Hallo Otubak!
Thank you for your answer.
So IÂ´m right with my assumption, polarization gives information about our position in regard to the plane of rotation of the spiraling two masses.

The elephant in the room here is the overall field strength. These solutions of Einstein's GR equations when solved within the circumstance of low field strength give the neat solutions that we are discussing. So that is fine if one is receiving GW energy well away from the source that is generating it. The analogy with EM waves is fine as far as it goes but there are a couple of crucial points to mention :

- Maxwell's equations are inherently linear across all magnitudes ( well, it is classical in that it has no QM effects like particle creation etc ) and involve full derivatives/integrals. GR is actually non-linear across all magnitudes. We may to choose to make linear approximations. The oft-quoted single tensor equation is mathematical symbology encapsulating a set of non-linear partial differential equations. Which is why they are generally a dog's breakfast to solve. Hence numerical relativity etc.

- Maxwell's equations assume a background coordinate system that doesn't muck about. GR is the mucking about of coordinates. The rulers and clocks keep misbehaving with respect to classical ideas. This is basically why

Quote:

... the exact appearance of the three-dimensional travelling pattern will depend on the details of their oscillation ...

applies. My highlight is to emphasise that the transverse nature ( with respect to propagation direction ) of the wave solutions is also an outcome of linearising. It doesn't have to be that way in strong fields. That's just the pattern it evolves to as the energy is propagated away from source. To the best of my knowledge - do please correct me if you know better - the near field may have longitudinal components rather like sound/density waves. Or possibly something weirder for which our usual descriptors ( polarity, transverse, longitudinal etc ) don't usefully assist.

Side Note : one can't localise energy in GR as we can with EM/QM. By that I mean the issue is tidal, or a comparison/differential b/w field strengths in separated locations. Different and equally legitimate frames will disagree as to 'where' the energy is. So if I sit in an accelerating frame looking at measurements made in another accelerating frame I attribute the energy, say labelled as kinetic, as 'over there'. The other frame looking back at me will say likewise about me. This is more than just the usual issue about what level do you choose as energy = zero.

Cheers, Mike.

( edit ) Note the heading of Markus PÃ¶ssel's page : 'The wave nature of simple gravitational waves' implying he is only talking of the easier-to-describe linear regime ones.

( late edit ) The total set of solutions is somewhat less well explored for GR than EM. I haven't yet found an explicable reference that is on par for my comprehension compared to say, a text on classical electrodynamics. The Maxwell equations are pretty easily formulated with correspondingly not too difficult visualisations as encompassed by Stoke's Theorem for instance, which in it's turn is just an example of the Fundamental Theorem of Calculus as extended to higher dimensions. My humble mentation/intuition suggests that something Stokesey ought apply in GR, integrate some little differentials, but what is the 'volume' surrounded by some sort of 'area' with an in/out bound flux of 'whatever' ? In other words what field aspect is conserved over all encasing summations ? With QM you have to get unitarity which roughly means conservation of probability. Bah, I'm just blathering ....

( later edit ) I've been thinking madly about this GW. So they exist, right ? Now because energy can't be localised as per GR field theory then it can't be quantised. Indeed in the paper they put a lower bound on the graviton wavelength ( huge ) and a corresponding/inverse upper mass ( very piddling ) on it. Let's jump the puddle here and say no gravitons. This means that melding GR with QM requires QM to change. I prefer that as QM is annoying. But it would imply one of several outcomes :

(a) QM and GR can't be joined. Too bad. We're too stupid, build a bridge, get over it etc.

(b) QM is basically classical underneath.

(c) There's a deeper formalism which degrades to either QM or GR depending on which way you move the scale slider. Here the formal correspondence b/w a Kerr black hole, as seen in the far field, with a fundamental particle at low energies assumes a 'spooky' significance.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

X-rays arrived 0.4[s] after gravitational waves
Hallo!
The Fermi gamma-ray space telescope did measure 0.4[s] after the GW150914 event a gamma-ray burst of 1[s] duration with a false alarm probability of 0.0022. See here for details.
If the gamma-rays have been emitted by the same event, they had a just 120.000[Km] longer path to us than the GWs. ThatÂ´s tremendously low compared to the 10^9 [ly] distant: 1,3E-17 . Is this due to deflections at dark matter on its path or is there another explanation, for example variations of electron density on the path? Unfortunately we donâ€™t have a time resolved spectrum of this event. For such the number of counts are much too low.

Hey Martin, yeah exactly, it does. Since both LIGO detectors are a bit rotated with respect to each other we can even pick up that polarisation (unless the signal comes from an unfortunate angle).

---

Just thinking of it, a pair of black holes has a huge number of degrees of freedom. Yet 3 of them seem to only affect 2 measurable parameters, signal amplitude and frequency:
* total mass and distance of BHs (both need to be changed by the same factor)
* centre of mass distance
* centre of mass velocity

So I'm wondering how scientists are going to make precise measurements here, probably by assuming that the centre of mass velocity does not deviate much from what you'd expect from cosmological redshift :)

Hallo!
First I like to say many thanks to all who answered on my questions.
But there came up some more for discussion:
1) Where does the energy of 2 solar masses came from, emitted mainly during merging of the 2 BHs?
2) Polarization of GWs
3) Spin of BHs
a) What does mean Spin = 0.67?
b) Effect of parallel/antiparallel spin direction on merging process?
c) What happens, if the axis of one or both spinning BHs is non perpendicular to the plane of rotation of the BHs?
4) Is Pi a constant in relativistic quantum physics?

To 1) : Here was some discussion about, where does the emitted energy of 2 solar masses came from. I like to put your attention on the well-known Hulse-Taylor Pulsar PSR B1913+16 See also. The reduction of orbital period, observed over a period of more than 30 years, can be explained by emission of GWs with an uncertainty of 0.2% only. The emitted energy has to come from orbital energy, as the masses of the neutron stars remain constant. This one can learn from the constant pulsar period of 59 millisecond. The increase of the pulsar period of 1.33 ppm per year can be explained as standard by transformation of rotation energy of the neutron star to electromagnetic radiation (dynamo effect see chapter emmission mechanics). It is not reported that this has changed within the last 30 Years. The pulsar period is variable with the orbital period in accordance with the SRT due to the high speed of 450Km/s = 0.15% of c in the periastron.

