Using the force constant in equations

[selfAdjoint]

You've got it! By God you've got it! Especially with the Alice reference snapper at the end! Can we do anything with what John Baez told us about E and F? By the way have you looked at putting A. Rivero's h from gravity paper into your natural units? It would seem a natural for that. (Sorry).

 

[marcus]

Thing Zero would be a quote from Einstein (Grundlage, 1916) ...

"The requirement of general covariance takes away from space and time the last remnant of physical objectivity."

I think the most puzzling thing about LQG is actually inherent already in Gen Rel that went before.

in Gen Rel you set up a machine to solve for the geometry ( the "metric" or measuring function defined on the manifold) that is in harmony with whatever matter or substance is supposed to be put in the manifold

and then you EQUATE two different metrics if one (along with its complement of matter) can be mooshed into the other (along with its matter)

so in the end instead of individual metrics you have families or fraternities or "equivalence classes" of metrics

and the gravitational field is an equivalence class of metrics, which are solutions to the einstein equation, in other words you LUMP TOGETHER all the spacetime geometries which the setup cranks out which are the same under mooshing.

At this stage in the proceedings, the points in the original manifold begin to seem somewhat unimportant----indeed Einstein quote up their says they are physically MEANINGLESS.

It is as if there is no space or spacetime, just a web of relationships between events which the manifold and its coordinates once-upon-a-time served as a convenience to describe. they were only there provisionally, so to speak

IMO this web of relationships is very abstract. You cannot, AFAIK, draw it.

Rovelli says it is rather like how Westerners as far back as Aristotle treated space (i.e. relationally) before Newton gave us the idea of an unconditional space that could exist of its own accord without any matter and material events. As Rovelli tells it, Leibniz preferred the relational, non-absolute, idea of space that is contingent on having stuff around in it, for it to be (for it is only relationships between stuff). Apparently Leibniz and Newton fought long and hard over this.

Geometry is the gravitational field, defined as an equivalence class of metrics on NO MANIFOLD IN PARTICULAR.

one begins with a smooth compact manifold and then one abstracts away from it.

the conceptual difficulty of LQG is that it gets away from Newton's idea of an independently existing space and that comes from a fundamental conceptual difficulty already in Gen Rel. it would be tempting to ignore Gen Rel but we cant. We do not know how to ignore it because Gen Rel has been successful and so far impossible to replace with an model based on unconditionally existing Newton-style space.

BTW there is a tough logical argument called the HOLE ARGUMENT which etera mentioned in his most recent post. I would not be trying to discuss this if etera had not mentioned it just now.
The Hole Argument which may appear to be a frightful hairy hard-to-understand thing, is explained in Rovelli section 2.2.5. He uses a picture.
The Hole Argument was used by Einstein to show that there is a logical "hole" in the idea of independently-existing 3d or 4d graph paper.

 

[marcus]

You've got it! By God you've got it! Especially with the Alice reference snapper at the end! Can we do anything with what John Baez told us about E and F? By the way have you looked at putting A. Rivero's h from gravity paper into your natural units? It would seem a natural for that. (Sorry).

selfAdjoint, I just saw your post, thank you so much for the note of approval and confirmation. I am very warmed and also encouraged

hope what i just posted (#317) does not re-muddle what had momentarily become clear. sometimes continuing to worry about something is counterproductive
I am still editing #317 to make it clearer, and often think the best would be to just have #315 and leave it at that without further explanation. Perhaps the extra explanation only raises doubts or reveals those in my own mind.

 

[selfAdjoint]

selfAdjoint, I just saw your post, thank you so much for the note of approval and confirmation. I am very warmed and also encouraged

hope what i just posted (#317) does not re-muddle what had momentarily become clear. sometimes continuing to worry about something is counterproductive
I am still editing #317 to make it clearer, and often think the best would be to just have #315 and leave it at that without further explanation. Perhaps the extra explanation only raises doubts or reveals those in my own mind.

