special and general relativity: einstein‘s new views of ... · special and general relativity:...

1C. Kiesling, MPI f. Physics, Munich, Einstein Forum, Lanzhou, China, 11.10.2005

Special and General Relativity:Einstein‘s New Views of Space and Time

Max-Planck-Institute for Physicsand

Ludwig-Maximilians-University, Munich

Christian Kiesling


(December 1916)

Einstein‘s book for the public audience:


„The mathematical Principles of Natural Philosophy"

Knowledge

The first modern experimentalist

HypothesisExperiment

TheoryGalileo Galilei

1564-1642

Newton‘sMechanics :The first moderntheory

Isaac Newton1642-1727


Newton‘s Law of Gravitation

Third Law of Kepler

Johannes Kepler (1571-1630)

M

311

26.673 10m

Gkg s

− ⎡ ⎤⎢ ⎥= × ⎢ ⎥⋅⎣ ⎦

2m MF GR⋅=

Gravitational constant

R

22 34T RGMπ=

revolution time


Newtonian Mechanics, Galilei-Transformation

train (system S‘)x x ′= +Δ

embankment (system S)

train 0v v= =S

x

S‘x ′

S

x

S‘x ′

Δ(analogously for und ) y z

Transformation of space coordinates:

train (still) at rest:

Transformation of velocity:

dxvdt

=

U

U ′Ball 0w w′ ′= =

w w ′=t t′ =

dxw

dt

′′ =

′


( )x x t′= +Δ

w v=S

x

S‘

v

x ′

S

x

S‘

v

x ′

0w ′ =

x x vt′= +

( )t vtΔ =

t t′ =w w v′= +

Δ(t)

(constant velocity)

velocity of the ball, measuredfrom the embankment:

Newtonian Mechanics, Galilei-Transformation


Test of the Galilei-Transformation

0w ′ >

S

x

S‘

v

x ′

S‘

v

x ′

S‘

v

x ′

w w v′= +x x vt′= +

Now the ball is rolling in the trainagainst the train‘s direction ofmovement:let w v′ = −

Then the ball has to be at restin system S:

S

x

0w w v v v′= + = − + =

0w =

q.e.d.

The ball is rolling in the train along the train‘s direction of movement:


The Principle of Relativity (A. Einstein, 1905)

( )F q E v B= ⋅ + ×

„The laws of Physics are independent of the frame of reference“(the frames of reference have constant relative velocities)

Fadenstrahlrohr

S

x

S‘

v

x ′

Example: the Lorentz force

A current-carrying wire generatesa magnetic field

An electron moving in a wire‘s B-Feldis pushed towards the wire No force on the electron

in system S‘ ???


S

x

S‘

v

x ′

torch light

Example: the speed of light in different systems of reference

cc−c = speed of light

(~300.000 km/sec)

S

S

c c v

c c v+

−

= += −

???

Experiment by Michelson & Morley (1887)

„Light propagates with the same speedin all systems of reference“

(compare, e.g., air planes flying fromEast to West against West to East)

S

S

c c

c c+

−

==

The Principle of Relativity (A. Einstein, 1905)„The laws of Physics are independent of the frame of reference“


The Special Theory of Relativity (A. Einstein, 1905)

w w v′ = −x x vt′ = −

2 21 /

x vtx

v c

−′ =−

2

2 2

/

1 /

t vx ct

v c

−′ =−

21

w vw

w vc

′ += ′

+

2

2 21 /

mcE

v c=

−

2 21 /

mvp

v c=

−Speed of light c now thesame in each frame of

referencec is also the largest

possible velocity

v c

y y′ =

z z′ =

0 :v = 2E mc=

Galilei-Transformationno longer valid for largevelocities

Instead: Lorentz-Transformation

w ′S

x

S‘

v

x ′

145 10mm

−Δ = ×

350 km/h:


Effects of the Special Theory of Relativity (I)

Matter (and anti-matter) from energy

Electron und anti-electron („Positron“) are producedfrom light (very energetic light particles, or „Photons“)

2E mc=

Direction of flightof the photon

Particle trackscurve in a magnetic field(Lorentz force)


Time-Dilation

2 21 /

tt

v c

′ΔΔ =

−

S

x

S‘

vx ′

U

U ′ Moving watches are running slower !

Surface of the earth

Boundary of atmosphere (~10 km height)

μ Lifetime of the muons: 2.2 sτ μ=

Range ~ 8 63 10 2.2 10cτ −= × × ×660m=

Examples: The twin paradox Muons from cosmic rays

Effects of the Special Theory of Relativity (II)


Contraction of length

2 21 /x x v c′Δ = Δ −S

x

S‘

vx ′

x ′Δ

xΔ

Effects of the Special Theory of Relativity (III)

Cavity seen by themoving electrons

Cavity seen by an electron at rest

Relativistic contraction of length (Lorentz-contraction)


… and how a relativistic bicycle rider wouldperceive his environment ….


