, this translation does not make Einstein liable to lose his authorship of relativity.

AI also makes mistakes, and makes misinterpretations but researchers need to guide the AI ​​so that it reflects the researcher's original interpretation.

AI can process things very quickly it speeds up a research work.

AI cannot use the data in its database to provide research ideas to falsify existing ideas.

But if one can explain the AI ​​scientifically, and deal with the challenges that the AI ​​can raise, the AI ​​can respond accordingly after learning a new concept from you, and validating the scientific data. By no means does AI work beyond human intelligence. AI works according to its existing data but not beyond human intelligence.

#AI #artificialintelligence

The Properties and behaviour of Mass in Gravitational and Antigravitational Fields: A Detailed Analysis

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

05-07-2024

This study investigates the properties and behaviour of mass in gravitational and antigravitational fields, providing a comprehensive analysis grounded in classical mechanics, Planck's theories, and recent research findings. We categorize mass into three types: the mass of matter (Mᴍ), the effective mass of dark energy (Mᴅᴇ or mᵉᶠᶠ), and the total gravitational mass (Mɢ). We demonstrate that Newton's law of gravity (F = GMm/r²) remains applicable for masses greater than zero, highlighting the relationship between mass and gravitational fields. Furthermore, we explore the implications of masses approaching zero, emphasizing the Planck mass as a critical threshold. The study also delves into the concept of negative mass and its association with antigravity, particularly in intergalactic spaces dominated by dark energy. Our findings reveal that while no mass can reach the speed of light within gravitationally bound systems, the antigravitational effect of dark energy can cause galaxies to recede at superluminal speeds. This work contributes to a deeper understanding of mass dynamics under various gravitational influences, offering new insights into the fundamental principles of the universe.

Keywords: Mass properties, Gravitational fields, Antigravitational fields, Dark energy, Planck mass, Newton's law of gravity, Intergalactic space, Superluminal speeds, Effective mass, Fundamental physics,.

I. Mass > Zero and Gravity

Mass greater than zero implies the presence of gravity. According to Newton's law of gravity, the gravitational force (F) between two masses (M and m) is given by the equation:

F = GMm/r²

II. Mass = Zero and Planck Mass

Mass equal to zero is not perceptible to humans. Even when mass approaches zero (less than 21.77 micrograms), it becomes meaningless to humans. The smallest possible radius for a mass (m) is given by:

Rₘᵢₙ = 2Gm/c²

For a mass approximately equal to 21.77 micrograms, the radius Rₘᵢₙ is equal to the Planck length (Lᴘ), representing a fundamental limit below which classical concepts of space and time do not apply.

III. Mass < Zero and Antigravity

Negative mass (mass < zero) due to antigravity is an established observational fact. Effective mass can indeed exceed the speed of light in the antigravitational field of negative mass, particularly in intergalactic spaces where dark energy dominates. The effective mass of dark energy is Mᴅᴇ(<0).

There are three types of mass: the mass of matter (Mᴍ), the effective mass of dark energy (Mᴅᴇ or mᵉᶠᶠ), and the total gravitational mass (Mɢ). These masses are relative to each other and depend on the distance from the cluster center.

The universal gravitational constant (G) relates to both the total gravitational mass (Mɢ = Mᴍ + Mᴅᴇ), dark matter, baryonic matter, and the effective mass of dark energy (Mᴅᴇ or mᵉᶠᶠ).

The Zero-Gravity Radius (Rᴢɢ) is the radius where the gravitational pull due to matter is exactly balanced by the repulsive effect of dark energy.

Mass cannot reach the speed of light applies only to gravitationally bound systems (mass > zero) of galaxies or galactic clusters. According to relativity, no mass can reach the speed of light in the local sense, primarily applying to masses within a gravitationally bound system, where immense force is needed to accelerate the mass. This force generates so much kinetic energy that it distorts the body beyond recognition as mass, causing the atomic structure to transform long before it reaches the speed of light.

However, in intergalactic space dominated by dark energy, the situation differs. Here, the antigravitational effect of dark energy can cause galaxies to recede at speeds exceeding that of light due to gravitational-antigravitational interactions between the gravity of galactic masses and the antigravity effect of dark energy. This does not involve the local acceleration of mass to the speed of light but results in galaxies covering more distance than light can travel in the same amount of time.