I believe very strongly that anybody who wants to get beyond the rubber sheet concept of GR has to come to grips with, and internalize, the hole argument. It took Einstein several years to think his way around it and come up with the idea of equivalence classes. Note that the geometry survives when the metric doesn't because any non-zero tensor, including the metric tensor, is mapped into something else by a diffeomorphism, but a zero tensor is invariant - zero in spite of all mooshing. So a tensor equation A = B, which is equivalent to (A -B) = 0 is unchanged too, the buzzword as you know is covariant - the two sides of the equation vary together and stay equal. So all laws of physics, quoth Einstein, must be tensor equations, because physics can never be dependent on which moosh we did last.

 

[Kea]

the lady vanishes

all that will be left is the Geometry that was on the space...
as the smile on the cat's face remains after the cat vanishes

Dear Marcus!

Hooray! And don't forget the mushrooms...and what else was it Alice had to eat?

And the bone and the dog:

"...Take a bone from a dog: what remains?"
Alice considered. "The bone wouldn't remain, of course, if I took it - and the dog wouldn't remain: it would come to bite me - and I'm sure I shouldn't remain!"
"Then you think nothing would remain?" said the Red Queen.
"I think that's the answer."
"Wrong, as usual," said the Red Queen: "the dog's temper would remain."


All the best
Kea

 

[marcus]

Dear Marcus!

Hooray! And don't forget the mushrooms...and what else was it Alice had to eat?

All the best
Kea

you are sweet, Kea
truly
the is mutual

 

[marcus]

what is thing one about quantum gravity

when I ask myself what is thing one, I think of the end of the author's preface in Rovelli "QG"---a preface is where the author's personal feelings are not out of place and where he can talk straight about how he sees the goals.
here is final paragraph of Rovelli preface:

"I have written this book thinking of a researcher interested in working in quantum gravity, but also of a good Ph.D. student or an open-minded scholar curious about this extraordinary open problem.

I have found the journey towards general relativistic quantum physics, towards quantum spacetime, a fascinating adventure.

I hope the reader will see the beauty I see, and that he or she will be able of completing the journey. The landscape is magic, the trip is far from being over."

I have broken the final paragraph up to help me focus on the second sentence.

there is a popular catchphrase "theory of everything". Rovelli does not use that phrase. He says the goal or destination of this collective journey is a
general relativistic quantum physics
that is because the whole of today's quantum physics---QFT, all the particles and fields, the standard model---is now built on the spacetime of special relativity, not general relativity. I take for granted that this cannot last because the spacetime of special relativity is not right (according to special relativity, space cannot expand or bend lightrays or form black holes---but it does)

as Galileo always liked to say "Eh! da space-a-time is-a curve! E pur si MUOVE and she bend a little too!"

So quantum physics built on the rigid foursquare Minkowski spacetime of special relativity cannot last. It has to be rebuilt on a new spacetime foundation and then it will be what he calls general relativistic quantum physics.

To provide the right spacetime foundation, Rovelli sees it necessary to construct what he calls, in his goal statement, a quantum spacetime.
That is a quantum version of the dynamic spacetime geometry of general relativity.

Because for Rovelli, as a relativist, gravity IS spacetime geometry, and the gravitational field is how geometry is described ( its continual give and take with the matter adrift in it)...because geometry is gravity, we can call this research goal "Quantum Gravity". But that is essentially a shorthand tag on the real goal which is to provide a dynamic quantum geometry and describe its interaction with matter so that a general relativistic quantum physics can be built on it.

 

[marcus]

I believe very strongly that anybody who wants to get beyond the rubber sheet concept of GR has to come to grips with, and internalize, the hole argument. It took Einstein several years to think his way around it and come up with the idea of equivalence classes. Note that the geometry survives when the metric doesn't because any non-zero tensor, including the metric tensor, is mapped into something else by a diffeomorphism, but a zero tensor is invariant - zero in spite of all mooshing. So a tensor equation A = B, which is equivalent to (A -B) = 0 is unchanged too, the buzzword as you know is covariant - the two sides of the equation vary together and stay equal. So all laws of physics, quoth Einstein, must be tensor equations, because physics can never be dependent on which moosh we did last.

