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Dec 20, 1999, 3:00:00 AM12/20/99

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I am relatively new to this group and struggling. I can't equal Oz in

questionts but I am trying.

questionts but I am trying.

What is the 8PI doing in Einstein's equation? What does it mean and how

did he derive it?

Jerry Freedman,Jr

--

Creation took 6 days because God didn't

have an installed base

Sent via Deja.com http://www.deja.com/

Before you buy.

Dec 20, 1999, 3:00:00 AM12/20/99

to

In article <83itm1$a6q$1...@nnrp1.deja.com>,

Jerry Freedman Jr <free...@netscout.com> wrote:

Jerry Freedman Jr <free...@netscout.com> wrote:

>What is the 8 PI doing in Einstein's equation?

It's just sitting there to make sure that in the Newtonian limit,

general relativity yields the inverse square force law for gravitational

attraction with the *usual constant of proportionality* - namely Newton's

gravitational constant G. If you didn't put an 8 pi there in Einstein's

equations, you'd get

F = - G m_1 m_2 / 8 pi r^2

in the Newtonian limit.

It's a lot like how some people put a 4 pi in Maxwell's equations.

Dec 20, 1999, 3:00:00 AM12/20/99

to

>What is the 8PI doing in Einstein's equation? What does it mean and how

>did he derive it?

It comes from the requirement that general relativity agree with

Newtonian gravity in the appropriate limit. Here's a partial

explanation. Note that I have no idea what your background is, so I

may say things that are insultingly obvious to you or things that are

ridiculously obscure -- or, most likely, some of each! Please don't

be insulted at the former, and go ahead and ask questions about the

latter.

One way to express Newton's law of gravity is

div g = 4 pi G rho.

Here g is the local acceleration due to gravity and rho is the

density. div is the divergence:

div g = dg_x / dx + dg_y / dy + dg_z / dz.

We want to compare this to the situation in general relativity.

It turns out that, if the curvature of spacetime is very small (weak

gravitational fields), and if some other assumptions hold (essentially

that time components of various things are much larger than

space components), you can calculate the accelerations of tiny

"test particles" floating around in spacetime to get an expression

for g, the acceleration due to gravity.

Note that you wouldn't dare *think* of g as being an acceleration --

in general relativity, gravity's not a force and doesn't cause an

acceleration! Rather, g is a measure of geodesic deviation -- that

is, the degree to which initially parallel paths of particles diverge

or converge. To nearby particles in nearly-flat spacetime, this looks

just like ordinary acceleration.

If you then calculate the divergence of g, you get

div g = (1/2) G_00,

where G_00 is one component of the Einstein tensor (the thing

on the left side of the Einstein equation).

Well, if you set the two expressions for div g equal to each other,

you get

(1/2) G_00 = 4 pi G rho.

Now rho is just T_00, a component of the stress-energy tensor. So the

above equation says G_00 = 8 pi G T_00. If you've already guessed in

some way the general form of the Einstein equation, i.e., that G and T

are proportional to each other, then you can conclude right away

that G = 8 pi T.

Of course, all I've really done here is to turn your one equation

into two. Instead of asking where the 8 pi in Einstein's equation

comes from, we now have to ask where the 4 pi in the Newtonian

equation (div g = 4 pi G rho) comes from, *and* where the 1/2

in div g = (1/2) G_00 comes from.

The 4 pi does have a simple explanation -- it's the same 4 pi as in

the expression for the surface area of a sphere. If you want

someone to expand on that, go ahead and ask and either I or

someone else will. I don't think I have a simple explanation for

the 1/2 in the other expression, which goes to show that I don't

understand that part of the calculation as well. Maybe someone

else can take care of that part.

-Ted

Dec 22, 1999, 3:00:00 AM12/22/99

to

Is there a reason not to set 8piG=1 in a Planck-like

unit system?

unit system?

Keith Ramsay

Dec 23, 1999, 3:00:00 AM12/23/99

to

Jacques Distler <dis...@golem.ph.utexas.edu> wrote:

>Toby Bartels <to...@ugcs.caltech.edu> wrote:

>>What is the 8 pi doing in Newton's law of gravitation?

>>That has a real geometrical answer, although I'm not sure exactly what it

>>is.

>Sure you are.

>4pi r^2 is the surface area of a 2-sphere surrounding the mass M. It's

>the same 4pi that appears in Gauss's Law in E&M. There, too, you have a

>choice of whether to put the 4pi in the Maxwell equation

>(div.E=4pi rho) or in Coulomb's law.

O, I understand the 4 pi well enough.

However, I don't understand the 1/2.

-- Toby

to...@ugcs.caltech.edu

Dec 24, 1999, 3:00:00 AM12/24/99

to

In article <83shu8$g...@gap.cco.caltech.edu>,

Toby Bartels <to...@ugcs.caltech.edu> wrote:

Toby Bartels <to...@ugcs.caltech.edu> wrote:

>Keith Ramsay <kra...@aol.commangled> wrote:

>>Is there a reason not to set 8piG=1 in a Planck-like

>>unit system?

>The only reason I can think is that it's confusing,

>since the most common approach by far is to set G = 1.

Right. Personally I think it's nicer to set 8 pi G = 1.

First of all, it's nice to write Einstein's equation as

simply

G = T

without the yucky 8 pi G getting in the way - or the added

confusion between Newton's constant G and the G in the above

equation, which stands for the Einstein tensor.

Secondly, in loop quantum gravity, it seems that the more useful

"Planck units" are those where 8 pi G = c = hbar = 1, not

the usual G = c = hbar = 1. To understand why, note first that

in the usual Planck units, the Planck length is

sqrt(G hbar / c^3)

When you see a square root, it's often a hint that some simpler

idea without a square root is lurking around the corner! This

suggests that perhaps more fundamental than the Planck length is

the "Planck area"

G hbar / c^3

And, lo and behold: in loop quantum gravity, area turns out to be

more fundamental than length! Spin network edges give area to

surfaces they poke through, and area is quantized. A spin network

edge labelled by the spin j gives an area equal to sqrt(j(j+1))

times

8 pi G hbar / c^3

to any surface it pokes through. Note the factor of 8 pi! This

comes directly from the 8 pi in the usual way of writing Einstein's

equation. So if we were smart, we would use modified Planck units

where 8 pi G = hbar = c = 1. Then the Planck area would become

8 pi G hbar / c^3 = 1

and everything would be nice and neat.