STR: The World has 4 Dimensions: Space (3) and Time (1)

x

z

tEx

Et

Ez

„Event E“

… the best is still to come !

theGeneral Theory of

Relativity


The extended („general“) Principle of Relativity

0dwdt

′≠

Sx

0dv

adt

= ≠

S‘x ′

Train is being accelerated:ball rolls backwards „by itself“

in a gravitationalfield

in an acceleratingrocket

… can one not always tell, which frame of referenceis the accelerated one ?

Einstein says: NO !!

(also in accelerated systems !!)„The laws of Physics are independent of the frame of reference“


Gravitation, Acceleration: The Principle of Equivalence

inertF m a= ⋅heavF m g= ⋅

inert heavm m=

heav

inert

ma gm

= ⋅

„Principle of Equivalence“

masses move oncurved trajectoriesin a GF

Straight linetrajectories:„free fall“

„No difference between accelerationand graviational field (GF)“:

General Relativity (GR)1915


Gravitation: Curved Trajectories in Space

Gravitational pull at the earth‘s surface: Mass M of Earth, distance R from the earth‘s center

2 29.81M m

g GR s

⎡ ⎤= = ⎢ ⎥

⎣ ⎦

Very different curvatures !

Free fall in the gravitational field:a curved trajectory !

Can they bededucedfrom a commonprinciple(„curvature of space-time“) ?

F m g= ⋅


ct

Space-Time has aunique (local) curvature

„Worldline“ of the ball

Space and Time are united (4-dimensional World)

world line of the bullet

GTR: also light „falls“ in thegravitaional field, here:

[ ]151.4 10 m−×

lightray

10m

Matter curvesspace-time !


Light rays also move on curved trajectories …Marvellous experiment to verify GTR:

Arthur Eddington (& A. Crommins)Isle of Principe (Gulfe of Guinea, close to the coast of West Africa)

Einstein‘s prediction: 1.75 arc seconds

Total eclipse onMai 29, 1919


An English Castle and the General Theory of Relativity …

… how it would look like, if the mass of planet Saturn, compressed to 10 m, stoodin front of it …

viewed through a „gravitational lens“


… and here a „real“ gravitational lens …

The little blue „cigars“ arethe gravitationally distortedpictures of a single galaxybehind a massive clusterof galaxies in theforeground.

(picture taken by theHubble Space Telescope)


GTR: the speed of clocks in a gravitational field(and in accelerated frames of reference)

L

Clock A sends one light pulse per second,received by clock B

1 1 /t L c=1. pulse needs time

2 2 /t L c=2. Pulse needs time

1L L< , because rocket has moved

2 1L L< , since rocket accelerates

Result: Clock A runs FASTER than clock B,(viewed from clock B).

GTR: same things happen for clocksin a gravitationsal field

Clocks on ground run slower thanclocks at high altitude


GTR: Without it GPS would not work …Satellite-based„Global Positioning System“

Principle of GPS:Determination of location via time measurement of signals from satelliteto location (A).

satellite sends:own position, own clock reading

runtime: comparison of satellite clockwith own clock in A


Erde

~ 26000km

[ ]3900 /v m s≈ 2 effects: STR und GTR

satellite flies „fast“according to STR, its clock is running slowerthan the clock in A (viewed from A)

A

2 2 101 1 / 0.835 10 st t

v ct

−′Δ −Δ = − − = ×Δ

satellite flies „high“According to GTR, its clock is running fasterthan the clock in A (viewed from A)

105.28 10 st tt

−′Δ −Δ =− ×Δ

6 times as big!

[ ] [ ]13.3 cm sl tΔ = ×ΔError without GTR: (correponds to 500 m in one hour of measuring time!!!)

GTR: Without it GPS would not work …


What Einstein did not foresee: The Expanding Universe

Edwin Hubble, 1929:All light from distant galaxiesis red-shifted („Doppler-effect“)

Einstein‘s equations of GRlead to a (non-static) expandingor contracting universe.

Einstein later:„The greatest blunder of my life“

The Universe expands !

1 172 8[ ]H kms Mpc− −= ±

He dismissed such a modeland introduced an additional„Cosmological Term“ intohis equations

Λ

v H d= ⋅ Hubble constant H


0( ) ( )

( )

r t r R t

RH t

RdR

Rdt

=

=

⎛ ⎞⎟⎜ = ⎟⎜ ⎟⎝ ⎠

From Einstein‘s equations of GR: 2 2

2

2

83 3

4 33 3

R G KcR R

R G pR c

πρ

πρ

⎛ ⎞ Λ⎟⎜ = − +⎟⎜ ⎟⎜ ⎟⎝ ⎠

Λ⎛ ⎞⎟⎜= − + +⎟⎟⎜⎝ ⎠

( )r t

ρG

p

Newton‘s constant

energy (matter) density

pressure

0r

curvature parameter


Dark Matter in the Galaxies

Andromeda nebula, distance about 2 Mio light years

Rotational velocity of stars about 200 km/sec, Independent of radius

Galactic disc apparentlyembedded in a “halo” ofnon-luminous matter

New formof matter ?


