I had to try plugging in some numbers â€“ I chose values for m1 and m2 such that Gm1m2 = 1, although I haven't had the chance to check on how realistic that is on such a small distance scale (probably not realistic at all, but it illustrates the general idea). I set R = 1^-30 for the minimum separation. In this (most basic) scenario the force of gravity is quite strong (no pun intended, mostly) at the distance where it's at the maximum â€“ close to the number 3x10^59, or the number 3 with 59 zeros trailing it! Here's a graph of what the modification looks like in a general way at that scale:
(click for full size)

I was relieved to see it came close to my earlier description of it shooting up and leveling off. From looking at it I got to wondering: When the more massive fundamental particles decay into the lightest ones, is this a type of Hawking radiation from micro black holes â€“ or from really deep and narrow gravity wells â€“ actually part well and part abstract-complex-domain-minimum-spacetime-interval-pixel thingy ? (note that on the graph, values for the x-axis start at r = 10^-30 instead of at the origin.)

Hmm, I had to try some numbers that might make a little more sense, like the radius and mass of a proton each for m1 and m2. I also added a plot of the standard equation. By the way, the masses for m1 and m2 in the previous post (such that Gm1m2 = 1) work out to be over a hundred thousand kilograms each (if I did that bit right) â€“ I don't think the density of neutron stars even comes close to that much mass per unit volume ... For R = 10^-15 meter and m1 = m2 = 1.671 621 58 Ã— 10^-27 kilogram, the gravitational force looks even less than exceedingly weak:

When I get some more time I'll try some numbers for something maybe more on the scale of quarks ...?

While checking on the sizes/masses for the quarks I happened across a short Wikipedia page on the Relativistic Breit-Wigner Distribution having to do with high energy physics, and the letter Î“ is used for the decay width (whose reciprocal is the mean lifetime of a particle), so I apologize for picking that letter â€“ my choice for it was of course from studying the effect that the 'gamma' from general relativity has on those equations (for transforming from one inertial frame of reference to another).

The masses of the individual quarks in a nucleus added together is quite a bit less than the mass of nucleus itself, and the reason is from additional energy present in the nucleus that holds it all together â€“ it's called the binding energy, and from E=mc^2 that energy is equivalent to mass and so the mass of a nucleus is the sum of the masses of the quarks plus the binding energy.

Mike, I guess your suggestion applies more to solutions of Einstein's field equations and black hole models?

Although reducing the number of fundamental interactions is a good thing, I'm a bit less optimistic about trying to show some mathematical equivalence between the exponential decay to gravity and the strong nuclear force â€“ but it seems like the idea of a 'minimum separation' might also mean that the electromagnetic interaction can suffer a similar decay at a different distance. Some reasons for such a decay might be that
a fair portion of the EM energy gets bound competing with a warped spacetime geometry caused by the quarks near point-like size?
maybe it would decay because the gravitational force is symmetric with only the system's center of mass while the electroweak force must accommodate charge, parity, and time symmetries, which are governed in the first place by the (warped) geometry of the local spacetime in which the quarks are enveloped ?
Or perhaps because a particle in a smaller region has a greater momentum and the exchange of electroweak bosons becomes less effective?

Anyway, for a chart showing the strength of gravity on the scale of quarks (where the distance could go to zero if quarks are point-like in size) , it's nice to be able to choose a value for the minimum separation that would give gravity a strength comparable to the strong interaction... since it's a naÃ¯ve formulation anyway, I used the mass of 3 quarks plus their binding energy (i.e. about the mass of a proton or neutron) and divided it evenly between m1 and m2 - so that's half of .938 GeV/c^2 times 1.783x10^-27 kg/GeV ... for the gravitational constant G, I used the value 6.673x10^-11 ... next I solved the modified equation for R (substituting R*sqrt(6)/2 for r, knowing that's where the maximum is) and got R = sqrt(2*G*m1*m2) / 3^.75 * sqrt(F_modified) ... then I plugged in a value of 10 newtons, which is the strength of the strong interaction (according to this Wikipedia page, where it's noted that a citation is necessary for the reference), and the result I got was that the constituents of a proton (quarks and gluons) will feel 10 newtons of force from gravity (with the modified equation) when R = 1.3402 x 10^-033 meters. Here's a graph of things at that scale:

Interesting to note that the distance for R is just a bit larger than the Plank length (at 1.6 x 10^-35 meters) - about 83 Plank lengths will fit within that distance.

