Transcript special relativity

Ta-Pei Cheng
talk based on …
Oxford Univ Press (2/ 2013)
Einstein’s Physics
Atoms, Quanta, and Relativity --- Derived,
Explained, and Appraised
2TOC
Albert Einstein
1879 – 1955
ATOMIC NATURE OF MATTER
1. Molecular size from classical fluids
2. The Brownian motion
QUANTUM THEORY
The book
explains his Physics
in equations
3. Blackbody radiation: From Kirchhoff to Planck
4. Einstein’s proposal of light quanta
5. Quantum theory of specific heat
6. Waves, particles, and quantum jumps
7. Bose–Einstein statistics and condensation
8. Local reality and the Einstein–Bohr debate
SPECIAL RELATIVITY
Today’s talk
provides
without
math details
some highlights
in
historical context
9. Prelude to special relativity
10. The new kinematics and E = mc2
11. Geometric formulation of relativity
GENERAL RELATIVITY
12. Towards a general theory of relativity
13. Curved spacetime as a gravitational field
14. The Einstein field equation
15. Cosmology
WALKING IN EINSTEIN’S STEPS
16. Internal symmetry and gauge interactions
17. The Kaluza–Klein theory and extra dimensions
3Atoms
Molecular size & Avogadro’s number
classical liquids with suspended particles
(4/1905) U
Zurich doctoral thesis: “On the determination of molecular dimensions”
(11 days later) the Brownian motion paper:
→ 2 equations relating P & N
viscosity and diffusion coefficients
Jean
Perrin
A to
While thermal forces change the direction and magnitude
Hydrodynamics Navier-Stokes equation, balance of osmotic and viscous forces
of the velocity of a suspended particle on such a small
E’s most cited publication!
time-scale that it cannot be measured, the overall drift
of such a particle is observable quantity.
x 2  2 Dt 
2 RT
t
N A 6 P
Fluctuation of a particle system
random walk as the prototype of discrete system
A careful measurement of this
zigzag motion through
a simple microscope would
allow us to deduce the
Avogadro number!
k   N
2
It finally convinced everyone, even the skeptics, of
the reality of molecules & atoms.
4Quanta 1
Einstein, like Planck, arrived at the quantum hypothesis thru BBR
Blackbody Radiation
(rad in thermal equilibrium) = cavity radiation
u(T ) &  (T , )
nd
Kirchhoff
(1860)
densities
Maxwell EM
radiation2 =law
a collection
of oscillators u = E2,=B2universal
~ oscillatorfunctions
energy kx2
) adiabatic
d
The ratio of oscillating energy to frequencyuis(Tan
; p = u/3
0  (T , )invariant
Stefan (1878) Boltzmann (1884):
Wien’s displacement law (1893):
u (T )  aT 4
 (T , )   3 f ( / T )
Wien’s distribution (1896):  (T , )   3e  / T fits data well…. until IR
Excellent fit
 3
of all the data
 (T , )   / T
Planck’s distribution (1900):
e
1
Wien = high ν of Planck
key: Wien 2 ->1 var
What is the physics? Planck found a relation   8c 3 2U  entropy dS  dU / T
What microstate counting W that can lead to this S via Boltzmann’s principle S=k lnW ?
Planck was “compelled” to make the hypothesis of energy quantization
  h
5Quanta 2
Einstein’s 1905 proposal of light quanta   h
was not a direct follow-up of Planck’s
  8c 3 2U and
Einstein used Planck’s calculation
invoked the equipartition theorem of stat mech
to derive the Rayleigh-Jeans law
  ν 2T
noted its solid theoretical foundation
showing
andclassical physics
BBR = clear challenge to
the problem of ultraviolet catastrophe
U  12 kT

