Solutions of a charged scalar field in five-dimensional helicoid solution with electromagnetic field

We study a charged and massive scalar field in the background of the Nutku-Ghezelbash-Kumar metric which is obtained by the addition of a time coordinate to the Nutku helicoid metric in a non-trivial way. The angular part of the Klein-Gordon equation can be written as a double confluent Heun equation. The radial equation cannot be solved in terms of a known function in its general form. However, in some special cases, the radial equation can also be written explicitly as a double confluent Heun equation. We study the full radial equation numerically and observe that the electromagnetic field parameter defines an effective cut-off on the range of the radial coordinate. Finally, we obtain a quasi-exact solution with an approximation.

Note:

14 pages, 8 figures. Matches the published version

Klein-Gordon equation
Gravitational instanton
Exact solutions
Numerical solutions
field theory: scalar: massive
dimension: 5
electromagnetic field
Heun equation
Klein-Gordon equation
background

References(33)

Figures(15)

[1]

Gravitation, Gauge Theories and Differential Geometry

Tohru Eguchi
(
- SLAC and
- Chicago U., EFI and
- Chicago U.
)
,
Peter B. Gilkey
(
- Princeton U., Math. Dept.
)
,
Andrew J. Hanson
(
- LBL, Berkeley and
- NASA, Ames
)

- Phys.Rept. 66 (1980) 213
•
DOI:
- 10.1016/0370-1573(80)90130-1

[2]

Gravitational instantons admit hyper-Kaehler structure

A.N. Aliev
(
- Feza Gursey Inst.
)
,
Y. Nutku
(
- Feza Gursey Inst.
)

- Class.Quant.Grav. 16 (1999) 189-210
•
e-Print:
- gr-qc/9805006
•
DOI:
- 10.1088/0264-9381/16/1/013

[3]

A Newman-Penrose formalism for gravitational instantons

E. Goldblatt
(
- New York U.
)

- Gen.Rel.Grav. 26 (1994) 979-997
•
DOI:
- 10.1007/BF02106666

[4]

Symmetries of type D+ D- gravitational instantons

E. Goldblatt
(
- New York U.
)

- J.Math.Phys. 35 (1994) 3029-3042
•
DOI:
- 10.1063/1.530502

[5]

A Newman-Penrose Calculator for Instanton Metrics

T. Birkandan
(
- Istanbul, Tech. U.
)

- Int.J.Mod.Phys.C 19 (2008) 1277
•
e-Print:
- 0711.0613
•
DOI:
- 10.1142/S0129183108012807

[6]

Gravitational instantons and minimal surfaces

Y. Nutku
(
- TUBITAK Res. Inst.
)

- Phys.Rev.Lett. 77 (1996) 4702-4703
•
DOI:
- 10.1103/PhysRevLett.77.4702

[7]

General minimal surface solution for gravitational instantons

- Phys.Rev.D 56 (1997) 1332-1333
•
DOI:
- 10.1103/PhysRevD.56.1332

[8]

Gravitational instantons derived from minimal surfaces

A.N. Aliev
(
- Feza Gursey Inst.
)
,
M. Hortacsu
(
- Istanbul, Tech. U. and
- Istanbul U.
)
,
J. Kalayci
(
- Istanbul, Tech. U. and
- Istanbul U.
)
,
Y. Nutku
(
- Feza Gursey Inst.
)

- Class.Quant.Grav. 16 (1999) 631-642
•
e-Print:
- gr-qc/9812007
•
DOI:
- 10.1088/0264-9381/16/2/024

[9]

Dirac equation in Euclidean Newman-Penrose formalism with applications to instanton metrics

Y. Sucu
(
- Akdeniz U.
)
,
N. Unal
(
- Akdeniz U.
)

- Class.Quant.Grav. 21 (2004) 1443-1451
•
DOI:
- 10.1088/0264-9381/21/6/011

[10]

Examples of Heun and Mathieu functions as solutions of wave equations in curved spaces

T. Birkandan
(
- Istanbul, Tech. U.
)
,
M. Hortacsu
(
- Istanbul, Tech. U. and
- Feza Gursey Inst.
)

- J.Phys.A 40 (2007) 1105-1116,
- J.Phys.A 40 (2007) 11203
•
e-Print:
- gr-qc/0607108
•
DOI:
- 10.1088/1751-8113/40/5/016

[11]

Dirac Equation in the Background of the Nutku Helicoid Metric

T. Birkandan
(
- Istanbul, Tech. U.
)
,
M. Hortacsu
(
- Istanbul, Tech. U.
)

- J.Math.Phys. 48 (2007) 092301
•
e-Print:
- 0706.2543
•
DOI:
- 10.1063/1.2785124

[12]

Comment on Dirac equation in the background of the Nutku helicoid metric

T. Birkandan
(
- Istanbul, Tech. U.
)
,
M. Hortacsu
(
- Istanbul, Tech. U.
)

- J.Math.Phys. 49 (2008) 054101
•
e-Print:
- 0711.0612
•
DOI:
- 10.1063/1.2913520

[13]

Exact helicoidal and catenoidal solutions in five- and higher-dimensional Einstein-Maxwell theory

A.M. Ghezelbash
(
- Saskatchewan U.
)
,
Vineet Kumar
(
- Saskatchewan U.
)

- Phys.Rev.D 95 (2017) 12, 124045
•
e-Print:
- 1704.01476
•
DOI:
- 10.1103/PhysRevD.95.124045

[14]

Minimal surfaces and generalized Einstein–Maxwell-dilaton theory

M. Butler
(
- Saskatchewan U.
)
,
A.M. Ghezelbash
(
- Saskatchewan U.
)

- Int.J.Mod.Phys.A 34 (2019) 12, 1950061
•
e-Print:
- 1810.13051
•
DOI:
- 10.1142/S0217751X19500611

[15]

M-branes on minimal surfaces

A.M. Ghezelbash
(
- Saskatchewan U.
)

- Eur.Phys.J.Plus 137 (2022) 2, 196
•
e-Print:
- 2407.07855
•
DOI:
- 10.1140/epjp/s13360-022-02395-2

[16]

Heun's Differential Equations

A. Ronveaux

[17]

Special Functions

SY Slavyanov
,
W. Lay

[18]

Heun Functions and Some of Their Applications in Physics

M. Hortacsu
(
- Istanbul, Tech. U.
)

- Proceedings of the 13th Regional Conference on Mathematical Physics, Antalya, Turkey, October 27-31,-2010, Edited by Ugur Camci and Ibrahim Semi 23-39. World Scientific,
- Adv.High Energy Phys. 2018 (2018) 8621573
•
e-Print:
- 1101.0471
•
DOI:
- 10.1142/9789814417532_0002