In terms of mass curving space, how big of a dimple does that make, why might it hold only three generations of particles, and how many dimples in what volume space are required to make a big enough dent to form a micro black hole? I may be awhile studying the Einstein field equations and their notable solutions.

At least it's getting a little easier to understand and appreciate why the experiments at the LHC are so important :)

Sorry to seem such an awful snob, I've just caught up with your posts this week. :-)

My basic idea is that historically we have deduced force laws from behaviours. Thus Newton, say, assumes a connection ( across distance scales ) between apples falling and the Moon orbiting the Earth, and in doing so reaches a general law which fortunately has been amply confirmed. What we lack is microscopic direct knowledge of gravity. However we can say that if we extrapolate ( ie. project the behaviour beyond the domain it has been experimentally verified for ) to small distances we wind up with the 'to infinity and beyond' singularity result that an unvarnished inverse square law gives.

This issue is not unique to gravity BUT has been largely solved with other forces because of field quantization - which, say, has solved the black body electromagnetic radiation problem at high frequencies ( ie. classic EM predicts infinite emission ). This means that higher energy quanta ( photons ) can only possess fixed amounts of energy which increase with frequency - there is no 'small change' in the currency at that end - hence limiting photons numbers at the ultraviolet end, and hence forestalling an infinite total across all frequency bands.

So if there was a corresponding principle for gravitons, that is assuming the field can be quantised, then silly infinities can be avoided. I assume here that an infinity is a sign of a broken theory, and that nature won't actually provide any infinities.

So qualitatively almost any law that restricts force magnitude at small distances will avoid singularities, but that hardly restricts the exact form of the law.

One other curious analogy between the large and the small is that black holes, once formed and 'settled', will only reveal charge, mass and angular momentum to the surrounding space. There is no structure within the hole that can be deduced by measurement without. So what do 'elementary' particles have as their properties? Charge, mass and angular momentum!! Admittedly we associate quantum numbers and the like to them as well, but those aren't 'measurable' directly but divined from interactions.

Cheers, Mike.

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

Sorry to seem such an awful snob, I've just caught up with your posts this week. :-)

Not at all â€“ visited the cafe and heard the news, glad you're okay!

Quote:

My basic idea is that historically we have deduced force laws from behaviours. Thus Newton, say, assumes a connection ( across distance scales ) between apples falling and the Moon orbiting the Earth, and in doing so reaches a general law which fortunately has been amply confirmed. What we lack is microscopic direct knowledge of gravity. However we can say that if we extrapolate ( ie. project the behaviour beyond the domain it has been experimentally verified for ) to small distances we wind up with the 'to infinity and beyond' singularity result that an unvarnished inverse square law gives.

This issue is not unique to gravity BUT has been largely solved with other forces because of field quantization - which, say, has solved the black body electromagnetic radiation problem at high frequencies ( ie. classic EM predicts infinite emission ). This means that higher energy quanta ( photons ) can only possess fixed amounts of energy which increase with frequency - there is no 'small change' in the currency at that end - hence limiting photons numbers at the ultraviolet end, and hence forestalling an infinite total across all frequency bands.

So if there was a corresponding principle for gravitons, that is assuming the field can be quantised, then silly infinities can be avoided. I assume here that an infinity is a sign of a broken theory, and that nature won't actually provide any infinities.

So qualitatively almost any law that restricts force magnitude at small distances will avoid singularities, but that hardly restricts the exact form of the law.

One other curious analogy between the large and the small is that black holes, once formed and 'settled', will only reveal charge, mass and angular momentum to the surrounding space. There is no structure within the hole that can be deduced by measurement without. So what do 'elementary' particles have as their properties? Charge, mass and angular momentum!! Admittedly we associate quantum numbers and the like to them as well, but those aren't 'measurable' directly but divined from interactions.

