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PhysicsGeneral relativityInteractive 3D · Interactive 2D 20 min

General relativity, hands-on

Place masses on spacetime, launch particles, spin an orbit into a rosette, bend light around a black hole, compare clocks from the ground to GPS and watch a gravitational wave go by.

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What you can change

Variables

  • Curvature: one mass (exact Schwarzschild embedding, compactness from a star to a black hole) or several masses to place and move; what the sheet shows (space or the pace of time); particles to launch by hand
  • Orbits: the shape of the orbit (periapsis, apoapsis) or its energy E and angular momentum L, also adjustable on the effective-potential graph; comparison with Newton, same start
  • Light: the impact parameter of a ray, a beam of parallel rays, comparison with “Newton’s light”; ray-traced view of a black hole, with accretion disc, tilt and frequency shifts
  • Time: clocks on the ground, at the top of a tower, on a plane, on the ISS, on a GPS satellite and in geostationary orbit; a probe lowered towards the horizon of a black hole
  • Gravitational waves: the masses of the black-hole pair, the polarisation (+, ×), the slow motion
  • Famous presets: Mercury, the Sun in 1919, Sgr A*, M87*, a neutron star, GW150914
Orders of magnitude

Key figures

42.98″
per century: the advance of Mercury’s perihelion due to general relativity
1.75″
deflection of light at the edge of the Sun, twice the “Newtonian” value
3 r_s
the innermost stable circular orbit (ISCO) around a black hole; 1.5 r_s for light
+38.6 µs
per day gained by a GPS clock: +45.7 µs (gravity) − 7.1 µs (speed)
11.6 km
the range error that would build up every day if GPS ignored relativity
10⁻²¹
the stretching of space measured by LIGO as GW150914 passed
Understand

What happens, step by step.

  1. 01

    Gravity is not a force, it is geometry

    For Einstein (1915), a mass curves spacetime, and a freely falling body follows the straightest possible line in that curved spacetime: a geodesic. Around a spherical mass, the exact solution is Karl Schwarzschild’s (1916). It brings out a length: the Schwarzschild radius r_s = 2GM/c², 2.95 km for the Sun, 8.9 mm for the Earth. As long as you stay far from r_s, you get Newton back; close to r_s, everything changes.

  2. 02

    What the sheet shows, and what it does not

    The funnel-shaped sheet is Flamm’s embedding: the geometry of space alone (at a fixed time), in the equatorial plane, drawn as a surface. The circles do have a circumference of 2πr, but the distance between two neighbouring circles is larger than the difference of their radii: space is stretched towards the centre. It is not a marble rolling on a sheet: for a slow object, it is mostly the curvature of time (clocks running slower lower down) that makes it fall. The sheet can also show that pace of time. With several masses there is no simple exact solution: the sheet then shows the gravitational potential, and the particles follow Newton, the weak-field limit of general relativity.

  3. 03

    Orbits that do not close

    For Newton, a lone planet around the Sun traces a closed ellipse. In general relativity, the closest point of the orbit (the perihelion) advances each orbit by 6πGM/(c²a(1 − e²)): 0.1035″ per orbit for Mercury, 42.98″ per century, the anomaly Le Verrier had brought to light in 1859 and that Einstein explained in November 1915. Near a black hole, the advance becomes huge, orbits trace rosettes, and there is no stable circular orbit below 3 r_s (the ISCO): the effective potential, which for Newton has an infinite centrifugal barrier, now has only a finite barrier, which the particle can cross.

  4. 04

    Light follows spacetime too

    A ray grazing the Sun is deflected by 4GM/(c²R) = 1.75″, twice as much as if light were a particle subject to Newton’s gravity. Measured during the eclipse of 29 May 1919 (Dyson, Eddington, Davidson), this deflection made Einstein famous. Closer to a compact mass, rays wind around: at 1.5 r_s light can go round in circles (the photon sphere); below an impact parameter of 2.6 r_s it is captured. Seen from afar, the black hole casts a shadow 5.2 r_s across, circled by a ring of light: this is what the Event Horizon Telescope photographed for M87* (2019) and Sgr A* (2022). A galaxy exactly behind another appears as a ring (an Einstein ring). Light is also delayed: this is the Shapiro effect, measured with radar echoes from Venus and Mercury.

  5. 05

    Time runs slower lower down

    A clock held at distance r from a mass ticks √(1 − r_s/r) times as fast as a far-away clock. On Earth the difference is tiny but measurable: 2.5 × 10⁻¹⁵ over the 22.5 m of a Harvard tower (Pound and Rebka, 1960), and optical clocks see it over 33 cm (2010). For GPS it is huge: at 20,200 km altitude, a satellite’s clock gains 45.7 µs a day because gravity is weaker and loses 7.1 µs because of its speed, a net +38.6 µs a day. Uncorrected, this would throw distances off by more than 11 km a day. So the clocks are set before launch to run 4.465 × 10⁻¹⁰ slow. Near a black hole the slowdown becomes spectacular, and light coming out of it arrives shifted towards the red.

  6. 06

    Ripples in spacetime

    Two black holes orbiting each other emit gravitational waves, lose energy, draw closer and merge. As the wave passes, distances shrink in one direction and stretch in the other, in turn (+ and × polarisations). On 14 September 2015, the two LIGO detectors recorded GW150914: two black holes of 36 and 29 solar masses, 410 Mpc away, whose signal rises from 35 to 250 Hz in 0.2 s (a “chirp”); 3 solar masses were converted into waves. The 4 km arm of a detector changed by about 4 × 10⁻¹⁸ m, a few thousandths of the diameter of a proton.

