Black Hole Calculator - Schwarzschild Radius & Hawking

A black hole calculator turns a single mass into the geometric, thermodynamic, and dynamical properties of a non-rotating uncharged Schwarzschild black hole: the Schwarzschild radius that defines the event horizon, the Hawking temperature of its thermal radiation, the surface gravity at the horizon, the Bekenstein-Hawking entropy, the lifetime an isolated black hole would take to evaporate, and the average density inside the horizon.

• • Stellar black hole homework: Compute the Schwarzschild radius, Hawking temperature, and lifetime of a 5-50 solar mass remnant from a recent supernova.
• • Supermassive black hole sizes: Check the event horizon diameter and surface gravity of M87*, Sgr A*, or any other supermassive candidate in solar masses.
• • Primordial black hole limits: Estimate when a primordial black hole of given mass would have evaporated and what Hawking temperature it reached.
• • Astrophysics concept review: See how the Schwarzschild radius, Hawking temperature, and entropy scale with mass for coursework on general relativity and black hole thermodynamics.

The black hole calculator covers the textbook non-rotating Schwarzschild solution of general relativity. It does not model rotating (Kerr) or charged (Reissner-Nordstrom) black holes, because those require additional inputs (angular momentum or charge) that change the geometry of the horizon and the formulas for Hawking radiation.

Every output comes from a single closed-form formula evaluated with NIST CODATA 2018 constants, so you can quote the same numbers a textbook or a research paper would use. The mass input is shared by every output, which is why the calculator stays useful for both a quick homework check and a research-style sanity check.

Once you have the Schwarzschild radius, the Gravitational Time Dilation Calculator uses the same mass to compute gravitational redshift and time dilation at a chosen radius, which is the next concept in any black hole physics unit.

The calculator takes the input mass, converts it to kilograms, then evaluates five closed-form formulas: the Schwarzschild radius r_s = 2GM/c^2, the Hawking temperature T_H = hbar*c^3/(8*pi*G*M*k_B), the surface gravity kappa = c^4/(4*G*M), the Bekenstein-Hawking entropy S = 4*pi*G*M^2*k_B/(hbar*c), and the evaporation lifetime t = 5120*pi*G^2*M^3/(hbar*c^4). Average density uses rho = M/((4/3)*pi*r_s^3).

• M: Black hole mass in kilograms (or solar masses, converted to kilograms).
• G: Gravitational constant, 6.67430e-11 m^3 kg^-1 s^-2 (NIST CODATA 2018).
• c: Speed of light, 299792458 m/s exactly (NIST CODATA 2018).
• hbar: Reduced Planck constant, 1.054571817e-34 J s (NIST CODATA 2018).
• k_B: Boltzmann constant, 1.380649e-23 J/K (NIST CODATA 2018).

All formulas are evaluated in one pass through a single pure JavaScript function. The unit selector only affects the mass-to-kg conversion; every output uses SI base units internally and is converted to the user-friendly unit (kilometres, kelvin, years) for display.

According to NIST CODATA 2018 - Fundamental Physical Constants, G, c, hbar, and k_B take the values used in the formulas above, which is the same fixed set evaluated by the calculator.

For circular orbits that dip close to the horizon, the Orbital Period Calculator takes the same mass as the central parameter and shows how a stable orbit shrinks as you approach r_s.

Four ideas hold the calculator together: a horizon defined by mass, a Hawking temperature that falls as 1/M, a Bekenstein-Hawking entropy that grows as the horizon area, and an evaporation lifetime that grows as M^3.

These four concepts share a common thread: every formula depends only on the mass. Once you know M, the geometry of the horizon, the temperature of the radiation it emits, the entropy it carries, and its evaporation time are all fixed by the same input.

Time dilation near a black hole comes in two forms, and the Time Dilation Calculator handles both the special-relativity gamma factor and the gravitational redshift factor at a chosen radius, which are the two corrections that enter Hawking's derivation.

Enter a mass in either kilograms or solar masses, then read the nine outputs the panel produces. The calculation runs on every change.

