Allowing for 'alternate geometries', can not Pi take on other numerical values other than what we have for our own flat planes in our 3d physical space?

An easy example of 'something different' is for when you wrap a 2d plane around a sphere (using an additional dimension, our 3rd dimension). For that example, you can then have triangles that have all three corners with 90 degree angles. Also, Pi would then be a variable dependant on the radius of the circle!

Hi Martin, true it is that the spacetime geometry may give you a different numerical value for pi when measuring a circle, depending on curvature caused by gravity. But how does gravity affect the values of the fundamental constants alpha (fine structure), h (Planck's), c (speed of light), and e (elementary charge) ?

The reason I ask is that (in this universe, with the values for the constants such as they are), if you take alpha times h times c, and divide that by twice the square of e (using cgs units), the result should be pi. That's ahc/2e^2 = pi. Using the following values (from here), the result is accurate to about 8 decimal places (same level as the precision of measurements in experiments to determine values of the various constants):
a = 7.297 352 570 x 10^-3 (dimensionless)
h = 6.626 068 85 x 10^-27 erg seconds
c = 2.997 924 58 x 10^10 centimeters per second
e = 4.803 204 27 x 10^-10 franklin (or square root ergs times square root centimeters)
result = ahc/2e^2 = 3.1415926181836104537698585493... (dimensionless)
pi = 3.1415926535897932384626433832795...
pi - result = 0.0000000035406182...

What does this representation of pi signify? A symmetry? Compactness? Would the value of the result change in a strong gravitational field?

Very recent astronomical results (still under discussion) say that alpha varies according to the direction of the astronomical observations. In opposite directions it is greater or smaller that here on the Earth.See here, from www.setiquest.org: Alpha
Tullio

Allowing for 'alternate geometries', can not Pi take on other numerical values...

... But how does gravity affect the values of the fundamental constants alpha (fine structure), h (Planck's), c (speed of light), and e (elementary charge) ?

... the various constants):
a ... (dimensionless)
h ... erg seconds
c ... centimeters per second
e ... franklin (or square root ergs times square root centimeters)
result = ahc/2e^2 = 3.1415926181836104537698585493... (dimensionless)
pi = 3.1415926535897932384626433832795...
pi - result = 0.0000000035406182...

What does this representation of pi signify? A symmetry? Compactness? Would the value of the result change in a strong gravitational field?

... where Alpha may vary according to location in space (but not necessarily time)...

All due to gravity?

Einstein famously said that our physics might only be a local phenomenon...

Hmm, so there's a North & South to the whole universe? Bit of a stretch considering the isotropy of the cosmic microwave background, no? - - although maybe there's an associated dipole with dark energy or dark matter?

As for the exploding black hole, am guessing yes, during the mergers of huge galaxies, where each has a supermassive black hole binary system at their center and plenty of stellar black holes to get caught between non-overlapping Schwarzschild radii while its own Schwarzschild radius overlaps two or more of the radii of the supermassive black holes, and the stellar mass black hole is subsequently stretched down to a minimum density where whatever isn't caught by the supermassive black holes gets released in a burst of gamma rays.

Could a black hole be spun up to such a ferocious extent by infalling matter to then grow and speed up and bulge outside its own event horizon?...

The tidal forces would be extreme as that lot developed...

Or would other effects limit such a scenario?

You get three changes to a black hole when matter falls in - mass, angular momentum and electric charge. What follows is pure classical GR and no quantum ....

Like most macroscopic objects electric charge disparity from neutral wouldn't last long, as something nearby of the opposite nett charge would electrically attract and come on in.

As the mass increases the surface area of the event horizon will grow.

As angular momentum increases the shape of the event horizon deviates from a sphere ( the no-rotation case ) to an oblate form. A discus has an oblate shape whereas an American/Aussie football is prolate.

Now I guess the question here is what happens as it really gets spun up via lots of stuff coming into the hole but only in one rotational sense. A book I have, Black Holes by J.P.Luminet ( a great case of nominative determinism ! ), suggests that the event horizon gets 'flicked off' and the central singularity is laid bare. What this essentially 'means' is that if conservation of angular momentum is to remain inviolate, and if gravity is a purely central force ( not contributing to tangential velocity change ) then in the extremal case of the tangential velocity at the event horizon radius being the speed of light - can things then be/remain captured?

So ... a 'naked' singularity? Now the debate really hots up. For the extreme case being discussed the central singularity isn't a point, but a torus - a donut with a hole! Apart from any singularity that remains in causal connection ( bare ) with the rest of the universe being a problem - it becomes an undetermined boundary with no exact values - there is a tantalising possiblity.

A clever punter could line up his space ship far away but astride the axis of rotation and shoot on in along said axis. He scoots through the hole in the donut missing the toroidal singularity and certain death. If the toroid is wide enough and he stays exactly on-axis then tidal forces may be tolerable. Well, I say 'punter' definitely whereas 'clever' remains to be demonstrated! Can you go to another universe this way? Or another part of the same one? Does the surface bounded by the torus, that you breach while going from the 'north' to the 'south' side, represent a stitching together of 'separate' realities not otherwise connected? [ Grab a toroidal type donut and stick your little pinky through it's central hole without touching the inner sides. Both the base and the tip of your finger lie in the same universe. It doesn't disappear into 'fat air' :-) ].

Mr Hawking creates a 'censorship' principle that says this naked singularity just won't happen, and when we are smarter ( ie. some theory of quantum gravity ) we'll know why ......

Quote:

Hmm, so there's a North & South to the whole universe? Bit of a stretch considering the isotropy of the cosmic microwave background, no?

There is a dipole in the measured CMB, represented as one area in the sky being slightly warmer with an area 180 degrees away being slightly cooler. This has been attributed to our peculiar motion with respect to the CMB field. But I guess we can't turn the Earth around to go some other way to test this! The isotropy otherwise quoted is after subtraction of this dipole pattern.

Quote:

.... dividing the circumferences of lots of circles by their respective radii, and coming up with an amazing constancy in that ratio ....

I was thinking of the Egyptians here. For a given circular object - judged to be so 'by eye' or by noting a fixed distance from some designated centre to each point on the edge - they would take two pieces of rope. One would measure the radius, and then be taken to the side and laid straight on the ground. Another would be wound around one circuit of the circumference and cut or marked appropriately. Then take it aside and lay it straight alongside the 'radius' rope. Repeat this process for lots of circles. You now have alot of pairs of ropes, each recording a single circle's radius/circumference instance. Now remembering that their math had only fractions then ( rational numbers or ratios of integers ) the idea was to then note that :

- you could scale any one rope pair unto another. That is : if you multiplied the length of one radius rope to become equal to some other second radius rope, the same multiplier could be used to scale up the circumference rope in the first pair to yield the circumference rope in the second pair. Hence the 'amazing constancy' .....

- you can't find a ratio of integers to perfectly represent the constancy, although there are plenty good approximates.

- whether you wanted PI or 2*PI or PI/4 etc for a given calculation didn't change these thoughts.

Remember the Egyptians built the pyramids! So these were not idle dabblers but serious engineers! Then there is the 'golden mean', but that's another story .....

Cheers, Mike.

( edit ) Sorry, I forgot to say the unspoken assumption that when the tangential velocity is the speed of light then there can be no radial component, speed of light being the universal maximum. So no radial component especially means no falling inwards. And while real material particles will only asymptotically approach the speed of light, that also means they will only asymptotically approach zero radial velocity component too .....

But as quantum mechanics tends to blur stuff, you can appreciate that these limits cases come into doubt if you apply it.

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

Could a black hole be spun up to such a ferocious extent by infalling matter to then grow and speed up and bulge outside its own event horizon?...

The tidal forces would be extreme as that lot developed...

Or would other effects limit such a scenario?

You get three changes to a black hole when matter falls in - mass, angular momentum and electric charge. What follows is pure classical GR and no quantum ....

Like most macroscopic objects electric charge disparity from neutral wouldn't last long, as something nearby of the opposite nett charge would electrically attract and come on in.

As the mass increases the surface area of the event horizon will grow.

As angular momentum increases the shape of the event horizon deviates from a sphere ( the no-rotation case ) to an oblate form. A discus has an oblate shape whereas an American/Aussie football is prolate.

Now I guess the question here is what happens as it really gets spun up via lots of stuff coming into the hole but only in one rotational sense. ... then in the extremal case of the tangential velocity at the event horizon radius being the speed of light - can things then be/remain captured?

... For the extreme case being discussed the central singularity isn't a point, but a torus - a donut with a hole!

... tidal forces ...

Mr Hawking creates a 'censorship' principle that says this naked singularity just won't happen, and when we are smarter ( ie. some theory of quantum gravity ) we'll know why ......

( edit ) Sorry, I forgot to say the unspoken assumption that when the tangential velocity is the speed of light then there can be no radial component, speed of light being the universal maximum. So no radial component especially means no falling inwards. And while real material particles will only asymptotically approach the speed of light, that also means they will only asymptotically approach zero radial velocity component too .....

But as quantum mechanics tends to blur stuff, you can appreciate that these limits cases come into doubt if you apply it.

Phew, that's a mind bender!

But then, also add in time dilation effects and... Is there not going to be some unholy shear effects from time and near-light-speed effects for the near light-speed rotating surface?

I suspect that something must break or limit before such a speedup...

But the big hole in the idea is... How do you get the hole in the centre in the first place? Why would a gravitational mass transform into a toroid?

Or is the assumption that in-falling mass accumulates at the light-speed circumference that then attracts the matter from inside and out? Would you not get a super-dense toroid with a gap and then a singularity in the middle still? (All rotating.)

## I hope that doesn't happen.

)

I hope that doesn't happen.

## RE: I hope that doesn't

)

Why not? As long as it's far enough away. ;-)

GruÃŸ,

Gundolf

Computer sind nicht alles im Leben. (Kleiner Scherz)

## RE: Allowing for 'alternate

)

Hi Martin, true it is that the spacetime geometry may give you a different numerical value for pi when measuring a circle, depending on curvature caused by gravity. But how does gravity affect the values of the fundamental constants alpha (fine structure), h (Planck's), c (speed of light), and e (elementary charge) ?

The reason I ask is that (in this universe, with the values for the constants such as they are), if you take alpha times h times c, and divide that by twice the square of e (using cgs units), the result should be pi. That's ahc/2e^2 = pi. Using the following values (from here), the result is accurate to about 8 decimal places (same level as the precision of measurements in experiments to determine values of the various constants):

a = 7.297 352 570 x 10^-3 (dimensionless)

h = 6.626 068 85 x 10^-27 erg seconds

c = 2.997 924 58 x 10^10 centimeters per second

e = 4.803 204 27 x 10^-10 franklin (or square root ergs times square root centimeters)

result = ahc/2e^2 = 3.1415926181836104537698585493... (dimensionless)

pi = 3.1415926535897932384626433832795...

pi - result = 0.0000000035406182...

What does this representation of pi signify? A symmetry? Compactness? Would the value of the result change in a strong gravitational field?

## Very recent astronomical

)

Very recent astronomical results (still under discussion) say that alpha varies according to the direction of the astronomical observations. In opposite directions it is greater or smaller that here on the Earth.See here, from www.setiquest.org:

Alpha

Tullio

## RE: RE: Allowing for

)

And... Alpha (Thanks tullio)

... where Alpha may vary according to location in space (but not necessarily time)...

All due to gravity?

Einstein famously said that our physics might only be a local phenomenon...

Keep searchin',

Martin

See new freedom: Mageia Linux

Take a look for yourself: Linux Format

The Future is what We all make IT (GPLv3)

## RE: And... Alpha (Thanks

)

Hmm, so there's a North & South to the whole universe? Bit of a stretch considering the isotropy of the cosmic microwave background, no? - - although maybe there's an associated dipole with dark energy or dark matter?

As for the exploding black hole, am guessing yes, during the mergers of huge galaxies, where each has a supermassive black hole binary system at their center and plenty of stellar black holes to get caught between non-overlapping Schwarzschild radii while its own Schwarzschild radius overlaps two or more of the radii of the supermassive black holes, and the stellar mass black hole is subsequently stretched down to a minimum density where whatever isn't caught by the supermassive black holes gets released in a burst of gamma rays.

## I think both WIMAP and Planck

)

I think both WIMAP and Planck are measuring the microwave background in high precision, so maybe it is not completely isotropic.

Tullio

## Could a black hole be spun up

)

Could a black hole be spun up to such a ferocious extent by infalling matter to then grow and speed up and bulge outside its own event horizon?...

The tidal forces would be extreme as that lot developed...

Or would other effects limit such a scenario?

Keep searchin',

Martin

See new freedom: Mageia Linux

Take a look for yourself: Linux Format

The Future is what We all make IT (GPLv3)

## RE: Could a black hole be

)

You get three changes to a black hole when matter falls in - mass, angular momentum and electric charge. What follows is pure classical GR and no quantum ....

Like most macroscopic objects electric charge disparity from neutral wouldn't last long, as something nearby of the opposite nett charge would electrically attract and come on in.

As the mass increases the surface area of the event horizon will grow.