I have bolded the first sentence of sA post as a reminder.
In Rovelli's book the figures for the hole argument are Figures 2.2, 2.3, and 2.4, on pages 65, 67, 69.
The section is 2.2.5 "General Covariance".
We need to go thru this and maybe get some alternative words and mental imagery to go along with it. But the best would be to have some online material like what is on "QG" pages 65-69. If you are using the draft Rovelli textbook then probably the pages will be different but the section will still be 2.2.5.

 

[marcus]

space, as a point set, goes away in General Relativity, and likewise LQG.
maybe mentioning this will amuse Kea
in either GR or LQG you start out with a smooth manifold that represents the idea of space----so, a continuum, a point set (with some add'l str'ctr)

the manifold helps you say what background independence (no prior metric) and diffeomorphism invariance mean, because at least you have diffeomorphisms!, and it helps with definitions

but then eventually you factor out the diffeomorphisms and the particular manifold, with its points, goes away. All that is left is the geometry or the gravitational field. (the relationships among the things you can measure about the disposition of the matter and spatial configuration of events etc.)

I think, in the end there is no point set representing space anymore

In this case the point set is obviated not by categories but simply by taking equivalence classes----you abstract the geometry making it independent of any particular point set or manifold implementation. this is a commonplace move mathematicians have used for ages to get rid of things that were annoyingly concrete or obnoxiously definite.

 

[marcus]

Richard asked about the appetites of black holes
If one has a black hole of mass M then how much should one feed it
so that it doesn't waste away but doesn't gain weight either.

what is the proper diet, to keep steady weight, for a black hole?


this is a good practical question which one knows from experience with pets, like a dog or cat that one wants to feed the right amount for them.

In natural units the temperature of a black hole is 1/M
and the power radiated per unit area of surface is therefore
(pi^2/60) (1/M)^4
by the StefanBoltzmann fourth power radiation law
and the surface area is (1/4pi)M^2
so the total power radiated by the black hole is

(pi/240)(1/M)^2

all I did was multiply the surface area by the power per unit area.
so that is the energy lost per unit time, by the hole.

Now remember that one natural energy unit is E5 food Calories, and that E50 time units is a year. So let's imagine a black hole with about the same food needs as a dog-----1000 Calories a day. the black hole wants E-2 natural energy units a day and 365E-2 energy units in a year. So it's food requirement is 3.65E-50. Let's solve for the mass M.

(pi/240)(1/M)^2 = 3.65E-50

(1/M)^2 = (240/pi) 3.65E-50 = 279E-50

1/M = 16.7 E-25

hmmm, I seem to have solved for the reciprocal of the mass instead of the mass, so i have to invert

M = 6E23 natural mass units

remember E8 is about a pound so one way of thinking of the hole is
6E15 pounds, that is 6 quadrillion pounds.

another way to imagine it is by its size, the radius in natural units is (1/4pi)M = (1/4pi) 6E23

radius = 0.48E23 natural length units.

for me that is almost incomprehensibly small---on the order of a tenbillionth of the width of my hand. hard to picture.

but if I had a black hole of a size I can visualize it would be much more massive and would have a very small appetite----then the appetite would be the hard thing to imagine

 

[nightcleaner]

Thank you Marcus

Little holes are hungry and if they don't get fed they evaporate.

Big holes are not very hungry. I would think there must be some average rate of infall per surface area of a hole. Can we calculate a size at which a black hole's infall rate equals its evaporation rate? This would be a threshold of stability for black holes.

Thanks,

Richard

 

[marcus]

Thank you Marcus

Little holes are hungry and if they don't get fed they evaporate.

Big holes are not very hungry. I would think there must be some average rate of infall per surface area of a hole. Can we calculate a size at which a black hole's infall rate equals its evaporation rate? This would be a threshold of stability for black holes.

yes, there are specially priviledged holes that live in swell rich surroundings where there is always lots to fall in, but out in generic middleamerica empty space most of what there is to fall into a hole is the ubiquitous CMB which is always shining on every square inch of a holes surface

this gives a "threshhold of stability" as you wished and it is real easy to calculate
the temperature of the CMB = E-31

temperature of a hole with mass M equals 1/M

the mass you need for stability is where the hole's temperature equals the surrounding space, so they are in balance (they trade glow back and forth and it is a fair exchange because they are equally radiant)

so you just have to solve
1/M = E-31

M = E31

the mass of the Earth is about E33, so this "stable" hole would be about a hundredth the mass of the earth

I really don't see why teachers don't show the kids natural units. it is so easy to use them. (actually the CMB temperature is 0.96E-31
so I was 4 percent off calling it a flat E-31, so M should be 4 percent bigger, so it's 1.04 E31, but who cares the main thing is to get fairly close order of magnitude answers without pain)

 

[selfAdjoint]

generic middleamerica empty space

Hey! Not nice from generic angeleno madhouse!

