Comments about "Numerical relativity" in arxiv
This document contains comments about the subject "Numerical relativity" in arxiv. https://arxiv.org/find
 The text in italics is copied from that url
 Immediate followed by some comments
In the last paragraph I explain my own opinion.
The following articles are discussed:
 1963 Gravitational Radiation from Point Masses in Keplerian Orbit by P.C Peters and J. Mathews.
 1964 Gravitational Radiation and the Motion of two Point Masses by P.C Peters
 2001 Numerical Relativity: A review by Luis Lehner
 2005 Evolution of Binary Black Hole Spacetimes by Frans Pretorius
 2013 A catalog of 174 binary blackhole simulations for gravitationalwave astronomy by Herald P.Pfeiffer et all
 2015 Accuracy and precision of gravitationalwave models of inspiraling neutron star – black hole
binaries with spin: comparison with numerical relativity in the lowfrequency regime by Prayush Kumar, Kevin Barkett, Swetha Bhagwat, Nousha Afshari, Duncan A. Brown, Geoffrey Lovelace, Mark A. Scheel, Béla Szilágyi
 2016 LidovKozai Cycles with Gravitational Radiation: Merging Black Holes in Isolated Triple Systems by Kedron Silsbee, Scott Tremaine
Reflections:
 General Reflection Numerical Relativity documents
 Reflection document 2: 1964
 Reflection document 3: 2001
 Reflection document 4: 2005
 Reflection document 5: 2013
 Reflection document 6: 2015
 Reflection document 7: 2016
1963 Gravitational Radiation from Point Masses in Keplerian Orbit
For a copy of this document by P.C. Peters and J. Mathews select:
http://gravity.psu.edu/numrel/jclub/jc/Peters_Mathews_PR_131_435_1963.pdf
This document starts with the text:

The linearized version of Einstein's general theory of relativity is strickingly similar to the classical electromagnetism. In particular, one might expect masses in arbitrary motion to radiate gravitational energy. The question has be raisewd, however, whether the energy so calculated has any physical meaning. We shall not concern ourselves with this question here; we shall take the point of view that the analogy with electromagnetic theory is a correct one and energy is actual radiated.

In actual fact this is a very important point to be concerned about. Physical electromagnetism and gravitation are two completely different phenomena. The radiation involved in electromagnetism are photons (light) and when gravity is considered this are gravitons. The point is that when an object of a certain mass is considerd the issue is how much energy is released in the form of photons and how much in the form of graviton's. IMO for a Sun sized object the first is much larger than the second. For a BH the second is larger than the first.
This raises the issue if a single BH has a final life time. When the BH is surrounded by a disc, the life time of the BH is "infinite" or endless because of infalling material resident in the disc.












1964 Gravitational Radiation and the Motion of two Point Masses
For a copy of this document by P.C. Peters select
http://gravity.psu.edu/numrel/jclub/jc/Peters_PR_136_B1224_1964.pdf
This document starts with the text:
At page B1225 (Left side) we read:

Clearly a consistent picture of gravitational radiation is required,

IMO the only way is when one coordinate system is used which is used for all the objects studied.

One approach to gravitational radiation is to consider only exact solutions of the nonlinear field equations of general relativity. All such solutions found so far correspond to unphysical systems.

I do not understand this reasoning. In general to try to find the exact solutions of the equations which describe the evolution of the planets around the Sun using Newton's Law do not exist. The best solutions are in the book "Astronomical Algotihms" by Jean Meeus. But these (very usefull) solutions are only valid for a short period of time. If you want to find better solutions you need a numerical method.

Therefore, one usually employs some approximation procedure in solving the field equations.

This raises the question if these solutions are correct over longtime periods.


At page B1228 of this document we read:

Therefore, the radiation of gravitational waves always yields a decrease in energy of the system.


This result is valid for any system, relativistic or nonrelativistic.

This raises the question what is a relativistic versus a non relativistic system?
I expect that a relativistic system is when you apply GR and a non relativistic system when you apply Newton's Law.
This says nothing about the system itself.




At page B1231 (Left side) we read:

The results of the previous sections can be applied to find the secular change in the elements of the relative orbit of two point masses resulting from gravitational radiation.

