Charles Francis
Oct12-06, 04:13 AM
I recently posted showing that replacing the affine connection in
general relativity with a teleconnection would lead to a
reinterpretation of measurements of cosmological redshift, in a manner
which I believe resolves a number of issues troubling cosmology. This
post is intended to describe a teleconnection in a friendly (non-
mathematical) way, so that readers can understand how it yields redshift
proportional to the square of the expansion parameter.
First observe that in classical gtr there is no local meaning to
expansion, because locally matter is measured relative to matter. We
establish length scales relative to local matter. If all matter locally
were to "shrink", including ourselves and all the matter contained our
measuring apparatus, then our fundamental length scales would "shrink"
in direct proportion, so that it would make no difference to the
numerical results of local distance measurements and there would be no
local means to detect the fact.
To talk about shrinking we have to make a comparison between the
coordinates we define here and now, using here and now clocks and
rulers, and the coordinates defined somewhere else. In practice the only
way we can make a direct comparison is to look at the light coming from
somewhere else and to deduce the distant coordinate system from the
properties we find in the light. We can only do this if we know how
light behaves.
In special relativity we define Minkowski coordinates using two way
reflection of light. This can be done locally, but sr does not tell us
how to extend coordinates indefinitely far from the origin. In general
relativity it is an assumption that light comes is parallel transported
as though it were sliding across a curved surface, but there is no
direct empirical evidence. In practice we can't slide rulers
indefinitely far into space. If we do slide a ruler, for example on
Pioneer, a short way into space we are no longer in direct contact with
it, and have to rely entirely on e.m. transmissions to read the ruler.
We can define 3-momentum from the Fourier transform of the probability
amplitude, and it is here assumed that this is a well defined, locally
conserved, property. Now, if momentum is a well defined property for Alf
on a space craft or another planet, then it is also a well defined
property for Beth on Earth, because Alf can communicate to Beth his
value of momentum. Alf's value of momentum does not have to be the same
as Beth's, but there has to be a way of converting his value to the
value Beth will use. At the time of emission of a photon passing from
Alf to Beth, Alf defines synchronous, conformally flat co-ordinates in 3
dimensions at constant cosmic time t. This is most naturally done in a
closed cosmos so that the universe can mapped onto a finite space, which
I will call Alf's map. Beth defines Beth's map in exactly the same way,
to the same scale, at the time of detection of the photon at cosmic time
t0.
For a closed universe in three dimensions both maps consist of the
interior of a sphere. Let a(t) be the radius of the universe, and let
a0=a(t0). If the universe expands during the time of travel of the
photon from Alf to Beth, then Beth's map is larger than Alf's map. The
maps can be placed in direct correspondence by enlarging Alf's map by a
factor a0/a(t). The teleconnection is defined such that photon momentum
is represented by an arrow of equal length and direction on Beth's map
and on Alf's enlarged map. This assumes that if momentum is well defined
at one place and time then it is well defined at other places and time
also and is justified empirically in so far as observation yields
precise values of cosmological redshift after allowing for dispersion
due to dust or other known factors. This is a fundamental assumption in
this model, of equal importance to the assumption of the constancy of
the speed of light in special relativity. Like that assumption, if it
were dropped we would be left, not with a different theory, but with no
known consistent theory (see e.g. Eppley and Hannah, 1977, for some of
the problems with the definition of momentum in curved space time).
A 4 dimensional map is found by considering all the times and positions
where Alf and Beth might be. This is a Penrose diagram in each space
direction, and is such that an arrow representing photon momentum is of
constant length and direction everywhere. This is equivalent to the
statement that we have plane wave motions for light in these
coordinates. Beth can compare the scales of her map to Fred's map by
studying at the red shift. There are two scaling effects. First Alf's
map has been enlarged by a factor a0/a(t). In addition, the scaling on
the map changes as you move from one point to another. That gives
another factor a0/a(t). Thus, the model predicts that the cosmological
redshift factor varies with the square of the expansion parameter
1+z=a0^2/a^2(t).
