# Inverse square vs inverse cube

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### Anon E. Mouse

Mar 17, 2012, 4:46:58 AM3/17/12
to
======================================================================
Moderator's note:

Since intensity of light from a distant star is about the energy flux
through an area around this star (area 4 pi r^2), due to energy-momentum
conservation you must have an inverse square law. If GRT effects have to
be taken into account, of course one has to calculate the area with the
metric at hand, but the relevant geometrical object still is a two
dimensional spacelike hypersurface rather than a three dimensional one.

HvH.
======================================================================

Recently replying to a question from Chalky I commented that the
inverse square law was derived from the area under the curve from a
point source.

Much of the day I have been troubled because, if the point source is a
distant star then the area under the curve is a sphere and not a
circle.

This seems significant. If the circle holds then the law is k/pi*r*r.
If the sphere is on the ball, then the relation is 4k/3*r*r*r.

I think Newton's Principia is the source of the inverse square law,
although Kepler had some similar ratios. Thing is, you can only
compute orbits in AU and solar masses, could it be that this
restriction exists because we have been using a formulation for
decrease in power with distance that was two dimensional instead of
three?

As far as I can tell the dissipation over distance relationship for
lux, gravitation, electric force, etc. should vary with 4/3 the
inverse cube of r. Can anyone refute this hypothesis? I find it very
troubling that I myself can't!

AAG

### Hendrik van Hees

Mar 17, 2012, 6:18:42 PM3/17/12
to
On 17/03/12 09:46, Anon E. Mouse wrote:
> ======================================================================
> Moderator's note:
>
> Since intensity of light from a distant star is about the energy flux
> through an area around this star (area 4 pi r^2), due to energy-momentum
> conservation you must have an inverse square law. If GRT effects have to
> be taken into account, of course one has to calculate the area with the
> metric at hand, but the relevant geometrical object still is a two
> dimensional spacelike hypersurface rather than a three dimensional one.
>
> HvH.
> ======================================================================

I think a clarification of this subject is at order. It all boils down
to the definition of luminosity in an expanding
Friedmann-Lemaitre-Robertson-Walker universe.

Consider a star or any object emitting electromagnetic radiation
isotropically from its surface (in the restframe of the object). Let the
luminosity, i.e., the total radiation energy emitted by this object per
unit of time.

Now we have to calculate the energy per unit time and unit area at a
certain distance from the object. To this end we consider the radiation
as a stream of photons with frequency f at the source. Their energy is
E=h f. Through the cosmological redshift this frequency gets lowered by
a factor 1/(1+z), where z is the usual definition of the red shift:
E0=E/(1+z), i.e., at the point of detection the frequency of the photons
is f0=f/(1+z).

Then let N be the number of photons per unit time at the surface. Also
this rate gets red shifted: N0=N/(1+f), i.e., L0=h f0 N0=h f/(1+z)
N/(1+z)=L/(1+z)^2.

The proper area of the sphere around the star intersecting the earth is
given by 4 pi (r a0)^2, where r is the dimensionless coordinate distance
of the source to the earth and a0 the FLRW scale factor at the time of
observation. Thus we have

l0=L0/(4 pi r^2 a0^2) = L/[4 pi r^2 a0^2(1+z)^2]=:L/(4 pi dL^2)

This defines the luminosity distance of the object. The energy density
scales with the square of this distance by definition and has the given
relation to the redshift, i.e.,

dL=r a0 (1+z).

--
Hendrik van Hees
D-60438 Frankfurt am Main
http://fias.uni-frankfurt.de/~hees/

### Nicolaas Vroom

Mar 18, 2012, 7:18:48 AM3/18/12
to
On 17 mrt, 23:18, Hendrik van Hees <h...@fias.uni-frankfurt.de> wrote:
> On 17/03/12 09:46, Anon E. Mouse wrote:
>
> > ======================================================================
> > Moderator's note:
>
> > Since intensity of light from a distant star is about the energy flux
> > through an area around this star (area 4 pi r^2), due to energy-momentum
> > conservation you must have an inverse square law. If GRT effects have to
> > be taken into account, of course one has to calculate the area with the
> > metric at hand, but the relevant geometrical object still is a two
> > dimensional spacelike hypersurface rather than a three dimensional one.
>
> > HvH.
> > ======================================================================
>
> The proper area of the sphere around the star intersecting the earth is
> given by 4 pi (r a0)^2, where r is the dimensionless coordinate distance
> of the source to the earth and a0 the FLRW scale factor at the time of
> observation. Thus we have
>
> l0=L0/(4 pi r^2 a0^2) = L/[4 pi r^2 a0^2(1+z)^2]=:L/(4 pi dL^2)
>
> This defines the luminosity distance of the object.

============= Moderator's note ===========================================

Of course, the functional form of the luminosity-red-shift relation
gives us insight into the actual dynamics and thus the "matter content"
of the universe. E.g., there are measurements of this relation in terms
of the distance modulus of Type Ia supernovae. Together with the
microwave background radiation fluctuations from WMAP, PLANCK etc. this
leads to the typical conclusions of the modern standard big bang model
("cold dark matter", CDM): ~70% dark energy, ~20% dark matter, ~5%
"standard-model matter". The somewhat disturbing conclusion is that we
know something definite only about roughly 5% of the matter content of
the universe ;-)).

HvH.
==========================================================================

This same issue is (more or less) discussed at my homepage.
go to:

http://users.telenet.be/nicvroom/friedmann's%20equation.htm

discussed:
#1 is of the form: k/r^2
#5 is of the form: k/r^2*(1+z^2). This is the one discussed
in the above posting.
#7 is of the form: k/r^2*(1+a*r+b*r^2)

The result of the calculations is that #7 gives the smallest error
between theory (Friedmann equation) and observation (SNLS data).
However that does not mean that #5 is wrong.
What you can also do is to start from #5 and modify that one
with an extra factor as a function of r.
The issue is when you do that the resulting parameters
of the Friedmann equation which describe the different
Flux Luminosity relations are quite different.

As I said: this is an interesting subject.

Nicolaas Vroom

Mar 20, 2012, 9:50:46 AM3/20/12
to
> Since intensity of light from a distant star is about the energy flux
> through an area around this star (area 4 pi r^2), due to energy-momentum
> conservation you must have an inverse square law. If GRT effects have to
> be taken into account, of course one has to calculate the area with the
> metric at hand, but the relevant geometrical object still is a two
> dimensional spacelike hypersurface rather than a three dimensional one.

This is an aside to the main discussion:

In cosmology, the luminosity distance is DEFINED as the distance one
gets by ASSUMING the inverse-square law. Since space can be
non-Euclidean and/or expanding (or contracting), this in general
differs from other ways of calculating distance (e.g. angular-size
distance, proper distance (instantaneous distance measured with a rigid
rule), light-travel-time distance, parallax distance, proper-motion
distance---all of these are the same in a static Euclidean space).
(Note for experts: to be precise, there are corrections made for the
fact that one is observing in a finite band.)

### Anon E. Mouse

Mar 20, 2012, 8:51:28 PM3/20/12
to
Expanding the polynomial of #7 gives a 4th degree polynomial which is
as you say a better fit for the data. When a fourth power is part of a
polynomial sum in the denominator its contribution to the final ratio
is very small.

The formulation I propose is 3rd order and not necessarily a
polynomial this result should be intermediary between #5 which would
understate the proper distance and #7 which should very slightly
overstate the proper distance. Supposing only that the most correct
relation be that which I am proposing.

The aperture area pi * r * r over (4/3 pi h*h*h) is quite simple to
calculate and has the virtue of being supported by the spherical
section geometric proof outlined in a very brief manner in a previous
posting.

I would be very pleased if someone in this field would attempt a small
sample analysis to asses the fitness of this function. I derived it
from purely geometric considerations and I have slowly come to believe
it may have merit.

Anthony A Gallistel

### Hendrik van Hees

Mar 21, 2012, 4:26:45 AM3/21/12
to
On 20/03/12 14:50, Phillip Helbig---undress to reply wrote:

> This is an aside to the main discussion:
>
> In cosmology, the luminosity distance is DEFINED as the distance one
> gets by ASSUMING the inverse-square law. Since space can be
> non-Euclidean and/or expanding (or contracting), this in general
> differs from other ways of calculating distance (e.g. angular-size
> distance, proper distance (instantaneous distance measured with a rigid
> rule), light-travel-time distance, parallax distance, proper-motion
> distance---all of these are the same in a static Euclidean space).
> (Note for experts: to be precise, there are corrections made for the
> fact that one is observing in a finite band.)

Sure, this I clarified in my other posting to this question. There, of
course, you have to take the Robertson-Walker metric and calculate the
very distance measure in question.

The same holds for the clock-synchronization thread. In the general
relativistic case, of course, clock synchronization in the most general
case of space times cannot be achieved globally but only locally, but I
guess this we should discuss in this other thread.

### Anon E. Mouse

Mar 31, 2012, 5:47:12 PM3/31/12
to
[[Mod. note -- Conservation of energy implies that the luminosity L of
the source (units of power, i.e., energy per unit time) must equal the
total power a surface S which completely surrounds the source. And in
the absence of absorbers, this must be true for *any* such surface S.
So... on laboratory scales (where spacetime is *very* well approximated
as being flat), if we consider a surface S which is a sphere of radius
r centered on the source, then clearly the average power crossing S is
just L / (4*pi*r^2).
-- jt]]

Can anyone refute this hypothesis? I find it very
> troubling that I myself can't!
>
> AAG

In the time since I made this original post I have been able to
determine that I can make a simple proof of the inverse cube law and I
can not make a rigorous proof, nor find same for the inverse square
law.

I have of course found many graphical illustrations that show that
spherical surfaces increase with the the square of the radius but of
course spherical volumes do not.

I recently went looking for experimental evidence that confirms the
square law relation and found this;

"I think the reason why the peak lux measurements do not follow the
inverse square law..."

"So, what can we learn from this exercise? Mainly, peak lux readings
for bright lights cannot be trusted at close distances. In this
example, only readings taken from greater than or equal to 3 meters
lights with larger reflectors, then I have no doubt that readings
taken at 3 meters would also fail to follow the inverse square
relation. Larger reflectors require measurements to be taken at
greater distances.

How much of a difference did it really make deleting the first few
points? Well, honestly not a lot. If I average the calculated peak lux
values for all of the points I end up with 112,228 lux whereas if I
only include the data from 3 to 6 meters I end up with 117,039 lux. If
I followed the convention of only measuring at 5 meters I would have
gotten a result of 121,500. This shows how averaging multiple readings
reduces error that can occur in any one measurement, especially for
bright lights at close distances."

(My comment)This last bit demonstrates that the use of the inverse
square law in this experiment overstates the source luminescence the
greater that actual distance.

I also found;

physics.fullerton.edu/.../Inverse%20Square%20Law%20Experiment

titled, "Inverse Square Law Experiment_2.txt"

or

"To measure and confirm the inverse square law fall off of l"

This experimental set up is better than the first example, however, in
the data analysis instructions it states, "Analysis of data: Plot the
data as a scatter plot in Excel with no line connecting the data, and
change the scale on both axis to log so it will be a log-log plot."

