Title: Gravitational Faraday Effect Produced by a Ring Laser
1Gravitational Faraday Effect Produced by a Ring
Laser
- James G. OBrien
- IARD Bi-Annual Conference
- University Of Connecticut
- June 13th, 2006
2History
- 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.
3The 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.
4Malletts Ring Laser
- In 2000, Dr. Mallett documented the gravitational
effects of a circulating laser.
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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
5Linear 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
6Classical 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.
7Classical Faraday Effect
8Foundations
- 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
9Required Calculations
Now having stated the givens, we are ready to
proceed by first calculating how Maxwells
equations are modified by the Gravity Field.
10Maxwells 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.
11Maxwell 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.
12Approximations 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
13Final 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.
14Incident 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.
15Coupling 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
16Total 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
17Solving
- 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.
18Etc 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)
19Polarization 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.
20Polarization 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)
21But 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
22Gravitational 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
23Evaluation
- 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.
24Current 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.
25Design
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.
26Conclusion
- 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!