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 = ± ∞ .
Communications in Mathematical Physics – Springer Journals
Published: Nov 4, 2017
It’s your single place to instantly
discover and read the research
that matters to you.
Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.
All for just $49/month
Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly
Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.
Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.
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.
“Hi guys, I cannot tell you how much I love this resource. Incredible. I really believe you've hit the nail on the head with this site in regards to solving the research-purchase issue.”
Daniel C.
“Whoa! It’s like Spotify but for academic articles.”
@Phil_Robichaud
“I must say, @deepdyve is a fabulous solution to the independent researcher's problem of #access to #information.”
@deepthiw
“My last article couldn't be possible without the platform @deepdyve that makes journal papers cheaper.”
@JoseServera
DeepDyve Freelancer | DeepDyve Pro | |
---|---|---|
Price | FREE | $49/month |
Save searches from | ||
Create lists to | ||
Export lists, citations | ||
Read DeepDyve articles | Abstract access only | Unlimited access to over |
20 pages / month | ||
PDF Discount | 20% off | |
Read and print from thousands of top scholarly journals.
Already have an account? Log in
Bookmark this article. You can see your Bookmarks on your DeepDyve Library.
To save an article, log in first, or sign up for a DeepDyve account if you don’t already have one.
All DeepDyve websites use cookies to improve your online experience. They were placed on your computer when you launched this website. You can change your cookie settings through your browser.
ok to continue