Using the force constant in equations

[marcus]

Well, honestly, nightcleaner, if you assume a creator you're going to get static from those who, like Laplace, have no need for that assumption.
...

some people are proud of their parentage
some of their name
some of the clothes they wear
some of belonging to a good country club
I can't help being proud of living in a culture, or at least marginally on the outskirts of a culture, in which Laplace said that thing to Napoleon.

has anyone seen the movie "Russian Ark" by the way?
the french aristocrat who goes with you through the Tsar winter palace
impressed me as a neat guy.

I found some stuff on web that suggests that he presented his celestial mechanics to napoleon (whom he had taught at the royal artillery college) around 1805 (the publication date for volume 4) and napoleon afterwards remarked that he had found no mention of god in the whole multivolume work. so we are talking about a bon mot that is just about 200 years back.

 

[marcus]

people give different versions the most common one in those i found being

"Sire, je n'avais pas besoin de cette hypothèse."

and some give the date 1799 when the first two volumes of the set were published, but it was pointed out that napoleon was in Egypt that year, so Laplace could not have presented the work then. and it is suggested that he presented it in 1805, when the fourth volume was published.

then, it is said, Napoleon remarked that in all the work on celestial mech he could find no mention of god, and Laplace replied
"Sire, i had no need of that hypothesis."

 

[nightcleaner]

"Have been enjoying your conversation with spicerack. Now I regret having butted in. It was better talk when it was just you and her"

Nonsense, Marcus, and it is I who must apologise for having stolen your wonderful thread here for my own personal vehicle.

I remember enjoying reading the Hawking book, but then there do seem to be some things which I don't remember very well at all, and more worrisome, some things that never happened that I seem to remember quite clearly. On the other hand, I do remember looking at the Bojowald papers you have linked, and am ashamed to say that I never understood a word. I do like the idea of an inverse big bang, and would never have figured out that it was in there. Thank you. And the AJL papers also?

Well the inversion thing comes out negative in the exponent. I could quibble that it is not an arithmetic expansion, but an exponential one? Again, the bounce is not really a bounce, is it? Not even an inflection, altho it looks like it would be from where we are standing. More like the geometric progression of the horizon than like hitting a wall.

Running along the hyperbolic x-axis the tanhx comes up smoothly from below approaching the origin, and the sine adds a little lift to counter the upward slope so we don't really have to work to climb it. Then the coshx swoops down to push us along and keep us from the steep slope to infinity that follows along the sinhx, making for easy peddling in the first quadrent. No bounce. Contrast this with the sinx and cosx, where it is all up and down hill. You have to have energy to cross that first ridge, and then you have to do it again and again. Luckily you can save some energy on the downslope to use on the way up the next hill. That saved, or recovered energy is the bounce part. It isn't required in the hyperbolic transition along x, which, I suppose, is the one that better fits the big bang as a horizon problem.

I have a horrible feeling we are all going to regret my trip to the bookstore to find "The First Three Minutes" by Stephan Hawking. And did you know, by synchronicity I swear, that our Paddy has also written a book called by the same title? I was amazed when I googled it that there were three pages of sponsored hits on the phrase. And, again, when on about page ten I realized that many of the hits, even maybe most of them, had to do with organized sports! I was flabberghasted.

Who would ever have thought that sports writers, coaches, inspirational speakers, and creationists would be so impressed with a book by an astrophysicist.

nc

 

[marcus]

people give different versions the most common one in those i found being

"Sire, je n'avais pas besoin de cette hypothèse."

and some give the date 1799 when the first two volumes of the set were published, but it was pointed out that napoleon was in Egypt that year, so Laplace could not have presented the work then. and it is suggested that he presented it in 1805, when the fourth volume was published.

then, it is said, Napoleon remarked that in all the work on celestial mech he could find no mention of god, and Laplace replied
"Sire, i had no need of that hypothesis."

How did we get started talking about Hawking in the same breath with Laplace?
I experience a slight embarrassment when hawking is mentioned because of the disproportion between his stature as a scientist and as a figure in the public imagination.

