Gravitational Faraday Effect Produced by a Ring Laser. James G. O’Brien IARD Bi-Annual Conference University Of Connecticut June 13 th , 2006. History.
James G. O’Brien
IARD Bi-Annual Conference
University Of Connecticut
June 13th, 2006
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.
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.
In 2000, Dr. Mallett documented the gravitational effects of a circulating laser.
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:
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.
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:
Now having stated the givens, we are ready to proceed by first calculating how Maxwell’s equations are modified by the Gravity 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.
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.
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
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.
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.
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.E’s, by making the following substitution:
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:
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.
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)
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.
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)
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:
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.
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.