Well that is weird. Something pulsing every 54minutes, with some period change also, consistent with a neutron star but might be something else like a white dwarf. Discovered by pure luck via the Australian Square Kilometre Array Pathfinder telescope and then followed up using MeerKAT. That is "the MeerKAT detections appear to arrive in phase with the ASKAP detections" orin other words it's a real signal from the stars. I'll bet that source object is, or was, in company and the current signal is the residue from interactions with other bodies.
A new category then, not a magnetar but a bludgetar. ;-)
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
I struggled with my first course in University calculus. It discouraged me from taking any more...
A Proud member of the O.F.A. (Old Farts Association). Be well, do good work, and keep in touch.® (Garrison Keillor) I want some more patience. RIGHT NOW!
I think the funny/weird metrics are exciting because one of them, or a group of them, may get around or even explain the 'paradox', for want of a better word, of the central singularity of black holes. That's where GR ought meet QM. I wonder what the Finsler et al geometries have to say about that. Watch this space I guess.
QM won't ever meet GR, because general relativity is non-renormalizable.
Whenever you quantize a classical theory, you will find divergences (like infinite integrals) that seem to blow the whole process out of the water. However, the quantum world is not at all like the classical world, where a particle remains the same unless it interacts with something else. In the quantum world, any particle is constantly undergoing "virtual" interactions, for example a bare electron travelling through space is constantly emitting and reabsorbing "virtual" photons -- virtual in the sense that they don't exist in the classical world, whereas they do in the quantum world. Add up all the virtual interactions, and you get rid of all the infinities.
As an extreme example, here is an electron absorbing a photon (left) with just one of a great many virtual interactions on the right. The circle in the right side represents a virtual electron-positron pair.
In general relativity, that turns out to be impossible. In the 4-dimenstional universe we live in, quantum general relativity cannot be renormalized to any order, even in an empty universe.
If you have any reference which suggests anything different, please provide a full citation, including page or section number.
Sure, bottom of page one to top of page two in the Introduction of Heefer's Paper :
Finsler geometry is just pseudo-Riemannian geometry without the quadratic restriction. In other words, it is the natural extension of pseudo-Riemannian geometry in which the squared line element ds2 is not restricted to be quadratic in the coordinate 1-forms dxi
Finsler geometry extends Riemannian geometry by allowing the metric to be any positively homogeneous and strongly convex function on the tangent bundle, rather than restricting it to a quadratic form. I think we may be 'talking past one other' in terms of definitions.
In the context of Finsler geometry, the terms "metric" and "fundamental tensor" refer to different but related concepts.
Finsler Metric: In Finsler geometry, the metric is a function F(x,y) that defines the length of a tangent vector y at a point on the manifold x. This function F generalizes the notion of distance in a way that is not necessarily quadratic in the coordinate differentials.
Fundamental Tensor: The fundamental tensor in Finsler geometry is derived from the Finsler metric. It is defined as the Hessian of 1/2 F2 with respect to the coordinate differentials, and this tensor generalizes the Riemannian metric tensor and reduces to it in the special case where F is quadratic in the coordinates.
In the context of Riemannian geometry, the metric and the fundamental tensor often refer to the same mathematical object, the Riemannian metric tensor. This all-too-subtle difference b/w Riemannian and Finslerian maths is a bit of a trap I think. Indeed the comment :
Even though it is the 2-homogeneous Finsler Lagrangian that enters in most (but not all) formulas in Finsler geometry, it is useful to independently define a similar 1-homogeneous object, the Finsler metric.
supports that distinction. HTH. Please do advise us of your thoughts ....
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
I think the funny/weird metrics are exciting because one of them, or a group of them, may get around or even explain the 'paradox', for want of a better word, of the central singularity of black holes. That's where GR ought meet QM. I wonder what the Finsler et al geometries have to say about that. Watch this space I guess.
QM won't ever meet GR, because general relativity is non-renormalizable.
Well that's the question. Does that non-renormalizability hold when you go to a Finsler geometry?
hadron wrote:
Whenever you quantize a classical theory, you will find divergences (like infinite integrals) that seem to blow the whole process out of the water.