You mentioned the book called The First Three Minutes. Was that not written in 1977 by Weinberg?

You might think Weinberg's book is out of date. But it is not so SPECULATIVE as hawking's popular writings. So Weinberg's book may well be on many people's shelf long after Hawking popular books are in landfill.

That doesn't mean there is anything wrong with speculation, just that it gets old faster than more factual substantive stuff.

and goodness knows, in science and cosmology in particular, even the FACTUAL stuff gets old pretty fast.

it is impressive that Steven Weinberg could write a book in 1977 that peoplel still want to read. he is a pretty impressive guy actually.

 

[marcus]

this morning went out in the garden and stood a while in a patch of bright sunlight by the apple tree, which is blossoming now (I tickled it with a feather yesterday apprehensive that bees might not come) and today there was a large black and yellow going around the apple flowers so things are working OK in that regard

I was feeling the solar constant 6E-117 and getting the warmth all the way into my bones---indoors it was cold---and thinking about the 14 psi pressure in the garden where psi is my jargon for E-107 natural units of pressure.

there were dewdrops on the leaves and they would have evaporated except for the 14 psi pressure (I mean 14E-107 natural)
so although I can't feel the pressure I can see the evidence. A pretty woman walked by the garden gate outside on the lane. She would be freeze-dried without this pressure. The leaves and I would be freezedried. We are all mostly moisture and other volatile liquids.

So I realized that part of the time when I'm out in the garden am am being appreciative of some basic quantities----the power per unit area of the sunlight, the force per unit area of the atmosphere, the temperature (E-29 natural at that hour of the day). I can say that I am glad those quantities are what they are, really glad, and if there were something to thank, I would give thanks, but since there isn't, I dont.

 

[marcus]

I was reading an article yesterday by a Madrid physicist Enrique Alvarez who has been, at least in former years, a string theorist. Now he seems
to be getting more interested in non-string quantum gravity and right in the middle of his paper I saw him define the Planck length with an 8pi.

It has been that way in most of the papers I've read recently whether by Bojowald, Padmanabhan, Alvarez. they don't apologize or remark on it. they just define the natural units the way like in this thread, with
|hbar|=|c|=|8piG|=1, and don't bat an eyelash.

this is more the way Kea said to go, forget that the Planck units were ever defined differently with |hbar|=|c|=|G|=1, and still appear that way in many a conventional textbook and handbook.

 

[nightcleaner]

Marcus
I respect your view, of course, and wouldn't dream of tickling your philosophical makeup, but...

Why is it not ok to thank a rock?

Thanks for being...

nc

 

[marcus]

Marcus
I respect your view, of course, and wouldn't dream of tickling your philosophical makeup, but...

Why is it not ok to thank a rock?

Thanks for being...

nc

hey, good idea nightcleaner!
it just hadnt occurred to me.
you earn your keep around here

 

[nightcleaner]

Marcus, I don't know if i have ever asked you for your evaluation of quantum consciousness and the idea of a self-organizing computational automaton universe.

Of course i do feel grateful for the rocks or else what would we have to stand upon? Not to mention the trees, the flowers, even the feathers of the birds that waft the seeds into being. The sheer detail of the phenomena around us is overwhelming once you get started on it. And shouldn't one be grateful for the beauty of the orbitals?

 

[marcus]

Marcus, I don't know if i have ever asked you for your evaluation of quantum consciousness and the idea of a self-organizing computational automaton universe.
...

you had better not ask me
I am a curmudgeon and set in my beliefs along these lines. I do not wish to elaborate on this. However I have a poetical streak. Here is a poem about rocks (and a hummingbird) which I seem to recall posting at PF before but forget when

Hummingbird Pauses by the Trumpet Vine
by Mary Oliver

Who doesn't love
roses, and who
doesn't love the lilies
of the black ponds

floating like flocks
of tiny swans,
and of course the flaming
trumpet vine

where the hummingbird comes
like a small green angel, to soak
his dark tongue
in happiness--

and who doesn't want
to live with the brisk
motors of his heart singing
like a Schubert,

and his eyes
working and working
like those days of rapture,
by van Gogh, in Arles?

