Relativity: History• 1879: Born in Ulm, Germany.• 1901: Worked at Swiss patent office.

– Unable to obtain an academic position.

• 1905: Published 4 famous papers.– Paper on photoelectric effect (Nobel prize).– Paper on Brownian motion.– 2 papers on Special Relativity. – Only 26 years old at the time!!

• 1915: General Theory of Relativity published.

• 1933: Einstein left Nazi-occupied Germany.– Spent remainder of time at Institute of

Advanced Study in Princeton, NJ.– Attempted to develop unified theory of gravity

and electromagnetism (unsuccessful).

Relativity

Thought Experiment - GedankenThe Special Theory of Relativity

Einstein asked the question “What would happed if I rode a light beam?”

• Would see static electric and magnetic fields with no understandable source.

• Electromagnetic radiation requires changing E and B fields.

• Einstein concluded that:

• No one could travel at speed of light.

• No one could be in frame where speed of light was anything other than c.

• No absolute reference frame

Einstein’s Postulates

• All inertial frames of reference are equivalent with respect to the laws of physics

• The speed of light in a vacuum always has the same value c, independent of the motion of the source or observer.

or• Nothing can move faster than the speed of light in a vacuum, which is the same with respect to all inertial frames

or• No experiment one can perform in a uniformly moving system in order to tell whether one is at rest or in a state of uniform motion. (No dependence on absolute velocity.)

Space-Time DiagramRequirement for 4 Dimensions

Definitions:

• Event: characterized by location (e.g., x,y,z) and time (t) at that location

• Space-time diagram: a coordinate system in which every point represents an event. 4 dimensions required.

ct

x

O

A (ct,x)

World line

• World line: trajectory of an event in the space-time diagram

Description of MotionIn a spacetime diagram, the motion of an object traces out aworld line.

For an object that moves at a constant velocity, a simple wayof measuring the velocity is to measure the positions of the object at two different times. Assume that the object movesfrom r1 at t1 to r2 at t2, the velocity of the object is then

)()( 1122

12

rtrtrr

v

We need a way of synchronizing the clocksat different locations!

Synchronization of Clocks

• Choose a reference clock and reset it to zero

• Generate a light pulse from the location of the reference clock• Set a local clock to the time that it takes for the light pulse to propagate from the location of the reference clock to the current location.

According the Einstein’s second postulate, no information canbe transmitted at a rate greater than the speed of light in vacuum. Since the speed of light is independent of inertial frames, it provides a natural (and ideal) way of sychronizing clocks.

The procedure can be described as follows:

Time Intervals: Simultaneous Events• Two events simultaneous in one reference frame are not

simultaneous in any other inertial frame moving relative to the first.

Two bolts seen simultaneously at C

Right bolt seen first at C’

Left bolt seen second at C’

Two lightning bolts strike A,B

Relativity of Simultaneity

ct

x

O

0v

A B Cx

ct

O

0v

A B C

vtxx i ct’

x’

Two events simultaneous in one inertial frame are not simultaneous in any other inertial frame moving relative to the first

ORClocks synchronized in one inertial frame are not synchronizedin any other inertial frame moving relative to the first

Light Clock• Light pulse bouncing between two mirrors

perpendicular to direction of possible motion

• A one way trip is one unit of time t = d/c

• Clearly moving light clock has longer interval between light round trips

Handy Light ClockHandy Light ClockConsider pulse of light bouncing between Consider pulse of light bouncing between

two mirrors (retroreflectors)two mirrors (retroreflectors)

dd

ttoo = d / c = d / c

Now Observe Same Clock movingNow Observe Same Clock moving

Thought Experiment Thought Experiment

Gedanken Gedanken ExperimentExperiment

Consider an inertial Consider an inertial frame of reference:frame of reference:

Elevator moving Elevator moving upward at a constant upward at a constant velocity,velocity, v v..

Moving Light ClockMoving Light Clock

Consider path of pulse of light in Consider path of pulse of light in moving frame of reference: Light moving frame of reference: Light ClockClock

dd

ttoo = d / c = d / c

ctctvtvt

Time Dilation calculatedTime Dilation calculated

Use Pythagorean Theorem:Use Pythagorean Theorem: (ct) (ct) 22 = d = d 22 + (vt) + (vt) 22

d d 22 = (ct) = (ct) 22 - (vt) - (vt) 22

d d 22/ c / c 22 = t = t 22 - (v - (v 22/ c / c 22)t )t 22

d / c = t [1 - (v d / c = t [1 - (v 22/ c / c 22)])]1/21/2

But d = ctBut d = ctoo , , SoSo

ddctct

vtvt

tto o = t = t [1[1- (v - (v 22/ c / c 22)] )] 1/21/2

The clock in the moving frame runs The clock in the moving frame runs slower.slower.

