🛰 Relativity in GPS: ~ How do satellites measure the location of smartphone? ~

🛰 Relativity in GPS:
~ How do satellites measure the location of smartphone?~

This slide deck explains how satellite navigation systems like GPS determine your location, starting from basic geometric concepts and leading up to the effects of special and general relativity.

If you've ever wondered, "Does relativity actually affect my daily life?" — this talk is for you. It's a journey that connects abstract physics with the familiar experience of using a map app on your phone.

Originally created for a VRChat physics meetup, this talk is accessible and engaging for anyone with even a slight curiosity about space, science, or technology.

Transcript

1. 🛰️RELATIVITY IN GPS HOW DO SATELLITES MEASURE THE LOCATION OF SMARTPHONES?
2. TODAY'S TOPIC 🛰️How does GNSS like GPS measure position? Case without considering relativistic effects (Geometric Optics) Effects of Special Relativity Effects of General Relativity 📱You'll understand what's happening when your smartphone shows your location on 🗾map apps!
3. ASSUMPTIONS Earth is a sphere Earth does not rotate Earth is electrically neutral Earth's mass distribution is uniform Earth has no atmosphere (it's a vacuum) No other celestial bodies (like the Moon) exist These assumptions are far from reality, but they simplify the problem! Surprisingly, this approximation works well
4. TECHNICAL TERMS GNSS (Global Navigation Satellite System) is the general term for satellite-based positioning systems 🇺🇸GPS (Global Positioning System) is one type of GNSS Others include 🇷🇺GLONASS, 🇪🇺Galileo, 🇨🇳 BeiDou, 🇯🇵Michibiki Today, we'll refer to it as GPS, which is most familiar to Japanese people It's like calling all game consoles "Famicom" - technically not correct, but commonly used
5. POSITION MEASUREMENT USING GEOMETRIC OPTICS
6. SIMPLE EXAMPLE OF DISTANCE MEASUREMENT Starting from point at time with velocity towards point Arriving at point at time What is the distance between and ? Distance between two points can be calculated as speed time! A t = 0 (s) v = 10 (m/s) B B t = 10 (s) A B ∣∣AB∣∣ = 10 (m/s) × 10 (s) = 100m ×
7. DISTANCE MEASUREMENT USING LIGHT Speed of light: Satellite transmits electromagnetic waves from point at time Smartphone receives the waves at point at time c = 3.0 × 10 (m/s) 8 P t ​ 0 Q t ​ + 0 Δt ∣∣PQ∣∣ = cΔt
8. POSITIONING WITH ONE SATELLITE Satellite transmits radio waves from point at time , smartphone receives it after Distance between points and is P₁ Q R₁ = cΔt Time: t₀ Time: t₀+Δt P ​ 1 t ​ 0 Δt P ​ 1 Q R ​ = 1 cΔt
9. WHERE IS THE SMARTPHONE? It's somewhere on the surface of sphere with center and radius ! P₁ R₁ = cΔt Time: t₀ Time: t₀+Δt Sphere S₁ (Radius R₁) The smartphone is somewhere on the surface of Sphere S₁! S ​ 1 P ​ 1 R ​ 1
10. POSITIONING WITH TWO SATELLITES Similarly measuring distance from two satellites Smartphone is somewhere on the surface of sphere And also on the surface of sphere P₁ P₂ Sphere S₁ (Radius R₁) Sphere S₂ (Radius R₂) Sphere S₁ (from Satellite 1) Sphere S₂ (from Satellite 2) S ​ 1 S ​ 2
11. NARROWING DOWN THE SMARTPHONE'S POSITION The intersection of two spheres is a circle It's somewhere on the circumference of circle ! By increasing the number of satellites, we can narrow down the position! P₁ P₂ Sphere S₁ (Radius R₁) Sphere S₂ (Radius R₂) Cross-section Circle C When two spheres S₁ and S₂ intersect, their intersection is a circle C. C
12. POSITIONING WITH THREE SATELLITES Getting sphere from the third satellite We know it's somewhere on the circumference of circle , which intersects at two points Now we're down to two possible points! P₃ S₃ Circle C Q₁ Q₂ The smartphone is either at point Q₁ or Q₂! S ​ 3 C
