The Notion of Time in Special Relativity

1. The Notion of Time in Special Relativity

2. t t x x The coordinate transformations from (t,x) to (t, x) are called Lorentz Transformations!

3. Galilean Transformations • Newton’s laws are conserved (invariant) under Galilean transformations • Coordinate • Velocity addition

4. t x v = c = 1 The worldline of an object is the path it follows in spacetime. Its slope can never be because v

5. t x World line of motionless object at x=2 in rest frame World line of motionless object in rest frame at x=0 Worldlines of objects that are motionless in the rest frame are parallel to the t axis (and perpendicular to the x-axis) Space axis (rest or “home” frame)

6. t x Worldline of object with v = 0.5 in rest frame Worldline of object with v = 0.1 in rest frame

7. t t x x The coordinate transformations from (t,x) to (t, x) are called Lorentz Transformations!

8. Coordinate Time • Coordinate time is the interval measured on clocks in different reference frames • Clocks are synchronized so that they agree on a particular starting point • Clocks are compared to each other in one particular reference frame Moving clock Clock at rest Rest (home) frame Moving (inertial) frame

9. Distance travelled by light beam = time measure in home frame on home clock Mirror Mirror D Light source Light source Light source Frame of reference in which light source is moving (“home” frame)

10. = time measure in home frame on moving clock Total distance travelled by light beam Mirror D Light source Frame of reference in which light source is at rest (“inertial / co-moving frame”)

11. How do and compare? The passage of coordinate time on a moving clock is slower, as measured by a clock at rest! Moving clocks appear to run slow!

12. A few things to note…. 1. When v=0, The longest time interval is measure in the rest frame! 2. When v=c, A clock moving at the speed of light doesn’t tick!

13. Proper Time • Proper time is the interval measured on a single clock that is present at an initial and final event in spacetime • This has geometric significance in Special Relativity

14. Proper Time • The proper time is the path length of a worldline • It has the same value in every reference frame! Same value = invariant!

15. Spacetime Interval • The spacetime interval is the time between events as measured on an inertial (co-moving) clock, and represents the shortest distance between the events in spacetime. • It has the same value in every reference frame! Same value = invariant!

16. t t x x The coordinate transformations from (t,x) to (t, x) are called Lorentz Transformations! The proper time and spacetime interval are invariant under Lorentz Transformations

17. A spaceship travels from Earth to Alpha Centauri (a distance of 4.4 lightyears). From the Earth, the ship is observed to reach its destination 6 years after launch. How fast is the ship moving, according to observers on the Earth? In the ship’s frame, how far has it moved during the journey? How long did the journey take, according to the ship’s clock? The ship is motionless in its own (inertial) frame, so it hasn’t moved at all! Inertial frame!

18. t t x x 4.08 y 6.0 y 4.4 y

19. Two events are observed to occur with separations in a rest (home) frame. In another (moving) frame, they are observed to occur with a spatial separation of . What is the time interval between the events in the other frame? Invariant! The events occur at the same time in the moving frame! They are now simultaneous!

20. t t x x 300 ns 400 ns 500 ns

21. The Metric • In Euclidean space, the length of a vector is • This is also the (shortest) distance between two points: the Euclidean metric y x

22. The Metric • In Minkowski space (Special Relativity), the length of a four-vector is • This is also the (shortest) distance between two points in spacetime! The Minkowski metric t x

23. ] • The metric tells us something about the fundamental geometry and invariance of the space • Euclidean: • Circle! • Minkowski: • Hyperbola!

24. t The length of a vector is invariant under rotations But their coordinates do not agree! Δr = 4 Δr = 4 Δr = 4 x

25. Think of the red line as the “rest” frame (inertial clock with v=0). y The vector pointing purely in the y-direction is like the worldline of a motionless observer, as measured in their own rest frame! Δr = 4 x

26. A frame of reference “moving” (as measured in the original frame) corresponds to a rotation of the vector. y These coordinates are measured by the red observer in the red (motionless) reference frame! Δr = 4 Δr = 4 x

27. But if we transform into the blue (moving) reference frame, the vectors look like this: y These coordinates are measured by the blueobserver in the blue (now motionless) reference frame! They are the same as the red coordinates from before (as measured in the red rest frame). Δr = 4 Δr = 4 x Now, as measured in the blue rest frame, the red frame looks like it is moving (at the same velocity in the opposite direction).

28. t The proper time in any frame of reference is: Vectors of the same length in hyperbolic spacetime are not the same length “on paper.” Light cone Δτ = 4 Moving at constant v as measured inred (“Earth”) coordinate system. At rest with respect to red coordinate system (“Earth”). Δτ = 4 x

29. t The proper time in any frame of reference is: Light cone Moving with –v as measured in blue (“ship”) coordinate system. Δτ = 4 At rest in blue (ship) coordinate system. Δτ = 4 x

30. y t Spacetime vectors (4-vecs) lie on a hyperbola and have the same “length”(proper time) regardless of velocity (but but different coordinates). The frame of reference where Δx = 0 is the rest frame. We can transform to the rest frame of another vector by Lorentz transforming to the corresponding velocity. x Spatial vectors lie on a circle, and have the same length regardless of rotation angle (but different coordinates). The “frame of reference” where Δx = 0 is the “rest” frame. We can always make another vector vertical (“at rest”) by rotating the coordinate system by the corresponding angle. x