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Extragalactic Astronomy Cosmology

Lecture SR1

4246 Physics 316

- Jane Turner
- Joint Center for Astrophysics
- UMBC NASA/GSFC
- 2003 Spring

Quiz 2 Revision Guide

- You should be able to-
- describe Hubbles key breakthroughs and use the

Hubble law - -describe the general approach of the Cosmic

Distance ladder ( an overview ) plus describe at

least some of the steps in detail - and note the

problems limiting use of some key standard

candles - You should be familiar with the use of Cepheid

variables - and SNe type 1a and what results from those have

told us - In addition, try to keep in mind some of the most

basic facts about galaxies which we learned a few

lectures ago

Quiz 2

Things which are not included in Quiz 2 (but will

be in the Mid-term exam) -Lives of stars Thi

ngs which will not be in any exam/quiz

-Luminosity functions of planetary nebulae/globu

lar clusters -Anything from the telescope session

-Fusion processes in stars

Mid-Term Exam

March 20 (Thursday), usual lecture room and time

(25 of final grade) Will cover the entire cour

se so far except items excluded

from all exams (already noted)

No math problems on GR (had no time for homework

s on this) but there will be some descriptive que

stions on GR. Will be both types for SR and the

rest of the course. Revision lecture on Tues M

arch 18 (BH-AGN and DM will wait until after th

e spring break)

Mid-Term Exam

Revision lecture on Tues March 18 Also will

give out project choices for the next half of

the semester List of options, you will be able

to select and mail in your choice the first week

after the break Also, student presentations wil

l be Tues April 22

Special Relativity-what is it?

Einstein tried to fit the idea of an absolute

speed for light into Newtonian mechanics. He

found the transformation from one reference frame

to another had to affect time-this led to the

theory of special relativity. In special relati

vity the velocity of light is special, inertial

frames are special. Anything moving at the speed

of light in one reference frame will move at the

speed of light in other inertial frames. Other

velocities are not preserved.

Have to worry about applicability of SR to accel

erating frames

Special Relativity-what is it?

So, special relativity is a theory which takes

into account the absoluteness of the speed of

light It is necessary to get calculation corre

ct where any velocities are even close to c

When velocities are s is an acceptable approximation to the right

answer Thats why people did not realize the ne

ed for SR for a long time, Newtonian mechanics

fit everyday life.

Special Relativity-what is it?

Special Relativity was constructed to satisfy

Maxwells Equations, which replaced the inverse

square law electrostatic force by a set of

equations describing the electromagnetic field.

Special Relativity

Einsteins postulates Ti

me dilation Length contraction

New velocity addition law

THE SPEED OF LIGHT PROBLEM

- Relativity tells us how to relate

measurements in different frames. - Galilean relativity
- Simple velocity addition law

vtotalvrunvtrain

Einsteins Postulates

Einstein threw away Galilean Relativity

Came up with two Postulates of Relativity

Einsteins Postulates

Postulate 1 The laws of nature are the

same in all inertial frames of reference

Postulate 2 The speed of light in a vacuum is

the

same in all inertial frames of reference

Lets start to think about the consequences of

these postulates We will perform thought expe

riments(Gedankenexperiment) For now, we will

ignore effect of gravity suppose we are

performing these experiments in the middle of

deep space

Time Dilation

Imagine a pulse of light from a bulb on a train

travelling at velocity v. A passenger on the t

rain sees the light hit a mirror and bounce back.

A person outside the train, at rest, sees the

light path to be longer.lets call them the

station master

E(see Hawley Holcomb page 175 - read chapters 6

7)

Frame of passenger Frame of Station Master

mirror

mirror

?tsm 2d/c

?tp 2H/c

H

d

v?tr

d2H2 (v? tsm /2) 2

?tsm ?tp /?1-(v2/c2)

The moving clock appears to run slowly

?tsm 2d/c d c ?tsm /2

?tp 2H/cHc ?tp /2

station master sees a longer time elapse than the

passenger

d2H2 (v? tsm /2) 2 4(d2-H2)/v2 ? tsm2

sub for d, H 4/v2 ? (c2 ?tsm2 /4) - (c2 ?tp2 /4

)? ? tsm2 c2/v2 (?tsm2 - ?tp2) ?tsm2 c2/v

2 (1 - ?tp2 /? tsm2) 1 1 - ?tp2 /?tsm2 v2/c2

?tp2 /?tsm2 1 - v2/c2

?tp2 1 - v2/c2 (?tsm2)

?tsm ?tp /?1-(v2/c2)

if v ?tp

The moving clock appears to run slowly

Time Dilation...

A moving clock appears to run more slowly !

Now, invert this, the station master has a bulb

mirror, the passenger sees this person as moving

at speed -v relative to them. The station masters

clock is a moving clock from the passengers

view. The passenger sees the station masters cl

ock running slowly. This is the Principal of

Reciprocity

No frame is preferred. Any clock at rest w.r.t an

inertial observer will show proper time-the

time between two events in the rest-frame in

which those events occurred.

