A Proof of Friedman’s Ergosphere Instability for Scalar Waves

A Proof of Friedman’s Ergosphere Instability for Scalar Waves Let $${(\mathcal{M}^{3+1},g)}$$ ( M 3 + 1 , g ) be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $${\mathscr{E}}$$ E and no future event horizon $${\mathcal{H}^{+}}$$ H + . In Friedman (Commun Math Phys 63(3):243–255, 1978), Friedman observed that, on such spacetimes, there exist solutions $${\varphi}$$ φ to the wave equation $${\square_{g}\varphi=0}$$ □ g φ = 0 such that their local energy does not decay to 0 as time increases. In addition, Friedman provided a heuristic argument that the energy of such solutions actually grows to $${+\infty}$$ + ∞ . In this paper, we provide a rigorous proof of Friedman’s instability. Our setting is, in fact, more general. We consider smooth spacetimes $${(\mathcal{M}^{d+1},g)}$$ ( M d + 1 , g ) , for any $${d\ge2}$$ d ≥ 2 , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $${\partial\mathscr{E}}$$ ∂ E of $${\mathscr{E}}$$ E on a small neighborhood of a point $${p\in\partial\mathscr{E}}$$ p ∈ ∂ E . This condition always holds if $${(\mathcal{M},g)}$$ ( M , g ) is analytic in that neighborhood of p, but it can also be inferred in the case when $${(\mathcal{M},g)}$$ ( M , g ) possesses a second Killing field $${\Phi}$$ Φ such that the span of $${\Phi}$$ Φ and the stationary Killing field T is timelike on $${\partial\mathscr{E}}$$ ∂ E . We also allow the spacetimes $${(\mathcal{M},g)}$$ ( M , g ) under consideration to possess a (possibly empty) future event horizon $${\mathcal{H}^{+}}$$ H + , such that, however, $${\mathcal{H}^{+}\cap\,\,\mathscr{E}=\emptyset}$$ H + ∩ E = ∅ (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira et al. in (Phys Rev D 89(12):124008, 2014). Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes. Applications of this estimate include a Morawetz-type bound for solutions $${\varphi}$$ φ of $${\square_{g}\varphi=0}$$ □ g φ = 0 with frequency support bounded away from $${{\omega}=0}$$ ω = 0 and $${{\omega}=\pm\infty}$$ ω = ± ∞ . http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Communications in Mathematical Physics Springer Journals

A Proof of Friedman’s Ergosphere Instability for Scalar Waves

Loading next page...
 
/lp/springer_journal/a-proof-of-friedman-s-ergosphere-instability-for-scalar-waves-MESiU8ExXn
Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag GmbH Germany
Subject
Physics; Theoretical, Mathematical and Computational Physics; Mathematical Physics; Quantum Physics; Complex Systems; Classical and Quantum Gravitation, Relativity Theory
ISSN
0010-3616
eISSN
1432-0916
D.O.I.
10.1007/s00220-017-3010-y
Publisher site
See Article on Publisher Site

Abstract

Let $${(\mathcal{M}^{3+1},g)}$$ ( M 3 + 1 , g ) be a real analytic, stationary and asymptotically flat spacetime with a non-empty ergoregion $${\mathscr{E}}$$ E and no future event horizon $${\mathcal{H}^{+}}$$ H + . In Friedman (Commun Math Phys 63(3):243–255, 1978), Friedman observed that, on such spacetimes, there exist solutions $${\varphi}$$ φ to the wave equation $${\square_{g}\varphi=0}$$ □ g φ = 0 such that their local energy does not decay to 0 as time increases. In addition, Friedman provided a heuristic argument that the energy of such solutions actually grows to $${+\infty}$$ + ∞ . In this paper, we provide a rigorous proof of Friedman’s instability. Our setting is, in fact, more general. We consider smooth spacetimes $${(\mathcal{M}^{d+1},g)}$$ ( M d + 1 , g ) , for any $${d\ge2}$$ d ≥ 2 , not necessarily globally real analytic. We impose only a unique continuation condition for the wave equation across the boundary $${\partial\mathscr{E}}$$ ∂ E of $${\mathscr{E}}$$ E on a small neighborhood of a point $${p\in\partial\mathscr{E}}$$ p ∈ ∂ E . This condition always holds if $${(\mathcal{M},g)}$$ ( M , g ) is analytic in that neighborhood of p, but it can also be inferred in the case when $${(\mathcal{M},g)}$$ ( M , g ) possesses a second Killing field $${\Phi}$$ Φ such that the span of $${\Phi}$$ Φ and the stationary Killing field T is timelike on $${\partial\mathscr{E}}$$ ∂ E . We also allow the spacetimes $${(\mathcal{M},g)}$$ ( M , g ) under consideration to possess a (possibly empty) future event horizon $${\mathcal{H}^{+}}$$ H + , such that, however, $${\mathcal{H}^{+}\cap\,\,\mathscr{E}=\emptyset}$$ H + ∩ E = ∅ (excluding, thus, the Kerr exterior family). As an application of our theorem, we infer an instability result for the acoustical wave equation on the hydrodynamic vortex, a phenomenon first investigated numerically by Oliveira et al. in (Phys Rev D 89(12):124008, 2014). Furthermore, as a side benefit of our proof, we provide a derivation, based entirely on the vector field method, of a Carleman-type estimate on the exterior of the ergoregion for a general class of stationary and asymptotically flat spacetimes. Applications of this estimate include a Morawetz-type bound for solutions $${\varphi}$$ φ of $${\square_{g}\varphi=0}$$ □ g φ = 0 with frequency support bounded away from $${{\omega}=0}$$ ω = 0 and $${{\omega}=\pm\infty}$$ ω = ± ∞ .

Journal

Communications in Mathematical PhysicsSpringer Journals

Published: Nov 4, 2017

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 12 million articles from more than
10,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Unlimited reading

Read as many articles as you need. Full articles with original layout, charts and figures. Read online, from anywhere.

Stay up to date

Keep up with your field with Personalized Recommendations and Follow Journals to get automatic updates.

Organize your research

It’s easy to organize your research with our built-in tools.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve Freelancer

DeepDyve Pro

Price
FREE
$49/month

$360/year
Save searches from
Google Scholar,
PubMed
Create lists to
organize your research
Export lists, citations
Read DeepDyve articles
Abstract access only
Unlimited access to over
18 million full-text articles
Print
20 pages/month
PDF Discount
20% off