Look! for most of the world
is waiting
or remembering--
most of the world is time

when we're not here,
not born yet, or died--
a slow fire
under the Earth with all

our dumb wild blind cousins
who also
can't even remember anymore
their own happiness--

Look! and then we will be
like the pale cool
stones, that last almost
forever.

 

[marcus]

I found the mary oliver poem on the web at
http://forwardmomentum.com/newsletter/forwardmomentum_news.htm
but it is also about 9/10 of the way down this page
https://www.physicsforums.com/archive/topic/t-27156_Greatest_Poem.html

 

[nightcleaner]

Beautiful, chilling.

Marcus, do you think it would be a good exercise to translate some of the formulations from my new Cambridge Handbook into the terms used in this thread?

nc

 

[marcus]

do you think it would be a good exercise to translate some of the formulations from my new Cambridge Handbook into the terms used in this thread?

Let's start a new thread to examine the cambridge handbook of physics formulas.

I don't know the handbook. May I have the honor? I would like to see the table of contents. Unfortunately it would probably be a lot of work for you to copy the TOC and type it in here.

When you include Applied Physics, then physics is a huge topic (solid state, fluid mechanics, turbulence, acoustics, plasma physics, ye gods the list is endless)
I would really like to see what the menu is. maybe I can find it on Amazon where i can take a look at the TOC

[edit] richard, I found out all I could easily find out about the Cambridge Handbook. I think now that it is something for you to learn from, but that we probably wouldn't be able to use as grist for our mill in this thread. but you can try me out on some that you think are promising.
as a rule, if you see a formula with hbar, or c in it, or the electron charge e, or Boltzmann k, then calculation with that formula may be facilitated working in natural units[/edit]

 

[nightcleaner]

Ok Marcus.

Just like me to learn to build a paper airplane and then suggest we immediately start colonies on Mars. Oh well. I'll go play in the new thread. Thanks,
nc

 

[marcus]

Thing One about LQG

...Just like me to learn to build a paper airplane and then suggest we immediately start colonies on Mars...

We are alike in that respect, except that I would prefer the colonies to be on the Jovian moon Callisto instead of Mars.

I was thinking lately about what would be Thing One to say about LQG, in a congenial company of nonspecialists.

It occurred to me that Thing Zero would be a quote from Einstein (Grundlage, 1916) that appears in Rovelli section 2.3.2

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

General covariance is nowadays often called diffeomorphism invariance. Diffeomorphisms are smooth mooshings of a manifold, where smooth means infinitely differentiable. Our main goal should be to outwit the jargon and find the simple idea. Jargon is the dragon guarding the gate.

a manifold is a continuum (selfAdjoint says he likes that word better). a manifold is equipped with coordinate charts. If transposing from one chart to another is a smooth mapping then we say the manifold is smooth.

Every mathematical introduction to LQG begins the same with a compact smooth manifold M. The author will then usually say that for convenience we think of M as looking like the 3d sphere. But, it is understood, without the sphere's geometry. A limp shapeless "bag" of a 3d sphere.

So M is S³ but deprived of its native S³ metric geometry.

We have to begin with M as our idea of space because Einstein began his
"allgemeine Relativitaetstheorie" with such a manifold M, without a metric, and required above all that the system for finding a metric in harmony with substance should be unaltered by mooshing the manifold.

The two primal features whether of the classical or quantum theory are background independence (which means NO PRIOR METRIC) and diffeomorphism invariance (which means MOOSHING DOESNT MATTER)

And well somewhere in the first paragraph they toss in a 4d manifold you can think of as R x M, and you can assume that the same applies to it:

1. no prior metric
2. mooshing mox nix (Der Musherei macht nichts bei der Mannigfaltigkeit!)

now something cruelly unexpected is going to happen

an idea which kindly old father Newton gave us of a space that is independent of material substance, this space, as a separate entity existing of its own accord, is going to go poof

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

 

[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.