Monochromatic plane-fronted waves in conformal gravity are pure gauge

Luca Fabbri; M. B. Paranjape

doi:10.1103/PhysRevD.83.104046

paper

Monochromatic plane-fronted waves in conformal gravity are pure gauge

arXiv:0812.2491 · doi:10.1103/PhysRevD.83.104046

Abstract

We consider plane-fronted, monochromatic gravitational waves on a Minkowski background, in a conformally invariant theory of general relativity. By this we mean waves of the form: $g_{μν}=η_{μν}+ε_{μν}F(k\cdotx)$, where $ε_{μν}$ is a constant polarization tensor, and $k_{μ}$ is a lightlike vector. We also assume the coordinate gauge condition $∣ g ∣ - 1/4 \partial_{τ} (∣ g ∣1/4 g^{σ τ}) = 0$ which is the conformal analog of the harmonic gauge condition $g^{μν} Γ_{μν}^{σ} = - ∣ g ∣ - 1/2 \partial_{τ} (∣ g ∣1/2 g^{σ τ}) = 0, w h er e$ \det[g_{μν}]\equivg $. R e q u i r in g a dd i t i o na l l y t h eco n f or ma l g a ug eco n d i t i o n$ g=-1 $s u r p r i s in g l y im pl i es t ha tt h e w a v es a r e b o t h t r an s v er se an d t r a ce l ess . A l t h o ug h t h e an s a t z f or t h e m e t r i c i se min e n tl y r e a so nab l e w h e n co n s i d er in g p er t u r ba t i v e g r a v i t a t i o na l w a v es, w es h o w t ha tt h e m e t r i c i sr e d u c ib l e t o t h e m e t r i co f M ink o w s k i s p a ce - t im e v iaa se q u e n ceo f coor d ina t e t r an s f or ma t i o n s w hi c h r es p ec tt h e g a ug eco n d i t i o n s, w i t h o u t an y p er t u r ba t i v e a pp r o x ima t i o n t ha t \in μν b es ma l l . T hi s im pl i es t ha tw e ha v e, in f a c t, e x a c tpl an e - w a v eso l u t i o n s; h o w e v er, t h ey a r es im pl y coor d ina t e / co n f or ma l a r t i f a c t s . A s a co n se q u e n ce, t h ey c a r r y n oe n er g y . O u r r es u l t d oes n o t im pl y t ha t co n f or ma l g r a v i t y d oes n o t ha v e g r a v i t a t i o na l w a v e p h e n o m e na . A d i f f er e n t, m or e g e n er a l i z e d an s a t z f or t h e d e v ia t i o n, t ak in g in t o a cco u n tt h e f o u r t h - or d er na t u r eo f t h e f i e l d e q u a t i o n, w hi c hha s t h e f or m$ g_{μν}=η_{μν}+B_{μν}(n\cdotx)G(k\cdotx)$, indeed yields waves which carry energy and momentum [P.D. Mannheim, Gen. Relativ. Gravit. 43, 703 (2010)]. It is just surprising that transverse, traceless, plane-fronted gravitational waves, those that would be used in any standard, perturbative, quantum analysis of the theory, simply do not exist.

5 pages, no figures, version published, substantial changes to the presentation, conclusions unaltered, title changed

Monochromatic plane-fronted waves in conformal gravity are pure gauge

Abstract

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