To 2) : At 22nd Feb. 16 was a panel discussion here in Berlin/Germany on the occasion of GW150914 with Alessandra Buonanno AEI/Golm and Bruce Allen AEI/Hannover as scientific speakers. I asked for the polarization of the GW waves. Bruce said, they were able to dig out the polarization information from the data as the two detectors are somewhat twisted and tilted against each, which gives the direction between the plane of rotation and the line of sight. This was necessary, to calculate the masses of the BHs and their distance to us, as the strength of the observed GW depends upon this angel. (Looking from on axis of rotation onto the mergers wouldnÂ´t show any GW to us, whereas the intensity would be maximal looking from in plane onto.)

To 3a : What does mean BH spin = 0.67? (From the GW150914:Factsheet) Is this the ratio of the radiuses of on axis to in plane of the event horizon? See here. Or what else? I didnÂ´t find anywhere a definition for.
To 3b) : There are two possibilities of spin orientation of the BHs merging: spin parallel or antiparallel. In GW150914 their spin was antiparallel, as one can see from the impressive simulation (0.23simulations from AEI. If the spin is antiparallel, the spacetime web becomes more and more stressed and thinned out, also if the difference in speed of the event horizons is zero, as the event horizons of the BHs comes closer and closer before merging. I assume, there must be a singularity in spacetime web at the point, the event horizons are meeting at starting merging. â€“ Never I heard about that before. - But itÂ´s clear, this si a very, very special point. So this process is relatively smooth, as both event horizons have the same direction of movement. If both spins are parallel, having the same direction of rotation, there is a point on the conjunction line between the centers of the BHs, at which the direction of movement snaps to the contrary. There is a commutating pole for the direction of movement, giving maximal â€œfrictionâ€. What happens at the point and moment of rupture of the spacetime web? I would think, the process of merging of BHs with parallel spin is much different from that with antiparallel spin. Can the later one be numerically simulated nowadays?
To 3c) : At higher rates of spinning the event horizon becomes elongated perpendicular to the axis of spinning. As seen before. If the axis of spin is non perpendicular to the plane of rotation of the BHs, the intersection of the event horizon with the plane of rotation of the BHs is no longer circular, depending upon the tilting angel of the spin axis. Because of this, I expect, there will be generated GWs with double of frequency than normal, especially in the final phase of approach of the circulating BHs and during merging.

To 4) : Pi is the proportional factor between the diameter and the circumference of a circle. Pi is an irrational and transcendental number. This is true only on a flat surface, not on a curved one, as in a curved spacetime web. In spherical geometry Pi can range down to 2. On the surface of our sun Pi is a few percent smaller, I learned from a scientist.
In Quantum Physics Pi plays a major role. For example the spin is a multiple of h-cross =h/(2*Pi). So, is Pi a constant in relativistic quantum physics or does it need a correction factor depending upon the warped spacetime? ( IÂ´m no physicist. I know just a tiny bit more than the word of quantum physic.)

1) Where does the energy of 2 solar masses came from, emitted mainly during merging of the 2 BHs?

Rather like the fiducial ice skater, much loved in angular discussions, when the arms are pulled in the rotation rate goes up. Now the angular momentum is conserved here, that's why the rate goes up. But what also goes up is the energy of rotation and varies somewhat like kinetic energy but not exactly. The skater has to do work when pulling their arms in and this goes into that energy. In the duo black hole case it is gravitational attraction doing the work, or if you like gravitational potential energy is being milked to provide kinetic energy, or things speed up when they fall towards each other ....

[ Some comments on the word 'moment'. This is context dependent. The original idea was moment = importance.

- so a moment in time is when something of interest/importance happened.

- the 'moment of motion' or 'quantity of motion' ( Newton's phrasing ) is the linear momentum ie. mass times velocity in the usual classical sense ( this can be consistently redefined elsewhere eg. for photons ). In fact Newton never said F = ma. He said F = dp/dt or that force is the rate of change of momentum. In the constant mass case you get F = ma.

- the 'moment of a force' is a lever or torque ie. the force component perpendicular to some radial line from some point about which rotation may take place.

- 'angular momentum' is linear momentum projected likewise perpendicular w.r.t. a similar radial line. The rate of change of angular momentum is proportional to a torque ( both with reference to a given pivot ).

- 'moment of inertia'* is a little more complex and helps account for real materials that have an extended nature ( not modeled as point like ) with possibly different mass densities within. IF it was a point mass then it would be that mass times the square of the radial distance, and is a scalar quantity. For extended bodies one can integrate the mass density over that body to achieve a point mass that acts as if it were at a certain radius. If you want a get a certain angular acceleration the moment of inertia will tell you how much torque to apply. So the moment of inertia is thus a proportional factor b/w angular momentum and angular acceleration, so this is reminiscent of F = ma but applied to a rotating case. A larger moment of inertia implies a greater difficulty ( read higher torque ) required to change the rotational velocity ie. a given angular acceleration. ]

Quote:

To 3a : What does mean BH spin = 0.67? (From the GW150914:Factsheet) Is this the ratio of the radiuses of on axis to in plane of the event horizon? See here. Or what else? I didnÂ´t find anywhere a definition for.

This is with respect to a maximal spinning black hole which would have a value of 1.0 by definition. For one reason and another a black hole's horizon ie. the surface of no return as seen from a distance, can't have a tangential velocity faster than light speed. This is a subtle point because the horizon is not a material body but a surface of interest to distant observers regarding information received. Thus any material body approaching the horizon of a spinning hole - and it is ever so crucial to re-state that this is a far field perspective - can't slide of sideways, as it were, with the horizon's rotation any faster than light speed. The BH spin could approach light speed if you just keep chucking in mass with suitable orbital angular momentum, and rotation rate will keep increasing. But it is like any approach to light speed for a material body in that no matter how much energy you pump in you will get ever closer to light speed but not quite ie. you need infinite energy ( that you can't provide ) to get there. Anyway 2/3 of that limit is quite vigorous, eh ? This is the sort of evidence that theorists will drool over for some time and is fulfilling their desire to 'test GR in the strong field regime'. :-)

Quote:

4) Is Pi a constant in relativistic quantum physics?

