ISP209s10 Lecture 7 -1-
Today
• HW#4 pushed back to 8:00 am Thursday
• Exam #1 is on Thursday Feb. 11
– Bring a calculator and a #2 pencil
– Allowed 1 page notes (front and back)
• E=mc2, General Relativity, and exam review
ISP209s10 Lecture 7 -2-
Comments on Exam 1
• 40 questions, multiple choice
• Bring #2 pencil and calculator
• Level of difficulty ! HW sets
• Extra credit opportunity if you bomb
– In LONCAPA, take a retake of a similar 40 point exam
– For each point higher than original in-class score, Iwill add .30 points to your original score
say you got a 30/40 on the exam, and a 40/40 on theretake. Then you would get (40-30)*0.3 = 3 extrapoints for an adjusted score of 33/40.
ISP209s10 Lecture 7 -3-
Special Relativity: Time, Length, Mass
• Time Dilation: “Moving clocks slow down”
– Time slows down
• Length Contraction: “Moving rulers shrink”
– Lengths in the direction of motion are shorter
than when at rest.
• Mass increases the faster you go (but you cannever reach “c” since mass would be infinite)
ISP209s10 Lecture 7 -4-
A few words about Energy
• Energy is the ability to do work. It comes intwo main types
– Kinetic energy: the energy of motion
– Potential energy: the energy of position
• Work = force x distance (it’s a scalarmeasured in Joules, J (same as Nm))
• Power is the rate work is done (it’s a scalarmeasured in J/s (same as a Watt W)
ISP209s10 Lecture 7 -5-
• Any increase in a system’s energy results in an
increase to its mass!
• The change in mass is equal to the change in
energy, divided by the square of the speed of light.
• The speed of light is very large, so the change in
mass is undetectable in ordinary situations.
• This idea (that mass can “change”) was very odd
at the time.
E = mc2: What it means
ISP209s10 Lecture 7 -6-
E = mc2 in action
Schematic picture of a chemical or nuclear reaction:
Start with some
initial mass (kg)
Something happens End up with some
final mass (kg)
Some fraction, f, is
converted to energy
ENERGY (Joules, J)
The amount of energy is E = mconverted c2
mconverted = (Minitial - Mfinal) = Minitial x fraction converted
ISP209s10 Lecture 7 -7-
Fraction of Energy Converted
• In most reactions not all the mass can be converted to
energy. Actually only a very small fraction (the exact value
of the fraction depends on the chemical/nuclear reaction).
Releasing a compressed
spring
1x10-15Mechanical
Burning coal1x10-10Chemical
Nuclear power plant0.001Fission
Power source of the Sun0.007Fusion
happens at particle
accelerators
1Matter-Antimatter Annihilation
ExampleFractionReaction
ISP209s10 Lecture 7 -8-
Some Samples
• A power plant generates 500 MW of electrical power and
700 MW of waste heat (plants always make more waste
heat than electrical power). How many Joules of energy
does the plant generate in 1 day? Data: 1 Watt = 1 Joule/s
dd
hr
hr
m
m
s1
246060
s
J100021
day ain seconds700MW)(500MW day) (1Energy
6!!!!!=
!+=
J 1.04E14 day) 1in (producedEnergy Electrical =
ISP209s10 Lecture 7 -9-
More on the power plant
Assume the power plant in the previous problem
burns 2.2 kg of oxygen and 1 kg of carbon from coal
to make 33 MJ of energy. How many kg of carbon
and oxygen will the plant use in a day?
( )
kgE701.1
kg 1.0kg 2.2J 33.E6
J .037E141 (kg) mass
kggeneratedenergy
energy wasteelectrical
kggeneratedenergy
producedenergy total (kg) mass
=
!"#$
%&
+
=
!"#$
%&
+=
!"#$
%&
=
ISP209s10 Lecture 7 -10-
How much of that mass was converted to energy?
( )kg
smE
JEm
c
EmcmE
converted
converted
3
2
2
2
converted
1016.1
83
1404.1 !"==
=#=
But we used more than 107 kg (10,000 metric tons), where did it all go?
Hint: The main byproduct of burning coal is C02 .
ISP209s10 Lecture 7 -11-
HW Help: How long will the Sun burn?
( )kgEs
mE
JE
f
cE
f
mm converted
seachburned 906.6007.0
83
2482.32
2
====
The sun generates 3.82E24 W ofpower by fusion of hydrogen intohelium. The fraction of massconverted for fusion is 0.007. Howmany kg of protons and electronsdoes the Sun use every second?
