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EEL 5930 sec. 5, Spring ‘05 Physical Limits of Computing

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### EEL 5930 sec. 5, Spring ‘05Physical Limits of Computing

### Review of Basic Physics Background

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

(Module 2)

Basic physical quantities & units

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

M. Frank, "Physical Limits of Computing"

Unit Prefixes

- See http://www.bipm.fr/enus/3_SI/si-prefixes.htmlfor 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
- Don’t get confused!

M. Frank, "Physical Limits of Computing"

Three “fundamental” quantities

M. Frank, "Physical Limits of Computing"

Some derived quantities

M. Frank, "Physical Limits of Computing"

Electrical Quantities

- We’ll skip magnetism & related quantities this semester.

M. Frank, "Physical Limits of Computing"

Information, Entropy, Temperature

- These are important physical quantities also
- But, are different from other physical quantities
- They are based on combinatorics and statistics
- But, we’ll 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).

M. Frank, "Physical Limits of Computing"

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.

M. Frank, "Physical Limits of Computing"

Some fundamental physical constants

- Speed of light c = 299,792,458 m/s
- Planck’s constant h = 6.6260755×1034 J s
- Reduced Planck’s constant = h / 2
- Remember this with the analogy: (h : 360°) :: ( : 1 radian)
- In fact, later we’ll see it’s valid to view h, as being these angles.
- Newton’s gravitational constant: G = 6.67259×1011 Nm2 / kg
- Boltzmann’s constant:k = kB = log e = 1.3806513×1023 J / K
- Others: permittivity of free space, Stefan-Boltzmann constant, etc. to be introduced later, as we go along.

M. Frank, "Physical Limits of Computing"

Physics that you should already know

- Basic Newtonian mechanics
- Newton’s laws, motion, energy, etc.
- Basic electrostatics
- Ohm’s law, Kirchoff’s laws, etc.
- Also helpful, but not prerequisite (we’ll introduce them as we go along):
- Basic statistical mechanics & thermodynamics
- Basic quantum mechanics
- Basic relativity theory

M. Frank, "Physical Limits of Computing"

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.

M. Frank, "Physical Limits of Computing"

Euler-Lagrange Equation

Note the over-dot!

Where:

- L(q, v) is the system’s 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.

or just

M. Frank, "Physical Limits of Computing"

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 (Newton’s 2nd law)

M. Frank, "Physical Limits of Computing"

Least-Action Principle

A.k.a.Hamilton’sprinciple

- 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.

M. Frank, "Physical Limits of Computing"

Hamilton’s Equations

Implicitsummationover i here.

- 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 Hamilton’s 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.

M. Frank, "Physical Limits of Computing"

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 φ(x) notation in place of qi.
- In local field theories, the Lagrangian L(φ) 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 φ, e.g.,

ℒ(x) = ℒ[φ(x), (∂φ/∂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.

M. Frank, "Physical Limits of Computing"

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 ≈ 3×108 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.

M. Frank, "Physical Limits of Computing"

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 observer’s state of motion!
- Maxwell’s 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...

M. Frank, "Physical Limits of Computing"

Relativity: Non-intuitive, but True

- How can the speed of something be a fundamental constant? Seemed broken...
- If I’m 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!(Notv+c.)
- Newton’s 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
- Einstein’s insight: The PoR is consistent w. Maxwell’s theory!
- But we must change the definition of space+time.

M. Frank, "Physical Limits of Computing"

Some Consequences of Relativity

- Measured lengths and time intervals in a system vary depending on the system’s velocity relative to observers.
- Lengths are shortened in direction of motion.
- Moving clocks run slower.
- Sounds paradoxical, but isn’t!
- Mass of moving objects is amplified.
- Energy and mass are really the same quantity measured in different units: E=mc2.
- Nothing (including energy, matter, information, etc.) can go faster than light! (SoL limit.)

M. Frank, "Physical Limits of Computing"

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!

M. Frank, "Physical Limits of Computing"

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 isn’t “true” non-locality.

M. Frank, "Physical Limits of Computing"

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 = Δx/Δt in another reference frame (x,t), under the following substitutions:

Where:

Note: our γ here is the reciprocal of the quantity denoted γ by other authors.

M. Frank, "Physical Limits of Computing"

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 γ.
- For lengths that are parallel to the direction of motion.
- Time dilation: (Poincaré, 1898)
- If time interval τ is measured between two co-located events in a given frame, a (larger) time t = τ/γ will be measured between those same two events in a relatively moving frame.
- Mass expansion: (Einstein’s fix for Newton’s F=ma)
- If an object has mass m0>0 in its rest frame, then it is seen to have the larger mass m = m0/γ in a relatively moving frame.

M. Frank, "Physical Limits of Computing"

Lorentz Transform Visualization

increasing t′

increasing t

increasing x′

increasing x

x=0

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 = Δx/Δt = 3/5γ = Δt′/Δt = 4/5vT = v/γ = Δx/Δt′ = 3/4

t = 0

The “tourist’s velocity.”

M. Frank, "Physical Limits of Computing"

Mixed-Frame Version of Lorentz Transformation

- Usual version (with c=1):
- 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 θ = arctan vT)

M. Frank, "Physical Limits of Computing"

Visualization of the Mixed Frame Perspective

t′

t

t′

StandardFrame #1

MixedFrame #1

t

x

x

In this example:

v = Δx/Δt = 3/5

vT = Δx/Δt′ = 3/4γ = Δt′ /Δt = 4/5

Note that (Δt)2 = (Δx)2 + (Δt′)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.

M. Frank, "Physical Limits of Computing"

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/γ
- Where m0 is the object’s mass in its rest frame.
- Energy of the moving object is thus Emoving = m0c2/γ.
- Kinetic energy Ekin :≡ Emoving − Erest= m0c2/γ − m0c2 = Erest(1/γ − 1)
- Substituting γ = (1−β2)1/2 and Taylor-expanding gives:

Higher-orderrelativistic corrections

Pre-relativistic kinetic energy ½ m0v2

M. Frank, "Physical Limits of Computing"

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 timeτ:
- 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 > 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 < 0 - Events are spacelike separated (s is imaginary)Not causally connected at all.

M. Frank, "Physical Limits of Computing"

Relativistic Momentum

- The relativistic momentum p = mv
- Same as classical momentum, except that m= m0/γ.
- 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.

M. Frank, "Physical Limits of Computing"

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