On space--time Killing tensors with a Segre characteristic ((11)(11))

Journal Article · Thu Jul 01 04:00:00 EDT 1976 · J. Math. Phys. (N.Y.); (United States)

DOI:https://doi.org/10.1063/1.523058· OSTI ID:7281934

Hauser, I; Malhiot, R J

We consider the set of all space--times which admit a Killing tensor K/sub ..cap alpha beta../ whose Segre characteristic is ((11)(11)) and whose eigenvalues lambda/sup 1/ and lambda/sup 2/ satisfy the condition that dlambda/sup 1/ X dlambda/sup 2/ is not null. We prove that there locally exist scalar fields phi/sup 3/, phi/sup 4/ such that lambda/sup 1/, lambda/sup 2/, phi/sup 3/, phi/sup 4/ constitute a chart, and dlambda/sup 1/ x dlambda/sup 2/ = dlambda/sup i/ x d phi/sup r/ = 0 for i=1, 2 and r = 3, 4. Also, relative to this coordinate system, the metric has the same general form as Carter's Hamilton--Jacobi separable metric except that his arbitrary functions of lambda/sup i/ are replaced by functions of lambda/sup i/, phi/sup 3/, phi/sup 4/. If R/sup ..cap alpha beta../(del/sub ..cap alpha../lambda/sup 1/)(del/sub ..beta../lambda/sup 2/) = 0, there exists a two-parameter Abelian group of isometries which commute with K/sub ..cap alpha beta../; also /phi/sup 3/,phi/sup 4/) can be chosen so that par.delta par.delta phi/sup 3/ and par.delta/par.delta phi/sup 4/ are the Killing vectors. R/sup ..cap alpha beta../(del/sub ..cap alpha../lambda/sup 1/)(del sub ..beta../lambda/sup 2/) = 0 is necessary and sufficient for Schroedinger equation separability in case (a) of Carter. A Newtonian analog of our results is discussed. (AIP)

• Research Organization: Department of Physics, Illinois Institute of Technology, Chicago, Illinois 60616
• OSTI ID: 7281934
• Journal Information: J. Math. Phys. (N.Y.); (United States), Journal Name: J. Math. Phys. (N.Y.); (United States) Vol. 17:7; ISSN JMAPA
• Country of Publication: United States
• Language: English

Related Subjects

657003* -- Theoretical & Mathematical Physics-- Relativity & Gravitation
71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
DIFFERENTIAL EQUATIONS
EINSTEIN FIELD EQUATIONS
EQUATIONS
FIELD EQUATIONS
FIELD THEORIES
GENERAL RELATIVITY THEORY
METRICS
SCHROEDINGER EQUATION
SPACE-TIME
TENSORS
WAVE EQUATIONS