 

[marcus]

Hey! Not nice from generic angeleno madhouse!

I try to overlook the existence of Los Angeles, and would consider it a lapse in good taste to mention that disquieting sprawl myself.
ordinary middleamerica folks can be all over the map geographically
and I use the term with tolerant affection, not scorn!

to clarify the confusing way i phrased it, what I meant was your average generic cross-section empty space which is our universe's heartland and knock we not.

 

[marcus]

moving finger has posted a challenge in Astronomy forum
https://www.physicsforums.com/showthread.php?t=70578
I think it is a false trichotomy, but don't have much urge to argue.

Here's a quick derivation of the formula for the evaporation time of a black hole with mass M


temperature = 1/M
power per unit area by Stefan-Boltzmann = (pi^2/60) (1/M)^4
area = M^2/4pi
multiplying the area by the power output per unit area we get the
total radiant power (luminosity) = (pi/240) (1/M)^2 = dM/dt

therefore dt/dM = (240/pi) M^2

we just need to integrate dm with the mass running from M down to 0.

integral from 0 to M of dt/dm
integral from 0 to M of (240/pi)m^2
(80/pi)M^3

and that equals the usual formula, or the barebones version of it which you get if you set hbar and c and k and the rationalized grav. const. 8piG = 1

so BH evaporation time is accessible to freshman calculus

 

[nightcleaner]

Hi Marcus

Just thought I'd point out that Cambridge Handbook of Physics Formulas, p 183, uses solar mass in the BH evaporation time formula. I'll try to transcribe in into LaTex here.

$$\tau_e = \frac{M^3}{(M_@)^3}x10^66$$ years

Well that's pretty good, except that it is 10^66 , and here the ampersand is to represent a circle with a dot in it (which means, in CHOP, one solar mass), and the equivalence sign here is just a wavy line in CHOP, to indicate "approximately equal".

"one solar mass" is also used in the CHOP formulae for Schwartzchild radius, Chandrasekhar limit, and black hole temperature.

I guess they follow this convention because a solar mass is a convenient unit when talking about black holes as cosmological objects.

Solar mass in Planck units? One solar mass in Wiki is 1.9891 x 10^30 kg

I have Planck mass as .434 x 10^-8 kg

I get 4.58x10^38 Planck units for solar mass.


nc

 

[marcus]

Wonderful! you hit Cambridge paydirt, or at least an isolated nugget. I will look at it and compare.

Hi Marcus

Just thought I'd point out that Cambridge Handbook of Physics Formulas, p 183, uses solar mass in the BH evaporation time formula. I'll try to transcribe in into LaTex here.

$$\tau_e \sim \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}$$

Well that's pretty good, except that it is 10^66 , and here the ampersand is to represent a circle with a dot in it (which means, in CHOP, one solar mass), and the equivalence sign here is just a wavy line in CHOP, to indicate "approximately equal".

"one solar mass" is also used in the CHOP formulae for Schwartzchild radius, Chandrasekhar limit, and black hole temperature.

I guess they follow this convention because a solar mass is a convenient unit when talking about black holes as cosmological objects.

