Transcript The Mathematics of Star Trek

The Mathematics of Star
Trek
Lecture 6: General
Relativity
Topics
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Captain Picard and Einstein’s Elevator
Principal of Equivalence
Bending of Light
General Relativity and Curved Spacetime
Einstein’s Equations of General Relativity
Mercury’s Orbit
Faster than Light (FTL) Travel
Impulse Drive, Tractor Beams, Deflector
Shields, and Cloaking Devices
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Captain Picard and Einstein’s
Elevator
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Suppose that just as a phaser
(light) beam is shot by a
Romulan Warbird at Captain
Picard’s 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!
Romulan Frame
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1.75
1.5
1.25
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0.75
0.5
0.25
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Picard Frame
10
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-20
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Captain Picard and Einstein’s
Elevator (cont.)
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If Picard’s acceleration is the same as
that due to gravity at the Earth’s 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 Earth’s 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
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Einstein’s 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
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In 1911, using the idea that light will be
bent in a gravitational field, along with
Newton’s 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.)
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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 much farther 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
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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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Einstein’s Equations of General
Relativity
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Mathematically, General Relativity is built
upon a set of ten coupled hyperbolicelliptic 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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Mercury’s Orbit
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Another way that the theory of General
Relativity was shown to be correct was
by answering a question about Mercury’s
orbit!
According to Newton’s 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,
Mercury should come back to its starting
point, which isn’t what happens.
It turns out that the perihelion (closest
point to the Sun) of the orbit of the planet
Mercury advances approximately two
degrees per century (image is from
Hyper Physics website).
All but approximately 40 arc seconds of
this advance can be accounted for via
classical Newtonian physics, such as the
force of gravity from other planets in the
solar system!
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Mercury’s Orbit (cont.)
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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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Mercury’s Orbit (cont.)
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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:
Mercury’s Orbit (cont.)
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Einstein suggested that the extra
advance of Mercury’s perihelion
could be due to the curvature of
spacetime near the Sun.
He predicted that the amount by
which the Mercury’s perihelion
should advance is given by:
where a is the length of the semimajor axis of Mercury’s 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
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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. 44 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.)
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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 3space!
It may be possible to traverse what
appears to be a huge distance (line-ofsight 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.)
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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.)
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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
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The same idea that works for warp drive,
namely warping spacetime, would allow for
travel at impulse speeds!
The crew wouldn’t 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 Newton’s Third Law doesn’t 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
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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.html
http://www-groups.dcs.st-and.ac.uk/~history/index.html
http://hubblesite.org/newscenter/newsdesk/archive/releases/2004/08/t
ext/
http://archive.ncsa.uiuc.edu/Cyberia/NumRel/EinsteinEquations.html#i
ntro
http://pds.jpl.nasa.gov/planets/captions/neptune/fullnep.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_Small.jpg
http://www.exn.ca/mini/startrek/warpdrive.cfm
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