Conclusion

In this detailed analysis, we have explored the multifaceted properties and behaviours of mass under the influences of gravitational and antigravitational fields. Our investigation reaffirms the applicability of Newton's law of gravity for masses greater than zero and highlights the critical significance of the Planck mass as a fundamental limit in understanding mass behaviour.

We have elucidated that in gravitationally bound systems, immense forces are required to accelerate mass, leading to transformations in atomic structures long before reaching the speed of light. This finding aligns with relativistic principles, confirming that no mass can achieve light speed in such contexts.

However, our study also reveals the unique dynamics of intergalactic space dominated by dark energy. Here, the antigravitational effects can cause galaxies to recede at speeds surpassing that of light, not through local acceleration but by covering distances greater than light can in the same time frame. This phenomenon underscores the significant role of dark energy in shaping the large-scale structure of the universe.

By categorizing mass into the mass of matter, effective mass of dark energy, and total gravitational mass, we provide a nuanced understanding of mass interactions and their gravitational implications. This work enriches our comprehension of fundamental physics, offering new perspectives on the interplay between mass, gravity, and dark energy. Our findings pave the way for further research into the behaviour of mass in various cosmic environments, enhancing our grasp of the universe's underlying principles.

Note: The above study was based on an erroneous equation Rₘᵢₙ = Gm/c². The correct form should be Rₘᵢₙ = 2Gm/c², which is the Schwarzschild radius (Rₛ). Setting Rₘᵢₙ to the Planck length Lᴘ, the mass m resolves to the Planck mass mᴘ≈21.77 µg. The study is corrected or modified accordingly.

#Massproperties, #Gravitationalfields, #Antigravitationalfields, #Darkenergy, #Planckmass, #Newtonslawofgravity, #Intergalacticspace, #Superluminalspeeds, #Effectivemass, #Fundamentalphysics,

Interpreting Photon Behaviour and Gravity: A Classical Mechanics Perspective Supported by Experimental Results.

Soumendra Nath Thakur

ORCiD: 0000-0003-1871-7803

04-07-2024

A1. A photon's speed can be expressed as Planck length divided by Planck time, ℓP/tP = c, which is approximately 3 × 10⁸ m/s.

A2. The path of a photon is bent due to the momentum exchange of the photon with the external gravitational field of massive bodies, and not due to curvature in spacetime.

A3. There is no question of relativity ruling out Newton's gravity as a force, with the relativistic interpretation of gravity as curvature of spacetime—which appears to be flawed.

A4. Any mass (M or m) is the property of gravity that generates a gravitational field around it. A single mass does not experience gravitational force unless there is another massive object within the gravitational influence of the mass (M or m). Generally, M or m represents the masses of two objects, where one mass (M) is more massive than the other mass (m). This interpretation is in accordance with Newton's Law. That is why the equation (F = GMm/r²) represents the force of gravitational attraction between two masses, M and m.

A5. According to relativity, no mass can reach the speed of light in a local sense. This statement primarily applies to mass within gravitationally bound systems, where immense force is needed to accelerate a mass. This force generates so much kinetic energy that it distorts the body beyond recognition as mass, causing the atomic structure to undergo transformation much before it reaches the speed of light. However, in intergalactic space dominated by dark energy, the situation differs. Here, the effect of dark energy, causing antigravity, may cause galaxies to recede at speeds exceeding that of light due to gravitational-antigravitational interactions between the gravity of galactic masses and the antigravity effect of dark energy. This does not involve the local acceleration of mass to the speed of light but rather results in galaxies covering more distance than light can travel in the same amount of time

Reference:

My earlier research titled, "Direct Influence of Gravitational Field on Object Motion invalidates Spacetime Distortion" provides a mathematical framework supporting the idea that the path of a photon is influenced by momentum exchange with an external gravitational field rather than by spacetime curvature. The research outlines the following key points:

The total energy of a photon under gravitational influence (Eg) remains equivalent to its intrinsic energy (E), ensuring energy conservation (Eg = E).

Changes in photon momentum (Δρ) exhibit symmetry, represented by Δρ = −Δρ.

The constant speed of electromagnetic waves (ℓₚ/tₚ = c) is maintained, highlighting the significance of energy conservation in gravitational interactions.

