Re: [asa] planck for the layman?

Merv -

Actually Planck himself, in the paper in which he introduced "his" constant
h, pointed out that it, together with the speed of light c and gravitational
constant G, defined a natural system of length, time & mass units:

L = sqrt(hG/c^3) ~ 10^-33 cm, T = L/c ~ 10^-43sec , M = Lc^2/G ~
10^-5gm.

(h there is Planck's h over 2*pi.) But this is simply dimensional analysis
& doesn't mean that you can't actually have lengths or times smaller than L
& T. The best way to show that clearly is as follows. (This follows a
paper of mine in American Journal of Physics 42, 1974, p.958.)

The quantum mechanical uncertainty principle says measurement of a time
interval t will be related to the uncertainty in the energy of the system DE
by t*DE > h .
In addition, the gravitational field of the clock will, according to general
relativity, influence the rate at which the clock runs. The fractional
change in an interval t in the vicinity of the clock will be on the order of
dt/t ~ gravitational potential/c^2 ~ G[M + DE/c^2]/Lc^2, where M is the
clock's mass, L its linear dimension, and the uncertainty in energy gives an
uncertainty in mass according to Einstein's formula.

Now in order for the clock to measure an interval t its parts must be able
to communicate with one another within that time, so L < ct. Thus dt/t >
G[M + DE/c^2]/c^3 > GDE/c^5 . But since DE > h/t we have dt > Gh/tc^5. The
measurement is of no value unless dt < t - i.e., unless t > sgrt(hG/c^5),
the Planck time defined above.

This does not mean, however, that space-time is quantized in the sense that
there are "atoms" of space & time. That may be the case but all this
argument shows is that the the concepts of length & time intervals lose
their meaning below the Planck scale. There are reasons to think that
space-time is still continuous, but it is not a metric space below this
scale.

Shalom
George
http://home.roadrunner.com/~scitheologyglm

----- Original Message -----
From: <mrb22667@kansas.net>
To: "asa" <asa@calvin.edu>
Sent: Tuesday, April 07, 2009 5:14 PM
Subject: [asa] planck for the layman?

>
> I'm needing help understanding some of the apparent ramifications that
> come from
> Planck's constant ... such as: somehow a quantized packet of energy leads
> to a
> "smallest packet" of space? --and of time as well? I think I remember
> this from
> a previous ASA discussion, but didn't understand it at the time & still
> don't now.
>
> Does this effectively cap our logarithmic romp towards smaller things in
> much
> the same way as Einstein's still expanding space caps the "think-big" end?
> If there is a good book on Planck for the layman, let me know.
>
> --Merv
> (It's enough to give a Euclidean-minded geometry teacher a headache.)
>
>
> To unsubscribe, send a message to majordomo@calvin.edu with
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Received on Tue Apr 7 20:56:50 2009