The Mathematics of Star Trek

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The Mathematics of Star Trek

• Lecture 5 Special Relativity

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Topics

• Galilean Relativity
• Electricity and Magnetism
• Maxwells Equations
• The Michelson-Morley Experiment
• Maxwells Equations Revisited
• The Lorentz Transformations
• Special Relativity

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Galilean Relativity

• The aim of science (of which mathematics is a
  part) is to describe and interpret reality.
  
• When people compare notes on an observation, they
  find points of common agreement, which ultimately
  make up what is physically or objectively real.
  
• Questions that need to be answered include
• Do my calculations agree with yours?
• Can I reproduce your results?

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Galilean Relativity (cont.)

• Addressing the question of objectivity in
  connection with motion, Gallileo came up with the
  following principle of relativity
  
• Two observers moving uniformly relative to one
  another must formulate the laws of nature in
  exactly the same way.
  
• In particular, no observer can distinguish
  between absolute rest and absolute motion by
  appealing to any law of nature hence there is no
  such thing as absolute motion, but only motion in
  relation to an observer.

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Galilean Relativity (cont.)

• From Galileos principle of relativity, we can
  conclude the following
  
• Any physical law must be formulated the same way
  by all observers.
  
• Anything formulated the same way by all observers
  is a physical law.

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Galilean Relativity (cont.)

• Suppose we have two Galilean observers, which
  well call the Romulan Commander (R) and Captain
  Picard (P).
  
• Each observer is on a spaceship with their own
  coordinate system.
  
• For simplicity, well assume motion in one space
  dimension.
  
• Adding in time, we get a spacetime vector for
  each observer in their reference frame!
  
• Such a vector is called an event.
• Romulan coordinates for an event (x, t)
• Picard coordinates for the same event (x, t)

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Galilean Relativity (cont.)

• Suppose that Picards ship, the Enterprise, is
  moving to the right at a constant velocity v and
  the Romulan ship, a warbird, is fixed in space.
  
• Further, assume that at time t t 0, the
  ships are at the same point in space.
  
• Using Galileos idea of relativity, if we know
  the coordinates of an event in one reference
  frame, we can figure out the coordinates of the
  event in the other reference frame!

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Galilean Relativity (cont.)

• If the Romulans coordinates for an event are
  (x,t), then Picards coordinates for this event
  will be
  
• (x, t) (x - vt, t). (1)
• If Picards coordinates for an event are (x,t),
  then the Romulans coordinates for this event
  will be
  
• (x, t) (x vt, t). (2)
• We call (1) and (2) Galilean transformations.

Romulan
Picard
velocity v
P-axis
x
0
R-axis
x
0
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Galilean Relativity (cont.)

• If Picard measures an objects velocity in his
  frame of reference as dx/dt w, what does the
  Romulan Commander see?
  
• Using Galilean transformation (2), the Romulan
  commander will see the objects velocity (in his
  reference frame) as
  
• w dx/dt d/dtx vt dt/dt (dx/dt
  v)(1) w v.

Romulan
Picard
velocity v
velocity w
P-axis
x
0
R-axis
x
velocity wv
0
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Electricity and Magnetism

• In 1820, Hans Christian Oersted (1777-1851)
  discovered that if an electric current is
  switched on or off in a wire near a compass
  needle, the needle will deflect.
• This showed that electricity and magnetism were
  related phenomena.
  
• The Oersted is a unit of magnetic field strength.

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Electricity and Magnetism (cont.)

• In 1826, after hearing about Oersteads
  experimental results Andre Marie Ampère
  (1775-1836) attempted to give a combined theory
  of electricity and magnetism.
• Ampère formulated a circuit force law and treated
  magnetism by postulating small closed circuits
  inside the magnetised substance.
  
• Ampère's theory, including Ampère's Law became
  fundamental for 19th century developments in
  electricity and magnetism.
  
• The standard unit of current is called the Ampère.

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Electricity and Magnetism (cont.)

• Charles Augustin de Coulomb (1737 1806)
  developed a theory of attraction and repulsion
  between bodies of the same and opposite
  electrical charge.
• He found that the force between a pair of charged
  particles obeys a law similar to Newtons Law of
  Universal Gravitation.
  
• Coulombs Law states that the force between a
  pair of charged particles proportional to the
  square of the distance between the charges.
  
• A fundamental unit of charge is the Coulomb.

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Electricity and Magnetism (cont.)

• Michael Faraday (1791-1867) was a self-taught
  experimentalist whose work led to deep
  mathematical theories of electricity and
  magnetism.
• For example, Faradays Law says that any change
  in the magnetic environment of a coil of wire
  will cause a voltage to be "induced" in the coil.
  One way to do this is to move a magnet near a
  coil of wire.
• Applications of Faradays Law include back-up
  generators and automobile engine ignition via
  sparkplugs and the Faraday Flashlight!
  
• A Farad is a unit of capacitance.