13. POSITIONING WITH FOUR SATELLITES Getting sphere from the fourth satellite With distance measurements from four satellites, we can determine the smartphone's position ! P₄ S₄ Q₁ Q₂ Only one of the two candidates passes! In this example, Q₁ is the correct answer! S ​ 4 Q
14. SUMMARY Receiving signals from one GPS satellite, we know the smartphone is on the surface of a sphere Receiving signals from four satellites, we can determine the position We can determine the smartphone's position from the intersection of spheres!
15. ADDITIONAL NOTES * In practice, we treat the Earth's surface as another sphere, and perform positioning with three satellites * For aircraft, positioning from the fourth satellite is important!
16. EFFECTS OF SPECIAL RELATIVITY
17. IMPORTANT CLAIM OF SPECIAL RELATIVITY Objects moving at high speeds appear to time slower to observers at rest This is a highly simplified explanation!
18. EXAMPLE OF SPECIAL RELATIVITY Consider a rocket moving at speed relative to Earth From Earth's perspective, the rocket clock appears to run slower! 12:00 13:00 12:00 12:59 Speed v 1 hour has passed From the perspective of Earth, the rocket's clock is 🚀The rocket's clock is 🌍The Earth's clock is slower! v
19. LORENTZ FACTOR When the rocket clock has advanced seconds, the Earth clock has advanced seconds This is called the Lorentz factor t′ t γ t = ′ ​ = ​ 1 − v /c 2 2 t γt
20. SIMPLE EXAMPLE OF TIME DILATION Let's consider a rocket with speed The Earth clock advances by 1 second while the rocket clock advances by 0.99 seconds From Earth's perspective, the rocket clock is 0.01 seconds ahead v = 4.2 × 10 (m/s) 7 Δt = 1 − ​ = γ t′ 1 − ​ ≃ 1 − ​ (3.0 × 10 ) 8 2 (4.2 × 10 ) 7 2 0.01(s)
21. EFFECTS OF SPECIAL RELATIVITY ON TIME Speed of GPS satellite Height of GPS satellite is about 20,000 km Speed of GPS satellite is about 3.9 km/s The Earth clock advances by 1 second while the GPS satellite clock is seconds ahead Δt ​ = sec 1 − ​ ≃ γ t′ 1 − 8.5 × 10−11 8.5 × 10−11
22. DAILY TIME DIFFERENCE 1 day is Daily GPS satellite clock delay: so a significant time difference occurs in distance measurement! GPS positioning requires correction for this time difference! 24(h) × 60(min) × 60(s) = 86400(s) Δt ​ = day 86400(s) × 8.5 × 10 ≃ −11 7.3(μs) cΔt ​ ≃ day 2.2(km)
23. EFFECTS OF GENERAL RELATIVITY
24. CLAIM OF GENERAL RELATIVITY Mass creates spacetime distortion This distortion is gravity Time passes slower in gravity Mysid, Wikipedia Commons (CC BY- SA 3.0).
25. HOW GRAVITY AFFECTS TIME Stronger gravity makes time pass slower Weaker gravity makes time pass faster Earth's surface is closer to Earth's center, so gravity is stronger, making the clock slower GPS satellite is at an altitude of 20,000 km, so gravity is weaker, making the clock faster
26. SCHWARZSCHILD METRIC The simplest metric to represent spacetime distortion (gravity) is the distance from the body's center Earth's gravity can be approximated by the Schwarzschild metric r ds = 2 − 1 − ​ c dt + ( c r 2 2GM ) 2 2 1 − ​ dr ( c r 2 2GM ) −1 2 + r (dθ + 2 2 sin θdϕ ) 2 2
27. WHY IS EARTH'S GRAVITY APPROXIMATED BY THE SCHWARZSCHILD METRIC? Assumptions of the Schwarzschild metric Body is spherically symmetric and stationary Body does not rotate Body is electrically neutral Today's Earth satisfies these conditions! The actual Earth has a negligible effect due to this approximation
28. FURTHER APPROXIMATION OF SCHWARZSCHILD METRIC The speed of the smartphone measuring position is very slow compared to the speed of light, so we can consider it almost stationary! We can ignore the spatial components of the Schwarzschild metric! Time dilation is only due to gravity! ds = 2 − 1 − ​ c dt ( c r 2 2GM ) 2 2