Solar mass in Planck units? One solar mass in Wiki is 1.9891 x 10^30 kg

nc

EDIT: put in \sim or \approx
for one squiggle or two squiggles

 

[marcus]

you might want to look at how I edited your LaTex formula


$$\tau_e \sim \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}$$

$$\tau_e \approx \frac{M^3}{(M_\odot)^3} \times 10^{66} \text{ years}$$

about their formula, it is just approximate, but looks very handy and useful.

we know that the evaporation time varies as the cube of the mass (that is how our formula went too)

so you can pick a convenient mass like the solar, and calculate the evap. time just for THAT, using some more basic messy derivation.

then suppose it comes out approximately E66 years (which it probably does)

from that point onwards you can use their simple formula which uses the solar mass as a point of reference.

if some hole is TWICE the solar mass, well it goes by cubes so the evaporation time would be EIGHT TIMES AS LONG and we know that the evap time for the solar mass is (approximately) E66 years, so it would be 8E66 years for a two solar mass hole

to verify the Cambridge formula, all I really need to do is check that it is (approx.) correct for one solar mass. we could do that, maybe i will

 

[marcus]

I get 4.58x10^38 Planck units for solar mass.

Yes! let's cube that and multiply by 80/pi

that will do it

when I cube 4.58 I get 96
so cubing 4.58E38 gives 96E114

multiplying 96 by 80/pi and E114 gives 2.44 E117

This is 2.1 X 10⁶⁷ years

But they say the evaporation time for a solar mass hole is a flat 10⁶⁶ years. I get a factor of 21 larger.

Well! their formula is off some!
We should write to them and point this out, maybe they can correct it in the next edition (or maybe they do not care and only want something that is correct to an an order of magnitude for something like this)

Also I will double check my formula to make sure IT is not off by a factor of two.

this is embarrassing. I can't find any mistake in my formula or arithmetic. Also I checked the Wiki article and they agree more or less with me, rather than Cambridge. Wiki gives some formulas in "common units" permitting one to estimate evaporation times. and they give an approximate time for the sun which is E67 years (rather than what Cambridge says E66).

 

[nightcleaner]

Hi Marcus

Yes I saw how you corrected my LaTex, thanks. Odot and curley brackets. Who would have thought. But I see you didn't put in the little squiggle for "Approximately equal."

nc

 

[marcus]

But I see you didn't put in the little squiggle for "Approximately equal."

I forgot that, earlier, but just saw your post and went back and put in a squiggle or a wavy equals sign.

I will try to get a link to some pages on LaTex that has these extra symbols. It is at some page of the PF thread on LaTex which however is now so long that it is laborious to look through it.

Oh good! The very first post of that thread has the links we need:
https://www.physicsforums.com/showthread.php?t=8997

here are some links given in that thread, I quote from Warren's post.

"A pdf file of the most useful LaTeX commands, symbols, and constructs is provided here:"

https://www.physicsforums.com/misc/howtolatex.pdf

"More symbol reference:"

http://amath.colorado.edu/documentation/LaTeX/Symbols.pdf

 

[nightcleaner]

Thanks Marcus this is great. I'll add these to my favorites list.

R.

 

[marcus]

Nightcleaner seems still to be at his cabin and we hear from him only sporadically. Maybe it is a question of batteries. Or on the other hand maybe he has been overwhelmed by the silence and cannot think of anything to say.

I have gotten quite comfortable with using units based on the force constant. A recent paper by Padmanabhan called them "rationalized Planck units", which may be a good designation. At any rate they now seem quite natural.

Out in the garden the sun is so bright I've taken to wearing a broad-brimmed straw hat. It is cold in the house and tempting to go out and let the sun warm my bones.

The raven nesting in the palm tree, or on lookout atop an even taller redwood, always calls three times
the pitch (which i determined by rushing indoors to compare it with keys on the piano) is the D beside middle C.
Accordingly the raven's frequency is (1/2)E-40 natural
or it might be D#, which would be just 6 percent higher----0.53E-40.

In natural terms the sunlight power per unit area is 6E-117. I finally got used to that. Think of it whenever, outdoors on a clear day, I see and feel the light.

have come to appreciate pressure 14E-107 which I can't feel but know is there. Know it's there because the fishpond would otherwise be simultaneously freezing and boiling, and we would have freeze-dried goldfish. That pressure of E-107 is nearly "pound per inch", so I know to what level, in "rationalized Planck" terms, I just inflated the tires.

Everything seems in order here on the immediate grounds, though zealots destroy the nation's moral standing and economy, still the sky is beautiful and all is well.

Not having heard from Richard the NC, I will suppose him still out in the woods, alone in his cabin, eating freeze-dried goldfish, since they are much easier to carry when you have to pack your food in.