Three Partial Differential Equations
in Curved Space and their Respective
Solutions
Gopinath Kamath
Abstract The partial differential equation in curved space whose solution is the
electrostatic potential A 0 is obtained following Whittaker (Whittaker, Proc R Soc
A 116:720–735, 1927) in: a. 2 + 1 dimensions with the metric g μν given by
(a) Deser, Jackiw and ‘tHooft (Deser et al., Ann Phys 152:220–235, 1984), and
Clement (Clement, Int J Theor Phys 24:267, 1985) and (b) by Banados, Teitelboim
and Zanelli (Banados et al., Phys Rev Lett 69:1849, 1992); and in b. 3 + 1
dimensions with the cylindrically symmetric metric due to Trendafilova and Fulling
(Trendafilova and Fulling, Eur J Phys 32:1663, 2011). While a series solution of
the partial differential equation is easily obtained for two of the examples cited in
2 + 1 and 3 + 1 dimensions, respectively, a sensible solution eludes us with the
Banados–Teitelboim–Zanelli (Banados et al., Phys Rev Lett 69:1849, 1992) black
hole metric, inviting an alternative to Whittaker’s approach (Whittaker, Proc R Soc
A 116:720–735, 1927) that is exact and sketched here briefly.
Keywords Partial differential equations in curved spaces · Whittaker’s method ·
2 + 1 and 3 + 1 dimensions
1 Introduction
The method of images has been a go-to for solutions to problems in electrostatics;
while the theoretical aspects of image charges have been discussed at length by
Muller-Kirsten [1], its application to several problems has been dealt with in detail
both by Jackson [2] and Muller-Kirsten [1]. Parenthetically, from the standpoint
of a stress-tensor the method has also been adopted for the Casimir effect [3] for
the parallel plate problem at zero and finite temperatures by Brown and Maclay
[4]. Going further, a matrix solution to the Maxwell’s equations in flat space that
yields both the static and time-dependent solutions has been presented by Moses [5]
G. Kamath ()
Department of Mathematics, Indian Institute of Technology Tirupati, Tirupati, India
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_46
491
in Curved Space and their Respective
Solutions
Gopinath Kamath
Abstract The partial differential equation in curved space whose solution is the
electrostatic potential A 0 is obtained following Whittaker (Whittaker, Proc R Soc
A 116:720–735, 1927) in: a. 2 + 1 dimensions with the metric g μν given by
(a) Deser, Jackiw and ‘tHooft (Deser et al., Ann Phys 152:220–235, 1984), and
Clement (Clement, Int J Theor Phys 24:267, 1985) and (b) by Banados, Teitelboim
and Zanelli (Banados et al., Phys Rev Lett 69:1849, 1992); and in b. 3 + 1
dimensions with the cylindrically symmetric metric due to Trendafilova and Fulling
(Trendafilova and Fulling, Eur J Phys 32:1663, 2011). While a series solution of
the partial differential equation is easily obtained for two of the examples cited in
2 + 1 and 3 + 1 dimensions, respectively, a sensible solution eludes us with the
Banados–Teitelboim–Zanelli (Banados et al., Phys Rev Lett 69:1849, 1992) black
hole metric, inviting an alternative to Whittaker’s approach (Whittaker, Proc R Soc
A 116:720–735, 1927) that is exact and sketched here briefly.
Keywords Partial differential equations in curved spaces · Whittaker’s method ·
2 + 1 and 3 + 1 dimensions
1 Introduction
The method of images has been a go-to for solutions to problems in electrostatics;
while the theoretical aspects of image charges have been discussed at length by
Muller-Kirsten [1], its application to several problems has been dealt with in detail
both by Jackson [2] and Muller-Kirsten [1]. Parenthetically, from the standpoint
of a stress-tensor the method has also been adopted for the Casimir effect [3] for
the parallel plate problem at zero and finite temperatures by Brown and Maclay
[4]. Going further, a matrix solution to the Maxwell’s equations in flat space that
yields both the static and time-dependent solutions has been presented by Moses [5]
G. Kamath ()
Department of Mathematics, Indian Institute of Technology Tirupati, Tirupati, India
© Springer Nature Switzerland AG 2021
M. B. Paranjape et al. (eds.), Quantum Theory and Symmetries, CRM Series in
Mathematical Physics, https://doi.org/10.1007/978-3-030-55777-5_46
491