Two answers :

- deemed to be so in and of itself in QM.

- unknown. Yet. This is the moot point in the melding of QM with GR. This would matter when the mass/energy density at small scale is significant. You solve that one and you're a hero for sure ! :-)

Yes, the Sun is missing an entire Imperial foot of circumference c/w Euclidean expectations at it's known radius.

Here's another interesting one : a really large flywheel rotating quite fast. An observer at the centre can utilise some measuring rod to measure PI. Give it to a colleague to step out the radius and that will yield some number of measuring rod lengths ie. so & so many when laid end to end. In this instance the velocity of the rod is always tangential to a radial line ie. at a right angle to the long axis of the rod. Hence there is no Lorentz contraction along that length. So the number or rod lengths for the radius is 'correct' as it were, or the same as if no rotation. Now get that assistant to go around the outer rim. He is now placing the measuring rod with it's long axis aligned with the tangential velocity vector, thus we have Lorentz contraction. Compared to the situation where the flywheel is non-rotating then more end to end rod lengths are required to go right around the rim. If you like the rods in that orientation are shorter than 'correct' ( the rod's rest frame length ). So what is PI while rotating ? It is more than the Euclidean amount. We would say space is negatively curved from the point of view of an observer sitting at the axis of rotation. Weird, huh ? But the deeper point is that any acceleration is absolutely measurable, and so you can call it an inertial acceleration or you can call it 'anti-gravity' ! But it isn't really as the flywheel is held together by EM forces ultimately ( so no 'free fall' ) and the actual ( rest ) mass of the flywheel isn't the issue here. We are talking of measurement distortion explicable at Special Relativity level brought about by the geometry of the scenario. It is an example of how counter-intuitive results appear if one takes SR as logically applied. There is also a time issue here and the assistant carrying the rod as he moves about the flywheel will have some dilatation ( his clock apparently slower ) with respect to the guy who stayed at the centre. In case you hadn't noticed it also brings out a key point in Relativity discussions : you are always comparing two viewpoints or reference frames whether you realise it or not ie. relating. There is no God's Eye View that reconciles. One must always account for the propagation delay of light and that requires locally based frames at the events of interest.

Cheers, Mike.

* Rather unusually for me I will recommend the following Wikipedia entry on the topic which is surprisingly lucid and accurate simultaneously ( for Wikipedia ). One reason why any of this rotation business is able to be explained by relatively simple math is the theorem of Emmy Noether which relates, in this case, rotations with preservations of certain calculable things as above. This says a great deal about the symmetry of the Universe within which we live, so much so that when we turn our head to some new position of view that act does not trigger any collapse of reality or morphing to a different paradigm. Which is nice. :-)

Indeed it reminds me of other 'moments'. For instance in statistics - deriving numbers that summarise groups of numbers - the plain total is the zeroeth moment, the average is the first moment, the variance is the second moment and I think the third moment is skew or skewness. In each case one is obtaining, effectively, higher differentials on a data set to characterise the spread of whatever quantity we are modelling by said numbers.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

To 3b) : There are two possibilities of spin orientation of the BHs merging: spin parallel or antiparallel. In GW150914 their spin was antiparallel, as one can see from the impressive simulation (0.23

The simulations are being expressed in different visual senses. The first is ( effectively ) ray tracing the light from the distant background starfield through the BH-BH merger neighbourhood to the viewing point. So in that regard would be rather like what a human observer would actually see. I note that the hypothesis of primordial origin of these holes would imply no significant accretion disks, so my earlier idea of two buzz saws meeting with quite a lot of EM fireworks wouldn't apply. The second is using selected 3D surfaces of constant_something to illustrate the situation which is really 4D, which is helpful because we humans are crap at 4D visuals ! Think of those colored warped sheets as projections ( 'shadows' perhaps ) into 3D space of 4D entities. This is analogous to 3D to 2D which we routinely do with '3D graphics', which in truth is another trick again of triggering our visual system to key features on the 2D plane of the computer monitor. Phew ! :-)

Quote:

If the spin is antiparallel, the spacetime web becomes more and more stressed and thinned out, also if the difference in speed of the event horizons is zero, as the event horizons of the BHs comes closer and closer before merging. I assume, there must be a singularity in spacetime web at the point, the event horizons are meeting at starting merging. â€“ Never I heard about that before. - But itÂ´s clear, this si a very, very special point. So this process is relatively smooth, as both event horizons have the same direction of movement.

Smooth is exactly right and indeed belies a deeper truth. A major base assumption of GR is that spacetime is a continuous manifold. This implies certain mathematical, and one hopes physical, properties. The exact mathematical definition of continuity is moderately tedious, but the key idea is that points which are close together will be similiar and the closer they are the more similiar they are. Here 'similiar' is with regard to whatever property is of interest. So this rules out awkward geometric things likes pointy bits, sharps edges and breaks. As it were. This enables differentiablity ie. every neighbourhood of every point has a well defined tangent ( line/plane/volume ) and to any number of derivatives too. Smooth means I can take a derivative as many times as I like ie. infinitely so and each derivative will be a continuous function ( this is a multidimensional comment ).

One important consequence is that if I get sufficiently close to any point in the manifold I can closely approximate it's nearby surroundings with a linear map, particularly the 'flat map' spacetime of Special Relativity. A linear map is rather like what we already do with Earth's geography ( ignoring elevation above sea level for the moment ). We map local geographies as if the Earth was globally flat when of course it is an actual sphere ( to a first approximation ). The map isn't the terrain. The map is a representation of the terrain. We know that the map and the terrain are not equal, but we know that they are close and the smaller the surroundings considered the better the approximation gets b/w the map and the terrain. Now if I have two such points where I have created a local linear map for each, and if those points are close enough, then I can created a third mapping : from linear map to linear map. Not from terrain to terrain, but from terrain representation to terrain representation. Those maps are linear by definition and thus easier to cope with correlating the two. But each linear map from adjacent points are mutually 'tilted' with respect to each other. That third mapping may account for that 'tilt' and thus, finally, we may closely correlate the surroundings of nearby manifold points.

