CPS713 Module on Numerical Simulation of the Collision of two Black Holes as part of Case Study (II) on CFD and Numerical Relativity Geoffrey Fox NPAC 111 College Place Syracuse NY 13244-4100 Abstract of Module on Numerical Simulation of the Collision of two Black Holes This describes the structure of Numerical Relativity as a set of differential equations but it does discuss state of the art solvers involving adaptive meshes Basic Motivation of General Relativity and its experimental tests Metric Tensor, its derivatives and Einstein's equations Initial value formulation and structure of elliptic and hyperbolic equations Examination of particular finite difference scheme for the Wave equation in three dimensions -- a study to understand large distances issues in solving numerical relativity References for CPS713 Module on Numerical Simulation of the Collision of two Black Holes Introductory References: "Solving Einstein's Equations on Supercomputers", Ed Seidel, NCSA,Illinois "Numerical Relativity", Ed Seidel(NCSA), Wai-Mo Suen(Washington University, St. Louis) More Technical Background for Discussion In Course: "A Description of the Initial Value Formulation of Vacuum General Relativity for the Non-Specialist", Mark Miller(NPAC,Syracuse University) "Gravitational Wave Extraction - A Benchmark?", R. Gomez, J. Winicour(Pittsburgh Univ.) "Pittsburgh Group Code : Gravitational Wave Extraction - A Benchmark? - A Summary", Mark Miller(NPAC) The Spirit of General Relativity as a Description of Gravitational Forces as the Structure of Space-Time We are familiar with Newton's Laws for the Interactions between Particles: "What" Causes this force? At the most fundamental level of Quantum gravity, we will we find "elementary" Gauge Particles -- The Gravitons which are analogue of Photons for Electromagnetism and Gluons for Quantum Chromodynamics Most Physicists believe all forces can be described by a unified theory at the Quantum level but this is still speculative. The Graviton is unusual as has Spin 2 (Corresponding to two indices in metric tensor gmn we will soon study) wheras other force quanta are spin 1 For instance, spin 1 Photon corresponds to single index vector field Am However we will study the classical theory which is sufficient for all observations General Relativity as a Theory of Distorted Space-Time Note that gravitational forces cause particles to move in vacuum but we are also familiar with particles moving "downwards" if placed on a nonflat surface Einstein's theory of General Relativity is a brilliant generalization of this to describe all gravity in terms of structure of Space-Time We do not say say the Earth creates a "Gravitational Force" which causes Apples to fall as noted by Newton Rather The Earth distorts space-Time and Apples move in this distorted Space-Time In Newton's law, gravitational force of Earth proportional to its Mass In Einstein's description, distortion of Space-Time is proportional to Mass of particle Everything (including you) distorts Space-Time; This distortion is insignificant except for very Heavy Bodies The Space-Time Structure Created by a Heavy Bowling Ball We can consider the distortion of Space-Time due to a Heavy Body as analogous to distortion of a FLAT rubber sheet into a DROOPING rubber sheet when a bowling ball is placed in the Middle The Path of a Marble in a Distorted Space-Time Now place a much lighter body (a marble) near the bowling ball The marble will roll to the "bottom" of the distortion in Space-Time Basic Notation for Numerical Formulation of Einstein's Equations Greek indices m,n .. denote Space-Time and run from 0 (time) to 3. Latin indices i,j,k .. denote Space and run from 1 to 3 (x y and z). Note indices ALWAYS balance in equations and there is difference between upper and lower indices For Instance matrix M ij is inverse of matrix M ij. Again M ij M jk = d i k is Expression of Inverse Condition The Metric Tensor in Einstein's Formulation of General Relativity-I The speed of light c is set to 1. Normally 3 X 1010 cm/second All quantities in General Relativity are a function of the four vector: The dynamical variables of the theory are the 10 independent components of the the metric tensor: For example, this describes distortion of rubber sheet by bowling ball g mn = g nm is SYMMETRIC If two Space-Time points are separated by four vector d m Square of distance between them is g mn d m d n The Metric Tensor in Einstein's Formulation of General Relativity-II Each of 16 (10 independent) components of 4 by 4 tensor g mn is a function of the