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.

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
D.
Grumiller
,
W.
Kummer
, and
D.
Vassilevich
, “
Dilaton gravity in two dimensions
,” hep-th/0205253 (Phys. Rept. 369 (2002), 327–430) and references therein.
A lucid introduction to string theory is available at 〈www.theory.caltech.edu/people/patricia (home page of Patricia Schwarz). More extensive texts on string theory are M. S. Green, J. H. Schwarz, and E. Witten, Superstring Theory (Cambridge U.P., Cambridge, 1987), Vols. I and II;
J. Polchinski, String Theory (Cambridge U.P., Cambridge, 1998), Vols. I and II.
3.
The original papers where this line element was first proposed are
G.
Mandal
,
A. M.
Sengupta
, and
S. R.
Wadia
, “
Classical solutions of two dimensional string theory
,”
Mod. Phys. Lett. A
6
,
1685
1692
(
1991
);
E.
Witten
, “
On string theory and black holes
,”
Phys. Rev. D
44
,
314
324
(
1991
).
4.
J. Harvey and A. Strominger, “Quantum aspects of black holes in string theory and quantum gravity,” in String Theory and Quantum Gravity ’92, edited by J. Harvey et al. (World Scientific, Singapore, 1993).
5.
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 Rμνξμξν⩾0 is obeyed. In two dimensions Rμν=1/2gμνR always and gμνξμξν=−1 (timelike condition).
Therefore convergence/focusing occurs within a finite λ only if R⩽0. For some details on this (particularly in two dimensions and in the context of stringy black holes), see
S.
Kar
, “
Stringy black holes and energy conditions
,”
Phys. Rev. D
55
,
4872
4879
(
1997
).
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