Modified Black Hole Spacetimes
Källan Berglund
Sejny Summer Institute July 2021
Interstellar
About me
Källan (Kay-lan) Berglund, pronouns she/her Honors in Astrophysics from Brown
5th year PhD Candidate Penn State, Institute for Gravitation and the Cosmos
Advisors: Martin Bojowald at Penn State and Karim Noui at the University of Paris Active in equity and inclusion on many fronts
Love social dancing, martial arts, and the outdoors
Interested in quantum gravity, black holes, and the nature of spacetime.
Outline
1. Overview 2. Background
3. Modified Spacetime 4. Black Hole Volume
5. Current Modified Volume Project
6. Future Quasinormal Mode Predictions 7. Summary
8. Discussion and Activities
Sky & Telescope
1) Overview
Use spacetime from modified constraint Recreate black hole volume calculations, with modification
Probe behavior near evaporation Future work: calculate quasinormal modes of gravitational wave signals (predictions testable with LISA in ~9
years) NASA JPL
Importance
Put a bound on the information capacity Connect String Theory and LQG through qualitative entropy-volume relationship Combine modified spacetime model with evaporation models
Ultimately, extend the modified spacetime to a full exterior solution
Narrow the scope of possible theories of quantum gravity
NASA
2) Background
Units: G = c = 1
Black hole: extremely dense point (singularity) warping spacetime such that light cannot
escape within the Schwarzschild radius Metric: mathematical object (tensor) that
contains the information about the structure of spacetime and is used to measure distances Constraint: physical restriction on the system Spacetime diagrams
Terminology
I will refer to black hole entropy, information capacity/content, and computability essentially interchangeably. They refer to the same property in different contexts.
Entropy: measure of disorder
Information content/capacity: amount of information that has been consumed or could be stored inside the horizon
Computability: the ability/time to calculate information on the interior
This property is relevant to the information paradox, a testing ground for the validity of many theories.
3) Effective line elements
Not “modified gravity,” as in modified actions (such as modifying the higher curvature term in scalar-tensor theories) Modify Hamiltonian constraint
Preserve covariance
Propagate to line element
Produces signature change boundary
Pixabay
Introducing the modification
Modify the Hamiltonian constraint by replacing the term quadratic in extrinsic curvature (K_phi) with an oscillating, adjustable function f_1
Where primes are derivatives w.r.t K_phi and delta is a parameter which controls the strength of the modification, usually chosen to be Planckian (0 classical limit)
(2)
Modified line element
Propagate the modification through to the line element by imposing the condition that covariance be maintained (Poisson brackets between constraints closed) Where N is the lapse function and N^x is the shift vector
Note that the beta function changes from + to -, causing the signature change
(2) (2)
Advantages: Consistency Conditions
Impose: Covariance (brackets closed)
Assume: Maintain the second-order nature of equations of motion
Implies: Preserved speed of gravitational waves relative to the speed of light (essentially the same, from LIGO observations)
LIGO
Distinctive Feature: Signature Change
Near r=0, the signature of the modified metric changes due to beta (2)
This replaces the singularity with a region of Euclidean space
The boundary of this region is the point of signature change, in the
high-curvature region near what would have been the singularity
Spacetime is modified outside the point of signature change as well.
4) Volume Calculation
Maximal spatial slicing (foliation)
Use as covariant volume definition for black hole interior
Finite, increasing change in volume Even in evaporating case, volume continues increasing
Scientific American
Hypersurfaces and Foliations
Imagine spacetime as a block of cheese
Cutting at an angle, there will be values on the slice with different time coordinates.
Cutting a uniform slice perpendicular to the time edge, all points on the slice will have the same time coordinate.
The latter is an example of a spatial hypersurface.
A foliation is a collection of hypersurfaces which span the entire space, slicing the entire block of cheese.
Getty Images
Hypersurface Deformations
Ex: Lorentz transformation/boost tilts the axis of a spacetime diagram.
In canonical gravity, expressions like the Hamiltonian and Diffeomorphism constraints generate hypersurface deformations.