Time Dilation

Hawley Holcomb page 177

This effect is called Time Dilation

The moving clock slows by a factor

The Lorentz factor

?

v/c

The shortest time for an event is that measured

by an observer in the same inertial frame as the

event is occurringthis is the proper time

Time Dilation -Example

The moving clock slows by a factor

The Lorentz factor

If we have a spacecraft traveling at v0.87c then

?2. An event taking 30s to an astronaut

on the spacecraft, appears to take 60s to an

outside observer in their own inertial frame

Length Lorentz Contraction

Measure length by comparison of an object to a

fixed standard ruler, where the two ends of an

object are measured at the same specific time

Consider two telephone poles beside our moving t

rain What is their separation ? The station ma

ster measures the time the front of the train

passes each pole, and then calculates the

distance between them to be ?xsm v ?tsm

Length Lorentz Contraction

?xsm v ?tsm The passenger sees the poles m

oving at -v So station master and passenger ag

ree the relative speed is v To the passenger, t

he dist between each pole passing the window is

?xp v ?tp We already have ?tsm ?tp /?1-(v2/

c2) Which gives us...

Length Lorentz Contraction

?xp ?tp ?1-(v2/c2) ?xsm ?tsm

?xp ?xsm ?1-(v2/c2)

or

if v 0.5c ?xp 0.87 x ?xsm

the passenger measures a shorter distance than

the station master The poles are in the frame o

f the station master, who sees them separated by

the maximum length anyone ever will, the proper

length

Length Lorentz Contraction

?xp ?xsm ?1-(v2/c2)

The passenger is taking a measurement of their

separation from a frame which has a relative

velocity, so sees a contraction in length

Note the contraction appears only in the

direction of relative motion, the heights of the

telephone poles would be seen as the same by both

observers!

Reciprocity Lorentz Contraction

Now consider the length of a car on the moving

train The passenger, moving with the car, sees

it at its proper length The station master s

ees length contraction and thus a moving car

appears shorter to them The passenger sees thin

gs in the station masters frame to be contracted,

the station master sees things in the passengers

frame to be contracted Reciprocity applies to l

ength as well as time effects

Example

For Concorde, travelling at twice the speed of

sound ? 1.000000000002 and length contraction1

0-8 cm

(out of 60m proper length) !!

Note the contraction appears only in the

direction of relative motion

Mass Increase

Similar arguments as for length contraction can

be used to relate moving mass to rest mass such

that moving mass rest mass/?1-(v2/c2) v

0.5c - moving mass1.15 x rest mass

Simultaneity

Consider an observer in a room. Suppose there is

a flash bulb exactly in the middle of the room.

Suppose sensors on the walls record when the

light rays hit the walls. Since the speed of li

ght is constant, rays will hit the walls at the

same time.Call these events A B.

Then perform the same experiment in a moving S

pacecraft, observed by somebody at rest

Simultaneity

Flash hits both front/back of train simult in the

train frame, seen by passenger

In station masters frame, light hits the back of

the train before the front The concept of eve

nts being simultaneous is different for observers

in different reference frames

The order of events

- Consider same experiment seen by three observers
- Moving astronaut thinks events A and B are

simultaneous - Observer at rest thinks A occurs before B

- What about a 3rd observer who is moving faster

than astronauts spacecraft? - 3rd observer sees event B before event A
- So, order in which events happen depends on frame

of reference.

Addition of Relativistic Velocities

Need a new formula for adding relativistic

velocities Suppose you see an astronaut moving

at vel V1 and she sees a second object moving re

lative to her at V2 -the Newtonian approx. says

the outside observer sees the 2nd object move at

(V1 V2) But once we take account of the way tim

e and distance depend on v, we find

No matter how close to c V1 and V2 are, Vadd

cannot exceed c because the speed of light is

absolute

Relativistic Doppler Formula

Classic Doppler effect seen when there is

relative motion, as the crests of the waves bunch

or stretch out Relativity adds the effect that

the frequency of the light (which is 1/time) is

smaller at the source than the receiver, due to

time dilation z 1 v (1v/c)/(1-v/c)

(Hawley Holcomb page 183)

Transverse Doppler Effect

A relativistic Doppler effect also occurs in the

direction perpendicular to the relative motion

The observation of a moving clock running slow m

eans the frequency of light in a moving frame

appears reduced Think of the frequency of ligh

t as a clock with a number of cycles per second,

if that clock in the moving frame runs slow, we

see fewer cycles completed per second

Freq reduction is like a redshift

Summary of Formulae

Lorentz Factor ? or ?