This mathematical presentation elucidates the behaviour of photons in strong gravitational fields, emphasizing their energy-momentum relationship and wavelength variations under gravitational influence. The findings contribute to a deeper understanding of quantum mechanics and the interplay between photons and gravity, enriching our comprehension of the universe's fundamental principles.

Fundamental Physical Constants and the Role of Dark Energy in the Expansion of the Universe

These are some of the physical constants of the universe, some of which are fundamental:

Gravitational constant, Planck constant, Planck length, speed of light, electron mass, proton mass, neutron mass, Boltzmann constant, Avogadro number, elementary charge, strong coupling constant, Bohr magneton, and the gas constant. Additionally, the six types of quarks are fundamental particles.

However, there are observable events where speeds appear to exceed the speed of light, such as the phase velocity of waves, Cherenkov radiation, and the recession of distant galaxies due to the increase of distance between galaxies driven by dark energy. These phenomena do not violate the constancy of the speed of light in a vacuum, which remains a fundamental constant and the ultimate speed limit for information and matter.

More Information on the Subject:

These empirical observations indicate that distant galaxies are receding faster than the speed of light due to the antigravitational effect of dark energy. This phenomenon occurs at intergalactic or galactic cluster scales, in regions well beyond the gravitational influence of individual galaxies. In these areas, only cosmic redshift occurs, not gravitational redshift, due to the absence of significant gravitational influences.

Observational calculations of motion and gravitational effects on these galaxies are derived using classical mechanics, specifically Newton's laws of gravitation, rather than relativistic interpretations of gravity as curved spacetime. These calculations consider gravity as a force and ignore the concept of distorted spacetime. As a result, the empirical observations, calculations, and applications of classical mechanics confirm that the effective motions and gravitational-antigravitational interactions between celestial bodies are due to forces, as per the classical interpretation of gravity.

This approach establishes that the galaxies are influenced solely by gravitational-antigravitational effects as forces, without involving relativistic gravity as spacetime distortion. Reference: 'Dark energy and the structure of the Coma cluster of galaxies'.

In this context, three masses characterize the structure of a regular cluster: the matter mass Mᴍ, the dark-energy effective mass Mᴅᴇ (which is negative), and the gravitating mass Mɢ = Mᴍ + Mᴅᴇ. Cosmic antigravity can be stronger than gravity both globally and locally on scales of approximately 1–10 Mpc. The local weak-field dynamical effects of dark energy can be described using Newtonian mechanics. Gravity dominates at distances R < Rᴢɢ, while antigravity is stronger at R > Rᴢɢ. A gravitationally bound system with mass Mᴍ can exist only within its zero-gravity sphere of radius Rᴢɢ. The matter content (dark matter and baryons) of the cluster is characterized by the mass Mᴍ(R) inside radius R.

Studies of nearby systems like the Local Group and the Virgo and Fornax clusters suggest their sizes are close to their zero-gravity radii. Around them, galaxy flows are observed; these systems are in gravity-dominated regions (R < Rᴢɢ), and the outflows occur at (R > Rᴢɢ). If these local systems have nearly maximal sizes, this may explain the apparent underdensity of the local universe.

Comparing Planck Length (Lᴘ) and Minimum Radius (Rₘᵢₙ): The Smallest Meaningful Scale in Physics

Soumendra Nath Thakur

01-07-2014

The smallest possible radius for a mass m is given by the equation: Rₘᵢₙ = (G/c²)·m (correctly 2Gm/c²). This equation describes the Schwarzschild radius, which is the radius of the event horizon of a non-rotating black hole. The Schwarzschild radius is the critical radius at which the escape velocity from the mass m becomes equal to the speed of light c. Any mass compressed within this radius will form a black hole. The Schwarzschild radius is derived from the concept of escape velocity. The escape velocity (vₑ) from a spherical mass m is given by the formula: vₑ = √2Gm/r. For the escape velocity to be equal to the speed of light (c): c = √2Gm/r → c² = √2Gm/r → r = 2Gm/c². This radius is called the Schwarzschild radius (Rₛ): Rₛ = 2Gm/c². 

