Gravitational Faraday Effect Gravitational Faraday Effect Produced by a Ring LaserProduced by a Ring Laser

James G. O’BrienJames G. O’BrienIARD Bi-Annual ConferenceIARD Bi-Annual Conference

University Of ConnecticutUniversity Of ConnecticutJune 13June 13thth, 2006, 2006

HistoryHistory

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

The idea of using a non-mechanical method of measuring the The idea of using a non-mechanical method of measuring the gravitational frame dragging was well documented in 1957 by N.L. gravitational frame dragging was well documented in 1957 by N.L. Balazs. His idea was to use a gravitational field to change the plane 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 of polarization of an incident light beam, due to a slowly rotating massive body. See below:massive body. See below:

Although in reality, as seen above, this presents many technical difficulties.

Change in Angle:

Mallett’s Ring LaserMallett’s Ring Laser

In 2000, Dr. Mallett documented the gravitational effects of a circulating In 2000, Dr. Mallett documented the gravitational effects of a circulating laser. 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:

a

a

aa

Linear CombinationsLinear Combinations

But there is another way to observe the But there is another way to observe the gravitational frame dragging effect, Light gravitational frame dragging effect, Light on Light.on Light.

After meeting with Francis Everett, Mallett After meeting with Francis Everett, Mallett suggested an attempt to combine Balazs suggested an attempt to combine Balazs into his own work.into his own work.

Of course, along the way, we see that the Of course, along the way, we see that the rabbit hole is deeper than we expect:rabbit hole is deeper than we expect:

Classical Faraday EffectClassical Faraday Effect

Recall the 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.

Classical Faraday EffectClassical Faraday Effect

FoundationsFoundations

Original Goal: To determine if and how the plane of polarization of an Original Goal: To determine if and how the plane of polarization of an incident beam is affected by a ring laser.incident beam is affected by a ring laser.

Thus, we turn to the foundations given by Mallett, and work in the linear Thus, we turn to the foundations given by Mallett, and work in the linear approximation for the gravitational field produced by the metric of approximation for the gravitational field produced by the metric of the ring laser:the ring laser:

Required CalculationsRequired Calculations

Where:Where:

Now having stated the givens, we are ready to proceed by first calculating how Maxwell’s equations are modified by the Gravity Field.

Maxwell’s Equations in G-FieldMaxwell’s Equations in G-Field

We see that the Modified Electromagnetic Fields are:We see that the Modified Electromagnetic Fields are:

Where we have reverted to the 3-space notation to see Maxwell equations more clearly.

Note: The vector g is a three dimensional representation of the off diagonal elements of the metric viz. the (0i) components.

Maxwell Continued…Maxwell Continued…

Thus, we see the Maxwell Equations are: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.

Approximations and ReductionsApproximations and Reductions

As we are working in the linear approximation, we can assume that the As we are working in the linear approximation, we can assume that the gravitational field produced by the ring laser is weak. Also, there gravitational field produced by the ring laser is weak. Also, there are no other electromagnetic sources (point charges, currents, etc), are no other electromagnetic sources (point charges, currents, etc), thusthus

Final form of MaxwellFinal form of Maxwell

We see that after writing the Maxwell equations in terms of only We see that after writing the Maxwell equations in terms of only BB and and DD yields equations of the form: 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.

Incident BeamIncident BeamLet the incident ray be plane polarized and traveling in the z-direction. 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: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.

Coupling of Field EquationsCoupling of Field Equations

Applying all of the previous to the Maxwell equations, we Applying all of the previous to the Maxwell equations, we are left with the following set of coupled equations: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:

Total Differential EquationsTotal Differential Equations

Using the previous, we arrive at the following, still coupled Using the previous, we arrive at the following, still coupled equationsequations::

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:

Solving…Solving…

With these new substitutions, we are led to the equations: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.

Etc etc…Etc etc…

Once the integral is known, we can back substitute into the Once the integral is known, we can back substitute into the expressions for expressions for BB and and DD. We can thus resolve the components of . We can thus resolve the components of the Electric Field using the standard forms of:the Electric Field using the standard forms of:

And setting the amplitudes as equal (polarization angle changes, not amplitude)

Polarization ShiftPolarization Shift

For once, a simple calculation shows the shift in polarization is: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.

Polarization Shift ContinuedPolarization Shift Continued

Evaluation of the integral yields: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)

But the Story Continues…But the Story Continues…

Original Goal was successfulOriginal Goal was successful

Admittedly, the effect we shown is VERY Admittedly, the effect we shown is VERY smallsmall

So can we remedy this?So can we remedy this?

As it turns out, there is a gravitational As it turns out, there is a gravitational analog of the consequence of this new analog of the consequence of this new “Faraday-Like” effect, as discussed “Faraday-Like” effect, as discussed earlier…earlier…

Gravitational Faraday EffectGravitational Faraday Effect

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

EvaluationEvaluation

Upon evaluation of the integral again, we see the result that: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.

Current WorkCurrent Work

The existence of this gravitational analog The existence of this gravitational analog allows us the possibility of terrestrial allows us the possibility of terrestrial experiments of gravitational frame experiments of gravitational frame dragging.dragging.

Dr. Chandra Roychoudhuri is currently Dr. Chandra Roychoudhuri is currently designing experimental apparatus to designing experimental apparatus to perform the experiment.perform the experiment.

DesignDesign

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.

ConclusionConclusion

Gravitational Frame Dragging may be able Gravitational Frame Dragging may be able to be tested in an easily controlled, to be tested in an easily controlled, terrestrial lab.terrestrial lab.This is due to the existence of a This is due to the existence of a gravitational analog of the Classical gravitational analog of the Classical Faraday Effect.Faraday Effect.David Cox and I would like to thank you all David Cox and I would like to thank you all for this wonderful opportunity and for your for this wonderful opportunity and for your attention!attention!