Time Dilation Observed!Time Dilation Observed!

Does this really work? Does this really work?

tto o = t = t [1[1- (v - (v 22/ c / c 22)] )] 1/2 1/2 t = t =ttoo

1.1. Mu-Mesons last longer before Mu-Mesons last longer before decaying if they are moving very fast. decaying if they are moving very fast.

by factor by factor = 1/ = 1/ [1[1- (v - (v 22/ c / c 22)] )] 1/21/2

2.2. Atomic Clocks run slower when Atomic Clocks run slower when moving. moving.

1 1 sec/1 000 000 sec at 675 mph.sec/1 000 000 sec at 675 mph.

Time Dilation: Derivation

In S’ frame, light travels up or down a distance D. In S frame, light travels a longer path along hypotenuse.

• Substitute t’ = D/c (proper time)

• Solve for t

Analyze laser “beam-bounce” in two reference frames

t 'D'

c

D

c

t D

c 2 v2

D

c

1

1 v 2 /c2

t '1

1 v 2 /c2t'

Time Dilation/Length Contraction: Muon Decay

• Why do we observe muons created in the upper atmosphere on earth? Proper lifetime is only = 2.2 s

travel only ~650 m at 0.99c

• Need relativity to explain!– Time Dilation: We see muon’s

lifetime as = 16 s.– Length Contraction: Muon sees

shorter length (by = 7.1)

LengthContraction

Muon’s frame

Earth’s frame

TimeDilation

Length Contraction

• Necessary consequence of postulates and for consistency of effects

• Can also derive in four dim. (ct, x, y, z) as rotation in a space-time plane preserving 4-D length, like rotation in a space-space plane preserve length

l2 x2 y2 z 23-D

4-D

PythagoreanTheorem

s2 ct 2 x 2 y2 z2 ct 2 l2

Relationship between Inertial Frames

x

ct

O

ct’

x’

O

Light Cone Unchanged

• If the speed of light is identical for all inertial frame observers, then the light cone must be unchanged.

Aberration of Light

• Discovered by Bradley in 1725 after seeing pennant on sailboat having direction intermediate to wind and boat motion.

Doppler Effect

Warp 0.92 (0.75c)

Relativity

Relativistic Increase in Mass• E = m0c2 = m0c2

• m = m0

v

E

v = c

E = m c2

22

0

1 cv

mm

/

Energy & Momentum

3-D Case 4-D Momentum

p m

v m0

v

E mc2 m0c2

p m0

u (E /c, px, py , pz )

p

2m0

2 u

2m0

2c 2

Energy and Momentum are separate in 3-D and have separate conservation laws.In 4-D are part of same vector and rotations preserve length (norm).

F

dp

dt

P dE

dt

F

v

F

dp

d

Rest Mass

• The rest mass m0 of a particle is an invariant. It is the length of the 4-D momentum vector.

p E /c, px , py , pz

p

2E 2 /c2 px

2 py2 pz

2 E 2 /c 2 p2

E 2 p2c2 m0c2 2

E 2 p2c2 m0c2 2

Einstein’s General Theory of Relativity predicts black holes

• Mass warps space resulting in light traveling in curved paths

Principle of Equivalence

A homogeneous gravitational field is completelyequivalent to a uniformly accelerated reference frame.

gi mm

It is impossible for us to speak of the absolute acceleration of the system of reference, just as the theory of special relativity forbids us to talk of the absolute velocity of a system.

Equivalence PrincipleConsider an observer in an elevator, in two situations:

1) Elevator is in free-fall. Although the Earth is exerting gravitational pull, the elevator is accelerating so that the internal system appears inertial!

2) Elevator is accelerating upward. The observer cannot tell the difference between gravity and a mechanical acceleration in deep space!

mi = mg

Uniformly Accelerating FrameUniformly Accelerating Frame

Light in Accelerating Frame of ReferenceLight in Accelerating Frame of Reference

acceleratioaccelerationn

Gravity?Gravity?

