We introduce an exactly solvable example of timelike geodesic motion and geodesic deviation in the background geometry of a well-known two-dimensional black hole spacetime. The effective potential for geodesic motion turns out to be either a harmonic oscillator or an inverted harmonic oscillator or a linear function of the spatial variable corresponding to the three different domains of a constant of the motion. The geodesic deviation equation also is exactly solvable. The corresponding deviation vector is obtained and the nature of the deviation is briefly discussed by highlighting a specific case.

## REFERENCES

1.

A few of the standard texts on general relativity include R. D’Inverno,

*Introducing Einstein’s Relativity*(Oxford U.P., Oxford, 1995);B. Schutz,

*A First Course in General Relativity*(Cambridge U.P., Cambridge, 1990);W. Rindler,

*Relativity*(Oxford U.P., Oxford, 2001);S. Weinberg,

*Gravitation and Cosmology*(Wiley, New York, 1971);C. W. Misner, K. S. Thorne, and J. A. Wheeler,

*Gravitation*(Freeman, San Francisco, 1972);L. Landau and E. M. Lifshitz,

*Classical Theory of Fields*(Pergamon, Oxford, 1962);R. M. Wald,

*General Relativity*(University of Chicago, Chicago, 1985).2.

For an extensive review of two-dimensional gravity, see

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*Superstring Theory*(Cambridge U.P., Cambridge, 1987), Vols. I and II;J. Polchinski,

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J. Harvey and A. Strominger, “Quantum aspects of black holes in string theory and quantum gravity,” in

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A good introduction to geodesics and geodesic deviation in particular is available in the text by R. D’Inverno referred to in Ref. 1.

6.

The statement made here is based on advanced topics such as the Raychaudhuri equation and geodesic focusing theorems (a good, though advanced discussion on these topics is available in the text by Wald referred to in Ref. 1). In a nutshell, convergence of timelike geodesics (that is, focusing) within a finite value of the affine parameter occurs only when the timelike convergence condition $R\mu \nu \xi \mu \xi \nu \u2a7e0$ is obeyed. In two dimensions $R\mu \nu =1/2g\mu \nu R$ always and $g\mu \nu \xi \mu \xi \nu =\u22121$ (timelike condition).

Therefore convergence/focusing occurs within a finite λ only if $R\u2a7d0.$ For some details on this (particularly in two dimensions and in the context of stringy black holes), see

S.

Kar

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