u   d  
0
Rayleigh-Jeans = the low frequency limit of the successful Planck’s distribution
The high frequency limit (Wien’s distribution ) = new physics
concentrate on
Statistical study of (BBR)wien
entropy change due to volume change: (BBR)wien ~ ideal gas
→ (BBR)wien= a gas of light quanta with energy of   h
Einstein arrived at energy quantization independently---- cited Planck only in 2 places
6Quanta 2
An historical aside:
The history of Rayleigh–Jeans law:
• June 1900, Rayleigh, applying the equipartition theorem to radiation, he
obtained the result of C1ν2T . Only a limit law? Intro cutoff ρ = C1ν2T exp(-C2ν/T)
• October–December 1900, The Planck spectrum distribution was discovered;
energy quantization proposed two months later
• March 1905, Einstein correctly derived the R-J law
  8c-3ν 2 kT
noted its solid theoretical foundation and the problem of ultraviolet catastrophe
• May 1905, Rayleigh returned with a derivation of C1. But missed a factor of 8
• June 1905, James Jeans corrected Rayleigh’s error…
But, explained away the incompatibility with experimental results by insisting that
the observed radiation was somehow out of thermal equilibrium.
• A.Pais: “It should really be called Rayleigh-Einstein-Jeans law”.
“Planck’s fortunate failure”?
7Quanta 3
The quantum idea
Planck 1900:   h is only a formal
relation, not physical (radiation not
inherently quantum: only during
transmission, packets of energy, somehow)
Einstein vs Planck
Einstein 1905: as P’s W-calculation unreliable…
E’s quantum “in opposition” to P’s quantum
 Einstein:
the quantum idea must represent new physics;
1906 Einstein came in agreement with Planck’s.
proposed photoelectric effect as test.
Also, gave a new derivation of Planck’s law
It clearly explained why energy quantization
can cure ultraviolet catastrophe
K max  h  W
The new physics must be applicable beyond
BBR: quantum theory of specific heat
Einstein’s photon idea was strongly resisted by the physics community for many years
because it conflicted with the known evidence for the wave nature of light
(Millikan 1916): “Einstein’s photoelectric equation . . . cannot in my judgment be looked upon at
present as resting upon any sort of a satisfactory theoretical foundation”, even though
“it actually represents very accurately the behavior” of the photoelectric effect”.
Planck did not accept Einstein’s photon for at least 10 years
8Quanta 4
(1909)
Light quanta = particles ?
Einstein applied his fluctuatio n theory of 1904 :


 E
ΔE stated:
 Equanta
 E carried
 kT Photon carries energy + momentum
1st time
T
by point-like particles
 ahdeep
 2 
priddle
 h/
Wave-Particle
Duality:
2
“pointons
of view
to radiation distributi
E ofvˆ so that h as
ΔEconversion
 vˆkT factor wave  particle
T
Newtonian emission theory”
2
Rayleigh - Jeans' 
Wein' s 
E 2
E 2
Planck' s distribution
W
2
RJ

c
2
2
3
8
2
E
vˆd
~ waves :
Δu 2  u 2
 E h ~ particles : Δu 2  uh
ΔE 2
Planck
 ΔE 2
W
 ΔE 2
RJ
Neither just as waves or just as particles, but a " fusion" of the two!
9 Quanta 5
1916–17, Einstein used Bohr’s quantum jump idea to construct a
microscopic theory of radiation–matter interaction: absorption and
emission of photons (A and B coefficients); he showed how Planck’s
Looking beyond Einstein: His discoveries in quantum theory:
spectral distribution followed. The central novelty and lasting feature is
Wave/particle
natureinofquantum
light and
quantum jumps
the introduction
of probability
dynamics
can all be accounted for in the framework of
quantum field theory
Modern quantum mechanics :
states = vectors in Hilbert space (superposition)
observables = operators (commutation relations)
Classical radiation field = collection of oscillators
Quantum radiation field = collection of quantum oscillators
En  (n  2 )h
• A firm mathematical foundation for Einstein’s photon idea
• Quantum jumps naturally accounted for by ladder operators
aˆ  n ~ n  1
The picture of interactions broadened QFT description:
Interaction can change not only motion, but also allows for
emission and absorption of radiation
→ creation and annihilation of particles
1
aˆ , aˆ   h