(Actually, here I'm ignoring an extra complication called the Immirzi

parameter... if we ever understand that, the story may change a bit.)

Note that this isn't the first time we've had to muck with fundamental

constants by throwing in geometrical factors that involve pi. Planck

screwed up and invented a constant h that was too big by a factor of

2 pi, forcing someone to invent hbar = h / 2 pi. I think the problem

was that Planck was thinking about wavelengths instead of frequencies,

which makes the circumference of the unit circle show up in a lot of

basic formulas.

Similarly, some systems of units in electromagnetism produce yucky

factors of 4 pi in Maxwell's equations - basically because this is

the area of the unit sphere. The modern attitude is that we shouldn't

stick geometrical factors in our fundamental laws of physics; instead,

we should let them appear on their own when we solve problems using

these laws. The 4 pi shows up when you derive the inverse square law

from Maxwell's equations, since the electric field spreads out in a

spherically symmetric way.

This time, the problem is the irritating factor of 8 pi in Einstein's

equation. Why did we ever screw up and feel the need to put *that* in

there? The factor of 4 pi in here is again due to the area of the unit

sphere. But what about the factor of 2? Well, as Ted Bunn explained,

it comes from the fact that the component G_{00} of the Einstein tensor

is not quite the same as the Laplacian of g_{00} - there's a factor of 2

relating them. There is probably some deep way of understanding *why*

this factor of 2 obtrudes, but I haven't figured it out yet.

It's tempting to define "Gbar" to be 8 pi G, even though the bar in

hbar referred to the fraction h / 2 pi, and we aren't dividing G by

8 pi here - we're multiplying!

Which reminds me of a puzzle... when did people start using a slash

or bar to stand for division? It's such an ingrained habit now that

it seems almost *obvious* that we should use this kind of symbol for

division... yet it wasn't always so.

Dec 29, 1999, 3:00:00 AM12/29/99

to

In article <1e3hour.165...@de-ster.demon.nl>, J. J. Lodder

<nos...@de-ster.demon.nl> writes

>John Baez <ba...@galaxy.ucr.edu> wrote:

>

>> Right. Personally I think it's nicer to set 8 pi G = 1.

>>

>> First of all, it's nice to write Einstein's equation as

>> simply

>>

>> G = T

>>

>> without the yucky 8 pi G getting in the way - or the added

>> confusion between Newton's constant G and the G in the above

>> equation, which stands for the Einstein tensor.

>

>Perhaps, but -please- don't.

>There is a consensus and a tradition that the 8\pi

>should be put in Einstein's equation, not in Newton's law.

>

>Starting to do it the other way round will inevitably split physicists

>into two camps, some folowing the old convention, some the new.

>One (perhaps not optimal) convention, followed by almost everybody,

>is far to be preferred to a situation with two different conventions,

>with the inevitable confusions.

<nos...@de-ster.demon.nl> writes

>John Baez <ba...@galaxy.ucr.edu> wrote:

>

>> Right. Personally I think it's nicer to set 8 pi G = 1.

>>

>> First of all, it's nice to write Einstein's equation as

>> simply

>>

>> G = T

>>

>> without the yucky 8 pi G getting in the way - or the added

>> confusion between Newton's constant G and the G in the above

>> equation, which stands for the Einstein tensor.

>

>There is a consensus and a tradition that the 8\pi

>should be put in Einstein's equation, not in Newton's law.

>

>Starting to do it the other way round will inevitably split physicists

>into two camps, some folowing the old convention, some the new.

>One (perhaps not optimal) convention, followed by almost everybody,

>is far to be preferred to a situation with two different conventions,

>with the inevitable confusions.

Too late. John already has me convinced, and I shall follow his

convention in future. Including 8pi indicates that there is some deeper

argument from which it results, which creates quite enough confusion

with only the standard convention.

>

>A second good reason for not changing is that the -literally correct-

>answer to the question posed in the subject is without any doubt:

>"Einstein did".

>

That is the way for science to stagnate, and it is an approach followed

far too often. Everyone working in and teaching maths and physics should

look for every simplification they can find.

>Even if somebody were to write an influential textbook,

>with G = T in it,

>the majority would (I believe) not desert Einstein.

>

It depends how many others like me want to write G = T but don't have

the guts until someone like John says so.

--

Regards

Charles Francis

cha...@clef.demon.co.uk

Dec 30, 1999, 3:00:00 AM12/30/99

to

J. J. Lodder wrote:

> Charles Francis <cha...@clef.demon.co.uk> wrote:

> snip G = 8pi G

> > It depends how many others like me want to write G = T but don't have

> > the guts until someone like John says so.

> Entropy always increases. :-(

>

> The present situation,

> (although perhaps marginally less than ideal),

> with almost everybody adhering to the -same- convention

> is far to be preferred to a situation with two competing conventions.

> Charles Francis <cha...@clef.demon.co.uk> wrote:

> snip G = 8pi G

> > It depends how many others like me want to write G = T but don't have

> > the guts until someone like John says so.

>

> The present situation,

> (although perhaps marginally less than ideal),

> with almost everybody adhering to the -same- convention

> is far to be preferred to a situation with two competing conventions.

Other than you perhaps stating you belong to the group "almost everyone"

can you provide a more rational statement justifying a preference to

adhere to a MORE COMPLEX, more archaic scientific expression?

Also, given the value of parallax and creative tension, aren't you

stating a preferrence for potential short-term gain through conflict

avoidance rather than seeking the BIG rewards that come through

encouraging conflict resolution?

Astrology held certain fascinations as people moved away from the

spiritual alchemies, but after a while those older, initiating notions

were excluded from astronomical descriptions. Considering the long

haul ahead and all that is emerging in the present era, your vote to

circle the wagons and forever hold to the old ways rings a similar

overly conservative note.

One would expect Einstein to have respected the transition and to not

make a big fuss over how far to press the simplification. To suggesting

carrying on that complexifying tradition, say, out to 2080 or into 2366

seems a bit unscientific.

As for avoiding competing (or, more like, augmenting conventions,

expressions and/or theories since they are all hooked together) that

sounds like the opposite prescription needed to maintain a vigorous and

healthy science.

I guess I sort of disagree with your opinion.

--

Best regards,

Ralph E. Frost

http://www.dcwi.com/~refrost/index.htm ..Feeling is believing..

