Schwarzschild Radius

12-07-2024

The Schwarzschild radius is a measure used in the context of black holes, representing the radius of the event horizon. The event horizon is the boundary beyond which nothing, not even light, can escape the gravitational pull of a black hole.

Equation for Schwarzschild Radius

The Schwarzschild radius (rₛ) is given by the formula:

rₛ = 2GM/c²

Where:

• G is the gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)

• M is the mass of the object

• c is the speed of light in a vacuum (3 × 10⁸ m/s)

Description

• Gravitational Constant (G): This is a fundamental constant that quantifies the strength of gravity in Newton's law of universal gravitation.

• Mass (M): The mass of the object for which we are calculating the Schwarzschild radius.

• Speed of Light (c): The speed at which light travels in a vacuum.

The Schwarzschild radius is significant because it provides a boundary around a black hole. If an object is compressed within this radius, it will form a black hole. For instance, the Schwarzschild radius for Earth is about 9 millimetres, meaning if you could compress all of Earth's mass into a sphere with a radius of 9 millimetres, it would become a black hole.

Explanation

The Schwarzschild radius calculated using relativistic principles approximately equals the Planck length when the mass involved is on the order of the Planck mass. This connection highlights the scale at which quantum effects and gravitational considerations become significant, as envisioned by Max Planck's work.

• Relativistic Principles: The Schwarzschild radius is derived from Einstein's theory of General Relativity, which provides a relativistic description of gravity.

• Planck Length: The Planck length (ℓp) is the scale at which quantum gravitational effects are believed to become significant. It is approximately 1.616 × 10⁻³⁵ meters.

• Planck Mass: The Planck mass (mᴘ) is the mass scale at which quantum gravitational effects are expected to be important. It is approximately 2.177 × 10⁻⁸ kilograms.

When substituting the Planck mass into the Schwarzschild radius equation:

rₛ = 2Gmᴘ/c²

Given that G = 6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻² and c = 3 × 10⁸ m/s

rₛ = 2 × 6.67430 × 10⁻¹¹ × 2.177 × 10⁻⁸/(3 × 10⁸)²

This yields a radius on the order of the Planck length (ℓp = 1.616 × 10⁻³⁵ meters).

Significance

This relationship shows that at the Planck scale, both quantum mechanical and relativistic gravitational effects are significant. Max Planck introduced these fundamental units to describe the scales where the effects of quantum gravity cannot be ignored. This is why the Planck length is often considered the smallest meaningful length scale, and the Planck mass represents the mass at which a particle's Schwarzschild radius is comparable to its Compton wavelength.

Exploring Minimum Radius and Gravitational Dynamics: A Critical Analysis of Mr. Berndt Barkholz's Propositions

11-07-2024

Abstract:

This study examines Mr. Berndt Barkholz's propositions concerning the minimum radius rₘᵢₙ = Gm/c² associated with a mass m where gravitational effects dominate. It discusses the orbital velocity condition v₀ = √Gm/r < c, highlighting its critical role in orbital mechanics and gravitational parameters. The derivation of rₘᵢₙ = Gm/c² is explored, emphasizing its significance in delineating regions of profound relativistic effects. The challenges in determining R or rₘᵢₙ  without external references like the Planck scale are addressed, emphasizing the need for specific observational or theoretical constraints. Furthermore, the paper evaluates the relationship m = 1.349×10²⁷ × rₘᵢₙ  proposed by Mr. Barkholz, urging for empirical or theoretical validation of m or rₘᵢₙ to substantiate claims about the smallest possible radius in gravitational theory. This analysis underscores the relevance and essentiality of Planck units, particularly when R < ℓᴘ, where ℓᴘ/tᴘ > vᴀᴠɢₘₐₓ > c.

Dear Mr. Berndt Barkholz,

Thank you for your detailed exploration of the minimum radius  and its implications for gravitational dynamics. Your approach provides a thought-provoking perspective that challenges conventional interpretations.

1. Minimum Radius rₘᵢₙ: You proposed rₘᵢₙ = Gm/c² as the smallest radius associated with a mass m, where gravitational effects dominate.

2. Orbital Velocity Condition: Your discussion on the orbital velocity condition v₀ = √Gm/r < c underscores the critical relationship between orbital mechanics and gravitational parameters. (R < Gm/c²).

3. Deriving rₘᵢₙ: From the orbital velocity condition, we derived rₘᵢₙ = Gm/c², highlighting its significance in delineating regions where relativistic effects become profound.

4. Finding R or rₘᵢₙ: Determining R or rₘᵢₙ without external references like the Planck scale requires specific observational or theoretical constraints pertinent to the physical system in question.

5. Determining m: Once R or rₘᵢₙ is established, m can be computed using rₘᵢₙ = Gm/c², adhering strictly to gravitational dynamics rather than external theoretical scales.

Acknowledging the rationality of your approach, it remains essential to substantiate the proposed relationship m = 1.349×10²⁷ × rₘᵢₙ  with empirical or theoretical values for either m or rₘᵢₙ. This clarity is crucial for validating the assertion that rₘᵢₙ represents the smallest possible radius without recourse to external scales.

Critical Consideration:

In the context of your analysis, it is important to note that Planck units are indeed relevant and essential, especially when considering the condition R < Lp. In such a scenario, the relationship ℓᴘ/tᴘ > vᴀᴠɢₘₐₓ > c becomes significant, where c = 3 × 10⁸ m/s.

In conclusion, your contributions open avenues for deeper exploration into gravitational theory. Clarifying the values of m or rₘᵢₙ would strengthen the scientific basis of your hypothesis and its implications for our understanding of gravitational phenomena.

Best regards,

Soumendra Nath Thakur

Equational Presentation of the Smallest Possible Radius for a Mass:

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.

#SchwarzschildRadius #PlanckMass #RestMass #QuantumGravity #PlanckLength

Maximum Speed of Electromagnetic Waves: Planck Length and Planck Time in Relation to the Speed of Light.

Soumendra Nath Thakur

09-07-2024

Max Planck formulated his expressions for the Planck scale in 1899, which was before the development of both relativity and quantum mechanics. The relationship between the Planck length (ℓₚ) and the Planck time (tₚ) to find the maximum speed of electromagnetic waves.

Given:

ℓₚ is the Planck length, which is the smallest possible perceptible length.

tₚ is the Planck time, which is the shortest possible meaningful time.

The speed of propagation vᵥᵥₐᵥₑ is the distance the wave travels in a given time, which is one wavelength in a time of one period. In equation form, it is written as: vᵥᵥₐᵥₑ = λ/T.

Since, vᵥᵥₐᵥₑ = λ/T = fλ, the maximum speed of electromagnetic waves, vᵥᵥₐᵥₑ(ₘₐₓ), can be given by the ratio of the Planck length to the Planck time:

vᵥᵥₐᵥₑ(ₘₐₓ) = ℓₚ/tₚ 

The Planck length (ℓₚ) is defined as:

ℓₚ = √ℏG/c³

The Planck time (tₚ) is defined as:

tₚ = √ℏG/c⁵

To find vᵥᵥₐᵥₑ(ₘₐₓ), substitute these definitions into the ratio:

vᵥᵥₐᵥₑ(ₘₐₓ) = √(ℏG/c³)/√(ℏG/c⁵) 

Simplify the expression:

vᵥᵥₐᵥₑ(ₘₐₓ) = √(ℏG/c³)/√(ℏG/c⁵) = √c²

vᵥᵥₐᵥₑ(ₘₐₓ) = c

Thus, the maximum speed of electromagnetic waves is:

vᵥᵥₐᵥₑ(ₘₐₓ) = c

This confirms that the maximum speed of electromagnetic waves is the speed of light (c), which aligns with our current understanding of physics.

#SpeedOfLight

Inverse-square law:

The inverse-square law is a scientific principle stating that the observed intensity of a physical quantity decreases in proportion to the square of the distance from its source. This phenomenon arises due to the geometric spreading of radiation from a point source in three-dimensional space.

• Intensity ∝ 1/distance²
• Intensity₁ × 1/distance₁² = Intensity₂ × 1/distance₂²