Again, do those integrals remain infinite when the metric is altered?
hadron wrote:
However, the quantum world is not at all like the classical world, where a particle remains the same unless it interacts with something else. In the quantum world, any particle is constantly undergoing "virtual" interactions, for example a bare electron travelling through space is constantly emitting and reabsorbing "virtual" photons -- virtual in the sense that they don't exist in the classical world, whereas they do in the quantum world. Add up all the virtual interactions, and you get rid of all the infinities.
Agreed, you can reorder infinite sums (or more exactly the limit of reordered partial sums may change) to suppress divergences.
hadron wrote:
As an extreme example, here is an electron absorbing a photon (left) with just one of a great many virtual interactions on the right. The circle in the right side represents a virtual electron-positron pair.
Where the background in these diagrams is spacetime with the usual special relativistic metric. Does the significance of these diagrams change with Finsler? I don't know .... mind you, the total tonnage of the list of things I don't know would sink fleets of battleships. ;-)
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
I neglected to mention in a previous post that I always use the Einstein summation convention, whereby a duplicated index in a product implies that the sum over the range of that index is to be carried out. Thus, for example, in an n-dimensional space, in the right-hand side of the expression
V = vidxi
it is implied that this stands for the summation of that term as i runs over its full range from 1 to n.
Mike Hewson wrote:
hadron wrote:
If you have any reference which suggests anything different, please provide a full citation, including page or section number.
Sure, bottom of page one to top of page two in the Introduction of Heefer's Paper :
Finsler geometry is just pseudo-Riemannian geometry without the quadratic restriction. In other words, it is the natural extension of pseudo-Riemannian geometry in which the squared line element ds2 is not restricted to be quadratic in the coordinate 1-forms dxi
I suddenly remember why, in 1982 or 83, I wrote to the editor of a well-respected journal, asking him to stop sending me papers on Finsler geometry for review. That statement you quote above is not merely misleading, it is utterly meaningless.
Just what is a line element ds? It is a linear differential 1-form; that is, at any point p, it is a linear map between the tangent space of a manifold at that point to the real number system. The "line element" ds2 is actually the tensor product of ds with itself, that is, it is a bilinear 2-form. When you integrate ds along any path between 2 points p and q, you get the distance along that path between them. It is meaningless to write something like ds2 = dx2 + dy + dz2, because, quite simply, you cannot (algebraically) add a 1-form and a 2-form.
Quote:
Finsler geometry extends Riemannian geometry by allowing the metric to be any positively homogeneous and strongly convex function on the tangent bundle, rather than restricting it to a quadratic form. I think we may be 'talking past one other' in terms of definitions.
In the context of Finsler geometry, the terms "metric" and "fundamental tensor" refer to different but related concepts.
Finsler Metric: In Finsler geometry, the metric is a function F(x,y) that defines the length of a tangent vector y at a point on the manifold x. This function F generalizes the notion of distance in a way that is not necessarily quadratic in the coordinate differentials.
Fundamental Tensor: The fundamental tensor in Finsler geometry is derived from the Finsler metric. It is defined as the Hessian of 1/2 F2 with respect to the coordinate differentials, and this tensor generalizes the Riemannian metric tensor and reduces to it in the special case where F is quadratic in the coordinates.
In the context of Riemannian geometry, the metric and the fundamental tensor often refer to the same mathematical object, the Riemannian metric tensor. This all-too-subtle difference b/w Riemannian and Finslerian maths is a bit of a trap I think. Indeed the comment :
Even though it is the 2-homogeneous Finsler Lagrangian that enters in most (but not all) formulas in Finsler geometry, it is useful to independently define a similar 1-homogeneous object, the Finsler metric.
supports that distinction. HTH.
We're not talking past each other at all. So far, my biggest objection has been to the mistake you made in trying to combine 1-forms and 2-forms algebraically in the same object. That simply cannot be done, as they operate on different spaces.
I have other objections to Finsler geometry as it is presented, mostly because of confusing or misleading mathematical statements. However, these are at a level which I think is far too advanced to be worth discussing here -- to do so would likely only add to the confusion. This is not a criticism of the geometry itself, but only of the way it is presented. But the bottom line is, as always, that whenever one stops scribbling equations on paper, and goes out to look at the real universe, the statements those scribblings make about the universe had better coincide with what we actually see.
I think the funny/weird metrics are exciting because one of them, or a group of them, may get around or even explain the 'paradox', for want of a better word, of the central singularity of black holes. That's where GR ought meet QM. I wonder what the Finsler et al geometries have to say about that. Watch this space I guess.