That's a pretty exhausting process but it has to be done. So if you take any two distinct points on a sphere, however close, you cannot use a straight-line vector to connect the two with that vector remaining entirely on the sphere. It has to 'cut the corner' as it were, leave the spherical surface by venturing into the surrounding 3D space. That's what we really mean by saying the sphere is curved. Or at least is an alternate definition of curved, based on local observations.

Imagine if I was a merely 2D enabled being on the sphere. I would only know information about points exactly on the sphere. I don't obviously know my space has 'curvature', I only know that two parameters are enough to define positions. All 'straight' arrows I might yield would have it's base point in my hand and maybe ( depending on how I hold it ) the tip of the arrow would be evident in the distance, but I can't sense the body of the vector between it's endpoints ( tip and tail ) because they are in some surrounding, and to be exact merely hypothecated, enclosing 'superspace'. I might look up a book written by a 2D Einstein who tells me there is some 'tilting' going on b/w nearby points in my 'curved' world.

You know the punchline. We are 3D enabled beings embedded in a 4D space or spacetime. You can't use 'straight' 4D vectors because of 'spacetime curvature', BUT you can approximate matters by this linear map business. And by stepping from linear map to linear map you can traverse the totality of the manifold. As you go you keep record of the 'tilts'. You can accumulate each little tilt in sequence to get a cumulative tilt for some complete journey. From that you can deduce, over all such possible journeys between all pairs of points in the manifold, the complete structure of the manifold. AND you only did that with local observations and not by venturing out into some 5th dimension or whatever.

Einstein's GR equations are the recipe for any universe which has the above manifold behaviour. The equations don't say you have to be in any particular universe/manifold. They do say that whichever universe you are in, the manifold must have this smoothness etc a/a.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

After much thought, I'd like to show this following still graphic from a merger video ( at 0:23 of 0:35 seconds ) :

which shows an en-face view of the mutual orbital plane of the BH pair a finger-snap before merger. The background starfield is distorted like an optic lens, which it is, but the unusual concept that gravity is doing the light bending not a mere refractive index born of electromagnetic interaction with traversing photons in some material medium. It has some really interesting features which may be appreciable without delving deeply into the mathematical morass of GR field equations. It may not be for that matter. Oh well.

Anyway what would be the obvious things to point out ?

- the broad feature that there is obviously a central volume where light paths go non-Euclidean. Depending upon your personal acuity/perception level this seems to have a threshold radius :

- it's easy to assess which is the bigger hole :

- the black holes near sides seem to 'repel' one another ie. are non circular with flattening adjacent :

- the lens shaped black features separate from each hole, away from the merging faces and also pretty well diametrically opposite with respect to the common centre of orbit :

- also some 'swirly bits' which appear to be continuous with brightened areas/layers just outside of the event horizons :

These would be obvious to a ( not too ) nearby astronaut. I would like to subsequently explain these things in a basic manner ! Let us see how I fare ..... :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

There also appears to be visible some wider distortion from each BH that is about twice the diameter of each BH but lagging (or leading?) by about 45 deg.

Well the first thing to think about is what do we really mean by a black hole and thus what might we expect light to do in the neighbourhood of one ?

We hear of the concept that a black hole captures everything and if any object got too close to one it would be swallowed whole, as it were, and no light emitted from that object will reach the distant universe. This is true but why ? Einstein came up with the ( now well tested predictions ) that the path that a photon travels may be altered by the presence of mass/energy.

Strictly speaking theory will predict that even a trivial amount of mass, or energy, will cause a deflection of light paths. But with the 'ordinary' densities that man has been historically accustomed to, one will strive hard to measure any effect of note. While the LIGO effort is an outstanding exception to this, one would be foolish to claim, say, that the subsequent evolution of the solar system would be materially affected by the reception of GW energy from black hole collisions of GW150914 variety ! So let's put this low field regime aside and focus upon regions of the universe where mass/energy is really crammed in, and in stellar amounts.

The word 'warp' comes to mind and in this we need some special care. Sure we can say that a photon's path in space is deflected from where it would have gone had mass not been there ( from now on it's fine to read 'mass and/or energy' when I just say mass ). So way away from anything - the hypothetical 'empty space' which is unaffected by anything, but in precise reality doesn't exist - these light paths define/exhibit 'a straight line', Euclidean/Pythagorean triangles and what-not. Slide a large mass in the way and, just like the famous Eddington eclipse expedition in 1919, one can see the source of light in a different position from where it would be viewed otherwise.

What Einstein asserted in General Relativity was that spacetime paths are 'bent'. What then is the difference b/w a bent spatial path and a bent spacetime path ? Time of course. So what is time when it is bent ? That can only having meaning if we say that the rate that time passes changes. The full horror is that one thinks of photons as having paths in four dimensions, one time and three spatial, and so when the scale of the time axis changes - it is the regularity of the marking along the time axis, in seconds say, that we mean by time rate - then that four dimensional path will alter. Even if the spatial components are fixed.

Maybe a 3D spatial analogy will help here. Suppose I measure the length, breadth and height of my house. But for some perverse reason I choose to assert that 2 metres along the height axis is identical to one metre in the length/breadth plane. In effect I have made my house less tall. But it is more than that. The volume will change : half as much in this instance. An angle from one corner of the base, say, to another corner on the roof will change : it's tangent will be half as much. Etc. All those things follow because I altered the regularity of the markings along an axis of measurement ie. I made 2 metres equal to one.

Here is a very key point in what follows :

Quote:

the rate that time passes is always slower in a stronger gravitational field than a weaker one

... where the weaker field may include the case where there is ( almost ) no field at all ie. our astronaut distantly viewing the black hole merger. This has been tested over a separation of a few hundred feet in altitude on Earth ( the lower clock ran slower with respect to the higher clock ). If the Eddington expedition had performed spectroscopy on the starlight of those stars studied, they would have noticed a lowering of the frequency of those spectral lines in addition to spatial path deflection.

Before they were called black holes, they were named as 'frozen stars' because at the event horizon ( to be discussed ) time appears to stop. Please spend some time ( pardon the pun ) thinking about time slowing and even stopping. It's a hard one and IMHO the most difficult cognitive bit about The Relativities, simply because you are never likely to see this actually happen around you. Fortunately we are not in the right neighbourhood for that to be so obvious to our naked senses ( unassisted by extremely accurate instruments ).