"It's not what the vision is, it's what the vision does." Peter Senge,

_The Fifth Discipline_

Dec 31, 1999, 3:00:00 AM12/31/99

to

In article <386B7BBC...@dcwi.com>, Ralph E. Frost

<ref...@dcwi.com> writes

<ref...@dcwi.com> writes

>Other than you perhaps stating you belong to the group "almost everyone"

>can you provide a more rational statement justifying a preference to

>adhere to a MORE COMPLEX, more archaic scientific expression?

>

>Also, given the value of parallax and creative tension, aren't you

>stating a preferrence for potential short-term gain through conflict

>avoidance rather than seeking the BIG rewards that come through

>encouraging conflict resolution?

Sigh.

You won't get anywhere with a bunch of guys who still cling to

'parsecs', 'angstroms', 'electron volts' and assorted other primitive

and incoherent units. Personally I'd prefer a coherent unit system and

the odd pi here and there, nothing wrong with pi as one can't really do

without it.

Of course then there are 'gods units' which seem to depend on precisely

which god is discussing what and I've never seen a convincing measure of

a gods unit for voltage or electric current and even gods units for

energy, time and distance seems to have some variable god-definitions.

In any case G = T is perfectly acceptable as a mathematical statement

and only needs have assorted constants included if you actually want to

calculate something real, which is as expected. Let's face it the gods

of physics (known here as 'moderators') frequently make a statement

covered by the comment "give or take a few two's, pi's and even not

uncommonly 'minusses' here and there". After all the laws of physics

don't really care about units, only some guy trying to build a bridge or

instrument does. In this case a standard set of physical units is helful

just in case someone gets the conversions wrong (which is of course

rocket science).

--

Oz

Jan 3, 2000, 3:00:00 AM1/3/00

to

Ralph E. Frost <ref...@dcwi.com> wrote:

> J. J. Lodder wrote:

>

> > Charles Francis <cha...@clef.demon.co.uk> wrote:

>

> > snip G = 8pi G

>

> > > It depends how many others like me want to write G = T but don't have

> > > the guts until someone like John says so.

>

> > Entropy always increases. :-(

> >

> > The present situation,

> > (although perhaps marginally less than ideal),

> > with almost everybody adhering to the -same- convention

> > is far to be preferred to a situation with two competing conventions.

>

> Other than you perhaps stating you belong to the group "almost everyone"

Einstein wrote 8pi in his eqn, all general relativity textbooks I have

ever seen use Einstein's convention. 'almost' was an understatement :-)

> can you provide a more rational statement justifying a preference to

> adhere to a MORE COMPLEX, more archaic scientific expression?

Of course not.

The fact is that the 4pi in Maxwell/Coulomb,

and the 8pi in Einstein/Newton has to appear -somewhere-,

since it has an unavoidable origin in geometry.

There can be no -rational- reason for putting them here rather than

there, since it is a matter of convention and convenience only.

It is not 'MORE COMPLEX',

it is complexity in one place rather than in another.

Some people will prefer one, some the other,

depending on which equations they happen to use most.

However, science is a cooperative process,

which depends on lots of communication.

It helps if everyone uses the same conventions.

[snip astrology]

> As for avoiding competing (or, more like, augmenting conventions,

> expressions and/or theories since they are all hooked together) that

> sounds like the opposite prescription needed to maintain a vigorous and

> healthy science.

Sure, express your electric fields in \sqrt(psi), for example,

(That is square root of pounds to the square inch,

a perfectly sensible electric field unit, was even used long ago)

and see what it gets you, apart from being ignored.

Why stay with archaic units like MKSA, when there is so much healthy and

invigorating conflict to be had when everybody uses his own system?

> I guess I sort of disagree with your opinion.

Indeed, entropy always increases, but not unavoidably so.

Human beings have the ability to impose some useful order, sometimes.

Wait and see, and hope for the best,

Jan

Jan 3, 2000, 3:00:00 AM1/3/00

to

In article <1e3q8n5.sr...@de-ster.demon.nl>, J. J. Lodder

<nos...@de-ster.demon.nl> writes

>Oz <O...@upthorpe.demon.co.uk> wrote:

>

>> Of course then there are 'gods units' which seem to depend on precisely

>> which god is discussing what and I've never seen a convincing measure of

>> a gods unit for voltage or electric current and even gods units for

>> energy, time and distance seems to have some variable god-definitions.

>

>Better refered to as Natural units,

>the only aspect of God's own units that everyone seems to agree on

>being that c should be one, (and dimensionless).

<nos...@de-ster.demon.nl> writes

>Oz <O...@upthorpe.demon.co.uk> wrote:

>

>> Of course then there are 'gods units' which seem to depend on precisely

>> which god is discussing what and I've never seen a convincing measure of

>> a gods unit for voltage or electric current and even gods units for

>> energy, time and distance seems to have some variable god-definitions.

>

>the only aspect of God's own units that everyone seems to agree on

>being that c should be one, (and dimensionless).

Bl**dy typical, even the gods don't agree on their own units.

>> After all the laws of physics don't really care about units,

>> only some guy trying to build a bridge or instrument does.

>

>But they do: the -form- the laws of physics take

>-does depend- on the unit system chosen.

Eh?? Explain convincingly.

--

Oz

Jan 3, 2000, 3:00:00 AM1/3/00

to

Oz wrote:

> In any case G = T is perfectly acceptable as a mathematical statement

> and only needs have assorted constants included if you actually want to

> calculate something real, which is as expected. Let's face it the gods

> of physics (known here as 'moderators') frequently make a statement

> covered by the comment "give or take a few two's, pi's and even not

> uncommonly 'minusses' here and there". After all the laws of physics> In any case G = T is perfectly acceptable as a mathematical statement

> and only needs have assorted constants included if you actually want to

> calculate something real, which is as expected. Let's face it the gods

> of physics (known here as 'moderators') frequently make a statement

> covered by the comment "give or take a few two's, pi's and even not

> don't really care about units, only some guy trying to build a bridge or

> instrument does. In this case a standard set of physical units is helful

> just in case someone gets the conversions wrong (which is of course

> rocket science).

> just in case someone gets the conversions wrong (which is of course

> rocket science).

Who am I to speak about "rational statements" anyway?

My comment was aimed more toward the conceptual advantage that I thought

I saw flash by wherein future discussions or education might begin

with G = T and then expand and relate the encapsulated terms as folks

need to, similarly to relate back to the Newtonian approximation.

Maybe next millenium.

