The Mathematics of Star Trek

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The Mathematics of Star Trek

• Lecture 5 General Relativity

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Topics

• Captain Picard and Einsteins Elevator
• Principal of Equivalence
• Bending of Light
• General Relativity and Curved Spacetime
• Einsteins Equations of General Relativity
• Mercurys Orbit
• Faster than Light (FTL) Travel
• Impulse Drive, Tractor Beams, Deflector Shields,
  and Cloaking Devices

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Captain Picard and Einsteins Elevator
Romulan Frame

• Suppose that just as a phaser (light) beam is
  shot by a Romulan Warbird at Captain Picards
  yacht Calypso, the Calypso accelerates forward,
  perpendicular to the phaser shot.
• In the Romulans frame of reference, the phaser
  beam travels in a straight line.
• Thinking of light as a particle, it can be shown
  with vectors and the equations of motion that
  Picard will see the light beam travel along a
  curved path!
• Mathematica example!

Picard Frame
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Captain Picard and Einsteins Elevator (cont.)

• If Picards acceleration is the same as that due
  to gravity at the Earths surface, then Picard
  will feel the same force pushing him back in his
  seat as he would due to the downward force of
  gravity at the Earths surface!
• Einstein argued that Picard (or someone in an
  elevator being accelerated upwards with the same
  acceleration) will never be able to perform any
  experiment to tell the difference between the
  reaction force due to accelerated motion and that
  due to the pull of gravity from some nearby heavy
  object outside the ship.
• Einstein concluded that whatever phenomena an
  accelerating object experiences would be the same
  as the phenomena an observer in a gravitational
  field experiences!

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The Principle of Equivalence

• Einsteins idea, known as the Principle of
  Equivalence, says that inertial mass and
  gravitational mass are the same and is the
  starting point for the theory of General
  Relativity!
• Here is one implication of the Principle of
  Equivalence
• Since Picard observes the light ray bending when
  he is accelerating away from it, it follows from
  Principle of Equivalence that the light ray would
  also bend in a gravitational field!
• Matter produces a gravitational field, so matter
  must bend the path of a light beam.

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Bending of Light

• In 1911, using the idea that light will be bent
  in a gravitational field, along with Newtons
  Laws, Einstein predicted that light passing by
  the outer edge of the Sun should be bent by an
  angle of approximately 0.85 seconds.
• In 1919, Sir Arthur Eddington led one of two
  expeditions (his went to Sobral Brazil) to
  observe the apparent position of stars on the sky
  near the Sun during a solar eclipse.
• During the eclipse, rays of light from stars
  passing close to the sun would be bent.
• The amount by which the light is bent could be
  deduced by comparing the stars relative
  positions to those at some other time of the
  year.
• Eddington found that the light bends exactly
  twice as much as that predicted by Einstein!

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Bending of Light (cont.)

• The phenomenon observed by Eddington is the same
  as that of gravitational lensing, where a cluster
  of galaxies can produce multiple magnified images
  of a galaxy twice as far away, which has been
  seen using the Hubble Space Telescope.
• A gravitational lens is a massive object that
  magnifies or distorts the light of objects lying
  behind it.
• For example, the powerful gravitational field of
  a massive cluster of galaxies can bend the light
  rays from more distant galaxies, just as a camera
  lens bends light to form a picture.

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General Relativity and Curved Spacetime

• In 1916, Einstein published a paper that
  introduced the world to his theory of General
  Relativity.
• Unable to incorporate gravity directly into the
  theory of Special Relativity, he was led to the
  idea that gravity is not a force, but a
  manifestation of the curvature of spacetime.
• Masses in space such as the Sun cause spacetime
  to be curved and the curved paths that we see
  objects (or light) following near these masses
  are simply the straightest possible paths in the
  curved spacetime!
• Adding in the idea that spacetime is curved, he
  was able to show that the predicted bending of
  light by the Sun during the 1919 eclipse was
  right!

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Einsteins Equations of General Relativity

• Mathematically, General Relativity is built upon
  a set of ten coupled hyperbolic-elliptic
  nonlinear partial differential equations, which
  can be represented symbolically as shown at the
  right. (Click on the image for a link to the
  equations!)
• The equations boil down to this CURVATURE (LHS)
  MATTER AND ENERGY (RHS).
• This theory is hard to work with, because
• The curvature of space is determined by the
  distribution of matter and energy in the
  universe.
• The distribution of matter and energy in the
  universe is determined by the curvature of space.

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Mercurys Orbit

• Another way that the theory of General Relativity
  was shown to be correct was by answering a
  question about Mercurys orbit!
• According to Newtons Laws, the planet Mercury
  moves around the Sun in an elliptical orbit with
  the Sun at one focus of the ellipse.
• After one revolution around the Sun, the Mercury
  should come back to its starting point, which
  isnt what happens.
• It turns out that the perihelion (closest point
  to the Sun) of the orbit of the planet Mercury
  advances approximately 0.012 degree per century
  (image is from Hyper Physics website).
• One way to account for this advance is due to the
  force of gravity from other planets in the solar
  system, but this only takes care of 80 seconds of
  arc.

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Mercurys Orbit (cont.)

