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Title: Gravitational Faraday Effect Produced by a Ring Laser


1
Gravitational Faraday Effect Produced by a Ring
Laser
  • James G. OBrien
  • IARD Bi-Annual Conference
  • University Of Connecticut
  • June 13th, 2006

2
History
  • Gravitational Frame Dragging was first
    introduced as a consequence of the General Theory
    of Relativity. It states that masses not only
    curve space and time, but rotating masses cause
    the very fabric of space and time to twist as
    well.

Current tests of the Frame Dragging Effect
include Gravity Probe B (2004), launched by NASA,
in conjunction with Stanford University under the
guidance of Francis Everett. This mechanical
method of testing the Frame Dragging Effect uses
ultra sophisticated gyroscope methods, and
telescope technology.
3
The Balazs Effect
  • The idea of using a non-mechanical method of
    measuring the gravitational frame dragging was
    well documented in 1957 by N.L. Balazs. His idea
    was to use a gravitational field to change the
    plane of polarization of an incident light beam,
    due to a slowly rotating massive body. See
    below

Change in Angle
Although in reality, as seen above, this presents
many technical difficulties.
4
Malletts Ring Laser
  • In 2000, Dr. Mallett documented the gravitational
    effects of a circulating laser.

a
a
a
a
Working in the linear approximation for the weak
gravitational field produced by the ring laser,
Mallett showed that if a massive, spinning
neutron were placed at the center, the precession
would be
5
Linear Combinations
  • But there is another way to observe the
    gravitational frame dragging effect, Light on
    Light.
  • After meeting with Francis Everett, Mallett
    suggested an attempt to combine Balazs into his
    own work.
  • Of course, along the way, we see that the rabbit
    hole is deeper than we expect

6
Classical Faraday Effect
  • Recall the Classical Faraday Effect

For an incident beam of light, when influenced by
a magnetic field, the plane of polarization
precesses (Classical Faraday Effect). Now, the
startling consequence is that if the light is
reflected, the polarization does not precess back
to its original state, but is instead amplified
in the new direction.
7
Classical Faraday Effect
8
Foundations
  • Original Goal To determine if and how the plane
    of polarization of an incident beam is affected
    by a ring laser.
  • Thus, we turn to the foundations given by
    Mallett, and work in the linear approximation for
    the gravitational field produced by the metric of
    the ring laser

9
Required Calculations
  • Where

Now having stated the givens, we are ready to
proceed by first calculating how Maxwells
equations are modified by the Gravity Field.
10
Maxwells Equations in G-Field
  • We see that the Modified Electromagnetic Fields
    are

Note The vector g is a three dimensional
representation of the off diagonal elements of
the metric viz. the (0i) components.
Where we have reverted to the 3-space notation to
see Maxwell equations more clearly.
11
Maxwell Continued
  • Thus, we see the Maxwell Equations are

Now, the above equations are still in terms of
both E and D as well as both H and B. Next, we
make some approximations and write the Maxwell
equations in terms of only D and B.
12
Approximations and Reductions
  • As we are working in the linear approximation, we
    can assume that the gravitational field produced
    by the ring laser is weak. Also, there are no
    other electromagnetic sources (point charges,
    currents, etc), thus

13
Final form of Maxwell
  • We see that after writing the Maxwell equations
    in terms of only B and D yields equations of the
    form

Which can be reduced after some labor since
div(g)0, leaving
And it is now clear as to the terms in which we
will need to solve these equations. Thus, we
turn our attention now to the incoming beam of
light.
14
Incident Beam
  • Let the incident ray be plane polarized and
    traveling in the z-direction. Recall that the
    ring laser is oriented in the x-y plane. Hence

More grinding shows that for an arbitrary vector
t, that
In lowest order terms (weak field). Note also in
the above is the first appearance of the
dimension a of the size of the ring laser.
15
Coupling of Field Equations
  • Applying all of the previous to the Maxwell
    equations, we are left with the following set of
    coupled equations

We can then eliminate the time differentials and
produce a set of full D.Es, by making the
following substitution
16
Total Differential Equations
  • Using the previous, we arrive at the following,
    still coupled equations

Now, assuming plane wave solutions for the
fields, along with some modification function due
to the gravity field, denoted by l(z), we see
then
17
Solving
  • With these new substitutions, we are led to the
    equations

Finally, after some more work, we arrive at the
pleasing result
Note, we arrived at the above equation only after
exploiting the fact that both l(z) and sigma are
small. Now we have a differential equation for
the modification to the plane waves, which can be
integrated immediately.
18
Etc etc
  • Once the integral is known, we can back
    substitute into the expressions for B and D. We
    can thus resolve the components of the Electric
    Field using the standard forms of

And setting the amplitudes as equal (polarization
angle changes, not amplitude)
19
Polarization Shift
  • For once, a simple calculation shows the shift in
    polarization is

Thus, we see that the change in angle is simply
the integral we calculated earlier. This result
makes sense since if we let l(z)0 then the
change in polarization angle is zero as expected.
Thus without further ado, we calculate the
change in polarization angle for the incident
beam caused by the ring laser.
20
Polarization Shift Continued
  • Evaluation of the integral yields

While at the limit where z increases without
bound (off to infinity), the shift is
Change in polarization due to the ring laser.
(Gravitational Analog of the Classical Faraday
Effect)
21
But the Story Continues
  • Original Goal was successful
  • Admittedly, the effect we shown is VERY small
  • So can we remedy this?
  • As it turns out, there is a gravitational analog
    of the consequence of this new Faraday-Like
    effect, as discussed earlier

22
Gravitational Faraday Effect
  • With a little more work, we can show that there
    does indeed exist a gravitational analog of the
    classical faraday effect. Let us now go back to
    the definitions of the incident beam, and let it
    incident from negative infinity. Then

23
Evaluation
  • Upon evaluation of the integral again, we see the
    result that

Which due to the sgn function, is positive
definite. Thus, we obtain in the large z region,
the previous result
Hence, no matter which way the beam is incident,
the change in polarization orientation is the
same (as seen in the classical case). Thus,
reflection of the light back through the ring
laser results in an amplification of the
precession angle.
24
Current Work
  • The existence of this gravitational analog allows
    us the possibility of terrestrial experiments of
    gravitational frame dragging.
  • Dr. Chandra Roychoudhuri is currently designing
    experimental apparatus to perform the experiment.

25
Design
Use of Confocal Lasers will be employed as the
incident beam in the ring laser as pictured.
This technique provides the highest amount of
finesse which allows for the maximum amount of
reflections without loss of intensity of the
incident beam. Then, hopes are to stack the ring
lasers in a helical pattern and allow for an
increase in polarization precession. Then, the
frame dragging effect can be measured by allowing
the wave to propagate over time, as opposed to a
huge space.
26
Conclusion
  • Gravitational Frame Dragging may be able to be
    tested in an easily controlled, terrestrial lab.
  • This is due to the existence of a gravitational
    analog of the Classical Faraday Effect.
  • David Cox and I would like to thank you all for
    this wonderful opportunity and for your attention!
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