Years Sun will last = (Total mass of the core/mass used per second) x(years/s)
Note: 1 year = 3.156E+7 s
Get E by using definition of power
(recall, 1 W = 1 J/s)
ISP209s10 Lecture 7 -12-
Einstein’s thought experiment
Here on earth, we feel the effects of gravity
ISP209s10 Lecture 7 -13-
Einstein’s Simple Question
(Deep in outer space)Would you be able to tell the
Difference between this situation
(where you NOT in a gravitational
Field) and accelerating upwards
at 9.8 m/s2 from when you are
On earth experiencing the earth’s
Gravity?
ISP209s10 Lecture 7 -14-
Another one of Einstein’s simple questions
The equivalence principle:
No experiment performed inside a
closed room can tell you whether
you are at rest in the presence of
gravity or accelerating in the
absence of gravity.
ISP209s10 Lecture 7 -15-
Another one of Einstein’s simple questions
Scott throws
a ball with an
initial horizontal
Velocity.
The ball follows the
curved path due to gravity
till it hits the ground
ISP209s10 Lecture 7 -16-
Another one of Einstein’s simple questions
Throw a ball on
a rocket in space that is
uniformly accelerating
at 9.8 m/s2 (= 1g)
Ball follows same
Trajectory it did when
I threw it on earth.
ISP209s10 Lecture 7 -17-
The Equiv. Principle => Gravity Bends Light!
NOTE: You would need a HUGE acceleration >> 1g to see this effect.
This is why we don’t see flashlight beams falling to the floor!ISP209s10 Lecture 7 -18-
General Relativity continued
• Main Postulate: Acceleration in one direction is like
gravity in the other direction. It is not possible
distinguish the two. “Equivalence Principle”
• What we perceive as gravity is really acceleration
resulting from space curved and stretched by mass
• Mass warps space
• Space and time are combined into a 4-dimensional
space-time
ISP209s10 Lecture 7 -19-
Pictorial
Gravity is actually the result of warped space. What we
perceive as acceleration (and hence say is due to a force) is
really just stretched and curved space.
ISP209s10 Lecture 7 -20-
Einstein Equation
ijijijij Tc
GgRgR
4
8
2
1 !" =##
• A rank-2 tensor equation that describes how space-
time is influenced by mass.
•Approximately, the left side is the curvature and
motion of space and the right side is the location and
motion of mass and energy.
• Rij is the Ricci tensor (curvature of spacetime), g is
the metric of space-time (specifies how space and time
are interrelated), G is the same constant in Newton’s
equation, etc.
ISP209s10 Lecture 7 -21-
Einstein Equations predict “Black holes”
A hole so deep that even light can not escape.
Growing experimental evidence…
www.ifa.hawaii.edu/~barnes/ast110_06/bhaq.html
A huge mass in a tiny spacetime region warps it
so strongly that it has “infinite curvature” aka a
“spacetime singularity” develops.
Einstein didn’t believe
in them. He thought it
was an “artifact” of the
mathematics.
ISP209s10 Lecture 7 -22-
Einstein Equation also predicts “Wormholes”
This could be the basis for a time machine. Wormholes emerge
as mathematical consequence of Einstein’s Equations. Unlike Black
Holes, there is ZERO evidence or hint that they are real.
ISP209s10 Lecture 7 -23-
Gravitational Time Dilation
• Mass stretches space, but since space and time areconnected (space-time) it also affects time.Mathematically, this interconnection of space and timeis described by the “metric of spacetime”.
• Near a mass, time runs more slowly. On the surface ofthe Earth this affect is only 10-9 s, but near a black holeit could be infinite!
• Why? As you travel through space you travel throughtime. Where space is stretched, time is stretched.
ISP209s10 Lecture 7 -24-
Proof of General Relativity I
• Bending of star light – the gravitational field of
the Sun bends star light by 1.75 arcseconds.
This was observed by A. Eddington in 1919
during an eclipse.
ISP209s10 Lecture 7 -25-
Proof of General Relativity II
Gravitational Lensing:
Routinely observed
and used to measure
the mass of distant
clusters of galaxies.
ISP209s10 Lecture 7 -26-
Real picture from the Hubble Telescope
Abel galaxy cluster
ISP209s10 Lecture 7 -27-
Paradoxes in Time Travel
• If time is a dimension like the other three, can wemove back and forth in time?
• Forward, yes! Think of the Twin Paradox. What aboutbackwards in time?
• If we can travel back in time, it would be possible forus to influence things so that we are not born.
• Two theories to resolve the paradox
– Travel back in time is not possible. End of story
– There are a very large number of parallel universes (stringtheories)
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