PH 301

Dr. Cecilia VogelLecture 5

Review

Outline Velocity transformation

NOT simple addition Spacetime

intervals, diagrams

Lorentz transformations order of events twin paradox

Velocity u = dx/dt u’ = dx’/dt’

( ' ')x x vt

'dx dx dtv

dt dt dt

'

'

dx dx dtv

dt dt dt

' /'

dtu u v

dt

What’s dt’/dt?

2' 'v

t t xc

21 '

'

dt vu

dt c

2

'

'1

u vu

u vc

SO…

Velocity Transformation

Note: Speed will never be bigger than c If u’ and v are <c, then u<c If |u’| or |v| =c, then u=c

speed of light the same

Pay attention to the sign of velocities Pay attention to order of frames

2

'

1

u vu

u vc

Using Velocity Transformation Step 1: Let u = answer you seek.

Step 2: u = velocity of A rel to B, so A and B are determined.

Step 3: Identify frame C -- what’s left? Step 4: Determine u’

u’= velocity of A rel to C If you have C rel to A, use opposite sign

Step 5: Determine v v = velocity of C rel to B If you have B rel to C, use opposite sign

Step 6: Plug in the numbers to compute u. Step 7: Check that your answer makes sense!

EXAMPLE A spaceship is approaching the

planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? u = velocity of shuttle relative to the ship

so… A is ________ and B is _____

Then C is _________

EXAMPLE A spaceship is approaching the

planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? u’ = velocity of A relative to C

so… u’ is velocity of ________ relative to _____ u’ = +0.95 c

EXAMPLE A spaceship is approaching the

planet Zorgon at a speed of 0.85c. A diplomatic shuttle is sent ahead to arrive at the planet earlier. With what velocity should the shuttle be launched relative to the ship, in order to approach the planet at a rate of 0.95c? v = velocity of C relative to B, so… v is

velocity of _______________ relative to _____

v =Ship relative to Zorgon is +0.85 c, so

EXAMPLEWith what velocity should the shuttle be

launched relative to the ship, in order to approach the planet at a rate of 0.95c?

u’ = +0.95 c, v = -0.85 c

2 2

' 0.95 0.85(0.95 )( 0.85 )

1 1

0.10.52

1 (0.95)(0.85)

u v c cu

u v c cc cc

c

EXAMPLE A proton is traveling at a speed of

0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab?

u = velocity of neutron relative to the labso… A is ________ and B is _____

Then C is _________

EXAMPLE A proton is traveling at a speed of

0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab?

u’ = velocity of A relative to Cso… u’ is velocity of ________ relative to _____

neutronu’ =proton

EXAMPLE A proton is traveling at a speed of

0.75c relative to the lab. A neutron is to collide with it at a relative speed of 0.90c. With what velocity should the neutron go relative to the lab?

v = velocity of C relative to B, so… v is velocity of _______________ relative to _____

lab

v =

proton

EXAMPLEWith what velocity should the

neutron go relative to the lab?

2 2

' 0.9 0.75( 0.9 )(0.75 )

1 1

0.150.46

1 (0.9)(0.75)

u v c cu

u v c cc cc

c

Space-time Considers time as a fourth

dimension. An event is given by a 4-

component vector: 3 space, 1 time

I can’t draw in 4 dimensions Let’s consider 1 space & 1 time

Space-time Diagrams An event is a

point on the diagram.

x

ct

A world-line is the path of an object on the diagram. The steeper the slope, the slower it’s going.

Wor

ld-li

ne o

f

light

Slope = 1, means speed c.

Classical Invariant

( )x x vt

In classical relativity, everyone measures the same distance between two events in space:

If

2 2 2x y z d Then

2 2 2x y z d

Invariant in Space-time In space-time, we let the 4-component

vector be (x, y, z, ict) So that the space-time interval,

2 2 2 2 2( )s x y z c t is invariant.

2 2 2 2 2( )s x y z c t

Spacetimerecall

special relativity (ch 2)An event is something that occurs at a particular place and time – at a particular point in spacetime

Spacetime graph Graphs an event as a point in 4-D spacetime

(x, y, z, t)We will consider 1-D space, 1-D time

2-D graphs are easier to draw!

Worldline Worldline of an object

Is the set of all spacetime points occupied by the objectAlthough t is ordinate, and x is abscissa, do not think of t(x)

Slope of worldlined(ct)/dx = c/(dx/dt)v/c = 1/slope

steeper slowervertical stopped|slope| = 1 |v|=c, worldline of lightgenerally, worldline |slope|>1

Transform WorldlinesA Spacetime graph is drawn from a particular reference frameIn the spacetime graph drawn from a different reference frame

the slope of the worldline of a massive object is different

according to the velocity transformation equation

the slope of the worldline of light is not different

slope is still +1

Spacetime FutureGiven a particular event at xo, yo, zo, to,

the points on a spacetime graph are divided into 3 regions: its future, its past, and its elsewhere

Spacetime Future of the eventthe set of all spacetime points such that

t> to, ANDd<ct

d = spatial distance between x, y, z and xo, yo, zo

t= t- to

Spacetime PastSpacetime Past of the event

the set of all spacetime points such that

t< to, ANDd<c|t|

NOTE:The event can be reached by a signal from its past

this event can be affected by events in the past

A signal from the event can reach points in the future

this event can affect events in its future

LightconeAn event’s lightcone

is the set of spacetime points such that

d=c|t|It is the boundary of the future and of the past

A signal from this event can only reach events in the lightcone

by traveling at the speed of light

This event can only be reached by a signal from an event in the lightcone

if the signal travels at speed of light

Lightcone, Past, and Futurelightcone|slope| =1

future

past

Elsewhere The Elsewhere Elsewhere of the event

consists of all the other spacetime points (other than lightcone, past, future)

d>c|t|The event cannot be reached by, nor can anything from the event reach, an event in its elsewhere

this event cannot affect nor be affected by events in its elsewhere

Elsewhere is not…This does NOT mean

that an event that is currently in our elsewhere can never affect us

that event may be in the past of future points on our worldline

It also does NOT meanthat an event that is currently in our elsewhere can never have been affected by us

that event may have been in the future of past points on our worldline

Elsewhere ExampleFor example

if the sun had disappeared 4 minutes ago

that event is in our elsewhere right nowd= 8c-min, c|t| = 4 c-min, d>c|t|

BUT, four minutes from now, that event will be in our past, and we will be gravely affected!

ourworldline

sundisappearing

worldlineof sunlight

- 4 min

+ 4 min

Transform lightconeThe lightcone of an event

is the same set of points in all reference frames

All observers agree onwhich events are in the event’s futureand its pastand its elsewhere