RELATIVITY

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Updated 2012Aug30Dr. Bill Pezzaglia

The End of Physics?

The following statement made by a Nobel prize winning physicist:

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Albert Abraham Michelson

“The most important fundamental laws and facts of physical science have all been discovered, and these are now so firmly established that the possibility of their ever being supplemented in consequence of new discoveries is exceedingly remote.”

1903(before relativity and quantum mechanics were invented)

A. Special Relativity

1. Galilean Relativity2. Michelson Morley experiment3. Einstein’s postulates

3 1. Galileo ’s relativity 4

Motion is relative (Galileo)there is no experiment one can do to determine absolute motion relative to “space”.

Laws of physics must hold in all reference frames which differ only by a constant velocity

Velocity Addition Formula:

2. Michelson-Morley Experiment 5

It was thought that light needed a medium in which to propagate, called the LuminferousÆther, and that the speed of light was relative to the Æther.

Hence if the earth is moving relative to the Æther, the speed of light would appear different (i.e. obeys the Galilean velocity addition formula).

Michelson-Morley experiment (1881, 1887) instead showed speed of light was unchanged by motion of earth, and there was NO Æther.

• Einstein Questioned: If you were moving at the speed of light, could you see yourself in a mirror held out in front of you?

• He concludes:Speed of light is thesame for all observers

• How is this possible?Nature conspires to distortspace and time

3. Einstein 6

3b Einstein ’s Relativity 7

1905 Einstein (26 years old) publishes theory of special relativity

It has 2 postulates:

1. Motion is relative (Galileo)there is no experiment one can do to

determine absolute motion relative to “space”.

2. The speed of light (in vacuum) is the same for ALL observers

At face value, these two statements seem inconsistent. To make it work he has to propose some changes

3c Relativistic Addition of Velocities 8

• Speeds add: V=

• Adding anything to speed of light gives speed of light

3d. Time Dilation 9

(Einstein)

•A moving clock will appear to run slower

2

2

1c

v

tt

−

=′

Paradoxically, two observers each moving with a clock, sees the OTHER clock running slowly

Muon Decay

• Muons don’t “live” long: half of any group decay after about a millionth of a second

• Frisch (one of my professors!) and Smith, 1963, on Mt. Washington in NH: measure rate of cosmic muons at the top and bottom of the mountain: Far more survive from top to bottom than lifetime should allow

• Traveling at close to c, the muon’s “internal clock”: runs slowly compared to the physicist’s clock – its time is stretched out

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GPS: Relativity and You

• 24 Global Positioning Satellites, each with an atomic clock

– Each always transmitting its precise clock time

– Position found by times and known satellite positions

• Times must be known to within 20 to 30 billionths of a second.

• Viewed from Earth: moving clocks run slowly, losing about 7 millionths of a seconds per day.

• General Relativity: clocks run slowly near a massive object. Farther from Earth, satellite clocks run faster, gaining about 45 millionths of a seconds a day.

• Without accounting for Relativity, GPS would not work!

11 3e Lorentz Contraction 12

1889 FitzGerald, 1892 Lorentz

•Propose a moving meter stick will appear to shrink in length

L’ = L [1-v 2/c2]1/2

•1905 Einstein deduces this from his postulates of relativity.

3f. Mass 13

(Einstein)

•A moving mass will appear more massive

2

2

0

1c

v

mm

−

=

Famous Equation: E=mc2

• "It followed from the special theory of

relativity that mass and energy are both…

are but different manifestations of the same thing –

a somewhat unfamiliar conception for the average

mind. Furthermore, the equation E is equal (to)

m c squared, in which energy is put equal to mass, multiplied with the square of the velocity of light, showed that

very small amounts of mass may be converted into a very large

amount of energy and vice versa. The mass and energy were in

fact equivalent, according to the formula mentioned above. This

was demonstrated by Cockcroft and Walton in 1932,

experimentally."

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Making E=mc2 “More Correct”

• Energy E in two forms:

– Rest mass m0

– Momentum p (Kinetic)

• Total energy is a Pythagorean sum:

• Photons have no rest mass energy, only momentum energy

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pc

20cm

E

( ) ( )2220 pccmE +=

Is Mass Equivalent to Energy?

• Consider a deuterium nucleus, composed of a proton and a neutron:

Md = 2.01355 amu

• The proton and neutron have mass

Mp = 1.00728 amu

Mn = 1.00866 amu

• Something’s funny! …

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It Doesn’t Add Up!

• Mp + Mn:

1.00728 amu

+ 1.00866 amu

= 2.01594 amu

• That’s MORE than the mass of the deuterium nucleus Md = 2.01355 amu! The proton and neutron are heavier separately than when they’re together!

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Mass IS Equivalent to Energy!

• 2.01594 amu - 2.01355 amu = .00239 amu

• Energy equivalent to the mass difference is that energy required to bind the proton and neutron together.

• The equivalent energy is 20 times higher than that of x-rays used by your dentist

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3g Where to find E = mc2?