It is very interesting that they use the words: "point masses" and not Black Holes. This implies that the relevence of this document applies to all collection of objects.

The equation of the realtive orbit of the motion is:

r = a(1e^2)/(1 + e*cos(psi))


This equation describes the movement of two objects using Newton's Law. See
reflection document 2: 1964 Gravitational Radiation and the Motion of two Point Masses



there are two parameters necessary to describe the orbit: the major axis a and the eccentricity e. In the Newton theory, they are constants of motion.

When you apply Newton's Law to simulate two objects the orbits are stable. The reason is because gravity acts instantaneous. For a simulation of the planets around the Sun that is sufficiant. The problem starts with Mercury.

In a previous paper (#15) the energy radiated from this system by gravitatinal waves was studied in detail

For a discussion in general about this important document go here:
P.C. Peters and J.Mathews, Physical Review, 131, 435




At page B1231 (Right side) we read:

Consider the case of circularly orbiting binary stars, for which we neglect deformation, mass flow, and other radiation processes.

I understand this approach, however if the inflow of mass (energy) is much larger than the outflow of gravitational energy than what is the point?


2001 Numerical Relativity: A review
For a copy of this document by Luis Lehner select https://arxiv.org/abs/grqc/0106072
This document starts at page 2 the text:

This new tool (computers) allows the study of systems which would otherwise be impossible (or extremely involved) analytically. Simulations not only are letting researchers tackle difficult problems but also allow for a nice
visualization of the outcome.

One strategy of science is to find the differential equations which describe the system being studied. The next step is to find the solution of these differential equations. This means a certain function which also describes this same system. The problem is in many cases such a function does not exist which encompasses the same physical space as the original problem.
At page 3 we read:

The main goal of numerical relativity is to provide the description of the spacetime
by solving Einstein equations numerically.

That is 100% correct.

This numerical implementation provides
the metric gab on, at least, some region of the manifold M ( M being an orientable, ndimensional
manifold of all physical events and gab a Lorentzian metric tensor).

That means when you consider a simulation in general the first thing that you should calculate/observe is metric gab and the manifold M. I have my doubts.
Next we read:

Although analytical extensions of nonglobally hyperbolic formulations can be obtained, the numerical treatment of such
situations is much more complex and has so far not been considered.

This keeps me thinking.
Next we read:

Hence, before attempting any computation one must carefully
 Choose appropriate form of equations and set of variables that govern the system
 Adopt a suitable reference frame with respect to which describe the system
 Define initial and/or boundary conditions

Issue 1 is very difficult because one has to take care that the calculated solution is a physical correct solution based on observations which are ambigious.
The answer for issue 2 is to use a reference frame that encompasses the physical space of all the objects studied.
The initial conditions for systems with binary BH's are tricky. See also next document Initial conditions
At page 5 we read:

Once the system of equations (1 out of 3) is chosen, as is the case with any simulation, care
must be taken with adopting
 a preferred set of suitable coordinates (so that from
the equivalence class of metric tensors defining the same geometry a single one is
obtained) and
 appropriate initial and boundary data for the problem under consideration.
Suitable

This immediate places a hugh burden on the physicist because generally speaking he must test all three systems if they all give the same solution.
Next we read in paragraph Suitable Coordinates:

These “ideal” coordinates satisfy the following properties
 Singularity avoidance properties (A) or amenability for singularity
excision (B):
Spacetimes containing singularities can be approached by either
 Simplification of variables:
 Degrees of Freedom:

The user is adviced to read the real text.
See also
At page 13 in paragraph "Inner Boundary Conditions" we read:

Cosmological censorship implies that singularities must be hidden inside the event horizons.
Moreover, the event horizon hides anything inside it; so, in principle,
an inner boundary could be chosen to lie inside the event horizon surrounding the singularity.

The problem with this text is that the reader:
 First, has to fully understand all the concepts that are in Black.
 Secondly, why does he need these concepts if he wants to simulate 2 Binary Black Holes surrounded by
a third Sun sized object using the Einstein Equations?