For this to be absolutely firm in our thinking we need to have a chin wag about :

Proper Time ... or ... Get Your Own Clock !

Any questions on this bit at least ? :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Hallo Otubak! Thank you for

)

Hallo Otubak!

Thank you for your answer.

So IÂ´m right with my assumption, polarization gives information about our position in regard to the plane of rotation of the spiraling two masses.

Kind regards and happy crunching

Martin

## The elephant in the room here

)

The elephant in the room here is the overall field strength. These solutions of Einstein's GR equations when solved within the circumstance of low field strength give the neat solutions that we are discussing. So that is fine if one is receiving GW energy well away from the source that is generating it. The analogy with EM waves is fine as far as it goes but there are a couple of crucial points to mention :

- Maxwell's equations are inherently linear across all magnitudes ( well, it is classical in that it has no QM effects like particle creation etc ) and involve full derivatives/integrals. GR is actually non-linear across all magnitudes. We may to choose to make linear approximations. The oft-quoted single tensor equation is mathematical symbology encapsulating a set of non-linear partial differential equations. Which is why they are generally a dog's breakfast to solve. Hence numerical relativity etc.

- Maxwell's equations assume a background coordinate system that doesn't muck about. GR is the mucking about of coordinates. The rulers and clocks keep misbehaving with respect to classical ideas. This is basically why

applies. My highlight is to emphasise that the transverse nature ( with respect to propagation direction ) of the wave solutions is also an outcome of linearising. It doesn't have to be that way in strong fields. That's just the pattern it evolves to as the energy is propagated away from source. To the best of my knowledge - do please correct me if you know better - the near field may have longitudinal components rather like sound/density waves. Or possibly something weirder for which our usual descriptors ( polarity, transverse, longitudinal etc ) don't usefully assist.

Side Note : one can't localise energy in GR as we can with EM/QM. By that I mean the issue is tidal, or a comparison/differential b/w field strengths in separated locations. Different and equally legitimate frames will disagree as to 'where' the energy is. So if I sit in an accelerating frame looking at measurements made in another accelerating frame I attribute the energy, say labelled as kinetic, as 'over there'. The other frame looking back at me will say likewise about me. This is more than just the usual issue about what level do you choose as energy = zero.

Cheers, Mike.

( edit ) Note the heading of Markus PÃ¶ssel's page : 'The wave nature of simple gravitational waves' implying he is only talking of the easier-to-describe linear regime ones.

( late edit ) The total set of solutions is somewhat less well explored for GR than EM. I haven't yet found an explicable reference that is on par for my comprehension compared to say, a text on classical electrodynamics. The Maxwell equations are pretty easily formulated with correspondingly not too difficult visualisations as encompassed by Stoke's Theorem for instance, which in it's turn is just an example of the Fundamental Theorem of Calculus as extended to higher dimensions. My humble mentation/intuition suggests that something Stokesey ought apply in GR, integrate some little differentials, but what is the 'volume' surrounded by some sort of 'area' with an in/out bound flux of 'whatever' ? In other words what field aspect is conserved over all encasing summations ? With QM you have to get unitarity which roughly means conservation of probability. Bah, I'm just blathering ....

( later edit ) I've been thinking madly about this GW. So they exist, right ? Now because energy can't be localised as per GR field theory then it can't be quantised. Indeed in the paper they put a lower bound on the graviton wavelength ( huge ) and a corresponding/inverse upper mass ( very piddling ) on it. Let's jump the puddle here and say no gravitons. This means that melding GR with QM requires QM to change. I prefer that as QM is annoying. But it would imply one of several outcomes :

(a) QM and GR can't be joined. Too bad. We're too stupid, build a bridge, get over it etc.

(b) QM is basically classical underneath.

(c) There's a deeper formalism which degrades to either QM or GR depending on which way you move the scale slider. Here the formal correspondence b/w a Kerr black hole, as seen in the far field, with a fundamental particle at low energies assumes a 'spooky' significance.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## X-rays arrived 0.4[s] after

)

X-rays arrived 0.4[s] after gravitational waves

Hallo!

The Fermi gamma-ray space telescope did measure 0.4[s] after the GW150914 event a gamma-ray burst of 1[s] duration with a false alarm probability of 0.0022. See here for details.

If the gamma-rays have been emitted by the same event, they had a just 120.000[Km] longer path to us than the GWs. ThatÂ´s tremendously low compared to the 10^9 [ly] distant: 1,3E-17 . Is this due to deflections at dark matter on its path or is there another explanation, for example variations of electron density on the path? Unfortunately we donâ€™t have a time resolved spectrum of this event. For such the number of counts are much too low.

Kind regards and happy crunching

Martin

## Hey Martin, yeah exactly, it

)

Hey Martin, yeah exactly, it does. Since both LIGO detectors are a bit rotated with respect to each other we can even pick up that polarisation (unless the signal comes from an unfortunate angle).

---

Just thinking of it, a pair of black holes has a huge number of degrees of freedom. Yet 3 of them seem to only affect 2 measurable parameters, signal amplitude and frequency:

* total mass and distance of BHs (both need to be changed by the same factor)

* centre of mass distance

* centre of mass velocity

So I'm wondering how scientists are going to make precise measurements here, probably by assuming that the centre of mass velocity does not deviate much from what you'd expect from cosmological redshift :)

## Hallo! First I like to say

)

Hallo!

First I like to say many thanks to all who answered on my questions.

But there came up some more for discussion:

1) Where does the energy of 2 solar masses came from, emitted mainly during merging of the 2 BHs?

2) Polarization of GWs

3) Spin of BHs

a) What does mean Spin = 0.67?

b) Effect of parallel/antiparallel spin direction on merging process?

c) What happens, if the axis of one or both spinning BHs is non perpendicular to the plane of rotation of the BHs?

4) Is Pi a constant in relativistic quantum physics?

To 1) : Here was some discussion about, where does the emitted energy of 2 solar masses came from. I like to put your attention on the well-known Hulse-Taylor Pulsar PSR B1913+16 See also. The reduction of orbital period, observed over a period of more than 30 years, can be explained by emission of GWs with an uncertainty of 0.2% only. The emitted energy has to come from orbital energy, as the masses of the neutron stars remain constant. This one can learn from the constant pulsar period of 59 millisecond. The increase of the pulsar period of 1.33 ppm per year can be explained as standard by transformation of rotation energy of the neutron star to electromagnetic radiation (dynamo effect see chapter emmission mechanics). It is not reported that this has changed within the last 30 Years. The pulsar period is variable with the orbital period in accordance with the SRT due to the high speed of 450Km/s = 0.15% of c in the periastron.

To 2) : At 22nd Feb. 16 was a panel discussion here in Berlin/Germany on the occasion of GW150914 with Alessandra Buonanno AEI/Golm and Bruce Allen AEI/Hannover as scientific speakers. I asked for the polarization of the GW waves. Bruce said, they were able to dig out the polarization information from the data as the two detectors are somewhat twisted and tilted against each, which gives the direction between the plane of rotation and the line of sight. This was necessary, to calculate the masses of the BHs and their distance to us, as the strength of the observed GW depends upon this angel. (Looking from on axis of rotation onto the mergers wouldnÂ´t show any GW to us, whereas the intensity would be maximal looking from in plane onto.)

To 3a : What does mean BH spin = 0.67? (From the GW150914:Factsheet) Is this the ratio of the radiuses of on axis to in plane of the event horizon? See here. Or what else? I didnÂ´t find anywhere a definition for.

To 3b) : There are two possibilities of spin orientation of the BHs merging: spin parallel or antiparallel. In GW150914 their spin was antiparallel, as one can see from the impressive simulation (0.23simulations from AEI. If the spin is antiparallel, the spacetime web becomes more and more stressed and thinned out, also if the difference in speed of the event horizons is zero, as the event horizons of the BHs comes closer and closer before merging. I assume, there must be a singularity in spacetime web at the point, the event horizons are meeting at starting merging. â€“ Never I heard about that before. - But itÂ´s clear, this si a very, very special point. So this process is relatively smooth, as both event horizons have the same direction of movement. If both spins are parallel, having the same direction of rotation, there is a point on the conjunction line between the centers of the BHs, at which the direction of movement snaps to the contrary. There is a commutating pole for the direction of movement, giving maximal â€œfrictionâ€. What happens at the point and moment of rupture of the spacetime web? I would think, the process of merging of BHs with parallel spin is much different from that with antiparallel spin. Can the later one be numerically simulated nowadays?

To 3c) : At higher rates of spinning the event horizon becomes elongated perpendicular to the axis of spinning. As seen before. If the axis of spin is non perpendicular to the plane of rotation of the BHs, the intersection of the event horizon with the plane of rotation of the BHs is no longer circular, depending upon the tilting angel of the spin axis. Because of this, I expect, there will be generated GWs with double of frequency than normal, especially in the final phase of approach of the circulating BHs and during merging.

To 4) : Pi is the proportional factor between the diameter and the circumference of a circle. Pi is an irrational and transcendental number. This is true only on a flat surface, not on a curved one, as in a curved spacetime web. In spherical geometry Pi can range down to 2. On the surface of our sun Pi is a few percent smaller, I learned from a scientist.

In Quantum Physics Pi plays a major role. For example the spin is a multiple of h-cross =h/(2*Pi). So, is Pi a constant in relativistic quantum physics or does it need a correction factor depending upon the warped spacetime? ( IÂ´m no physicist. I know just a tiny bit more than the word of quantum physic.)

Kind regards and happy crunching

Martin

## Hi Martin !

)

Hi Martin ! :-)

Rather like the fiducial ice skater, much loved in angular discussions, when the arms are pulled in the rotation rate goes up. Now the angular momentum is conserved here, that's why the rate goes up. But what also goes up is the energy of rotation and varies somewhat like kinetic energy but not exactly. The skater has to do work when pulling their arms in and this goes into that energy. In the duo black hole case it is gravitational attraction doing the work, or if you like gravitational potential energy is being milked to provide kinetic energy, or things speed up when they fall towards each other ....

[ Some comments on the word 'moment'. This is context dependent. The original idea was moment = importance.

- so a moment in time is when something of interest/importance happened.

- the 'moment of motion' or 'quantity of motion' ( Newton's phrasing ) is the linear momentum ie. mass times velocity in the usual classical sense ( this can be consistently redefined elsewhere eg. for photons ). In fact Newton never said F = ma. He said F = dp/dt or that force is the rate of change of momentum. In the constant mass case you get F = ma.

- the 'moment of a force' is a lever or torque ie. the force component perpendicular to some radial line from some point about which rotation may take place.

- 'angular momentum' is linear momentum projected likewise perpendicular w.r.t. a similar radial line. The rate of change of angular momentum is proportional to a torque ( both with reference to a given pivot ).

- 'moment of inertia'* is a little more complex and helps account for real materials that have an extended nature ( not modeled as point like ) with possibly different mass densities within. IF it was a point mass then it would be that mass times the square of the radial distance, and is a scalar quantity. For extended bodies one can integrate the mass density over that body to achieve a point mass that acts as if it were at a certain radius. If you want a get a certain angular acceleration the moment of inertia will tell you how much torque to apply. So the moment of inertia is thus a proportional factor b/w angular momentum and angular acceleration, so this is reminiscent of F = ma but applied to a rotating case. A larger moment of inertia implies a greater difficulty ( read higher torque ) required to change the rotational velocity ie. a given angular acceleration. ]

This is with respect to a maximal spinning black hole which would have a value of 1.0 by definition. For one reason and another a black hole's horizon ie. the surface of no return as seen from a distance, can't have a tangential velocity faster than light speed. This is a subtle point because the horizon is not a material body but a surface of interest to distant observers regarding information received. Thus any material body approaching the horizon of a spinning hole - and it is ever so crucial to re-state that this is a far field perspective - can't slide of sideways, as it were, with the horizon's rotation any faster than light speed. The BH spin could approach light speed if you just keep chucking in mass with suitable orbital angular momentum, and rotation rate will keep increasing. But it is like any approach to light speed for a material body in that no matter how much energy you pump in you will get ever closer to light speed but not quite ie. you need infinite energy ( that you can't provide ) to get there. Anyway 2/3 of that limit is quite vigorous, eh ? This is the sort of evidence that theorists will drool over for some time and is fulfilling their desire to 'test GR in the strong field regime'. :-)

Two answers :

- deemed to be so in and of itself in QM.

