1 The two postulates in Special Relativity - part 2

Op zondag 3 juli 2016 15:41:50 UTC+2 schreef Gregor Scholten:

>	Nicolaas Vroom wrote:

> >	My understanding is that the speed will increase when the photons approach the earth. In fact when you send a light signal from the sun towards the earth, first the speed will decrease and than increase.

SKIP

>	at least in that way that one should be able to find a coordinate system in which the speed of light, measured with respect to that coordinate systems, shows the behaviour you described.

What this means in some sense that the behaviour of light (of photons) is very complex. However this raises a serious issue.

If you want to simulate a star cluster or a galaxy what you need are the positions of each of the objects involved (by preference) in a coordinate system at a sequence of identical events (t1,t2,t3) at regular intervals in the past. Using these positions of all the objects and a model you can calculate the positions t4,t5 and t6 in the future.

However it is not that simple. When an observer measures the positions of the objects at t1 he does not measure the position at t1 but the position at ta in the past of t1. The problem is because he uses a light signal from the star and this signal uses time to reach the observer at t1. The same for the signal measured from that same object at t2 which shows the position at tb and for t3 which shows the position at tc. That means you have to do some type of forward transformation based on the values measured at ta,tb,tc to calculate the positions at t1,t2 and t3. This transformation is for each object different because the (average) differences involved are different.

What the above mentioned discussion shows is that such a forward transformation is extremely tricky, because the speed of light is not constant.

In order to compare the calculated positions at t4,t5, and t6 with the actual observations you have to perform some type of backward transformation which is also a function of the speed of light.

These transformations (using photons) is not what I want to discuss. The issue is the model or better the physical model. What is involved? Specific what I want to know if photons (the speed of light) is involved. I have my doubts. The same with almost every form of radiation. Except that there is a constant (small amount) decrease in mass, because stars emit photons. When you study Newton's Law, photons are not considered to study the movement of the objects (except to measure the positions as outlined before). Newton's Law uses masses and these masses have to be calculated as an intrinsic part of the model, based on the calculated positions. Newton's Law is based on forces, gravity and gravitons. These forces act instantaneous, which when gravitons are considered as part of this process, is not correct, because the propagation is time dependent. That means you have to modify Newton's law in order to take these delays in consideration.

What is also important, and that is true for all models, it does not matter if the objects involved are stars, neutron stars or blackholes: the basic laws that describe the behaviour (movement) are the same.

GR IMO also causes a problem specific when the Lorentz transformations are considered as part of this (physical) model, because these transformations are based around photons, the speed of light.

Maybe quantum gravity is the answer, but I have no idea how to incorporate that.

The mathematics to handle the pre-processing and the post-processing to calculate the initial conditions (and the predictions) of the model are generally speaking identical however they are different in the details. Gravitational waves do not show the lensing effect, photons do. As such they clearly should be handled separately. This is particular important when the evolution of the universe is studied.

It is also wrong to start your model directly from the observations and consider the pre-processing based on photons and the time delay of the gravitons (or the gravitational forces) as identical. You can do that (more or less) for the object from which the observations are done, but is wrong for all the other objects. That is also why the handling of the photons and the gravitons should be discussed completely separately.

Just some thoughts.

Nicolaas Vroom.

2 The two postulates in Special Relativity - part 2

In article <020a5a39-45cc-4237-85f3-dd5687e970ad@googlegroups.com>, Nicolaas Vroom writes:

>

If you want to simulate a star cluster or a galaxy what you need are the positions of each of the objects involved (by preference) in a coordinate system at a sequence of identical events (t1,t2,t3) at regular intervals in the past. Using these positions of all the objects and a model you can calculate the positions t4,t5 and t6 in the future. However it is not that simple. When an observer measures the positions of the objects at t1 he does not measure the position at t1 but the position at ta in the past of t1. The problem is because he uses a light signal from the star and this signal uses time to reach the observer at t1. The same for the signal measured from that same object at t2 which shows the position at tb and for t3 which shows the position at tc. That means you have to do some type of forward transformation based on the values measured at ta,tb,tc to calculate the positions at t1,t2 and t3. This transformation is for each object different because the (average) differences involved are different.

Consider also that the speed of gravity is finite. However, to second order, if I recall correctly, the Newtonian approach of an infinite speed and positions at a single time is the same as a more relativistic approach (finite speed of gravity but positions also extrapolated from the Newtonian case). This error, while small enough to be negligible, is probably larger than any effects of a non-constant speed of light.

>	I have my doubts. The same with almost every form of radiation. Except that there is a constant (small amount) decrease in mass, because stars emit photons.

Completely negligible.

>	These forces act instantaneous, which when gravitons are considered as part of this process, is not correct, because the propagation is time dependent. That means you have to modify Newton's law in order to take these delays in consideration.

See above.

3 The two postulates in Special Relativity - part 2

No. the light has a very simple behavior: in a local (small) region it moves at constant speed c and doesn't change direction. And as for your remark one post ago: "I doubt how important the last is. What is the point when local implies a very small region, while in general what we want is to study the evolution of the universe." Well, the universe consists of very small regions, you just need a lot of them. And there is no action at a distance, so local theories are all we need.

The complex thing you are referring to is not the behavior of light, but how the small regions are stitched together in space-time to form the total universe. This is as complex as the Einstein equation classically, and in quantum theory even more.

...

>	However it is not that simple. When an observer measures the positions of the objects at t1 he does not measure the position at t1 but the position at ta in the past of t1. The problem is because he uses a light signal from the star and this signal uses time to reach the observer at t1.

An observer who does this, will indeed need complicated analysis to understand the result. But that is not because his light signals behaved in a complicated way, it is due to the strange scenery of curved space-time through which the light had to travel. The photons (classically) used the simplest possible behavior (wherever you are, go straight ahead). It is just that they travel through a labyrinth of potential wells, holes, worm-holes, and perhaps also some reasonably flat regions.

-- Jos

4 The two postulates in Special Relativity - part 2

>>	at least in that way that one should be able to find a coordinate system in which the speed of light, measured with respect to that coordinate systems, shows the behaviour you described.

>

What this means in some sense that the behaviour of light (of photons) is very complex. However this raises a serious issue. If you want to simulate a star cluster or a galaxy what you need are the positions of each of the objects involved (by preference) in a coordinate system at a sequence of identical events (t1,t2,t3) at regular intervals in the past.

A remark: obviously, what you denote here t1, t2 or t3 are not events, but times. You have a four-dimensional coordinate system (t,x,y,z), and an event (single point in spacetime) is defined by setting a certain value for each of the four coordinates: (t,x,y,z) = (t1,x1,ty1,z1). By setting a certain value for only the time coordinate, you define a three-dimensional spacelike hypersurface.

>

Using these positions of all the objects and a model you can calculate the positions t4,t5 and t6 in the future. However it is not that simple. When an observer measures the positions of the objects at t1 he does not measure the position at t1 but the position at ta in the past of t1. The problem is because he uses a light signal from the star and this signal uses time to reach the observer at t1. The same for the signal measured from that same object at t2 which shows the position at tb and for t3 which shows the position at tc. That means you have to do some type of forward transformation based on the values measured at ta,tb,tc to calculate the positions at t1,t2 and t3. This transformation is for each object different because the (average) differences involved are different.

The usual way to simulate interactions where changes in the interactions-carrying field (electromagnetic field, gravitational field, ...) propagate with limited speed only is to simulate the interacting-carrying field itself. To do so, you dicretize space and time: you define a lattice in space with a discrete set of points, and calculate the field values at these points in dicrete time steps.

To incorporate objects that couple to the field - electric charges in case of electromagnetic field oder mass-carrying bodies in case of gravitational field - you can describe those objects by a matter field.

This numerical field theory approach is very different from your former numerical mechanics approach, where you use discret time steps, too, but allow continuous positions for the objects.

>	What the above mentioned discussion shows is that such a forward transformation is extremely tricky, because the speed of light is not constant.

Even if the speed of light would be constant, e.g. in a simulation of a system with electromagnetic interactions instead of a system with gravity, a numerical mechanics approach would be very complicated, because in Electrodynamics, the field of a moving charge is not simply a retarded Coulomb field, but is described by a Lienard-Wiechert potential which is much more complicated. Adding a variable propagation speed for changes in field would make things even worse.

A numerical field theory approach is much more straight-forward. Although is requires much CPU power.

>	These transformations (using photons) is not what I want to discuss. The issue is the model or better the physical model. What is involved? Specific what I want to know if photons (the speed of light) is involved.

Since GR is not quantized, it does not know photons. If you wanted to add quantized Electrodynamics to you simulation, photons would be involved.

Concerning the speed of light: this has little to do with photons. In Relativity, the quantity c, i.e. the "speed of light", is not only the propagation speed of electromagnetic waves, but rather the speed of all massless particles and of all waves that obey a wave equation of type \partial_mu \partial^mu A(x) = 0. The terminus "speed of light" is only due to historical reasons.

>	When you study Newton's Law, photons are not considered to study the movement of the objects

Neither they are when you study the laws of GR. In GR, the quantity c is involved, as propagation speed of changes in gravitational field, but the involvement of this quantity does not imply considering photons or even classical (non-quantized) electromagnetic waves.

>	Newton's Law is based on forces, gravity and gravitons.

No, not on gravitons. There aren't any gravitons in Newtonian theory. Gravitons are assumed to occur in quantum gravity, when quantizing a gravitational field with finite propagation speed, i.e. like the gravitational field of GR. On the one hand, there would be real gravitations, which would be quantized gravitational waves, and on the other hand virtual gravitons, appearing when applying perturbation theory on scattering processes intermediated by gravitational field.

One could quantize Newtonian gravity, but this wouldn't in any way yield gravitons: there are no dynamical degrees of freedom for the gravitational field in Newtonian theory, since gravitational field is fully determined by matter distribution. To yield field quanta, like photons, graviton, W and Z bosons or gluons, in quantization procedure, there are dynamical degrees of freedomg required, which are related to finite propagation speed.

>	These forces act instantaneous

What exludes the occurence of gravitons.

>	which when gravitons are considered as part of this process, is not correct

Because gravitons can only occur if the gravitational field has dynamical degrees of freedom, what implies that gravity is not instantaneous.

>	because the propagation is time dependent. That means you have to modify Newton's law in order to take these delays in consideration.

No, that means that you can either use Newton's theory and accept that there won't be gravitons in quantized theory, or use a theory with dynamical degrees of freedom for the gravitational field, like GR, and expect gravitons occurding as soon as you manage to implement the quantization procedure.

Or as another alternative, you skip quantization of the gravitational field and consider a classical (non-quantized) gravitation field, like the Newtonian field or the GR field.

>	What is also important, and that is true for all models, it does not matter if the objects involved are stars, neutron stars or blackholes: the basic laws that describe the behaviour (movement) are the same.

In GR, it makes a difference, due to the central singularities in black holes. For stars or neutron stars, you can use GR field equations in a numerical field theory approach, but for black holes, you cannot, since field equations fail at the central singularities. Therfore, black holes require a different approach.