Well said. In the same vein as Feynman's style with regard to explaining difficult things, you show there's an art to science :)

I think at some fundamental level that the manifestation of any measurable (â€œphysicalâ€?) quantity invokes all the abstract rules of math like a filter for what's physically realizable (stemming from symmetries induced between competing orders of similar oppositions and potentials) and what isn't, from a resolution of all the possibilities. This in turn invokes the idea of a minimum separation (or fundamentally minimal quantized resolution) because beyond that point means two previously separate objects become competing possibilities for the same volume of space at the same time â€“ back to the minimal quantized â€œsquare-oneâ€? so to speak.

I took a look at the Wikipedia page on blackbody radiation and noticed another type of factor that works similar to the one I mentioned previously. I copied the form of the equation for a blackbody's apparent temperature when it's directly receding or approaching the observer with a relativistic velocity. For this 'Gamma' factor (I also forgot the letter Gamma is used for the Christoffel symbols, and probably in lots of other places, so I'll stick with it), which I called Gamma_2, still using R for the minimum separation, I came up with Gamma_2 = sqrt[(r-R)/(r+R)]. It produces a similar effect, but weaker than the first Gamma (now Gamma_1 on the following graph).

I have a lot more to learn about math as well. I tried to broaden the peak by applying Gamma_1 to the numerator and Gamma_2 to the denominator of Newton's equation (forming essentially a third Gamma, G3 = G1/G2), and it didn't quite work as expected, but interestingly produced the same modified curve with a strength that peaks well above the unmodified equation (at the same distance r) â€“ here's the graph:

After some research - Lee Smolin'sThe Trouble With Physics - it seems the gravitational field has been quantised. Just not successfully. The problem can be ( mainly ) reduced to 'background dependence'.

With EM the assumption is some fixed spacetime, or at least one that doesn't vary with EM behaviour. So some distribution of masses, and movement thereof, produces a ( variable ) metric of spacetime. Light then maps out whatever that metric is - say solar eclipses revealing shifts in apparent distant star positions. Or that pair of orbiting neutron stars causing pulsar ( EM ) signal delay when one of the pair occults the other in a certain phase of their mutual orbit.

Not so with the gravitons, the hypothecated quanta of gravitational fields. Whereas two photons will pass by each other with no mutual effect ( linear superposition of fields if you like ), this can't happen with gravitons. By definition they are packets of gravitational field energy, thus have an equivalent mass, hence that must enter into the determination of the metric.

Imagine you travel to work each day along a certain road from SuburbA to CityB. If the actual road's path changed on a continuous basis you might cope with that. Just adjust your driving habits ( and reflexes ! ) to suit. This is what a photon does when transiting spacetime. Now imagine that the road changes it's geometry because you are driving along it! That is, your presence alters the evolution of your trip to work. This is like a graviton.

Or some freeways we have DownUnda, for that matter. :-)

So now two gravitons passing by will affect each other. How do they affect each other? Like any other pair of lumps of mass/energy - by exchanging gravitons!! :-)

Whoops. The rabbit hole just opens up and sucks you down ....

It's easy to see why, in one's calculations of quantum gravity, infinities abound and readily overtake you. Even on a good day. So the graviton field is non-linear, superposition wise, and currently non-solvable in a rational way. The field has 'self energy' if you like. Note that GR does not assume some background like EM. It is the background!

Photons are 'carriers' of the EM field, as are gluons for the strong nuclear force, and vector bosons for the weak force. They act between particles that have the 'charge' appropriate to each force - electric charge, color, and err ... weak isospin ( I think ?? ). But as a graviton has energy/mass it is both a carrier and a charge. Hmmmmm.

This stuff is the main reason why grand unification(s) fail, so far, because of this special character of gravity that EM, strong and weak forces lack. String theory needs to contain/constrain this behaviour too.

Cheers, Mike.

( edit ) My other questions are : if I'm nearby the exterior of a black hole then nothing can get out it because nothing can exceed the speed of light. Gravitons included. So how does the mass within the event horizon let me know, via gravitons, that it is there? If it can't then I won't feel it's gravity will I? So how can a black hole pull stuff in? Does this not imply that if gravitons exist they must be superluminal?

Interestingly some earlier attempts at 'unified field theories' predicted tachyons - faster than light particles. These were seen as fatal to such theoretical efforts, on various grounds. If gravitons were tachyonic it would explain while we will never see them, possibly prevent the self energy property ( 'cos also they'll never see each other? ) and allow black holes too.

