P1X*Dynamics & Relativity:

Newton & Einstein

Chris ParkesOctober 2005

Special Relativity

Postulates

Time Dilation

Length Contraction

Lorentz Transformation

Relativistic Energy

http://ppewww.ph.gla.ac.uk/~parkes/teaching/DynRel/DynRel.html

Part II - “When a man sits with a pretty girl for an hour, it

seems like a minute. But let him sit on a hot stove for a minute and it's longer than any hour.

That's relativity.”

2005 Centenary

Read the textbook Phenomena close to speed of light, high energy

                                             

Michelson Morley Interferometer

Ether was a hypothetical substance through which light travelled•The universe was thought to be a stationary frame of reference full of ether.

So, if the earth is traveling at v with the ether then light should have one velocity c+v (w.r.t. ether).In the perpendicular direction the velocity would just be c

In 1887 Michelson & Morley set up this apparatus in which a light beam is split into two perp. Beams, then

recombined.

By observing interference between the two beams, they showed that the

speed of light was constant

Maxwell’s eqns of electromagnetism (1865)Contain a velocity c for light (emag. waves)

Einstein’s Postulates• First Postulate

– Laws of physics are the same in any inertial frame of reference (principle of relativity)

• See before, co-ordinate axes moving at constant velocity• No preferred frame of reference, no frame is more

“correct”- cannot tell if you are moving at constant velocity

• Second Postulate– The speed of light in vacuum is the same in all inertial

frames• Independent of the motion of the source• at rest I measure speed of light c (~3x108 ms-1)• If it is the headlight of a train moving at u

Newton: speed =c+u Einstein: speed still =c !!!

c

v

Newton’s laws are the low velocity limit of Einstein’s special relativity

Co-ordinate Transformations Revisited

v

vtxx

dtdx

dtdx

'

'

dtdy

dtdy

yy'

'

0

ut

Frame SFrame S’(x’,y’)

x’

yV of S’ wrt S

0 x

uvv xx '

Now, consider point moving, differentiate

Obtain Galilean velocity transformation(as before)

i.e. despite looking convincing this is wrong at high velocities. It gives us the

ucv problem for our train headlight.

We have assumed time is the same for our observer in S (station)and S’ (train) t=t’

No tt’, and the velocities must be defined as:dtdxv '

'' dtdxv

(and similarly with distances…)

Time Intervals: Simultaneous Events• Two events simultaneous in one reference frame are not

simultaneous in any other inertial frame moving relative to the first.

Two bolts seen simultaneously at C

Right bolt seen first at C’

Left bolt seen second at C’

Two lightning bolts strike A,B

S-frame : simultaneous lighting strikesS’-frame : right bolt hit first

Time Dilation• Time interval is shortest in a reference frame where a clock

is at rest. Moving clocks appear slower to an observer.

i.e. an observer looking at a moving clock, measures a longer time on her watch than on the moving clockLets prove this:

v

Stationary observer Moving w.r.t. observer

light

mirror

In both frames (S,S’) observers agree light is travelling at cbut they disagree on the distance (path) the light has travelled

S’S

Frame S: Time takenc

dt

2

speed

distance0

d

This is called the proper time (measured in stationary frame)This is the shortest possible time interval, moving observers measure a longer time

Time Dilation continuedFrame S’ (moving observer):

L L

vt

d

Distance travelled by light =2LBy Pythagoras,

So time taken2

222

222

22

2

2

vtdL

ct

c

Lt

vtdL

Now,

20

222

20

22222

220

2

2

2

2

2

2

2

2

0200

)1(

22

2

2

2

0

tctc

tctvtc

vtctct

ctdd

ct

c

dt

cv

vtctct

So, (By substituting for d in expression above)

Hence, time measured by moving observer2

2

10

cv

tt

Define velocity relative to light speedAnd time “correction” factor

cv

21

1

Hence, can rewrite time as 0tt

Time appears slower

(to stationary observer)

for moving system

t>t0

(So,<1, >1 always)

Time Dilation Example:b-quark decay

• An unstable b quark is produced• The b quark travels ~ 4mm

before decaying• It is travelling at 0.99c (=9)• Hence

4mm

Image reconstructed byDELPHI particle physics

Experiment at CERN

pssms

m

v

st 13103.1

10399.0

004.0 1118

However, the average lifetime (at rest)of a b-quark is 1.5 ps (1 pico-s is 1x10-12 s)

So why did we measure 13ps ? Time dilation

pspstt 5.135.190

Length Contraction• Lengths are longest in a reference frame where the object is at rest. Moving Objects appear shorter to

an observer.