Spacelike foliations are the type of hypersurfaces we are interested in here.
Ex: evolving in time from one slice of cheese to the next is a deformation.
diys.com
Lie Algebroids
Lie algebroids are fiber bundles on a base manifold.
Ex: the base manifold and the union of tangent bundles at every point.
Limit: If the base manifold is a single point, then the algebroid is an algebra.
Spacetime diffeomorphism can be represented as hypersurface deformations classically and mapped onto Lie Algebroids, but we need to reverse this process and adapt it for modified spacetimes. (7)
Developing this mathematical tool could allow us to extend the modified spacetime to a solution for the exterior of the black hole.
Maximally slicing a black hole
This paper constructs a foliation out of maximal spatial hypersurfaces, defined by the following requirement:
(3) Slices have infinite volume, for an eternal black hole
Slices appear to converge at the boundary r = 3/2 M
How big is a black hole?
Accounting for formation (4), slices terminate (no longer infinite) on the left.
We are only interested in the interior.
Examine change in volume from one slice to the next: finite, and increasing with Eddington-Finkelstein null
coordinate v = r + t
On the volume inside old black holes
Now accounting for evaporation (5) Mass is now a function of advanced Eddington-Finkelstein time
Volume still increasing, despite evaporation!
(same equation, but with initial mass)
5) My project: What Changes with Modification?
r = 3/2 M stays in low-curvature region
Signature change stays in high-curvature region Near evaporation, high-curvature region extends towards horizon
How will the two interact approaching evaporation?
Will r = 3/2 M boundary deform? Will they collide?
How can we use this to probe evaporation?
Worksheet 1: Foundations
Reflect on the questions from worksheet 1.
Free-write some questions about the material so far.
Prepare page 2 with anonymous questions for collection and class discussion.
Discuss your responses in small groups while I collect page 2 and look over the questions.
Next I will go into some detail on calculation procedure and speculative outcomes.
Worksheet 1 Questions
1) What is the general topic and goal of this project?
2) How was this modified spacetime constructed and why?
3) How is black hole volume being defined? How does it behave over time?
4) What questions could this project answer?
5) Free-write questions about the presentation:
Current Work: Calculation
Using triad variables,
I calculated the poisson brackets for the equations of motion,
In order to determine the effect of the modification on the lapse and shift.
I will propagate the modification through the volume calculations And examine the behavior at late-stage evaporation.
Constraint Equations
(2)
(1)
Equations of Motion
Lapse and Shift
Lapse function describes movement between spatial slices, and shift vector describes displacement within a slice.
Compare equations of lapse and shift with and without modification, to
determine the effect of the modification.
Use the modified lapse and shift to recreate the volume calculations.
Then analyze modified volume for the various inquiries mentioned earlier.
Possible Outcome
What if the maximal spatial slices pass the point of signature change and enter the Euclidean region?
Then there would be infinite options of infinite length paths in the Euclidean space, which could cause the definition of volume to break down near
evaporation.
Phase Transitions
We can conceptualize spatial slices as going through a phase transition when crossing the point of signature change.
The hypersurfaces are now free to move anywhere in the four dimensions.
This is not exactly time evolution because our concept of time comes to a stop.
This concept of phase transition can be related to quantum information theory via entropy.
Study.com
Computability
A reservoir of volume is needed to build quantum qubits.
Signature change provides space in the Euclidean region.
But it also removes time.
Evolution of states requires nature to do a computation.
So what does it mean for something to happen around signature change?
6) Future Work: Quasinormal Modes Collaboration
Collaborate with Karim Noui at Université Paris Diderot (6) Bypass the need for a modified line element on the
exterior by conducting a novel derivation of the QNM equations directly from the modified constraints.
Requires more than spherical symmetry, so additional work is required to figure out the correct gauge condition (lapse and shift).
Introduce a small perturbation in all canonical fields, then expand the constraints and solve resulting linear PDE’s
LIGO
Prediction
Calculate gravitational wave quasinormal mode signals detectable by LISA in
approximately nine years.