four vector Einstein's equations are nonlinear equations for the metric g and its derivatives usually constructed in terms of so called Christoffel Symbol G Why Study General Relativity Numerically Newtonian gravity is linear (weak field) limit of General Relativity Tests of General Relativity -- Small deviations from Newton's laws in planetary orbits Energy loss in Binary Neutron Stars from Gravitational wave radiation Bending of light in strong gravitational fields New experiments will directly observe gravitational waves due to interactions with sensitive detectors on earth LIGO in Louisiana and Washington State(Hanford) VIRGO in Europe Can estimate that Computations of Gravitational Waves from black hole collision would take 100,000 hours on a Cray-YMP 10 Hours on a Teraflop Machine Estimate that LIGO will sensitivity to detect black hole gravitational waves around the year 2005 Some Tests of General Relativity Einstein first published equations in 1915 with three predictions: Bending of Light as it passed massive objects such as stars Experimentally verified in 1919 by observation of star field near the sun during an eclipse There is a small (43 seconds of arc per century -- with 3600 seconds of arc as one degree) shift in Mercury's perihelion not accounted for by Newtonian Gravity Einstein verified that General relativity predicted this soon after publication of equations "Blue" shift (frequency or energy change) of light as it falls down a gravitational field In 1960, (5 years after Einstein's death) Pound and Rebka observed the two parts in 1012 frequency shift as photons fell 74 feet of the Jefferson tower of the physics building at Harvard They used Mossbauer effect More Tests of General Relativity Observed time delay of radar signals bounced off planets or spacecraft during superior conjunction so that signals are affected by close passage to sun Observed decrease in period of binary pulsar discovered by Hulse and Taylor(1975) Consistent with energy decrease due to radiation of gravitational waves predicted by general relativity A laser monitors distance between Moon and Earth showing that they both have same acceleration due to sun with a precision of 7 parts per ten trillion(1010) Tests Equivalence Principle Equivalence Principle Says that a uniform gravitational field is the same as constant acceleration If you are in a room with no windows, there is no way for you to tell that you were on earth and subject to gravitational field as opposed to hypothesis that you were far from any field but accelerating at 9.8 meters/sec2 Alternatively, Equivalence Principle says that the gravitational mass (the mass m1 that appears in Newton's law of gravity) F=Gm1m2/r122 is equal to inertial mass m1 appearing in Newton's law of motion F=m1 x acceleration In general relativity, the Equivalence Principle is built in Initial Value Formulation of General Relativity We formulate General Relativity in fashion one would expect for solving wave equations in three spatial dimensions This is called foliation and slices Space-Time into Spacial Surfaces labelled by particular time values t and coordinatized by (x,y,z) Projection of Einstein's Equations onto Spacial Surfaces The Einstein equations for g mn can now be written in terms of a reduced 3 by 3 symmetric metric matrix q ij q ij has 9 components of which 6 are independent Structure of Einstein's Equations in Initial Formulation There are 10 dynamical equations for 6 independent components of q ij 4 of these are elliptic constraint equations which must be satisfied by initial data -- analogous to initial conditions on positions and velocities in Newton's equations. 6 of the equations are true dynamical equations describing time evolution of the q ij Linearization of Time Evolution Equations for q ij When Solving a second order time evolution equation such as One typically converts this to a pair of first order in time equations by transformation: We perform the analogous step in general relativity, introducing a new symmetric matrix K ij which is called the extrinsic curvature and related to but not exactly equal to single time derivative of q ij Structure of Numerical Relativity Equations in terms of 3 by 3 matrices q and K K is a Symmetric 3 by 3 matrix And we now have 14 equations: Coodinate and Foliation Choices in General Relativity One of the fascinating features of Numerical Relativity is the freedom to choose coordinates This can allow equations to be smooth and appropriate for finite differences in one choice and very irregular and requiring adaptive