Engineering
Schwarzschild Black Hole Radius, Gravitational Time Dilation & Hawking Temp
Calculate the Schwarzschild event horizon radius R_s = 2*G*M / c^2, general-relativistic gravitational time dilation 1 / SQRT(1 - R_s / r), and Hawking radiation temperature.
Nonrotating, uncharged black-hole model
The Schwarzschild model describes an ideal nonrotating, uncharged black hole. Its areal horizon radius is 2GM/c^2. The coordinate description does not imply that locally measured light stops moving at the horizon. The rotating or charged cases require different expressions.
For a static clock outside the horizon, dt/dtau=1/SQRT(1-Rs/r) compares Schwarzschild time, normalized to a clock at infinity, with the local proper time. It is also the factor 1+z for a photon received at infinity from that static emitter, not z itself. A freely falling clock is a different comparison.
The static-clock factor grows without bound as r approaches Rs from outside. At or inside the horizon this static-observer formula is not applicable; it does not return a physical static-clock rate there. All calculations use unrounded Rs.
| Relativistic Black Hole Metric | Formula | 10 Solar-Mass Black Hole (r = 45 km) | 1 Solar Mass / Inside Horizon |
|---|---|---|---|
| Schwarzschild Radius R_s (km) | ((2 * G * M) / c^2) / 1000 | 29.5334 km | 2.9533 km (1 M_sun) |
| Photon Sphere Radius (1.5 * R_s) | 1.5 * R_s | 44.3001 km (unstable light orbit) | 4.4300 km |
| Time Dilation Factor at r = 45 km | 1 / SQRT(1 - R_s / r) | dt/dtau = 1.7057 | Static-clock formula not applicable at or inside Rs |
Schwarzschild Event Horizon Radius R_s in Kilometers
Evaluates 2 * G * (M_suns * M_sun_kg) / c^2 in meters and divides by 1,000.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(((2 * B4 * (B2 * B3)) / (B5 ^ 2)) / 1000, 4)))Gravitational Time-Dilation / Redshift Factor with Event-Horizon Guard
B6 is the areal radius in kilometers, not height above the horizon. The check includes equality with the horizon and returns a descriptive result rather than attempting a square root outside its domain.
=IF(COUNT(B2:B6)<>5,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<=0,B6<=0),"Invalid inputs",IF(B6 <= (((2 * B4 * (B2 * B3)) / (B5 ^ 2)) / 1000), "No static observer at or inside horizon", ROUND(1 / SQRT(1 - ((((2 * B4 * (B2 * B3)) / (B5 ^ 2)) / 1000) / B6)), 4))))Hawking temperature in nanokelvin
The semiclassical Schwarzschild prediction is hbar*c^3/(8*pi*G*M*kB). This section displays nanokelvin to avoid rounding stellar-mass temperatures to zero. hbar is the reduced Planck constant, not h; the constants supplied here include rounded values. This is a theoretical temperature, not a measured stellar black-hole signal.
=IF(COUNT(B2:B5,B7:B8)<>6,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<=0,B7<=0,B8<=0),"Invalid inputs",ROUND((B7*B5^3/(8*PI()*B4*B2*B3*B8))*1000000000,4)))Photon-sphere areal radius in kilometers
In this Schwarzschild model the photon sphere is at 1.5Rs. Its circular null orbits are unstable; it is not a stable parking orbit for a massive spacecraft.
=IF(COUNT(B2:B5)<>4,"Invalid inputs",IF(OR(B2<=0,B3<=0,B4<=0,B5<=0),"Invalid inputs",ROUND(1.5*((2*B4*B2*B3)/B5^2)/1000,4)))Try the example
Use the original values or change them to test your understanding.