Introduction to numerical relativity

Luca Baiotti 梅乙亭 瑠嘉

Osaka University, Institute for Laser Engineering Plan of the lectures • Overview of general relativity • Gravitational waves • Gravitational-wave detection • Numerical relativity • 3+1 decomposition • Einstein equations • gauge conditions • gravitational-wave extraction • non-vacuum spacetimes: • general-relativistic magnetohydrodynamics • Some applications: binary neutron-star merger General Relativity

• General Relativity is Einstein’s theory of gravity (1915). • At the beginning of the XX century there were no observational inconsistencies with Newton’s theory of gravitation. • But Newtonian gravity is inconsistent with the principle of causality of special relativity: nothing can move faster than the speed of light. • Indeed Newton’s inverse square law implies action at a distance: If one object moves, the other one knows about it instantaneously due to the change in the gravitational force, no matter how big their separation is. • The work of Einstein to make gravity consistent with special relativity brought to a major revision of how we think of space and time. It started with a simple principle, now known as the equivalence principle. The equivalence principle

• The equivalence principle in few common words: “All things fall in the same way”.

• Or slightly more wordy: “all objects have the same acceleration in a gravitational field” (e.g. a feather and bowling ball fall with the same acceleration in the absence of air friction).

• The fact that “All things fall in the same way” is true because the “inertial” mass that enters Newton’s law of motion, F=ma, is the same as the “gravitational” mass that enters the gravitational-force law. The principle of equivalence is really a statement that inertial and gravitational masses are equal to each other for any object. Einstein’s equivalence principle

• Einstein reasoned that a uniform gravitational field in some direction is indistinguishable from a uniform acceleration in the opposite direction. • And so postulated his version of the equivalence principle, which puts gravity and acceleration on an equal footing: • “we [...] assume the complete physical equivalence of a gravitational field and a corresponding acceleration of the reference system." General relativity and the equivalence principle

• Einstein started to think of the path of an object as a property of spacetime itself, rather than being related with the specific properties of the object.

• The idea is that gravity is a manifestation of the fact that objects in free fall follow geodesics, in curved spacetimes.

• What are geodesics?

• We know in our ordinary experience (flat spacetime) that in the absence of any forces, objects follow straight lines, and we also know that straight lines are the shortest possible paths that connect two points in such conditions.

• The generalization of the notion of a “straight line” valid also in curved spacetimes is called geodesic. Ants on an apple

• A famous story to simply illustrate the idea of general relativity is the "Parable of the Apple" by Misner, Thorne, and Wheeler [Gravitation (1973)].

• The parable tries to explain the nature of gravitation in terms of the curvature of spacetime. The spacetime of the parable is the two-dimensional curved surface of an apple. Ants on an apple

• The tale goes like this. One day a student, reflecting on the difference between Einstein's and Newton's views about gravity, noticed ants running on the surface of an apple. • By advancing alternately and of the same amount the left and right legs, the ants seemed to take the most economical path; “wow, they are going along geodesics on this surface!” • The student followed the path of an ant tracing it and then cutting with a knife a small stripe around the trace: “Indeed when put on a plane the path is a straight line!” • Each geodesic may be regarded as a path (world line) of a free particle on this surface (taken as a two-dimensional spacetime). Ants on an apple

• Then the student looked at two ants going from the same spot onto initially divergent paths, but then, when approaching the top (near the dimple) of the apple, the paths crossed and continued into different directions! • The reason of the curved trajectories is: • According to Newton, gravitation is acting at a distance from a center of attraction (the dimple). • According to Einstein, the local geometry of the surface around the dimple is curved. Ants on an apple

• Comments

• Einstein interpretation dispenses with any action-at-a-distance.

• Although the surface of the apple is curved, if you look at any local spot closely (with a magnifying glass), its geometry looks like that of a flat surface (the Minkowski spacetime of the special relativity).

• The interaction of spacetime and matter is summarized in John Archibald Wheeler's favorite words, spacetime tells matter how to move, and matter tells spacetime how to curve.

• This reciprocal influence (matter ↔ spacetime) makes Einstein’s field equation non-linear and so very hard to solve. Ants on an apple

• Summary of the parable:

• 1) objects follow geodesics and locally geodesics appear straight;

• 2) over more extended regions of space and time, geodesics originally receding from each other begin to approach at a rate governed by the curvature of spacetime, and this effect of geometry on matter is what was called “gravitation”;

• 3) matter in turn warps geometry. Einstein equations and general relativity • Important quantities in general relativity: • the metric (the metric tensor g), which may be regarded as a machinery for measuring distances: 2 µ ν dS = gµν dx dx • curvature, expressed by the Riemann curvature tensor Rα = Γα Γα + Γα Γσ Γα Γσ βµν βν,µ − βµ,ν σµ βν − σν βµ 1 (where Γ α = g ασ ( g + g g ) are the Christoffel symbols) βµ 2 βσ,µ σµ,β − βµ,σ