At page 50 we read in paragraph: Qualitative studies of Binary Neutron Star Spacetimes

An approach that has
been exploited to gain insight into the behavior of binary neutron star systems
assumes the system is in quasiequilibrium. Under this approach, the system
is assumed to radiate negligible amounts of energy and the system can be, in
some sense, approximated by obtaining equilibrium configurations at different
separations.

Why is this also not true for Black Holes ?


At page 54 in paragraph Dynamical GR  quasiequilibrium NS: we read:

This approach, called ‘matter without matter’ does, a
priori, a better job to describe the spacetime since gravitational radiation is not
neglected (although its back reaction on the sources is).

IMO it makes sense to call gravitational radiation "matter".
See also: General Reflection "Numerical Relativity" documents




At page 56 in paragraph 9. Working together: Complement with other approaches

In the description of binary systems, some distinct phases can be recognized. The first
one, is an adiabatic or inspiraling phase, where the members of the binary orbit around
each other while the separation between them slowly decreases as energy is carried
away by gravitational radiation.

What is the physical explanation?








2005 Evolution of Binary Black Hole Spacetimes
For a copy of this document by Frans Pretorius select https://arxiv.org/abs/grqc/0507014
This document starts with the text:

I. Introduction: One of the more pressing, unsolved problems in general relativity today is to understand the structure of spacetime describing the evolution and merger of binary black hole systems.

This raises immediate the question: How do we know that?

Binary black holes are thought to exist in the universe, and the gravitational waves emitted during a merger event are expected to be one of the most promising sources for detection by gravitational wave observatories (LIGO, VIRGO, TAMA,
GEO 600, etc.).

Of course if you detect gravitational waves you can ask yourself what is the case. And then of course if you have some model which produces gravitational waves you can claim "Eureka", but that is no prove that the physical processes which are behind your model are also in this case the cause of what you have detected. There could also other processes be involved.
A little further on we read:

The full 3D problem has, for many reasons, proven to be a more challenging undertaking, and only recently has progress
been made in the ability of numerical codes to evolve binary system

The cause of the gravitational waves can easily be a system in which 3 BH's are involved.


At the right hand side of page 1 we read:





The code has several features of note, some or all of which may be responsible for its stability properties:

This raises the question why is this necessary?
Consider a system which consists of more objects, Are the same features allowed? The problem is that you can hide the true solutions.


Initial conditions
At the left hand side of page 2 we read:

We use scalar field gravitational collapse to prepare initial data that will evolve towards a binary black hole system.
What does this mean? When you start with a collection of objects (a cloud) which does not "rotate" it will collapse towards one point. To take care that it collapses towards to points is tricky except when you start with two clouds of objects. The question is why.


Specifically, at t = 0 we have two Lorentz boosted scalar field profiles, and choose initial amplitude, separation and boost parameters to approximate the kind of orbit that the black holes (which form as the scalar field collapses) will have.

The same comment. Why these special initial conditions?
below III results we read:

we chose initial data parameters such that the black holes would merge within roughly one orbit — see Fig. 1 and Table I.

The same more or less as above: Why do you start with these special initial conditions? Specific the fact that the BH's should merge?




At the left hand side of page 3 we read (below Fig2) :

The loss of mass (and similarly increase in alpha) with time after the merger is due to accumulating numerical error.

"Strange" explanation. Maybe the explanation is the strength of the gravitational field. See the function psi4 discussed next.




At the right hand side of page 3 we read:

To estimate the gravitational waves emitted by the binary we use the NewmanPenrose scalar psi4,

How do we know that this is correct? What you need is the strength of the gravitational field.

To estimate the total energy E emitted in gravitational waves, we use the following formula.
See origianal document.

How do you know that this is correct? The strength of the gravitational field should be "embedded" in the Einstein equation. This is equivalent like the gravitational force on a point particle using Newton's Law.
At the right hand side of page 4 we read:

The binary merged within approximately 1 orbit, leaving behind a blackhole of mass Mf approx 1.9 M0

This is a very short simulation.
The simulation starts with two BH's of equal mass M0. How do we know that the simulation is correct that during the merging we loose 0.1 M0?
What is the result when we perform a simulation of 2 orbits? Are the results the same?














