Time Travel Research Center © 2005 Cetin BAL - GSM:+90 05366063183 - Turkey / Denizli
Deviation of Light near the Sun
|
![]() |
(1) |
and this quantity does not depend on the frame in which it is evaluated.
In other words all observers agree on the value of
computed by
Formula (1), although the values of
and
differ from
one system of reference to another.
Be careful: unless otherwise indicated, distances will be measured in
units of time, as is often done in astronomy. We have chosen to do so in
writing Equation (1). On the contrary if distances s are expressed in
conventional units, for instance in centimeters, then one should pass from
the latter to our distance s expressed in seconds via the formula
cm/s.
(Expressing distances and times with the same unit would amount to taking
the speed of light equal to unity.)
In general relativity the property of invariance of the proper interval
with respect to a change of the coordinates remains valid but only locally,
i.e. under the condition of staying in a sufficiently small region of
spacetime (its size depends on the accuracy of the measurements). The main
novelty concerns the expression of the proper time as given by Formula (1).
The coefficients entering this formula depend now on the point of spacetime
under consideration and the resulting expression takes the name of
metrics. In fact the whole structure of spacetime, and especially its
curvature, is included in the local expression of
and in the form of its coefficients.
We are interested here in the structure of spacetime around the sun. In
order to describe the physics locally we consider two nearby events
separated by infinitesimal amounts of the time and space coordinates
,
,
and
.
If space were flat, the metrics would have the form
which is usually written (and a little sloppily) by convention as
![]() |
(2) |
By working in spherical coordinates, in a plane containing the center of the sun (this choice removes one spatial coordinate), that formula becomes
where r denotes the distance to the center and
an azimutal angle in the plane of the orbit (see the figure below).
But spacetime around a center of attraction of mass
(for instance a black hole or the vicinity of the sun) is not flat. It is
characterized by the Schwarzschild metric
![]() |
(3) |
The story is really fantastic: the whole structure of spacetime is embodied in this "simple" formula (3).Even the famous black hole lurks behind those apparently innocuous symbols.
One question: in which units is expressed the mass
in that formula? It is seen that
has the dimension of a length, a quantity that we measure here in seconds.
Therefore
will also be measured in seconds. The formula allowing to transform grams in
seconds is
where
.
The metric, that is (exactly) the formula expressing at a givent point of
spacetime the temporal interval between two nearby events, reveals the
presence of curvature as soon as the expression deviates from Formula (2)
corresponding to flat euclidian space. That metric will allow us to find the
properties of the motion of a test particle free from acceleration. Actually
both special and general relativity teach us that between two given events
and
a freely moving body follows the path for which the time interval
is maximum. Equivalently one can say that a freely moving particle follows a
geodesic of spacetime as a geodesic is precisely defined by this property of
maximazing the time interval.
Definition of a geodesic: the geodesic between two eventsand
is the wordline for which the interval of proper time between
and
is maximum.
That property of maximazing the proper time will allow us to derive the equations of a geodesic. It will also yield the expressions of the energy and angular momentum of a particle in orbit around the center of attraction.
Let us apply the principle of maximisation of the proper time interval in
the following manner. Suppose that a free spatial ship (whose rockets are
turned off) falls radially, therefore along a straight path, towards the
central attractive mass. Imagine that three successive flashes, with nearby
time and space coordinates, are emitted inside the spaceship. We observe
those three events in some external frame. In that latter frame the event
consists in the emission of a flash at time
when the spatial engine is located at radius
.
The flash
is emitted at time
when the cabin is at radius
.
The flash
is emitted at time
when the cabin is at radius
.
The quantity
is assumed to be small. We then assume that we vary the intermediate
coordinates of
.
The principle of maximal aging says that the geodesic starting from
and ending at
will pass through Event
such that the proper time interval
![]() |
(4) |
is maximum. Here
measures the interval over the first spacetime segment
,
which connects
to
and
measures the time interval over the second segment
,
which connects
to
.
