Black Hole Lifetime Calculator by Temperature and Speed

Black Hole Lifetime Equation:

\[ t = \frac{5120 \pi (G M / c^2)^3 (c^5 / (G^2 \hbar))}{T^4 v^3} \]

Estimated Black Hole Lifetime:

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1. What is the Black Hole Lifetime Equation?

The Black Hole Lifetime equation estimates how long a black hole of given mass would take to evaporate via Hawking radiation, considering its temperature and relative speed. This calculation is based on quantum field theory in curved spacetime.

2. How Does the Calculator Work?

The calculator uses the Black Hole Lifetime equation:

\[ t = \frac{5120 \pi (G M / c^2)^3 (c^5 / (G^2 \hbar))}{T^4 v^3} \]

Where:

\( G \) — Gravitational constant (6.67430 × 10⁻¹¹ m³ kg⁻¹ s⁻²)
\( M \) — Black hole mass (kg)
\( c \) — Speed of light (299,792,458 m/s)
\( \hbar \) — Reduced Planck constant (1.0545718 × 10⁻³⁴ J s)
\( T \) — Temperature (K)
\( v \) — Speed (fraction of light speed)

Explanation: The equation accounts for the Hawking radiation process where black holes lose mass over time, with the rate depending on their temperature and relative motion.

3. Importance of Black Hole Lifetime Calculation

Details: Understanding black hole lifetimes is crucial for astrophysics, quantum gravity research, and studying the ultimate fate of black holes in our universe.

4. Using the Calculator

Tips: Enter mass in kilograms, temperature in Kelvin, and speed as a fraction of light speed (0 to 1). All values must be positive.

5. Frequently Asked Questions (FAQ)

Q1: Why does speed affect black hole lifetime?
A: Relative motion increases the effective temperature of Hawking radiation, accelerating evaporation.

Q2: What's the typical lifetime of a stellar-mass black hole?
A: For a 5 solar mass black hole (≈10³¹ kg) at 2.7K (CMB temperature), lifetime exceeds the current age of the universe.

Q3: How accurate is this calculation?
A: It's a theoretical estimate - actual evaporation may differ due to environmental factors and unknown physics.

Q4: Can primordial black holes evaporate in our lifetime?
A: Very small primordial black holes (mass < 10¹² kg) could be evaporating now, potentially detectable by gamma-ray bursts.

Q5: What happens at the end of a black hole's life?
A: Theoretical models suggest a final explosive evaporation, but the exact process remains uncertain.