( edit ) It gets worse. If superluminal gravitons exist then causality - cause preceding effect - gets trashed. Such gravitons, because they beat the photons, arrive earlier and can affect a massive particle with an electric charge ( no shortage of those ). We don't see this, do we? Do we receive a radio transmission before any effect from the accelerating charges in your local station's transmitter can reach us? Or do we?

Have the gravitons, weak as they are, been arriving just before the radio frequency photons all along and we never noticed - because their effect is small ( EM vs gravity strength ) and if they are only just superluminal then any breakage of 'cause and effect' is very slight? Is the 'speed of light' actually the 'speed of gravitons'? Like a two horse race with only a nose between a win and a place - we triggered on the leading horse but ( not knowing it was there ) mistook it for the second?

Come on Mike. Lie down for while, over here ..... :-)

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

## I had to try plugging in some

)

I had to try plugging in some numbers â€“ I chose values for m1 and m2 such that Gm1m2 = 1, although I haven't had the chance to check on how realistic that is on such a small distance scale (probably not realistic at all, but it illustrates the general idea). I set R = 1^-30 for the minimum separation. In this (most basic) scenario the force of gravity is quite strong (no pun intended, mostly) at the distance where it's at the maximum â€“ close to the number 3x10^59, or the number 3 with 59 zeros trailing it! Here's a graph of what the modification looks like in a general way at that scale:

(click for full size)

I was relieved to see it came close to my earlier description of it shooting up and leveling off. From looking at it I got to wondering: When the more massive fundamental particles decay into the lightest ones, is this a type of Hawking radiation from micro black holes â€“ or from really deep and narrow gravity wells â€“ actually part well and part abstract-complex-domain-minimum-spacetime-interval-pixel thingy ? (note that on the graph, values for the x-axis start at r = 10^-30 instead of at the origin.)

## Hmm, I had to try some

)

Hmm, I had to try some numbers that might make a little more sense, like the radius and mass of a proton each for m1 and m2. I also added a plot of the standard equation. By the way, the masses for m1 and m2 in the previous post (such that Gm1m2 = 1) work out to be over a hundred thousand kilograms each (if I did that bit right) â€“ I don't think the density of neutron stars even comes close to that much mass per unit volume ... For R = 10^-15 meter and m1 = m2 = 1.671 621 58 Ã— 10^-27 kilogram, the gravitational force looks even less than exceedingly weak:

When I get some more time I'll try some numbers for something maybe more on the scale of quarks ...?

## While checking on the

)

While checking on the sizes/masses for the quarks I happened across a short Wikipedia page on the Relativistic Breit-Wigner Distribution having to do with high energy physics, and the letter Î“ is used for the decay width (whose reciprocal is the mean lifetime of a particle), so I apologize for picking that letter â€“ my choice for it was of course from studying the effect that the 'gamma' from general relativity has on those equations (for transforming from one inertial frame of reference to another).

The masses of the individual quarks in a nucleus added together is quite a bit less than the mass of nucleus itself, and the reason is from additional energy present in the nucleus that holds it all together â€“ it's called the binding energy, and from E=mc^2 that energy is equivalent to mass and so the mass of a nucleus is the sum of the masses of the quarks plus the binding energy.

Mike, I guess your suggestion applies more to solutions of Einstein's field equations and black hole models?

Although reducing the number of fundamental interactions is a good thing, I'm a bit less optimistic about trying to show some mathematical equivalence between the exponential decay to gravity and the strong nuclear force â€“ but it seems like the idea of a 'minimum separation' might also mean that the electromagnetic interaction can suffer a similar decay at a different distance. Some reasons for such a decay might be that

a fair portion of the EM energy gets bound competing with a warped spacetime geometry caused by the quarks near point-like size?

maybe it would decay because the gravitational force is symmetric with only the system's center of mass while the electroweak force must accommodate charge, parity, and time symmetries, which are governed in the first place by the (warped) geometry of the local spacetime in which the quarks are enveloped ?

Or perhaps because a particle in a smaller region has a greater momentum and the exchange of electroweak bosons becomes less effective?

