EEL%204930%20

1
EEL 4930 6 / 5930 5, Spring 06Physical Limits
of Computing
http//www.eng.fsu.edu/mpf

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

2
Physical Limits of ComputingCourse Outline
Currently I am working on writing up a set of
course notes based on this outline,intended to
someday evolve into a textbook

• Course Introduction
• Moores Law vs. Modern Physics
• Foundations
• Required Background Material in Computing
  Physics
• Fundamentals
• The Deep Relationships between Physics and
  Computation

• IV. Core Principles
• The two Revolutionary Paradigms of Physical
  Computation
• V. Technologies
• Present and Future Physical Mechanisms for the
  Practical Realization of Information Processing
• VI. Conclusion

3
Part II. Foundations

• This first part of the course quickly reviews
  some key background knowledge that you will need
  to be familiar with in order to follow the later
  material.
• You may have seen some of this material before.
• Part II is divided into two chapters
• Chapter II.A. The Theory of Information and
  Computation
• Chapter II.B. Required Physics Background

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Chapter II.B. Required Physics Background

• This chapter covers All the Physics You Need to
  Know, for purposes of this course
• II.B.1. Physical Quantities, Units, and
  Constants
• II.B.2. Modern Formulations of Mechanics
• II.B.3. Basics of Relativity Theory
• II.B.4. Elementary Quantum Mechanics
• II.B.5. Thermodynamics Statistical Mechanics
• II.B.6. Solid-State Physics

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Section II.B.1Physical Quantities, Units, and
Fundamental Constants

• Unit prefixes, Physical quantities, Dimensional
  identities, Planck units, Physical constants

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Order-of-Magnitude Prefixes
Factor Prefix Abbr. English Factor Prefix Abbr. English
101 deca da ten 10-1 deci d tenth
102 hecto h hundred 10-2 centi c hundredth
103 kilo k thousand 10-3 milli m thousandth
106 mega M million 10-6 micro µ millionth
109 giga G billion 10-9 nano n billionth
1012 tera T trillion 10-12 pico p trillionth
1015 peta P quadrillion 10-15 femto f quadrillionth
1018 exa E quintillion 10-18 atto a quintillionth
1021 zetta Z sextillion 10-21 zepto z sextillionth
1024 yotta Y septillion 10-24 yocto y septillionth
Source BIPM, http//www.bipm.org/en/si/prefixes.h
tml
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Binary Power Multiples
Factor Decimal Value Prefix Abbr. Approx.
210 1,024 kibi Ki 1.02 k
220 1,048,576 mebi Mi 1.05 M
230 1,073,741,824 gibi Gi 1.07 G
240 1,099,511,627,776 tebi Ti 1.10 T
250 1,125,899,906,842,624 pebi Pi 1.13 P
260 1,152,921,504,606,846,976 exbi Ei 1.15 E
270 1,180,591,620,717,411,303,424 zebi Zi 1.18 Z
280 1,208,925,819,614,629,174,706,176 yobi Yi 1.21 Y
Source NIST, http//physics.nist.gov/cuu/Units/bi
nary.html
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Fundamental Physical Quantities

• All other physical quantities can be defined in
  terms of these
• However, other sets of fundamental units are
  possible

Dimensional Symbol Denotes Typical variable names Some Units
L Physical distance, length, position, radius d, ?, x, r Ångstrom (Å), inch (in), foot (ft), yard (yd), meter (m), mile (mi), astronomical unit (au), light-year (ly), parsec (pc)
T Physical time, time interval t, T, t second (s), minute (min), hour (hr), day (dy), year (yr)
M Physical mass m, M atomic mass unit (amu), gram (g), pound mass (lbm)
Q Electric charge q, Q Coulomb (C), Franklin (Fr)
S Entropy, uncertainty, extropy, information S, H, X, I Joules/Kelvin (J/K), kilocalorie/Kelvin (kcal/K), bit (b), nat (n), dit (d)
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Some derived quantities
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Electrical Quantities

• Well skip magnetism related quantities this
  semester.

