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.
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