Anyway, for a chart showing the strength of gravity on the scale of quarks (where the distance could go to zero if quarks are point-like in size) , it's nice to be able to choose a value for the minimum separation that would give gravity a strength comparable to the strong interaction... since it's a naÃ¯ve formulation anyway, I used the mass of 3 quarks plus their binding energy (i.e. about the mass of a proton or neutron) and divided it evenly between m1 and m2 - so that's half of .938 GeV/c^2 times 1.783x10^-27 kg/GeV ... for the gravitational constant G, I used the value 6.673x10^-11 ... next I solved the modified equation for R (substituting R*sqrt(6)/2 for r, knowing that's where the maximum is) and got R = sqrt(2*G*m1*m2) / 3^.75 * sqrt(F_modified) ... then I plugged in a value of 10 newtons, which is the strength of the strong interaction (according to this Wikipedia page, where it's noted that a citation is necessary for the reference), and the result I got was that the constituents of a proton (quarks and gluons) will feel 10 newtons of force from gravity (with the modified equation) when R = 1.3402 x 10^-033 meters. Here's a graph of things at that scale:

Interesting to note that the distance for R is just a bit larger than the Plank length (at 1.6 x 10^-35 meters) - about 83 Plank lengths will fit within that distance.

In terms of mass curving space, how big of a dimple does that make, why might it hold only three generations of particles, and how many dimples in what volume space are required to make a big enough dent to form a micro black hole? I may be awhile studying the Einstein field equations and their notable solutions.

At least it's getting a little easier to understand and appreciate why the experiments at the LHC are so important :)

## Sorry to seem such an awful

)

Sorry to seem such an awful snob, I've just caught up with your posts this week. :-)

My basic idea is that historically we have deduced force laws from behaviours. Thus Newton, say, assumes a connection ( across distance scales ) between apples falling and the Moon orbiting the Earth, and in doing so reaches a general law which fortunately has been amply confirmed. What we lack is microscopic direct knowledge of gravity. However we can say that if we extrapolate ( ie. project the behaviour beyond the domain it has been experimentally verified for ) to small distances we wind up with the 'to infinity and beyond' singularity result that an unvarnished inverse square law gives.

This issue is not unique to gravity BUT has been largely solved with other forces because of field quantization - which, say, has solved the black body electromagnetic radiation problem at high frequencies ( ie. classic EM predicts infinite emission ). This means that higher energy quanta ( photons ) can only possess fixed amounts of energy which increase with frequency - there is no 'small change' in the currency at that end - hence limiting photons numbers at the ultraviolet end, and hence forestalling an infinite total across all frequency bands.

So if there was a corresponding principle for gravitons, that is assuming the field can be quantised, then silly infinities can be avoided. I assume here that an infinity is a sign of a broken theory, and that nature won't actually provide any infinities.

So qualitatively almost any law that restricts force magnitude at small distances will avoid singularities, but that hardly restricts the exact form of the law.

One other curious analogy between the large and the small is that black holes, once formed and 'settled', will only reveal charge, mass and angular momentum to the surrounding space. There is no structure within the hole that can be deduced by measurement without. So what do 'elementary' particles have as their properties? Charge, mass and angular momentum!! Admittedly we associate quantum numbers and the like to them as well, but those aren't 'measurable' directly but divined from interactions.

Cheers, Mike.

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

## RE: Sorry to seem such an

)

Not at all â€“ visited the cafe and heard the news, glad you're okay!

Well said. In the same vein as Feynman's style with regard to explaining difficult things, you show there's an art to science :)

I think at some fundamental level that the manifestation of any measurable (â€œphysicalâ€?) quantity invokes all the abstract rules of math like a filter for what's physically realizable (stemming from symmetries induced between competing orders of similar oppositions and potentials) and what isn't, from a resolution of all the possibilities. This in turn invokes the idea of a minimum separation (or fundamentally minimal quantized resolution) because beyond that point means two previously separate objects become competing possibilities for the same volume of space at the same time â€“ back to the minimal quantized â€œsquare-oneâ€? so to speak.

I took a look at the Wikipedia page on blackbody radiation and noticed another type of factor that works similar to the one I mentioned previously. I copied the form of the equation for a blackbody's apparent temperature when it's directly receding or approaching the observer with a relativistic velocity. For this 'Gamma' factor (I also forgot the letter Gamma is used for the Christoffel symbols, and probably in lots of other places, so I'll stick with it), which I called Gamma_2, still using R for the minimum separation, I came up with Gamma_2 = sqrt[(r-R)/(r+R)]. It produces a similar effect, but weaker than the first Gamma (now Gamma_1 on the following graph).

