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Orbits in Black Hole Spacetimes - Math Problem Example

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This paper calculates the orbits of bodies and light rays in both the Schwarzschild and the Reissner-Nordstrom spacetime metric. The Schwarzschild metric is a solution of the Einstein equations that describes the spacetime outside a spherical star as well as the exterior of a nonrotating black hole…
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Orbits in Black Hole Spacetimes
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Download file to see previous pages Sections three and four of this paper calculate the same orbits and light ray bending near a charged black hole, described by a Reissner-Nordstrom spacetime.
The precession of the orbit is found the difference of this and 2π, or 2πε, which is the angle amount that the axis of the orbit rotates after each orbit is traced out by the planet Mercury. This angle of precession is equal to:
The light rays of distant stars are bent by the Sun, and this is described by a Schwarzschild spacetime outside the sun. This calculation is similar to the calculation of the previous section, except that a null geodesic is used here. For a null geodesic,
To find the angle of deflection (= ) of a light ray near the Sun, 1 is defined as the angle between the observer and the Sun, and 2 is the angle between the star that emitted the light rays and the Sun; summing the angles 1 and 2 gives the angle of deflection. The two values of (which is the angle to the light ray, taking the Sun as the origin) are taken to be -1 and + 2 when essentially r goes to infinity and u goes to 0.
The smallest value of b is for light rays just grazing the Sun so that b is effectively the radius of the Sun (b = 6.96E8m) (Hartle 211). This deflection can be found using data from the Sun (Halliday and Resnick A5):
The last two equations (for a = 2 and a = 3) are unchanged, since the - and -dependence of the metric was not changed. Because of this, many of the results of section 1 are unchanged here. For = /2 we again get:
In this section, we recalculate the bending of light rays (calculated in section 2 with a Schwarzschild metric) by using a Reissner-Nordstrom spacetime metric instead of the Schwarzschild spacetime metric. ...Download file to see next pagesRead More
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