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Maxwells Equations

• Using the work of Ampere, Coulomb, and Faraday,
  James Clerk Maxwell (1831-1879) found that the
  laws of electricity and magnetism were related to
  each other mathematically.
• Maxwells Equations are a set of four partial
  differential equations relating electric fields
  and magnetic fields, due for example to a current
  flowing through a wire.

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Maxwells Equations (cont.)

• In addition to showing that electricity and
  magnetism are related mathematically, Maxwells
  equations can be used to show that
  electromagnetic waves exist.
• These waves travel through space at a constant
  speed, which turns out to be the speed of light!
  
• From this, people deduced that light itself is an
  electromagnetic wave!
  
• This ended the debate over whether light is a
  particle or a wave.

• Equations (3) - (6) are Maxwells Equations.
• Electric field E(t,x,y,z) and magnetic field
  H(t,x,y,z) are vector functions.
  
• ?(t,x,y,z) is a scalar function that describes
  the electric charge density and J(t,x,y,z) is a
  vector function known as the current density.

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Maxwells Equations (cont.)

• In the Star Trek universe, sensors are devices
  that are used to gather information about a
  planet, starship, life-form, etc.
  
• A primary example of this is the tricorder.
• One place where Maxwells equations are used in
  real life is to design sensors, such as
  magnetometers.

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Maxwells Equations (cont.)

• A magnetometer is a device that is used to detect
  changes in a magnetic field.
  
• Magnetometers have applications in areas such as
  industry, biomedicine, oceanography, space
  exploration, and law enforcement.
  
• The Galileo Orbiter used a magnetometer to map
  the structure and dynamics of Jupiter's
  magnetosphere.
  
• Magnetometers are used to detect metallic weapons
  such as handguns.

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Maxwells Equations (cont.)
Electric Field
Magnetic Field
HFSS Model of Magnetometer Component
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The Michelson-Morley Experiment

• If light is a wave, it should travel in some
  medium (think of water waves).
  
• Since Aristotles time, people had called this
  medium the ether.
  
• Between 1881 and 1887, Albert Michelson and
  Edward Morley set up a series of increasingly
  more accurate experiments to detect the effects
  of the ether on light.
• Using Galilean relativity, if a beam of light
  travels in the same direction that the earth is
  traveling around the sun (at 30 km/sec) and
  another beam travels in another direction, the
  light beams should have different velocities.

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The Michelson-Morley Experiment (cont.)

• The Michelson-Morley Experiment uses an
  interferometer to measure the change in velocity
  of light beams traveling in different
  directions.
• Unfortunately, Michelson and Morley were unable
  to detect any difference in the speeds of the
  light beams.
  
• One possible reason is that there is no ether!
• Another possibility is that the speed of light is
  the same for all Galilean observers, no matter
  what their relative motion!
  
• If this is true, then light doesnt transform
  properly under the Galilean transformations!

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Maxwells Equations Revisited

• Returning to Maxwells equations, lets assume
  that we are in empty space, so that ? 0 and J
  0.
  
• Further, suppose that our electric field depends
  only on t and x.
  
• With these assumptions, it follows that the
  electric field is a scalar field, E, which
  satisfies the wave equation that we saw earlier
  in this course!

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Maxwells Equations Revisited (cont.)

• Recall that according to Galileo, any physical
  law must be formulated the same way by all
  observers.
  
• Thus, if Maxwells equations are a physical law,
  then they should have the same form for both the
  Romulan commander and Captain Picard (in the
  setting as above).
• Similarly, the wave equation should look the same
  in both reference frames.

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Maxwells Equations Revisited (cont.)

• In the Romulans coordinates, the wave equation
  is the same as what we saw above
  
• Using the Galilean transformation (2), with
  ?(t,x) E(t, x-vt), in Picards coordinates,
  the wave equation becomes

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The Lorentz Transformations

• In 1904, mathematical physicist Hendrick Lorentz
  (1853-1928) proposed a modification to the
  Galilean transformations to mathematically fix
  the problem with Maxwells equations and Galilean
  Relativity.
• With the Romulan commander and Picard as above,
  the Lorentz transformations are as follows

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The Lorentz Transformations (cont.)

• If the Romulan commander sees an event with
  coordinates (x, t), then Picard will see the
  event as

• If Picard sees an event with coordinates (x,
  t), then the Romulan commander will see the
  event as
  

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Special Relativity

• The Lorentz transformations were introduced to
  mathematically address the problems that arose
  from trying to apply Galilean relativity to the
  Michelson-Morley experiment and Maxwells
  equations.
• In 1905, starting with the hypothesis that the
  speed of light is constant for all Galilean
  observers, Albert Einstein (1879 - 1955) was able
  to show that the Lorentz transformations must
  replace the Galilean transformations.
• This work is known as the Theory of Special
  Relativity.

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Implications of Special Relativity

• Here are some of the things that follow from
  Special Relativity, again with Picard moving at a
  velocity v, relative to the Romulan commander.
  