I have a lot more to learn about math as well. I tried to broaden the peak by applying Gamma_1 to the numerator and Gamma_2 to the denominator of Newton's equation (forming essentially a third Gamma, G3 = G1/G2), and it didn't quite work as expected, but interestingly produced the same modified curve with a strength that peaks well above the unmodified equation (at the same distance r) â€“ here's the graph:

Back to the books for me :)

## After some research - Lee

)

After some research - Lee Smolin's The Trouble With Physics - it seems the gravitational field has been quantised. Just not successfully. The problem can be ( mainly ) reduced to 'background dependence'.

With EM the assumption is some fixed spacetime, or at least one that doesn't vary with EM behaviour. So some distribution of masses, and movement thereof, produces a ( variable ) metric of spacetime. Light then maps out whatever that metric is - say solar eclipses revealing shifts in apparent distant star positions. Or that pair of orbiting neutron stars causing pulsar ( EM ) signal delay when one of the pair occults the other in a certain phase of their mutual orbit.

Not so with the gravitons, the hypothecated quanta of gravitational fields. Whereas two photons will pass by each other with no mutual effect ( linear superposition of fields if you like ), this can't happen with gravitons. By definition they are packets of gravitational field energy, thus have an equivalent mass, hence that must enter into the determination of the metric.

Imagine you travel to work each day along a certain road from SuburbA to CityB. If the actual road's path changed on a continuous basis you might cope with that. Just adjust your driving habits ( and reflexes ! ) to suit. This is what a photon does when transiting spacetime. Now imagine that the road changes it's geometry because you are driving along it! That is, your presence alters the evolution of your trip to work. This is like a graviton.

Or some freeways we have DownUnda, for that matter. :-)

So now two gravitons passing by will affect each other. How do they affect each other? Like any other pair of lumps of mass/energy - by exchanging gravitons!! :-)

Whoops. The rabbit hole just opens up and sucks you down ....

It's easy to see why, in one's calculations of quantum gravity, infinities abound and readily overtake you. Even on a good day. So the graviton field is non-linear, superposition wise, and currently non-solvable in a rational way. The field has 'self energy' if you like. Note that GR does not assume some background like EM. It is the background!

Photons are 'carriers' of the EM field, as are gluons for the strong nuclear force, and vector bosons for the weak force. They act between particles that have the 'charge' appropriate to each force - electric charge, color, and err ... weak isospin ( I think ?? ). But as a graviton has energy/mass it is both a carrier and a charge. Hmmmmm.

This stuff is the main reason why grand unification(s) fail, so far, because of this special character of gravity that EM, strong and weak forces lack. String theory needs to contain/constrain this behaviour too.

Cheers, Mike.

( edit ) My other questions are : if I'm nearby the exterior of a black hole then nothing can get out it because nothing can exceed the speed of light. Gravitons included. So how does the mass within the event horizon let me know, via gravitons, that it is there? If it can't then I won't feel it's gravity will I? So how can a black hole pull stuff in? Does this not imply that if gravitons exist they must be superluminal?

Interestingly some earlier attempts at 'unified field theories' predicted tachyons - faster than light particles. These were seen as fatal to such theoretical efforts, on various grounds. If gravitons were tachyonic it would explain while we will never see them, possibly prevent the self energy property ( 'cos also they'll never see each other? ) and allow black holes too.

( edit ) It gets worse. If superluminal gravitons exist then causality - cause preceding effect - gets trashed. Such gravitons, because they beat the photons, arrive earlier and can affect a massive particle with an electric charge ( no shortage of those ). We don't see this, do we? Do we receive a radio transmission before any effect from the accelerating charges in your local station's transmitter can reach us? Or do we?

Have the gravitons, weak as they are, been arriving just before the radio frequency photons all along and we never noticed - because their effect is small ( EM vs gravity strength ) and if they are only just superluminal then any breakage of 'cause and effect' is very slight? Is the 'speed of light' actually the 'speed of gravitons'? Like a two horse race with only a nose between a win and a place - we triggered on the leading horse but ( not knowing it was there ) mistook it for the second?

Come on Mike. Lie down for while, over here ..... :-)

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