15. General Relativity & Spacetime
CollegeEinstein's greatest insight: gravity isn't a force pulling objects together — it's the curvature of spacetime itself, caused by mass and energy.
The equivalence principle and curved spacetime
Einstein's equivalence principle states that being in a gravitational field is locally indistinguishable from being in an accelerating reference frame — there is no local experiment that can tell them apart. Extending this insight, general relativity (1915) describes gravity not as a force but as the curvature of four-dimensional spacetime caused by mass and energy; objects simply follow the straightest possible paths, called geodesics, through that curved geometry.
Around a spherical, non-rotating mass, the geometry is described by the Schwarzschild solution, which defines the Schwarzschild radius — the size to which a mass must be compressed for even light to be unable to escape (a black hole):
Real, measurable predictions
General relativity is not merely philosophical — it makes precise, testable predictions that Newtonian gravity gets wrong. It correctly predicts the extra 43 arcseconds per century of perihelion precession in Mercury's orbit (a discrepancy Newtonian mechanics couldn't explain), the bending of starlight around the Sun (confirmed during the 1919 solar eclipse), gravitational time dilation (clocks run slower in stronger gravity), and gravitational waves — ripples in spacetime from accelerating masses, directly detected by LIGO in 2015.
Worked example — the Schwarzschild radius of the Sun
Find the Schwarzschild radius of the Sun (M = 1.989×10³⁰ kg), and compare it to the Sun's actual radius (6.96×10⁸ m).
- Apply rs = 2GM/c² with G = 6.674×10⁻¹¹ and c² = 9.00×10¹⁶.
- Numerator: 2 × 6.674×10⁻¹¹ × 1.989×10³⁰ = 2.655×10²⁰.
- Divide: rs = 2.655×10²⁰ / 9.00×10¹⁶ ≈ 2 950 m ≈ 2.95 km.
- Compare to the Sun's real radius of about 696,000 km — the Sun would need to be crushed down by a factor of roughly 235,000 to become a black hole.
Answer: about 2.95 km — since the Sun's actual radius is vastly larger than its Schwarzschild radius, the Sun is nowhere near dense enough to be a black hole, and never will be (it lacks the mass to end its life that way).
Test your understanding
According to general relativity, why do planets orbit the Sun?
Not because the Sun "pulls" them with a force in the Newtonian sense, but because the Sun's mass curves the spacetime around it, and planets simply follow the straightest available path (a geodesic) through that curved geometry — which happens to be a near-elliptical orbit.
What historical problem with Mercury's orbit did general relativity solve?
Mercury's elliptical orbit slowly rotates (precesses) over time faster than Newtonian gravity, accounting for perturbations from other planets, could explain — by about 43 arcseconds per century. General relativity's correction to Newtonian gravity for strong, close-in fields predicted exactly this leftover precession.
Why do GPS satellite clocks need both special and general relativistic corrections?
Special relativity says the satellites' motion relative to Earth's surface makes their clocks run slightly slower. General relativity says their weaker gravity (being farther from Earth's mass) makes their clocks run faster. The general relativistic effect wins out, for a net gain of about 38 microseconds per day, which must be corrected for or GPS positions would drift by kilometres daily.
What is a gravitational wave, and how was one first directly detected?
A gravitational wave is a ripple in the curvature of spacetime itself, radiated outward by accelerating massive objects (most dramatically, merging black holes or neutron stars). LIGO first directly detected one in September 2015, from two black holes (about 29 and 36 solar masses) merging roughly 1.3 billion light-years away, confirming a century-old prediction of general relativity.
How the ideas connect
Every key idea in this chapter, branching from the core concept — use it to see the whole picture at a glance.
The key facts, visualised
Worked problems, step by step
Follow each solution line by line, then try to reproduce it on paper before moving on.
Example 1Find the Schwarzschild radius of the Sun (M = 2e30 kg). Use r_s = 2GM/c^2, G = 6.67e-11, c = 3e8.
- Numerator = 2 x 6.67e-11 x 2e30 = 2.67e20
- Denominator = c^2 = 9e16
- r_s = 2.67e20 / 9e16 = ~2,960 m
Example 2Light from a distant star passes near the Sun and appears shifted. What does general relativity predict?
- Mass curves spacetime around it
- Light follows the curved geometry
- So the star's apparent position shifts (gravitational lensing)
Now you try
Work each one out first, then tap to reveal the worked answer.