α µν • the Ricci tensor R µ ν = R µ αν and the curvature (Ricci) scalar R = g R µ ν . • covariant derivative: a derivative that takes into account the curvature of the spacetime Einstein equations and general relativity

• From these quantities the path of any particle can be calculated. This is how "geometry tells matter how to move". • The other direction ("matter tells spacetime how to curve") requires to know the distribution of matter (mass/energy/momentum), described through the stress-energy tensor T. • After many years of thinking, Einstein reached a satisfactory form for the equations relating geometry and matter:

Einstein_tensor = constant x T

• The Einstein tensor (usually called G) is a tensor in 4D spacetime that has the wanted properties of: • being a symmetric tensor (it must because the stress-energy tensor is symmetric) • having vanishing (covariant) divergence (it must because the stress-energy tensor has vanishing divergence) • the weak-field limit of the Einstein equations gives the Newtonian Poisson equation (from the comparison to which the value of the above constant is found) The fundamental equations

The Einstein equations: Einstein Ricci tensor Matter and other tensor Curvature scalar fields (spacetime)

Metric (measure of spacetime distances) 4-velocity

Rest-mass density Internal energy density Pressure

i.e.  six, second-order-in-time, second-order-in-space, coupled, highly- nonlinear, quasi-hyperbolic, partial differential equations (PDEs)  four, second-order-in-space, coupled, highly-nonlinear, elliptic PDEs Some verified predictions of general relativity

• Gravitational redshift • change in the frequency of light moving in regions of different curvature • measured in experiments and taken into account by the GPS • Periastron shift • first measured in the perihelion advance of Mercury (the GR prediction coincides with the “anomalous” advance, if computed in Newtonian theory) • binary pulsar (strong fields, so larger shifts) • Bending of light • first measured in the bending of photons traveling near the Sun • gravitational lensing • ... Yet unverified (or only partially verified) predictions of general relativity: black holes

• A black hole is literally a region from which light cannot escape. • The boundary of a black hole is called the event horizon. When something passes through the event horizon, it can no longer communicate in any way with the world outside. However, it is not a concrete surface, there is no local measurement that can tell an observer that he/she is on the event horizon. • General relativity predicts that black holes can form by gravitational collapse. Once a star has burned up its supply of nuclear fuel, it can no longer support itself against its own weight and it collapses. It will reach a critical density at which an explosive re-ignition of burning occurs, called a supernova. • There are several candidate objects that are thought to be black holes (because they are very compact), but there has been no direct observation of black holes up to now. • The definitive way to detect black holes is through gravitational waves (more on this later), because gravitational waves from black holes are distinct from those of other objects. Yet unverified (or only partially verified) predictions of general relativity: gravitational waves

• In general relativity, disturbances in the spacetime curvature (the “old” “gravitational field”) propagate at the speed of light as gravitational radiation or gravitational waves (also known as gravity waves, but this term was already in use in fluid dynamics with a different meaning, so I recommend to avoid it) .

• Gravitational waves are a strong point of general relativity, which solves the action-at-a- distance problem of Newtonian gravity.

• Analogously to light, which is produced by the motion of electric charges, gravitational waves are produced by the motion of ... anything (mass-energy). When you wave your hand, you make gravitational waves.

• The point is that the amplitude of gravitational radiation is very small. Weakness of gravitational waves An approximate calculation (quadrupole approximation) gives: 1 G ∂3Q ∂3Q L = ij ij GW 5 c5 ￿ ∂t3 ∂t3 ￿ (the luminosity in gravitational waves is proportional to the third time derivative of the quadrupole moment of the mass distribution). The quadrupole moment of a system is approximately equal to the mass M of the part of the system that moves, times the square of the size R of the system. This means that the 3rd-order time derivative of the quadrupole moment is 3 2 2 ∂ Qij MR Mv ∂t3 ￿ T 3 ￿ T where v is the velocity of the moving parts and T is the time scale for a mass to move from one side of the system to the other. The time scale (or period) is proportional to the inverse of the square root of the mean density of the system 3 T R /(GM) so the luminosity in gravitational waves￿ (energy emitted in GWs per unit time) is: ￿ 1 G M v2 1 G4 M 2 L ￿ ￿ GW ￿ 5 c5 T ￿ 5 c5 R ￿ ￿ Weakness of gravitational waves

Approximate luminosity in gravitational waves:

1 G M v2 1 G4 M 2 L ￿ ￿ GW ￿ 5 c5 T ￿ 5 c5 R ￿ ￿ So intense sources are compact, massive and move at relativistic speeds.