29. TIME DILATION DUE TO EARTH'S GRAVITY Here is the distance from Earth's center The Earth's center advances by 1 second while the point at distance advances by the right- hand side Faster in stronger gravity dτ = − ​ = c2 ds2 ​ dt ≃ 1 − ​ c r 2 2GM 1 − ​ dt ( c r 2 GM ) r r
30. TIME DIFFERENCE BETWEEN EARTH AND SATELLITE Let's denote the Earth's radius as Let's denote the altitude of the GPS satellite as Distance from Earth's center of the GPS satellite is Time difference between the Earth clock and the GPS satellite clock: R h R + h Δt = 1 − ​ − ( c (R + h) 2 GM ) 1 − ​ ( c R 2 GM )
31. VALUES USED IN CALCULATION Substitute the following values: G = 6.674 × 10 (m kg s ) −11 3 −1 −2 M = 5.972 × 10 (kg) 24 R = 6.378 × 10 (m) 6 h = 2.0 × 10 (m) 7 c = 3.0 × 10 (m/s) 8
32. CALCULATION RESULT Daily time difference between the GPS satellite clock and the Earth clock: 1 day, the GPS satellite clock is 45.5 ahead of the Earth clock Larger than the time delay due to special relativity! Δt ​ ≃ day 45.5(μs) μs
33. TOTAL TIME DIFFERENCE Time delay due to special relativity is (delay) Time advance due to general relativity is (advance) Adding both: 1 day, the GPS satellite is 38.2 ahead of the Earth clock! −7.3μs +45.5μs Δt ​ = total 45.5 − 7.3 = +38.2μs μs
34. WHAT HAPPENS? 1 day, the clock is 38.2 ahead 35 days, the distance is about 400 km (Tokyo- Osaka distance) If you're heading to Tokyo from Tokyo, you'll end up in Osaka! μs
35. HOW ARE WE CORRECTING IT? 1 day, we intentionally slow down the clock by 38.2 ! Resulting in a balance Strictly, we use clocks with different clock frequencies Theoretical calculation is elegant But the solution is surprisingly simple! GNSS is a combination of elegance and simplicity! μs
36. SUMMARY GPS measures the time it takes for light to reach from satellites to measuring devices 1 satellite: sphere surface, 2 satellites: circle circumference, 3 satellites: 2 points, 4 satellites: 1 point, and the more satellites, the more we can narrow down the position Accurate positioning requires accurate time correction Time delay due to special relativity Time advance due to general relativity Relativity is everywhere in our lives!
37. REFERENCES International Committee on Global Navigation Satellite Systems, Detailed explanation of why positioning becomes more accurate by increasing the number of satellites Many illustrations, easy to understand GNSS, How it Works and Applications
38. REFERENCES Neil Ashby, , DOI: 10.12942/lrr-2003- 1 We mainly referred to this survey paper for the method of correcting GNSS clocks due to relativistic effects More advanced topics such as geoid correction are explained in detail Relativity in the Global Positioning System
39. REFERENCES 内山龍雄， ，岩波書店 (1989) We referred to this textbook for the basic concepts of relativity Compact summary of basic knowledge from special relativity to general relativity Famous opening statement "If you can't understand this, you should probably give up on learning relativity." ...I don't think so 相対性理論
40. LT SPEAKER RECRUITMENT We're recruiting LT speakers at the physics meeting! Any genre is OK! If there's no application, the organizer will hold a Jaiyan Ressairoku (LT event) under the guise of LT... If you're interested, join the Discord server of the physics meeting!
41. ANNOUNCEMENT Next meeting is scheduled for May 3 We're all going to watch the physics video together We're also looking for proposals for "I want to watch this video with everyone!"