Only applies to lengths in direction of travel

lightmirror

L0

vvt

Stationary observer Moving w.r.t. observerS’SFrame S: Time taken for light to bounce back and forth

c

Lt 0

0

2

speed

distance

L

L0 is the proper length (measured in stationary frame)This is the longest possible length, moving observers measure a shorter length

Frame S’ (moving observer):Time taken for light to travel from source to mirror = t1

Corresponding distance travelledHence,

111 ctvtLd

vcLtLvtct 111

Length Contraction continued

• Length contraction(remember >1)

Time taken for light to travel from source to mirror = t2

Corresponding distance travelledHence,

Frame S’ (moving observer) cont.:

222 ctvtLd

vcLtLvtct 222

So, total time vcL

vcLttt 21

22

)1(

2

2))((

)()(

2

2

22

cL

c

L

vcLc

vcvcvcLvcL

c

v

But,from time dilation cL

cLtt 0222

0

0LL

Moving object appears to shrink by a factor in the direction of travel

L

vt2

Including S’ origin movement

If S’ movement in x direction, then y,z co-ords unaffected

Lorentz Transformations

• Covered special cases– Time dilation,

– length contraction

• General form of how to relate two frames S, S’

vt

Frame S

Frame S’(x’,y’)

x’

y

V of S’ wrt S

0x

•Transform (x,y,z,t) in S to (x’,y’,z’,t’) of S’

A distance x’ in S’ is seen as x’/ w.r.t. S

)(''' vtxxvtxvtx xx

',' zzyy

0

–S’ moving with velocity v along x axis

Spatial Transform

Same as Gallilean transforms

if =1, i.e.=0, low velocity approx.

Lorentz Tranformations : time

Lorentz Transformations Summary:

A distance x in S is seen as x/ w.r.t. S’

Including S origin movement (velocity –v w.r.t. S)

)('

'

vtxx

vtx x

but from spatial transform

)('

'

'

)1('

'

2

2

2

2

21

cxv

cv

cv

x

x

tt

xtt

xvtvt

xvtxvtvt

vtxvt

Hence,

since 2

2

22

2

2211

1

12 11cv

cv

cv

)(' vtxx

)(' 2cxvtt

'

'

zz

yy

Moves from Frame S to a

Frame S’ travelling at velocity v along x axis

Thus,

To move from S’ to S, just reverse sign on ve.g. )'( 2

'c

vxtt )''( vtxx

(tt’for low velocity)

Lorentz Velocity Transform• Particle moving at speed v in S• Frame S’ moving at speed u wrt S’• What speed (v’) is it moving at in S’ ?

Return to our velocity question…

Velocity in frame S dtdxv Velocity in frame S’ '

'' dtdxv

Now, )(' udtdxdx )(' 2c

udxdtdt from Lorentz transforms

Hence,

dtdx

cu

udtdx

dt

dx

21'

'

or

21'

c

xuvx uv

xv

and2'1

'

c

xuvx uv

xv

Is speed of light a constant? Put vx=c (e.g. c for velocity of headlight of train, u is speed of train)

cvcucu

cu

c

uc

cucucx

1

)1(

11 2' i.e. if it moves with c in one frame

it moves with c in all frames second postulate

…and low velocity limit?uvv xx ' back to Gallilean transform

11 2

' c

uv x

So,denominator

Spacetime So, for moving systems as measured by stationary system:Time got longer, distance got shorter.But more than this - from Lorentz transforms we know:Space co-ords in one frame depends on both space and time in other frameAnd time in one frame depends on both space and time in otherSpace & time have become intertwined - ‘joined’ in spacetime.Space and time mix between frames.We have to consider spacetime 4-vector co-ordinates (x,y,z,t)

Spacetime distance is conserved between frames (i.e. a rotation)-the increased time co-ord compensates the reduced spatial co-ord.-Define co-ords as (x,y,z,ct)3D space distance2 is: 4D space-time distance2 is:

22222222 )()( ctzyxictzyx

ict

z

y

x

ict

z

y

x

222 zyx

z

y

x

z

y

x

x

ct

light

Minkowski spacetime

That Equation• Mass is a form of energy

– Can be used to create new particles A->B+C

• Rest mass m0

vmvmp 0

•E = mc2

•m = m0

m0c2

also

420

222

420

220

222

420

420

2420

22 )1(

cmcpE

cmcmvE

cmcmcmE

since 2

2

2

2

2

2

2

2

2

2

2

2

22

11

)1(11

1

1)1(

cv

cv

cv

cv

cv

cv

More useful to express in terms of momentum:

Relativity Top 5• Laws of physics are the same in any inertial frame of

reference• The speed of light in vacuum is the same in all inertial

frames• Moving clocks appear slower to an observer.• Time difference is shortest when the clock is at rest in the

reference frame

• Moving Objects appear shorter to an observer• Object is longest when the object is at rest in the reference

frame

• space and time get mixed spacetime– Transform (x,y,z,t) in S to (x’,y’,z’,t’) of S’

0tt 21

1

where cv

Postulate 1

Postulate 2

TimeDilation

LengthContraction

0LL

LorentzTransformations