The Wonder
Room.

1. Turn gravitational bending off. Find the one background source.
2. Turn it on and center Source alignment at 0. Predict what the telescope will see, then check.
3. Double the lens strength from 1 to 2. The ring radius grows by √2, about 1.41×, in this model.

The image is calculated from the thin point-mass lens equation β = θ − Mθ/|θ|² for a small extended source. Coordinates and mass strength are relative units. The side view enlarges deflection angles; moving dots illustrate direction, not a measured journey time. Real galaxies have extended mass distributions and structure.

ESA: how an Einstein ring forms · NASA/JPL: see a real Einstein ring

1. Select GPS altitude and Circular orbit. Record the gravity, motion, and net clock rates.
2. Set elapsed time to 24 hours. Turn corrections off for Satellite A. Compare its range bias with the calculated position error.
3. Turn corrections on. Then repeat with a common clock offset and explain why the result differs.
4. Try Low orbit, then a clock held at the same altitude. Which effect changed?

Weak-field clock model relative to a stationary surface clock on a non-rotating spherical Earth: gravity/day = GM(1/R − 1/r) × 86400/c²; motion/day = −v² × 86400/(2c²); v = √(GM/r) for circular orbit. Here R = 6,371 km and GM = 3.986004418 × 10¹⁴ m³/s². Units are converted to μs. The held-clock option uses v = 0.

The ranging model uses pᵢ = |sᵢ − x| + b − cδtᵢ, with receiver clock bias b in distance units. The solver estimates two position coordinates and b. Real GPS uses a reference time scale, clock frequency offsets and broadcast corrections, precise ephemerides, and propagation corrections. Earth rotation, orbital eccentricity, atmospheric delays, measurement noise, and changing geometry during flight are omitted here. A corrected zero is a result of this idealized model, not a claim that real GPS has zero error.

Ohio State: relativity and GPS clocks · FAA: how GPS uses time and distance · NIST: GPS receivers and relativity