• Our Sun, in which fusion reactions exchange the mass of Hydrogen for energy in the form of light and solar wind

• The conversion of food mass into energy to make your body run

• The conversion of the mass of gasoline into energy to make your car run

19 B. General Relativity

1. Equivalence Principle

2. Bending of light

3. Bending of space, black holes

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There are 3 ways to think about mass

1. Definition of Mass 21

1. Inertial Mass F=ma

2. Passive Gravitational Mass F=mg

3. Active Gravitational Mass2r

GMg =

The “Weak Equivalence principle” says that inertial mass equals passive gravitational mass

1b. Galileo’s Experiment at Pisa

• 1590 Galileo’s Principle :All bodies fall at the same rate, regardless of mass

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• 1907 Weak EEP(Einstein Equivalence Principle)All bodies will follow same path, independent of internal structure (e.g. mass or composition)

1c. The Equivalence PrincipleReference at rest with Gravity is indistinguishable to a reference frame which is accelerating upward in gravity free environment.

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The apple accelerating downward due to gravity looks the same as an apple at rest in space, with the floor accelerating upward towards it.

2a. Bending of Starlight (Date?)

• Newton: Light is NOT affected by gravity• Einstein: Elevator example shows light must be

affected by gravity.• Predicts starlight will be bent around sun!• 1919 Measured by Eddington!

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Time is distorted by gravity

2b. Gravitational Redshift 25

1. Time runs slower in a gravity field (your feet are aging slower than your head)

2. Photons leaving the sun lose energy pulling away and are “redshifted”

3. If the mass is big enough, the escape speed becomes the speed of light, which means light cannot escape, i.e. it is redshifted out of existence (“black hole”)

2cr

GM≅∆λλ

3a. Curved Space

Einstein argued that mass curves space, and “gravity” is simply particles following the curves of space

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3b. Curvature and Black Holes 27

Einstein’s General Theory of Relativity

Gravity is “curved space”

Big curvature makes a Black Hole that you can fall in and never get out

For example, when people throw things into my tuba, they are never seen again.

3c. Schwarzschild RadiusIf any mass is compressed into a size smaller than the “Schwarzschild Radius”, it will become a black hole

This can happen during a supernovaexplosion, or later by additional massfalling on a neutron star.

Anything that comes closer than the Schwarzschild Radius, will fall in andnever escape.

Or, light from a black hole isredshifted out of existance

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12

1

1

2

−−

=∆

rc

GMλλ

3d. Observing Black Holes

They are black, how do we see them?Find one in a binary system. As mass falls into the black hole there will be some radiation released

29 3e. Jets

Not all the material falls intothe hole. Some is ejectedat very high energiesout “jets” along theaxis of the black hole.

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3f. Galactic Jets

Chandra X-Ray TelescopeSees jets coming out of galaxy M87, suggesting there is a BIG black hole at the center.

31 3g. Radio Lobes from galaxy Centaurus A

Again, suspect a big black hole in the center

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C. Cosmology

1. Newton’s static universe2. Einstein’s curved universe3. The expanding universe

33 1. Newtonian Cosmology

Newton proposed that the universe must be infinite to be balanced; a finite universe would collapse due to gravity

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But:• 1823 Olber shows universe is not infinite

• 1920 Eddington shows Newton’s Universeis Unstable and would collapse

• 1929 Inconsistent with Hubble’s Law (expanding universe)

2. Curved Finite Universe 35

1917 Einstein proposes universe:•Is finite, curved like a ball(this fixes Olber’s Paradox)

•But gravity would still collapse it

•Proposes negative pressure(cosmological constant) prevents collapse

•Later calls this his “biggest blunder”

2b Curved Space 36

If we live in a (positively) curved space, then no matter what direction we look, we might see that back of our head!

Its like living on the surface of a big ball.

This would explain why sky is dark at night.

3a Big Bang Theory 37

•1922 Friedmann, and 1927 Father Lemaitre show that another solution to Einstein’s equation would be that the universe is expanding from kinetic energy leftover from a “big bang” creation. No need for negative pressure.

•This would also explain the observed “redshifts”

3b Open vs Closed Universe 38

CLOSED UNIVERSE: Just like a ball thrown upward will fall back to earth due to gravity, we might expect the universe will slow down, and collapse.

OPEN UNIVERSE: If you throw a ball upward fast enough, it won’t fall back, but it certainly will slow down due to gravity

We expect that the universe must be decelerating due to gravity. Whether the universe is open or closed is only a question of how fast it is slowing down.

3c The Universe is Accelerating? 39

1998 Measurements of distant supernova (i.e. in the distant past) were 20% fainter than expected.

Interpretation: universe WAS slowing down for first half of lifetime

BUT, since then, it has been ACCELERATING

This is a big surprise

References

• Some material “borrowed” from Phil Kesten, Santa Clara University

• Bassett & Edney, “Introducing Relativity”, Icon Books (2002), p. 56

• Hartle, “Gravity”, Addison Wesley (2003)• D’Inverno, “Introducing Einstein’s Relativity”,

Clarendon Press (1992)• Misner, Thorne & Wheeler, “Gravitation”, Freeman &

Company (1973)

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