--
--
Watch two synchronized clocks tick at different rates!
The closer to the black hole, the slower time flows! At the event horizon, the near clock appears to stop completely from the distant observer's perspective.
Time Dilation: γ = 1/√(1 - Rs/r) - Depends ONLY on r/Rs ratio
Tidal Forces: Δg = 2GMh/r³ - Depends on mass (∝ 1/M²)
At the horizon (1.0 Rs) of different black holes:
The Mind-Blow: Same time dilation, WILDLY different tidal forces! This is why you could survive crossing into Sgr A* but would be vaporized by a stellar black hole!
Time dilation depends ONLY on the ratio r/Rs, not absolute distance!
This is why general relativity is about CURVED SPACETIME, not just gravity! The geometry is what matters, not the absolute scale.
The event horizon is shown as the glowing RED circle (1.0 Rs)
At the event horizon (r = 1.000 Rs): γ → ∞
Time dilation approaches infinity as you approach the horizon!
What this means:
• From YOUR perspective falling in: You cross the horizon in finite proper time (seconds to hours) and don't notice anything special happening (except tidal forces!). No barrier, no wall, just empty space.
• From a DISTANT observer's perspective: You appear to slow down asymptotically and FREEZE at the red circle, taking infinite coordinate time to cross. You get infinitely redshifted and fade to black. They NEVER see you cross!
Try this: Slide down toward 1.0 Rs and watch the clocks diverge!
• 1.5 Rs: γ = 1.73x (cyan photon sphere)
• 1.1 Rs: γ = 3.3x
• 1.05 Rs: γ = 4.6x
• 1.01 Rs: γ = 10x
• 1.001 Rs: γ = 31x - Near clock almost stops!
This is the ultimate observer-dependent reality! Two observers see completely different things happening to the same event. There is no "absolute" answer to "did they cross the horizon?"
Miller's Planet near Gargantua: 1 hour on planet = 7 years in orbit
Time dilation: γ ≈ 61,320x slower!
Using our formula γ = 1/√(1 - Rs/r), that puts Miller's planet at approximately r ≈ 1.00000026 Rs
That's INSANELY close to the event horizon - within 0.000026% of it!
But here's the twist: Since Gargantua is a supermassive black hole (~100 million M☉), that tiny Rs fraction is still thousands of kilometers in absolute distance - enough to survive the gentle tidal forces!
The "little maneuver" cost them 23 years because they were hovering so close to the event horizon where time nearly stops.
Science advisor: Kip Thorne (Nobel Prize 2017 for gravitational wave detection)
Schwarzschild Radius: Rs = 2GM/c²
The radius at which escape velocity equals the speed of light!
Average Density: ρ = 3c⁶/(32πG³M²)
Surprisingly, LARGER black holes have LOWER density! A supermassive black hole (10⁹ M☉) has the density of water.
Surface Gravity: g = GM/Rs² = c⁴/(4GM)
Inversely proportional to mass - smaller black holes have higher surface gravity!
Hawking Temperature: TH = ℏc³/(8πGM kB)
Black holes emit thermal radiation and slowly evaporate. Smaller = hotter!
Tidal Forces (Spaghettification): Δg = 2GMh/r³
Differential gravity across your body. At the event horizon of a stellar mass black hole, you'd be torn apart!
Photon Sphere: 1.5 Rs - Light can orbit in circular paths
Innermost Stable Circular Orbit (ISCO): 3.0 Rs - Last stable orbit for matter
Event Horizon: 1.0 Rs - The point of no return. Time stops here from outside perspective!
Singularity: r = 0 - Where spacetime curvature becomes infinite and physics breaks down
Not a vacuum cleaner: Black holes have the same gravitational pull as any object of equal mass. If the Sun became a black hole, Earth's orbit wouldn't change!
You can never see someone cross the event horizon: Time dilation means they appear to freeze and redshift forever.
Inside is weird: Past the horizon, "forward in time" means "toward the singularity" - you can't avoid it any more than you can avoid tomorrow!
Supermassive black holes are gentle: The tidal forces at the horizon of Sagittarius A* (4 million M☉) are survivable!