1. Spacetime Structure

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

3. Failure of Galilean Transformations
Maxwell’s equations are not invariant under Galilean transformations!
Michelson-Morley experiment showed that the speed of light is c=2.99×108 m/s in all inertial reference frames!
As such, Maxwell’s Equations are also not conserved under Galilean transformations
Need new type of coordinate transformation – and therefore geometry -- that makes it so!

4. Spacetime Geometry
Space and time are not separate quantities, but are parts of the same framework (manifold)
Michelson-Morley experiment showed that the speed of light is c=2.99×108 m/s in all inertial reference frames!
As such, Maxwell’s Equations are also not conserved under Galilean transformations
Need new type of coordinate transformation – and therefore geometry -- that makes it so!

5. Spacetime Diagrams

6. Natural (SR) Units
It is convenient to set c = 1 when working in special relativity!
This means we can express distance in time units, or vice versa!
So, in natural units, 1 second is equivalent to this unit of distance
Similarly,
Bottom line: we can use distance and time interchangeably, because they can be expressed in the same units! (same with energy and masss!)

7. Spacetime Structure Spacetime is a four-dimensional manifold
Points in spacetime are specified by the coordinates
Spacetime diagrams are a two-dimensional representation consisting of the t coordinate and the x (direction of relative motion)

8. Spacetime Structure and Spacetime Diagrams

9. Light cone represented by 45º lines (x=t) in all reference frames
v = c = 1 t x
Time axis
(rest or “home” frame)
Light cone represented by
45º lines (x=t) in all reference frames
Origin (t=0,x=0) of rest/home frame coord. system
Space axis (rest or “home” frame)

10. An event is something that happens at a specific point in spacetime x
An event is something that happens at a specific point in spacetime
Event at (t=3,x=1)
Event at (t=2,x=2)
Event at (t=1,x=3)
Event at (t=0,x=0)

11. Regions of Spacetime
Since nothing can travel faster than light, the spacetime diagram can be broken into three regions:
Timelike region: the collection of spacetime points (events) that can be influenced by an event at (0,0).
Causally connected to origin
Lightlike region: the collection of spacetime points (events) that lie along the light cone.
Spacelike region: the collection of spacetime points (events) that cannot be influenced by an event at (0,0)
Causally disconnected to origin

12. t x
Timelike region
Lightlike “region”
Spacelike region

13. Lines joining causally-connected events have slopes ≥𝟏x
Timelike region
Lightlike “region”
Lines joining causally-connected events have slopes ≥𝟏
Causally connected
Causally connected
Spacelike region
NOT causally connected

14. Slices of constant time are horizontal lines (parallel to the x-axis)
All x at specific t (photograph!)
t = 3
t = 2
t = 1
t = 0

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

16. 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)

17. Worldline of object with v = 0.5 in rest framex
Worldline of object with v = 0.1 in rest frame

18. D =½ T We define distance by the round-trip
travel-time of a pulse of light (“radar method”) t x
D =½ T T
Mirror D x

19. t
The line joining the reflection points defines the spatial axis in the rest frame! 
+3t 
+2t +t x -t 
-2t
If light signal goes out some fixed distance in a time
t and reflects, then it will come back in the
same amount of time. 
Sends light signal
-3t

20. Moving Reference Frames

21. t t x
Line which represents something
moving at velocity v (spaceship as
seen from the Earth) is equivalent
to the time axis of the observer inside
the spaceship!!

22. If light signal goes out a distance D in a time
The line joining the reflection points defines the spatial axis in the moving reference frame!
+3t  x
+2t 
+t x
-t
If light signal goes out a distance D in a time
t and reflects, then it will come back in the
same amount of time according to the moving frame (ship)!
-2t 
-3t 
Sends light signal

23. t x t x
“Photographs” in rest frame
(events in all space at fixed time) t4 t3 t2 t1
Lines of constant t are parallel to x-axis!

24. t x t x
“Photographs” in moving frame
t4
t3
t2
t1
Lines of constant t are parallel to x-axis!

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