[[Mod. note -- Actually, you can indeed (and researchers, myself included, routinely do) use a numerical field theory approach to simulate black holes; you just have to do it in a suitable way. There are two common approaches (as well as some other approaches which used to be common but are no longer as widely used): * "Excision", where you omit ("excise") a certain neighborhood of the black hole's singularity from your computational domain. * The "puncture method", where your numerical fields are the *difference* between the actual fields and an analytical model of the singularity.

5 The two postulates in Special Relativity - part 2

What I try to do is to see what is involved when you want simulate a star cluster (a set of visible objects) Starting point is a set of pictures of the objects involved at a sequence of times t1, t2 and t3 etc. Using these observations you can calculate (in principle) the positions of the objects. The problem is, when you consider t1, what you get is not the position at t1 but the position at ta, a certain moment in the past. This moment is different for each object involved. The primary reason is the speed of light. To get the position at t1 you have to perform a certain amount of pre-processing. The same for t2 and t3.

>

To incorporate objects that couple to the field - electric charges in case of electromagnetic field oder mass-carrying bodies in case of gravitational field - you can describe those objects by a matter field. This numerical field theory approach is very different from your former numerical mechanics approach, where you use discret time steps, too, but allow continuous positions for the objects. A numerical field theory approach is much more straight-forward. Although is requires much CPU power.

I will come back below.

>

> >	These transformations (using photons) is not what I want to discuss. The issue is the model or better the physical model. What is involved? Specific what I want to know if photons (the speed of light) is involved.

You answer the question your self below. What is involved is the propagation speed of the gravitational field. I prefer that to call this the quantity cg. IMO what you do not need in the simulations is the quantity c i.e. the speed of light. The speed of light is only required in what I call the pre- processing and the post-processing in order to handle the observations.

In fact in order to simulate the movements of objects electromagnetic fields are of no importance, except if interstellar clouds are involved.

When you study the evolution of the universe the laws involved also show this division. In order to observe the universe and space expansion light and the speed of light is involved. The cosmic micro wave background radiation also falls in this cathegory. To study the evolution (sec) the friedman equations are used, which should be based on (incorporate) the parameter cg.

>

Since GR is not quantized, it does not know photons. If you wanted to add quantized Electrodynamics to you simulation, photons would be involved. Concerning the speed of light: this has little to do with photons. In Relativity, the quantity c, i.e. the "speed of light", is not only the propagation speed of electromagnetic waves, but rather the speed of all massless particles and of all waves that obey a wave equation of type \partial_mu \partial^mu A(x) = 0. The terminus "speed of light" is only due to historical reasons.

IMO "you" should only indentify the speed of electromagnetic phenomena with the quantity c. The law E= mc^2 is from a mathematical point simple, but very difficult from a physical point of view.

> >	When you study Newton's Law, photons are not considered to study the movement of the objects

>	Neither they are when you study the laws of GR. In GR, the quantity c is involved, as propagation speed of changes in gravitational field, but the involvement of this quantity does not imply considering photons or even classical (non-quantized) electromagnetic waves.

The problem is (if I'am allowed to mention this here) is the relation between SR and GR. SR is based (very roughly) on light and clocks. GR is based on moving masses and the parameter sg. From an intrinsic "?" point of view these laws have nothing in common. The problem is when you want to do GR you need light and clocks to do your pre and post processing. IMO the pre processing should also be done such that "inside" GR there are no moving clocks involved.

> >	Newton's Law is based on forces, gravity and gravitons.

>	No, not on gravitons. There aren't any gravitons in Newtonian theory.

I fully agree with you.

>

Gravitons are assumed to occur in quantum gravity, when quantizing a gravitational field with finite propagation speed, i.e. like the gravitational field of GR. On the one hand, there would be real gravitations, which would be quantized gravitational waves, and on the other hand virtual gravitons, appearing when applying perturbation theory on scattering processes intermediated by gravitational field.

I like your comments. For me the problem is when do you speak of GR and when of quantum gravity. I more or less thought that GR always involves a gravitational field which propagates and what propagates are the gravitons. For an electrical field this are the photons. In that sense Newton's theory which act instantaneous does not involve gravitons but when you modify it (make it finite) you "need" gravitons.

>

One could quantize Newtonian gravity, but this wouldn't in any way yield gravitons: there are no dynamical degrees of freedom for the gravitational field in Newtonian theory, since gravitational field is fully determined by matter distribution. To yield field quanta, like photons, graviton, W and Z bosons or gluons, in quantization procedure, there are dynamical degrees of freedomg required, which are related to finite propagation speed.

I have simulated the forward movement of Mercury and it works when you introduce cg based on Newton's Law. Not only that, you can also study how this forward angle evolves in due time. (Horse shoe curve)

> >	These forces act instantaneous

>	What exludes the occurence of gravitons.

I fully agree with you. The forces in Newton's law act instantaneous. That means cg is infite. The problem is they are "wrong". That means in some way or an other you have to modify the laws and make cg finite. That allows for the introduction of gravitons.

> >	That means you have to modify Newton's law in order to take these delays in consideration.

>

No, that means that you can either use Newton's theory and accept that there won't be gravitons in quantized theory, or use a theory with dynamical degrees of freedom for the gravitational field, like GR, and expect gravitons occurding as soon as you manage to implement the quantization procedure. Or as another alternative, you skip quantization of the gravitational field and consider a classical (non-quantized) gravitation field, like the Newtonian field or the GR field.

> >	What is also important, and that is true for all models, it does not matter if the objects involved are stars, neutron stars or blackholes: the basic laws that describe the behaviour (movement) are the same.

>

In GR, it makes a difference, due to the central singularities in black holes. For stars or neutron stars, you can use GR field equations in a numerical field theory approach, but for black holes, you cannot, since field equations fail at the central singularities. Therfore, black holes require a different approach.

The issue is, what the difference is in behaviour, between a cluster of 10 stars of 50m0 with the same cluster of 10 BH with the same mass. Simpler is two stars and two BH's with the same mass. Will they merge? IMO the two stars will not merge. The two BH's will in general only merge when there is infalling matter.

>	[[Mod. note etc See, for example, arXiv:1203.5166

In this document mainly the technical details of the simulations are discussed but not the physical implications. (For the neutron stars mergers this is different after page 11) At page 2 they write: "(in a clean vacuum environment)," Why? Space around BH's is not empty.

At page 5 they write: "Gauge conditions and constraint damping terms contain parameters chosen by trial and error, and mesh structures are tuned based on user experience." Seems to me dangerous.

At page 9 they write: "there were difficulties for the SpEC code to obtain robust and automatic mergers". Is that wrong?

At page 12 they write: "Two groups are investigating the effects of electro-magnetic fields surrounding BH-BH binaries." Ofcourse when electro-magnetic fields are involved the whole picture can drastically change,

6 The two postulates in Special Relativity - part 2

Not really; as was already pointed out, light's behavior is as simple as possible: at each point along its path, the light just goes "straight ahead" with speed c (in vacuum). Here "straight ahead" simply means to move in the direction of its 4-momentum, so the result is that the light follows a (null) geodesic in spacetime.

>	However this raises a serious issue.

Not really. You give a long and involved description which I think boils down to: Light from each star in a distant star cluster or galaxy is deflected and slowed by the other stars, making it very difficult to determine the star positions, as the light does not propagate in a straight line with constant speed (as seen by us).

In principle, one could perform a complicated self-consistent simulation of all this.

In practice that is not needed, as the deflections and delays are all negligible [#] except in certain situations: the light passes close to a black hole.

[#] Light is deflected by the sun at most by the minuscule angle of 1.75 seconds of arc -- presumably other stars deflect light comparably. While this gives a considerable displacement over astronomical distances, seeing the light selects a combination of initial direction and deflection that result in the light reaching your telescope. As the deflectors are quite close to the source star (compared to the distance to earth), the image remains in the cluster or galaxy; in virtually all cases the amount of light reaching us that has been deflected significantly is tiny compared to the amount of light that was not deflected. Light is delayed by the sun by at most a few microseconds, which is completely negligible for all astronomical sources (exposures are usually minutes to hours or longer).

Compared to stars, black holes can deflect and delay by vastly larger amounts, because they can have much larger "masses", and the light can approach much closer to the "mass". That's why the LIGO team does a full and complete simulation using GR -- their gravitational radiation also follows null geodesics.

Tom Roberts

7 The two postulates in Special Relativity - part 2

I doubt if the calculation of this (null) geodesic is that simple in practice.

> >	However this raises a serious issue.

>

Not really. You give a long and involved description which I think boils down to: Light from each star in a distant star cluster or galaxy is deflected and slowed by the other stars, making it very difficult to determine the star positions, as the light does not propagate in a straight line with constant speed (as seen by us).

That is correct. I call this pre-processing of the observations. The same is required after your simultion to compare the results again with observations. I call that post-processing.

>	In principle, one could perform a complicated self-consistent simulation of all this.

Correct. That is why IMO the behaviour of light is complex.

>	In practice that is not needed, as the deflections and delays are all negligible [#] except in certain situations: the light passes close to a black hole.

The issue is not so much in practice. It is in primary in principle.

>	[#] Light is deflected by the sun at most by the minuscule angle of 1.75 seconds of arc -- presumably other stars deflect light comparably.

This deflection is the reason that I have a trouble with the idea: 'that light goes "straight ahead"'. IMO it does not always. The same is that the speed is c (in vacuum). Is there a vacuum around a star? I doubt that.

>	Compared to stars, black holes can deflect and delay by vastly larger amounts, because they can have much larger "masses", and the light can approach much closer to the "mass".

The primary reason of this posting is to discuss the actual simulation of the movement of the objects involved. That means the positions of the objects are know and electromagnetic fields (assuming the objects have no charge) can be neglected. IMO also the parameter c (speed of light) can de neglected. What is important is the parameter cg (speed of gravitational radiation) as pointed by Gregor Scholten.

Nicolaas Vroom

8 The two postulates in Special Relativity - part 2

9 The two postulates in Special Relativity - part 2

10 The two postulates in Special Relativity - part 2

11 The two postulates in Special Relativity - part 2

That is 100% correct, insofar they have to do with the observations, based on the speed of light. In this discussion I would like to start from the assumption that all the initial conditions are known.

> >	IMO also the parameter c (speed of light) can de neglected. What is important is the parameter cg (speed of gravitational radiation) as pointed by Gregor Scholten.

>	Both are, to much more accuracy than you will ever need, 299,792,458 kilometres per second.

How do you know that the speed of gravity is 299792458 m/sec? Did you actual perform an experiment to test that? Of course you can claim that the speed of gravity and the speed of light are the same, but physical they are completely different phenomena. In fact, this difference, is the topic of this thread. IMO the speed of gravity can be declared as being a constant. The speed of light IMO not, the reason is gravity.

See for more detail my posting of 25 july.

