Some new results on the validity of Huygens' principle for the scalar wave equation on a curved space-time

2005 ◽  
pp. 138-142
Author(s):  
J. Carminati ◽  
R. G. McLenaghan
Keyword(s):  
Wave Equation ◽  
Space Time ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Curved Space ◽  
Huygens Principle

scholarly journals THE SCALAR WAVE EQUATION IN A NON-COMMUTATIVE SPHERICALLY SYMMETRIC SPACE-TIME

2008 ◽  
Vol 05 (01) ◽  
pp. 33-47 ◽  
Author(s):  
ELISABETTA DI GREZIA ◽  
GIAMPIERO ESPOSITO ◽  
GENNARO MIELE
Keyword(s):  
Asymptotic Behavior ◽  
Wave Equation ◽  
Rest Mass ◽  
Physical Space ◽  
Scalar Fields ◽  
Space Time ◽  
Asymptotic Behavior Of Solutions ◽  
Scalar Wave ◽  
Null Infinity ◽  
Scalar Wave Equation

Recent work in the literature has studied a version of non-commutative Schwarzschild black holes where the effects of non-commutativity are described by a mass function depending on both the radial variable r and a non-commutativity parameter θ. The present paper studies the asymptotic behavior of solutions of the zero-rest-mass scalar wave equation in such a modified Schwarzschild space-time in a neighborhood of spatial infinity. The analysis is eventually reduced to finding solutions of an inhomogeneous Euler–Poisson–Darboux equation, where the parameter θ affects explicitly the functional form of the source term. Interestingly, for finite values of θ, there is full qualitative agreement with general relativity: the conformal singularity at spacelike infinity reduces in a considerable way the differentiability class of scalar fields at future null infinity. In the physical space-time, this means that the scalar field has an asymptotic behavior with a fall-off going on rather more slowly than in flat space-time.

An explicit determination of the empty space-times on which the wave equation satisfies Huygens' principle

1969 ◽  
Vol 65 (1) ◽  
pp. 139-155 ◽  
Author(s):  
R. G. McLenaghan
Keyword(s):  
Wave Equation ◽  
Wave Equations ◽  
Empty Space ◽  
Space Time ◽  
Necessary Condition ◽  
Scalar Wave ◽  
Wave Space ◽  
Huygens Principle ◽  
Necessary And Sufficient

AbstractThe validity of Huygens' principle in the sense of Hadamard's ‘minor premise’ is investigated for scalar wave equations on curved space-time. A new necessary condition for its validity in empty space-time is derived from Hadamard's necessary and sufficient condition using a covariant Taylor expansion in normal coordinates. A two component spinor calculus is then employed to show that this necessary condition implies that the plane wave space-times and Minkowski space are the only empty space-times on which the scalar wave equation satisfies Huygens' principle.

The scalar wave equation in a Schwarzschild space-time

1979 ◽  
Vol 367 (1731) ◽  
pp. 503-525 ◽  
Keyword(s):  
Wave Equation ◽  
Rest Mass ◽  
Flat Space ◽  
Cauchy Data ◽  
Space Time ◽  
Scalar Wave ◽  
Spatial Infinity ◽  
Null Infinity ◽  
Scalar Wave Equation ◽  
Near Future

This paper studies the asymptotic behaviour of solutions of the zero rest mass scalar wave equation in the Schwarzschild space-time in a neighbourhood of spatial infinity which includes parts of future and past null infinity. The behaviour of such fields is essentially different from that which occurs in a flat space-time. In particular fields which have a Bondi-type expansion in powers of ' r -1 ’ near past null infinity do not have such an expansion near future null infinity. Further solutions which have physically reasonable Cauchy data probably fail to have Bondi-type expansions near null infinity.

scholarly journals THE EASTWOOD–SINGER GAUGE IN EINSTEIN SPACES

2009 ◽  
Vol 06 (04) ◽  
pp. 583-593 ◽  
Author(s):  
GIAMPIERO ESPOSITO ◽  
RAJU ROYCHOWDHURY
Keyword(s):  
Wave Equation ◽  
Fourth Order ◽  
Space Time ◽  
Order Equation ◽  
Scalar Wave ◽  
Scalar Wave Equation ◽  
Fourth Order Equation ◽  
Vector Wave ◽  
Vector Wave Equation ◽  
Einstein Spaces

Electrodynamics in curved space-time can be studied in the Eastwood–Singer gauge, which has the advantage of respecting the invariance under conformal rescalings of the Maxwell equations. Such a construction is here studied in Einstein spaces, for which the Ricci tensor is proportional to the metric. The classical field equations for the potential are then equivalent to first solving a scalar wave equation with cosmological constant, and then solving a vector wave equation where the inhomogeneous term is obtained from the gradient of the solution of the scalar wave equation. The Eastwood–Singer condition leads to a field equation on the potential which is preserved under gauge transformations provided that the scalar function therein obeys a fourth-order equation where the highest-order term is the wave operator composed with itself. The second-order scalar equation is here solved in de Sitter space-time, and also the fourth-order equation in a particular case, and these solutions are found to admit an exponential decay at large time provided that square-integrability for positive time is required. Last, the vector wave equation in the Eastwood–Singer gauge is solved explicitly when the potential is taken to depend only on the time variable.

scholarly journals Regularity at space-like and null infinity

2008 ◽  
Vol 8 (1) ◽  
pp. 179-208 ◽  
Author(s):  
Lionel J. Mason ◽  
Jean-Philippe Nicolas
Keyword(s):  
Wave Equation ◽  
Vector Field ◽  
Flat Space ◽  
Space Time ◽  
Scalar Wave ◽  
Null Infinity ◽  
Field Methods ◽  
Scalar Wave Equation ◽  
Schwarzschild Metric ◽  
Delay Space

AbstractWe extend Penrose's peeling model for the asymptotic behaviour of solutions to the scalar wave equation at null infinity on asymptotically flat backgrounds, which is well understood for flat space-time, to Schwarzschild and the asymptotically simple space-times of Corvino–Schoen/Chrusciel–Delay. We combine conformal techniques and vector field methods: a naive adaptation of the ‘Morawetz vector field’ to a conformal rescaling of the Schwarzschild metric yields a complete scattering theory on Corvino–Schoen/Chrusciel–Delay space-times. A good classification of solutions that peel arises from the use of a null vector field that is transverse to null infinity to raise the regularity in the estimates. We obtain a new characterization of solutions admitting a peeling at a given order that is valid for both Schwarzschild and Minkowski space-times. On flat space-time, this allows larger classes of solutions than the characterizations used since Penrose's work. Our results establish the validity of the peeling model at all orders for the scalar wave equation on the Schwarzschild metric and on the corresponding Corvino–Schoen/Chrusciel–Delay space-times.

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