Soumendra Nath Thakur
10-07-2024
Abstract:
This text explores the smallest possible radius for a mass using the Schwarzschild radius equation, emphasizing the distinction between rest mass and relativistic mass. By setting the Schwarzschild radius to the Planck length, the mass resolves to the Planck mass (approximately 21.77 μg). This demonstrates the fundamental limit where quantum gravitational effects become significant.
Keywords: Schwarzschild radius, Planck mass, Rest mass, Quantum gravity, Planck length,
In the context of the Schwarzschild radius equation, the mass m typically refers to the rest mass (or invariant mass) of an object, not the relativistic mass. The Schwarzschild radius is derived from general relativity and applies to objects with rest mass, describing the radius at which the escape velocity equals the speed of light.
1. Schwarzschild Radius Equation (Smallest Possible Radius for a Mass):
Rₛ = 2Gm/c²
2. Expressing the Smallest Possible Radius:
Rₘᵢₙ = 2G/c²·m =Rₛ
3. Setting Rₘᵢₙ to the Planck Length (Lᴘ):
Lᴘ = 2G/c²·m
4. Solving for m (Planck Mass):
m = (√ℏG/c³)·c²/2G = √ℏc/G = mᴘ ≈ 21.77 μg
Conclusion:
The smallest possible radius for a mass is given by:
Rₘᵢₙ = 2G/c²·m =Rₛ
This holds true when:
Rₘᵢₙ = Lᴘ
Clarification on Mass m:
• Type of Mass: The mass m in the Schwarzschild radius equation is the rest mass (invariant mass).
• Planck Mass: In the context of setting Rₘᵢₙ to the Planck length (Lᴘ), the mass m resolves to the Planck mass (mᴘ), which is approximately 21.77 µg.
The rest mass m considered in the Schwarzschild radius equation is equivalent in value to the Planck mass mᴘ when the smallest possible radius Rₘᵢₙ is set to the Planck length Lᴘ. This equivalence underscores the fundamental limit where quantum gravitational effects become significant.
Conclusion:
The equational presentation of the smallest possible radius for a mass, as described by the Schwarzschild radius equation and its connection to the Planck length, highlights the pivotal role of rest mass (invariant mass) in gravitational physics. Setting the smallest radius to the Planck length corresponds to the Planck mass, approximately 21.77 μg, signifying the boundary where quantum gravitational effects become prominent. This alignment underscores the fundamental interplay between mass, radius, and the onset of quantum gravity at microscopic scales.
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