Nichols, David A. and Owen, Robert and Zhang, Fan and Zimmerman, Aaron and Brink, Jeandrew and Chen, Yanbei and Kaplan, Jeffrey D. and Lovelace, Geoffrey and Matthews, Keith D. and Scheel, Mark A. and Thorne, Kip S. (2011) Visualizing spacetime curvature via framedrag vortexes and tidal tendexes: General theory and weakgravity applications. Physical Review D, 84 (12). p. 124014. ISSN 05562821. http://resolver.caltech.edu/CaltechAUTHORS:20120106113050494

PDF
 Published Version
See Usage Policy. 5Mb 
Use this Persistent URL to link to this item: http://resolver.caltech.edu/CaltechAUTHORS:20120106113050494
Abstract
When one splits spacetime into space plus time, the Weyl curvature tensor (vacuum Riemann tensor) gets split into two spatial, symmetric, and tracefree tensors: (i) the Weyl tensor’s socalled electric part or tidal field Ɛ_(jk), which raises tides on the Earth’s oceans and drives geodesic deviation (the relative acceleration of two freely falling test particles separated by a spatial vector ξ^k is Δa_j=Ɛ_(jk)ξ^k), and (ii) the Weyl tensor’s socalled magnetic part or (as we call it) framedrag field B_(jk), which drives differential frame dragging (the precessional angular velocity of a gyroscope at the tip of ξ^k, as measured using a local inertial frame at the tail of ξ^k, is ΔΩ_j=B_(jk)ξ^k). Being symmetric and tracefree, Ɛ_(jk) and B_(jk) each have three orthogonal eigenvector fields which can be depicted by their integral curves. We call the integral curves of Ɛ_(jk)’s eigenvectors tidal tendex lines or simply tendex lines, we call each tendex line’s eigenvalue its tendicity, and we give the name tendex to a collection of tendex lines with large tendicity. The analogous quantities for B_(jk) are framedrag vortex lines or simply vortex lines, their vorticities, and their vortexes. These concepts are powerful tools for visualizing spacetime curvature. We build up physical intuition into them by applying them to a variety of weakgravity phenomena: a spinning, gravitating point particle, two such particles sidebyside, a plane gravitational wave, a point particle with a dynamical currentquadrupole moment or dynamical massquadrupole moment, and a slowmotion binary system made of nonspinning point particles. We show that a rotating current quadrupole has four rotating vortexes that sweep outward and backward like water streams from a rotating sprinkler. As they sweep, the vortexes acquire accompanying tendexes and thereby become outgoing currentquadrupole gravitational waves. We show similarly that a rotating mass quadrupole has four rotating, outwardandbackward sweeping tendexes that acquire accompanying vortexes as they sweep, and become outgoing massquadrupole gravitational waves. We show, further, that an oscillating current quadrupole ejects sequences of vortex loops that acquire accompanying tendex loops as they travel, and become currentquadrupole gravitational waves; and similarly for an oscillating mass quadrupole. And we show how a binary’s tendex lines transition, as one moves radially, from those of two static point particles in the deep near zone, to those of a single spherical body in the outer part of the near zone and inner part of the wave zone (where the binary’s mass monopole moment dominates), to those of a rotating quadrupole in the far wave zone (where the quadrupolar gravitational waves dominate). In Paper II we will use these vortex and tendex concepts to gain insight into the quasinormal modes of black holes, and in subsequent papers, by combining these concepts with numerical simulations, we will explore the nonlinear dynamics of curved spacetime around colliding black holes. We have published a brief overview of these applications in R. Owen et al. Phys. Rev. Lett. 106 151101 (2011). We expect these vortex and tendex concepts to become powerful tools for general relativity research in a variety of topics.
Item Type:  Article  

Related URLs: 
 
Additional Information:  © 2011 American Physical Society. Received 27 August 2011; published 6 December 2011. We thank John Belcher, Larry Kidder, Richard Price, and Saul Teukolsky for helpful discussions. Our simulations have been performed using the Spectral Einstein Code (SpEC) [46]. This research was supported by NSF Grants No. PHY0601459, No. PHY0653653, No. PHY0960291, No. PHY0969111, No. PHY1005426, PHY1068881 and CAREER Grant No. PHY0956189, by NASA Grants No. NNX09AF97G and No. NNX09AF96G, and by the Sherman Fairchild Foundation, the Brinson Foundation, and the David and Barbara Groce fund.  
Group:  TAPIR  
Funders: 
 
Classification Code:  PACS: 04.25.dg, 04.25.Nx, 04.30.w  
Record Number:  CaltechAUTHORS:20120106113050494  
Persistent URL:  http://resolver.caltech.edu/CaltechAUTHORS:20120106113050494  
Usage Policy:  No commercial reproduction, distribution, display or performance rights in this work are provided.  
ID Code:  28701  
Collection:  CaltechAUTHORS  
Deposited By:  Jason Perez  
Deposited On:  06 Jan 2012 21:16  
Last Modified:  17 Dec 2014 18:40 
Repository Staff Only: item control page