In order to avoid varying all quantities at the same time, we assume in
this experiment that the locations of the radii
,
and
are fixed and that only the time
,
at which the second flash is emitted, is allowed to change. According to
Formula (3) the interval of proper time over the first segment
is given by its square
![]() |
(5) |
from which we deduce
![]() |
(6) |
The lapse of time over Segment
between the events
and
is
,
and therefore the proper time duration
is given by
![]() |
(7) |
from which we deduce
![]() |
(8) |
To make the total time interval
maximum with respect to a variation
of the time
,
we write
![]() |
(9) |
Deducing
and
from Equations (6) and (8) and letting quite naturally
and
,
we easily get
![]() |
(10) |
The left side of that equation depends only on parameters characterizing
the first segment A (which connects
to
).
The right side depends only on parameters related to the second segment B (which
connects
to
).
We have discovered in Equation (10) a quantity that is the same for both segment. This quantity is thus a constant of the motion for the free particle under consideration. For good physical reasons (especially to recover the formulae of special relativity), one is led to identify that constant of motion as the ratio of the energy of the particle to its mass. We write this very important result under the form
![]() |
(11) |
an expression in which we have returned to the differential notation for
the intervals
and
.
Incidentally we may notice that with the units we have chosen, energy
and mass
are expressed in the same unit (for instance the centimeter).
We have applied the principle of maximazing the proper time interval by
varying the time of the intermediate event E2. We
now perform the same operation but this time we vary the angle
of that intermediate event. We recall that
measures the direction of the moving particle with respect to some direction
chosen as the origin. We call it the azimuth.
We consider again three events consisting in the emission of flashes
inside a spaceship floating freely in space. The first segment
connects Event
to Event
.
The second segment
connects
to
.
The azimutal angle of the first event is fixed at
.
The angle of the last one is fixed at
.
The intermediate azimuth is taken as the variable
.
Again in order not to vary everything at the same time, we assume that the
radius
at which the second flash is emitted stays constant.
We follow the same chain of reasoning as in the previous section. From
the metric (3), the time interval
over the first segment is given by its square
![]() |
(12) |
and the interval
over the second by
![]() |
(13) |
from which we get
![]() |
![]() |
![]() |
(14) |
![]() |
![]() |
![]() |
(15) |
By writing
one easily obtains, similarly to Formula (10)
![]() |
(16) |
after having written quite naturally
and
.
The left side, which contains only terms that are specific to the first
segment, is equal to the right side, which contains only terms relative to
the second segment. We thus exhibit another constant of motion, namely
(by shifting back to the differential notation), a quantity that turns out
to be identified with the ratio of the angular momentum
of the particle to its mass
,
which we write as
![]() |
(17) |
Technically speaking in order to determine the trajectory of a moving
body free from acceleration we apply the following strategy. Knowing the
energy
and the angular momentum
of the particle of mass
(
and
depend on the initial conditions) we can follow the position of that
particle by computing the increments of its spacetime coordinates
,
and
as the proper time
itself advances. Algebraically for each increment
of the proper time we compute (or the computer calculates) the corresponding
increments
,
and
of the coordinate of the mobile body. The squares of the increments
and
are extracted from Equations (11) and (17) in the following form:
![]() |
![]() |
![]() |
(18) |
![]() |
![]() |
![]() |
(19) |
We notice that the expression of
is missing. We get it by transporting the values of
and
into the metric equation (3) and solving it for
.
This yields
![]() |
(20) |
By dividing both sides of Equations (20) and (19) we directly arrive to
the equation of the orbit in polar coordinates as
![]() |
(21) |
The preceding treatment is apparently not relevant to the case of a
photon. In fact the calculation of the trajectory was done by incrementing
the proper time but this latter concept has no meaning for a photon since
the interval between two events that are located on the wordline of a photon
is always equal to zero (as at light velocity
,
the interval
vanishes).