Comment:

This claim was made in response to my statement that the Planck length (Lᴘ) is the smallest perceptible length. Therefore, to evaluate this claim, the value of the relativistic mass (m) was calculated using the relation Rₘᵢₙ = Lᴘ, to find m = Lᴘ⋅c²/G, when radius Rₘᵢₙ is equal to the Planck length (Lᴘ), the calculated mass found to be approximately 21.77 micrograms:

m = {(1.616255 × 10⁻³⁵) × (2.998 × 10⁸)²}/6.67430 × 10⁻¹¹ = 21.77 micrograms

Thereafter I realize that this derived mass is also very close to the Planck mass (Mᴘ), which is approximately 21.76 micrograms.

Therefore, the relativistic mass (mᵣₑₗ) is actually the Planck mass (Mᴘ).

mᵣₑₗ = 21.77 µg ≈ 21.76 µg

A claim involves the idea that the smallest possible radius for a mass m is given by the equation:

Rₘᵢₙ = (G/c²)·m

where:

• G is the gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
• c is the speed of light in a vacuum (2.998 × 10⁸ m/s)
• m is the mass in question

1. Derivation and Verification

To understand what this equation implies, let's break it down:

• Gravitational Constant (G): This is a fundamental physical constant that measures the strength of gravity.

• Speed of Light (c): This is the maximum speed at which all energy, matter, and information in the universe can travel.

• Mass (m): This is the mass of the object being considered.

The equation suggests that there is a minimum radius below which a given mass m cannot be compressed. This radius is determined by the constants G and c, as well as the mass m.

Let's calculate Rₘᵢₙ for a specific mass to see if it is smaller than the Planck length 1.616255 × 10⁻³⁵ meters.

Calculation:

First, we rearrange the equation to find Rₘᵢₙ:

Rₘᵢₙ = (G/c²)·m

Plugging in the values for G and c:

G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²

c = 2.998 × 10⁸ m/s

Example Mass Calculation:

Let's calculate Rₘᵢₙ for a mass of 1 kg:

Rₘᵢₙ = (6.67430 × 10⁻¹¹)/(2.998 × 10⁸)² × 1

Simplifying:

Rₘᵢₙ = 6.67430 × 10⁻¹¹/8.988004 × 10¹⁶

Rₘᵢₙ ≈ 7.426 × 10⁻²⁸ meters

Comparison with Planck Length:

The Planck length is 1.616255 × 10⁻³⁵ meters

7.426 × 10⁻²⁸ meters > 1.616255 × 10⁻³⁵ meters

Conclusion:

The calculated Rₘᵢₙ for a mass of 1 kg is much larger than the Planck length.

Thus, for any realistic mass, Rₘᵢₙ as given by the equation is much larger than the Planck length. This implies that Rₘᵢₙ does not suggest a scale smaller than the Planck length, and for any mass m, Rₘᵢₙ will generally be larger than the Planck length, indicating that the Planck length remains the smallest meaningful scale in physics according to current understanding.

2. Let's calculate Rₘᵢₙ for a mass of 1 gram (which is 0.001 kg) using the equation:

Rₘᵢₙ = (G/c²)·m

where:

• G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²
• c = 2.998 × 10⁸ m/s
• m = 0.001 kg

Calculation:

First, substitute the known values into the equation:

Rₘᵢₙ = (6.67430 × 10⁻¹¹)/(2.998 × 10⁸)² × 0.001

Simplifying:

Calculate the denominator:

c² = (2.998 × 10⁸)² = 8.988004 × 10¹⁶

Now calculate the fraction:

G/c² = (6.67430×10⁻¹¹)/(8.988004×10¹⁶) ≈ 7.426×10⁻²⁸ meters/kilogram

Finally, multiply by the mass m:

Rₘᵢₙ = 7.426 × 10⁻²⁸ × 0.001 = 7.426 × 10⁻³¹ meters

Comparison with Planck Length:

The Planck length is 1.616255 × 10⁻³⁵ meters

7.426 × 10⁻³¹ meters > 1.616255 × 10⁻³⁵ meters

Conclusion:

For a mass of 1 gram (0.001 kg), the calculated Rₘᵢₙ is 7.426 10⁻³¹ meters, which is still significantly larger than the Planck length.

Therefore, even for a small mass like 1 gram, Rₘᵢₙ does not suggest a scale smaller than the Planck length. The Planck length remains the smallest meaningful scale in physics according to current understanding.

• Guinness World Records mentions, 'The Plank Length'. The smallest possible size for anything in the universe is the Planck length, which is 1.6 x 10⁻³⁵ m.