QM won't ever meet GR, because general relativity is non-renormalizable.
Well that's the question. Does that non-renormalizability hold when you go to a Finsler geometry?
hadron wrote:
Whenever you quantize a classical theory, you will find divergences (like infinite integrals) that seem to blow the whole process out of the water.
Again, do those integrals remain infinite when the metric is altered?
1. In any theory of gravity, there is no fixed background space: space itself is dynamic, and in fact, space itself is the thing that has to be quantized.
2. What is a Hamiltonian in any theory of gravity? In any theory in which the background space is fixed (including both quantum theory and Newtonian physics), a Hamiltonian is a time evolution operator. In any theory of gravity, time becomes a dynamical variable, so the notion of a Hamiltonian as a time evolution operator becomes a bit murky. Using the proper time of an object under observation doesn't work, because gravitational time dilation is an established fact.
Mike Hewson wrote:
hadron wrote:
However, the quantum world is not at all like the classical world, where a particle remains the same unless it interacts with something else. In the quantum world, any particle is constantly undergoing "virtual" interactions, for example a bare electron travelling through space is constantly emitting and reabsorbing "virtual" photons -- virtual in the sense that they don't exist in the classical world, whereas they do in the quantum world. Add up all the virtual interactions, and you get rid of all the infinities.
Agreed, you can reorder infinite sums (or more exactly the limit of reordered partial sums may change) to suppress divergences.
It's not a matter of merely reordering the sums. In gravitational perturbation theory, there are infinitely (countable or uncountable? I do not know) many independent parameters to be specified. It is thus impossible to fix all the parameters needed to achieve a successful renormalization. The paragraphs under the heading https://en.wikipedia.org/wiki/Quantum_gravity#Quantum_mechanics_and_general_relativity describe all of this better than I can.
Mike Hewson wrote:
hadron wrote:
As an extreme example, here is an electron absorbing a photon (left) with just one of a great many virtual interactions on the right. The circle in the right side represents a virtual electron-positron pair.
Where the background in these diagrams is spacetime with the usual special relativistic metric. Does the significance of these diagrams change with Finsler? I don't know .... mind you, the total tonnage of the list of things I don't know would sink fleets of battleships. ;-)
Be careful here; in flat Finsler space, is it the metric tensor or the fundamental tensor which has physical meaning? It had better be the latter, because special relativity is the best verified theory we have. There can be no doubt about its validity, I think, Or, is it possible that investigating Finsler space is an interesting thing to do, physical space really is described by a bilinear quadratic form, ie. a Riemannian metric, and that studying Finsler spaces is simply a fascinating mathematical exercise?
I think the funny/weird metrics are exciting because one of them, or a group of them, may get around or even explain the 'paradox', for want of a better word, of the central singularity of black holes. That's where GR ought meet QM. I wonder what the Finsler et al geometries have to say about that. Watch this space I guess.
Cheers, Mike.
I agree Mike and you and I could spend 24 hours reading and trying to think of our own theories on these subjects and all we have seen and heard and studied our mainly entire lives.
It started with me on black and white tv watching NASA when I was 5 years old and for years would dream about seeing planets at night.
Just wish we could discover a way to make us live another 200 years to actually see beyond what we dream of right now.
3am here and I was just reading about a local Apollo astronaut in his final flight at 90 years old just north of me.
Aww, it was Bill Anders! Doing what he loved to do probably. I remember that Earthrise scene like it was yesterday. Darn.
To this day I am awestruck by that photo -- to realize just how insignificant this planet is in the vast realm of space.
Alas, we may be on the verge of destroying it. Even if we stopped dumping CO2 into the atmosphere today, how long would it take for the planet to stop warming up, then cool down to pre-industrial levels?
And there is no Planet B.
Mike Hewson wrote:
As for evolution of thinking and theories I reckon astrophysics is the place to be.
Cheers, Mike.
Nonsense ;) The place to be is, and has been since 1982, in quantum reality and quantum philosophy. That was when Alain Aspect's test of Bell's theorem showed the world that classical reality isn't real at all.
We haven't really come very far since then, but I am sure we are on the verge of a paradigm shift in our thinking.
