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Details of Grant 

EPSRC Reference: EP/Y021487/1
Title: Rigidity Problems in Holography and Relativity
Principal Investigator: Shao, Dr C A
Other Investigators:
Researcher Co-Investigators:
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Department: Sch of Mathematical Sciences
Organisation: Queen Mary University of London
Scheme: Standard Research - NR1
Starts: 01 January 2024 Ends: 31 December 2024 Value (£): 77,733
EPSRC Research Topic Classifications:
Mathematical Analysis Mathematical Physics
EPSRC Industrial Sector Classifications:
No relevance to Underpinning Sectors
Related Grants:
Panel History:
Panel DatePanel NameOutcome
07 Sep 2023 EPSRC Mathematical Sciences Small Grants Panel September 2023 Announced
Summary on Grant Application Form
The proposed short project addresses outstanding problems in mathematical relativity pertaining to questions of rigidity, or uniqueness. Broadly speaking, the research lies within two main themes: (1) holography and the AdS/CFT correspondence, and (2) black hole rigidity.

First, the AdS/CFT correspondence - an especially influential research direction toward the question of reconciling relativity and quantum mechanisms - posits a correspondence between the gravitational dynamics of asymptotically Anti-de Sitter (aAdS) spacetimes and a conformal field theory (CFT) on their boundaries. In particular, this theory serves as a realisation of the holographic principle, since the boundary theory is in one dimension less than that of the spacetime. Although these ideas have led to numerous novel insights in physics (e.g. black hole entropy, superconductors), there are very few rigorous mathematical statements and theorems pertaining to the AdS/CFT correspondence. As a result, there is a timely need to develop a mathematical foundation for these recent physical advancements, and the first theme of the proposed project addresses precisely this need.

In my recent work, I have formulated this correspondence as a rigidity problem for the Einstein-vacuum equations (a system of elaborate partial differential equations) from the conformal boundary of aAdS spacetimes, and I have proved that this correspondence indeed holds under appropriate geometric conditions on the boundary. The project builds upon this milestone by addressing a number of new questions arising from the above-mentioned result:

- What are the optimal conditions needed on the boundary for the above-mentioned correspondence result to hold? Do these conditions confirm the assumptions adopted by the theoretical physics community?

- Does a similar correspondence hold for aAdS spacetimes satisfying the Einstein equations coupled to matter fields (e.g. electromagnetic or fluid)?

Second, the black hole rigidity conjecture - which roughly posits that the only stationary, vacuum, black hole spacetimes are the Kerr spacetimes (modelling rotating black holes) - is pivotal to understanding the late-time and asymptotic behaviours of the universe. In particular, an affirmation of black hole rigidity would provide strong evidence in favour of the final state conjecture, i.e. that asymptotically flat vacuum spacetimes eventually decouple into Kerr black holes and gravitational radiation. Though a proof of this final state conjecture is currently well out of reach, the conjecture itself already closely informs how physicists interpret observational data on gravitational waves from LIGO.

The black hole rigidity conjecture itself has an extensive history, starting from the 1970s, and has seen significant partial progress in recent years. However, the full conjecture remains wide open, and it is believed that new mathematical ideas will be needed to resolve this. Thus, the second theme of the project aims to build expertise toward this by tackling some related but more tractable questions:

- Do the current state-of-the-art rigidity results also apply to extremal black hole spacetimes, which have the minimal possible mass to angular momentum ratio? In other words, do black hole rigidity results hold for the entire range of black hole parameters?

- In asymptotically AdS settings, do similar rigidity results hold for the Kerr-AdS black hole spacetimes (that is, the direct aAdS analogues of Kerr spacetimes), and under what additional assumptions? In particular, such a characterisation of Kerr-AdS black holes would also be especially relevant toward AdS/CFT.
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