2013 A catalog of 174 binary blackhole simulations for gravitationalwave astronomy
For a copy of this document by Herald P.Pfeiffer et all select https://arxiv.org/abs/1304.6077
This document starts with the text at page 1:

For widely separated binaries, postNewtonian (PN) calculations provide accurate gravitational waveforms.
However, numerical simulations of the full Einstein equations are needed during the late inspiral, merger, and
ringdown.

IMO it is the other way around. How further away the objects in any simulation (assuming the objects are point masses) are, how more important the full Einstein equations are, because time becomes an issue. Time meaning gravitation radiation propagation issues. The shorter the distances Newton's Law is sufficiant.
When objects are not considered point masses than the details of the physical processes are an issue, but that is also the case with Newton's Law.
To be more precise: The full Einstein equations should be used to demonstrate that Binary Black holes merge in the first place, Newton's Law does not do that.
At page 2 in the paragraph Techniques we read:

The simulations are computed using the Spectral Einstein Code (SpEC)

This is the same as the next document.










2015 Accuracy and precision of gravitationalwave models of inspiraling neutron star – black hole
binaries with spin: comparison with numerical relativity in the lowfrequency regime
For a copy of this document by Prayush Kumar, Kevin Barkett, Swetha Bhagwat, Nousha Afshari, Duncan A. Brown, Geoffrey Lovelace, Mark A. Scheel, Béla Szilágyi select https://arxiv.org/abs/1507.00103
This document starts with the text:
At page 3 we read:

We construct our NR waveforms using the Spectral Einstein Code (SpEC) (#50)

For the document #50 see: http://www.blackholes.org/SpEC.html
The problem with this url is that it gives very little technical background what SpEC is.
To get some understanding try this Power Point Presentation by Mark Scheel:
http://www.tapir.caltech.edu/~romgr/slides/Mark_Scheel.pdf
Mark Scheel is one of the principle maintainers of the code.
The url also contains a list of 132 articles starting from the year 2000 until 2016. The articles are interesting but it is very difficult to get an umbrella overview what SpEC is.
 The year 2000 document is interesting because it mentions that solutions of the Einstein equations are very difficult to obtain. See: https://arxiv.org/abs/grqc/0005056 "Black hole evolution by spectral methods" by Lawrence E. Kidder, Mark A. Scheel, Saul A. Teukolsky, Eric D. Carlson, Gregory B. Cook
 The year 2012 document is a very interesting Review Article.
See: https://arxiv.org/abs/1203.5166 "Numerical simulations of compact object binaries" by Harald P. Pfeiffer. The document contains a literature list of 204 articles.












2016 LidovKozai Cycles with Gravitational Radiation: Merging Black Holes in Isolated Triple Systems
For a copy of this document by Kedron Silsbee, Scott Tremaine select https://arxiv.org/abs/1608.07642
This document starts with the text:

The merger time for two black holes on a circular orbit due to gravitationalwave emission is: etc
Peters, P. C. 1964, Physical Review, 136, 1224

It is amazing that this behaviour is already described in 1964
For a discussion go here: 1964 Gravitational Radiation and the Motion of two Point Masses
A prerequisite of the "1964" document is:
1963 Gravitational Radiation from Point Masses in Keplerian Orbit by P.C Peters and J. Mathews.
At page 2 of this "2016" document we read:

In this paper, we describe an alternative formation channel based on orbital evolution in an isolated triple system containing a blackhole binary and an external companion.

Systems with triple BH's are the most realistic combination to demonstrate BH merging. A better name in this case is a collision. The primary cause, why collisions happen in systems with 3 objects or more, has "almost nothing" to do with Energy loss caused by gravitational radiation, but more with the almost chaotic behaviour of the trajectories caused by the interferences (forces) between the objects. Such collisions can easily be demonstrated using Newton's Law.