Nicolaas Vroom

12 The two postulates in Special Relativity - part 2

>>>	IMO also the parameter c (speed of light) can de neglected. What is important is the parameter cg (speed of gravitational radiation) as pointed by Gregor Scholten.

>>	Both are, to much more accuracy than you will ever need, 299,792,458 kilometres per second.

>	How do you know that the speed of gravity is 299792458 m/sec? Did you actual perform an experiment to test that?

It follows from GR that the "speed of gravity", i.e. the speed with which changes in the gravitational field do propagate, is equal to the speed of light. So, if you intend to program a simulation based on GR, you have to implement this. Of coure, you can as well feel free to reject GR, but then you have to find an alternative theory of gravity to found your simulation on.

>	Of course you can claim that the speed of gravity and the speed of light are the same, but physical they are completely different phenomena.

As already pointed out, in Relativity (SR as well as GR), the speed of light is not only the speed of light in the sense of a quantity that is special to electrodynamics, but a much more general quantity that is related to the structure of spacetime. So, the speed of light does not only concern electrodynamic phenomena, but as well completely different phenomena.

Take a point P in spacetime. There are two spacetime regions, one of them composed by points that are time-like seperated from that point P, and the other one composed by points that are space-like seperated from P. Now take a signal that is travelling along the boundary between both regions. The speed of such a signal is what we call the speed of light (and the boundary itself we call the light cones).

>	In fact, this difference, is the topic of this thread.

Is it? The title of this thread rather let me think that the topic would be the two postulates of SR?

>	IMO the speed of gravity can be declared as being a constant. The speed of light IMO not, the reason is gravity.

If we follow GR, the speed of gravity is in the same sense constant or non-constant like the speed of light:

- Measured with respect to a local inertial frame, the speed of light is constant, and since the speed of gravity is equal to the speed of light, the speed of gravity measured with respect to that local inertial frame is constant, too.

- Measured with respect to a general coordinate system, the speed of light may be variable (non-constant), and the speed of gravity as well, since it is equal to the speed of light.

Both, changes in electromagnetic field and changes in gravitational field, propagate along null-geodesics, i.e. worldlines in spacetime of length zero (spacetime length, i.e. proper time, not spatial length!). So, the speed is the same for both, no matter if measured with respect to a local inertial frame or with respect to a general coordinate system.

And again: of cource, you can feel free to reject GR and try to develop a theory of gravity in which speed of gravity is different from the speed of light. One property of such a theory is alreay quite easy to foresee: the principle of relativity will probably break down. Imagine a source that emits electromagnetic signals as well as gravitational signals. There will be a frame of reference S in which the speed of the electromagnetic signals as well as the gravitational signals is the same in all directions. And there will be another frame S' in which either the speed of the EM signals or the speed of the gravitational signals (or even both) is direction-dependent. So, the frame S is distinguished as a preferred frame of reference, enabling to distinguish absolutely between rest and uniform movement.

13 The two postulates in Special Relativity - part 2

14 The two postulates in Special Relativity - part 2

>

This article requires carefull study. At page 33 we read: "Black holes contain physical spacetime singularities, regions where the gravitational tidal field (curvature) becomes infinite. It is crucial, but hardly easy, to choose a computational technique that avoids encountering those singularities." The issue where are these supposed mathematical singularities? IMO inside the BH at the center of the BH. When you use Newton's law you can also get infinite result when the distance between two objects is zero. However physical you will never reach this condition because at the distance r=3D0 the two objects do not exist any more. Thus why worry?

Because we are not in Newtonian Gravity, but in GR. GR permits - or better say: even forces - the condition of a singularity being reached physically. Imagine a star collapsing to a black hole. According to GR, the collaps of the star's matter reaches the state of a singularity, i.e. of infinite density, within finite proper time.

>

At page 33 we also read: "Finally, different formulations of Einstein's equations behave very differently when implemented numerically, and we numerical relativists had to find suitable formulations that generate stable solutions" I like this honesty, but I'am worried. The issue is if the Einstein equations have solutions. The fact if they are stable is of less importance. When the situation you describe is not stable than the solutions should reflect that.

This is not about described situations being unstable, but the *simulation* of such situations being unstable. I once programmed a simulation of the time evolution of a wave function, ruled by Schroedinger equation, based on solving the Schroedinger equation numerically. Several times, I found the simulated wave function rapidly diverging to infinity. This happened when I chosed the time steps too rough, what caused to numerical procedure to become unstable. The situation which I was describing itself isn't unstable, though.

>

At page 33 we also read: "The structure of the 3 + 1 decomposition is familiar from Maxwell's equations, which similarly consist of a set of constraint equations for the electric (E) and magnetic (B) fields," IMO the physical behaviour of the (E) and (B) fields (because they are inter-twined) is completely different as the gravitational (or matter) field.

According to GR, your opinion is wrong. In the 3+1 decomposition of GR's equations, the bevaviour of the gravitational field is quite similar to the electromagnetic field.

>	IMO, a gravitational field is also simpler as an E/B field. The cause of the gravitational field are objects with masses mn and possitions xn and velocities vn at a sequence of time events tn.

That might be true for Newtonian Gravity. But in GR, this is surely not the case. The gravitational field is rather more complicated than the electromagnetic field there.

>	In such a system all objects are like blackholes.

You mean because you can handle celestial bodies like mass points? You can do that in Newtonian Gravity, but not in GR.

To program a simulation based on Newtonian Gravity, you can apply a numerical mechanics approach where you simulate the movement of particles that are sufficiently described by positions, but in GR, this does not work pretty well. You are rather obliged to apply a numerical field theory approach, where space is discretized and you calculate the participating fields (gravitational field and at least one matter field that describes the celestial bodies) at the discrete space points.

>	The fact that some emit light is of almost no importance.

Indeed. But the fact that the gravitational field has dynamical degrees of freedom in GR is of great importance.

>	Also in such a system baryonic matter and nonbaryonic matter is handled at equal footing.

One matter field might be sufficient for baryonic and nonbaryonic matter in first order, yes. But besides, you need to simulate the gravitational field, too.

>	In such a system the speed of light is not considered.

Little correction: the propagation of electromagnetic signals is not considered. The speed of light, however, is considered, since it is the speed with which changes in gravitational field do propagate.

>	There is no issue if this speed is constant or not. The most(?) important parameter is cg the speed of gravity propagation. IMO this speed can be considered constant.

In GR, cg is equal to c. Measured with respect to a general coordinate system, c may be variable, and so may be cg.

>	In such a system you can use Newton's Law.

If you prefer to program a simulation based on Newtonian Gravity, you can Newtonian Gravity, that's right (trivially). But in the article you are referring to, they talk about a simulation based on GR. Not only that, they talk about the simulation of a black hole merger. A simulation based on Newtonian Gravity would be highly inappropriate for such a situation.

>	However this law does not handlethe movement of Mercury

Or the processes that run during a black hole merger.

>	At page 34 we read: "and showed that only about 0.1% of the total mass of the blackholes is radiated away in the collision as gravitational waves"

See you? If you would simulate a black hole merger based on Newtonian Gravity, there wouldn't be any gravitational waves at all, since Newtonian Gravity does not know gravitational waves. So, obviously, Newtonian Gravity is highly inappropriate here.

>	I can understand that after any collision there is a loss in total mass, but not as gravitational waves.

If you program a black hole merger based on Newtonian Gravity, there does not occur any emission of gravitational waves, that's true. But in a simulation based on GR, such an emissions does occur.

>	At page 35 we read: "In particular, some formulations satisfy criteria that guarantee stable or otherwise well-behaved solutions, while others do not." I find this remark disturbing. See above question 1. Suppose the two objects (BHs?) do not merge, but move apart. Is that wrong?

It is quite obvious that this would be wrong. If two bodies (no matter if black holes or other celestial bodies) orbit each other initially due to some attractive force, and then start to move apart, without influx of energy from outside, then energy conversation must be violated.

>	At page 35 we read: "After decades of effort and anticipation, the combination of the above techniques enabled the first successful simulations of binary black hole inspiral and merger," Based on which previous observations?

This questions does not make sense: simulations are based on theories (and on numerical methods), not on observations. Except in that sense that the theories are based on observations themselves. In this case, the theory is GR, and the observations are those observations that have been the motivation to delevelop GR.

>	At page 36 we read: "For symmetric binaries, the emissions (gravitational waves) from the two companions cancel each other, but for asymmetric binaries they do not." I do not understand. IMO the emission of gravitational energy for any body of mass M is the same. When they merge (2M) it doubles.

Gravitational waves can interfer, just like EM waves. Imagine two sources of EM waves, e.g. two accelerated charges. The emitted EM waves can interfer destructively, making the emitted energy lower than twice the energy that would be emitted if only one of the two charges were present.

For the emission of EM radiation, one distinguishes between coherent and incoherent emission. For incoherent emission, interference effects are neglectable, and the intensities (energy per time and surface) just sum up. For coherent emission, on the other hand, destructive interference can lower the total intensity.

15 The two postulates in Special Relativity - part 2

I will not reject GR. I try to understand what is involved in the movement of objects (as simple as possible)

> >	Of course you can claim that the speed of gravity and the speed of light are the same, but physical they are completely different phenomena.

>

As already pointed out, in Relativity (SR as well as GR), the speed of light is not only the speed of light in the sense of a quantity that is special to electrodynamics, but a much more general quantity that is related to the structure of spacetime. So, the speed of light does not only concern electrodynamic phenomena, but as well completely different phenomena.

All of this is important to handle observations, but IMO less important for the actual simulations.

>

Take a point P in spacetime. There are two spacetime regions, one of them composed by points that are time-like seperated from that point P, and the other one composed by points that are space-like seperated from P. Now take a signal that is travelling along the boundary between both regions. The speed of such a signal is what we call the speed of light (and the boundary itself we call the light cones).

I fully agree with you. The issue is why do you need a light cone inorder to understand the movement of objects (planets). Ofcourse you need light to make observations, but that does not mean you need the speed of light in order to describe the trajectories of the objects. (Except if the physical quantities change of the objects studied as a function or speed)

> >	In fact, this difference, is the topic of this thread.

>	Is it? The title of this thread rather let me think that the topic would be the two postulates of SR?

> >	IMO the speed of gravity can be declared as being a constant. The speed of light IMO not, the reason is gravity.

>

If we follow GR, the speed of gravity is in the same sense constant or non-constant like the speed of light: - Measured with respect to a local inertial frame, the speed of light is constant, and since the speed of gravity is equal to the speed of light, the speed of gravity measured with respect to that local inertial frame is constant, too.

16 The two postulates in Special Relativity - part 2

>>>	How do you know that the speed of gravity is 299792458 m/sec? Did you actual perform an experiment to test that?

>>

It follows from GR that the "speed of gravity", i.e. the speed with which changes in the gravitational field do propagate, is equal to the speed of light. So, if you intend to program a simulation based on GR, you have to implement this. Of coure, you can as well feel free to reject GR, but then you have to find an alternative theory of gravity to found your simulation on.