Jan 4, 2000, 3:00:00 AM1/4/00

to

Toby Bartels <to...@ugcs.caltech.edu> wrote:

> >The fact is that the 4pi in Maxwell/Coulomb,

> >and the 8pi in Einstein/Newton has to appear -somewhere-,

> >since it has an unavoidable origin in geometry.

> >There can be no -rational- reason for putting them here rather than

> >there, since it is a matter of convention and convenience only.

>

> I disagree. Maxwell's equations and Einstein's equations

> are the fundamental equations that describe the phenomena under consideration.

> Coulomb's law and Newton's law are specific examples of the phenomena,

> and the geometric stuff is particular to these examples.

> It's more rational to have pi appear in formulae about the specific examples

> to which the geometry is relevant than in the fundamental equations.

> You can argue that the need for consistent conventions overrides this,

> but it *is* a valid consideration.

This boils down to mere words.

I would consider your rational reasons

as rationalizations for your preferences.

Where one sees 'valid' another may disagree.

But the point was that people will never agree on what is 'rational',

so a fortunate situation where (almost) everybody happens to use the

same conventions should not be spoiled for illusory conveniences of

some.

Changing the convention (by some) will merely result

in yet another tribal marker for a subgroup of physisists.

Best,

Jan

Jan 6, 2000, 3:00:00 AM1/6/00

to

J. J. Lodder <j...@de-ster.demon.nl> wrote:

>This boils down to mere words.

>I would consider your rational reasons

>as rationalizations for your preferences.

>Where one sees 'valid' another may disagree.

You make it sound as if I began with a preference

and then cast around for reasons to support it.

OTC, I have never seen an independent argument

(that is, one which doesn't appeal to convention)

as to why 4 pi belongs in Maxwell's equations and not in Coulomb's law;

I have never seen a geometric reason for 4 pi in Maxwell's equations.

There is a difference; one preference is more rational than the other,

because it has independent reasons, and the other does not.

The need to maintain consistent conventions may override the reasons,

but the difference is there.

>But the point was that people will never agree on what is 'rational',

>so a fortunate situation where (almost) everybody happens to use the

>same conventions should not be spoiled for illusory conveniences of

>some.

I'm not attacking this point, only the claim that

the two conventions have nothing to decide between them

other than the question of which is standard now.

-- Toby

to...@ugcs.caltech.edu

Jan 7, 2000, 3:00:00 AM1/7/00

to

In article <84vsgl$7...@gap.cco.caltech.edu>,

Toby Bartels <to...@ugcs.caltech.edu> wrote:

Toby Bartels <to...@ugcs.caltech.edu> wrote:

>I have never seen an independent argument

>(that is, one which doesn't appeal to convention)

>as to why 4 pi belongs in Maxwell's equations and not in Coulomb's law;

>I have never seen a geometric reason for 4 pi in Maxwell's equations.

>There is a difference; one preference is more rational than the other,

>because it has independent reasons, and the other does not.

Perhaps I should remind people what these independent reasons were:

they were mentioned earlier in this thread, but people may have

forgotten!

If we don't put a 4 pi in Maxwell's equations, we'll get a 4 pi

in Coulomb's law when we derive it from Maxwell's equations. But

this is not surprising or upsetting. We *expect* a factor of 4 pi

to show up in a calculation involving spherical symmetry, since 4 pi

is the area of the sphere. And indeed, this is precisely why the

4 pi shows up when we derive Coulomb's law. Everything makes sense,

so we are happy.

On the other hand, we can put in a 4 pi in Maxwell's equations to

cancel the 4 pi that shows up when we derive Coulomb's law from

these equations. But this is a bit like a shell game - at least

if you accept that Maxwell's equations are more fundamental than

Coulomb's law.

In particular, if there's a 4 pi in Maxwell's equations, lots of

students will wonder why. To which we can only reply: "Historically,

Coulomb's law came first and did not have a 4 pi in it; we need a

4 pi in Maxwell's equations because we want to cancel the 4 pi

that shows up when we derive Coulomb's law, to retain backwards

compatibility with that older convention." There are probably also

lots of people out there wondering why Einstein's equations have an

8 pi in them. Again, the main reason is backwards compatibility.

As Jan keeps reemphasizing and Toby fully admits, in practice it may

actually be better to keep backwards compatibility than to go for a

more logically satisfying setup. But that doesn't mean we can't

ponder, just for fun, what a more logically satisfying setup would

be. Nobody is suggesting that we set up a picket line in front of

the International Standards Organization world headquarters.

Jan 7, 2000, 3:00:00 AM1/7/00

to

Toby Bartels <to...@ugcs.caltech.edu> wrote in message

news:84vsgl$7...@gap.cco.caltech.edu...

> I have never seen an independent argument

> (that is, one which doesn't appeal to convention)

> as to why 4 pi belongs in Maxwell's equations and not in Coulomb's law;

Well, obviously we should split the difference and put a 2pi in each.

[Sci.physics.research moderator's note: You mean a 2 sqrt(pi), right? -TB]

Jan 7, 2000, 3:00:00 AM1/7/00

to

Oz <O...@upthorpe.demon.co.uk> wrote:

> >As discussed before, defining an independent force unit

> >introduces an additional constant (with a dimension) into F = ma.

>

> It's got one already called "m".

There is no limit to the number of superfluous extra constants

one can introduce into the laws of physics, such as:

F = g^{-1} m a, with F in Kg(F), g the standard acceleration,

m the mass in Kg(M), and a the usual acceleration.

-I- dindn't invent -that one- :-)

Jan

Jan 8, 2000, 3:00:00 AM1/8/00

to

In article <8557ee$639$1...@rosencrantz.stcloudstate.edu>, thus spake John

Baez <ba...@galaxy.ucr.edu>

Baez <ba...@galaxy.ucr.edu>

[Sci.physics.research moderator's note: Quoted text deleted. Please

trim quoted material judiciously, or else I'll do it injudiciously,

as shown here. -TB]

4pi in Coulomb's law is uncontentious because it appears from the

spherical symmetry of Maxwell's equations, and arises naturally from a

theory based in exchange particles, and in which the inverse square law

has a relation to spherical symmetry. But there is much reason to think

that gravity is not such a theory, and that the inverse square law has

greater relation to the hyperbolic metric of Minkowsky space, rather

than the Pythogorean metric of Euclidean space, so the appearance of pi

can be a major distraction to the student seeking an intuitive

understanding of the formulae, and feeling that the text book has not

given a proper explanation.

