EEL 5930 sec. 5, Spring

1
EEL 5930 sec. 5, Spring 05Physical Limits of
Computing
http//www.eng.fsu.edu/mpf

• Slides for a course taught byMichael P. Frankin
  the Department of Electrical Computer
  Engineering

2
Review of Basic Physics Background

• (Module 2)

3
Basic physical quantities units

• Unit prefixes
• Basic quantities
• Units of measurement
• Planck units
• Physical constants

4
Unit Prefixes

• See http//www.bipm.fr/enus/3_SI/si-prefixes.html
  for the official international standard unit
  prefixes.
• When measuring physical things, these prefixes
  always stand for powers of 103 (1,000).
• But, when measuring digital things (bits bytes)
  they often stand for powers of 210 (1,024).
• See also alternate kibi, mebi, etc. system at
  http//physics.nist.gov/cuu/Units/binary.html
• Dont get confused!

5
Three fundamental quantities
6
Some derived quantities
7
Electrical Quantities

• Well skip magnetism related quantities this
  semester.

8
Information, Entropy, Temperature

• These are important physical quantities also
• But, are different from other physical quantities
• They are based on combinatorics and statistics
• But, well wait to explain them till we have a
  whole lecture on this topic later.
• Interestingly, there have been attempts to
  describe all physical quantities entities in
  terms of information (e.g., Frieden, Fredkin).

9
Unit definitions conversions

• See http//www.cise.ufl.edu/mpf/physlim/units.txt
  for definitions of the above-mentioned units,
  and more. (Source Emacs calc software.)
• Many mathematics applications have built-in
  support for physical units, unit prefixes, unit
  conversions, and physical constants.
• Emacs calc package (by Dave Gillespie)
• Mathematica
• Matlab - ?
• Maple - ?
• You can also do conversions using Google or using
  other web-based calculators.

10
Some fundamental physical constants

• Speed of light c 299,792,458 m/s
• Plancks constant h 6.626075510?34 J s
• Reduced Plancks constant ? h / 2?
• Remember this with the analogy (h 360) (?
  1 radian)
• In fact, later well see its valid to view h,?
  as being these angles.
• Newtons gravitational constant G
  6.6725910?11 N m2 / kg
• Boltzmanns constant k kB log e
  1.380651310?23 J / K
• Others permittivity of free space,
  Stefan-Boltzmann constant, etc. to be introduced
  later, as we go along.

11
Physics that you should already know

• Basic Newtonian mechanics
• Newtons laws, motion, energy, etc.
• Basic electrostatics
• Ohms law, Kirchoffs laws, etc.
• Also helpful, but not prerequisite (well
  introduce them as we go along)
• Basic statistical mechanics thermodynamics
• Basic quantum mechanics
• Basic relativity theory

12
Generalized Classical Mechanics
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Generalized Mechanics

• Classical mechanics can be expressed most
  generally and concisely in the Lagrangian and
  Hamiltonian formulations.
• Based on simple functions of the system state
• The Lagrangian Kinetic minus potential energy.
• The Hamiltonian Kinetic plus potential energy.
• The dynamical laws can be derived from either of
  these energy functions.
• This framework generalizes to be the basis for
  quantum mechanics, quantum field theories, etc.

14
Euler-Lagrange Equation
Note the over-dot!
or just

• Where
• L(q, v) is the systems Lagrangian function.
• qi Generalized position coordinate w. index
  i.
• vi Generalized velocity coordinate i,
• or (as appropriate)
• t Time coordinate
• In a given frame of reference.

15
Euler-Lagrange example

• Let q (qi) (with i ? 1,2,3) be the ordinary
  x, y, z coordinates of a point particle with mass
  m.
• Let L ½mvi2 - V(q). (Kinetic minus potential.)
• Then, ?L/?qi - ?V/?qi Fi
• The force component in direction i.
• Meanwhile, ?L/?vi ?(½mvi2)/?vi mvi pi
• The momentum component in direction i.
• And,
• Mass times acceleration in direction i.
• So we get Fi mai or F ma (Newtons 2nd law)

16
Least-Action Principle
A.k.a.Hamiltonsprinciple

• The action of an energy quantity means the
  integral of that quantity over time.
• The trajectory specified by the Euler-Lagrange
  equation is one that locally extremizes the
  action of the Lagrangian
• Among trajectories s(t)between specified
  pointss(t0) and s(t1).
• Infinitesimal deviations from this trajectory
  leave the action unchanged, to 1st order.