particle
behavior
10Quanta 6
The riddle of wave–particle duality in radiation fluctuation
elegantly resolved in QFT
“three-man paper” of (Born, Heisenberg, and Jordan 1926):
forgotten history
The same calculation of fluctuation of a system of waves,
but replacing classical field by operators
Ae
i j
i
 i
 aˆ j  e j  aˆ j  e j
with
aˆ j  , aˆ k   h jk
Noncommuti ng aˆ  bring about an extra term
Δu 2  u 2  uh
Alas, Einstein never accepted this beautiful resolution
as he never accepted the new framework of quantum mechanics
11Quanta 7
Local reality & the Einstein-Bohr debate
Orthodox interpretation of QM (Niels Bohr & co): the attributes of a physical object
(position, momentum, spin, etc.) can be assigned only when they have been measured.
Local realist viewpoint of reality (Einstein,…): a physical object has definite attributes
whether they have been measured or not.
…. QM is an incomplete theory
The orthodox view (measurement actually produces an object’s property)
the measurement of one part of an entangled quantum state would instantaneously
produce the value of another part, no matter how far the two parts have been separated.
Einstein, Podolsky & Rosen (1935) : a thought experiment highlighting this
“spooky action-at-a-distance” feature ; the discussion and debate of “EPR paradox”
have illuminated some of the fundamental issues related to the meaning of QM
• Bell’s theorem (1964) : these seemingly philosophical questions could lead to
observable results. The experimental vindication of the orthodox interpretation has
sharpened our appreciation of the nonlocal features of quantum mechanics. Einstein’s
criticism allowed a better understanding of the meaning of QM.
• Nevertheless, the counter-intuitive picture of objective reality as offered by QM
still troubles many, leaving one to wonder whether quantum mechanics is ultimately
a complete theory
12SR 1
Special Relativity
Maxwell’s equations:
Contradict relativity? 2 inertial frames x’ = x - vt
EM wave – c
get velocity add’n rule
u’ = u - v
The then-accepted interpretation: Max eqns valid only in the rest-frame of ether
Q: How should EM be described for sources and observers
moving with respect to the ether-frame?
“The electrodynamics of a moving body”
1895 Lorentz’s theory (a particular dynamics theory of ether/matter)
could account all observation stellar aberration, Fizeau’s expt… to O(v/c)
v
t
'

t

x
[
+
a
math
construct
‘local
time
’]
x'  x  vt
2
c
Michelson-Morley null result @ O(v2/c2)  length contraction
Lorentz transformation Maxwell ‘covariant’ to all orders (1904)
Einstein’s very different approach ..
13SR 2
Special Relativity
Einstein’s very different approach ..
The magnet-conductor thought expt
constructive theory
vs
theory of principle
Relativity = a symmetry in physics
Physics unchanged under some transformation
How to reconcile (Galilean) relativity u’ = u - v
with the constancy of c?
Resolution: simultaneity is relative
Time is not absolute, but frame dependent
t'  t
Relation among inertial frames
Correctly given by Lorentz transformation,
with Galilean transformation as low v/c approx
Case I: moving charge in B (ether frame)
Lorentz force (per unit charge)
f
e
 v B
Case II: changing B induces an E via
Faraday’s law, resulting exactly the
same force. yet such diff descriptions
▪ Invoke the principle of relativity
This equality can be understood naturally
as two cases have the same relative motion
▪ Dispense with ether
14SR 3
Special Relativity 1905
From “no absolute time” to the complete theory in five weeks
The new kinematics
allows for an simple derivation of the Lorentz transformation.
All unfamiliar features follow from t ' . t
time dilation, length contraction, etc.
Transformation rule for EM fields, radiation energy,..
Lorentz force law from Max field equations
Work-energy theorem to mass-energy equivalence E = mc2
10yr
15SR 4
Special Relativity
Geometric formulation
Even simpler perspective
Hermann Minkowski (1907)
Essence of SR:
time is on an equal footing as space.
To bring out this, unite them in a single math structure, spacetime
Emphasizes the invariance of the theory: c → s
s 2  c 2 t 2  x 2  y 2  z 2  x g x 
metric
 1



