Spacetime Structure

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

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!

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!

Spacetime Diagrams

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!)

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)

Spacetime Structure and Spacetime Diagrams

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)

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)

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

t x Timelike region Lightlike “region” Spacelike region

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

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

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

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)

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

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

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

Moving Reference Frames

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!!

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

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!

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

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