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Hamiltons Equations

• The Hamiltonian is defined as H vipi - L.
• Equals Ek Ep if L Ek - Ep and vipi 2Ek
  mvi2.
• We can then describe the dynamics of (q, p)
  states using the 1st-order Hamiltons equations
• These are equivalent to (but often easier to
  solve than) the 2nd-order Euler-Lagrange
  equation.
• Note that any Hamiltonian dynamics is what we
  might call bi-deterministic
• Meaning, deterministic in both the forwards and
  reverse time directions.

Implicitsummationover i here.
18
Field Theories

• Here the space of indexes i of the generalized
  coordinates is continuous, thus uncountable.
• Usually it forms some topological space T, e.g.,
  R3.
• We often use f(x) notation in place of qi.
• In local field theories, the Lagrangian L(f) is
  the integral of a Lagrange density function L(x)
  where the point x ranges over the entire space T.
• This L(x) depends only locally on the field f,
  e.g.,
• L(x) Lf(x), (?f/?xi)(x), (x)
• All successful physical theories can be
  explicitly written down as local field theories!
• Thus, there is no instantaneous action at a
  distance.

19
Special Relativity and the Speed-of-Light Limit
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The Speed-of-Light Limit

• No form of information (including quantum
  information) can propagate through space at a
  velocity (relative to its local surroundings)
  that is greater than the speed of light, c
  3108 m/s.
• Some consequences
• No closed system can propagate faster than c.
• Although you can define open systems that do, by
  definition
• No given chunk of matter, energy, or momentum
  can propagate faster than c.
• The influence of all of the fundamental forces
  (including gravity) propagates at (at most) c.
• The probability mass associated with a quantum
  particle flows in an entirely local fashion, at
  no faster than c.

21
Early History of the Limit

• The principle of locality was first anticipated
  by Newton
• He wished to get rid of the action at a
  distance aspects of his law of gravitation.
• The fact of the finiteness of the speed of light
  (SoL) was first observed experimentally by Roemer
  in 1676.
• The first decent speed estimate was obtained by
  Fizeau in 1849.
• Weber Kohlrausch derived a constant velocity of
  c from empirical electromagnetic constants in
  1856.
• Kirchoff pointed out the match with the speed of
  light in 1857.
• Maxwell showed that his EM theory implied the
  existence of waves that always propagate at c in
  1873.
• Hertz later confirmed experimentally that EM
  waves indeed existed
• Michaelson Morley (1887) observed that the
  empirical SoL was independent of the observers
  state of motion!
• Maxwells equations are apparently valid in all
  inertial reference frames!
• Fitzgerald (1889), Lorentz (1892,1899), Larmor
  (1898), Poincaré (1898,1904), Einstein (1905)
  explored the implications of this...

22
Relativity Non-intuitive, but True

• How can the speed of something be a fundamental
  constant? Seemed broken...
• If Im moving at velocity v towards you, and I
  shoot a laser at you, what speed does the light
  go, relative to me, and to you? Answer both c!
  (Not vc.)
• Newtons laws were the same in all frames of
  reference moving at a constant velocity.
• Principle of Relativity (PoR) All laws of
  physics are invariant under changes in velocity
• Einsteins insight The PoR is consistent w.
  Maxwells theory!
• But we must change the definition of spacetime.

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Some Consequences of Relativity

• Measured lengths and time intervals in a system
  vary depending on the systems velocity relative
  to observers.
• Lengths are shortened in direction of motion.
• Moving clocks run slower.
• Sounds paradoxical, but isnt!
• Mass of moving objects is amplified.
• Energy and mass are really the same quantity
  measured in different units Emc2.
• Nothing (including energy, matter, information,
  etc.) can go faster than light! (SoL limit.)

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Three Ways to Understand the c limit

• Energy of motion contributes to mass of object.
• Mass approaches ? as velocity ? c.
• Infinite energy would be needed to reach c.
• Lengths, times in a faster-than-light moving
  object would become imaginary numbers!
• What would that even mean?
• Faster than light in one reference frame ?
  Backwards in time in another reference frame
• Sending information backwards in time violates
  causality, leads to logical contradictions!