>	I will not reject GR.

Let's note that.

>>>	Of course you can claim that the speed of gravity and the speed of light are the same, but physical they are completely different phenomena.

>>

As already pointed out, in Relativity (SR as well as GR), the speed of light is not only the speed of light in the sense of a quantity that is special to electrodynamics, but a much more general quantity that is related to the structure of spacetime. So, the speed of light does not only concern electrodynamic phenomena, but as well completely different phenomena.

>	All of this is important to handle observations, but IMO less important for the actual simulations.

In the actual simulations, you want to simulate the gravitational field. Or at least the movement of celestial bodies under the regime of gravity. So, for the gravitational field, c is relevant, because it is the propagation velocity of changes in the gravitational field. Of course, you can neglect that if you restrict yourself to consider the Newtonian limit.

>>

Take a point P in spacetime. There are two spacetime regions, one of them composed by points that are time-like seperated from that point P, and the other one composed by points that are space-like seperated from P. Now take a signal that is travelling along the boundary between both regions. The speed of such a signal is what we call the speed of light (and the boundary itself we call the light cones).

>	I fully agree with you. The issue is why do you need a light cone inorder to understand the movement of objects (planets).

Because the movement of celestial bodies is ruled by gravity. And the speed of light is relevant for the gravitational field. Once again: as long as you restict yourself to the Newtonian limit, you do not need the speed of light.

>	Ofcourse you need light to make observations, but that does not mean you need the speed of light in order to describe the trajectories of the objects.

That is trivial. But as already pointed out: in Relativity, c is not only the speed with which electromagnetic waves propagate, but rather a general quantity that concerns all laws of physics, e.g. the laws of mechanics, i.e. the equations that rule the movement of bodies, or the laws of field theory: the field equations of all interaction-carrying field (EM field, gravitational field, nuclear fields, ...) incorporate the quantity c.

>>>	IMO the speed of gravity can be declared as being a constant. The speed of light IMO not, the reason is gravity.

>>

If we follow GR, the speed of gravity is in the same sense constant or non-constant like the speed of light: - Measured with respect to a local inertial frame, the speed of light is constant, and since the speed of gravity is equal to the speed of light, the speed of gravity measured with respect to that local inertial frame is constant, too.

>	This document http://arxiv.org/abs/gr-qc/0403060 points out that you can also start from a theory where c and cg are different.

However, such theories are different from GR. And above, you claimed that you do not intend you reject GR.

>>

- Measured with respect to a general coordinate system, the speed of light may be variable (non-constant), and the speed of gravity as well, since it is equal to the speed of light. Both, changes in electromagnetic field and changes in gravitational field, propagate along null-geodesics, i.e. worldlines in spacetime of length zero (spacetime length, i.e. proper time, not spatial length!). So, the speed is the same for both, no matter if measured with respect to a local inertial frame or with respect to a general coordinate system.

>	For me the most important issue to understand is why do you need c in order to describe the movement of objects i.e. Einstein equations.

Einstein field equations incorporate that changes in the gravitational field propagate with c. Not only that, Einstein field equations are based on a four-formalism that is founded on the concept of spacetime, where c is the connector quantity between space and time. So, any simulation based on calculations that incorporate Einstein field equations incorporates the quantity c.

>	SR is primarily based around the speed of light, moving clocks, length contraction and simultaneity For me the question is why do you need these concepts in order to describe the movement of the stars in our galaxy and the galaxy's in the Universe.

The concepts of SR considered on its own are hardly needed, at least as long as the stars move with velocities << c, but Einstein field equations are needed.

>	(Assuming no electric or magnetic fields) How important are the lorentz transformations?

Except in SR limit, Lorentz transformations are rather unimportant in GR. However, Lorentz transformations formulas are not the only relativistic equations where c occurs. Einstein field equations contain c as well.

>	When you study this document: http://arxiv.org/pdf/gr-qc/9810065 they use a grid. That means you have one coordinate system, which makes everything much simpler.

It gets rid of coordinate transformations, but not of Einstein field equations (which contain c).

17 The two postulates in Special Relativity - part 2

This does not explain why physicists use the concept of singularity. Singularities IMO are only mathematical constructs, inside the radius of the BH. That means you should stay outside this radius.

> >	The article amazes me because it discusses a BH encounter with a neutron star (P37). IMO these type of mergers do "not" produce gravitational waves

>	According to our best understanding of the Einstein equations, you're mistaken: BH-NS encounters do indeed produce gravitational waves. (In fact, any time you accelerate massive objects in a non-sphericall -symmetric fashion you produce gravitational waves.)

In principle they do not have to be massif.

> >	and they are extremely difficult, because the merging process is "slow". Instead "you" study BH BH mergers which are "fast".

>

For various technical reasons it is indeed harder to simulate BH-NS mergers in detail than it is to simulate BH-BH mergers in detail. But that's not because of the time scales. Rather, the problem is that the usual means used to avoid singularities have trouble handling matter (like the NS), and the usual numerical schemes that can handle matter can't handle singularities.

I can imagine when you want to simulate BH-NS like in Figure 5 page 37 this is extremely tricky. To simulate BH-BH mergers is "much" simpler. Still I do not understand the issue of singularities for BH's.

> >	This article requires carefull study. At page 33 we read: "Black holes contain physical spacetime singularities, regions where the gravitational tidal field (curvature) becomes infinite.

etc

>

In fact, using the usual numerical-relativity techniques it's already quite a hard problem to just simulate a single BH sitting undisturbed in an otherwise-empty spacetime. It took many years of research before techniques were developed to perform simulations of this type which could run for long times without crashing or suffering rapidly-growing numerical instabilties.

The physical issue is what is the difference between a gravitational field of a star with mass m0 and a BH with mass m0.

Both at r=0 have a singularity?

> >

At page 33 we also read: "Finally, different formulations of Einstein's equations behave very differently when implemented numerically, and we numerical relativists had to find suitable formulations that generate stable solutions" I like this honesty, but I'am worried. The issue is if the Einstein equations have solutions.

>

Proving the *existence* and *uniqueness* of solutions to the Einstein equations (for some finite nonzero time interval) is a separate (hard) mathematical problem, which numerical relativists customarily ignore. I.e., numerical relativists typically *assume* that such solutions exist and take it as our task to try to find numerical approximations to these solutions.

Which make it very difficult to test if the solutions are physical correct.

> >	The fact if they are stable is of less importance. When the situation you describe is not stable than the solutions should reflect that.

>	Suppose the true solution to the Einstein equations is f(t,x,y,z) , while the output of our numerical computation is f(t,x,y,z) + g(t,x,y,z) where g represents the errors (inaccuracies) in our computation.

Only the sum is known. In a sense this means you should start with simulations where the solution of the Einstein equations are known.

>

It turns out that for lots of otherwise-plausible numerical schemes, the error g grows exponentially with time! This means that pretty quickly the numerical output will be dominated by the error term g , and our results will bear little or no resemblence to the actual solution f . (And, our calculation may crash due to (e.g.) floating-point overflow when g gets big enough.)

I had similar problems when I try to simulate stable galaxy rotation curves.

>	The problem which Baumgarte and Shapiro are discussing here is that of trying to prevent this exponential-growth-of-the-error from happening, i.e., the problem of designing a numerical scheme (including a formulation of the Einstein equations)

How difficult this is I think I got an idea when you try to compute the equations 15.30, 15.27 15.25 15.18/19/20 and 15.17 in that order which are shown in the book "Introducing Einstein's Relativity". In these exercises (in refers order) you know the solution. (Perihelion of Mercury)

>	While BHs may have (be surrounded by) dynamically-important magnetic fields, this article is primarily focused on simulations of "vacuum" BHs (with no electromagnetic fields)

> >	That is why IMO what the article should show is the description/discussion of the Gravitational field.

>	That would be considerably more complicated; that level of technical detail wouldn't be appropriate for an article in Physics Today.

18 The two postulates in Special Relativity - part 2

>

For me the most important issue to understand is why do you need c in order to describe the movement of objects i.e. Einstein equations. SR is primarily based around the speed of light, moving clocks, length contraction and simultaneity For me the question is why do you need these concepts in order to describe the movement of the stars in our galaxy and the galaxy's in the Universe. (Assuming no electric or magnetic fields) How important are the lorentz transformations?

A remark concerning SR and Lorentz transformation, since this seems to a crucial issue for you:

On the one hand, Lorentz transformation is the transformation from one inertial frame to the other. So, when considering only one single frame, one could assume that Lorentz transformation is of little relevance (like you seem to assume). However, that is not correct: SR does not only tell us that inertial frames are transformed to one another by Lorentz transformation, but also tells us that the laws of physics are the same in all inertial frames.

This requires that the laws of physics are Lorentz-covariant, i.e. keep their form when transforming from one inertial frame to the other. Take e.g. Maxwell equations. Using the four-potential A^mu and the Lorentz gauge, one can write the Maxwell equations as

\partial_mu \partial^mu A^nu = j^nu

where j^nu is the electric four flux density. Now transform these equations from inertial frame to the other: the four flux density j^nu is transformed, the four potential A^nu is transformed, and the derivatives \partial_mu, \partial^mu are transformed:

j^nu -> j'^nu = L^nu_mu j^mu A^nu -> A'^nu = L^nu_mu A^mu partial_mu -> partial'_mu = L^nu_mu partial_nu partial^mu -> partial'^mu = L^mu_nu partial^nu

where L_mu^nu is the Lorentz transformation matrix. Due to the Lorentz-covariance of the Maxwell equations, the new equations in the new inertial frame:

\partial'_mu \partial'^mu A'^nu = j'^nu

must be satisfied. This is only possible because the Maxwell equations have a special form, namely a Lorenz covariant form. Take as counter-example the non-relativistic Schroedinger equation: after Lorentz transformation of its terms into a new inertial frame, the Schroedinger equation is no longer satisfied.

So now, when considering the upper form of the Maxwell equations more in detail, we see the quantity c occuring: partial_mu is an abbreviation for partial / (partial x^mu), where x^mu is defined by

x^mu = (ct, x, y, z)

So, the term \partial_mu \partial^mu is nothing but

-c^2 partial^2/(partial t^2) + partial^2/(partial x^2) + partial^2/(partial y^2) + partial^2/(partial z^2)

what contains the quantity c (as c^2). The same is true for any other equation that satisfied the requirements of Lorentz covariance: the quantity c is always present.

So, even if you restrict yourself to consider only one single frame of reference, you need to incorporate the quantity c when doing calculations, e.g. solving field equations or equations of motion for particles or bodies.

Now, when we change from SR to GR, Lorentz covariance is only relevant in the SR limit. In general, we have to consider general covariance instead. But requirement of covariance still yields the occurance of c in all equations.

19 The two postulates in Special Relativity - part 2

I fully understand. IMO the laws of physics should be a description of the physical reality. The parameters used should also correspond to the (measurable) physical reality.

>

But what about the actual physical world (universe) in which we live? The question of the speed at which gravitational waves (or other effects) propagate is ultimately one which must (can hopefully) be answered by experimental/observational measurement and analysis. Since general relativity mathematically hard-wires the speed of gravity be identical to the speed of light, the previous sentence's analyses can't be done using solely general relativity. Rather, other relativistic gravity theories must be used.

What I try to do is to discuss the movement of objects (simulations) independent of observations. As such the equations that describe these movements should be strictly based on the parameter cg and not on the parameter c except when the movement is influenced by electromagnetic effects.

>	There's a very clear and readable discussion of this in section 7.4 ("Speed of gravitational waves") of the superb (open-access!) paper Clifford M. Will "The Confrontation between General Relativity and Experiment" Living Reviews in Relativity 17 (2014), 4 http://www.livingreviews.org/lrr-2014-4

20 The two postulates in Special Relativity - part 2

GR (as all laws) should be a description of the physical reality. When a star collapses, somewhere it stops to contract, meaning that the density physical does not reach infinity i.e a singularity. As such a BH does not contain a singularity.

For a colliding star the same issue exists. The distance never reaches zero, as such during the collision and merging there is no singularity issue,

21 The two postulates in Special Relativity - part 2

You can only reject something if you "fully" understand it.

>	In the actual simulations, you want to simulate the gravitational field. Or at least the movement of celestial bodies under the regime of gravity. So, for the gravitational field, c is relevant, because it is the propagation velocity of changes in the gravitational field.

Please let us call this speed cg.

>	Of course, you can neglect that if you restrict yourself to consider the Newtonian limit.

Under Newton's law the speed cg is infinite (instantaneous)

>	Because the movement of celestial bodies is ruled by gravity. And the speed of light is relevant for the gravitational field. Once again: as long as you restict yourself to the Newtonian limit, you do not need the speed of light.

Consider a world in which all objects are darkmatter object. You could also consider a world with only BH's. Why do you need the speed of light?

Consider also that you know all the present positions.

> >	Ofcourse you need light to make observations, but that does not mean you need the speed of light in order to describe the trajectories of the objects.

>	That is trivial. But as already pointed out: in Relativity, c is not only the speed with which electromagnetic waves propagate, but rather a general quantity that concerns all laws of physics, e.g. the laws of mechanics, i.e. the equations that rule the movement of bodies, etc.

"I know what Relativity says", but physical photons and gravitons are completely different "objects" and IMO you should try to study them independently.

At page 170 of th earlier book discussed by Ray d'Inverno we read: "The non-linearity reveals itself physically in the following way: the gravitational field produced by some source contains energy and hence, by SR, mass, and this mass in turn is itself a source of a gravitational field: that is to say that the gravitational field is coupled to itself. This non-linearity means that the equations are very difficult to solve in general." This raises immediate two questions: How do you than know that the equations are correct and how do you calculate the parameters of the equations. A different question is what does it mean: that you have solved the equations. In many cases there are no continuous solutions. The best you can do is to use a numerical solution and solve the equation step by step in a timewise manner. What is the most difficult part to simulate a billiard game: the collision between the balls. That means we temporarily exclude collisions.

IMO why celestial problems are so difficult is because of lack of information about the actual gravitational field. The actual gravitational field here is some type of "superposition" of the subset gravitational fields caused by all the objects considered. The problem is that each of these fields we observe here is caused by the objects in the past. The same for all actual fields for each of the objects considered. This interaction (delay) is what everything makes so difficult.

> >	This document http://arxiv.org/abs/gr-qc/0403060 points out that you can also start from a theory where c and cg are different.

>	However, such theories are different from GR. And above, you claimed that you do not intend you reject GR.

Reading all about numerical solutions and how difficult this is, you start thinking are there no different, more simpler solutions to do a correct simulation of the evolution of the planets.

> >	For me the most important issue to understand is why do you need c in order to describe the movement of objects i.e. Einstein equations.

>

Einstein field equations incorporate that changes in the gravitational field propagate with c. Not only that, Einstein field equations are based on a four-formalism that is founded on the concept of spacetime, where c is the connector quantity between space and time. So, any simulation based on calculations that incorporate Einstein field equations incorporates the quantity c.

I agree, however the issue when you consider numerical solutions, when you consider that the initial positions and velocities are known, to what extend do you need the quantity c and cg. IMO only cg. The quantity c is only used as part of the observations or when there are electromagnetic fields involved.

A typical case where you need c is when you want to calculate the Schwarzschild Radius. See chapter 16.9 of the mentioned book.

Ofcourse you could claim that they have the same strength and are mathematical indistinquishable, but physical they are not.

For example it is possible that you can solve gravitational fields in one coordinate system and without moving clocks. IMO such a consideration would make the equations simpler.

> >	(Assuming no electric or magnetic fields) How important are the lorentz transformations?

>	Except in SR limit, Lorentz transformations are rather unimportant in GR. However, Lorentz transformations formulas are not the only relativistic equations where c occurs. Einstein field equations contain c as well.

That means when you want to simulate the movements of the planets using GR you do not need the Lorentz transformations.

> >	When you study this document: http://arxiv.org/pdf/gr-qc/9810065 they use a grid. That means you have one coordinate system, which makes everything much simpler.

>	It gets rid of coordinate transformations, but not of Einstein field equations (which contain c).

But does it also get rid of moving clocks?

Nicolaas Vroom.

22 The two postulates in Special Relativity - part 2

>>	More precisely, the physical implication is that any singularity or singularities *inside* the BH, can't affect the (singularity-free) exterior region *outside* the BH.

>	This does not explain why physicists use the concept of singularity. Singularities IMO are only mathematical constructs, inside the radius of the BH.

If we follow GR, singularities are not only mathematical constructs, but exist unavoidably. Of course, one can hope that in a theory of quantum gravity, singularities can be avoided, e.g. by the gravitational collaps stopping at some finite density, like Planck density (10^94 g/cm^3), but if one programs a simulation based on GR, one has to implement the statements of GR, not of a theory of quantum gravity.

>>

In fact, using the usual numerical-relativity techniques it's already quite a hard problem to just simulate a single BH sitting undisturbed in an otherwise-empty spacetime. It took many years of research before techniques were developed to perform simulations of this type which could run for long times without crashing or suffering rapidly-growing numerical instabilties.

>	The physical issue is what is the difference between a gravitational field of a star with mass m0 and a BH with mass m0. Both at r=0 have a singularity?

No. The black hole has a singularity at r = 0 (according to GR), but the the ordinary star does not. The ordinary star has a non-zero radius and a finit mass density.

If you describe the movement of the ordinary star in terms of a mechanical approach, you can handle the star as a mass point in some sense, i.e. you do not need to take its non-zero radius into account, but as soon as you want to consider the star's gravitational field, you can no longer apply this description. You need to incorporate the star's finite size then.

>

The following document is also interesting: https://arxiv.org/abs/gr-qc/0507014 Evolution of Binary Black Hole Spacetimes by Frans Pretorius The system studied consists of 2BH of equal mass M0. At page 2 we read: " We use scalar field gravitational collapse to prepare initial data that will evolve towards a binary black hole system." That means they use (it seems) special initial conditions such that the two BH's will merge.

No, that does not mean that. The use initial conditions that result in a collaps to two black holes, which do not immediately merge. Such initial conditions can e.g. be two ordinary stars that each undergo a collaps to a black hole. The merger of the two black holes is something that happens in the later evolution of the binary black hole system.

>>>

At page 34 we read: "Unlike Maxwell's equations, however, Einstein's equations are nonlinear, and so they introduce a new set of phenomena and challenges." Also we read: "In finite-difference applications, the spacetime continuum is represented as a discrete lattice or grid," and: One class is initial data problems" To start from the correct initial conditions in any simulation is a difficult issue. Consider two objects of identical mass which revolve around each other in a circle. The question (1) is how do they behave? Is this a stable configuration?

>>	Again, the details of how to compute this are rather complicated. Continuing the Maxwell-equations analogy, etc.

>	The book by Ray d'Inverno mentioned above in paragraph 15.3 explains the "Advance of the perihelion of Mercury" The impression I get is that it is very difficult to simulate more (all?) planets using GR.

You should note that paragraph 15.3 depicts a mechanical approach: the equations describe the movement of a body (Mercury), not the evolution of fields. The configuration of the gravitational field is presumed as known there (namely as being the Schwarzschild solution), so it needn't be calculated.

In the simulation of a black hole merger as well as in the Maxwell-equations analogy, things are totally different: the evolution of the gravitational (or electromagnetic) field itself is what needs to be calculated, not the movement of bodies under the regime of a known field configuration.

>>>	At page 35 we read: "In particular, some formulations satisfy criteria that guarantee stable or otherwise well-behaved solutions, while others do not." I find this remark disturbing. See above question 1. Suppose the two objects (BHs?) do not merge, but move apart. Is that wrong?

>>	For these simulations the initial conditions are set up so that the two initial BHs form a bound system. It's thus mathematically guaranteed that the two BHs will eventually merge.

>	This is the same (more or less) as the example above. For example you can use Newton's Law to give the two BH's initial conditions such that they move in a circle (using Newton's Law) The issue is what happens when you use GR?

The first difference that occurs that you can no longer apply a mechanical approach like in Newtonian theory or for the advance of the Mercury perihelion. You need to calculate the field configuration applying a field theory approach.

For the advance of the Mercury perihelion, although it belongs to GR, the mechanical approach is sufficient because the field configuration is known (from an analytical field theory approach applies by Karl Schwarzschild in 1917 that yielded Schwarzschild solution). For the blach hole merger, the field configuration is not known analytically, therefore a numerical field theory approach is required.

>	Will they merge?

According to the result of the simulation we are talking about: yes. Accordint to a mechanical approach comparable to the approach for the advance of the Mercury perihelion: there is no such approach, therefore is question does not make sense.

>	(If yes why don't the planets of our solar system merge)

In fact, according to GR, the planets of our solar system do merge: during their movement around the sun, they emit gravitational waves and therefore loose energy, making them fall into the sun spirally. However, the intensity of the emitted gravitational waves is very low, so the energy loss per time unit is very low, too, and by this, it takes some 10^24 years until all planets have fallen into the sun.

In a close binary black hole system, the energy loss due to gravitational wave emissions is much higher, so the merger is very much quicker.

>

Some of this is described in: https://arxiv.org/abs/gr-qc/050701 "Evolution of Binary Black Hole Spacetimes" by Frans Pretorius Page 4 :"V Conclusion" is a good starting point of this document! At page 1 we read: "The code has several features of note, some or all of which may be responsible for its stability properties:" Generally speaking when you use Newton's Law there are no stability issues in the sense of software issues.

For Newton's law, numerical mechanics is sufficient. For GR, you need to apply a numerical field theory approach (except situations that are as simple as the advance of the Mercury perhelion). That's very different.

>	It is possible that the system you try to simulate is not stable. For example you can try a comet which collides with the sun.

For that, you can assume the gravitational field of the comet as being neglectable, so that you only need to consider the gravitational field of the sun, for which you can assume the known Schwarzschild solution, like for the advance of the Mercury perihelion (where you neglect the gravitational field of Mercury), enabling you to apply a numerical mechanics approach.

For a black holer merger, however, you need to consider the gravitational field of both black holes, requiring a field theory approach.

>	In that case your software should show the same results. When you try to solve Einstein's equations and there are stability issues with the software you have to be very carefully, because it is possible that the system itself is non stable.

There are good reasons to assume that a two-body system, even with taking a limited speed of gravity into account, is stable.

>	At page 2 we read: "We use scalar field gravitational collapse to prepare initial data that will evolve towards a binary black hole system." This sentence worries me, because it gives the impression that the code is modified to force that the 2 BH's will merge.

That impression is wrong, though. As already discussed, the initial data are prepared to make two black holes, not a black hole merger.

>>	Observations of actual astrophysical BHs and their inspiral/mergers are a whole separate topic, which I'm not going to get into here.

>	But this is actual very important. I mean are we sure that BH's actual merge?

No, we aren't. However, we are sure that according to GR, they do. And we have recent observations of gravitational waves that match the predictions that GR makes concerning gravitational waves emitted during a black hole merger.

>	When you simulate binary systems using Newton's Law they are stable. (No energy loss)

But we know that Newton's law is not fully correct. We know that GR is more correct than Newton's theory.

>	When there is an increase in mass they start to merge.

You mean in Newton's theory? Then your statement is not correct. In Newton's theory, it depends on the way the mass is increased. Imagine two celestial bodies orbitting each other. Imagine the mass of one body is increased by infalling comets. If the comets fall in that direction that is opposite to the orbitting direction, they slow down the body when hitting it, resulting in the orbit becoming closer, like in a merger scenarion. If the comets fall in the other direction, however, the body is accelerated, so that the orbit becomes wider.

>	When the opposite the average distance will increase. Such a binary system is not stable, but is not caused by computational (error) constraints.

You mean you consider the merger itself as instability? When talking about instabilities related to the simulation of a black hole merger, one means something different. The merger itself is no instability. Instability means that the results of the numerical computation become more and more different from the analytical solution, the longer the simulation runs.

>>	But if you have that same pair of BHs falling together in an ingoing spiral (where the initial state is that they're in a very far apart almost-circular orbit around each other), then on the order of 4% of the total mass is radiated in gravitational waves!

>	Again what is the difference with this example when you have two stars under identical initial conditions?

You mean two stars that are no black holes but have radii >> their Schwarzschild radii? Then the difference is that the two stars behave different from two black holes at the latest when their surfaces touch each other.

23 The two postulates in Special Relativity - part 2

>>

Gravitons are assumed to occur in quantum gravity, when quantizing a gravitational field with finite propagation speed, i.e. like the gravitational field of GR. On the one hand, there would be real gravitations, which would be quantized gravitational waves, and on the other hand virtual gravitons, appearing when applying perturbation theory on scattering processes intermediated by gravitational field.

>	I like your comments. For me the problem is when do you speak of GR and when of quantum gravity. I more or less thought that GR always involves a gravitational field which propagates and what propagates are the gravitons.

Then you thought something wrong. A field itself does not propagate. A field is everywhere at every time, in that sense that at any point (x,t) in spacetime, there is a value of this field, let's call it A(x,t). What may propagate are *changes* in the field.

Let's consider the electromagnetic field. Imagine an electric charge q being at rest at a position x0. In a wide range around the charge, the configuration of the electromagnetic field is that of an electrostatic field caused by the charge, i.e. the magnetic field is zero and the electric field at a point x is E(x) = q / (4 pi eps0 |x - x0|^2). Since that field configuration is static, there does not propagate anything.

Now imagine the charge is accelerated for a short time. After the acceleration phase, the charge is moving with constant velocity away from it initial position x0. In a small range around the position x0, the electromagnetic field already "knows" that the charge is moving away from x0. The field configuration in this range has already changed: the electric field at some point x in this range has a new value E(x,t) = q / (4pi eps0 |x - x'(t)|^2), where x'(t) is the position of the charge at the time t, and the magnetic field is non-zero.

Outside this range, the electromagnetic field still has its old configuration, it does knot "know" yet that the charge has been accelerated. So, there are two regions, one where the field configuration is already modified and one where it is not yet. Between both, there is a boundary at which the field configuration is currently changing from the old to the new configuration. This boundary is a thin spherical shell of radius (t - t0) * c, where t is the current time and t0 is the time at which the charge was accelerated (we assume that the duration of the acceleration phase is << t - t0).

The spherical shell is expanding, with the speed c. This shell, where the field the field is changing its configuration from the old one to the new one, is what propagates. To get an impression how the electric field looks like in this shell, take this image:

24 The two postulates in Special Relativity - part 2

We don't know that. So far, there isn't any observation that yields any information about what exactly happens to a collapsing star after it has deceeded the Schwarzschild radius. So, the statement of GR that the collaps reaches a singularity state within finite proper time is not in contradiction to any current observation. Assumed, we will ever make an obversation that gives evidence of the collaps stopping at a finite density (even though it is hard to imagine how such an observation could look like), this would mean a falsification of GR, but currently, no such observation has been made yet.

Many physics do not feel pretty happy with singularities occuring in GR, though, an hope that a theory of quantum gravity will yield a way to prevent singularities, but as long as one programs a simulation based on GR and not on quantum gravity, one has to implement what GR tells us.

>>>	At page 33 we also read: "Finally, different formulations of Einstein's equations behave very differently when implemented numerically, and we numerical relativists had to find suitable formulations that generate stable solutions"

>

>>

This is not about described situations being unstable, but the *simulation* of such situations being unstable. I once programmed a simulation of the time evolution of a wave function, ruled by Schroedinger equation, based on solving the Schroedinger equation numerically. Several times, I found the simulated wave function rapidly diverging to infinity. This happened when I chosed the time steps too rough, what caused to numerical procedure to become unstable. The situation which I was describing itself isn't unstable, though.

>	IMO there are different problems: 1. The situation can be unstable, 2. the equations are unstable, 3. the numerical solution can be unstable

In the article you are referring to, they talk about the numerical solution becoming unstable. The situation is assumed to be stable.

Unstabilities in sets of interacting bodies that are unstabilities of the situation are only know from cases where three of more bodies are involved. Take e.g. a planet in the gravitational field of two nearby suns (two suns + one planet = three bodies), the trajectory of the planet is known to easily become unstable then. In a two-body system, the situation shouldn't become unstable.

Concerning your distinction between category 1 and 2: that doesn't make sense. The situation being unstable is equivalent to the equations being unstable. Equations are part of the theoretical description of the situation. If it follows from the theory that situation is stable, then the equations are stable, too. What could happen is that the theory predicts that the situation (and as well the equations) becomes unstable, but from obversation, we find out that the situation does not become unstable. Then, the theory is in contradiction to observation, and by this, falsified. This does not belong to a category "situation is stable, but equations are unstable", but to the category "theory is proven to be wrong".

>	4. or any combination Step size problems belong in cathegory 3. The problems I had with implementing the equations outlined in chapter 15.3 of the book "Introducing Einstein's relativity" belong in cathegory 2. In these cases the solution is known.

As we have seen, category 2 does not make sense. Maybe you could describe what your problem has been, and what the solution has been, so that we can analyze to what category it belongs in fact?

As one can read here:

25 The two postulates in Special Relativity - part 2

>>	In the actual simulations, you want to simulate the gravitational field. Or at least the movement of celestial bodies under the regime of gravity. So, for the gravitational field, c is relevant, because it is the propagation velocity of changes in the gravitational field.

>	Please let us call this speed cg.

You can, if you want, distinguish in GR between the propagation speed of electromagnetic signals, let's call it cl (l for light), and the propagation speed of gravitational signals, cg, although theirs values are always the same. However, you have to define a third quantity then, the connector quantity between space and time, let's call it cs (s for spacetime). So, we have cs, cg and cl.

>>	Of course, you can neglect that if you restrict yourself to consider the Newtonian limit.

>	Under Newton's law the speed cg is infinite (instantaneous)

Another remark to your approach you described in a parallel post where you modified Newton's law by making cg finite. Newton's law can be written as

\vec H(\vec r, t) = - (M G / |\vec r - \vec r0(t)|^2) \vec e_{r,r0} (1)

where \vec H is the strength of the gravitational field at the point \vec r at time t, M the mass of the gravitational centre, \vec r0 the position of the gravitational centre, and \vec e_{r,r0} the unit vector from the position of the gravitational centre to the point \vec r.

Now your approach is that you replace the actual position \vec r0(t) of the gravitational centre at time t by the retarded position

[\vec r0]_ret = \vec r0(t_ret)

i.e. the position for a earlier time t_ret:

\vec H(\vec r, t) = - [(M G / |\vec r - \vec r0|^2) \vec e_{r,r0}]_ret

= - (M G / |\vec r - \vec r0(t_ret)|^2) [\vec e_{r,r0}]_ret (2)

The trouble with this approach is the the non-retarded version (1) is the solution of a field equation

\div \vec H(\vec r, t) = - rho(\vec r, t) (3)

where \div is the divergence that can be written in Cartesian coordinates as

\div \vec H = \partial_x H_x + \partial_y H_y + \partial_z H_z

and rho is the mass density. In some sense, equation (3) itself can be considered as Newton's law. For equation (2) however that is constructed in your approach, you didn't show that it is a solution of a field equation. Thefore your approach is incomplete.

To get an impression how a field equation that yields a retarding effect, i.e. a finite speed of gravity, could look like, let's have a look on Electrodynamics, i.e. on Maxwell equations:

\div \vec E = \rho_el (4a) \div \vec B = 0 (4b) \rot \vec E = - \partial_t \vec B (4c) \rot \vec B = \vec j + \partial_t \vec E (4d)

(4a) looks very similar to (3), and is nothing but the Coulomb law from Electrostatics. What makes the retarding are equations (4c) and (4d), that contain the magnetic field besides the electric field. So, for a retarded gravitational field, one might assume that is not only the gravitational field \vec H, but as well some "gravitomagnetic" field.

Second, the solutions of the Maxwell equations for an electric charge that is moving is not simply a electrostatic Coulomb field with retarding like

\vec E(\vec r, t) = - [(q / (4 pi eps0 |\vec r - \vec r0|^2)) \vec e_{r,r0}]_ret (5)

that would be comparable to the gravitational field (2) from your approach. E.g. for a uniformly moving charge, equation (5) would yield an electric field with a direction towards the retarded position of the charge. However, in fact, Maxwell equation yield a solution for the electric field of a moving charge where the direction of the field is towards the *actual* position of the charge.

So, your approach is probably inconsistent.

>>	Because the movement of celestial bodies is ruled by gravity. And the speed of light is relevant for the gravitational field. Once again: as long as you restict yourself to the Newtonian limit, you do not need the speed of light.