- unknown. Yet. This is the moot point in the melding of QM with GR. This would matter when the mass/energy density at small scale is significant. You solve that one and you're a hero for sure ! :-)

Yes, the Sun is missing an entire Imperial foot of circumference c/w Euclidean expectations at it's known radius.

Here's another interesting one : a really large flywheel rotating quite fast. An observer at the centre can utilise some measuring rod to measure PI. Give it to a colleague to step out the radius and that will yield some number of measuring rod lengths ie. so & so many when laid end to end. In this instance the velocity of the rod is always tangential to a radial line ie. at a right angle to the long axis of the rod. Hence there is no Lorentz contraction along that length. So the number or rod lengths for the radius is 'correct' as it were, or the same as if no rotation. Now get that assistant to go around the outer rim. He is now placing the measuring rod with it's long axis aligned with the tangential velocity vector, thus we have Lorentz contraction. Compared to the situation where the flywheel is non-rotating then more end to end rod lengths are required to go right around the rim. If you like the rods in that orientation are shorter than 'correct' ( the rod's rest frame length ). So what is PI while rotating ? It is more than the Euclidean amount. We would say space is negatively curved from the point of view of an observer sitting at the axis of rotation. Weird, huh ? But the deeper point is that any acceleration is absolutely measurable, and so you can call it an inertial acceleration or you can call it 'anti-gravity' ! But it isn't really as the flywheel is held together by EM forces ultimately ( so no 'free fall' ) and the actual ( rest ) mass of the flywheel isn't the issue here. We are talking of measurement distortion explicable at Special Relativity level brought about by the geometry of the scenario. It is an example of how counter-intuitive results appear if one takes SR as logically applied. There is also a time issue here and the assistant carrying the rod as he moves about the flywheel will have some dilatation ( his clock apparently slower ) with respect to the guy who stayed at the centre. In case you hadn't noticed it also brings out a key point in Relativity discussions : you are always comparing two viewpoints or reference frames whether you realise it or not ie. relating. There is no God's Eye View that reconciles. One must always account for the propagation delay of light and that requires locally based frames at the events of interest.

Cheers, Mike.

* Rather unusually for me I will recommend the following Wikipedia entry on the topic which is surprisingly lucid and accurate simultaneously ( for Wikipedia ). One reason why any of this rotation business is able to be explained by relatively simple math is the theorem of Emmy Noether which relates, in this case, rotations with preservations of certain calculable things as above. This says a great deal about the symmetry of the Universe within which we live, so much so that when we turn our head to some new position of view that act does not trigger any collapse of reality or morphing to a different paradigm. Which is nice. :-)

Indeed it reminds me of other 'moments'. For instance in statistics - deriving numbers that summarise groups of numbers - the plain total is the zeroeth moment, the average is the first moment, the variance is the second moment and I think the third moment is skew or skewness. In each case one is obtaining, effectively, higher differentials on a data set to characterise the spread of whatever quantity we are modelling by said numbers.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## RE: To 3b) : There are two

)

The simulations are being expressed in different visual senses. The first is ( effectively ) ray tracing the light from the distant background starfield through the BH-BH merger neighbourhood to the viewing point. So in that regard would be rather like what a human observer would actually see. I note that the hypothesis of primordial origin of these holes would imply no significant accretion disks, so my earlier idea of two buzz saws meeting with quite a lot of EM fireworks wouldn't apply. The second is using selected 3D surfaces of constant_something to illustrate the situation which is really 4D, which is helpful because we humans are crap at 4D visuals ! Think of those colored warped sheets as projections ( 'shadows' perhaps ) into 3D space of 4D entities. This is analogous to 3D to 2D which we routinely do with '3D graphics', which in truth is another trick again of triggering our visual system to key features on the 2D plane of the computer monitor. Phew ! :-)

Smooth is exactly right and indeed belies a deeper truth. A major base assumption of GR is that spacetime is a continuous manifold. This implies certain mathematical, and one hopes physical, properties. The exact mathematical definition of continuity is moderately tedious, but the key idea is that points which are close together will be similiar and the closer they are the more similiar they are. Here 'similiar' is with regard to whatever property is of interest. So this rules out awkward geometric things likes pointy bits, sharps edges and breaks. As it were. This enables differentiablity ie. every neighbourhood of every point has a well defined tangent ( line/plane/volume ) and to any number of derivatives too. Smooth means I can take a derivative as many times as I like ie. infinitely so and each derivative will be a continuous function ( this is a multidimensional comment ).

One important consequence is that if I get sufficiently close to any point in the manifold I can closely approximate it's nearby surroundings with a linear map, particularly the 'flat map' spacetime of Special Relativity. A linear map is rather like what we already do with Earth's geography ( ignoring elevation above sea level for the moment ). We map local geographies as if the Earth was globally flat when of course it is an actual sphere ( to a first approximation ). The map isn't the terrain. The map is a representation of the terrain. We know that the map and the terrain are not equal, but we know that they are close and the smaller the surroundings considered the better the approximation gets b/w the map and the terrain. Now if I have two such points where I have created a local linear map for each, and if those points are close enough, then I can created a third mapping : from linear map to linear map. Not from terrain to terrain, but from terrain representation to terrain representation. Those maps are linear by definition and thus easier to cope with correlating the two. But each linear map from adjacent points are mutually 'tilted' with respect to each other. That third mapping may account for that 'tilt' and thus, finally, we may closely correlate the surroundings of nearby manifold points.

That's a pretty exhausting process but it has to be done. So if you take any two distinct points on a sphere, however close, you cannot use a straight-line vector to connect the two with that vector remaining entirely on the sphere. It has to 'cut the corner' as it were, leave the spherical surface by venturing into the surrounding 3D space. That's what we really mean by saying the sphere is curved. Or at least is an alternate definition of curved, based on local observations.

Imagine if I was a merely 2D enabled being on the sphere. I would only know information about points exactly on the sphere. I don't obviously know my space has 'curvature', I only know that two parameters are enough to define positions. All 'straight' arrows I might yield would have it's base point in my hand and maybe ( depending on how I hold it ) the tip of the arrow would be evident in the distance, but I can't sense the body of the vector between it's endpoints ( tip and tail ) because they are in some surrounding, and to be exact merely hypothecated, enclosing 'superspace'. I might look up a book written by a 2D Einstein who tells me there is some 'tilting' going on b/w nearby points in my 'curved' world.