Jan 9, 2000, 3:00:00 AM1/9/00

to

John Baez wrote:

..snip the helpful clarification..

..snip the helpful clarification..

> In particular, if there's a 4 pi in Maxwell's equations, lots of

> students will wonder why. To which we can only reply: "Historically,

> Coulomb's law came first and did not have a 4 pi in it; we need a

> 4 pi in Maxwell's equations because we want to cancel the 4 pi

> that shows up when we derive Coulomb's law, to retain backwards

> compatibility with that older convention." There are probably also

> lots of people out there wondering why Einstein's equations have an

> 8 pi in them. Again, the main reason is backwards compatibility.

>

> As Jan keeps reemphasizing and Toby fully admits, in practice it may

> actually be better to keep backwards compatibility than to go for a

> more logically satisfying setup. But that doesn't mean we can't

> ponder, just for fun, what a more logically satisfying setup would

> be. Nobody is suggesting that we set up a picket line in front of

> the International Standards Organization world headquarters.

Who are you calling "Nobody"? ;-)

Science is a forward seeking enterprise. Maintaining "backward"

compatibility is an economic procedural choice. It's got it's place but

I don't think, given that we're sort of entering a new era in LONG-TERM

pulications, that the operating system upgrade should be postponed.

Maintaining backward compatibilty, in the traditional manner at the

level of _fundamental_ research, actually is in violation of several of

the rules of order embedded into the scientific method. (I've misplaced

my copy of the rulebook, so I can't quite quote chapter and verse.

Perhaps some else one can help out.)

There is, for example, the OTHER way of maintaining backward

compatibility.

In practice, it is NOT better to forever accumulate backward

compatibility along the traditional line. Lord know we do it and will

continue to, but it's not better. I think there is a rule that says

"when there is a big split in the scientific paradigm and then lots of

little subfractionations around that crevasse, then that's an indication

that the PRIOR traditional backward compatibility patch is the CAUSE and

not the CURE to problem". I forget where I read that. The sack

ultimately fills up. The stress at the fault lines ultimately exceed

allowable limits.

Sometimes just pondering the alternatives releases the stresses and

avoids a more uncomfortable tectonic shift. Group therapy sometimes

gets some good things out of the sack.

One advantage of pondering things through the lens of an potentially

"more logically satisfying setup", is _potentially_ the change in

conceptual or symbolic partitioning leads to some slight insight that is

more efficient, overall. Let's face it. Whatever the distinction is

between the "two camps" in physics, it has got to be pretty durn

subtle. If one NEVER looks at things via the alternative backward

compatibility route, I can guarantee one will NEVER see things anything

different.

It's a non-linear, rock-hopping, stream-crossing type of fun.

Short term, yes, it's best to keep with the old.

Ahem, but fundamental research in basic science is not about

"short-term", now is it, fellows?

--

Best regards,

Ralph E. "Nobody" Frost

Jan 10, 2000, 3:00:00 AM1/10/00

to

Ralph E. Frost <ref...@dcwi.com> wrote:

> Science is a forward seeking enterprise. Maintaining "backward"

> compatibility is an economic procedural choice. It's got it's place but

> I don't think, given that we're sort of entering a new era in LONG-TERM

> pulications, that the operating system upgrade should be postponed.

It should be postponed untill the vast majority has been convinced it

actually --is-- an upgrade, and not merely useless additional confusion.

Why don't you start a rationalized calendar reform,

far more irrationality out of the window that way,

and it will keep you busy far longer :-)

> Maintaining backward compatibilty, in the traditional manner at the

> level of _fundamental_ research, actually is in violation of several of

> the rules of order embedded into the scientific method. (I've misplaced

> my copy of the rulebook, so I can't quite quote chapter and verse.

> Perhaps some else one can help out.)

You appear (not :-) to have the wrong rule book.

The first rule for -any- new theory is that it should reproduce

what is known already, in some suitable limit.

v << c, \hbar -> 0, n >>1 and all that.

(Only extreme Kuhnians, not Thomas Kuhn himself,

and some postmodern sociologists of science seem to deny it.)

In other words, new theories, even revolutionairy ones,

should be 'backwards compatible',

Jan

Jan 11, 2000, 3:00:00 AM1/11/00

to

In article <38778208...@dcwi.com>, Ralph E. Frost

<ref...@dcwi.com> writes

<ref...@dcwi.com> writes

>If one NEVER looks at things via the alternative backward

>compatibility route, I can guarantee one will NEVER see things anything

>different.

That's entirely my point (although I don't think I ever specifically

said it).

I have (for a variety of reasons) gone through a number of these unit

changes. I would offer the following comments:

1) It's a lot easier than you think IFF it's rigidly enforced and the

old units are thoroughly banned. Once NO publications accept units in

anything other than strict SI in multiples of 1000 the conversion will

be rapid. Yes, that even goes for measuring star distances in yottam.

The basic units will rapidly become quite ordinary, within a few years

the old ones will seem archaic. It's equivalent to going to a country

with a different currency, within a week you become used to prices and

costs and rarely make a conversion (except when considering the cost of

purchases you intend to bring home).

2) The interdisciplinery barriers tend to fall away. When engineers used

to dealing in nm talk to physicist talking in angstroms there is always

some level of incomprehension, that will go. This probably applies to

many groups of scientists working in different fields. This was

particularly marked in the change from cgs to mks in electrical work of

all kinds.

3) There will always be a problem with time. sec/m/h/d/y. I would

suggest the use of seconds for technical documents as this wouldn't fase

the smart guys reading them. IIRC JB suggested easy mental conversions

some years ago. For less technical publications the others will always

be 'satisfactory'.

4) It ain't going to happen. This is a poor reflection on science and

technologists in general.

--

Oz

Jan 11, 2000, 3:00:00 AM1/11/00

to

Oz <O...@upthorpe.demon.co.uk> wrote:

> 3) There will always be a problem with time. sec/m/h/d/y. I would

> suggest the use of seconds for technical documents as this wouldn't fase

> the smart guys reading them. IIRC JB suggested easy mental conversions

> some years ago. For less technical publications the others will always

> be 'satisfactory'.

Also historical accident.

Originally decimal time (based on the day) was part of the metric

system, as was the decimal degree (grad).

A few decimally divided clocks have been built.

(now valuable and very rare museum pieces)

It never took hold, even in France.

Decimal time was dropped, together with with the revolutionary calender,

when a certain Napoleon Buonaparte needed the pope to hold a crown for

him. The pope insisted on the Christian calendar being restored.

Astronomers avoid the problem by calculating in Julian Day,

which is just a running count of (decimally devided) days.

Written, Julian Day 2451555.6309,

Jan

Jan 12, 2000, 3:00:00 AM1/12/00

to

..

>

> 4) It ain't going to happen. This is a poor reflection on science and

> technologists in general.

Overly speculative and not founded in fact. Things change non-linearly.

They just change really slowly at the start.

Don't cut bait until you've fished in earnest. There are some good

'sales' opportunities opening up.

Also, don't cut science folks and technologists so short. After all, if

THEY don't do it, who will?

--

Best regards,

Ralph E. Frost

Jan 12, 2000, 3:00:00 AM1/12/00

to

In article <1e44zas.45...@de-ster.demon.nl>,

J. J. Lodder <j...@de-ster.demon.nl> wrote:

J. J. Lodder <j...@de-ster.demon.nl> wrote:

>Ralph Frost wrote:

>> I don't think, given that we're sort of entering a new era in LONG-TERM

>> publications, that the operating system upgrade should be postponed.

>It should be postponed until the vast majority has been convinced it

>actually --is-- an upgrade, and not merely useless additional confusion.

Has there ever been a case where a change in conventions was made only

*after* the vast majority was convinced that it was an improvement?

It seems that either:

1) some people make the change and then convince others that

it's an improvement - so it catches on,

2) some people make the change and fail to convince other

that it's an improvement - so it doesn't catch on, or

3) some board or committee with authority to decide these matters

goes ahead and decides, and then tells everyone else what to do.

I think somehow this discussion has convinced me that in the

next edition of "Knots and Quantum Gravity" I'm going to use

units where 8 pi G = 1 instead of G = 1.

Jan 18, 2000, 3:00:00 AM1/18/00

to

In <85ii76$ais$1...@rosencrantz.stcloudstate.edu> ba...@galaxy.ucr.edu

(John Baez) writes:

(John Baez) writes:

>I think somehow this discussion has convinced me that in the

>next edition of "Knots and Quantum Gravity" I'm going to use

>units where 8 pi G = 1 instead of G = 1.

Did the question ever get answered as to where the 8pi comes from in

comparing Einstein's equation vs Newton's? We noted that a 4pi factor

enters in when we go from the electrostatic force law to the Maxwell

3-D field equations, and intuitively I've always thought of this as a

kind of integrated total solid angle that enters in when you go

between differential 3-D geometry on one side (unit vector basis,

volume normalized to "1"), and a sort of 1-D scalar equation on the

other. With gravity, sort of by (horrid) analogy, perhaps you have the

same kind of thing between Newton's law (which sort of looks like the

electrostatic law), and 4-D rather than 3-D differential geometry, so

now you have an extra dimention to worry about. So perhaps then you

have to integrate through a unit hypersphere and obtain a total 4-D

"solid angle," if you want to use the same constant of proportionality

"G" as appears in the 1-D equation (Newton's).

Now, these integrated angles all go up by a factor of 2 per dimention.

The integrated circle gets you 2 pi radians (I presume Maxwell's laws

in Flatland would use this), a sphere is 4 pi steradians, and (by

golly) a hypersphere or indeed any 4-D object in radial coordinates

should need 8pi hyper-steradians. The only problem is that we're

equating one tensor with another, actually, in the Einstein equation.

Fudge. What is it about the Riemann or Einstein tensor that makes one

behave as though the other had to be converted by an angle factor? Is

it simply the differential description of space on one side vs. the

integrated description on the other?

Jan 18, 2000, 3:00:00 AM1/18/00

to

John Baez <ba...@galaxy.ucr.edu> wrote:

> In article <1e44zas.45...@de-ster.demon.nl>,

> J. J. Lodder <j...@de-ster.demon.nl> wrote:

>

> >Ralph Frost wrote:

>

> >> I don't think, given that we're sort of entering a new era in LONG-TERM

> >> publications, that the operating system upgrade should be postponed.

>

> >It should be postponed until the vast majority has been convinced it

> >actually --is-- an upgrade, and not merely useless additional confusion.

>

> Has there ever been a case where a change in conventions was made only

> *after* the vast majority was convinced that it was an improvement?

Yes, the international conference on weights and measure

does not change conventions like the definition of the meter

until there is a broad consensus (among those in the know)

that this is -a good thing-.

> It seems that either:

>

> 1) some people make the change and then convince others that

> it's an improvement - so it catches on,

>

> 2) some people make the change and fail to convince other

> that it's an improvement - so it doesn't catch on, or

>

> 3) some board or committee with authority to decide these matters

> goes ahead and decides, and then tells everyone else what to do.

You forget the most likely outcome:

4) The originally united community is split into two tribes,

each stubbornly clinging to it's own convention.

Compare those electrical engineers who isist on recognizing

'our kind of people' by writing -jw, where 'the others' write i\omega.

> I think somehow this discussion has convinced me that in the

> next edition of "Knots and Quantum Gravity" I'm going to use

> units where 8 pi G = 1 instead of G = 1.

You want your own tribe with your own tribal marker?

You can try,

Jan

Jan 18, 2000, 3:00:00 AM1/18/00

to

In article <85ii76$ais$1...@rosencrantz.stcloudstate.edu>, thus spake John

Baez <ba...@galaxy.ucr.edu>

Baez <ba...@galaxy.ucr.edu>

>I think somehow this discussion has convinced me that in the

>next edition of "Knots and Quantum Gravity" I'm going to use

>units where 8 pi G = 1 instead of G = 1.

Please don't. It really is only appropriate for Euclidean not Minkowsky

metric, and sets of all sorts of confusions, wondering if Einstein had

thought of something that has been missed from the text books.

--

Regards

Charles Francis

cha...@clef.demon.co.uk

[Moderator's note: Unnecessary quoted text trimmed. -MM]

Jan 18, 2000, 3:00:00 AM1/18/00

to

Oz wrote:

>

> In article <386B7BBC...@dcwi.com>, Ralph E. Frost

> <ref...@dcwi.com> writes

>

> You won't get anywhere with a bunch of guys who still cling to

> 'parsecs', 'angstroms', 'electron volts' and assorted other primitive

> and incoherent units. Personally I'd prefer a coherent unit system and

> the odd pi here and there, nothing wrong with pi as one can't really do

> without it.