What makes gravitational-wave astronomy challenging are the factors:

G 60 2.7 10− cgs units c5 ￿ × 4 G 74 8.2 10− cgs units c5 ￿ × So even the gravitational waves from the most intense sources are very weak. Additionally intense sources are far from the Earth, consequently the signal is also weaker because of the distance (proportionally to the inverse of the distance). Hints of gravitational waves • In fact, gravitational waves have not been detected yet, even if there are strong indications that gravitational waves exist as predicted by general relativity. • Such an indirect evidence comes from binary pulsars: Two neutron stars orbiting each other, at least one of which is a pulsar. The curvature is large, so it is an interesting place to study general relativistic effects. (Incidentally, their orbit may precess by 4 degree per year!) • The spin axis of pulsars is not aligned with their magnetic axis, so at each rotation they emit pulses of radio waves and we can detect them from the Earth. Pulsars behave as very accurate clocks. Hints of gravitational waves

• Actually such measurements are very precise and confirm that the orbit is shrinking with time at a rate in complete agreement with the emission of gravitational waves predicted by general relativity. Why study gravitational waves? Why study gravitational waves? GSFC/NASA radio far-IR mid-IR near-IR optical x-ray gamma-ray

??? GWs It has happened over and over in the history of astronomy: as a new “window” has been opened, a “new” universe has been revealed. The same will happen with GW-astronomy.

Large-scale Cryogenic Gravitational-wave Telescope Some gravitational-wave sources o Binary black-holes o Binary neutron stars o Mixed binary systems o Extreme-Mass-Ratio Inspirals (EMRI) o Gravitational collapse (supernovae, neutron stars) o Deformed compact stars, including Low Mass X-ray Binaries (LMXB) and pulsars o Cosmological stochastic background o ... Numerical Relativity

Numerical relativity is the science of simulating (solving) general- relativistic dynamics on computers.

It is often the only way to solve the Einstein equations without approximations and so to investigate the dynamics of astrophysical compact objects and to model sources of gravitational waves, like GRBs, supernovae, gravitational collapse, black-hole formation and evolution, compact stars, ... Gravitational-wave templates for detectors

TAMA-Tokyo GEO-Hannover LIGO-Livingston LIGO-Hanford VIRGO-Cascina

and LCGT will measure ΔL/L ~ h < 10-21 with S/N~1.

Knowledge of the waveforms can h compensate for the very small S/N (matched-filtering) and so enhance detection and allow for source- post-Newtonian Numerical Perturbative characterization expansions relativity analysis possible. The fundamental equations The fundamental equations standard numerical relativity aims at solving are the Einstein equations: Einstein Ricci tensor Matter and other tensor Curvature scalar fields (spacetime)

Metric (measure of spacetime distances) 4-velocity

Rest-mass density Internal energy density Pressure i.e.  six, second-order-in-time, second-order-in-space, coupled, highly- nonlinear, quasi-hyperbolic, partial differential equations (PDEs)  four, second-order-in-space, coupled, highly-nonlinear, elliptic PDEs Solving the einstein equations on computers

General relativity states that our World is a 4D and curved spacetime and the Einstein equations describe its dynamics.

How to solve the Einstein equations numerically?

Prominently, there is no a priori concept of “flowing of time” (we are not involved in thermodynamics here), time is just one of the dimensions, and on the same level as space dimensions...

There is a successful recipe, though. First step: foliate the 4D spacetime We have the illusion to live in 3D and it is easier to tell computers to perform simulations (time-)step by (time-)step. So, given a manifold describing a spacetime with 4-metric we want to foliate it via space-like, three-dimensional hypersurfaces: . We label such hypersurfaces with the time coordinate t. Define therefore:

(“the direction of time”) Also assign a normalization such that

(As mostly in numerical relativity, the signature is here -+++) The function α is called the ”lapse” function and it is strictly positive for spacelike hypersurfaces: Let’s also define: i) the unit normal vector to the hypersurface

so that

ii) and the spatial metric Second step: decompose 4D tensors By using we can decompose any 4D tensor into a purely spatial part (hence in ) and a purely timelike part (hence orthogonal to and aligned with ). The spatial part is obtained by contracting with the spatial projection operator, defined as while the timelike part is obtained by contracting with the timelike projection operator:

The two projectors are obviously orthogonal: The 3D covariant derivative of a spatial tensor is then defined as the projection on of all the indices of the the 4D covariant derivative:

