1. What is Black Hole Temperature Calculator?

Definition: This calculator computes the Hawking temperature ($T$) of a black hole, which is the theoretical temperature at which a black hole emits Hawking radiation, based on the black hole's mass ($M$).

Purpose: It is used in theoretical physics to understand the thermal properties of black holes, which are related to their quantum behavior and eventual evaporation through Hawking radiation.

2. How Does the Calculator Work?

The calculator uses the following equation to compute the Hawking temperature:

Where:

• $T$: Hawking temperature (K)
• $\hslash$: Reduced Planck constant ($1.054571800\times {10}^{-34}\text{J·s}$)
• $c$: Speed of light ($299,792,458\text{m/s}$)
• $G$: Gravitational constant ($6.67408\times {10}^{-11}{\text{m}}^3{\text{kg}}^{-1}{\text{s}}^{-2}$)
• $M$: Mass of the black hole (kg)
• $k_B$: Boltzmann constant ($1.380649\times {10}^{-23}\text{J/K}$)

Unit Conversions:

• Mass ($M$):
  • 1 kg = 1 kg
  • 1 Solar Mass = $1.989\times {10}^{30}\text{kg}$
  • 1 Earth Mass = $5.972\times {10}^{24}\text{kg}$
• Temperature ($T$):
  • 1 K = 1 K
  • °C = K - 273.15
  • °F = (K - 273.15) × 9/5 + 32

Steps:

• Enter the mass of the black hole ($M$) with its respective unit (kg, solar masses, or Earth masses).
• Convert the mass to kilograms.
• Calculate $T$ using the Hawking temperature equation.
• Convert the result to the selected unit for display (K, °C, or °F).
• If the temperature is less than 0.0001, display it in scientific notation; otherwise, display it with 10 decimal places due to the extremely small temperatures involved.

3. Importance of Black Hole Temperature Calculation

Calculating the Hawking temperature is crucial for:

• Understanding Black Hole Evaporation: The temperature determines the rate at which a black hole emits Hawking radiation, leading to its eventual evaporation over extremely long timescales.
• Quantum Gravity Research: The Hawking temperature connects quantum mechanics and general relativity, providing insights into the nature of quantum gravity.
• Astrophysical Phenomena: Helps in studying the thermal effects of black holes on their surroundings, such as the emission of radiation detectable in principle.

4. Using the Calculator

Example: Calculate the Hawking temperature of a black hole with a mass of 1 solar mass ($M=1.989\times {10}^{30}\text{kg}$).

1. Enter the mass as 1 solar mass.
2. The calculator computes:
   • $T=\frac{\hslash c^3}{8\pi GM{k}_B}=\frac{(1.054571800\times {10}^{-34})\times (299,792,458{)}^3}{8\times 3.14159265359\times (6.67408\times {10}^{-11})\times (1.989\times {10}^{30})\times (1.380649\times {10}^{-23})}$
   • $T\approx \frac{2.841\times {10}^{22}}{4.614\times {10}^{18}}\approx 6.172\times {10}^{-8}\text{K}$
   • Since $6.172\times {10}^{-8}<0.0001$, the result is displayed in scientific notation: $6.1720\times {10}^{-8}\text{K}$
   • In °C: $6.172\times {10}^{-8}-273.15\approx -273.1499999383\text{°C}$, which is greater than 0.0001 in magnitude, so displayed as $-273.1499999383\text{°C}$

5. Frequently Asked Questions (FAQ)

Q: What is the Hawking temperature?
A: The Hawking temperature ($T$) is the theoretical temperature at which a black hole emits Hawking radiation due to quantum effects near the event horizon, as predicted by Stephen Hawking.

Q: Why is the Hawking temperature so low for large black holes?
A: The temperature is inversely proportional to the black hole's mass ($T\propto \frac{1}{M}$). Larger black holes have lower temperatures because their event horizons are farther from the singularity, reducing the quantum effects that produce Hawking radiation.

Q: Why is scientific notation used for small temperatures?
A: For temperatures less than 0.0001, scientific notation is used to make the extremely small values more readable and to avoid displaying long strings of zeros.

Black Hole Temperature Calculator© - All Rights Reserved 2025