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Fundamental Physical Constants

• The speed of light in vacuum c 299,792,458
  m/s
• This value is exact, by the very definition of
  the meter!
• The energy of frequency (Plancks constant) h
  6.626075510?34 J s
• The energy of angular frequency (reduced Plancks
  constant) ? h / 2?
• Remember this with the analogy (h 1 circle)
  (? 1 radian)
• In fact, later well see why its valid to view
  h,? as being these angles.
• The force of gravity (Newtons gravitational
  constant) G 6.6725910?11 N m2 / kg
• The force of electricity (Coulombs law
  constant) kC 8.9876?109 N m2 / C2
• The e-fold of uncertainty (Boltzmanns
  constant) k kB Loge 1.380651310?23 J /
  K
• Others permittivity of free space,
  Stefan-Boltzmann constant, etc. to be introduced
  later, as we go along.

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Constants and Laws

• Each of the fundamental constants arises as a
  simple proportionality constant in some
  fundamental physical law
• Newtons Law. Bodies exert an attractive force
  proportional to their masses and the inverse
  square of their separation
• Fgrav G(m1m2/d2)
• Coulombs Law. Electrical charges exert a force
  proportional to their charges and the inverse
  square of their separation
• Felec kC(q1q2/d2)
• Constancy of the speed of light (Maxwell). The
  distance traversed by an electromagnetic
  disturbance (e.g., light) is proportional to the
  elapsed time
• ? ct
• Boltzmanns Relation. The amount of energy added
  to a system as heat is proportional to its
  temperature times the increase in its information
  capacity.
• dE kBT d(LogW)
• Quantization principle (Planck, Einstein).
  Electromagnetic disturbances come in discrete
  units (quanta) having an energy proportional to
  their frequency
• E hf

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The Unification of Physical Quantities

• By letting fundamental constants be
  dimensionless, we can establish identities
  between different physical dimensions
• If we let G be dimensionless, we get
• ML3/T2 (mass rate of acceleration of volume?)
• If we let kC be dimensionless, we get
• Q2ML3/T2 (charge squared mass ? volume
  acceleration?)
• This is how charge units such as the
  Franklin/statcoulomb/electrostatic unit are
  defined
• With both G and kC dimensionless, we get QM
  (chargemass)
• If we let c be dimensionless, we get
• TL (timelength), and EM (energymass).
• With G dimensionless also, we get ML.
• If we let kB be dimensionless, we get
• TE/S (temperature energy per unit
  uncertainty)
• If we let h be dimensionless, we get
• E1/T (energyinverse time, or frequency)
• p1/L (momentuminverse length, or spatial
  frequency)
• If c is also dimensionless, we get M1/L.
• and if G is also dimensionless, we get 1/LML ?
  L1 (dimensionless!)

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Plancks Natural Units

• Max Planck realized in 1898 that letting G,c,h be
  dimensionless implied that all of the basic
  quantities L,T,M were really dimensionless
  quantities also
• I.e., any length, time, or mass quantity
  corresponds to a unique pure number determined by
  G, c and h!
• Thus, length/time/mass have associated natural
  units for measuring them
• The units discovered by Planck are
• The Planck length LP (G?/c3)1/2 1.6?10-35 m
• The Planck time tP (G?/c5)1/2 5.4?10-44 s
• The Planck mass mP (?c/G)1/2 22 µg
• Its thought that these units (or something close
  to them) will be seen to have fundamental
  significance
• in the presumed eventual unified theory of
  everything

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Physics that you should already know

• Well assume you already know these well, and
  wont review them
• 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

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Section II.B.2Modern Formulations of Mechanics

• Euler-Lagrange Equation, Least-Action Principle,
  Hamiltons Equations, Field Theories

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Chapter II.B. Required Physics Background

• This chapter covers All the Physics You Need to
  Know, for purposes of this course
• II.B.1. Physical Quantities, Units, and
  Constants
• II.B.2. Modern Formulations of Mechanics
• II.B.3. Basics of Relativity Theory
• II.B.4. Quantum Mechanics
• II.B.5. Thermodynamics Statistical Mechanics
• II.B.6. Solid-State Physics

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Generalized Mechanics

• Classical mechanics (Newtons Laws, etc.) can be
  expressed in the most general and concise way
  using the Lagrangian and Hamiltonian
  formulations.
• Developed by Lagrange and Hamilton in the 1800s.
• Each of these is based on a simple energy-valued
  function of the systems instantaneous state
• The Lagrangian Kinetic minus potential energy.
• The Hamiltonian Kinetic plus potential energy.
• The dynamical laws for the system can be derived
  from either of these energy functions,
• due to general principles of dynamics.
• The Lagrangian (or Hamiltonian) gives the laws of
  physics!
• This framework generalizes to be the basis for
  quantum mechanics, quantum field theories, etc.

