# Geodesics in the Morris-Thorne Wormhole Spacetime

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The simplest wormhole geometry is given by the line element , see [2]. The parameter defines the size of the throat of the wormhole, and represents the proper length radius.

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Light rays and objects in free motion in four-dimensional spacetimes follow lightlike or timelike geodesics. In general, these geodesics must be computed numerically. However, in the Ellis wormhole spacetime, there is an analytic solution of the geodesic equation in terms of elliptic integral functions. Because of the spherical symmetry and staticity of the metric, it suffices to consider geodesics in the hypersurface . This two-dimensional surface can be embedded in the three-dimensional Euclidean space. The corresponding embedding function reads with .

In this application, you can change the throat size , the initial position of the observer, and the initial angle of the geodesic with respect to the local reference frame of the observer.

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Contributed by: Thomas Müller (July 2010)
Open content licensed under CC BY-NC-SA

## Details

A detailed discussion about analytic geodesics in the Morris–Thorne wormhole spacetime can be found in [1].

The metric described in [2] was first mentioned in [3]. Hence, it should be called Ellis wormhole instead. See also the apology in [4], Ref. 14.

References

[1] T. Müller, "Exact Geometric Optics in a Morris–Thorne Wormhole Spacetime," Physical Review D, 77(4) 2008. doi: 10.1103/PhysRevD.77.044043.

[2] M. S. Morris and K. S. Thorne, "Wormholes in Spacetime and Their Use for Interstellar Travel: A Tool for Teaching General Relativity," American Journal of Physics, 56(5), 1988 pp. 395–412.

[3] H. G. Ellis, "Ether Flow through a Drainhole: A Particle Model in General Relativity," Journal of Mathematical Physics, 14, 1973 pp. 104–118; 1974 Errata: 15, p. 520.

[4] O. James, E. von Tunzelmann, P. Franklin, and K. S. Thorne, "Visualizing Interstellar's Wormhole," American Journal of Physics 83, 2015 pp. 483–499.

## Permanent Citation

Thomas Müller

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