If the relation is k/h*h then the plot should be log / linear and not
log log. Using log log equates the square of the distance and the
inverse square of the lux. Which I believe will reduce to the fourth
rout of the proper distance. But I can only say for certain that this
method of analysis does not seem correct to me.

a deeply disturbed,

AAG

### Anon E. Mouse

Mar 31, 2012, 5:47:29 PM3/31/12
to

> As far as I can tell the dissipation over distance relationship for
> lux, gravitation, electric force, etc. should vary with 4/3 the
> inverse cube of r. Can anyone refute this hypothesis? I find it very
> troubling that I myself can't!
>
> AAG

I finally located some seemingly reliable data from an inverse square
law experiment measuring luminosity over distance. The greatest
deviation they report is for the shortest distance, a finding others
also report.

Using their data I computed the projected luminosity for 3 feet based
on 6 feet and projected 221 compared with an actual measured 220. For
short distances the cube law fails even worse than the square law, but
a not mentions that the power of the meter is 16 at 4 feet. I can
convert this to an proportional magnification cone through the
objective to "measure" the effective aperture at short range, but that
is not the sort of data manipulation I would like to use to test my
hypothesis. A source distance less than the focal length of the meter
would cause a lack of focus and a under reporting of luminosity
progressively increasing as the source length falls below the focal
length which is what this data set shows.

I very much appreciate the contributions others have made to this
quandary I have fallen into. I think I shall find in the end that the
fitness of the square law is fine and Herr Hendricks methods are
precisely correct.

Still, I have had the experience of having a fiber optic cable
technician tell me that in this special case the signal falls as the
cube of the distance and he was bitter about it for two reasons,
firstly, this caused the need for more repeaters in a given run, and
second, because the higher ups in his company would not admit the
cubed law relation that the tech's kept experiencing, even if it was a
single length of cable with no splices or repeaters. The db loss he
claimed was always cubic. I think this is the direction I in which I
may go. I have not found and good lab data, perhaps product spec data
may show a more certain relation.

I will also re-examine the spherical volume "proof" I outlined. If I
did not correctly differentiate then that could be a source of error
in my concept.

Best Regards,

AAG

### mpalenik

Apr 16, 2012, 9:56:35 PM4/16/12
to
On Mar 31, 4:47 pm, "Anon E. Mouse" <agall...@gmail.com> wrote:
> > As far as I can tell the dissipation over distance relationship for
> > lux, gravitation, electric force, etc. should vary with 4/3 the
> > inverse cube of r. Can anyone refute this hypothesis? I find it very
> > troubling that I myself can't!
>
> > AAG
>
> I finally located some seemingly reliable data from an inverse square
> law experiment measuring luminosity over distance. The greatest
> deviation they report is for the shortest distance, a finding others
> also report.
>

So, again, I'm pretty late to the party, but I just find it really
weird that no one actually even remotely attempted to address his
relatively simple question (with the exception of one sentence in the
moderator's note) and instead started discussing FRW cosmology, which
as far as I can tell, is not related even remotely to his original
question.

First of all, the geometric reason that 1/r^2 makes sense is this:
You can think of field lines, or radiation, or whatever you're looking
at, as spreading out as you get farther away from the source. They
don't expand to fill the volume, but rather, as they move outward,
they cover the surface of larger and larger spheres. Think of it like
if you were to explode a bomb inside a spherical cavity. What would
the density of the shrapnel be on the surface of that sphere? It
would go as 1/r^2, since the surface area of the cavity goes as
http://en.wikipedia.org/wiki/Flux

As for evidence that there actually is a 1/r^2 law:

1) There are no stable orbits for 1/r^3 forces. By applying
conservation of angular momentum, you can write an effective potential
of kinetic energy + potential, and show that it has no minimum forces
that go as 1/r^3.
2) A 1/r^2 force law is consistant with Kepler's laws of planetary
motion (i.e. it makes the same predictions)
3) As the moderator said, energy-momentum conservation requires a 1/
r^2 law (light, as it moves away, spreads out over larger and larger
spherical shells with surface area 4pi*r^2, NOT larger and larger
volumes)
4) I believe there's a reference in the Feynman lectures to very
accurate measurements made of electric fields for the purposes of
proving the masslessness of photons that shows that they go as 1/r^2.