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Euler-Lagrange Equation
Note the time-derivativeover-dot!
or just

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

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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 pot. energy)
• 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 (Newtons 2nd
  law)

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Least-Action Principle
A.k.a.Hamiltonsprinciple

• The action of an energy quantity means the
  integral of that quantity over time.
• Dimensions of action ET ML2/T 1
• The trajectory specified by the Euler-Lagrange
  equation locally extremizes (usually minimizes)
  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.
• Physical systems always take the path of least
  action

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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 the 2nd-order
  Euler-Lagrange equation.
• But sometimes are easier to solve than it.
• 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.
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Lagrangian Formulation of Field Theories

• Here, the space of indices i of the generalized
  coordinates is continuous (thus uncountable).
• Usually it forms some topological space T, e.g.,
  R3.
• So, 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) L f(x), (?f/?xi)(x), (x)
• All successful physical theories (so far) can be
  explicitly written down as local field theories!
• Thus, there is no instantaneous action at a
  distance.

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Section II.B.3Basics of Relativity Theory

• Special Relativity, The Speed-of-Light Limit,
  General Relativity, Black Holes

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Chapter II.B. Required Physics Background

• This chapter covers All the Physics You Need to
  Know, for purposes of this course
• II.B.1. Physical Quantities, Units, and
  Constants
• II.B.2. Modern Formulations of Mechanics
• II.B.3. Basics of Relativity Theory
• II.B.4. Quantum Mechanics
• II.B.5. Thermodynamics Statistical Mechanics
• II.B.6. Solid-State Physics

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Subsection II.B.3.aSpecial Relativity and the
Speed-of-Light Limit

• Speed-of-Light Limit, Relativistic Effects,
  Lorentz Transformation, Relativistic
  Energy/Momentum

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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 of the limit
• No closed system can propagate faster than c.
• Although you can define open systems that do, by
  definition
• No given piece of matter, energy, or momentum can
  propagate faster than c.
• All conserved physical quantities flow through
  space in a local fashion.
• The influence of all of the fundamental physical
  forces (including gravity) propagates at (at
  most) c.
• The probability mass associated with a quantum
  particle (or configuration of a multiparticle
  system) flows through space (or configuration
  space) in an entirely local fashion, at no faster
  than c.

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Early History of the Limit

• The principle of locality was first anticipated
  by Newton
• He expressed a desire 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
  value, c, from empirical electromagnetic
  constants (e0, µ0) in 1856.
• Kirchoff pointed out the match with the empirical
  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
  (Earths) 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...

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Relativity Non-intuitive, but True

• How can the speed of something be a fundamental
  constant? Seems broken at first
• If Im moving at some large velocity v towards
  you, and I shoot a laser pulse at you, at what
  speed does the pulse travel, relative to me, and
  to you?
• The answer to both is exactly c! (Not vc or
  anything else!)
• Newtons laws were the same in all frames of
  reference moving at a constant velocity.
• Einsteins Principle of Relativity (PoR) All
  laws of physics are invariant under changes in
  velocity
• Einsteins insight The PoR is perfectly
  logically consistent w. Maxwells theory of a
  constant SoL!
• But, in showing consistency, we must change our
  definition of the concepts of space and time!

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Some Consequences of Relativity

• Measured lengths and time intervals in a system
  vary depending on the systems velocity
• relative to the observers making the measurements
• In particular, relativity yields the following
  effects
• The length of any moving object is shortened
• in its direction of motion.
• All physical processes in a moving object run
  more slowly,
• Faster motion through space ? Slower motion
  through time!
• The mass of any moving object is amplified.
• Energy and mass are really the same quantity,
  measured in different units! A realization that
  led to ?
• The conversion between units is given by Emc2.
• Nothing (including energy, matter, information,
  etc.) can travel 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.
• Due to the form of the equations, lengths and
  times in a faster-than-light moving object would
  become imaginary numbers!
• What would that even mean, physically?
• 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!

