Flashcards in 3. Spacetime and Special Relativity Deck (30)

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1

##
Space, Time and Motion in Newtonian Physics

Outline

###
-absolute Euclidian space

-absolute time agreed on by all observers

-absolute free motion of a particle in the absence of forces

-these three points lead to the concept of an inertial frame which describes the laws of physics

2

## Geodesics in Euclidian Space

### -straight lines

3

## Relativity

### -each type of relativity describes a way of relating different reference frames / observers

4

##
Galilean Relativity

Definition

### -in all inertial reference frames, the laws of physics should agree

5

##
Galilean Relativity

Position

###
-have reference frame O and reference frame O' moving along the x axis with relative velocity v such that at t'=t=0, O=O'

-Galilean transformations:

t = t'

y = y'

z = z'

x = x' + vt

6

##
Galilean Relativity

Velocity

###
-differentiate the position Galilean transforms:

dy/dt = dy'/dt'

dz/dt = dz'/dt'

dx/dt = dx'/dt' + v

7

##
Galilean Relativity

Speed of Light

###
-in the frame O:

w = (dx/dt, dy/dt, dz/dt)

-and in frame O' :

w' = (dx'/dt', dy'/dt', dz'/dt')

-then:

w = w' + v

-what if w'=c, speed of light

-then:

w = w' + v = c + v > c

-for v>0

8

##
Space, Time and Motion in Special Relativity

Outline

###
-Einstein's principle of relativity: "all laws of physics are the same in all inertial frames"

-speed of light principle: "the speed of light is the same in all inertial frames

9

##
Special Relativity

Position Derivation

###
-start with the most general possible linear transformation:

x' = Ax + Bt

t' = Cx + Dt

-where A, B, C and D are at most functions of v

-using conditions:

x'=0, x=vt

x=0, x'=-vt'

10

##
Special Relativity

Position in terms of Av and Ev

###
y' = y

z' = z

x' = Av (x - vt)

t' = Av (Ev x + t)

11

##
Special Relativity

Lorentz Transformation Derivation

###
-suppose there is a third frame O''

-O' is moving with speed v with respect to O and O'' moves with speed w with respect to O'

-find an expression for {x'',t''} in terms of {x,t}

-all of these relations should have the same structure since they are the same transformation

12

##
Special Relativity

Lorenzt Transformations

###
y' = y

z' = z

x' = [x - vt] \ √[1 - v²/c²]

t' = [-xv²/c² + t]\√[1 - v²/c²]

13

## Minkowski Metric

###
-spacetime requires a non-Eulician metric e.g. Minkowski metric

gm = -c²dtxdt + dxxdx + dyxdy + dzxdz

14

## Minkowski Metric and the Lorentz Transformation

### -the Minkowski metric is invariant under the Lorentz transformation

15

## Lorentz Factor Definition

###
γ = 1 / √[1 - v²/c²]

γ≥1

16

## Lorentz Transforms in Terms of the Lorentz Factor

###
t = γ (t' + v/c² x')

x = γ(x' + vt')

y = y'

z = z'

17

##
Special Relativity

Speed of Light

###
-consider a particle moving with speed w'=dx'/dt' along x' axis in frame O' and w=dx/dt in frame O:

-using the Lorentz transforms to find expressions for dx' and dt'

=>

w = [w' + v] / [1 + w' v/c²]

-suppose w'=c, then w=c

=>

-the Lorentz transforms are consistant with the speed of light principle

18

## Time in Special Relativity

###
t=t'

-each inertial frame has its own time

19

##
Special Relativity

Relativity of Simultaneity and Temporal Order

###
-consider two events, suppose in frame O' they are separated by time interval Δt' and their x' coordinates differ by Δx'

-then Δt and Δx in frame O can be determined using the Lorentz transforms

=>

Δt = γv/c² Δx'

-assuming v>0, this is >0 f0r Δx'>0, =0 for Δx'=0 and <0 for Δx'<0

-hence, the order of events is different in different inertial frames

20

##
Special Relativity

Time Dilation Effect

###
-consider a standard clock at rest in O', denote τ as the time of this clock in O' (the proper time since the clock is at rest in this reference frame)

-then for this clock, Δx'=0 and Δt'=Δτ

-using the Lorentz transforms,

Δt = γΔτ

-since γ≥1, Δt ≥ Δτ

-similarly, any standard clock at rest in O will appear to run slower when observed in frame O'

-the rate of a moving clock slows down compared to time in the inertial frame where the clock is observed

21

##
Special Relativity

Relativity of Spatial Order

###
-consider two events, suppose in frame O' they are separated by time interval Δt' and their x' coordinates differ by Δx'

-then Δx in frame O is given by the Lorentz transform

=>

Δx = γ(Δx' + vΔt')

-suppose Δx'=0, then Δx=γvΔt'

-assuming v>0, this is >0 for Δt'>0, =0 for Δt'=0 and <0 for Δt'<0

=>

-spatial order of events is different in different inertial frames

22

##
Special Relativity

Length Contraction Effect

###
-consider a bar at rest in frame O, denote l as its length measured in this frame (proper length) and suppose it is aligned with the x axis:

l = xb - xa = Δx

-it doesn't matter when the ends of the rod, xa and xb, are measured in frame O since the bar is at rest in that frame

-in frame O' the bar is moving so the coordinates need to be measured simultaneously e.g. at Δt'=0:

l' = xb' - xa' = Δx'

-using the Lorentz transforms:

Δx=γΔx' or l'=l/γ

-since γ≥1, l' ≤ l

=>

-the length contraction effect, in frame O', the coordinate grid of frame O appears contracted along the x-axis

-each inertial frame has its own space / coordinate grid

23

## Spacetime in Special Relativity

###
-special relativity insists that absolute space and absolute tie don't exits, to describe any physical process an inertial frame must first be specified

-spacetime is absolute, whichever frame is used to measure space and time, the spacetime description will be the same

24

## Spacetime Interval General Formula

###
ds² = -c²dt² + dx² + dy² + dz²

-or separating into space and time components:

ds² = -c²dt² + dl²

25

## Normalising Minkowski Spacetime Coordinates

###
ds² = -c²dt² + dx² + dy² + dz²

-time coordinate is not normalised, let:

xo = ct

x1 = x

x2 = y

x3 = z

=>

ds² = -dxo² + dx1² + dx2² + dx3²

26

## What are the three types of spacetime interval?

###
1) space-like

2) time-like

3) light-like

27

## Describe the future and past light cones diagram

###
-the observer is at the centre

-a flat surface with the observer at the centre describes a hypersurface of simultaneity, a surface on which time is constant

-the vertical axis represents time

-the past light cone is a cone down the negative time axis

-the future light cone is a cone up the positive time axis

-the cones make a 45' angle with the hypersurface

-light can only travel on the surface of the cone

-events are represented by points

-any event that has a chance of effecting you in the present lies in your past light cone, any event that you can affect lies in your future time cone

28

## Space-Like Spacetime Intervals

###
ds² = -c²dt² + dl²

-if ds² > 0, there exists a frame where dt=0

=>

ds² = dl²

-this represents lines on the hypersurface of simultaneity

29

## Time-Like Spacetime Intervals

###
ds² = -c²dt² + dl²

-if ds² < 0, there exists a frame where dl=0

=>

ds² = -c²dt²

30