25
The c limit in quantum physics

• Sometimes you see statements about non-local
  effects in quantum systems. Watch out!
• Even Einstein made this mistake.
• Described a quantum thought experiment that
  seemed to require spooky action at a distance.
• Later it was shown that this experiment did not
  actually violate the speed-of-light limit for
  information.
• These non-local effects are only illusions,
  emergent phenomena predicted by an entirely local
  underlying theory respecting the SoL limit..
• Widely-separated systems can still maintain
  quantum correlations, but that isnt true
  non-locality.

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The Lorentz Transformation
Actually it was written down earlier e.g., one
form by Voigt in 1887

• Lorentz, Poincaré All the laws of physics
  remain unchanged, relative to the reference frame
  (x',t') of an object moving with constant
  velocity v ?x/?t in another reference frame
  (x,t), under the following substitutions

Where
Note our ? here is the reciprocal of the
quantity denoted ? by other authors.
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Some Consequences of the Lorentz Transform

• Length contraction (Fitzgerald 1889, Lorentz
  1892)
• An object having length ? in its rest frame
  appears, when measured in a relatively moving
  frame, to have the (shorter) length ??.
• For lengths that are parallel to the direction of
  motion.
• Time dilation (Poincaré, 1898)
• If time interval t is measured between two
  co-located events in a given frame, a (larger)
  time t t/? will be measured between those same
  two events in a relatively moving frame.
• Mass expansion (Einsteins fix for Newtons
  Fma)
• If an object has mass m0gt0 in its rest frame,
  then it is seen to have the larger mass m m0/?
  in a relatively moving frame.

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Lorentz Transform Visualization
x0
x'0
Original x,t(rest) frame
Line colors
Isochrones(space-like)
t'0
Isospatials(time-like)
New x',t'(moving) frame
Light-like
In this example v ?x/?t 3/5? ?t'/?t
4/5vT v/? ?x/?t' 3/4
t 0
The tourists velocity.
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Mixed-Frame Version of Lorentz Transformation

• Usual version (with c1)
• Letting (xA,tA)(x, t') and (xB,tB)(x', t),
  andsolving for (xA,tA), we get
• Or, in matrix form
• The Lorentz transform is thus revealed as a
  simple rotation of the mixed-frame coordinates!

(Where ? arctan vT)
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Visualization of the Mixed Frame Perspective
t'
t
t'
StandardFrame 1
MixedFrame 1
t
x
x
In this example v ?x/?t 3/5 vT ?x/?t'
3/4 ? ?t' /?t 4/5 Note that (?t)2 (?x)2
(?t')2by the PythagoreanTheorem!
Rememberthe slogan My space isperpendicularto
your time.
x
t'
t
x'
StandardFrame 2
MixedFrame 2
x'
x'
Note the obvious complete symmetryin the
relation between the two mixed frames.
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Relativistic Kinetic Energy

• Total relativistic energy E of any object is E
  mc2.
• For an object at rest with mass m0, Erest
  m0c2.
• For a moving object, m m0/?
• Where m0 is the objects mass in its rest frame.
• Energy of the moving object is thus Emoving
  m0c2/?.
• Kinetic energy Ekin Emoving - Erest
  m0c2/? - m0c2 Erest(1/? - 1)
• Substituting ? (1-ß2)1/2 and Taylor-expanding
  gives

Higher-orderrelativistic corrections
Pre-relativistic kinetic energy ½ m0v2
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Spacetime Intervals

• Note that the lengths and times between two
  events are not invariant under Lorentz
  transformations.
• However, the following quantity is an invariant
  The spacetime interval s, where
• s2 (ct)2 - xi2
• The value of s is also the proper time t
• The elapsed time in rest frame of object
  traveling on a straight line between the two
  events. (Same as what we were calling t'
  earlier.)
• The sign of s2 has a particular significance
• s2 gt 0 - Events are timelike separated (s is
  real) May be causally connected.
• s2 0 - Events are lightlike separated (s
  is 0) Only 0-rest-mass signals may connect
  them.
• s2 lt 0 - Events are spacelike separated (s
  is imaginary) Not causally connected at all.

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Relativistic Momentum

• The relativistic momentum p mv
• Same as classical momentum, except that m m0/?.
• Relativistic energy-momentum-rest-mass
  relation E2 (pc)2 (m0c2)2If we use units
  where c 1, this simplifies to just E2 p2
  m02
• Note that if we solve this for m02, we get
• m02 E2 - p2
• Thus, E2 - p2 is another relativistic invariant!
  
• Later we will show how it relates to the
  spacetime interval s2 t2 - x2, and to a
  computational interpretation of relativistic
  physics.