These measurements could either rule out or support certain modifications.
These novel results would limit the scope of possible theories of quantum gravity.
NASA
7) Summary
Recreate volume calculations with modified spacetime Put a bound on the information capacity
Connect String Theory and LQG through qualitative entropy-volume relationship Explore effect of signature change on volume
Probe near-evaporation stage
Combine modified spacetime model with evaporation models (argument for large-volume remnant)
Next project: predict testable modified quasinormal mode signals
Worksheet 1: Check-in
Reflect on the questions from worksheet 1.
Free-write some questions about the additional material.
Prepare page 3 with anonymous questions for collection and class discussion.
Discuss your responses in small groups while I collect page 3 and look over the questions.
Worksheet 1 Questions
1) What is the general topic and goal of this project?
2) How was this modified spacetime constructed and why?
3) How is black hole volume being defined? How does it behave over time?
4) What questions could this project answer?
5) Free-write questions about the presentation:
Worksheet 2: Making Connections
Reflect on the questions from worksheet 2.
Discuss your responses in small groups.
Continue to consider these questions throughout the following activities.
Nature
Worksheet 2 Prompts
Philosophers: information theory and the nature of observations, as well as the time question of signature change in our modified spacetimes.
Mathematicians: understanding these non-Riemannian geometries, as well as the representation of spacetime symmetries as a Lie algebroid.
Computer scientists: spacetime simulations and information theory arguments about the
computability/information content of black holes, and the relationship I have been exploring with black hole volumes.
Physicists: black hole interiors, behavior of late-stage evaporation, and connections to other theories of quantum gravity.
1) What research questions do you have surrounding this project?
2) In what ways does this topic connect to your work and expertise?
3) Is there a part of this project you would like to contribute to? If so, how do you imagine your involvement?
Worksheet 3: Research Questions
Reflect on the questions from worksheet 3.
Form groups around questions of interest.
Discuss your chosen question in small groups.
Depending on interest, we may shuffle groups and repeat, so you can discuss additional
questions.
nichq.org
Worksheet 3 Research Ideas
How can different modified versions of the Lie algebroid describing hypersurface deformations be related to spacetime geometry in 4D?
How can we redefine time to make sense of dynamics and phase transitions at, and beyond, the point of signature change?
In what specific ways does the beta function affect the dynamics between the horizon and signature change?
How can we formalize the volume relationship to entropy, computability, and information content in order to compare with other results?
If/when the volume slicing crosses the point of signature change, what will be our new understanding of volume?How does this new concept of volume relate to phase transitions, entropy, and quantum information?
Worksheet 4: Readings
Look over the optional reading recommendations.
Form groups around papers of interest.
Skim and discuss your chosen paper in small groups.
Depending on interest, we may shuffle groups and repeat, so you can discuss additional papers.
Bibliography
1. Bojowald, Martin. (2006). Quantum Riemannian Geometry and Black Holes.
2. Bojowald, Martin & Brahma, Suddhasattwa & Yeom, Dong-han. (2018). Effective line elements and black-hole models in canonical (loop) quantum gravity.
PhysRevD.98.046015.
3. Frank Estabrook, Hugo Wahlquist, Steven Christensen, Bryce DeWitt, Larry Smarr, and Elaine Tsiang. (1973). Maximally Slicing a Black Hole. PhysRevD.7.2814
4. Marios Christodoulou and Carlo Rovelli. (2015). How big is a black hole?
PhysRevD.91.064046
5. Marios Christodoulou and Tommaso De Lorenzo. (2016). Volume inside old black holes.
PhysRevD.94.104002
6. Jibril Ben Achour, Frédéric Lamy, Hongguang Liu, and Karim Noui. (2018). Polymer Schwarzschild black hole: An effective metric. EPL (Europhys. Lett.) 123, 20006.
7. Christian Blohmann, Marco Cezar Barbosa Fernandes, Alan Weinstein (2010).
Groupoid symmetry and constraints in general relativity arXiv:1003.2857