Time Dilation in Gravitational Field• Clock lower down runs slower

tB 1gh

c2

tA 1

c2

tA

eB / c 2

tB e A / c 2

tA

tB e / c 2

tA

Is it General Relativity right?• The orbit of Mercury is explained by

Relativity better than Kepler’s laws

• Light is measurably deflected by the Sun’s gravitational curving of spacetime.

• Extremely accurate clocks run more slowly when being flown in aircraft & GPS satellites

• Some stars have spectra that have been gravitationally redshifted.

If we apply General Relativity to a collapsing stellar core, we find that it can be sufficiently

dense to trap light in its gravity.

Several binary star systems contain black holes as evidenced by X-rays emitted

Cygnus X-1

must have a mass of about 7

times that of the Sun

Other black hole candidates include:

• LMC X-3 in the Large Magallenic Cloud orbits its companion every 1.7 days and might be about 6 solar masses

• Monoceros A0620-00 orbits an X-ray source every 7 hours and 45 minutes and might be more than 9 solar masses.

• V404 Cygnus has an orbital period of 6.47 days which causes Doppler shifts to vary more than 400 km/s. It is at least 6 solar masses.

Supermassive black holes exist at the centers of most galaxies

Supermassive black holes exist at the centers of most galaxies

Primordial black holes may have formed in the early universe

• The Big Bang from which the universe emerged might have been chaotic and powerful enough to have compressed tiny knots of matter into primordial black holes

• Their masses could range from a few grams to more massive than planet Earth

• These have never been observed• Mathematical models suggest that these might

evaporate over time.

How big is a black hole?

Matter in a black hole becomes much simpler than elsewhere in the universe

• No electrons, protons, or neutrons• Event horizon

– the shell from within light cannot escape

• Schwarzschild radius (RSch)– the distance from the center to the event horizon

• gravitational waves– ripples in spacetime which carry energy away from the black

hole

• The only three properties of a black hole– mass, angular momentum, and electrical charge

Structure of Schwarzschild Black Hole

Structure of a Kerr (Rotating)

Black Hole

In the Erogoregion, nothing can remain at rest as spacetime here is being pulled around the black hole

Structure of Kerr (Rotating Black hole

Falling into a black hole is an infinite voyage as gravitational tidal forces pull spacetime in such a way

that time becomes infinitely long

Black Hole Evaporation: Caused by virtual particles

Black holes evaporate

Virtual particles that appear in pairs near a event horizon may not be able to mutually annihilate each other if only one manages to survive a trip along the event horizon.

Summary• Special Relativity yields:

– Lost of universal simultaneity– Time dilation of moving systems– Length Contraction of moving objects– Equivalence of Mass and Energy– Integrated 4-Dimensional space-time

• General Relativity / Equivalence Principle– Curved Space-Time– Time Dilation in gravitational potential (curved time)– Bending of light and all inertial paths (no gravity)– Black Holes– Matter/Energy tells spacetime how to curve,

spacetime tells matter/energy how to move

Appendix: 4-D Vectors• Summary discussion of four dimensional (4-D) vectors

• Have vector algebra just like 3-D vectors but have 4 components instead of 3:

Transformations that leave length unchanged are the familiar:

1) Translations - displacements in space or time

2) Rotations - angular rotations

3) Velocity boosts (Lorentz transformation) which are equivalent to rotations in a space-time plane

3 D :x (x,y,z)

4 D :x (ct, x, y,z)

Lorentz TransformationsLeave 4-D vectors length Invariant

ct ' ct vc

x

x ' x vc

ct

y 'y

z'z

ct ct' vc

x'

x x ' vc

ct '

y y'

z z'

Examples of 4-D VectorsEasiest way to see that 4-D vector transforms like the prototype under Lorentz transformations is to construct them that way!

x (ct,x,y,z)

u

d

dx

dt

d(c,vx ,vy ,vz ) (c,vx,vy ,vz )

p m0

u m0(c,vx ,vy ,vz ) (E /c, px , py , pz )

a

d

du

d2x

d 2

Velocity Composition

ux 'ux v

1 uxv /c2

uy 'uy

(1 uxv /c2 )

uz 'uz

(1 uxv /c 2)

3-D Velocity 4-D Velocity

Lorentz Transformation

u

2c 2

Change of velocityis simply a rotation through and angle