Check

What the scene simplifies

  • The black hole does not spin (Schwarzschild metric, not Kerr) and all orbits lie in its equatorial plane. Real black holes spin; M87* and Sgr A* do too.
  • Curvature: the single-mass sheet is the exact embedding of the spatial part of the metric (Flamm’s paraboloid, and the interior of a uniform-density star); the sheet’s third dimension has no physical meaning. A static star cannot be more compact than 9/8 r_s (Buchdahl limit): below that, the setting is only an illustration. With several masses, the sheet shows the Newtonian potential and the particles follow Newton: there is no simple exact many-body solution, and the masses stay fixed.
  • Test particles: their mass does not deform spacetime. On the single-mass sheet they move according to the time of a far-away observer: near the horizon they slow down and fade; the calculation removes them at 1.01 r_s.
  • Orbits: Newton and general relativity start from the same point with the same velocity and angular momentum; Newton advances with its own time, general relativity with the particle’s proper time. The “Mercury” case keeps the real eccentricity but not the distance (the real advance would be invisible).
  • Light: the beam’s rays come in parallel from infinity and are traced in a plane; “Newton’s light” is a corpuscle launched at speed c, to show the factor of 2. The ray-traced view computes light geodesics in the same metric; the accretion disc is thin and its emission schematic.
  • Time: clocks on circular orbits, Earth with its equatorial radius and only the J2 term in the geoid potential; the plane flies at 10 km above the equator at 250 m/s. The GPS budget follows Ashby (2003): the orbits’ eccentricity (± 23 ns), the Sagnac effect and tides are not shown.
  • Gravitational waves: the waveform is schematic (inspiral at leading post-Newtonian order, then the ringing of the final black hole, joined by hand), not the measured signal; the cosmological redshift (z ≈ 0.09) is ignored; the ring’s deformation is magnified by a factor of order 10²⁰.
Sources

Further reading

  1. Einstein, Die Feldgleichungen der Gravitation, Sitzungsberichte der Preußischen Akademie der Wissenschaften, 844-847 (1915)

    The field equations, 25 November 1915. See also Erklärung der Perihelbewegung des Merkur, 831-839 (1915).

  2. Einstein, Die Grundlage der allgemeinen Relativitätstheorie, Annalen der Physik 354 (7), 769-822 (1916)

    The complete exposition of the theory.

  3. Schwarzschild, Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie, Sitzungsberichte, 189-196 (1916)

    The exact solution around a spherical mass; English translation arXiv:physics/9905030.

  4. Dyson, Eddington & Davidson, Philosophical Transactions of the Royal Society A 220, 291-333 (1920)

    The deflection of light measured during the eclipse of 29 May 1919 (Sobral and Príncipe).

  5. Pound & Rebka, Apparent Weight of Photons, Physical Review Letters 4, 337 (1960)

    The gravitational frequency shift over 22.5 m: (2.57 ± 0.26) × 10⁻¹⁵ measured for 2.46 × 10⁻¹⁵ predicted.

  6. Shapiro, Fourth Test of General Relativity, Physical Review Letters 13, 789 (1964)

    The delay of light passing near the Sun (radar echo).

  7. Hafele & Keating, Around-the-World Atomic Clocks, Science 177, 166-170 (1972)

    Atomic clocks flown eastwards and westwards on airliners.

  8. Chou, Hume, Rosenband & Wineland, Optical Clocks and Relativity, Science 329, 1630 (2010)

    The slowing of time measured over a 33 cm height difference.

  9. Ashby, Relativity in the Global Positioning System, Living Reviews in Relativity 6, 1 (2003)

    The budget of GPS clocks: 4.4647 × 10⁻¹⁰, i.e. +38.6 µs per day.

  10. Everitt et al. (Gravity Probe B), Physical Review Letters 106, 221101 (2011)

    Geodetic precession (−6,601.8 ± 18.3 mas/yr) and frame dragging (−37.2 ± 7.2 mas/yr).

  11. Abbott et al. (LIGO & Virgo), Observation of Gravitational Waves from a Binary Black Hole Merger, PRL 116, 061102 (2016)

    GW150914: 36 + 29 solar masses, 410 Mpc, from 35 to 250 Hz, peak strain 1.0 × 10⁻²¹.

  12. Event Horizon Telescope Collaboration, First M87 Event Horizon Telescope Results. I, ApJ Letters 875, L1 (2019)

    The ring of M87*: 42 ± 3 µas. For Sgr A*: ApJ Letters 930, L12 (2022), 51.8 ± 2.3 µas.

  13. Will, The Confrontation between General Relativity and Experiment, Living Reviews in Relativity 17, 4 (2014)

    Review of the tests: Mercury’s perihelion (42.98″ per century), deflection and delay of light, redshift.

  14. Hartle, Gravity: An Introduction to Einstein’s General Relativity, Addison-Wesley (2003)

    Chapters 9 and 10: Schwarzschild geodesics, effective potential, precession, deflection of light; chapter 6: GPS.

  15. Carroll, Spacetime and Geometry, Addison-Wesley (2004)

    Chapter 5: the Schwarzschild solution, orbits, ISCO, Flamm’s embedding.

  16. Misner, Thorne & Wheeler, Gravitation, W. H. Freeman (1973)

    Chapters 23 and 25: uniform-density stars, embedding, orbits in the Schwarzschild field.

  17. Taylor, Wheeler & Bertschinger, Exploring Black Holes, 2e éd. (2018)

    Clocks near a black hole, orbits and light, in Schwarzschild coordinates and in a local frame.