extragalactic astronomy & cosmology lecture sr1 jane turner joint center for astrophysics umbc...

Extragalactic Astronomy & Cosmology

Lecture SR1Jane Turner

Joint Center for AstrophysicsUMBC & NASA/GSFC

2003 Spring

[4246] Physics 316

Jane Turner [4246] PHY 316 (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

Things which will not be in any exam/quiz

-Luminosity functions of planetary nebulae/globular 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 course so far except items excluded from all exams (already noted)

No math problems on GR (had no time for homeworks on this)but there will be some descriptive questions on GR.

Will be both types for SR and the rest of the course.

Revision lecture on Tues March 18

(BH-AGN and DM will wait until after the 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 will 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 relativity 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 accelerating frames



So, special relativity is a theory which takes into account the absoluteness of the speed of light

It is necessary to get calculation correct where any velocities are even close to c

When velocities are << c then Newtonian mechanics is an acceptable approximation to the right answer

That’s why people did not realize the need for SR for a long time, Newtonian mechanics fit everyday life.



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

Einstein’s postulates

Time 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 : vtotal=vrun+vtrain


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

Let’s start to think about the consequences of thesepostulates

We will perform “thought experiments”(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 train 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

dH

vtr

tp =2H/c

d2=H2+ (v tsm /2) 2

tsm =2d/c

tsm = tp /[1-(v2/c2)]

The moving clock appears to run slowly


tsm =2d/cd = c tsm /2

tsm = tp /[1-(v2/c2)] if v< c then tsm > tp

The moving clock appears to run slowly

d2=H2+ (v tsm /2) 2

4(d2-H2)/v2 = tsm2

sub for d, H 4/v2 (c2 tsm

2 /4) - (c2 tp2 /4) = tsm

2 c2/v2 (tsm

2 - tp2) = tsm

2 c2/v2 (1 - tp

2 / tsm2) = 1

1 - tp2 /tsm

2 = v2/c2

tp2 /tsm

2 = 1 - v2/c2

tp =2H/cH=c tp /2

tp2 = 1 - v2/c2 (tsm

2)

station master sees a longer time elapse than the passenger


Time Dilation...

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 clock running slowly. This is the “Principal of Reciprocity”

A moving clock appears to run more slowly !

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

This effect is called Time Dilation

The moving clock slows by a factor

The Lorentz factor

v/c

Hawley & Holcomb page 177

The shortest time for an event is that measured by an observer in the same inertial frame as the event is occurring…this is the “proper time”


Time Dilation -Example

The moving clock slows by a factor

The Lorentz factor

If we have a spacecraft traveling at v=0.87c 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 trainWhat is their separation ?

The station master measures the time the front of the train passes each pole, and then calculates the distance between them to be

xsm = v tsm



xsm = v tsm

The passenger sees the poles moving at -v

So station master and passenger agree the relative speed is v

To the passenger, the dist between each pole passing the window is xp = v tp

We already have tsm = tp /[1-(v2/c2)]Which gives us...



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 of the station master, who sees them separated by the maximum length anyone ever will, the “proper length”



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 sees length contraction and thus a moving car appears shorter to them

The passenger sees things in the station masters frame to be contracted, the station master sees things in the passengers frame to be contracted

Reciprocity applies to length as well as time effects


Example

For Concorde, travelling at twice the speed of sound =1.000000000002 and length contraction=10-8 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 mass=1.15 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 light is constant, rays will hit the walls at the same time.Call these events A & B.

Then perform the same experiment in a moving Spacecraft, 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 events 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 relative 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 time 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 = √[ (1+v/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 means the frequency of light in a moving frame appears reduced

Think of the frequency of light 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

Relativistic Doppler z + 1 = √[ (1+v/c)/(1-v/c) ]

Relativistic Addition of Velocities

lengthmoving =lengthrest [1-(v2/c2)]

Lorentz Contraction

timemoving = timerest [1-(v2/c2)]Time Dilation

Lorentz Factor or

Mass massmoving =massrest/[1-(v2/c2)]


Derivation of E=mc2

Start with mass increase formulamassmoving =massrest/[1-(v2/c2)]

m =m0[1-(v2/c2)]-0.5

use a mathematical approximation, where << 1(1+)-0.5 ≈ 1-0.5 and substitute = -v2/c2

(1+ -v2/c2)-0.5 ≈ 1-0.5(-v2/c2) Our substitution means which can be simplified to we are dealing (1-v2/c2)-0.5 ≈ 1+v2/2c2 the case v << cm =m0(1+v2/2c2)


Derivation of E=mc2

m =m0(1+v2/2c2)expand m =m0 + m0v2/2c2

multiply both sides by c2

mc2 =m0c2 + m0v2/2 m0v2/2 is the kinetic

energymc2 is the total energy of the object (E)

what about an object which is not moving, so the kinetic energy term is zero, then the total energyis not zero, as there is the term m0c2

i.e. even when the vel is zero, and object 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)

BaKrnnU 14489235 31 ++→+


From web site ofGeorgia 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

υ22 4 41 ++→ +eHeH


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 4109 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

“LightCone”

Only plot one dimension of space for simplicity

Any point is an event, a line/curve connecting points is a worldline


SPACE-TIME DIAGRAMS

“LightCone”

time axis often renormalized and plotted as ct, so it has same dimensions as space axis


SPACE-TIME DIAGRAMS

“LightCone”

light beam follows a world line ct=x , using x versus ct - this is a line at 450

object B traveling at v<c has worldline > 450

object C would have to travel at v>c so impossible


SPACE-TIME DIAGRAMS

“LightCone”

Inertial Observers

Accelerated Observer


SPACE-TIME DIAGRAMS

“LightCone”


SPACE-TIME DIAGRAMS

“LightCone”

future

past

elsewhereelsewhere

Event A constrained to liewithin cones defined by lines equiv to v=c (45o)


SPACE-TIME DIAGRAMS

“LightCone”

in general r2= x2+ y2

s= √(ct)2 - (x)2

defines a spacetime interval

How do we define a ‘separation’ between two events on a worldline?


SPACE-TIME DIAGRAMS

“LightCone”

in general r2= x2+ y2

s= √(ct)2 - (x)2

a separation in Minkowski spacetime, a spacetime interval

-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

“LightCone”

s= √(ct)2 - (x)2 a separation in Minkowski spacetime

interval2 = (dist traveled by light in time t)2 - (dist between events)2

s2 > 0 timelike (light had more than enough time to travel between events)

s2 = 0 null (or lightlike, light had exactly enough time to travel between events)

s2 < 0 spacelike (not enough time for light to travel between the events)


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 Relativity 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

extragalactic astronomy & cosmology lecture sr1 jane turner joint center for astrophysics umbc...

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