1


g   
1 




1


s = a spacetime length
(c as the conversion factor)
Lorentz-transformation = rotation → SR features
4-tensor equations are automatically relativistic
SR: The arena of physics is the
4D spacetime
Einstein was initially not impressed,
calling it
“superfluous learnedness”
.. until he tried to formulate
General relativity (non-inertial frames)
= Field theory of gravitation
Gravity = structure of spacetime
SR = flat spacetime
GR = curved spacetime
16GR 1
Why does GR principle automatically
bring gravity into consideration?
How is gravity related to spacetime?
“Gravity disappears in a free fall frame”
The Equivalence Principle (1907)
played a key role in the formulation of
general theory of relativity
starting from Galileo Remarkable
empirical observation
All objects fall with the same acceleration
a ↑
=
↓ g
accelerated frame = inertial frame w/ gravity
Einstein: “My happiest thought”
EP as the handle of going from SR to GR
From mechanics to electromagnetism… → light deflection by gravity, time dilation
with such considerations...Einstein proposed a geometric theory of gravitation in 1912
gravitational field = warped spacetime
Note: A curved space being locally flat, EP incorporated in GR gravity theory in a fundamental way.
17GR 2
gravitational field = warped spacetime
Source
Sourceparticle
particle
Field eqn
Einstein
field eqn
1915
gN
T
G 
Newton’s constant
Curved Field
spacetime
G  g N T
energy momentum tensor
curvature tensor = nonlinear
2nd
derivatives of [gμν]
metric tensor [gμν] =
rela. grav. potential
Test
Test particle
particle
Eqn of
Geodesic
motion
Eqn
The Einstein
equation
10 coupled PDEs
solution = [gμν]
Metric = gravi pot
Curvature = tidal forces
In the limit of test particles moving with non-relativistic velocity
in a static and weak grav field
Einstein → Newton (1/r 2 law explained!)
ie new realms of gravity
18GR 3
GR = field theory of gravitation
3 classical tests
Grav redshift
Bending of light
Precession of planet orbit
In relativity, space-dep → time-dep, GR → gravitational wave
Indirect, but convincing, evidence thru decade-long observation of
Hulse-Taylor binary pulse system
Black Holes = full power and glory of GR
Gravity so strong that even light cannot escape
Role of space and time is reversed: lightcones tip over across the horizon
Alas, Einstein
never believed
the reality of BH
19 cosmo
Cosmology
(Einstein 1917)
The 1st paper on modern cosmology
The universe = a phys system
the constituent elements being galaxies
Gravity the only relevant interaction
GR = natural framework for cosmology
Spatial homogeneity & isotropy
(the cosmological principle) →
Robertson-Walker metric : k, a(t)
In order to produce a static universe he found
a way to introduce a grav repulsion in the
form of the cosmology constant Λ
G  g   g N T
Easier to interpret it as a vacuum energy:
constant density and negative pressure →
repulsion that increases w/ distance. –
significant only on cosmological scale
Λ = a great discovery
key ingredient of modern cosmology
Inflation theory of the big bang: a large Λ→
the universe underwent an explosive
derivatives a (t )  0 superluminal expansion in the earliest mo
Λ = dark energy → the U’s expansion to
Expanding Universe
accelerate in the present epoch
Einstein equation
GR provide the framework !
Still, Einstein missed the chance of its
prediction before the discovery in late 1920’s
The concordant ΛCDM cosmology
20 sym
Einstein and the symmetry principle
General relativity
Before Einstein, symmetries were generally regarded
as mathematical curiosities of great value to
crystallographers, but hardly worthy to be included
Rotation symmetry
among the fundamental laws of physics. We now
 principle
 is
understand that a symmetry

ˆ
vector transformatio n  A   A '  R A
not only an organizational device,




but alsoFamethod
to
discover
ma  0  F 'm'new
a '  dynamics.
0
curved spacetime with moving basis vectors
spacetime –dependent metric [g] = [g(x)]
general coord transf = spacetime dependent
ˆR
ˆ ( x)
R
 
   
 


