The big idea: gravity is not a pull but curved space-time
Newton said masses pull each other. Einstein, in 1915, gave a different picture. Space-time means the three directions of space joined with time. Mass and energy bend space-time. Objects that are free to move simply follow the straightest possible path on this bent surface. We see that path as falling or orbiting.
The rubber-sheet picture is only an aid. Real space-time has four dimensions, and time bends too.
The equivalence principle
Imagine you are in a closed lift. If the lift is far in space and speeds up upward, you feel pushed to the floor. That feeling is the same as standing on Earth. If you drop a ball, it falls in both cases. Einstein's equivalence principle: the effects of gravity and of acceleration cannot be told apart in a small closed room.
In free fall (a falling lift) you feel weightless, like astronauts in orbit. They are falling around the Earth all the time. This is why all objects fall with the same acceleration g, whatever their mass.
Light bends and clocks slow down
If the lift picture is true, a light beam crossing a lift that is accelerating would look bent. So gravity must bend light. In 1919, during a solar eclipse, astronomers saw that stars close to the Sun appeared slightly shifted, as predicted.
Gravitational time dilation: clocks run slower where gravity is stronger. A clock on a tall mountain runs very slightly faster than a clock at sea level.
Gravitational lensing
A very massive object, like a galaxy or a cluster of galaxies, bends light from things far behind it. The same source may appear as two or more images, as an arc, or as a full circle called an Einstein ring when everything lines up exactly. Astronomers use lensing to spot invisible dark matter and very faint, distant galaxies.
Black holes
If a lot of mass is squeezed into a very small space, space-time becomes so steep that nothing can escape from inside a certain distance, not even light. That distance is the event horizon. For a non-spinning black hole its radius is the Schwarzschild radius, r = 2GM / c². For the Sun's mass it is about 3 km. A black hole can form when a very heavy star ends its life. We do not see it directly, but we see gas heating up around it and stars orbiting it. In 2019 the first image of a black hole's shadow was released.
Try it
In the 3D, go to the free-play step. Set mass to zero and watch the ball go straight. Slowly raise the mass: predict whether the orbit gets faster or slower, then check. At home, press a heavy ball into a stretched cloth and roll a marble past it. Watch the marble curve.
Key formulas and definitions
- Schwarzschild radius: r = 2GM / c²
- Sun: r ≈ 3 km. Earth: r ≈ 9 mm
- G = 6.67 × 10⁻¹¹ N m² kg⁻², c = 3 × 10⁸ m/s
- Equivalence principle: gravity and acceleration look the same in a small closed room
Worked examples
1. Find the Schwarzschild radius of the Earth (M = 6 × 10²⁴ kg).
r = 2GM/c² = 2 × 6.67×10⁻¹¹ × 6×10²⁴ / (9×10¹⁶) = 8.0×10¹⁴ / 9×10¹⁶ ≈ 8.9×10⁻³ m, about 9 mm. Earth would need to be squeezed to the size of a marble.
2. A black hole has a mass of 10 Suns. Its radius for one Sun is about 3 km. What is its horizon radius?
Radius is proportional to mass. So r = 10 × 3 km = 30 km.
3. An astronaut in the International Space Station feels weightless. Does gravity vanish there?
No. Gravity at that height is about 90% of its value on the ground. The astronaut and station are both in free fall around Earth, so there is no floor pushing back and the astronaut feels weightless.
Common mistakes
- Thinking a black hole sucks everything in like a vacuum cleaner. From far away its pull is the same as any star of the same mass.
- Believing the rubber-sheet picture is exactly what space looks like. It is only a 2D helper for a 4D idea.
- Saying astronauts float because there is no gravity. They float because they are in free fall.
- Mixing up the two theories: special relativity has no gravity; general relativity includes gravity.