Soumendra Nath Thakur Dear Soumendra, what you calculate there above (and imho a little obfuscating, complicated formulated) is just Planck-mass, as is well known from standard expression root of hBar times c over G.

So ... what do you want me/us to tell here? All the expressions are exactly known for both value and an exact integer geometry in iSpace theory (btw.):

2.17646473349017217708713820147795038047920*10^-8 VAs^3/m^2 [Kg]

Thanks and kind regards,

Christian

Also adding to my above comment, from a integer geometric point of view, no-one (not even iSpace-theory) is yet able to predict microscopic geometry relations in full detail (which imho are likely integer-fractal btw), so such are by no means "triangles" (like in graphic cards of PCs), but could be square cell based (iSpace-Pi4 based), hexagonal cell based (iSpace-Pi3 based), a form of continuum with cells of apollonic packaging of base circles (like oranges in a sales pile in a shop), or any other base geometry not mentioned here like "true" (4D) continuum or string theory based 10D spacetime or mixed integer/continuum one as assumed by 10D spacetime of iSpace theory (6D aka 2 times 3D course grained subspace) + 3D (percieved continuos yet fine grained discrete) space + 1D (fully discrete) time etc. etc.

Please check it: Planck length is 1.616*10-35 meter.

Your equation shows me that Rmin varies with m but I have a different equation for radius and mass of matter.

Dear Christian G. Wolf Ittipat Roopkom

This claim was made in response to my statement that the Planck length (Lᴘ) is not the smallest perceptible length but Rₘᵢₙ. 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 realise 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

Best regards,

Soumendra Nath Thakur

Soumendra Nath Thakur while deriving properly from your model nothing but Plsnvk-mass (so you indeed model physically did something right!) it’s not micrograms (not wrong, but unusual to refer to so), it’s just Planckmass as in the books (derived by 4-5 digits limitation of current best Newtonian G measurement precision).

In iSpace theory however, there is predicted an exact integer geometry and hence also value for G snd with this and likewise exact Planck constant h and SI or IQ definition of exact speed of light c we (trivially!) get all Planck-units based constants exactly (like Planck-mass, Planck-length and Planck-time) via the well known relations in the books.

Everxthing including intimate intrinsic iSpace-theoretical revealed inter-constant relations are completely exact - and such from quantum to cosmological level over typical 10^42 (!) orders of magnitude (or even more). Exact is exact.

Just have a deep look on iSpace theory to fully understand why all such is necessarily so based on actual identified base quanta Volt, Ampere, Second and Meter. Its all freely available on my RG home ave very easy to understand, as basically first order multiplicative only (yet fully exact).

Best and kind regards,

Christian

Ittipat Roopkom „Your equation shows me that Rmin varies with m but …“ NO - wrong as shown by me as above m is not „some“ mass but the PlanckMass - and there is only one - as both c and G are constant (it’s wrong to assume G would be a cosmological running constant, while it contains a time factor (proven by iSpace theory) which compensated exactly for (so the argument is not wrong per-se, but all the concluding results are!).

Constants of nature verifiable do NOT vary!

And such is both trivially and easy to see and understand once willing to take the minimal time to follow iSpace theoretical result papers as freely available on my RG home, showing so proof-style in all of iSpace-IQ, iSpace-SI and with this so unavoidable also MKS/A-SI units.

I should deeply comprehend the iSpace theory. I will try to understand it, as it may connect to my theory about time and gravity in some way.

Soumendra Nath Thakur „the smallest perceptible length …“ is not (as blindly assumed by most re-praying unjustified mainstream wrong scientific “wisdom“) Planck-length - Planck base units are while „operable“ not quanta as wrongly assumed, but iSpace Volt, Ampere, Second and Meter quantized base units are (in form of all their SI time-human artefact freed-of form in iSpace-IQ unit system, with iSpace prefix:

1) the Quantum of Ampere:

GoldenRatio iSpaceAmpere (and Ampere!)

2) the Quantum of Volt:

2*Pi3*6961 iSpaceVolt (and Volt!)

