Calculator
Example data table
| Model | Mass | a* | q | Approx. temperature trend |
|---|---|---|---|---|
| Schwarzschild | 1 M☉ | 0 | 0 | Extremely cold, far below a kelvin |
| Kerr | 10 M☉ | 0.8 | 0 | Lower temperature than non-spinning case |
| Reissner–Nordström | 1012 kg | 0 | 0.6 | Hotter than stellar masses, reduced by charge |
| Kerr–Newman | 1011 kg | 0.3 | 0.4 | Spin and charge both suppress temperature |
Formula used
The Hawking temperature is set by the surface gravity κ at the outer horizon:
- T = ħ κ / (2π kB c)
- For a non-spinning, uncharged black hole: T = ħ c³ / (8π G M kB)
For spinning and charged models, this tool applies a standard horizon factor using:
- s = √(1 − a*² − q²) (within the chosen model)
- Outer horizon radius: r+ = (GM/c²)(1 + s)
- Area: A = 4π (r+² + a²) with a = a* (GM/c²)
Charge uses q = Q / Qext, where Qext = √(4π ε₀ G) M. The evaporation time shown is the common Schwarzschild approximation.
How to use this calculator
- Select a black hole model that matches your scenario.
- Enter the mass and choose the mass unit.
- For spinning or charged cases, set a* and q values.
- Click Calculate to view results above the form.
- Use Download CSV or Download PDF for saving outputs.
Technical article
This calculator implements Hawking’s result by linking temperature to horizon surface gravity. It reports temperature in kelvin and translates the same scale into energy and frequency, so you can compare black hole radiation with laboratory and astrophysical backgrounds.
1) Hawking temperature scaling with mass
For a stationary, uncharged horizon the temperature follows T = ħc³/(8πGMkB). A one-solar-mass black hole is about 6.2×10−8 K, while 10 M☉ drops near 6×10−9 K. Smaller masses rise dramatically.
2) Why astrophysical black holes look “cold”
The cosmic microwave background is about 2.725 K, far above stellar-mass Hawking temperatures. That means typical astrophysical black holes absorb more ambient radiation than they emit thermally. Only sufficiently small primordial-mass candidates would radiate strongly compared with cosmic backgrounds.
3) Spin effects in the Kerr model
Rotation reduces the surface gravity at the outer horizon. The tool uses the dimensionless spin a* = cJ/(GM²) and applies a horizon factor that approaches zero as a*→1. For the same mass, increasing a* lowers temperature and increases horizon area.
4) Charge effects in Reissner–Nordström
Electric charge also suppresses surface gravity. The normalized fraction q = Q/Qext uses Qext = √(4π ε₀ G)M, the extremal scale where the horizon becomes degenerate. As q approaches 1, the computed temperature trends toward zero.
5) Kerr–Newman: combined spin and charge
When both rotation and charge are present, the horizon condition requires a*² + q² < 1. The calculator returns r+ and r−, showing how the horizons separate with decreasing extremality. Near extremality, small parameter changes can noticeably shift temperature.
6) Geometry outputs that support auditing
Beyond temperature, the page reports outer/inner radii, area A = 4π(r+² + a²), and Bekenstein–Hawking entropy S = kBc³A/(4Għ). These derived quantities help check consistency: larger area generally corresponds to lower temperature in comparable models.
7) Energy and frequency interpretations
The same temperature scale is shown as kBT in joules, kBT in eV, and a thermal frequency kBT/h in hertz. For example, a 1011 kg Schwarzschild mass implies T≈1.2×1012 K, corresponding to high-energy emission scales.
8) Evaporation time as a scaling indicator
The evaporation time displayed uses the common Schwarzschild estimate t ≈ 5120πG²M³/(ħc⁴). It is useful for order-of-magnitude comparisons because of the strong M³ dependence. Detailed lifetimes depend on graybody factors and available particle species.
FAQs
1) What does this calculator compute?
It computes Hawking temperature from mass and the selected horizon model, then reports r±, area, surface gravity, entropy, kBT, and a thermal frequency for interpretation.
2) What inputs are required?
Mass is always required. Spin a* is required for Kerr and Kerr–Newman. Charge fraction q is required for Reissner–Nordström and Kerr–Newman. Use a* and q between 0 and just below 1.
3) What does the horizon condition mean?
To keep an event horizon, the model requires a*2 + q2 < 1. Values near 1 represent near-extremal black holes where the temperature becomes very small.
4) Why are stellar-mass temperatures tiny?
Temperature scales as 1/M. For masses near a solar mass, T is around 10−8 K, far below the 2.725 K cosmic microwave background, so ambient radiation dominates.
5) Are the charge outputs realistic astrophysically?
Large net charge is usually neutralized quickly by surrounding plasma. The charge options are mainly for theoretical exploration and to study how extremality modifies horizon structure and temperature.
6) Is the evaporation time exact?
No. It is a standard Schwarzschild scaling estimate. It is excellent for trends (t ∝ M³) but not a precision lifetime because graybody factors and particle physics details are neglected.
7) How do I export results?
After calculating, use Download CSV for a structured table or Download PDF for reporting. Both exports include inputs and computed outputs, supporting reproducible records.