Inertia Retention in Stars
A hypothesis on dark matter
James Moynihan, B.Eng., Ph.D.
Abstract: This paper indicates how stars may accumulate
inertia causing the space-time curvature around them to be greater than that
merited by their actual mass. This paper
may obviate the need to search for new particulate constituents of dark matter.
The Transfer of Momentum
A Thought Experiment
In Figure 1
A photon is
released from atom A at point p1 at time, t1 in some
local frame of reference. The energy of
this photon is determined at time t1 and is equal to,
The
magnitude of the momentum of this photon is
But momentum
is a vector and the direction of the photon’s momentum transfer is not known
until the photon is received by one of the many atoms which can receive it at
time, t2.
In essence
at time t1 < t < t2 for the photon,
The
momentum can remain infinitesimally close to zero.
At time, t2
atom B at point p2 receives the photon. Only then are both the magnitude and direction
of p known. Only at time t2
can momentum p leave atom A’s region.
The Ideal Star
This is not
meant to be a description of a real star.
It is only meant to demonstrate a concept.
The ideal
star is spherical. It has mass, ms.
It converts mass to energy and emits this energy as photons of radiation
at its surface.
At the
surface of the ideal star photons carry away packets of energy,
They should
also carry away momentum of magnitude,
But because
the direction of their momentum is not known at time of release, the photons
cannot take away their momentum. The
inertia remains at the star.
It has to
remain at the star until the direction of momentum transport is known. This is
only known when the photons are captured by other atoms. Most photons wander off through space forever
without recapture.
The
momentum of un-captured photons has to remain at the star.
Yet the mass
used to generate these photons has been lost.
Therefore
the inertia and mass of the star cannot be identical.
 |
Figure 1
The mass-to
energy conversion process acts like a momentum barrier. The photons leave with their energy but leave
their momentum behind.
Incidentally,
the ratio of the energy released to the momentum impacted away is
In linear
motion the momentum of the star is
which is
normally the same as
However, in
this case the momentum of the star is unchanged by the energy conversion
process: the star loses mass but does not lose momentum.
The
velocity of the star also remains unchanged since its surface is spherical,
with the same forces acting on all sides.
These
requirements can be satisfied by defining the inertia of the star to be the
current mass of the star plus the mass of all the matter the star has turned
into energy during its history. If a
mass mconv is lost in the energy conversion process then
Implications of Inertia Retention
The star
can produce a maximum energy of 
using all of its current mass.
However if
the star has converted a quantity of mass mh to energy during its history
then its inertia will be,
, where 
is the energy the star has radiated away in the past.
The extra
inertia
is equal to
the inertia of all the matter the star has turned into radiation in the past.
The effects
of the extra inertia could be mistaken for the effects of an unseen dark matter when the star is interacting
with another star. However since the
mass of the star is still ms the analysis of interactions which do
not involve motion and acceleration of the star itself should not reveal any
unexpected results.
Dark matter
Observations
indicate that galaxies seem to hold approximately nine times more hidden or
dark matter than visible matter. Given
this ideal star model this would mean 
for stars in the galaxies and this might indicate that the
stars in the galaxies have processed approximately ten times more matter into
energy than they now hold.
This
indicates that for the total population of stars 
And if
stars have been burning at the same rate for most of the history of the
universe then this would imply that most of the matter in galaxies resides in
stars with a lifetime equal to approximately one ninth of the age of the
universe.
This may be
verifiable.
Also, older
galaxies should exhibit more evidence of dark matter than young ones. This may also be verifiable.
Recent Observations
This image
reproduced from <http://hubblesite.org/newscenter/newsdesk/archive/releases/2006/39/image/a>
shows a galaxy cluster formed from the collision of two smaller clusters. The hot visible gas is shown in red but
analysis of gravitational lensing effects gives the impression that most of the
matter is dark and lies in the regions highlighted in blue.
Figure 2 STSc1-PRC2006-39a
The theory presented
here would suggest that this dark matter is only the accumulated inertia of the
two clusters separated from their respective normal, bright, or baryonic matter
by the gravitational attraction between the two colliding clumps of ordinary
matter. In other words, if it were not
for the ordinary Newtonian gravitational attraction between the two red clumps
of matter then the red clump on the left would continue on its way leftwards
surrounded by its halo of inertia and so would the smaller bullet-shaped
cluster on the right. The two “dimples”
of inertia are immune to the gravitational attraction effect and diminish like
a vortex into the void.
Conclusion
The
analysis here may suggest an alternative to the search for the particle identity
of the elusive dark matter.
This
concept may also be useful in explaining the concept of dark energy since, as
the universe ages the average ratio of momentum to mass has to increase.
Glossary
E Energy
h Plank’s constant
f Frequency of photon
λ Wavelength of photon
c Speed of light
p Momentum
v Velocity
Eh Energy released by star in its history
Is Inertia of star
Ih Inertia equivalent of Eh
ms
Mass of star
mh Mass converted to energy in star’s
history
The Author
James
Moynihan is a retired electronic engineer; he is a graduate of the