Well that is weird. Something
)
Well that is weird. Something pulsing every 54 minutes, with some period change also, consistent with a neutron star but might be something else like a white dwarf. Discovered by pure luck via the Australian Square Kilometre Array Pathfinder telescope and then followed up using MeerKAT. That is "the MeerKAT detections appear to arrive in phase with the ASKAP detections" or in other words it's a real signal from the stars. I'll bet that source object is, or was, in company and the current signal is the residue from interactions with other bodies.
A new category then, not a magnetar but a bludgetar. ;-)
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
Gentlemen, Thank you for
)
Gentlemen,
Thank you for giving me my morning head spin!
I struggled with my first course in University calculus. It discouraged me from taking any more...
A Proud member of the O.F.A. (Old Farts Association). Be well, do good work, and keep in touch.® (Garrison Keillor) I want some more patience. RIGHT NOW!
Mike Hewson wrote:I think the
)
QM won't ever meet GR, because general relativity is non-renormalizable.
Whenever you quantize a classical theory, you will find divergences (like infinite integrals) that seem to blow the whole process out of the water. However, the quantum world is not at all like the classical world, where a particle remains the same unless it interacts with something else. In the quantum world, any particle is constantly undergoing "virtual" interactions, for example a bare electron travelling through space is constantly emitting and reabsorbing "virtual" photons -- virtual in the sense that they don't exist in the classical world, whereas they do in the quantum world. Add up all the virtual interactions, and you get rid of all the infinities.
As an extreme example, here is an electron absorbing a photon (left) with just one of a great many virtual interactions on the right. The circle in the right side represents a virtual electron-positron pair.
In general relativity, that turns out to be impossible. In the 4-dimenstional universe we live in, quantum general relativity cannot be renormalized to any order, even in an empty universe.
hadron wrote:If you have
)
Sure, bottom of page one to top of page two in the Introduction of Heefer's Paper :
Finsler geometry is just pseudo-Riemannian geometry without the quadratic restriction. In other words, it is the natural extension of pseudo-Riemannian geometry in which the squared line element ds2 is not restricted to be quadratic in the coordinate 1-forms dxi
Finsler geometry extends Riemannian geometry by allowing the metric to be any positively homogeneous and strongly convex function on the tangent bundle, rather than restricting it to a quadratic form. I think we may be 'talking past one other' in terms of definitions.
In the context of Finsler geometry, the terms "metric" and "fundamental tensor" refer to different but related concepts.
Finsler Metric: In Finsler geometry, the metric is a function F(x,y) that defines the length of a tangent vector y at a point on the manifold x. This function F generalizes the notion of distance in a way that is not necessarily quadratic in the coordinate differentials.
Fundamental Tensor: The fundamental tensor in Finsler geometry is derived from the Finsler metric. It is defined as the Hessian of 1/2 F2 with respect to the coordinate differentials, and this tensor generalizes the Riemannian metric tensor and reduces to it in the special case where F is quadratic in the coordinates.
In the context of Riemannian geometry, the metric and the fundamental tensor often refer to the same mathematical object, the Riemannian metric tensor. This all-too-subtle difference b/w Riemannian and Finslerian maths is a bit of a trap I think. Indeed the comment :
Even though it is the 2-homogeneous Finsler Lagrangian that enters in most (but not all) formulas in Finsler geometry, it is useful to independently define a similar 1-homogeneous object, the Finsler metric.
supports that distinction. HTH. Please do advise us of your thoughts ....
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
hadron wrote: Mike Hewson
)
Well that's the question. Does that non-renormalizability hold when you go to a Finsler geometry?
Again, do those integrals remain infinite when the metric is altered?
Agreed, you can reorder infinite sums (or more exactly the limit of reordered partial sums may change) to suppress divergences.
Where the background in these diagrams is spacetime with the usual special relativistic metric. Does the significance of these diagrams change with Finsler? I don't know .... mind you, the total tonnage of the list of things I don't know would sink fleets of battleships. ;-)
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
I neglected to mention in a
)
I neglected to mention in a previous post that I always use the Einstein summation convention, whereby a duplicated index in a product implies that the sum over the range of that index is to be carried out. Thus, for example, in an n-dimensional space, in the right-hand side of the expression
V = vidxi
it is implied that this stands for the summation of that term as i runs over its full range from 1 to n.