Nevertheless it happens that by letting the mass of the particle tend
towards zero, one arrives at the right results. Thus for
our equation (21) takes the form
![]() |
(22) |
That equation will allow us to determine the deviation of ligth rays passing near the sun.
It is necessary to specify the parameters found in the formulae. First
the angular momentum of the moving particle at infinity is equal by
definition to the product of its linear momentum
by what is called the impact parameter
,
which represents the distance between the center of attraction (the sun in
the present case) and the initial direction of the velocity of the particle
(see the figure).
In other words
![]() |
(23) |
In addition it is known that the momentum
of a photon is equal to its energy
(with the units that were chosen). It results at once from this formula that
![]() |
(24) |
If the ratio
is equal to the impact parameter
Equation (22) writes as
![]() |
(25) |
Formula (25) will allow us to determine the change in the direction of a
light pulse caused by the gravitational field of the sun. To achieve this
aim we have to sum up the successive infinitesimal increments
of the azimuthal angle
along the path. This means that we have to carry out the integration of
when r varies from the minimum distance denoted
(
is the radius of the sun if the light ray grazes its surface). We should
still multiply that quantity par 2 to account for both symmetrical "legs" of
the trajectory (the photon first approaches the Sun then recedes from it).
It is necessary to stipulate a further point, namely the relation
existing between the two quantities
and
that we have introduced and that are not independent. The point
corresponds to the place where the light photon is closest to the sun. There
the photon moves tangentially. Since at that point there is no radial
component, we can write that the derivative
vanishes. It suffices to take the element
from Equation (25) to find immediately
![]() |
(26) |
so that this same equation (25) becomes
![]() |
(27) |
The form of the expression dictates to us to pose
where
varies between 1 and 0. The last equation (27) then becomes
![]() |
![]() |
![]() |
|
or | |||
![]() |
![]() |
![]() |
(28) |
Consequently the infinitesimal variation
of the azimuth is given in terms of the variation
of
by
![]() |
![]() |
![]() |
|
![]() |
![]() |
(29) |
The presence of the term
in Expression (29) encourages us to make the change of variable
which leads to
![]() |
(30) |
By observing that
we end up with the final equation of the trajectory under the form
![]() |
(31) |
with
It is interesting to emphasize that so far there have been no approximation. This is quite rewarding.
The small value of the term
will allow us to make an approximation and in this way will make us able to
complete the integration. In conventional units the mass of the sun is
grams
and its radius is
centimeters.
By using the factor
cm/g
which makes it possible to transform grams into centimeters, we get
In Equation (31) we can thus use the classical approximation
to arrive at
![]() |
(32) |
Therefore the total variation of the azimuth
along the path of the photon is
![]() |
![]() |
![]() |
(33) |
![]() |
![]() |
(34) | |
![]() |
![]() |
(35) |
The first term
gives the total change in the azimuthal angle of the photon where there is
no Sun present, since in that case the photon follows a straight path. But
the second term gives the additional angle of deflection
with respect to this straight line (see the figure)
![]() |
![]() |
![]() |
(36) |
or in conventional units | |||
![]() |
![]() |
![]() |
(37) |
Numerically at the surface of the sun (with the values of the mass and
the radius given above) one finds
radian,
or ( knowing that
radians equal 180 degrees and that there are 60 minutes of arc in one degree
and 60 seconds of arc in one minute of arc)
Hiçbir yazı/ resim izinsiz olarak kullanılamaz!! Telif hakları uyarınca bu bir suçtur..! Tüm hakları Çetin BAL' a aittir. Kaynak gösterilmek şartıyla siteden alıntı yapılabilir.
The Time Machine Project © 2005 Cetin BAL - GSM:+90 05366063183 -Turkiye/Denizli
Ana Sayfa /
index /Roket bilimi /
E-Mail /CetinBAL/Quantum Teleportation-2
Time Travel Technology /Ziyaretçi Defteri /UFO Technology/Duyuru