3. The Planck length, 1.616255 × 10⁻³⁵ meters, is often referred to as the smallest possible size for anything in the universe. This notion arises from theoretical considerations in quantum gravity and the limits of our current physical theories, particularly where quantum mechanics and general relativity intersect.

Comparing with Rₘᵢₙ:

To reiterate, Rₘᵢₙ as proposed, is given by:

Rₘᵢₙ = (G/c²)·m

Let's see if there could be a mass for which Rₘᵢₙ might approach or even be smaller than the Planck length.

Finding the mass where Rₘᵢₙ = Planck length:

Given:

Rₘᵢₙ = (G/c²)·m = Lᴘ

where Lᴘ is the Planck length 1.616255 × 10⁻³⁵ meters.

Rearrange to solve for m:

m = Lᴘ⋅c²/G

Plugging in the constants:

Lᴘ = 1.616255 × 10⁻³⁵ meters.

G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²

c = 2.998 × 10⁸ m/s

Calculation:

m = {(1.616255 × 10⁻³⁵) × (2.998 × 10⁸)²}/6.67430 × 10⁻¹¹

Calculate c²:

(2.998 × 10⁸)² = 8.988004 × 10¹⁶

Now, multiply Lᴘ by c²:

(1.616255 × 10⁻³⁵) × (8.988004 × 10¹⁶) = 1.453266 × 10⁻¹⁸

Finally, divide by G:

m = 1.453266 × 10¹⁸/6.67430 × 10⁻¹¹ ≈ 2.177 × 10⁻⁸ kg

This mass is approximately 2.177 × 10⁻⁸ kilograms or about 21.77 micrograms.

4. Summary of Discussion

Initial Equation and Concept

• Erroneous Equation: Rₘᵢₙ = G/c²·m

• This equation is intended to describe the smallest possible radius for a mass m, which is related to the Schwarzschild radius.

• Correct Schwarzschild Radius Equation: Rₛ = 2Gm/c²

• This is the radius of the event horizon of a non-rotating black hole, where the escape velocity equals the speed of light.

Derivation and Verification

• Setting Rₘᵢₙ to Planck Length Lᴘ:

• Erroneous Equation: G/c²·m = Lᴘ

• Solving for m:

m = Lᴘ·c²/G

• Since Lᴘ = √ℏG/c³:

m = √ℏG/c³ = mᴘ

• This resolves to the Planck mass mᴘ ≈ 21.77 μg.

• Modified Equation: Rₘᵢₙ = 2G/c²·m = Rₛ

• Setting Rₘᵢₙ to Planck Length Lᴘ:

2G/c²·m = Lᴘ

• Solving for m:

• m = Lᴘ·c²/2G

• Since Lᴘ = √ℏG/c³:

m = (√ℏG/c³)·c²/2G = √ℏc/G = mᴘ

• This also resolves to the Planck mass mᴘ ≈ 21.77 μg.

Conclusion

• The multiplier 2 in the equation Rₘᵢₙ = 2G/c²·m = Rₛ does not affect the value of m when Rₘᵢₙ is set to the Planck length Lᴘ. In both the erroneous and modified equations, the mass mᴘ.

• Final Statement: To maintain coherence, the correct form of the equation is Rₘᵢₙ = 2G/c²·m = Rₛ. Setting Rₘᵢₙ to the Planck length Lᴘ, the mass m indeed resolves to the Planck mass mᴘ ≈ 21.77 𝜇g. This consistency underscores the importance of using the correct form of the equation.

This summary confirms that using either form of the equation, the mass m equals the Planck mass when Rₘᵢₙ is set to the Planck length.

Conclusion:

For a mass of approximately 21.77 micrograms, the Rₘᵢₙ given by the proposer's formula equals the Planck length. For any mass greater than this, Rₘᵢₙ will be larger than the Planck length. For any mass smaller than this, Rₘᵢₙ would theoretically be smaller than the Planck length, but physical interpretations at such small scales are not well-defined by our current understanding of physics.

The assertion that the Planck length is the smallest meaningful length scale holds because it represents a fundamental limit below which the classical ideas of space and time cease to be applicable, marking a boundary of our current physical theories.

#planckmass #Plancklength, #Minimumradius, #Quantumgravity, #Gravitationalconstant, #Speedoflight, #Fundamentallimitsinphysics,