All the 4D tensors in the Einstein equations can be projected straightforwardly onto the 3D spatial slice. In particular, the 3D connection coefficients:

the 3D Riemann tensor: and the 3D contractions of the 3D Riemann tensor, i.e. the 3D Ricci tensor the 3D Ricci scalar: and . We have restricted ourselves to 3D hypersurfaces: have we lost some information about the 4D manifold (and so of full GR)? Not really, such information is contained in a quantity called extrinsic curvature, which describes how the 3D hypersurface is embedded (“bent”) in the 4D manifold. Another way of saying this is that the information present in (the 4D Riemann tensor) and “missing” in (the 3D version) can be found in another spatial tensor: precisely the extrinsic curvature. The extrinsic curvature is defined in terms of the unit normal to as: and this can be shown to be equivalent also to: where is the Lie derivative along . This also expresses that the extrinsic curvature can be seen as the rate of change of the spatial metric. Properties of the Lie derivative Recall that the Lie derivative can be thought of as a geometrical generalization of a directional derivative. It evaluates the change of a tensor field along the flow of a vector field. For a scalar function this is given by:

For a vector field , this is given by the commutator:

= Xµ∂ V ν V µ∂ Xν µ − µ For a 1-form ,this is given by: µ ν = X ∂µων + ωµ∂µX As a result, for a generic tensor of rank this is given by:

= Xα∂ T µ T α ∂ Xµ + T µ ∂ Xα α ν − ν α α ν The extrinsic curvature measures the gradients of the normal vectors and, since these are normalized, they can only differ in direction. Thus the extrinsic curvature provides information on how much the normal direction changes from point to point and so on how the hypersurface is deformed.

Hence it measures how the 3D hypersurface is “bent” with respect to the 4D spacetime.

Consider a vector at one position P and then parallel- parallel transport it to a new location P + δ P . transport The difference in the two vectors is proportional to the extrinsic curvature and this can either be positive or negative. Third step: decompose the Einstein equations Next, we need to decompose the Einstein equations in the spatial and timelike parts.

To this purpose it is useful to derive a few identities.

Gauss equations: decompose the 4D Riemann tensor projecting all indices:

Codazzi equations: take 3 spatial projections and a timelike one: Ricci equations: take 2 spatial projections and 2 timelike ones:

Another important identity which will be used in the following is

and which holds for any spatial vector . We must also consider the projections of the stress-energy tensor (the right-hand-side of the Einstein equations).

Let’s define the double timelike projection of the stress-energy tensor as:

Similarly, the momentum density (i.e. the mass current) will be given by the mixed time and spatial projection:

j = γαnβT µ − µ αβ And similarly for the space-space projection. Now we can decompose the Einstein equations in the 3+1 splitting.

We will get two sets of equations:

1) the “constraint” equations, which are fully defined on each spatial hypersurfaces (and do not involve time derivatives) 2) the “evolution” equations, which instead relate quantities (the spatial metric and the extrinsic curvature) between two adjacent hypersurfaces. The constraint equations (I) We first time-project twice the left-hand-side of the Einstein equations to obtain

Doing the same for the right-hand-side, using the Gauss equations contracted twice with the spatial metric and the definition of the energy density we finally reach the form of the equation, which is called Hamiltonian constraint equation

Note that this is a single elliptic equation (not containing time derivatives), which should be satisfied everywhere on the spatial hypersurface . The constraint equations (II) Similarly, with a mixed time-space projection of the left-hand- side of the Einstein equations we obtain

Doing the same for the right-hand-side, using the contracted Codazzi equations and the definition of the momentum density, we reach the equations called the momentum constraint equations which are also 3 elliptic equations. The 4 constraint equations are the necessary and sufficient integrability conditions for the embedding of the spacelike hypersurfaces in the 4D spacetime . Before proceeding to the derivation of the evolution equations, we must ensure that when going from one hypersurface Σ 1 at time t to another Σ 2 at time t + δ t all the vectors originating on Σ 1 end up on Σ 2 : we must land on a single hypersurface. The most general of such vectors that connect two hypersurfaces is

where is any spatial “shift” vector. Indeed we see that

µ µ so that the change in t along t is δ t = t µ t =1 and so it is the same for all points, which consequently end∇ up all on the same hypersurface. With this definition we can revise the Lie derivative along the unit normal . Since

the definition of the extrinsic curvature: can now be rewritten as

(*)

NOTE: that the Ricci equations (*) are just a manipulation of the definition of the Riemann tensor and not pieces of the Einstein equations. The evolution part of the Einstein equations

We can now express the last piece of the 3+1 decomposition and so derive the evolution part of the Einstein equations. As for the constraints, we need suitable projections of the two sides of the Einstein equations and in particular the two spatial ones:

Using the Ricci equations one then obtains:

where . Last step: select a coordinate basis

So far the treatment has been coordinate independent, but in order to write computer programs we have to specify a coordinate basis. Doing so can also be useful to simplify equations and to highlight the “spatial” nature of and .

