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Black Holes & Schwarzschild Radius Reference cheat sheet - grade 11-12

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Physics Grade 11-12

Black Holes & Schwarzschild Radius Reference Cheat Sheet

A printable reference covering black holes, event horizons, Schwarzschild radius, escape velocity, gravitational time dilation, and accretion disks for grades 11-12.

TopicsPhysicsRelativityGravityEnergyTimeOrbitsSpace ExplorationStudy Guide
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This cheat sheet summarizes the physics of black holes, with a focus on the Schwarzschild radius and the event horizon. Students need it to connect Newtonian gravity, relativity, and observable effects such as time dilation and accretion. It is especially useful for comparing black holes with stars, planets, and other massive objects.

The reference keeps the most important equations, constants, and ideas in one printable place.

The central formula is the Schwarzschild radius, rs=2GMc2r_s = \frac{2GM}{c^2}, which gives the event horizon radius for a non-rotating, uncharged black hole. Escape velocity reaches the speed of light when ve=cv_e = c, making the event horizon a boundary beyond which light cannot escape. Gravitational time dilation near a Schwarzschild black hole is described by Δtfar=Δtnear1−rsr\Delta t_{\text{far}} = \frac{\Delta t_{\text{near}}}{\sqrt{1 - \frac{r_s}{r}}}.

Key ideas include mass, radius, density, tidal forces, photon orbits, and the difference between the event horizon and the singularity.

Key Facts

  • The Schwarzschild radius of a non-rotating, uncharged black hole is rs=2GMc2r_s = \frac{2GM}{c^2}.
  • Newtonian escape velocity is ve=2GMrv_e = \sqrt{\frac{2GM}{r}}, and setting ve=cv_e = c gives the Schwarzschild radius.
  • The event horizon is located at r=rsr = r_s, where the escape velocity equals the speed of light cc.
  • For a Schwarzschild black hole, gravitational time dilation is Δtfar=Δtnear1−rsr\Delta t_{\text{far}} = \frac{\Delta t_{\text{near}}}{\sqrt{1 - \frac{r_s}{r}}} for r>rsr > r_s.
  • The photon sphere of a Schwarzschild black hole is at r=32rsr = \frac{3}{2}r_s, where light can orbit in an unstable circular path.
  • The innermost stable circular orbit for matter around a Schwarzschild black hole is r=3rsr = 3r_s.
  • The average density inside the event horizon can be estimated by ρ=M43πrs3\rho = \frac{M}{\frac{4}{3}\pi r_s^3}, even though the real interior is not uniform.
  • Tidal force effects increase when gravitational field strength changes rapidly with distance, approximately following Δg≈2GMΔrr3\Delta g \approx \frac{2GM\Delta r}{r^3}.

Vocabulary

Black hole
A region of spacetime where gravity is so strong that nothing inside the event horizon can escape.
Schwarzschild radius
The radius rs=2GMc2r_s = \frac{2GM}{c^2} of the event horizon for a non-rotating, uncharged black hole.
Event horizon
The boundary at r=rsr = r_s beyond which light and matter cannot escape to distant observers.
Singularity
A predicted central point or region where classical general relativity gives infinite density and stops giving usable physical answers.
Accretion disk
A hot, rotating disk of gas and dust that can form around a compact object as matter spirals inward.
Gravitational time dilation
The slowing of time near a massive object compared with time far away, described near a Schwarzschild black hole by Δtfar=Δtnear1−rsr\Delta t_{\text{far}} = \frac{\Delta t_{\text{near}}}{\sqrt{1 - \frac{r_s}{r}}}.

Common Mistakes to Avoid

  • Confusing the event horizon with the singularity, because the event horizon is a boundary at r=rsr = r_s while the singularity is predicted at the center.
  • Using diameter instead of radius in rs=2GMc2r_s = \frac{2GM}{c^2}, because the formula gives the radius from the center to the event horizon, not the full width.
  • Forgetting to use SI units, because G=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2, MM must be in kg\text{kg}, and cc must be in m/s\text{m/s}.
  • Thinking a black hole pulls harder than any object of the same mass at the same distance, because outside the event horizon the gravitational field depends mainly on mass and distance.
  • Substituting r=rsr = r_s into the time dilation formula as if it gives a normal finite result, because 1−rsr\sqrt{1 - \frac{r_s}{r}} becomes 00 at the event horizon for a distant observer.

Practice Questions

  1. 1 Calculate the Schwarzschild radius of a black hole with mass M=10M⊙M = 10M_{\odot} using M⊙=1.99×1030 kgM_{\odot} = 1.99 \times 10^{30}\ \text{kg}, G=6.67×10−11 N⋅m2/kg2G = 6.67 \times 10^{-11}\ \text{N}\cdot\text{m}^2/\text{kg}^2, and c=3.00×108 m/sc = 3.00 \times 10^8\ \text{m/s}.
  2. 2 Find the escape velocity at r=2.0×107 mr = 2.0 \times 10^7\ \text{m} from an object with mass M=5.0×1030 kgM = 5.0 \times 10^{30}\ \text{kg} using ve=2GMrv_e = \sqrt{\frac{2GM}{r}}.
  3. 3 A spaceship hovers at r=4rsr = 4r_s from a Schwarzschild black hole. Use Δtfar=Δtnear1−rsr\Delta t_{\text{far}} = \frac{\Delta t_{\text{near}}}{\sqrt{1 - \frac{r_s}{r}}} to find how much time a distant observer measures if the ship measures Δtnear=1.00 h\Delta t_{\text{near}} = 1.00\ \text{h}.
  4. 4 Explain why the Sun would not suddenly pull harder on Earth if it were replaced by a black hole with the same mass, assuming Earth stayed at the same orbital distance.