Regards
--
Charles Francis
general relativity with a teleconnection would lead to a
reinterpretation of measurements of cosmological redshift, in a manner
which I believe resolves a number of issues troubling cosmology. This
post is intended to describe a teleconnection in a friendly (non-
mathematical) way, so that readers can understand how it yields redshift
proportional to the square of the expansion parameter.
First observe that in classical gtr there is no local meaning to
expansion, because locally matter is measured relative to matter. We
establish length scales relative to local matter. If all matter locally
were to "shrink", including ourselves and all the matter contained our
measuring apparatus, then our fundamental length scales would "shrink"
in direct proportion, so that it would make no difference to the
numerical results of local distance measurements and there would be no
local means to detect the fact.
To talk about shrinking we have to make a comparison between the
coordinates we define here and now, using here and now clocks and
rulers, and the coordinates defined somewhere else. In practice the only
way we can make a direct comparison is to look at the light coming from
somewhere else and to deduce the distant coordinate system from the
properties we find in the light. We can only do this if we know how
light behaves.
In special relativity we define Minkowski coordinates using two way
reflection of light. This can be done locally, but sr does not tell us
how to extend coordinates indefinitely far from the origin. In general
relativity it is an assumption that light comes is parallel transported
as though it were sliding across a curved surface, but there is no
direct empirical evidence. In practice we can't slide rulers
indefinitely far into space. If we do slide a ruler, for example on
Pioneer, a short way into space we are no longer in direct contact with
it, and have to rely entirely on e.m. transmissions to read the ruler.
We can define 3-momentum from the Fourier transform of the probability
amplitude, and it is here assumed that this is a well defined, locally
conserved, property. Now, if momentum is a well defined property for Alf
on a space craft or another planet, then it is also a well defined
property for Beth on Earth, because Alf can communicate to Beth his
value of momentum. Alf's value of momentum does not have to be the same
as Beth's, but there has to be a way of converting his value to the
value Beth will use. At the time of emission of a photon passing from
Alf to Beth, Alf defines synchronous, conformally flat co-ordinates in 3
dimensions at constant cosmic time t. This is most naturally done in a
closed cosmos so that the universe can mapped onto a finite space, which
I will call Alf's map. Beth defines Beth's map in exactly the same way,
to the same scale, at the time of detection of the photon at cosmic time
t0.
For a closed universe in three dimensions both maps consist of the
interior of a sphere. Let a(t) be the radius of the universe, and let
a0=a(t0). If the universe expands during the time of travel of the
photon from Alf to Beth, then Beth's map is larger than Alf's map. The
maps can be placed in direct correspondence by enlarging Alf's map by a
factor a0/a(t). The teleconnection is defined such that photon momentum
is represented by an arrow of equal length and direction on Beth's map
and on Alf's enlarged map. This assumes that if momentum is well defined
at one place and time then it is well defined at other places and time
also and is justified empirically in so far as observation yields
precise values of cosmological redshift after allowing for dispersion
due to dust or other known factors. This is a fundamental assumption in
this model, of equal importance to the assumption of the constancy of
the speed of light in special relativity. Like that assumption, if it
were dropped we would be left, not with a different theory, but with no
known consistent theory (see e.g. Eppley and Hannah, 1977, for some of
the problems with the definition of momentum in curved space time).
A 4 dimensional map is found by considering all the times and positions
where Alf and Beth might be. This is a Penrose diagram in each space
direction, and is such that an arrow representing photon momentum is of
constant length and direction everywhere. This is equivalent to the
statement that we have plane wave motions for light in these
coordinates. Beth can compare the scales of her map to Fred's map by
studying at the red shift. There are two scaling effects. First Alf's
map has been enlarged by a factor a0/a(t). In addition, the scaling on
the map changes as you move from one point to another. That gives
another factor a0/a(t). Thus, the model predicts that the cosmological
redshift factor varies with the square of the expansion parameter
1+z=a0^2/a^2(t).
Regards
--
Charles Francis