ˆ
 RF  ma   0



Differentiation results in a non-tensor



dA'  Rˆ dA
Tensors have def transf property
Tensor equations are automatically
rotational symmetric.
Special relativity
R̂ = Lorentz transformation
4-tensor eqns are auto relativistic
Spacetime-independent R̂
Global symmetry
Local symmetry
ˆ 0
dR
Must replace by covariant differentiation
d  D  d      g 
DA '  Rˆ DA 
.
.
SR  GR with d  D
gravity is brought in
symmetry → dynamics
21gauge1
Einstein & unified field theory
Gauge principle:
the last 30 years of his life , strong conviction:
Regard ψ transf as more basic, as it can be gotten
by changing U(1) from global to be local.
GR + ED → solving the quantum mystery?
Was not directly fruitful, but his insight had
fundamental influence on effort by others:
Gauge theories and KK unification, etc.
But both made sense only in modern QM
d DdA
brings in the compensating field A ,
the gauge field
Gauge invariance of electrodynamics
E, B → A, Φ invariant under
Given A , Maxwell derived by SR+gauge
in quantum mechanics must + wf transf
i ( r ,t )
History:
  
A'  A   r, t 
 ' (r , t )  e
'    t  r, t 
 (r , t )
U(1) local transformation
Transformation in the internal charge space
“changing particle label”
Such local symmetry in a charge space is
now called gauge symmetry
ie the simplest
Electrodynamics as a gauge interaction
Gauge principle can be used to extend
consideration to other interactions
Inspired by Einstein’s geometric GR
1919 H Weyl attempt GR+ED unification via
Local scale symmetry [g’ (x)]= λ(x)[g(x)]
Calling it eichinvarianz
1926 V Fock, after the advent of QM,
discovered phase transf of ψ(x)
F London: drop “i” is just Weyl transf
Weyl still kept the name: gauge transf
22gauge2
Particle physics
Special relativity, photons, & BoseEinstein statistics = key elements
But Einstein did not work
directly on any particle phys theory.
Yet, the influence of his ideas had
been of paramount importance to
the successful creation of
the Standard Model of particle physics
Symmetry principle
as the guiding light.
ED is a gauge interaction based on
abelian (commutative) transf.
1954 CN Yang + R Mills extend it to
non-abelian (non-commutative)
Much richer, nonlinear theory, can
describe strong & weak interactions
Straightforward extension of QED ?
Quantization and renormalization of Yang-Mills th
extremely difficult. Furthermore, the truly relevant
degrees of freedom for strongly interacting particle
are hidden (quark confinement).
The applicability of gauge sym to weak int was
doubted because the symmetry itself is hidden
(spontaneous sym breaking due to Higgs mech)
1970’s renaissance of QFT → SM’s triumph
The Standard Model is a good example of
a theory of principle:
the gauge symmetry principle → dynamics,
as well as a constructive theory : discoveries of
* quarks and leptons,
* the sym groups of SU(2)xU(1) & SU(3)
follow from trial-and-error theoretical prepositions
and experimental checks
SM is formulated in the framework of QM
Holy grail of modern unification = [GR + QM]
23KK
Q: What is the charge space?
What’s the origin of gauge symmetry?
*Gauge transf = coord transf in extra D
Internal charge space = extra D
Kaluza-Klein theory
unification of GR+Maxwell
1919 Th Kaluza : 5D GR
extra dimension w/ a particular geometry [g]kk
GR5kk = GR4 + ED4
The Kaluza-Klein miracle!
In physics , even a miracle requires an explanation
1926 O Klein explained in modern QM
Foreshadowed modern unification theories.
GR + SM
the compactified space = multi-dimensional
*Compactified extra D → a tower of KK states
the decoupling of heavy particles
simplifies the metric to [g]kk
Einstein’s
influence lives on!
24
Summary of a summary
The fundamental nature of
Einstein’s contribution
illustrated by Planck unit system
h -- c -- gN
Natural units, not human construct
Dimensions of a fundamental theory
i.e. quantum gravity (GR + QM)
form an unit system of
mass/length/time
Fundamental
constants
nature
c 5 of these19
2
M shown
 1.2  10
GeV
P c  as conversion
factors
gN
connecting disparate phenomena
h: Wave & Particle (QT)
c: Space & Time (SR)
gN: Mass/energy & Geometry(GR)
lP 
g N
3
c
All due to
 1.6  10 33 cm
Einstein’s
essential
!
g N contribution
 44
tP 
c
5
 5.4  10
s
These PowerPoint slides are posted @
www.umsl.edu/~chengt/einstein.html