Today • HW#4 pushed back to 8:00 am Thursday Exam #1 is on...

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


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, I

will 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.


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)


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)


• Any increase in a system’s energy results in anincrease to its mass!

• The change in mass is equal to the change inenergy, divided by the square of the speed of light.

• The speed of light is very large, so the change inmass is undetectable in ordinary situations.

• This idea (that mass can “change”) was very oddat the time.

E = mc2: What it means


E = mc2 in action

Schematic picture of a chemical or nuclear reaction:

Start with someinitial mass (kg)

Something happens End up with somefinal mass (kg)

Some fraction, f, isconverted to energy

ENERGY (Joules, J)

The amount of energy is E = mconverted c2

mconverted = (Minitial - Mfinal) = Minitial x fraction converted


Fraction of Energy Converted• In most reactions not all the mass can be converted to

energy. Actually only a very small fraction (the exact valueof the fraction depends on the chemical/nuclear reaction).

Releasing a compressedspring

1x10-15Mechanical

Burning coal1x10-10Chemical

Nuclear power plant0.001Fission

Power source of the Sun0.007Fusion

happens at particleaccelerators

1Matter-Antimatter Annihilation

ExampleFractionReaction


Some Samples

• A power plant generates 500 MW of electrical power and700 MW of waste heat (plants always make more wasteheat than electrical power). How many Joules of energydoes 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 =


More on the power plant

Assume the power plant in the previous problemburns 2.2 kg of oxygen and 1 kg of carbon from coalto make 33 MJ of energy. How many kg of carbonand 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

=

!"#$

%&

+

=

!"#$

%&

+=

!"#$

%&

=


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 .


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)


Einstein’s thought experiment

Here on earth, we feel the effects of gravity


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 upwardsat 9.8 m/s2 from when you are On earth experiencing the earth’sGravity?


Another one of Einstein’s simple questions

The equivalence principle:

No experiment performed inside aclosed room can tell you whetheryou are at rest in the presence ofgravity or accelerating in theabsence of gravity.



Scott throwsa ball with an initial horizontalVelocity.

The ball follows the curved path due to gravitytill it hits the ground



Throw a ball ona rocket in space that isuniformly acceleratingat 9.8 m/s2 (= 1g)

Ball follows sameTrajectory it did whenI threw it on earth.


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!


General Relativity continued

• Main Postulate: Acceleration in one direction is likegravity in the other direction. It is not possibledistinguish the two. “Equivalence Principle”

• What we perceive as gravity is really accelerationresulting from space curved and stretched by mass

• Mass warps space• Space and time are combined into a 4-dimensional

space-time


Pictorial

Gravity is actually the result of warped space. What weperceive as acceleration (and hence say is due to a force) isreally just stretched and curved space.


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 andmotion of space and the right side is the location andmotion of mass and energy.• Rij is the Ricci tensor (curvature of spacetime), g isthe metric of space-time (specifies how space and timeare interrelated), G is the same constant in Newton’sequation, etc.


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 itso strongly that it has “infinite curvature” aka a“spacetime singularity” develops.

Einstein didn’t believe in them. He thought it was an “artifact” of themathematics.


Einstein Equation also predicts “Wormholes”

This could be the basis for a time machine. Wormholes emergeas mathematical consequence of Einstein’s Equations. Unlike Black Holes, there is ZERO evidence or hint that they are real.


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.


Proof of General Relativity I

• Bending of star light – the gravitational field ofthe Sun bends star light by 1.75 arcseconds.This was observed by A. Eddington in 1919during an eclipse.


Proof of General Relativity II

Gravitational Lensing:Routinely observedand used to measurethe mass of distantclusters of galaxies.


Real picture from the Hubble Telescope

Abel galaxy cluster


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 (string

theories)

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