finite element in another choice Choice is called Choice of Gauge: The Lapse and Shift in Gauge Transformations The Lapse describes how much time elapses between two spacial surfaces The Shift functions N i describe the tangential shift as you move from the t 0 to t 0+ dt spacial surface Choose Lapse and Shift to: Avoid Unphysical Singularities (One cannot avoid physical black hole singularity Choice of spherical polar coordinates has singularity at radius r=0. Keep Numerical Algorithm Stable Insure Consistency of Constraints Geometrical Picture for Lapse and Shift Gauge Transformations a and N i are depicted below for two adjacent spacial surfaces t and t + dt Notation for Einstein's Equations in Initial Value Formulation Define Inverse q ij of 3 by 3 matrix q ij: Define the spacial restriction of Christoffel Symbol by: Note summation convention over m in above formula. Get full Christoffel Symbol by replacing q ij by g mn and makes all indices Greek running from 0 to 3. The Four Elliptic Constraint Equations in Initial Value Formulation of Einstein's Equations Note these are very nonlinear and complicated Written out as Fortran code, they will be very complicated Need high level (symbolic Algebra) front end The Twelve Hyperbolic Evolution Equations in Initial Value Formulation of Einstein's Equations 6 First Order in Time Evolution Equations for q ij 6 First Order in Time Evolution Equations for K ij A benchmark Numerical Relativity problem The numerical relativity problem has several key features Our first example is based on work of University of Pittsburgh group involving R. Gomez and J.Winicour R. Gomez, R. Isaacson and J. Winicour, Journal Computational Physics, 98,11(1992) R. Gomez, J. Winicour, "Gravitational Wave Extraction - A Benchmark ?" This studies the wave-like features of Einstein's equations where key numerical problem is reliable extraction of Gravitational waves This is nontrivial as gravitational waves can only be extracted after one has evolved a "long way" from the black holes It is difficult numerically to reliable evolve oscillating solutions through long distances See examples of convection equations in Hirsch Characteristic Surfaces and Key Features of Pittsburgh Approach Key features of Pittsburgh Approach Use Characteristic Surfaces instead of spacial surfaces Spacial: fixed time Characteristic: fixed t+r or t-r Compactification -- Map infinity (where waves sought) to finite point) Numerical Formulation of Three Dimensional Wave Equation in Polar Coordinates The 3D Wave Equation is: We will study this as "essential issues" in wave extraction are exhibited More generally take position and partial derivative dependent coefficients of second derivatives and add such forcing terms Transform to Spherical Polar Coordinates Introduce characteristic variables: Compactification and Computational Variables for Three Dimensional Wave Equation Introduce new computational variables (x,h,z) Note x=1 corresponds to radial variable r equal to infinity. These three variables are used as well as u (=t-r) replacing time as fourth independent variable Replace F by G as dependent variable: Final Computational Formulation of Pittsburgh Benchmark We must solve: In Compact Domain: With Boundary Conditions given in diagram: Note we assume that you have used other techniques (represntations) to integrate upto: Final Computational Formulation of Pittsburgh Benchmark -- Diagram This illustrates formalism on previous page Discretization of Computational Formulation of 3D Wave Equation We take simple equally paced gridpoints in u, x, h, and z. We have already seen for convection equation subtly in time discretization of this type of equation. A stable explicit method can be obtained by a carefully chosen integral formulation described in Miller's memo. This involves integrating over a box PQRS with sides parallel to the characteristic axes u=constant, v=constant Finite Volume Integral Formulation of Differencing Equations We find these pictures for box PQRS in (t,r) (u,v) (u,x) spaces Stable Finite Difference Form of Discretized Pittsburgh Wave Equations-I Alternatively we can view as a brilliant choice of finite difference scheme Which is justified a posteriori by the Von Neumann stability analysis given in Pittsburgh memo which shows all eigenvalues have unit modulus as required for Stable algorithm preserving magnitude of amplitude. This clever choice is: Giving: Stable Finite Difference Form of Discretized Pittsburgh WaveEquations-II Further we similarily: Which gives final numerical equations: