Can an alternative measure of Newtonian potential
energy remove the Schwarzschild singularity
(or at least move it to r=0)?

A shell of matter gives rise to a potential which
causes a redshift in the frequency of light.

Redshifts do not add together - they multiply,
for instance
1 + z = (1 + z_1) (1 + z_2), not 1 + z_1 + z_2.

It is fundamental to general relativity that
energy and frequency transform identically in a
gravitational field - it would be inconsistent
to assume differing transformations.

Thus the calculation of gravitational potential
energy should be obtained not from addition
over the shells, but from a process reflecting
the multiplicative nature of the redshift.

It can be shown that a continuous multiplication
of a function can be defined in the same way
that integration is defined to represent a
continuous summation.

If u is the integral of a function, the
continuous multiplication of the function gives
U = U_0 exp(u / U_0) as the product where U_0
is an initial value.

For u = -GMm/R as the Newtonian gravitational
potential energy obtained from integration
over the shells, continuous multiplication
gives the product U = U_0 exp(u / U_0) where
the initial value can be surmised to be the
rest energy U_0 = mc^2 from Mach's principle,
which postulates that rest energy is potential
energy due to displacement of the mass, m, from
distant matter.

In the presence of a gravitational field the rest
energy diminishes from U_0 = mc^2 to
U = mc^2 exp(-GM/Rc^2).

This can be compared to general relativity which
has the gauge property that the dimensions of
time, length and mass vary with the scale factor
sigma = sqrt(1 - 2GM/Rc^2)
as
T = T_0 / sigma
L = L_0 sigma
M = M_0 / sigma^3
respectively, transforming the rest energy from
E_0 = mc^2 far from the gravitational field to
E = E_0 sigma
in the presence of the field.

U and E are almost indistinguishable for weak
gravitational fields, so the question becomes
how to get the exponential scale factor associated
with U to replace the scale factor sigma which
appears in the Schwarzschild solution.

My guess would involve an exponential map of the
metric to produce the multiplicative process.

A file with more details (shells.pdf)
is available at
http://sites.google.com/site/revisingnewton/

Colin Walker wrote on Sun, 28 Jun 2009 13:05:09 +0000:

Can an alternative measure of Newtonian potential energy
remove the Schwarzschild singularity (or at least move it
to r=0)?

The Schwarzschild singularity in GR is already at r=0.


--
http://www.canonicalscience.org/

On Jun 28, 5:05*am, Colin Walker wrote:
Can an alternative measure of Newtonian potential
energy remove the Schwarzschild singularity
(or at least move it to r=0)?

Given that Newtonian gravitation and GR are different theories, I'm
going with "no". Furthermore, displacing the singularity by a
coordinate change doesn't change the physics.


A shell of matter gives rise to a potential which
causes a redshift in the frequency of light.

Redshifts do not add together - they multiply,
for instance
1 + z = (1 + z_1) (1 + z_2), not 1 + z_1 + z_2.

It is fundamental to general relativity that
energy and frequency transform identically in a
gravitational field *- it would be inconsistent
to assume differing transformations.

Since when?

In GR, energy is simply the result of the metric being invariant under
an infintesimal change in the t coordinate. Frequency isn't any more
special.

[snip rest]

On Jul 4, 8:56*pm, Eric Gisse wrote:
On Jun 28, 5:05*am, Colin Walker wrote:

Can an alternative measure of Newtonian potential
energy remove the Schwarzschild singularity
(or at least move it to r=0)?

Given that Newtonian gravitation and GR are different theories, I'm
going with "no". Furthermore, displacing the singularity by a
coordinate change doesn't change the physics.

Sorry for using the wrong term - I should have
said Schwarzschild radius or absolute event
horizon instead of singularity.
What I am proposing is not a coordinate
transformation but a field transformation.
Different field, different physics.



It is fundamental to general relativity that
energy and frequency transform identically in a
gravitational field *- it would be inconsistent
to assume differing transformations.

Since when?

In GR, energy is simply the result of the metric being invariant under
an infintesimal change in the t coordinate. Frequency isn't any more
special.

A reference for the gauge property is
Gravitation and Relativity by M.G. Bowler,
published in 1976.
Some elementary problems in GR can be solved
using dimensional analysis.

For example, a photon has energy E = h f
where h is Planck's constant and f is
photon frequency.
By breaking down h into its dimensions of
time, length and mass, it is clear that
h = h_0 so Planck's constant does not change
in a gravitational field.
Doing the same with frequency (1/T) and
energy shows that they both vary as sigma.
My point is that if the redshift of frequency
is computed using multiplication then so
should the calculation of energy.

This approach has limited application but
is sufficient for a simple idealized free-fall
example.
By contrast, the gravitation constant varies
as the 8th power of sigma, G = G_0 sigma^8
making dimensional analysis difficult for
more complicated problems.

On Jul 1, 6:06*pm, CarlBrannen wrote:
On Jun 28, 6:05 am, Colin Walker wrote:

Can an alternative measure of Newtonian potential
energy remove the Schwarzschild singularity
(or at least move it to r=0)?

Hmmm. You might try using the Newtonian equivalent of the
Schwarzschild equations of motion. A paper which gives them,
along with the (somewhat simpler) equations of motion for
the Gullstrand-Painleve coordinates of the Schwarzschild
metric is at:http://www.brannenworks.com/Gravity/BranGravArXiv.pdf

Thanks for the link to your paper.
The modern role of force has been a source
of confusion for me, and I was unaware of
Gullstrand-Painleve coordinates.

I think you have provided the means to answer
my question because GP coordinates are entirely
determined by the escape velocity.

The river model of black holes by
Hamilton and Lisle 2006 arXiv:gr-qc0411060v2
gives a metric
ds^2 = -dt^2 + (dr + beta dt)^2
+ r^2(dtheta^2 + sin^2theta dphi^2)
where dt is in the frame of an object which is
in free fall from infinity and where beta is
the speed of that object, or the escape velocity.

They say beta 'can also be considered to be a
more general function of radius'. I take this to
mean that it can be chosen freely.

The escape velocity in GR can be found by
increasing the energy E = E_0 sigma by the
Lorentz factor 1/sqrt(1 - v^2/c^2) to get
c sqrt(1 - sigma^2)
with sigma = sqrt(1 - 2GM/Rc^2)
which is also the Newtonian result.

With an exponential scale factor, the escape
velocity would be
v = c sqrt[1 - exp(-2GM/Rc^2)]
which determines Gullstrand-Painleve
coordinates with beta = v.

Thanks again

Colin