In the spirit of the 3+1 formalism, the natural choice for the coordinate unit vectors is: i) three purely spatial coordinates with unit vectors: ii) one coordinate unit vector along the vector : As a result: i.e. the Lie derivative along is a simple partial derivative i.e. the space covariant components of a timelike vector are zero; only the time component is different from zero i.e. the zeroth contravariant component of a spacelike vector are zero; only the space components are nonzero Putting things together and bearing in mind that : Recalling that the spatial components of the 4D metric are the components of the 3D metric ( g ij = γ ij ) and that γ α 0 =0 (true in general, for any spatial tensor), the contravariant components of the metric g µ ν = γ µ ν n µnν in a 3+1 split are −

Similarly, since , the covariant components are

Note that (i.e. are inverses) and thus they can be used to raise/lower the indices of spatial tensors. We can now have a more intuitive interpretation of the lapse, shift and spatial metric. Using the expression for the 4D covariant metric, the line element is given by

It is now clearer that: •the lapse measures proper time between two adjacent hypersurfaces

•the shift relates spatial coordinates between two adjacent hypersurfaces

•the spatial metric measures distances between points on every hypersurface We can now have a more intuitive interpretation of the lapse, shift and spatial metric. Using the expression for the 4D covariant metric, the line element is given by

coordinate line normal line It is now clearer that: •the lapse measures proper time between two adjacent hypersurfaces

•the shift relates spatial coordinates between two adjacent hypersurfaces

•the spatial metric measures distances between points on every hypersurface Recap: the steps taken so far... The 3+1 or ADM (Arnowitt Deser Misner) formulation

First step: foliate the 4D spacetime in 3D spacelike hypersurfaces leveled by a scalar function: the time coordinate. This determines a normal unit vector to the hypersurfaces.

Second step: decompose 4D spacetime tensors in spatial and timelike parts using the normal vector and the spatial metric.

Third step: rewrite Einstein equations using such decomposed tensors. Also selecting two functions, the lapse and the shift, that tell how to relate coordinates between two slices: the lapse measures the proper time, while the shift measures changes in the spatial coordinates.

Fourth step: select a coordinate basis and express all equations in 3+1 form. Gauge conditions

NOTE: the lapse, and shift are not solutions of the Einstein equations but represent our “gauge freedom”, namely the freedom (arbitrariness) in which we choose to foliate the spacetime.

Any prescribed choice for the lapse is usually referred to as a ”slicing condition”, while any choice for the shift is usually referred to as ”spatial gauge condition”.

While there are infinite possible choices, not all of them are equally useful to carry out numerical simulations. Indeed, there is a whole branch of numerical relativity that is dedicated to finding suitable gauge conditions. Different recipes for selecting lapse and shift are possible: i) make a guess (i.e. prescribe a functional form) for the lapse, and shift: e.g. geodesic slicing obviously not a good idea Choosing the right temporal gauge Suppose you want to follow the gravitational collapse to a black hole and assume a simplistic gauge choice (geodesic slicing):

That would lead to a code crash as soon as a singularity forms! No chance of measuring gws!

One needs to use smarter temporal gauges! In particular we want time to progress at different rates at different positions in the grid: “singularity avoiding slicing” (e.g. maximal slicing).

Some chance of measuring gravitational waves! Different recipes for selecting lapse and shift are possible: i) make a guess (i.e. prescribe a functional form) for the lapse, and shift: e.g. geodesic slicing obviously not a good idea ii) fix the lapse, and shift by requiring they satisfy some condition: e.g. maximal slicing for the lapse which has the desired “singularity-avoiding” properties.

Good idea mathematically, but unfortunately this leads to elliptic equations which are computationally too expensive to solve at each time. iii) determine the lapse, and shift dynamically by requiring that they satisfy comparatively simple evolution equations.

This is the common solution. The advantage is that the equations for the lapse and shift are simple time evolution equations.

A family of slicing conditions that works very well to obtain both a strongly hyperbolic evolution equations and stable numerical evolutions is the Bona-Masso slicing:

where and is a positive but otherwise arbitrary function. Choosing the right spatial gauge (i.e. β(xμ) )

A high value of the metric components means that the distance between numerical grid points is actually large and this causes problems. In addition, large gradients my be numerically a problem. Choosing a “bad” shift may even lead to coordinates singularities. A popular choice for the shift is the hyperbolic “Gamma-driver” condition.

i i where B ∂ t β and acts as a restoring force to avoid large oscillations≡ in the shift and the driver tends to keep the Gammas constant.

Overall, the “1+log” slicing condition and the “Gamma- driver” shift condition are the most widely used both in vacuum and non-vacuum spacetimes. Excising parts of the spacetime with singularities apparent horizon

The region of spacetime inside a horizon (yellow region) is causally disconnected from the outside (blue region).

So a region inside a horizon may be excised from the numerical domain.

