General Relativity in a Fish Tank
Matthew J. Wahl, Northport High School; John No� and Harold Metcalf, Laser Teaching
Center, Department of Physics and Astronomy, Stony Brook University
Einstein's theory of general relativity predicts that a gravitational field can warp
space-time. This warp creates a skewed path (or paths) in space along which light will
travel. Therefore, gravitational fields can bend light. Einstein's theory was
confirmed in 1919, during a total solar eclipse, when a star whose light passed close
by the darkened sun was observed at a slightly different location than usual. A
similar effect takes place on a much smaller scale when light travels through an
optical medium with a continually changing index of refraction. This happens in nature
in mirages, for example, since heated air has a lower index than cooler more dense
air. The path the light follows turns out to be the one that requires the least
possible time, a principle that was first recognized by the French mathematician
Fermat in the 1600's. Thus when light travels through a gradient medium it follows a
curved path which can represent warped space-time.
The initial phase of this project has involved creating a Gradient Index of Refraction
(GRIN) tank by carefully adding a layer of water (index n =
1.33) on top of a layer of corn syrup (n = 1.48). The two liquids gradually diffuse
into one another, creating a gradient mixture in which the refractive index changes in
the vertical direction but not the horizontal plane. A laser beam shined into the
gradient mixture continually bends downwards, towards the direction of increasing
refractive index. Thus a beam that is initially pointed above the horizontal will
follow a parabolic curve, reach a maximum height, and then turn downwards. The light
is simply following the path that takes the least time, compared to other paths that
it could follow, but doesn't. (An alternative but less elegant explanation of the
motion can be derived from Snell's law of refraction, by dividing the medium into
horizontal layers.)
Three ways of measuring indices of refraction were used. The first method measures the
horizontal displacement of a laser beam as it passes through a medium (held in a tank
with parallel sides) at an angle. This method proved very accurate and effective for a
uniform medium but is not useful for the GRIN tank. The second method involves
measuring the intensity of light reflected from the inside surface of the tank wall,
at the particular angle (Brewster's angle) where the reflection of suitably polarized
light from the outside surface drops to zero. A complicated formula derived from
Fresnel's Equations and Snell's Law was used to compute the index of the liquid from
the measured intensity of the reflected light relative to the incident light. The
final method involved measuring the vertical drop of an initially horizontal beam as
it crossed a much thinner, specially made tank. The analysis of these results involves
calculus and is still in progress.
Future experiments can be done to model the behavior of light around different
gravitational fields. For example, a spherical index gradient should bend light
around its center, simulating how light would behave in the gravitational field of a
black hole.
We thank the Simon's Foundation for its funding our research this summer, and
Prof. Erlend Graf for providing
the thin tank.
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