1. 1 Choose a mass unit: Pick solar masses for stellar or supermassive black holes; pick kilograms for primordial or Earth-mass candidates.
2. 2 Enter the mass: Type the mass value. Defaults to 10 solar masses, a textbook size for a stellar black hole from core collapse.
3. 3 Read the Schwarzschild radius: The headline output is the event horizon radius in kilometres; the diameter below it is twice that value.
4. 4 Check the Hawking temperature: Compare the Hawking temperature against the cosmic microwave background (2.725 K) to see whether the black hole is hotter or colder than the present-day Universe.
5. 5 Scan the surface gravity and density: Surface gravity in Earth g is huge even for supermassive holes; density falls fast as mass grows, which is the classic paradox about supermassive black holes being less dense than water.
6. 6 Compare lifetime against the Universe: The evaporation lifetime in years can be compared to the current age of the Universe (about 1.38e10 years) to decide whether the black hole has already evaporated.

For a quick textbook check, leave the mass unit on solar masses and enter 1. The radius reads about 2.95 km, the Hawking temperature about 6e-8 K, and the lifetime about 2e67 years. Switch to kilograms and enter 1.989e30 to confirm the same outputs (rounding aside). For a primordial black hole that would have evaporated by today, enter 1e11 kg and look for a lifetime below 1.38e10 years.

To read the Hawking temperature as a thermal population, the Boltzmann Factor Calculator uses the same k_B to turn T_H into exp(-E/(k_B*T)) for an energy E near the Hawking temperature.

The black hole calculator packages six textbook formulas behind one input, so homework, exam prep, and research-style sanity checks share the same numbers.

• • Six outputs from one mass: Schwarzschild radius, Hawking temperature, surface gravity, Bekenstein-Hawking entropy, evaporation lifetime, and density all share the same mass input.
• • NIST CODATA 2018 constants: G, c, hbar, k_B, and the solar mass are read straight from NIST CODATA 2018, so the outputs match published values to four or more digits.
• • Covers 21 orders of magnitude: From Planck-mass candidates (1e-8 kg) to the heaviest known supermassive holes (1e13 kg), the same formulas stay valid within their respective domains.
• • Solar or SI units: The mass selector lets you quote M87* in solar masses without first dividing by 1.989e30, while keeping kg available for primordial and textbook SI exercises.
• • Cross-validation friendly: Worked examples match M87*, a 10-solar-mass stellar hole, and the textbook one-solar-mass reference, so the calculator can be cross-checked against lecture notes.

For one-off homework the calculator gives you a defensible numerical answer in seconds; for research-style checks it gives a consistent set of black hole properties from a single mass without re-typing the constants.

Three input factors change every output and the classification band labels where the input sits, while three assumptions and caveats keep the numbers honest about the regimes where they apply.

• • The calculator assumes a non-rotating uncharged Schwarzschild black hole; rotating (Kerr) or charged (Reissner-Nordstrom) holes have a smaller horizon, a different Hawking spectrum, and an extra input (angular momentum or charge).
• • The evaporation lifetime is the standard Hawking estimate for an isolated black hole and ignores accretion, mergers, and the cosmic background; for a real astrophysical black hole the relevant timescale is set by accretion, not by Hawking radiation.
• • Near or below the Planck mass the formulas are extrapolations; quantum gravity is expected to modify both the Hawking temperature and the final evaporation stage, so the results should be read as order-of-magnitude.

Hawking radiation has never been observed directly, so the temperature and lifetime outputs are theoretical predictions that combine general relativity with quantum field theory in curved spacetime.

According to NASA - Black Holes, isolated black holes emit Hawking radiation as a thermal black-body and evaporate on a timescale that scales as M^3, well below the cosmic microwave background for any astrophysical mass.

When the calculation moves from black-body emission to kinetic theory, the Mean Free Path Calculator gives the average distance a thermal particle travels between collisions, the kinetic-theory tool used to characterise the radiation field in equilibrium with the horizon.