I suddenly remember why, in 1982 or 83, I wrote to the editor of a well-respected journal, asking him to stop sending me papers on Finsler geometry for review. That statement you quote above is not merely misleading, it is utterly meaningless.
Just what is a line element ds? It is a linear differential 1-form; that is, at any point p, it is a linear map between the tangent space of a manifold at that point to the real number system. The "line element" ds2 is actually the tensor product of ds with itself, that is, it is a bilinear 2-form. When you integrate ds along any path between 2 points p and q, you get the distance along that path between them. It is meaningless to write something like ds2 = dx2 + dy + dz2, because, quite simply, you cannot (algebraically) add a 1-form and a 2-form.
We're not talking past each other at all. So far, my biggest objection has been to the mistake you made in trying to combine 1-forms and 2-forms algebraically in the same object. That simply cannot be done, as they operate on different spaces.
I have other objections to Finsler geometry as it is presented, mostly because of confusing or misleading mathematical statements. However, these are at a level which I think is far too advanced to be worth discussing here -- to do so would likely only add to the confusion. This is not a criticism of the geometry itself, but only of the way it is presented. But the bottom line is, as always, that whenever one stops scribbling equations on paper, and goes out to look at the real universe, the statements those scribblings make about the universe had better coincide with what we actually see.
Mike Hewson wrote: hadron
)
1. Yes; Finsler geometry is still classical
2. See 1.
See also https://en.wikipedia.org/wiki/Quantum_gravity#Nonrenormalizability_of_gravity which is short, but describes the problem fairly well.
Also problematic are:
1. In any theory of gravity, there is no fixed background space: space itself is dynamic, and in fact, space itself is the thing that has to be quantized.
2. What is a Hamiltonian in any theory of gravity? In any theory in which the background space is fixed (including both quantum theory and Newtonian physics), a Hamiltonian is a time evolution operator. In any theory of gravity, time becomes a dynamical variable, so the notion of a Hamiltonian as a time evolution operator becomes a bit murky. Using the proper time of an object under observation doesn't work, because gravitational time dilation is an established fact.
It's not a matter of merely reordering the sums. In gravitational perturbation theory, there are infinitely (countable or uncountable? I do not know) many independent parameters to be specified. It is thus impossible to fix all the parameters needed to achieve a successful renormalization. The paragraphs under the heading https://en.wikipedia.org/wiki/Quantum_gravity#Quantum_mechanics_and_general_relativity describe all of this better than I can.
Be careful here; in flat Finsler space, is it the metric tensor or the fundamental tensor which has physical meaning? It had better be the latter, because special relativity is the best verified theory we have. There can be no doubt about its validity, I think, Or, is it possible that investigating Finsler space is an interesting thing to do, physical space really is described by a bilinear quadratic form, ie. a Riemannian metric, and that studying Finsler spaces is simply a fascinating mathematical exercise?
Mike Hewson wrote: Hi Magic!
)
I agree Mike and you and I could spend 24 hours reading and trying to think of our own theories on these subjects and all we have seen and heard and studied our mainly entire lives.
It started with me on black and white tv watching NASA when I was 5 years old and for years would dream about seeing planets at night.
Just wish we could discover a way to make us live another 200 years to actually see beyond what we dream of right now.
3am here and I was just reading about a local Apollo astronaut in his final flight at 90 years old just north of me.
https://komonews.com/news/local/plane-crash-jones-island-san-juan-county-sheriffs-office-wdfw-us-coast-guard
Cheers
Aww, it was Bill Anders!
)
Aww, it was Bill Anders! Doing what he loved to do probably. I remember that Earthrise scene like it was yesterday. Darn.
As for evolution of thinking and theories I reckon astrophysics is the place to be.
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
Mike Hewson wrote:Aww, it
)
To this day I am awestruck by that photo -- to realize just how insignificant this planet is in the vast realm of space.
Alas, we may be on the verge of destroying it. Even if we stopped dumping CO2 into the atmosphere today, how long would it take for the planet to stop warming up, then cool down to pre-industrial levels?
And there is no Planet B.
Nonsense ;) The place to be is, and has been since 1982, in quantum reality and quantum philosophy. That was when Alain Aspect's test of Bell's theorem showed the world that classical reality isn't real at all.
We haven't really come very far since then, but I am sure we are on the verge of a paradigm shift in our thinking.