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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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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 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.
Jane Turner [4246] PHY 316 (2003 Spring)
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.
Jane Turner [4246] PHY 316 (2003 Spring)
Special Relativity
Einstein’s postulates
Time dilation
Length contraction
New velocity addition law
Jane Turner [4246] PHY 316 (2003 Spring)
THE SPEED OF LIGHT PROBLEM
“Relativity” tells us how to relate measurements in different frames.
Galilean relativity
Simple velocity addition law : vtotal=vrun+vtrain
Jane Turner [4246] PHY 316 (2003 Spring)
Einsteins Postulates
Einstein threw away Galilean Relativity
Came up with two “Postulates of Relativity”
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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.
Jane Turner [4246] PHY 316 (2003 Spring)
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”
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
Length: Lorentz Contraction
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...
Jane Turner [4246] PHY 316 (2003 Spring)
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 of the station master, who sees them separated by the maximum length anyone ever will, the “proper length”
Jane Turner [4246] PHY 316 (2003 Spring)
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!
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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.
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)]
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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 ++→+
Jane Turner [4246] PHY 316 (2003 Spring)
From web site ofGeorgia State University
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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!
Jane Turner [4246] PHY 316 (2003 Spring)
Particle production in a particle accelerator Can reproduce conditions similar to early universe in modern
particle accelerators…
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
SPACE-TIME DIAGRAMS
“LightCone”
time axis often renormalized and plotted as ct, so it has same dimensions as space axis
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
SPACE-TIME DIAGRAMS
“LightCone”
Inertial Observers
Accelerated Observer
Jane Turner [4246] PHY 316 (2003 Spring)
SPACE-TIME DIAGRAMS
“LightCone”
Jane Turner [4246] PHY 316 (2003 Spring)
SPACE-TIME DIAGRAMS
“LightCone”
future
past
elsewhereelsewhere
Event A constrained to liewithin cones defined by lines equiv to v=c (45o)
Jane Turner [4246] PHY 316 (2003 Spring)
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?
Jane Turner [4246] PHY 316 (2003 Spring)
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
Jane Turner [4246] PHY 316 (2003 Spring)
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)
Jane Turner [4246] PHY 316 (2003 Spring)
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