Curved Spacetime Simulator – Orbits, Light Bending & Gravitational Time Dilation

Gravity reimagined: exploring curved spacetime

Newton described gravity as a force pulling masses together. Einstein described it as geometry: mass and energy curve spacetime, and objects in free fall simply follow the straightest possible paths through that curved spacetime. Explore both pictures, and see where they agree and where they differ.

Under 15Imagine a trampoline with a bowling ball in the middle: it makes a dip. Roll a marble nearby and it curves round the ball. Einstein said gravity works a bit like that: heavy things like the Sun "bend" space and time, and planets curve round them. Even light bends near heavy objects! (The trampoline is just a helpful picture: real space doesn't sit on anything.) Start with the 3D grid, then try launching planets in the orbit lab.

20+ · going deeperThe field equations G_μν = (8πG/c⁴) T_μν relate curvature to energy–momentum. Outside a static spherical mass the solution is Schwarzschild: ds² = −(1 − r_s/r)c²dt² + (1 − r_s/r)⁻¹dr² + r²dΩ². Slow particles fall mainly because of the g_tt (time) part; light bending gets equal contributions from g_tt and the spatial curvature, giving twice the Newtonian deflection. The orbit-lab correction term 3GML²/(c²r⁴) (per unit mass) comes from the effective potential of the Schwarzschild geodesic equation.

1 · Discover 2 · Spacetime grid 3 · Newton vs Einstein 4 · Orbits 5 · Light bending 6 · Time dilation 7 · Challenges Glossary

1Discover: a short guided tour

Step 1. Play with the 3D grid to get a feel for "mass curves space". Step 2. Compare it with Newton's force law, which works brilliantly in everyday gravity. Step 3. Launch planets and find circular, elliptical and escape orbits. Step 4. See that light, which has no mass, is also bent by gravity: a key prediction of Einstein, confirmed in 1919. Step 5. Find out why clocks on GPS satellites tick faster than clocks on the ground.

The rubber-sheet grid is an analogy, not a picture of real spacetime. Real curvature involves time as well as space, and objects don't "sink" into anything. In weak gravity and at low speeds, Newton and Einstein give almost the same answers; the differences show up near very dense objects or for light.

2The spacetime grid (illustrative analogy)

Central mass: Small orbiting object Slowly rotate the view

Drag to look around. With zero mass the grid is flat and the small object would travel in a straight line. Add mass and the grid dips; the object's path curves round. The dip shape follows the Newtonian potential (∝ −M/r), drawn as an embedding-style picture for intuition.

3Newtonian gravity vs general relativity

Mass 1:

Mass 2:

Separation r:

Newton's law of gravitation: F = G m₁ m₂ / r², with G = 6.674 × 10⁻¹¹ N m²/kg². Both masses feel the same size force, in opposite directions. Here: . (Tiny for everyday masses! Gravity only becomes noticeable when one mass is planet-sized.)

4Orbital motion

Central mass: Start distance: Start speed: Start direction (from radial):

Add exaggerated GR precession

Orbit type

Circular / escape speed here

Units: distances in AU (Earth–Sun distance), mass in Suns, speeds in km/s. Earth starts at 1 AU moving at 29.8 km/s, giving a near-circular orbit. The basic mode uses an accurate Newtonian integrator (velocity Verlet). An orbiting body keeps "falling" towards the star but moves sideways fast enough to keep missing it.

The GR option adds the leading relativistic correction to the force (∝ 1/r⁴), multiplied about a million times so you can see the orbit's ellipse slowly rotate (perihelion precession). This is an illustration: real precession is tiny, 43″ per century for Mercury.

5Light bending & gravitational lensing

Mass of the object: Impact parameter of the highlighted ray:

Deflection α = 4GM/(c²b)

A ray passing a mass at closest distance b is bent by α = 4GM / (c² b): twice what you would get by treating light as a Newtonian particle. Rays closer to the object bend more. For the Sun, a ray grazing its edge bends by 1.75 arcseconds, measured during the 1919 solar eclipse. The rays here are traced with the weak-field approximation, with the mass hugely exaggerated so the bending is visible.

Gravitational lensing

Drag on the picture to move a background galaxy behind a foreground mass (the cross). Its light reaches us along several bent paths, so we see arcs, double images and, when perfectly lined up, an Einstein ring. This uses the standard thin, point-mass lens equation, β = θ − θ_E²/θ.

Lens mass (Einstein radius θ_E): Show the galaxy's true position

6Gravitational time dilation

A clock held at rest at distance r from a spherical mass M runs at the rate dτ/dt = √(1 − 2GM / (r c²)) compared with a clock very far away (the Schwarzschild result for static clocks). The clock farther from the mass ticks faster. Near Earth the effect is tiny, but GPS has to correct for it: satellite clocks gain about 45 μs per day from weaker gravity (and lose about 7 μs from their orbital speed). Without these corrections GPS positions would drift by kilometres every day.

The animated clocks are sped up and the rate difference exaggerated when it's too small to see; the numbers underneath are the real values.

7Interactive challenges

Glossary

Spacetime: The combination of 3 dimensions of space and 1 of time into a single 4-dimensional geometry.
Geodesic: The straightest possible path through curved spacetime. Freely falling objects and light follow geodesics.
Free fall: Motion under gravity alone, with no other forces. You feel weightless in free fall.
Escape speed: The minimum launch speed to coast away forever: v = √(2GM/r).
Gravitational lensing: The bending of light from a distant source by a mass in between, producing distorted, multiple or ring-shaped images.
Einstein ring: The ring image formed when source, lens and observer line up exactly.
Schwarzschild radius: r_s = 2GM/c². If a mass is squeezed inside it, it becomes a black hole.
Perihelion precession: The slow rotation of an orbit's ellipse. GR explains Mercury's extra 43″ per century.