• Astronomers guessed that the last 40 seconds of
  arc might be due to another (unknown) planet in
  our solar system, which they named Vulcan.
• This is how the planet Neptune was discovered!
• Neptune was the first planet located through
  mathematical predictions rather than through
  systematic observations of the sky!
• After the discovery of Uranus in 1781,
  astronomers noted that Uranus was not faithfully
  following its predicted path.
• Uranus seemed to accelerate in its orbit before
  1822 and to slow after that.

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Mercurys Orbit (cont.)

• One possible explanation was that the gravity of
  an undiscovered planet was affecting the orbit of
  Uranus.
• Starting in 1841, British astronomer John Couch
  Adams and the following summer, French astronomer
  Urbain Jean Joseph Le Verrier without knowledge
  of each other independently calculated where the
  new planet should be.
• At first, neither was taken seriously, but by
  1846, based on their work, Neptune was discovered!

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Experimental Verification Mercurys Orbit
(cont.)

• Einstein suggested that the extra advance of
  Mercurys perihelion could be due to the
  curvature of spacetime near the Sun.
• He predicted that the amount by which the
  Mercurys perihelion should advance is given by
• where a is the length of the semi-major axis of
  Mercurys elliptical orbit, e is the eccentricity
  of the ellipse, c is the speed of light, and T is
  the period of revolution.
• HW See if this formula works!

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Faster Than Light (FTL) Travel

• One of the implications of curved spacetime is
  the idea that what we perceive as a straight line
  need not be the shortest path between two points!
• Krauss gives an explanation of how this might
  occur in two-dimensions!
• Consider a piece of elastic material, as shown on
  p. 36 of our textbook.
• If the material is laid flat and a circle drawn
  on the sheet, the shortest path between two
  opposite points on the circle, A and B, would be
  a straight line through the center.

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FTL Travel (cont.)

• If the center is pushed down and the material
  stretched, then the shortest path may be along
  the circle.
• The sheet has been curved (in 3-space), but to a
  bug walking on the line from A to B, it thinks
  it is moving in a straight line.
• This is the idea we want to extend to spacetime -
  we are the bug and cannot perceive the curve of
  spacetime in 3-space!
• It may be possible to traverse what appears to be
  a huge distance (line-of-sight wise) by finding a
  shorter route through spacetime!
• If spacetime itself can be manipulated, then
  objects can travel locally at low velocities, yet
  an accompanying expansion or contraction of space
  could allow huge distances to be traversed in
  short time intervals!

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FTL Travel (cont.)

• An example of this idea has been developed by
  physicist Miguel Alcubierre, who has shown that
  mathematically warp drive could be possible in
  the theory of general relativity.
• According to Alcubierre, a spacetime
  configuration can be created in which a
  spacecraft could traverse a distance between two
  points in an arbitrarily short period of time.
• Throughout this journey, the spacecraft would
  move with respect to its local surroundings at
  speeds much less than the speed of light!
• Therefore clocks on the ship would stay
  synchronized with the outside world.

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FTL Travel (cont.)

• The idea works like this warp spacetime so it
  expands behind the ship and contracts in front of
  it, thus propelling the ship along with the space
  surrounding the ship (like a surfboard and surfer
  on a wave).
• The spaceship will never travel faster than the
  speed of light, as the light near the ship will
  be carried along with the ship!
• In this theory, it would be possible to arrange
  for the huge gravitational fields needed to be
  somewhere far from the spaceship or star bases or
  planets the ship may travel to as a destination,
  thus avoiding problems with slow clocks.

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Impulse Drive, Tractor Beams, Deflector Shields,
and Cloaking Devices

• The same idea that works for warp drive, namely
  warping spacetime, would allow for travel at
  impulse speeds!
• The crew wouldnt be subjected to large
  accelerations, so inertial dampers would no
  longer be needed!
• Warping spacetime could also be used to move a
  planet contract space behind the asteroid and
  expand space in front of it!
• This could be how the tractor beam really works
  if so, then Newtons Third Law doesnt apply any
  more!
• Two other applications of warping spacetime might
  be deflector shields and cloaking devices!
• Deflector shields are force fields that prevent
  phaser beams (light rays) from hitting a
  starship.
• Cloaking devices make a ship invisible.
• In each case, space would be warped to either
  deflect (bend) the light rays away from the ship
  or cause the light rays to bend around the ship
  (cloak it).

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References

• Relativity The Special and General Theory by
  Albert Einstein
• The Geometry of Spacetime by James Callahan
• The Physics of Star Trek by Laurence Krauss
• http//hyperphysics.phy-astr.gsu.edu/hbase/hph.htm
  l
• http//www-groups.dcs.st-and.ac.uk/history/index.
  html
• http//hubblesite.org/newscenter/newsdesk/archive/
  releases/2004/08/text/
• http//archive.ncsa.uiuc.edu/Cyberia/NumRel/Einste
  inEquations.htmlintro
• http//pds.jpl.nasa.gov/planets/captions/neptune/f
  ullnep.htm
• http//members.aol.com/nogravityguy/book02.htm
• http//marge.uvm.edu/Sdempse/images/TV_Movies/Star
  _Trek/borgtrac.gif
• http//www.thasos.ukgateway.net/images/Ent_Warp_Sm
  all.jpg
• http//www.exn.ca/mini/startrek/warpdrive.cfm