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.

Astro Matt
June 2006

Laser Teaching Center