You know the punchline. We are 3D enabled beings embedded in a 4D space or spacetime. You can't use 'straight' 4D vectors because of 'spacetime curvature', BUT you can approximate matters by this linear map business. And by stepping from linear map to linear map you can traverse the totality of the manifold. As you go you keep record of the 'tilts'. You can accumulate each little tilt in sequence to get a cumulative tilt for some complete journey. From that you can deduce, over all such possible journeys between all pairs of points in the manifold, the complete structure of the manifold. AND you only did that with local observations and not by venturing out into some 5th dimension or whatever.

Einstein's GR equations are the recipe for any universe which has the above manifold behaviour. The equations don't say you have to be in any particular universe/manifold. They do say that whichever universe you are in, the manifold must have this smoothness etc a/a.

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## After much thought, I'd like

)

After much thought, I'd like to show this following still graphic from a merger video ( at 0:23 of 0:35 seconds ) :

which shows an en-face view of the mutual orbital plane of the BH pair a finger-snap before merger. The background starfield is distorted like an optic lens, which it is, but the unusual concept that gravity is doing the light bending not a mere refractive index born of electromagnetic interaction with traversing photons in some material medium. It has some really interesting features which may be appreciable without delving deeply into the mathematical morass of GR field equations. It may not be for that matter. Oh well.

Anyway what would be the obvious things to point out ?

- the broad feature that there is obviously a central volume where light paths go non-Euclidean. Depending upon your personal acuity/perception level this seems to have a threshold radius :

- it's easy to assess which is the bigger hole :

- the black holes near sides seem to 'repel' one another ie. are non circular with flattening adjacent :

- the lens shaped black features separate from each hole, away from the merging faces and also pretty well diametrically opposite with respect to the common centre of orbit :

- also some 'swirly bits' which appear to be continuous with brightened areas/layers just outside of the event horizons :

These would be obvious to a ( not too ) nearby astronaut. I would like to subsequently explain these things in a basic manner ! Let us see how I fare ..... :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal

## Hey, awaiting the exploration

)

Hey, awaiting the exploration with interest! :-)

There also appears to be visible some wider distortion from each BH that is about twice the diameter of each BH but lagging (or leading?) by about 45 deg.

Which way are they orbiting in any case?!

Keep searchin',

Martin

See new freedom: Mageia Linux

Take a look for yourself: Linux Format

The Future is what We all make IT (GPLv3)

## Well the first thing to think

)

Well the first thing to think about is what do we really mean by a black hole and thus what might we expect light to do in the neighbourhood of one ?

We hear of the concept that a black hole captures everything and if any object got too close to one it would be swallowed whole, as it were, and no light emitted from that object will reach the distant universe. This is true but why ? Einstein came up with the ( now well tested predictions ) that the path that a photon travels may be altered by the presence of mass/energy.

Strictly speaking theory will predict that even a trivial amount of mass, or energy, will cause a deflection of light paths. But with the 'ordinary' densities that man has been historically accustomed to, one will strive hard to measure any effect of note. While the LIGO effort is an outstanding exception to this, one would be foolish to claim, say, that the subsequent evolution of the solar system would be materially affected by the reception of GW energy from black hole collisions of GW150914 variety ! So let's put this low field regime aside and focus upon regions of the universe where mass/energy is really crammed in, and in stellar amounts.

The word 'warp' comes to mind and in this we need some special care. Sure we can say that a photon's path in space is deflected from where it would have gone had mass not been there ( from now on it's fine to read 'mass and/or energy' when I just say mass ). So way away from anything - the hypothetical 'empty space' which is unaffected by anything, but in precise reality doesn't exist - these light paths define/exhibit 'a straight line', Euclidean/Pythagorean triangles and what-not. Slide a large mass in the way and, just like the famous Eddington eclipse expedition in 1919, one can see the source of light in a different position from where it would be viewed otherwise.

What Einstein asserted in General Relativity was that spacetime paths are 'bent'. What then is the difference b/w a bent spatial path and a bent spacetime path ? Time of course. So what is time when it is bent ? That can only having meaning if we say that the rate that time passes changes. The full horror is that one thinks of photons as having paths in four dimensions, one time and three spatial, and so when the scale of the time axis changes - it is the regularity of the marking along the time axis, in seconds say, that we mean by time rate - then that four dimensional path will alter. Even if the spatial components are fixed.

Maybe a 3D spatial analogy will help here. Suppose I measure the length, breadth and height of my house. But for some perverse reason I choose to assert that 2 metres along the height axis is identical to one metre in the length/breadth plane. In effect I have made my house less tall. But it is more than that. The volume will change : half as much in this instance. An angle from one corner of the base, say, to another corner on the roof will change : it's tangent will be half as much. Etc. All those things follow because I altered the regularity of the markings along an axis of measurement ie. I made 2 metres equal to one.

Here is a very key point in what follows :

... where the weaker field may include the case where there is ( almost ) no field at all ie. our astronaut distantly viewing the black hole merger. This has been tested over a separation of a few hundred feet in altitude on Earth ( the lower clock ran slower with respect to the higher clock ). If the Eddington expedition had performed spectroscopy on the starlight of those stars studied, they would have noticed a lowering of the frequency of those spectral lines in addition to spatial path deflection.

Before they were called black holes, they were named as 'frozen stars' because at the event horizon ( to be discussed ) time appears to stop. Please spend some time ( pardon the pun ) thinking about time slowing and even stopping. It's a hard one and IMHO the most difficult cognitive bit about The Relativities, simply because you are never likely to see this actually happen around you. Fortunately we are not in the right neighbourhood for that to be so obvious to our naked senses ( unassisted by extremely accurate instruments ).

For this to be absolutely firm in our thinking we need to have a chin wag about :

Proper Time ... or ... Get Your Own Clock !

Any questions on this bit at least ? :-)

Cheers, Mike.

I have made this letter longer than usual because I lack the time to make it shorter ...

... and my other CPU is a Ryzen 5950X :-) Blaise Pascal