See also
Following is a list with "Comments in Wikipedia" about related subjects
General Reflection "Numerical Relativity" documents
When you study the litterature the emphasis is on binary Black Holes, gravitational waves, waveforms and LIGO.
LIGO is important to detect gravitational waves, that is true. As a consequence you can study waveforms, also true. However Numerical Relativity is much more than gravitational waves. Numerical Relativity means solving The full Einstein Field Equations using actual examples based on observations like our Solar system without simplifications. That is demonstrated to be very difficult, often leading to instabilities. This is troublesome specific when the same simulations using Newton's Law are stable.
For some technical detail read: VB Mercury numerical.htm for comments about a program which only tries to simulate the trajectory of the planet Mercury.
In the litterature most examples studied are Black Hole's. A BH is nothing more than a very compact object. IMO when you consider it as a point mass its behaviour is identical as a Neutron star or any sun sized object. BH's are supposed to contain a singularity, which require special mathematical considerations. The question is, what is the physical meaning of a singularity?
All objects emit radiation, in this case I mean gravitational radiation. Gravitational radiation means energy loss which mathematical means mass loss. The issue is how much grmass loss is there compared to mass increase caused by infalling material. This is an important question because the main stream opinion is that two binary BH's holes will merge and in principle each has a small life span. (That does not mean that two BH can not collide, but than the cause is different).
The two articles from 1963 and 1964 discussed in this document do not mention this issue, which I think is a mistake (with hindsight) because many more recent articles refer to these articles.
Reflection document 2: 1964 Gravitational Radiation and the Motion of two Point Masses
In this document the equation

r = a(1e^2)/(1 + e*cos(psi))
DT
is used. This equation describes the movement of two binary objects.
IMO this equation can only be used to calculate the initial conditions, that means initial positions and velocities for simulations using either Newton's Law or GR.
Using Newton's Law this is rather straight forward but with GR the full initial conditions depend on the speed of gravity propagation or c. In fact the gravitational field depents on its history. The problems involved are explained in: Historical Overview #4 which is dedicated to Numerical Relativity.
Reflection document 3: 2001 Numerical Relativity: A review
This is a very clear review. Very interseting to read.
One main objection is the word singularity. This concept is used many times in the article without any good definition. What means that a Black Hole as a singularity? Why is it not enough when you study a BH that it has a certain mass and a radius. I expect that this is more than enough for most simulations.
Reflection document 4: 2005 Evolution of Binary Black Hole Spacetimes
What is missing in the document is real time data. Maybe when you try "hard" you can retrieve this.
What you need is the distance between the two BH's versus the revolution time versus the speed of light.
A much more general comment is: Why should BH's merge? The issue is can we uberhaupt speak of two BH's? as if you can have two objects surounded by almost empty space. In reality what you expect is that each BH is surrounded by some stars and smaller objects.
A different important issue is that this simulation only involves one revolution of the two BH's. IMO such a simulation is too short to make any claim about the behaviour of BH's in general.
At the same time what this means, is how difficult numerical relativity (in order to simulate actual configurations) using Einstein's equation in reality is.
Reflection document 5: 2013 A catalog of 174 binary blackhole simulations for gravitationalwave astronomy
Why this catalog? IMO it does not seem very usefull.
The problem is that all the simulations could based on the same wrong assumptions. IMO what you need is one good description of one simulation specific a descriptions of the initial parameters (conditions).
Reflection document 6: 2015 Accuracy and precision of gravitationalwave models of inspiraling neutron star – black hole
binaries with spin: comparison with numerical relativity in the lowfrequency regime
When you read this document, often the word waveform is used. I ask my self the question what has the concept of waveform to do with numerical relativity, to simulate the movement of two objects. In this case a BH and a neutron star. I doubt if any. More study is required.
The answer is may be find in the review article by Harold P. Pfeiffer: The main emphasis in the simulations is in the gravitational waves emitted by the binary Black Holes or neutron stars. These waves are important for the LIGO experiment. The emphasis is much less on the trajectories of the binary objects, which are very difficult to simulate.
The document is interesting because it gives a long list with current state of the art arxiv litterature.
Reflection document 7: 2016 LidovKozai Cycles with Gravitational Radiation: Merging Black Holes in Isolated Triple Systems
For a program in Visual Basic which tries to solve Triple objects using Newton's Law read this: VB BHmerger operation . With 2 rotating BH's and when a third BH is involved, BH merging becomes "simple"
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Created: 11 October 2016
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