This is successfully done in pure spacetime evolutions since the work In practice, the actual of Nadëzhin Novikov Polnarev (1978). excision region is a “legosphere” (black region) and Baiotti et al. [PRD 71, 104006, is placed well inside the (2005)] and other groups [Duez et al., apparent horizon (which is PRD 69, 104016 (2004)] have shown found at every time step) and that it can be done also in non-vacuum is allowed to move on the grid. simulations. Excising parts of the spacetime with singularities: the moving-puncture method

There is an alternative to explicit excision. Proposed independently by Campanelli et al., PRL96, 111101 (2006) and Baker et al., PRL96, 111102 (2006), it is nowadays a very popular method for moving--black-hole evolution. It consists in using coordinates that allow the punctures (locations of the singularities) to move through the grid, but do not allow any evolution at the puncture point itself (i.e., the lapse is forced to go to zero at the puncture, though not the shift vector, hence the “frozen” puncture can be advected through the domain). The conditions that have so far proven successful are modifications to the so-called 1+log slicing and Gamma-driver shift conditions. In practice, such gauges take care that the punctures are never located at a grid point, so that actually no infinity is present on the grid. This method has proven to be stable, convergent, and successful. A few such gauge options are available, with parameters allowed to vary in determined ranges (but no fine tuning is necessary). This mechanism works because it implements an effective excision (“excision without excision”). It has been shown that in the coordinates implied by the employed gauges the singularity is actually always outside the numerical domain. In practice, the ADM equations are essentially never used! The ADM equations are perfectly all right mathematically but not in a form that is well suited for numerical implementation.

Indeed the system can be shown to be weakly hyperbolic (namely, its set of eigenvectors is not complete) and hence “ill- posed” (namely, as time progresses, some norm of the solution grows more than exponentially in time).

In practice, numerical instabilities rapidly appear that destroy the solution exponentially.

However, the stability properties of numerical implementations can be improved by introducing certain new auxiliary functions and rewriting the ADM equations in terms of these functions. Let’s inspect the 3+1 evolution equations again:

Think of the above system as a second-order system for γ ij . In this equation there are mixed second derivatives, since contains mixed derivatives in addition to a Laplace operator acting on . µ Instead, we would like to have only terms of the type ∂ ∂µγij because without the mixed derivatives the 3+1 ADM equations could be written in a such way that they behave like a wave equation for . We care to have wave equations, like because wave equations are manifestly hyperbolic and mathematical theorems guarantee the existence and uniqueness of the solutions (as seen in previous lectures of this school).

We can make the equations manifestly hyperbolic in different ways, including: i) using a clever specific gauge; ii) introducing new variables, which follow additional equations (imposed by the original system).

These methods aim at removing the mixed-derivative term. Generalized Harmonic Coordinate (GHC) formulation Let choose a clever gauge that makes the ADM equations strongly hyperbolic.

The generalized harmonic formulation is based on a generalization of the harmonic coordinates:

When such condition is enforced in the Einstein equations, the principal part of the equations for each metric element becomes a scalar wave equation, with all nonlinearities and couplings between the equations relegated to lower order terms. The harmonic condition is known to suffer from pathologies. However, alternatives can be found that do not suffer from such pathologies and still have the desired properties of the harmonic formulation. In particular, the generalized harmonic coordinates have the form:

where the a are a set of source functions.

The and the equations for their evolution must be suitably chosen. The second method: introducing new variables New evolution variables are introduced to obtain from the ADM system a set of equations that is strongly hyperbolic. A successful set of new evolution variables is:

φ : conformal factor

γ˜ ij : conformal 3-metric K : trace of extrinsic curvature

A ˜ ij : trace-free conformal extrinsic curvature Γ˜ i :“Gammas” And the ADM equations are then rewritten as:

tγ˜ij = 2αA˜ij , where ∂ D − Dt ≡ t − Lβ 1 φ = αK , Dt −6

4φ TF l A˜ = e− [ α + α (R S )] + α KA˜ 2A˜ A˜ , Dt ij −∇i∇j ij − ij ij − il j ￿ ￿ 1 1 K = γij α + α A˜ A˜ij + K2 + (ρ + S) , Dt − ∇i∇j ij 3 2 ￿ ￿ 2 Γ˜i = 2A˜ij∂ α + 2α Γ˜i A˜kj γ˜ij∂ K γ˜ijS + 6A˜ij∂ φ Dt − j jk − 3 j − j j ￿ 2 ￿ ∂ βl∂ γ˜ij 2γ˜m(j∂ βi) + γ˜ij∂ βl . − j l − m 3 l ￿ ￿ These equations are also known as the BSSN-NOK equations or more simply as the conformal traceless formulation of the Einstein equations. Although not self evident, the BSSN-NOK equations are strongly hyperbolic with a structure which is resembling the 1st-order in time, 2nd-order in space formulation

scalar wave equation

conformal traceless formulation

The BSSN-NOK equations are nowadays the most widely used form of the Einstein equations and have demonstrated to lead to stable and accurate evolution of vacuum (binary-- black-holes) and non-vacuum (neutron-stars) spacetimes. Constraints of BSSN-NOK

