## Periodic Reporting for period 1 - QuEST (Quantum Energy Conditions and Singularity Theorems)

Reporting period: 2017-09-01 to 2019-08-31

In the context of general relativity, a singularity is defined as a boundary point of spacetime beyond which no extension is possible, and a singularity theorem is a proof that singularities are inevitable under certain conditions. Although singularities were discovered early in the history of general relativity in solutions to Einstein’s field equations, the question of whether singularities occur in the physical universe (e.g. inside black holes or at the big bang) remains open and continues to intrigue both mathematicians and physicists. Work on singularities can help answer questions on the origin of our Universe and give insights into a complete theory that would unify gravity with other forces.

Fifty years ago, Hawking and Penrose developed the first general singularity theorems in classical General Relativity. These theorems showed that singularities exist in any spacetime that satisfies certain properties. Some of these properties are mild assumptions on spacetime geometry but others depend on matter content and are more problematic. For example, Hawking assumed a Strong Energy Condition (SEC) asserting positivity of the effective energy density (EED). Unfortunately quantised matter as described by quantum field theory (QFT) allows states that violate these energy conditions and so it is necessary to try to prove singularity theorems under weaker assumptions. The main goal of the research project was to establish mathematically rigorous singularity theorems for quantized matter fields based on quantum energy inequalities (QEIs) that constrain local averages of quantities like the EED.

Final period conclusions: During the project considerable progress was made towards deriving singularity theorems for quantised matter, and work is nearly complete on a Hawking-type singularity theorem using hypotheses satisfied by a quantum field theory. Along the way, we have derived a classical singularity theorem for the Einstein-Klein-Gordon system, a QSEI for nonminimally coupled scalar fields and a new method for proving singularity theorems (of both types) with weakened energy conditions. Additional results were derived during the duration of the project or are expected in the near future and they are discussed below.

Fifty years ago, Hawking and Penrose developed the first general singularity theorems in classical General Relativity. These theorems showed that singularities exist in any spacetime that satisfies certain properties. Some of these properties are mild assumptions on spacetime geometry but others depend on matter content and are more problematic. For example, Hawking assumed a Strong Energy Condition (SEC) asserting positivity of the effective energy density (EED). Unfortunately quantised matter as described by quantum field theory (QFT) allows states that violate these energy conditions and so it is necessary to try to prove singularity theorems under weaker assumptions. The main goal of the research project was to establish mathematically rigorous singularity theorems for quantized matter fields based on quantum energy inequalities (QEIs) that constrain local averages of quantities like the EED.

Final period conclusions: During the project considerable progress was made towards deriving singularity theorems for quantised matter, and work is nearly complete on a Hawking-type singularity theorem using hypotheses satisfied by a quantum field theory. Along the way, we have derived a classical singularity theorem for the Einstein-Klein-Gordon system, a QSEI for nonminimally coupled scalar fields and a new method for proving singularity theorems (of both types) with weakened energy conditions. Additional results were derived during the duration of the project or are expected in the near future and they are discussed below.

The project had four main outcomes.

1. We have proved a singularity theorem for the classical Einstein-Klein-Gordon theory in which the SEC can be violated, thus preventing the use of Hawking's original theorem. We first derived lower bounds on local averages of the effective energy density (EED) for solutions to the Klein–Gordon equation, which were then used to prove a singularity theorem. This shows that all solutions of this theory with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. The required initial contraction was calculated for cosmological applications. (Publication [1].)

2. We have derived a mathematically rigorous quantum strong energy inequality (QSEI) for nonminimally coupled scalar fields valid in general spacetimes. As had been anticipated, these QSEIs depend on the state of interest. The state-dependence of these bounds in Minkowski spacetime for thermal (KMS) states was analyzed, and it was shown that the lower bounds grow more slowly in magnitude than the EED itself as temperature increases. The lower bounds are therefore of lower energetic order than the EED, and qualify as nontrivial state-dependent QEIs. (Publication [2].)

3. We have developed a new method of proving singularity theorems with weakened energy conditions that avoids the Raychaudhuri equation but instead makes use of index form methods. These results improve over existing methods and can be applied to hypotheses inspired by QEIs. In that case, quantitative estimates of the initial conditions required for our singularity theorems to apply were made. (Reference [3]; currently under peer-review.)

4. Finally, we have made progress towards the first derivation of a semiclassical singularity theorem, combining the methods of part 3 with the QSEI bound of part 2. This joint work of the ER and the supervisor is expected to appear as a pre-print in the near future.

A short summary of parts 1, 2 and 4 has been produced as a conference proceedings article [4] for the Proceedings of the 15th Marcel Grossman meeting.

References:

[1] PJ Brown, CJ Fewster and E-A Kontou, A singularity theorem for Einstein-Klein-Gordon theory. General Relativity and Gravitation 50 (2018) 121 (24pp). DOI: 10.1007/s10714-018-2446-5. arXiv:1803.11094

[2] CJ Fewster and E-A Kontou, Quantum strong energy inequalities. Phys. Rev. D 99 (2019) 045001 (17pp). DOI: 10.1103/PhysRevD.99.045001 arXiv:1809.05047

[3] CJ Fewster and E-A Kontou, A new derivation of singularity theorems with weakened energy hypotheses (27pp). arXiv:1907.13604

[4] PJ Brown, CJ Fewster and E-A Kontou, Classical and quantum strong energy inequalities and the Hawking singularity theorem (6pp). arXiv:1904.00419

1. We have proved a singularity theorem for the classical Einstein-Klein-Gordon theory in which the SEC can be violated, thus preventing the use of Hawking's original theorem. We first derived lower bounds on local averages of the effective energy density (EED) for solutions to the Klein–Gordon equation, which were then used to prove a singularity theorem. This shows that all solutions of this theory with sufficient initial contraction at a compact Cauchy surface will be future timelike geodesically incomplete. The required initial contraction was calculated for cosmological applications. (Publication [1].)