Velocity v/c Gamma value 0 1 0.1

1.005 0.87 2 0.9 2.29 0.99 7.1 0

.999 22.4

Summary of Formulae

Lorentz Factor ? or ?

timemoving timerest ?1-(v2/c2)

Time Dilation

Lorentz Contraction

lengthmoving lengthrest ?1-(v2/c2)

Mass massmoving massrest/?1-(v2/c2)

Relativistic Addition of Velocities

Relativistic Doppler z 1 v

(1v/c)/(1-v/c)

Derivation of Emc2

Start with mass increase formula

massmoving massrest/?1-(v2/c2)

m m01-(v2/c2)-0.5 use a mathematical appro

ximation, where ? bstitute ? -v2/c2 (1 -v2/c2)-0.5 1-0.5(-v2

/c2) Our substitution means which can be simpli

fied to we are dealing (1-v2/c2)-0.5 1v2/2c2

the case v

Derivation of Emc2

m m0(1v2/2c2) expand m m0 m0v2/2c2 multip

ly both sides by c2 mc2 m0c2 m0v2/2 m0v2

/2 is the kinetic energy mc2 is the total energy

of the object (E) what about an object which i

s not moving, so the kinetic energy term is zero,

then the total energy is not zero, as there is th

e term m0c2 i.e. even when the vel is zero, and o

bject has energy due to its rest mass, E m0c2

more often written E mc2

II EXAMMASS TO ENERGY

- Nuclear fission (e.g., of Uranium)
- Nuclear Fission the splitting up of atomic

nuclei - E.g., Uranium-235 nuclei split into fragments

when smashed by a moving neutron. One possible

nuclear reaction is - Mass of fragments slightly less than mass of

initial nucleus neutron - That mass has been converted into energy

(gamma-rays and kinetic energy of fragments)

From web site of Georgia State University

- Nuclear fusion (e.g. hydrogen)
- Fusion the sticking together of atomic nuclei
- Much more important for Astronomy than fission
- e.g. power source for stars such as the Sun.
- Explosive mechanism for particular kind of

supernova - Important example hydrogen fusion.
- Ram together 4 hydrogen nuclei to form helium

nucleus - Spits out couple of positrons and neutrinos

in process

- Mass of final helium nucleus plus positrons and

neutrinos is less than original 4 hydrogen

nuclei - Mass has been converted into energy (gamma-rays

and kinetic energy of final particles) - This (and other very similar) nuclear reaction is

the energy source for - Hydrogen Bombs (about 1kg of mass converted into

energy gives 20 Megaton bomb) - The Sun (about 4?109 kg converted into energy per

second)

EXAMPLES OF CONVERTING ENERGY TO MASS

- Particle/anti-particle production
- Energy (e.g., gamma-rays) can produce

particle/anti-particle pairs - Very fundamental process in Nature shall see

later that this process, operating in early

universe, is responsible for all of the mass that

we see today!

- Particle production in a particle accelerator
- Can reproduce conditions similar to early

universe in modern particle accelerators

Spacetime

From SR we found time intervals, space

separations and simultaneity are not absolute

space and time have to be considered together to

understand events - so we need to consider

4-dimensional spacetime Difficult to think in

4D, but we can make nice spacetime diagrams!

First developed in 1908 by Hermann Minkowski

SPACE-TIME DIAGRAMS

Only plot one dimension of space for simplicity

Light Cone

Any point is an event, a line/curve connecting

points is a worldline

SPACE-TIME DIAGRAMS

Light Cone

time axis often renormalized and plotted as ct,

so it has same dimensions as space axis

SPACE-TIME DIAGRAMS

object B traveling at v 450

Light Cone

object C would have to travel at vc so impossible

light beam follows a world line ctx , using x

versus ct - this is a line at 450

SPACE-TIME DIAGRAMS

Inertial Observers

Accelerated Observer

Light Cone

SPACE-TIME DIAGRAMS

Light Cone

SPACE-TIME DIAGRAMS

Event A constrained to lie within cones defined b

y lines equiv to vc (45o)

future

Light Cone

elsewhere

elsewhere

past

SPACE-TIME DIAGRAMS

How do we define a separation between two

events on a worldline?

Light Cone

in general ?r2 ?x2 ?y2 ?s v?(c?t)2 - (?x)2?

defines a spacetime interval

SPACE-TIME DIAGRAMS

in general ?r2 ?x2 ?y2 ?s v?(c?t)2 - (?x)2?

a separation in Minkowski spacetime, a spacetime

interval

Light Cone

-ve sign because time cannot be treated like a

spatial dimension ?s called a spacetime interval

and is invariant - all observers agree on the

quantity

SPACE-TIME DIAGRAMS

?s v?(c?t)2 - (?x)2? a separation in

Minkowski spacetime

interval2 (dist traveled by light in time ?t)2

- (dist between events)2 ?s2 0 timelike (ligh

t had more than enough time to travel between

events) ?s2 0 null (or lightlike, light had exa

ctly enough time to travel between events)

?s2 travel between the events)

Light Cone

Summary

We have learned the special theory of relativity

relates observations made in inertial frames to

one another, because inertial frames are special,

we call it the Special Theory Special Relativit

y showed us we had to discard the concepts of

absolute space time, space time are

inextricably linked Special Relativity brings

mechanics and electromagnetics into consistency

and provides a model for situations where

velocities approach the speed of light