In addition to the (6+6+3+1+1=17) hyperbolic evolution equations to be solved from one time slice to the next, there are the usual 3+1=4 elliptic constraint equations:

NOTE: most often these equations are not solved but only monitored to verify that and 5 additional constraints are introduced by the new variables: Extraction of gravitational waves Wave-extraction techniques Computing the waveforms is the ultimate goal of a large portion of numerical relativity. There are several ways of extracting GWs from numerical relativity codes, the most widely used of which are:

•Weyl scalars (a set of five complex scalar quantities, describing the curvature of a 4D spacetime) •perturbative matching to a Schwarzschild background

Both are based of finding some gauge invariant quantities or the perturbations of some gauge-invariant quantity, and to relate them to the gravitational waveform. Wave-extraction techniques In both approaches, “observers” are placed on nested 2-spheres and they calculate there either the Weyl scalars or decompose the metric into tensor spherical-harmonics to calculate the gauge-invariant perturbations of a Schwarzschild black hole.

Once the waveforms are calculated, all the related quantities: the radiated energy, momentum, and angular momentum can be derived simply. Summary The ADM equations are ill posed and not suitable for numerical integrations Alternative formulations (BSSN-NOK, GHC) have been developed and shown to be strongly hyperbolic and hence well- posed and effective to solve Einstein equations Getting a good formulation of the Einstein equations will work only in conjunction with good gauge conditions. “1+log” slicing “Gamma-driver” conditions work well in a number of conditions Any astrophysical prediction needs the calculation of “realistic” initial data and hence the solution of elliptic equations Gravitational waves are present in simulations and can be extracted with great accuracy. To know more on these issues

A first course in general relativity, Bernard Schutz, Cambridge University Press (2009)

Gravitation, C. Misner K. Thorn J.A. Wheeler, Ed. W. H. Freeman (1973)

Numerical Relativity, Thomas Baumgarte and Stuart Shapiro, Cambridge University Press (2010)

Introduction to 3+1 Numerical Relativity, Miguel Alcubierre, Oxford University Press (2008) When the spacetime is not vacuum

The additional set of equations to solve is:

i.e. 4 conservation equations for the energy and momentum,

i.e. conservation of baryon number and equation of state (EOS) (microphysics input). GRMHD equations

The evolution of the magnetic field obeys Maxwell’s equations: 1 µν ∗F µν = !µνσρF ∗F = 0 σρ ∇ν 2 that give the divergence-free condition: √γB￿ = 0 ∇ · ￿ ￿ and the equations for the evolution of the magnetic field: ∂ √γB￿ = α￿v β￿ √γB￿ ∂t ∇ × − × ￿ ￿ ￿￿ ￿ ￿ ￿￿ for a perfect fluid with infinite conductivity (ideal MHD approximation) Energy-momentum tensor of ideal MHD 1 T µν =(ρ + ρε + b2)uµuν + p + b2 gµν bµbν 2 − ￿ ￿

where ρ is the rest-mass density ε is the specific internal energy u is the four-velocity p is the gas pressure vi is the Eulerian three-velocity of the fluid (Valencia formulation) W the Lorentz factor b the four-vector of the magnetic field Bi the three-vector of the magnetic field measured by an Eulerian observer Recap: the fundamental equations Let’s recall the equations we are dealing with:

+1 + 3 + 1 This is not yet “real” astrophysics but our approximation to “reality”. Still very crude but it can be improved: microphysics for the

µν EOS, viscosity, ∗F =0, (Maxwell eqs. : induction, zero div.) ∇ν radiation transport,...

The complete set of equations is solved with the Cactus/Carpet/Whisky codes. Cactus/Carpet/Whisky

Cactus (www.cactuscode.org) is a computational “toolkit” developed at the AEI/CCT and provides a general infrastructure for the solution in 3D and on parallel computers of PDEs (e.g. Einstein equations).

Whisky (www.whiskycode.org), developed at the AEI, Osaka University, Southampton, LSU, ..., for the solution of the relativistic hydrodynamics and magnetohydrodynamics equations in arbitrarily curved spacetimes.