2. We have derived a mathematically rigorous quantum strong energy inequality (QSEI) for nonminimally coupled scalar fields valid in general spacetimes. As had been anticipated, these QSEIs depend on the state of interest. The state-dependence of these bounds in Minkowski spacetime for thermal (KMS) states was analyzed, and it was shown that the lower bounds grow more slowly in magnitude than the EED itself as temperature increases. The lower bounds are therefore of lower energetic order than the EED, and qualify as nontrivial state-dependent QEIs. (Publication [2].)

3. We have developed a new method of proving singularity theorems with weakened energy conditions that avoids the Raychaudhuri equation but instead makes use of index form methods. These results improve over existing methods and can be applied to hypotheses inspired by QEIs. In that case, quantitative estimates of the initial conditions required for our singularity theorems to apply were made. (Reference [3]; currently under peer-review.)

4. Finally, we have made progress towards the first derivation of a semiclassical singularity theorem, combining the methods of part 3 with the QSEI bound of part 2. This joint work of the ER and the supervisor is expected to appear as a pre-print in the near future.

A short summary of parts 1, 2 and 4 has been produced as a conference proceedings article [4] for the Proceedings of the 15th Marcel Grossman meeting.

References:

[1] PJ Brown, CJ Fewster and E-A Kontou, A singularity theorem for Einstein-Klein-Gordon theory. General Relativity and Gravitation 50 (2018) 121 (24pp). DOI: 10.1007/s10714-018-2446-5. arXiv:1803.11094

[2] CJ Fewster and E-A Kontou, Quantum strong energy inequalities. Phys. Rev. D 99 (2019) 045001 (17pp). DOI: 10.1103/PhysRevD.99.045001 arXiv:1809.05047

[3] CJ Fewster and E-A Kontou, A new derivation of singularity theorems with weakened energy hypotheses (27pp). arXiv:1907.13604

[4] PJ Brown, CJ Fewster and E-A Kontou, Classical and quantum strong energy inequalities and the Hawking singularity theorem (6pp). arXiv:1904.00419

All the completed projects described in the previous section of the report represent progress beyond the state of the art. Specifically:

* Our index-form method is a simpler and much more general way to prove singularity theorems under weakened conditions than previous methods.

* No classical or quantum inequality bounds were previously known for theories violating the strong energy condition.

* When completed, our current project will be the first singularity theorem for a matter described by quantum fields under general conditions.

This is theoretical work without direct technological applications in view. However, it attracts significant public interest, reflected by the audiences at the two public outreach events during this project. The first was an event “Travels in time, fiction and physics” at the Festival of Ideas in York, a popular festival featuring over 150 events every year including a wide range of public talks from experts. The ER coordinated the event, which included presentations from the ER, the supervisor and two historians of science from the University of York as well as discussion with the audience. The event was highly successful with over 160 attendees and positive feedback.

The second event was a public talk “Can we create wormholes using quantum fields?” by the ER as part of the “Pint of Science” series - a worldwide science festival which brings researchers to local pubs to present their scientific discoveries. There are 600 events every year across the UK. The event sold out and received highly positive feedback.

During the fellowship the ER delivered over 15 talks in invited seminars as well as international and local conferences. For example, she spoke at the 15th Marcel Grossman meeting, GR22, APS meeting and delivered seminars at the Perimeter Institute, Penn State University and the University of Nottingham. She also gave a talk at the North British Mathematical Physics Seminar.

As part of the fellowship, the two-day workshop “Energy conditions in quantum field theory and gravity” was organized at the University of York by the ER and the supervisor. The workshop was considered very successful with a total of 35 participants and 10 invited speakers, all of whom are international experts in the field.

* Our index-form method is a simpler and much more general way to prove singularity theorems under weakened conditions than previous methods.

* No classical or quantum inequality bounds were previously known for theories violating the strong energy condition.

* When completed, our current project will be the first singularity theorem for a matter described by quantum fields under general conditions.

This is theoretical work without direct technological applications in view. However, it attracts significant public interest, reflected by the audiences at the two public outreach events during this project. The first was an event “Travels in time, fiction and physics” at the Festival of Ideas in York, a popular festival featuring over 150 events every year including a wide range of public talks from experts. The ER coordinated the event, which included presentations from the ER, the supervisor and two historians of science from the University of York as well as discussion with the audience. The event was highly successful with over 160 attendees and positive feedback.

The second event was a public talk “Can we create wormholes using quantum fields?” by the ER as part of the “Pint of Science” series - a worldwide science festival which brings researchers to local pubs to present their scientific discoveries. There are 600 events every year across the UK. The event sold out and received highly positive feedback.

During the fellowship the ER delivered over 15 talks in invited seminars as well as international and local conferences. For example, she spoke at the 15th Marcel Grossman meeting, GR22, APS meeting and delivered seminars at the Perimeter Institute, Penn State University and the University of Nottingham. She also gave a talk at the North British Mathematical Physics Seminar.

As part of the fellowship, the two-day workshop “Energy conditions in quantum field theory and gravity” was organized at the University of York by the ER and the supervisor. The workshop was considered very successful with a total of 35 participants and 10 invited speakers, all of whom are international experts in the field.