As angular momentum increases the shape of the event horizon deviates from a sphere ( the no-rotation case ) to an oblate form. A discus has an oblate shape whereas an American/Aussie football is prolate.

Now I guess the question here is what happens as it really gets spun up via lots of stuff coming into the hole but only in one rotational sense. A book I have, Black Holes by J.P.Luminet ( a great case of nominative determinism ! ), suggests that the event horizon gets 'flicked off' and the central singularity is laid bare. What this essentially 'means' is that if conservation of angular momentum is to remain inviolate, and if gravity is a purely central force ( not contributing to tangential velocity change ) then in the extremal case of the tangential velocity at the event horizon radius being the speed of light - can things then be/remain captured?

So ... a 'naked' singularity? Now the debate really hots up. For the extreme case being discussed the central singularity isn't a point, but a torus - a donut with a hole! Apart from any singularity that remains in causal connection ( bare ) with the rest of the universe being a problem - it becomes an undetermined boundary with no exact values - there is a tantalising possiblity.

A clever punter could line up his space ship far away but astride the axis of rotation and shoot on in along said axis. He scoots through the hole in the donut missing the toroidal singularity and certain death. If the toroid is wide enough and he stays exactly on-axis then tidal forces may be tolerable. Well, I say 'punter' definitely whereas 'clever' remains to be demonstrated! Can you go to another universe this way? Or another part of the same one? Does the surface bounded by the torus, that you breach while going from the 'north' to the 'south' side, represent a stitching together of 'separate' realities not otherwise connected? [ Grab a toroidal type donut and stick your little pinky through it's central hole without touching the inner sides. Both the base and the tip of your finger lie in the same universe. It doesn't disappear into 'fat air' :-) ].

Mr Hawking creates a 'censorship' principle that says this naked singularity just won't happen, and when we are smarter ( ie. some theory of quantum gravity ) we'll know why ......

There is a dipole in the measured CMB, represented as one area in the sky being slightly warmer with an area 180 degrees away being slightly cooler. This has been attributed to our peculiar motion with respect to the CMB field. But I guess we can't turn the Earth around to go some other way to test this! The isotropy otherwise quoted is after subtraction of this dipole pattern.

I was thinking of the Egyptians here. For a given circular object - judged to be so 'by eye' or by noting a fixed distance from some designated centre to each point on the edge - they would take two pieces of rope. One would measure the radius, and then be taken to the side and laid straight on the ground. Another would be wound around one circuit of the circumference and cut or marked appropriately. Then take it aside and lay it straight alongside the 'radius' rope. Repeat this process for lots of circles. You now have alot of pairs of ropes, each recording a single circle's radius/circumference instance. Now remembering that their math had only fractions then ( rational numbers or ratios of integers ) the idea was to then note that :

- you could scale any one rope pair unto another. That is : if you multiplied the length of one radius rope to become equal to some other second radius rope, the same multiplier could be used to scale up the circumference rope in the first pair to yield the circumference rope in the second pair. Hence the 'amazing constancy' .....

- you can't find a ratio of integers to perfectly represent the constancy, although there are plenty good approximates.

- whether you wanted PI or 2*PI or PI/4 etc for a given calculation didn't change these thoughts.

Remember the Egyptians built the pyramids! So these were not idle dabblers but serious engineers! Then there is the 'golden mean', but that's another story .....

Cheers, Mike.

( edit ) Sorry, I forgot to say the unspoken assumption that when the tangential velocity is the speed of light then there can be no radial component, speed of light being the universal maximum. So no radial component especially means no falling inwards. And while real material particles will only asymptotically approach the speed of light, that also means they will only asymptotically approach zero radial velocity component too .....

But as quantum mechanics tends to blur stuff, you can appreciate that these limits cases come into doubt if you apply it.

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

## RE: RE: Could a black

)

Phew, that's a mind bender!

But then, also add in time dilation effects and... Is there not going to be some unholy shear effects from time and near-light-speed effects for the near light-speed rotating surface?

I suspect that something must break or limit before such a speedup...

But the big hole in the idea is... How do you get the hole in the centre in the first place? Why would a gravitational mass transform into a toroid?

Or is the assumption that in-falling mass accumulates at the light-speed circumference that then attracts the matter from inside and out? Would you not get a super-dense toroid with a gap and then a singularity in the middle still? (All rotating.)

Keep searchin',

Martin

See new freedom: Mageia Linux

Take a look for yourself: Linux Format

The Future is what We all make IT (GPLv3)