>

> In article <386B7BBC...@dcwi.com>, Ralph E. Frost

> <ref...@dcwi.com> writes

>

> >Other than you perhaps stating you belong to the group "almost everyone"

> >can you provide a more rational statement justifying a preference to

> >adhere to a MORE COMPLEX, more archaic scientific expression?

..> >adhere to a MORE COMPLEX, more archaic scientific expression?

> You won't get anywhere with a bunch of guys who still cling to

> 'parsecs', 'angstroms', 'electron volts' and assorted other primitive

> and incoherent units. Personally I'd prefer a coherent unit system and

> the odd pi here and there, nothing wrong with pi as one can't really do

> without it.

except in those rare instances where you can substitute

circumference

--------------

diameter

Jan 19, 2000, 3:00:00 AM1/19/00

to

In article <85k4lg$if5$1...@nntp3.atl.mindspring.net>,

Steven B. Harris <sbha...@ix.netcom.com> wrote:

Steven B. Harris <sbha...@ix.netcom.com> wrote:

>In <85ii76$ais$1...@rosencrantz.stcloudstate.edu> ba...@galaxy.ucr.edu

>(John Baez) writes:

>>I think somehow this discussion has convinced me that in the

>>next edition of "Knots and Quantum Gravity" I'm going to use

>>units where 8 pi G = 1 instead of G = 1.

> Did the question ever get answered as to where the 8pi comes from in

>comparing Einstein's equation vs Newton's?

Yeah, I think Ted Bunn explained it. The 4 pi comes from the

area of the sphere, just like electromagnetism. The extra factor

of 2 comes from the fact that h_{00} is -2 times the Newtonian

gravitational potential phi. What's h_{00}? Well, h_{00}

measures the amount of "time dilation due to the gravitational

field". Technically, h_{00} is the difference between the

time-time component of the actual spacetime metric and that

of flat Minkowski spacetime.

Here's how it works. In the Newtonian limit, Einstein's equations

say that

Laplacian(h_{00}) = - 8 pi G rho

On the other hand, according to Newtonian gravity we have

Laplacian(phi) = 4 pi G rho

These are consistent, thanks to that extra factor of 2.

(The minus sign is related to the fact that we're using a

convention where the time-time component of the metric

is negative.)

By the way, I think this factor of 2 is secretly the same as

the 2 you see in the formula for the Schwarzschild solution: the

spacetime metric for a spherical object of mass M is described

by a formula involving the quantity 2M.

I suppose now someone is gonna ask where the 2 comes from in

the formula h_{00} = -2 phi.

[Sci.physics.research moderator's note: Consider it asked. -TB]

Jan 20, 2000, 3:00:00 AM1/20/00

to

J. J. Lodder <j...@de-ster.demon.nl> wrote at last:

>John Baez <ba...@galaxy.ucr.edu> wrote:

>>I think somehow this discussion has convinced me that in the

>>next edition of "Knots and Quantum Gravity" I'm going to use

>>units where 8 pi G = 1 instead of G = 1.

>You want your own tribe with your own tribal marker?

Well, he's got to print *some*thing.

He either prints the tribal marker of your tribe,

or he prints the tribal marker of my tribe.

Which do you think he should choose:

the tribe he agrees with or the tribe he disagrees with?

(Of course, he could print the tribal marker of Gerard's tribe

and include all the factors of G/c^2 :-).)

-- Toby

to...@ugcs.caltech.edu

Jan 20, 2000, 3:00:00 AM1/20/00

to

ba...@galaxy.ucr.edu (John Baez) says...

>Yeah, I think Ted Bunn explained it. The 4 pi comes from the

>area of the sphere, just like electromagnetism. The extra factor

>of 2 comes from the fact that h_{00} is -2 times the Newtonian

>gravitational potential phi.

I thought that the source of this 2 is the square-root appearing

in the proper time formula:

dT = square-root(g_{mu,nu} dx^mu dx^nu)

In weak, time-independent gravitational fields,

g_{mu,nu} takes the form

g_{0,0} = 1 + h

g_{1,1} = g_{2,2} = g_{3,3} = -1

g_{mu,nu} = 0 (for mu unequal to nu)

So the formula for proper time becomes (using

(v/c)^2 = (dx^1/dx^0)^2 + (dx^2/dx^0)^2 + (dx^3/dx^0)^2

dT = square-root(1+h - (v/c)^2) dt

For A and v small, we can expand the square-root to get

dT = (1 + 1/2 h - 1/2 (v/c)^2) dt

The equations of motion for a test particle are obtained

by maximizing the integral of (1 + 1/2 h - 1/2 (v/c)^2) dt.

Since the constant 1 is irrelevant for the equations of

motion, this is the same as maximizing the integral of

D dt, where

D = 1/2 h - 1/2 (v/c)^2

But, in this limit (small A and small v), Newtonian

mechanics can be used, so we should *also* be able

to obtain the equations of motion by minimizing the

integral of L dt, where

L = 1/2 mv^2 - m Phi

(where Phi is the Newtonian gravitational potential).

This suggests that D and L are linearly related. The

only way for this to be true is if

h = 2 Phi/c^2

Daryl McCullough

CoGenTex, Inc.

Ithaca, NY

Jan 20, 2000, 3:00:00 AM1/20/00

to

In article <85k4lg$if5$1...@nntp3.atl.mindspring.net>,

Steven B. Harris <sbha...@ix.netcom.com> wrote:

>

Steven B. Harris <sbha...@ix.netcom.com> wrote:

>

>Now, these integrated angles all go up by a factor of 2 per dimention.

>The integrated circle gets you 2 pi radians (I presume Maxwell's laws

>in Flatland would use this), a sphere is 4 pi steradians, and (by

>golly) a hypersphere or indeed any 4-D object in radial coordinates

>should need 8pi hyper-steradians.

>The integrated circle gets you 2 pi radians (I presume Maxwell's laws

>in Flatland would use this), a sphere is 4 pi steradians, and (by

>golly) a hypersphere or indeed any 4-D object in radial coordinates

>should need 8pi hyper-steradians.

Actually, in 4-D, the "area" is 2 pi^2. In general, for d dimensions,

it's

2 pi^(d/2) / Gamma(d/2)

If the concept doesn't make your head hurt, feel free to apply this

formula to spheres of nonintegral dimension too.

Exercise: Prove this.

