Talk presented at the 30th Midwest Relativity Meeting (Virtual), University of Notre Dame, IN, USA, October 23, 2020, https://doi.org/10.6084/m9.figshare.13699531

1. Tendex and Vortex Lines
around Spinning
Supermassive Black Holes
Ashkbiz Danehkar, Postdoc
Department of Astronomy, University of Michigan
danehkar@umich.edu
The 30th
Midwest Relativity Meeting, University of Notre Dame, Indiana, USA, October 23, 2020
Image
Credit:
J.
Bergeron,
Sky
&
Telescope
Magazine
Image
Credit:
J.
Bergeron,
Sky
&
Telescope
Magazine

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Motivation for Tendex and Vortex
Observational Background
X-ray Observations of AGN (1999-present)
Chandra X-ray Observatory (1999-present)
XMM-Newton, X-ray Multi-Mirror Mission (2000-present)
NuSTAR, Nuclear Spectroscopic Telescope Array (2012-present)
Gierlinski + 1999
Lonanov 2007
Credit: Roen Kelly @ Astronomy

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Observational Background
AGN Classification: Radio-loud & Radio-quiet AGN
Dermer & Giebles 2016

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Observational Background
Evidence for Unified AGN Outflow Model
Correlation between outflow kinematics and physical conditions
Tombesi + 2013
(Ultra-fast outflows)
(Warm Absorbers)

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Observational Background
Evidence for Unified AGN Outflow Model
Implication of SMBH Angular Momentum for Unified AGN Outflow Model
Danehkar +

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Problems
Radio-quiet and radio-loud AGN
Garofalo + 2010
●
Radio-quiet AGN
– Compact outflows
– Weak (or no) radio emission
●
Radio-loud AGN
– Strong radio source
– Extended jets
– Typically in elliptical massive
galaxies that may evolve
from recent mergers (binary SMBH?)
●
Observation Tests:
– If correlation between BH Spin and Outflow kinematics?
– If binary SMBH present in radio-load galaxies with extended jets?
– If BH spinning prograde to accetion disk (a<0) in radio-load galaxies?

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Weyl Curvature
Vacuum Riemann tensor
Einstein’s 70th birthday, Institute for Advanced Study, 1949
Weyl, Mathematische Zeitschrift, 2, 384, 1918

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Weyl Curvature
●
Gravitoelectric & Gravitomagnetic fields
– Names coined by Kip Thorne (IAU, 97, 255, 1982)
– Thorne et al. Black holes: The membrane paradigm (Yale
University, 1986)
●
Gravitoelectric Tensor
– Newtonian Tidal Force
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Gravitomagnetic Tensor
– Frame-dragging vortex & Gravitational Waves
●
Bianchi Identities
– Constraints for gravitoelectric & gravitomagnetic
(see e.g. Relativistic Cosmology, Ellis, Maartens, &
MacCallum, Cambridge, 2012)
Kip Thorne’s 60th birthday, Caltech, 2000

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Weyl Curvature
Tendex and Vortex Lines
●
Visualization of Gravitoelectric & Gravitomagnetic tensors
– Nichols et al. PRD 84, 124012, 2011; PRD 86, 104028, 2012
●
Tidal Tendex Line
– Tendex coined by David Nichols (tendere: ‘to stretch’)
– integral curves of eigenvectors of gravitoeletric tensor
– Owen el al. PRL 106, 151101, 2011
– Zhang et al. PRD 86, 084049, 2012
●
Frame-dragging Vortex Line
– integral curves of eigenvectors of gravitomagnetic tensor
eigenvector eigenvalue
eigenvector eigenvalue
Owen el al. PRL 106, 151101, 2011