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The c limit in quantum physics

• Sometimes you see statements about non-local
  effects in quantum systems. Watch out!
• Even Einstein made this mistake.
• He described a quantum thought experiment that at
  first appeared to require spooky action at a
  distance.
• Later, it was shown that such experiments did not
  actually violate the speed-of-light limit for
  information.
• All of the apparently non-local quantum effects
  can be explained away as mere illusions
• emergent phenomena that are predicted by an
  entirely local underlying theory fully respecting
  the SoL limit..
• Widely-separated systems can still maintain
  quantum correlations (entanglement),
• but that isnt true non-locality, in the usual
  sense.

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Coordinate Systems

• We can think of a coordinate system (or frame of
  reference) as a set of interlocking stacks of
  planes (egg carton)
• One stack of planes for each dimension x,y,z,t.
• All points in a given plane have the same value
  for the given coordinate

A plane with equal values of twould be called an
isochrone. A plane with equal values of xwould
be called an isospatial. In geometry, such a
stack of planes is called a 1-form a 1-form
is related to a vector, but it transforms
differently when the coordinate system changes.
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The Galilean Transformation

• According to Galileo, the coordinates in a frame
  moving at velocity v are t' t, x' x - vt.
• The isochrones remain unchanged.
• The isospatials of the moving frame get skewed
  (tilted)
• But they remain the same distance apart in the
  original x dimension

Galilean boostby velocity 1/4
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Why Galilean Coordinate Transformations Dont Work

• Under the Galilean transformation, if some entity
  travels at a constant velocity (e.g., 1 space
  unit per time unit) in the original frame, then
  it will not do so in the new (relatively moving)
  frame.
• E.g., in the figure below, the green trajectory
  has speed 1 in the old coordinate system, and ¾
  in the new one while the purple one goes from
  speed -1 to -5/4.
• Lorentz, Poincaré, and Einstein asked, how can we
  modify the Galilean transformation in a way that
  will fix this?
• So that a certain speed (namely, c) will remain
  the same in all reference frames.

Galilean boostby velocity 1/4
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Fixing the Problem

• Clearly, the only way to fix the problem is to
  skew the isochrones also.
• Like collapsing an egg crate or dish barrel
  dividers ?
• It turns out that this is a good analogy, b/c the
  planes also get compressed closer together as the
  angles are skewed
• In the figure on the right, notice that the green
  and purple trajectories still have speeds of 1
  and -1 in the skewed system.
• Since the isochrones must be skewed, events that
  are simultaneous in a moving coordinate system
  (i.e., on the same isochrone) are NOT
  simultaneous from a stationary observers point
  of view, and vice-versa.
• This is called the relativity of simultaneity.

Planes for increasingtime (t) coordinates
Lorentz boostby velocity 1/4
Planes for increasingspace (x) coordinates
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The Lorentz Transformation
Usually credited to Lorentz (1904), but actually
it was written down earlier e.g., Larmor
(1897), and a variant form by Voigt (1887)

• Lorentz, Poincaré Maxwells laws remain
  unchanged in the reference frame (x',t') of an
  object moving with constant velocity v ?x/?t in
  another reference frame (x,t), if we postulate
  the following transformation between the
  coordinate systems

where
Note Our ? symbol in these slides denotes the
reciprocal of the quantity thats called ? by
most authors. We do this so that we can write ß2
?2 1.
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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 (Larmor 1897, 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-Fitzgerald Length Contraction

• Consider a limousine moving at 3/5 lightspeed!
• Galileo Lorentz

v 0.6 c
Length in own frame still 10 units
In a snapshot, theback of the limoappears
slightlyolder than the front!
Time ?
Time ?
Apparent length in ground frame 8 units
Length 10 units
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A Cool View of Time Dilation

• The time dilation law can be rewritten in the
  form
• ?x/?t (or v ßc) is the rate at which the moving
  object covers distance through space
• c?t'/?t (or ?c) is the rate at which the object
  traverses a distance of c?t' along its own
  personal time dimension t'
• c?t/?t c is the rate at which distance through
  real time is covered.
• Thus, we can say, we always travel at the speed
  of light, when all travel is counted.
• Normally, all of this travel is through time,
  which is why we age at one year per year.
• But, if some of our travel is through space, then
  the portion of our travel that takes us along our
  personal time dimension is lowered
• Its exactly as if personal time we experience is
  a dimension thats perpendicular to the space
  that we move through, and the total distance
  traveled is given by the Pythagorean theorem!