>	Consider a world in which all objects are darkmatter object. You could also consider a world with only BH's. Why do you need the speed of light?

We do not need cl. But cs and cg.

>>>	Ofcourse you need light to make observations, but that does not mean you need the speed of light in order to describe the trajectories of the objects.

>>	That is trivial. But as already pointed out: in Relativity, c is not only the speed with which electromagnetic waves propagate, but rather a general quantity that concerns all laws of physics, e.g. the laws of mechanics, i.e. the equations that rule the movement of bodies, etc.

>	"I know what Relativity says", but physical photons and gravitons are completely different "objects" and IMO you should try to study them independently.

In GR, you should not try to study gravitons at all, since gravitons as quantum gravity objects are incompatible with GR. Accepting a limited lack of consistency, you can add photons to GR.

If you didn't really intend to refer to quantum theory, but rather used "photons" and "gravitons" as other words for electromagnetic field and gravitational field: both fields can be studied independly. However, in any case, one has to deal with a quantity that is typically denoted c. In the upper definition, this quantity is neither cl nor cg, but cs.

>	At page 170 of th earlier book discussed by Ray d'Inverno we read: "The non-linearity reveals itself physically in the following way: the gravitational field produced by some source contains energy and hence, by SR, mass, and this mass in turn is itself a source of a gravitational field

That argument is incompatible with GR. It is rather a little fruitfull attempt to combine SR and Newtonian Gravity. Since we are talking about a simulation based on GR, such approaches are little helpful.

In GR, the source of the gravitational field is NOT mass, but the stress-energy tensor T^ij, a tensor of rank 2. In many cases, this tensor can be written as

T^ij = diag(rho, p, p, p)

where rho is energy density and p is pressure.

>	that is to say that the gravitational field is coupled to itself. This non-linearity means that the equations are very difficult to solve in general." This raises immediate two questions: How do you than know that the equations are correct

The equations that come out when trying to combine SR and Newtonian Gravity are probably not correct. The equations of GR are checked to be correct by comparing predictions of GR with observations. Those predictions come from special cases where the field equations have been succesfully solved analytically. These cases are, amongst others:

- Schwarzschild solution - Gravitational waves - Cosmological solutions (Friedmann-Lemaitre-Robertson-Walker metric)

>	A different question is what does it mean: that you have solved the equations. In many cases there are no continuous solutions. The best you can do is to use a numerical solution and solve the equation step by step in a timewise manner.

And after doing so, you have solved the equations, numerically. In the cases where you have found an analytical solution, you have solved the equations analytically.

>

IMO why celestial problems are so difficult is because of lack of information about the actual gravitational field. The actual gravitational field here is some type of "superposition" of the subset gravitational fields caused by all the objects considered. The problem is that each of these fields we observe here is caused by the objects in the past.

That problem also occurs e.g. in Electrodynamics. However, there it is easier to solve because Electrodynamics is linear: the EM field configuration caused by two electric charges is just a linear combination of two EM field configurations that are each caused by a single charge. GR, on the other hand, is non-linear: the gravitational field configutation caused by two celestial bodies is not a simple superposition, but much more complicated. Therefore, the non-linearity of GR makes the problem even harder.

>>>	This document http://arxiv.org/abs/gr-qc/0403060 points out that you can also start from a theory where c and cg are different.

>>	However, such theories are different from GR. And above, you claimed that you do not intend you reject GR.

>	Reading all about numerical solutions and how difficult this is, you start thinking are there no different, more simpler solutions to do a correct simulation of the evolution of the planets.

If you want to program a simulation that is in agreement with GR, your possibilities to imagine simpler solutions are limited. Solutions where cl, cg and cs have different values are excluded.

>>>	For me the most important issue to understand is why do you need c in order to describe the movement of objects i.e. Einstein equations.

>>

Einstein field equations incorporate that changes in the gravitational field propagate with c. Not only that, Einstein field equations are based on a four-formalism that is founded on the concept of spacetime, where c is the connector quantity between space and time. So, any simulation based on calculations that incorporate Einstein field equations incorporates the quantity c.

>

I agree, however the issue when you consider numerical solutions, when you consider that the initial positions and velocities are known, to what extend do you need the quantity c and cg. IMO only cg. The quantity c is only used as part of the observations or when there are electromagnetic fields involved. A typical case where you need c is when you want to calculate the Schwarzschild Radius.

You're wrong. The quantity c in the formula

rs = 2 G M / c^2

for the Schwarzschild radius is rather cs than cl. In GR, the meaning of the Schwarzschild radius is that is some kind of geometrization of mass: a mass M "is" a length of G M / c^2, so that rs = 2 M. Since this geometrization does not concern the propagation of EM signals (cl) nor or gravitational signals (cg), the relevant quantity is cs.

>	Ofcourse you could claim that they have the same strength

I guess you wanted to write the same value instead the same stength? A speed does not have a strength.

>	For example it is possible that you can solve gravitational fields in one coordinate system and without moving clocks. IMO such a consideration would make the equations simpler.

This sounds as if you think that Einstein field equations can be written either in a way where only one coordinate system is involved or in a way where several coordinate systems (and moving clocks) are involved. Let's have a look on the field equations:

R_ij + 1/2 R g_ij = T_ij

The quantities R_ij, g_ij and T_ij with two indices i,j are tensors of 2nd level, the indices i and j run each from 0 to 3. In total, each of the three tensors has 4 x 4 = 16 components, yielding 16 equations, that can be reduced to 10 due to symmetries of the tensors. Now write the equations in a single coordinate system, i.e. express the quantities occuring in the equation in terms that refer to that coordinate system. There are still 10 equations then.

Now, let's assume you want to write the field equations in a way that refers not only to one single coordinate system, but to two different coordinate systems. The only way to do so is to write the equations in the first coordinate system, yielding 10 equations, and then again in the second coordinate system, yielding another 10 equations, so that you have 10 + 10 = 20 equations in total. And if you want to write the field equations in a way that refers to three coordinate systems, you have 30 equations in the end. And 40 equations for four coordinate systems, and so on. Of course, you can do that, but it makes little sense, since all information is present in the first 10 equations. All additional equations are redundant.

So, writing the field equations in a way that involves only one coordinate system instead of N coordinate systems make the equations indeed simpler, by reducing the number of equations from 10 * N to 10, but that is trivial. Imagine you want to solve Newtonian equation of motion, m a = F, for a single body. You can, if you want, write this equation in one inertial frame S, and then again in a second inertial frame S', and then solve the equation in S and once again in S'. You double the number of equations by this from one to two, although they both express exactly the same, and calculate two solutions that or both the same. Skipping this and solve only one equation is simpler, so your statement is correct in some sense, but that's trivial.

And of course, any physicist who tries to solve Einsteins field equations writes these equations as 10 equations in a single coordinate system, not as 10 * N equations referring to N coordinate systems. So, you cannot make anything simpler here than it already is usually.

What is involved in Einstien field equations, however, is the property that they are generally covariant, i.e. that they do not only apply in the coordinate system in which they are currently written for being solved, but as well in any other coordinate system. If this property is what you are talking about: solving the equations in one coordinate system does not change anything of this property. So, there isn't any way to make the equations simpler by removing that property. If that is what you wanted to say.

Concerning moving clocks: I guess you intend to refer to special-relativistic time dilation when talking about moving clocks? On the one hand, time dilation is more complicated in GR than in SR, since time dilation does not only occurd due to movement, but also du to gravity. On the other hand, you do not need to consider this when solving Einstein field equations. You can, if you want, use the found solution to calculate the proper times of clocks, but you are not obliged to do so.

>>>	(Assuming no electric or magnetic fields) How important are the lorentz transformations?

>>	Except in SR limit, Lorentz transformations are rather unimportant in GR. However, Lorentz transformations formulas are not the only relativistic equations where c occurs. Einstein field equations contain c as well.

>	That means when you want to simulate the movements of the planets using GR you do not need the Lorentz transformations.

Exactly.

Even in a special-relativistic theory, like Electrodynamics, you do not need Lorentz transformations when considering the movement of bodies that couple to the particular interaction. Considering e.g. the movement of electric charges in an EM field does not require Lorentz transformations (but dealing with Lorentz-covariant equations like Maxwell equations). It's like Einsteins field equations described above: you can do all the calculations within one single inertial frame, you do not need to duplicate the equations by considering several inertial frames.

>>>	When you study this document: http://arxiv.org/pdf/gr-qc/9810065 they use a grid. That means you have one coordinate system, which makes everything much simpler.

>>	It gets rid of coordinate transformations, but not of Einstein field equations (which contain c).

>	But does it also get rid of moving clocks?

Under assumption that you mean the requirement to consider the proper time of moving bodies when saying "moving clocks": yes. You can calculate the proper time of the involved bodies, if you want, but it is not necessarily needed.

26 The two postulates in Special Relativity - part 2

The whole issue in some sense is that you have to make testable predictions, and singularities and infinities are not part of these predictions.

> >	The physical issue is what is the difference between a gravitational field of a star with mass m0 and a BH with mass m0. Both at r=0 have a singularity?

>	No. The black hole has a singularity at r = 0 (according to GR), but the the ordinary star does not. The ordinary star has a non-zero radius and a finit mass density.

IMO a BH also has a non-zero radius and a finite mass density

> >	That means they use (it seems) special initial conditions such that the two BH's will merge.

>

No, that does not mean that. The use initial conditions that result in a collaps to two black holes, which do not immediately merge. Such initial conditions can e.g. be two ordinary stars that each undergo a collaps to a black hole. The merger of the two black holes is something that happens in the later evolution of the binary black hole system.

And what causes this to happen? IMO when both (normal and BH) collect more mass (Energy) than they radiate they will never merge

> >	The book by Ray d'Inverno mentioned above in paragraph 15.3 explains the "Advance of the perihelion of Mercury" The impression I get is that it is very difficult to simulate more (all?) planets using GR.

>	You should note that paragraph 15.3 depicts a mechanical approach: the equations describe the movement of a body (Mercury), not the evolution of fields. The configuration of the gravitational field is presumed as known there (namely as being the Schwarzschild solution), so it needn't be calculated.

The issue is here to calculate the movement of all the planets using GR. My impression is here that this is very difficult

>

The first difference that occurs that you can no longer apply a mechanical approach like in Newtonian theory or for the advance of the Mercury perihelion. You need to calculate the field configuration applying a field theory approach. For the advance of the Mercury perihelion, although it belongs to GR, the mechanical approach is sufficient because the field configuration is known (from an analytical field theory approach applies by Karl Schwarzschild in 1917 that yielded Schwarzschild solution). For the blach hole merger, the field configuration is not known analytically, therefore a numerical field theory approach is required.

This is in line with my assumption that this is very difficult. It is also in line with the doubt I have if two binary BH always will merge. Anyway the behaviour of two objects is a physical phenomena. How sure are we that a binary star system will merge?

> >	Will they merge?

>	According to the result of the simulation we are talking about: yes.