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Weyl Curvature
Tendex and Vortex Lines of Slowly Spinning SMBH
Danehkar, IJMPD, 2020, arXiv:2006.13287 [gr-qc]
●
Visualization of Eab & Hab around a slow Kerr BH
– slow Kerr metric (Zhang et al. PRD 86, 084049, 2012)
– gravitoelectric tensor
– gravitomagnetic tensor (Zhang et al. PRD 86, 084049, 2012)

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Weyl Curvature
Tendex and Vortex Lines of Fast Spinning BH
●
Visualization of Eab & Hab around a fast Kerr BH
Tidal Tendex Line Frame-dragging Vortex Line
Zhang et al. PRD 86, 084049, 2012

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Weyl Curvature
Tendex and Vortex Lines of Binary BH
●
Visualization of Eab & Hab around binary BHs
Tidal Tendex Line Frame-dragging Vortex Line
●
Spectral Einstein Code (SpEC)
– https://www.black-holes.org/code/SpEC.html
– SpEC is not publicity available
●
New version: SpECTRE
– https://github.com/sxs-collaboration/spectre
– SpECTRE is still under development by the SXS (Simulating eXtreme Spacetimes) Collaboration, and not yet ready
– Updates on SpECTRE code: https://icerm.brown.edu/programs/sp-f20/w3/ (next week talk, Oct 28)
Owen el al. PRL 106, 151101, 2011

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Weyl Curvature
Tendex and Vortex Lines of Binary BH
●
Visualization of Eab & Hab around binary BHs
– Gravitational Wave Simulations by SpEC
Owen el al. PRL 106, 151101, 2011
LIGO detection of gravitational waves, 2016

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Weyl Curvature
Tendex and Vortex Lines of Exact Solutions
●
The Einstein Toolkit (https://einsteintoolkit.org/)
– Cactus Framework and Computational Toolkit (http://cactuscode.org/)
– Cactus Thorns (http://svn.einsteintoolkit.org/cactus/)
– Recent Tutorial: https://icerm.brown.edu/programs/sp-f20/w1/
●
Einstein Toolkit Thorn: EinsteinAnalysis/WeylScal4
– calculates Weyl scalars in the Einstein Toolkit
– developed using Mathematica notations
– converted to Thorn using Kranc (http://kranccode.org/)
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Thorn for gravitoelectric and gravitomagnetic tensors in the Einstein Toolkit
– can be made by Mathemtica scripts and converted to Thorn using Kranc
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Kranc: Mathematica program turns tensorial equations into a thorn for the Cactus Computational Toolkit
– can be written in LaTex notations and converted to NRPy+
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NRPy+: Python-based code generation for numerical relativity that can turn tensorial formulas into a
thorn for the Cactus Einstein Toolkit
transverse wave component for GW simulations of mergers
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VisIt
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ParaView
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Python packages (yt, etc)

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Summary
Tendex and Vortex Lines around SMBHs
●
Observational Motivations:
– Observational Evidence for a Unified AGN Outflow Model
– Possible correlation between SMBH angular momentum and AGN outflows
– Physical mechanism behind radio-quiet and radio-loud AGN: binary SMBH in radio-loud?
●
Gravitoelectric and Gravitiomagnetic tensors
are visualized using their eigenvectors and eigenvalues:
called Tidal Tendex and Frame-dragging Vortex Lines.
– Visualization of Eab and Hab for a slowly spinning BH can be easily done in weak-
field limit using analytically obtained linear solutions
– Visualization of Eab and Hab for a fast spinning BH can be done in strong fields
using approximation solutions, but requires higher resolutions and computations
– Visualization of Eab and Hab for exact solutions and binary BHs are very complex,
but can be implemented using a new Thorn made by either Kranc or NRPy+ for
the Einstein Toolkit. These simulations require supercomputers. As its simulation
outputs (HDF5) are very big, they also require very large storage devices.

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Image
Credit:
J.
Bergeron,
Sky
&
Telescope
Magazine
Image
Credit:
J.
Bergeron,
Sky
&
Telescope
Magazine
Thank you for your attention
Thank you for your attention