Or, dividing through by c2,
1 total rate of travel (Speed of Light,
always!)
? rate of travel through movers
personal time
ß rate of travel through space
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Lorentz Transform Visualization
x0
x'0
New x',t'(moving) frame
Line colors
Isochrones(space-like)
t'0
Isospatials(time-like)
Light-like
Original x,t(rest) frame
In this example v ?x/?t 3/5? ?t'/?t
4/5vT v/? ?x/?t' 3/4
t 0
The tourists velocity.
42
Lorentz Transform Animation

• In this animation,
• We see the instantaneous coor-dinate system of
  an observer moving along a non-uniform
  trajectory.
• Small dots are events
• E.g., supernova explosions
• Large dots are events that the traveler
  experiences in equal amounts of his own time.
• E.g., birthday parties he has, each time hes
  aged a year
• Note that dots on fast-moving parts of the
  trajectory seem farther apart until he matches
  velocities
• This is due to the time dilation effect

Future lightcone
Increasing t' coordinates
Past light cone
Increasing x' coordinates
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The Twin Paradox
?x 4 lyr

• One of the many relativistic effects that appears
  paradoxical at first
• but really isnt.
• Suppose I get in a rocket fly to a star 4
  light-years away with v 4/5 c.
• Then I turn around and return at v 4/5 c.
• Each way takes 5 years, so 10 years go by back on
  Earth
• But since ? 3/5, the traveler only ages 6
  years.
• The paradox asks
• Since we can also say that it is the Earth that
  is moving at 4/5 c relative to the traveler, why
  is it the traveler that ages less?

?t 10 yr
v 4/5 c
v 4/5 c
44
Resolution of the Paradox

• The reason for the asymmetry is that
• the traveler is the one who has changed between
  coordinate systems!
• when he turned around!
• Remember the skewed isochrones in the moving
  frame?
• During our 3-year outbound trip,
• we consider only 3? 9/5 1.8 years to go by
  back on Earth.
• Yes, Earth is also time-dilated relative to the
  traveler!
• Similarly, during our 3-year return trip,
• we only see another 1.8 years go by on Earth.
• But, when we turn around to head back,
• we change reference frames,
• The change in reference frame means that whenwe
  go to the new frame, we suddenly consider a date
  on Earth thats 10 - 3.6 6.4 years later than
  before to be now.
• Another example of this
• If I walk at 3 m/s towards a galaxy 100 million
  light-years away, the points I label now in
  that galaxy suddenly are points a year later in
  the galaxys history
• If I walk in the other direction, my now shifts
  to points a year earlier.

1.8yrs
6.4yrs
1.8yrs
45
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 by
  angle ?!

(Where ? arctan vT)
46
Visualization of the Mixed Frame Perspective
StandardFrame 1
MixedFrame 1
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
World-line of stationary object, x 0
x'
StandardFrame 2
MixedFrame 2
Note the obvious complete symmetryin the
relation between the two mixed frames!
Lightlike trajectories
47
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
48
Spacetime Intervals

• Note that the lengths and times between two
  events are not invariant under Lorentz
  transformations.
• However, the following quantity is a Lorentz
  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) The events 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) The events are not causally
  connected at all.

49
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.

50
Topics not yet Covered

• Other topics to be added to special relativity
  section, eventually
• Mathematical conveniences
• Tensors.
• Index notation.
• Covariant and contravariant tensors.
• 4-vectors.
• Spacetime metrics.
• Momenergy
• (or, the relativistic energy-momentum 4-vector)

51
Subsection II.B.3.bSome Tidbits on General
Relativity

• Equivalence Principle, Curved Spacetime, Black
  Holes

SLIDES NOT YET WRITTEN