I agree with you. But you have to know the details of what is physical involved. Is there preference for binary BH's to merge? compared to stars to merge? Specific when equal masses are considered.

>

In fact, according to GR, the planets of our solar system do merge: during their movement around the sun, they emit gravitational waves and therefore loose energy, making them fall into the sun spirally. However, the intensity of the emitted gravitational waves is very low, so the energy loss per time unit is very low, too, and by this, it takes some 10^24 years until all planets have fallen into the sun. In a close binary black hole system, the energy loss due to gravitational wave emissions is much higher, so the merger is very much quicker.

The issue is to compare equal masses.

> >>	Observations of actual astrophysical BHs and their inspiral/mergers are a whole separate topic, which I'm not going to get into here.

> >	But this is actual very important. I mean are we sure that BH's actual merge?

>	No, we aren't. However, we are sure that according to GR, they do. And we have recent observations of gravitational waves that match the predictions that GR makes concerning gravitational waves emitted during a black hole merger.

I my simulation I assume that a third object is involved. What is wrong with that?

> >	When there is an increase in mass they start to merge.

>

You mean in Newton's theory? Then your statement is not correct. In Newton's theory, it depends on the way the mass is increased. Imagine two celestial bodies orbitting each other. Imagine the mass of one body is increased by infalling comets. If the comets fall in that direction that is opposite to the orbitting direction, they slow down the body when hitting it, resulting in the orbit becoming closer, like in a merger scenarion. If the comets fall in the other direction, however, the body is accelerated, so that the orbit becomes wider.

I fully agree with your objections but that does not mean that my simulation is wrong. In fact you partly agree with me. (50%?)

Nicolaas Vroom

27 The two postulates in Special Relativity - part 2

28 The two postulates in Special Relativity - part 2

29 The two postulates in Special Relativity - part 2

>>>	This does not explain why physicists use the concept of singularity. Singularities IMO are only mathematical constructs, inside the radius of the BH.

>>	If we follow GR, singularities are not only mathematical constructs, but exist unavoidably.

>	The whole issue in some sense is that you have to make testable predictions, and singularities and infinities are not part of these predictions.

First, it is not forbidden for a theory to make non-testable predictions besides testable predictions. Second, the collaps to a singularity indeed is a testable prediction: if a collapsong star stops collapsing at a non-zero radius and finite density, it is, in principle, possible to observe that. By this, the prediction that the collaps does not stop before reaching infinite density can be falsified.

>>>	The physical issue is what is the difference between a gravitational field of a star with mass m0 and a BH with mass m0. Both at r=0 have a singularity?

>>	No. The black hole has a singularity at r = 0 (according to GR), but the the ordinary star does not. The ordinary star has a non-zero radius and a finit mass density.

>	IMO a BH also has a non-zero radius and a finite mass density

According to GR, the black hole itself has a non-zero radius, namely its Schwarzschild radius, but the matter of the preceding star has collapsed to a singularity in the centre of the black hole.

>>>	That means they use (it seems) special initial conditions such that the two BH's will merge.

>>

No, that does not mean that. The use initial conditions that result in a collaps to two black holes, which do not immediately merge. Such initial conditions can e.g. be two ordinary stars that each undergo a collaps to a black hole. The merger of the two black holes is something that happens in the later evolution of the binary black hole system.

>	And what causes this to happen?

While orbitting each other, the two black holes loose kinetic energy due to emission of gravitational waves. Due to that, the orbits become closer and closer, until the black holes merge. From the analytical point of view, there is no doubt that this happens. The reason why they run numerical simulations of this process is not to verify that a merger happens, but rather to analyze the details of the process, which are beyond currently known analytical approaches.

>	IMO when both (normal and BH) collect more mass (Energy) than they radiate they will never merge

Remember my argument of infalling comets from a parallel post: matter that falls into one of both black holes out of the environment will sometimes accelerate the orbit velocity and sometimes slow it down. In the average, changes in the orbit velocity caused by infalling matter will compensate each other, so infalling matter has no influence on the black holes doing or not doing merge. The emission of gravitational waves, on the other hand, does have an influence: it causes the loss of kinetic energy and therefore the orbits becoming closer and closer.

>>>	The book by Ray d'Inverno mentioned above in paragraph 15.3 explains the "Advance of the perihelion of Mercury" The impression I get is that it is very difficult to simulate more (all?) planets using GR.

>>	You should note that paragraph 15.3 depicts a mechanical approach: the equations describe the movement of a body (Mercury), not the evolution of fields. The configuration of the gravitational field is presumed as known there (namely as being the Schwarzschild solution), so it needn't be calculated.

>	The issue is here to calculate the movement of all the planets using GR. My impression is here that this is very difficult

This issue seperates in two different issues:

1) calculate the movements of planets in a known gravitational field configuration (like the movement of Mercury in the Sun's gravitational field described by Schwarzschild solution). For this issue, it is sufficient to solve the geodesic equations for the trajectory of the planet.

2) calculate the movements of celestial bodies taking the gravitational fields of the bodies themselves into account (like in a binary black hole system). Here, the gravitational field configuration is not known and needs to be calculated, too. So, one has to solve a set of coupled equations, namely geodesic equations and field equations. For a numerical approach, one can replace solving the geodesic equations for the trajectories of the bodies by solving field equations for matter fields that describe the bodies. The approach is a pure field theory approach then, not a mixture of field theory and mechanics.

Issue 2) is even more difficult than issue 1).

Another remark concering your approach to introduce a finite speed of gravity to Newtonian Gravity. You said in some other post that in your simulation, the advance of the Mercury perihelion occured then. The interesting question is now: did you neglect the gravitational field of Mercury itself and take only the Sun's gravitational field into account? Or did you take the gravitational field of Mercury into account, too, yielding Sun and Mercury both orbit a common centre of mass (that might be located inside the Sun, but not in the Sun's centre)?

If you neglected the gravitational field of Mercury, the usual approach would be to consider the Sun resting at the position r = 0 in a spherical coordinate system (r,theta,phi). The gravitational field of the Sun would be static then, and a finite speed of gravity wouldn't have any influence. So, if this was your approach, your result that the advance of the Mercury perihelion occurs due to the finite speed of gravity would probably be wrong.

If you, on the other hand, took the gravitational field of Mercury into account, so that Sun and Mercury orbit a common centre of mass, then a finite speed of gravity could indeed have some effect, due to the non-uniform movement of the Sun. However, those effects should depend on the mass of Mercury: the lower the mass of Mercury, the closer is the common centre of mass to the centre of the Sun, and the lower are the changes in the Sun's position. So, the advance of the Mercury perihelion yielded by your simulation should depend of the mass of Mercury, in contradiction to GR, where the advance of the perihelion is independent from Mercury's mass.

>>

The first difference that occurs that you can no longer apply a mechanical approach like in Newtonian theory or for the advance of the Mercury perihelion. You need to calculate the field configuration applying a field theory approach. For the advance of the Mercury perihelion, although it belongs to GR, the mechanical approach is sufficient because the field configuration is known (from an analytical field theory approach applies by Karl Schwarzschild in 1917 that yielded Schwarzschild solution). For the blach hole merger, the field configuration is not known analytically, therefore a numerical field theory approach is required.

>	This is in line with my assumption that this is very difficult. It is also in line with the doubt I have if two binary BH always will merge.

What makes you think in would be in line with that doubt? That a binary black hole systemv will merge is out of doubt from the analytical point of view in GR. The numerical simulations are just run to evaluate the details of the merger process, not to verify that it happens at all, since that is already clear analytically.

For comparison, take numerical simulations of Ising model that describes ferromagnetism. It is well known that a ferromagnatic solid body prefers to be magnetized under its Curie temperature and de-magnetizes above its Curie temperature, so, there is a phase transition occuring at Curie temperature.

In the framework of Ising model, this phase transition can be derived from an analytical consideration. However, there is a high number of numerical simulations based on Ising model, using Metropolis algorithm. The aim of those simulations is not to verify the occurence of a phase transition (this is already verified analytically), but to examine the details of the phase transition process.

>	Anyway the behaviour of two objects is a physical phenomena. How sure are we that a binary star system will merge?

Within the framework of GR, we are very sure, from the analytical point of view, that a binary star system emits gravitational waves, causing loss of kinetic energy, and by this, making the orbits closer and closer. What results in a merger, sooner or later.

>>>	Will they merge?

>>	According to the result of the simulation we are talking about: yes.

And from the analytical point of view, too.

>	I agree with you. But you have to know the details of what is physical involved. Is there preference for binary BH's to merge? compared to stars to merge?

Both tend to merge due to emission of gravitational waves that causes loss of kinetic energy.

>	Specific when equal masses are considered.

Let's consider two extreme cases:

1) Two celestial bodies with very different masses, like the Sun and Mercury. The centre of mass is located in the Sun, and the Sun does only little orbitting movement. So, the emission of gravitational waves from the Sun does not have a very high intensity. Mercury, on the other hand, does very much more orbitting movement, but its mass is very low. So, Mercury does not emit gravitational waves with high intensity, too. In total, the sytem Sun-Mercury does not emit gravitational waves very strongly.

2) Two celestial bodies of similar (or equal) mass, e.g. a binary star system. The centre of mass is located in the middle between both stars, and both perform much orbitting movement. Since their masses are high, too, they should emit gravitational waves of quite high intensity.

So, roughly estimated, a system of two bodies of similar or equal mass should yield a higher intensity of gravitational waves emission than a system of bodies where the one body has much more mass than the other. So, for equal masses, the loss of kinetic energy is higher, and the merger is faster.

>>>>	Observations of actual astrophysical BHs and their inspiral/mergers are a whole separate topic, which I'm not going to get into here.

>>>	But this is actual very important. I mean are we sure that BH's actual merge?

>>	No, we aren't. However, we are sure that according to GR, they do. And we have recent observations of gravitational waves that match the predictions that GR makes concerning gravitational waves emitted during a black hole merger.

>	I my simulation I assume that a third object is involved. What is wrong with that?

If we do not restrict ourselves to consider GR, we cannot exclude from the recent observations of gravitational waves that those waves originated from a process that involved a third object (or a 4th, 5th, ... object). One could imagine that there might be some theory that predicts that a system of three bodies produces waves with the observed characteristics.

If we took Newtonian Gravity, we would immediately know that there were no gravitational waves, since gravitational waves do not exist in Newtonian Gravity. So, Newtonian Gravity is in contradiction to the observation of gravitational waves. If we, one the other hand, take the approach you described, where you modify Newtonian Gravity by introducing a finite speed of gravity, we have the problem, that this approach has a lack of consistency (e.g. because you do not indicate a field equation), so it is not decidable whether it yields gravitational waves or not.

If we, on the other hand, restrict ourselves to consider GR, we can be quite sure the characteristics of the observed gravitational waves are in good compliance with the characterics GR predicts for a merger of two black holes. It isn't likely that a system where a third object (of non-neglectable mass) was involved would produce a similar characteristic.