Hint: Int[dx exp(-x^2)] = Sqrt(pi)

--

======================================================================

Kevin Scaldeferri Calif. Institute of Technology

The INTJ's Prayer:

Lord keep me open to others' ideas, WRONG though they may be.

Jan 21, 2000, 3:00:00 AM1/21/00

to

J. J. Lodder wrote:

> Yes, the international conference on weights and measure

> does not change conventions like the definition of the meter

> until there is a broad consensus (among those in the know)

> that this is -a good thing-.

>

THAT makes sense. The question I am wondering is whether you would like

that attitude extended, say, to these lofty peaks so that no uprisings

can start?

How many board members of the International Union are participating in

s.p.r.?

--

Best regards,

Ralph E. Frost

http://www.dcwi.com/~refrost/index.htm ..Feeling is believing..

[Moderator's note: Quoted text trimmed. -MM]

Jan 21, 2000, 3:00:00 AM1/21/00

to

In article <865o41$c...@gap.cco.caltech.edu>, Toby Bartels

<to...@ugcs.caltech.edu> writes

>

>Well, he's got to print *some*thing.

>He either prints the tribal marker of your tribe,

>or he prints the tribal marker of my tribe.

>Which do you think he should choose:

>the tribe he agrees with or the tribe he disagrees with?

>(Of course, he could print the tribal marker of Gerard's tribe

>and include all the factors of G/c^2 :-).)

<to...@ugcs.caltech.edu> writes

>

>Well, he's got to print *some*thing.

>He either prints the tribal marker of your tribe,

>or he prints the tribal marker of my tribe.

>Which do you think he should choose:

>the tribe he agrees with or the tribe he disagrees with?

>(Of course, he could print the tribal marker of Gerard's tribe

>and include all the factors of G/c^2 :-).)

Of course he could print all three, and thus make everyone happy and

allow his readers to:

1) Make up their own minds.

2) Not be fazed when the see the expression of another tribal marker.

He could also take the opportunity for a Feynman-like digression at that

point.

--

Oz

Jan 21, 2000, 3:00:00 AM1/21/00

to

Steven B. Harris <sbha...@ix.netcom.com> wrote:

>Now, these integrated angles all go up by a factor of 2 per dimention.

>The integrated circle gets you 2 pi radians (I presume Maxwell's laws

>in Flatland would use this), a sphere is 4 pi steradians, and (by

>golly) a hypersphere or indeed any 4-D object in radial coordinates

>should need 8pi hyper-steradians.

Sadly, this is not so.

For example, the integrated 0sphere gets you 2, not pi.

IIRC, the 3sphere has a surface volume that has a pi^2 in it.

>The only problem is that we're

>equating one tensor with another, actually, in the Einstein equation.

>Fudge. What is it about the Riemann or Einstein tensor that makes one

>behave as though the other had to be converted by an angle factor? Is

>it simply the differential description of space on one side vs. the

>integrated description on the other?

Nothing about the Einstein tensor involves angles.

This is why there is no 8 pi in Einstein's equation,

which reads "Guv = Tuv" in rationalised units.

Angles show up in Newton's approximate equation,

which involves the factor 1/(8 pi) (aka "G").

At least, that's the way people like John and me do it.

-- Toby

to...@ugcs.caltech.edu

Jan 21, 2000, 3:00:00 AM1/21/00

to

In article <867fq3$16...@edrn.newsguy.com>, da...@cogentex.com (Daryl

McCullough) wrote:

McCullough) wrote:

>ba...@galaxy.ucr.edu (John Baez) says...

>

>>Yeah, I think Ted Bunn explained it. The 4 pi comes from the

>>area of the sphere, just like electromagnetism. The extra factor

>>of 2 comes from the fact that h_{00} is -2 times the Newtonian

>>gravitational potential phi.

>I thought that the source of this 2 is the square-root appearing

>in the proper time formula:

>

> dT = square-root(g_{mu,nu} dx^mu dx^nu)

Yes, that is exactly where it comes from! (I'm starting to

lose track of the minus signs, though.)

Daryl went on to derive this relation from the above using Lagrangian

mechanics. My favorite hand-waving way to do it actually involves quantum

mechanics! It involves taking a lot more on faith than the Lagrangian

argument, but it is more intuitive for a particle guy like me, and it's

the only way I can remember to derive it easily in my head.

A particle's wave function has a phase that goes around like exp(iEt).

Now in a relativistic theory we ought to include the mass, so if the

particle's energy of motion is negligible, this is exp(imt) where c = 1.

Well, in GR this really ought to be proper time: exp(im tau).

Replacing t with tau ought to be equivalent to the effect of a Newtonian

gravitational potential in the appropriate limit. The gravitational

potential gives the particle a gravitational energy m phi. So we have

exp[i(m + m phi) t] = exp(im tau)

(1 + phi) t = tau

phi = (tau/t) - 1

~= sqrt(g^00) - 1 = sqrt(1 + h^00) - 1

~= (1/2) h^00

h^00 ~= 2 phi

Hmmm, I got the same sign that Daryl did.

--

Matt McIrvin http://world.std.com/~mmcirvin/

Jan 23, 2000, 3:00:00 AM1/23/00

to

Oz <O...@upthorpe.demon.co.uk> wrote:

>Toby Bartels <to...@ugcs.caltech.edu> wrote:

>>Well, he's got to print *some*thing.

>>He either prints the tribal marker of your tribe,

>>or he prints the tribal marker of my tribe.

>>Which do you think he should choose:

>>the tribe he agrees with or the tribe he disagrees with?

>>(Of course, he could print the tribal marker of Gerard's tribe

>>and include all the factors of G/c^2 :-).)

>Of course he could print all three, and thus make everyone happy and

>allow his readers to:

>1) Make up their own minds.

>2) Not be fazed when the see the expression of another tribal marker.

Well, he could certainly *mention* all 3 at some point,

but I think it would be silly to *use* all 3 throughout the book.

That's what I meant.

-- Toby

to...@ugcs.caltech.edu

Jan 24, 2000, 3:00:00 AM1/24/00

to

Toby Bartels <to...@ugcs.caltech.edu> wrote:

[different unit systems in the same book]

> Well, he could certainly *mention* all 3 at some point,

> but I think it would be silly to *use* all 3 throughout the book.

> That's what I meant.

I once saw a textbook that did just that,

but it required colour printing.

SI-believers see the red, the others don't,

Jan

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