3) the Quantum of time (!):

1/6961 iSpaceSecond (not SI, as itBHA^n)

4) the Quantum of length:

GoldenRatio iSpaceMeter (not SI, itBHA^n)

Thats the true quanta, everything else can be derived from - and exactly so by the very nature of exact input in very simple integer geometric multiplicative equations of course results in exact output as well (geometry and hence arbitrary choosable digit precision is available, all coded in exact (error-free by design) Wolfram Mathematica toolkit library of iSpace theory.

Christian G. Wolf

How can anyone appreciates a statement that states Planck length, Planck units as blindly assumed by most re-praying unjustified mainstream wrong scientific “wisdom“?

Nor this calculation or the conclusion are 'blindly assumed',

"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.

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."

I cannot appreciate this 'blindly assumed ' statement as scientific statement.

-

Soumendra Nath Thakur Well, sorry but for whatever yot try - you will not ever convince me as I know what I know from absolutely precise, exact derivation of iSpace integer geometry and esp. so human artefact free iSpace-IQ unit system - the one enabling for desired inter-constant zero unit pressure and with this likewise zero error bar values.

Planck units are NOT primary - proven - to check by anyone able to read an equation.

It seems to me, most really need to let loose of their „education induced believe rail“ here.

As with any breakthrough science like iSpace can and should be considered, there is a time before and after, changing many myths (like it’s impossible to calculate aby constant, or all known constants are independent - while in fact most are not at all.

If course everyone is free to ignore scientific progress (and massive one, in case of iSpace theory) and continue to persue any model one sees best fit for their own roadmap.

Kind regards,

Christian

Soumendra Nath Thakur You‘re aware that the root of hBar times c over G is just Plank-mass and exactly so !?! Nothing to argue here, as it’s absolute beginners level.

All the rest above is really just algebra to me.

Foreseeable algebra, that is.

Aside - you invited me to do an RG style review, commenting on your paper or discussion thread. So now dont come and tell you don’t like the outcome (of RG peer-review replacement, nothing else such is).

Christian G. Wolf

While the content of the discussion is very clear and does not favour or dislike any outcome, the question of replacing RG peer-review is irrelevant to the original discussion.

It's a discussion.

A claim was made in response to my statement that the Planck length (Lᴘ) is not the smallest perceptible length but Rₘᵢₙ.

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

Soumendra Nath Thakur your derivation sequence is not said to be wrong by me, what is imho confusing (as stated above) to use available CODATA G precision of 4-5 digits to suggest to the reader that you have discovered some (new) physical relation (as you don‘t, it’s all just algebraically rewritten well known mainstream Planck-unit math).

Your relation assumed is not approximate, but exact within the error bar of Newtonian G. As iSpace theory is able to calculate such with zero error bar and hence zero unit pressure from its integer geometric theoretic model, it’s proven the relation is an exact one - namely Planck-mass - indeed.

In other words: if one calculates a mass from a physical relation, and then either does not recognize it’s just Planck-mass or (for some reasons beyond me) intentionally avoids to hint to, it should be Planck-mass and instead name it some 21.77 (correct rounded 4-digit) mass in the milligram range (what should tell us that physically, anyway?) it’s not clear to me what’s going on and why such is done.

The “found“ relation is an exact one per definition of Planck-units and carries the well known name Planck-mass, of which as by definition of its creating equation there can only be one:

PlanckMass = Root[(h / (2*Pi)) * c) / G]

or in exact iSpace constants of nature library Mathematica notation with its typical i Prefix naming convention (nothing to do with complex numbers i):

imPl = Root[(ih / (2*Pi)) * ic) / iG]

imPl == 2.17646473349017217708713820147795038047920*10^-8 Volt Ampere Second^3 / Meter^2 (VAs^3/m^2 or [Kg] *)

or when desired to be converted to milligram

imPl == Convert[21.7646473349017217708713820147795038047920 Milli Gram, Volt Ampere Second^3 / Meter^2] (* [mg] *)

both of which above equality relations yield True and hence are prooven to not just be a numeric approximation only but an exact one (to arbitrary digit precision).

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.

• 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.

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

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

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.

Summary:

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.

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

2G/c²·m = Lᴘ

Since Lᴘ = √ℏG/c³:

m = (√ℏG/c³)·c²/2G = √ℏc/G = mᴘ

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

• 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.