• All clocks on Picards ship will appear to be
  ticking more slowly to the Romulan commander.
  
• If Picards measures a time interval of T0, then
  the Romulan commander will see the time interval
  T as

• All rulers on Picards ship will appear shorter
  in length to the to the Romulan commander.
  
• If Picards ruler has length L0 in his frame,
  then the Romulan commander will see the rulers
  length L as
  

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Implications of Special Relativity (cont.)

• As the Enterprises velocity v increases, its
  mass will increase, approaching infinity as v
  approaches c.
  
• If the Enterprise has rest mass M0, then at
  velocity v, it will have relativistic mass M

• Another result that follows from Einsteins
  theory of Special Relativity is E mc2 which
  relates energy to mass via the speed of light.
  
• This equation is the key to making nuclear
  reactors and nuclear weapons work.

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Hermann Minkowskis Contribution to Special
Relativity

• In 1907, Hermann Minkowski (1864-1909) developed
  a new view of space and time and laid the
  mathematical foundation of the theory of
  relativity.
• Minkowski realised that the work of Lorentz and
  Einstein could be best understood in a
  non-Euclidean space (in particular Hyperbolic
  space).
• He considered space and time, which were formerly
  thought to be independent as vectors with four
  components - three for space and one for time.
  
• This space-time continuum provided a framework
  for all later mathematical work in relativity.

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Hyperbolic Geometry

• Geometry is the study of figures in a space of a
  given number of dimensions and of a given type.
  
• Euclidean geometry, which is often taught in high
  school, is based on a thirteen volume book called
  the Elements written in 300 B.C.
  
• Starting with five postulates (axioms), Euclid
  (325 - 265 B.C.) shows how basic properties of
  triangles, parallels, parallelograms, rectangles,
  squares, circles, etc. follow.

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Hyperbolic Geometry (cont.)

• Here are Euclids Postulates
• A straight line segment can be drawn joining any
  two points.
  
• Any straight line segment can be extended
  indefinitely in a straight line.
  
• Given any straight line segment, a circle can be
  drawn having the segment as radius and one
  endpoint as center.
  
• All right angles are congruent.
• If two lines are drawn which intersect a third in
  such a way that the sum of the inner angles on
  one side is less than two right angles, then the
  two lines inevitably must intersect each other on
  that side if extended far enough.
• Postulate 5 is equivalent to what is known as the
  parallel postulate Given any straight line and
  a point not on it, there "exists one and only one
  straight line which passes" through that point
  and never intersects the first line, no matter
  how far they are extended.

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Hyperbolic Geometry (cont.)

• In 1823, Janos Bolyai (1802-1860) and Nikolai
  Lobachevsky (1792-1856) independently realized
  that entirely self-consistent "non-Euclidean
  geometries" could be created in which the
  parallel postulate did not hold.
• For example, we get hyperbolic geometry by
  keeping Euclids first four postulates the same
  and replacing the parallel postulate with
  
• For any infinite straight line and any point not
  on it, there are many other infinitely extending
  straight lines that pass through and which do not
  intersect.

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Hyperbolic Geometry (cont.)

• In hyperbolic geometry, we find that
• The sum of angles of a triangle is less than 180
  degrees.
  
• Triangles with the same angles have the same
  areas.
  
• Not all triangles have the same angle sum
• There are no similar triangles in hyperbolic
  geometry.

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Hyperbolic Geometry (cont.)

• One way to visualize hyperbolic geometry in the
  plane is via the Poincaré disk. In this model,
  
• A line is represented as an arc of a circle
  (diameters are permitted) whose ends are
  perpendicular to the disk's boundary.
  
• Two arcs which do not meet correspond to parallel
  rays.
  
• Arcs which meet with an angle of 90 degrees
  correspond to perpendicular lines.
  
• Arcs which meet on the boundary are a pair of
  limits rays.
  
• The artist M. C. Escher's used hyperbolic
  geometry to create the pattern Circle Limit III.

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References

• The Geometry of Spacetime, James J. Callahan,
  Springer Verlag (2000).
  
• Hyper Physics http//hyperphysics.phy-astr.gsu.e
  du/hbase/hph.html
  
• St. Andrews' University History of Mathematics
  http//www-groups.dcs.st-and.ac.uk/history/index.
  html
  
• Wikipedia (Michelson-Morley Experiment)
  http//en.wikipedia.org/wiki/Michelson-Morley_expe
  riment
  
• Eugenii Katzs Homepage (Famous Scientists)
  http//chem.ch.huji.ac.il/eugeniik/history/oerste
  d.htm
  
• Math World http//mathworld.wolfram.com/
• Euclidean and Non-Euclidean Geometries, Marvin
  Jay Greenberg, W.H. Freeman and Company (1979).
  
• Doug Dunhams Homepage http//www.d.umn.edu/ddu
  nham/isis4/section6.html