Cactus Whisky

Carpet (www.carpetcode.org) provides box-in-box adaptive mesh refinement with vertex-centered grids. Carpet: a mesh-refinement driver

Carpet (www.carpetcode.org) is a driver mainly developed by E. Schnetter [CQG 21, 1465 (2004)], which has removed the limitation of using uniform 3D grids. Carpet follows a (simplified) Berger-Oliger [J. Comput. Phys. 53, 484 (1984)] approach to mesh refinement, that is: • refined subdomains consist of a set of cuboid (= rectangular parallelepiped) grids • refined subdomains have boundaries aligned with the grid lines • the refinement ratio between refinement levels is constant While the refined meshes are not automatically moving on the grid, they can be activated and deactivated during the evolution, obtaining a progressive fixed mesh refinement or even a moving-grid mesh refinement. The Einstein Toolkit

Many codes for numerical relativity are publicly available from the recent project of the Einstein Toolkit (einsteintoolkit.org) , which aims at providing computational tools for the community. It includes: • spacetime evolution code • GRHydro code (not completely tested yet; see also the Whisky webpage) • GRMHD code (in the near future) • mesh refinement • portability • simulation factory (an instrument to manage simulations) High Resolution Shock Capturing schemes An important issue for hydrodynamics, is that, for compressible fluids, discontinuous solutions (shocks) are produced even from smooth initial data. And they must be treated carefully by the numerical schemes.

Actually, a generic problem arises when a Cauchy problem described by a set of continuous PDEs is solved in a discretised form: the numerical solution becomes piecewise constant. It is like having discontinuities between each point end its neighbours. Riemann problem

Definition: in general, for a hyperbolic system of equations, a Riemann problem is an initial-value problem with initial condition given by:

where UL and UR are two constant vectors representing the “left” and “right” state.

For hydrodynamics, a (physical) Riemann problem is the evolution of a fluid initially composed of two states with different and constant values of velocity, pressure and density. High Resolution Shock Capturing schemes

In HRSC methods, each discretisation-produced discontinuity is considered a local Riemann problem.

There are powerful numerical methods to solve Riemann problems, both exactly and approximately. Our strengths: • High-order (up to 8th) finite-difference techniques for the field equations • Flux conservative form of HD and MHD equations with constraint transport or hyperbolic divergence-cleaning or evolution of the vector potential for the magnetic field; HRSC methods with different solvers (HLLE, Roe, Marquina) and reconstruction methods (linear, PPM) • Multiple options for the wave extraction (Weyl scalars, gauge-invariant pertbs) • AMR with moving grids • Accurate measurements of BH properties through apparent horizons (IH) •Use excision (matter and/or fields) if needed Our weaknesses: • Idealized (analytic) EoSs (work in progress to include more detailed EoSs) • Single-fluid description: no superfluids nor crusts • Only inviscid fluid so far (not necessarily bad approximation) • Radiation and neutrino transport totally neglected (work in progress) • Very coarse resolution; far from regimes where turbulence/dynamos develop BINARY NEUTRON-STAR MERGERS

• Baiotti, Giacomazzo, Rezzolla 2008, PRD 78, 084033 • Baiotti, Giacomazzo, Rezzolla 2009, CQG 26, 114005 • Giacomazzo, Rezzolla, Baiotti, 2009, MNRAS 399, L164-L168 • Rezzolla, Baiotti, Giacomazzo, Link, Font 2010, CQG 27, 114105 THE INITIAL GRID SETUP FOR THE BINARY MASS 1.6X2, IDEAL-FLUID EOS Visualization by Giacomazzo, Kaehler, Rezzolla

Matter dynamics high-mass binary Merger

Collapse to BH

soon after the merger the torus is formed and undergoes oscillations

Waveforms: polytropic EOS high-mass binary Merger Collapse to BH

The full signal from the inspiral to the formation of a BH has been computed Imprint of the EOS: ideal-fluid vs polytropic EoS

Reasonable to expect that for any realistic EOS, the GWs will be between these two extreme cases. GWs will work as Rosetta stone to decipher the NS interior.

After the merger, a BH is produced After the merger, a BH is produced over a timescale comparable with the over a timescale larger or much dynamical one. larger than the dynamical one. Waveforms: ideal-fluid EoS

low-mass binary Conservation properties of the code low-mass binary

conservation of energy conservation of angular momentum Imprint of the EOS: frequency domain

The pre-merger dynamics are very similar; the post-merger phase is Contributions from the very different merger Imprint of the EOS: frequency domain

The pre-merger dynamics are very similar; the post-merger phase is Contributions from the very different bar-deformed HMNS Imprint of the EOS: frequency domain

The pre-merger dynamics are very similar; the post-merger phase is Contributions from the very different collapse to BH Extending the work to MHD We have considered the same models also when an initially poloidal magnetic field of ~108 or ~1012 G is introduced.

The magnetic field is added by hand using the vector potential: A = A r2[max(P P , 0)]n φ b − cut where A b and P cut = 0 . 04 max( P ) are two constants defining respectively the strength and× the extension of the magnetic field inside the star. n=2 defines the profile of the initial magnetic field.

The initial magnetic fields are therefore fully contained inside the stars: i.e. no magnetospheric effects. Note that the torus is much less dense and a large plasma outflow is starting to be launched. Validations are needed to confirm these results.

Typical evolution for a magnetised binary Ideal fluid,M =1.65 M ,B= 1012 G − ⊙