Distant Supernovae

Supernovae arefarther away thanexpected frompresent rate of expansion of theUniverse

Expansionaccelerates !

White dwarfRed giant

Crab nebula,discovered in 1054by Chinese astronomers


The Cosmic Microwave Background Radiation

…originating about 380.000 years afterthe Big Bang , 3000 °K

red-shifted over 13 Gy:2.72 °K

very isotropic (< 10-5)

discovered in 1964by A. Penzias andR. Wilson


Expansion History of the Universe

13.7GyBig Bang


A New Form of Energy:

Dark Energy

CMB measurements:curvature of space-timeis close to zero(„flat Universe“):

„critical“ energy density3

crit. 4 protons/mρ ∼

crit.M

ρρ Λ

Ω = = Ω +Ω

1.02 0.02Ω = ±


:KR R=[ ]151.4 10 m−×

photon

gravitationalfield of Earth

10m

ctz

KR2

Mg G

R=

light cannot escape from such a stronggravitational field „black hole“

Examples

[ ]1SR cm=Earth:

[ ]1.5SR km=Sun:

View into the center of theMilky Way

83.6 10M ×∼ solar masses

„Schwarzschild-Radius“

Speculation: a massive Black Hole is loomingin the center of the Milky Way

2

2MR G

c=

SR

Gravity in the Extreme: Black Holes


Black Holes

The Center of our Milky Way

… within a few light yearsseveral 10.000 stars,circulating the galacticcenter (on Kepler-orbits).

The revolution times allowmeasuring the mass in thecenter

Result: a super-massiveBlack Hole in the center

the speculation : mostgalaxies have Black Holesin their centers


electriccharge

0

0

0

±1

electriccharge

+2/3

-1/3

0

-1

„particles“: Spin ½(Fermions)

TheThe MicroMicro--CosmosCosmos: : ElementaryElementary ParticlesParticles

Masses of particles (in GeV):

0.1 1.5 175 0

0.1 0.5 5 0

>0 >0 >0 91

0.0005 0.1 1.8 80

„fields“:Spin 1(Bosons)

„Standard Model“of Particle Physics

remarkable symmetry !

Leptons


The Fundamental Forces of Nature

Electroweak Interaction,Prediction of the Z0

“Grand Unified Theories”, prediction of Leptoquarks

Theory Of Everything ?

Fieldquantum

Range

Effects Gravity Radioactivity Electricity Nuclear Force

10-38 10-5 10-2 1

∞ ∞10-15 cm 10-13 cm

“Graviton” W + , W - Photon γ Gluon g

Gravitation Weak Electromag. Strong

Strength


General Relativity and Quantum Mechanics

"Quantum mechanics is certainly imposing. But an inner voice tells me that it is not yet the real thing. The theory says a lot, but does not really bring us closer to the secret of the „Old One“. I, at any rate, am convinced thatHe is not playing at dice."

Einstein and Quantum Mechanics:

Macro-Cosmos (Universe):

Micro- Cosmos (particle physics):(or Gravitiy in extreme cases !) probabilistic theory

Quantum Mechanics

deterministic theory

General Relativity

xx p hΔ ⋅Δ ≥Heisenberg‘sUncertainty Relation

Both theories mutually incompatible


Combine QM and GR: Superstring Theory

Relativistic Quantum Mechanics („Quantum Field Theory“):

particles (quarks, leptons) are „points“,have no spacial extension

Superstring Theory: particles are „extended strings“(open or closed)

time

Size of strings: 34~ 10 [ ]m−SST only consistent in10 space-time dimensions!

(6 space dimensions must be„curled up“ to describe thereal world)

Can we measure such shortdistances?

with present accelerators: 1810 [ ]m−>

time

e+

e−

μ+

μ−


… The next Generation of Particle Colliders

LHC @CERN: 14 TeVpp 2007

ILC

>2015

: 500 GeVe e+ −

Do supersymmetric particles exist?

Are there extra dimensions of space?

What is Dark Matter ?

Einstein‘s dream of Unified ForcesDo all forces become one?

Can we solve the Mystery of Dark Energy?

How did the Universe come to be?

Where did all the antimatter go? … many open questions,waiting to be answered……..


proton-protoncollisionsat the LHC

(starting in2007)

The next setof answers:

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