I wanted to publish this so that I would have copyright protection. Unfortunately I was not able to copy the equations from microsoft equation editor, but I will continue to work on this so that I might add the math bit at some future time.

The Gravitational Red Shift and Lensing Effect

Michael F. Golden

Introduction:

The interpretation of space-time curvature has produced what is currently the most accurate method for the calculation of gravitational phenomenon. However, when the theory of General Relativity is extended to both the extreme large and small gage, the theory fails to produce observable predictions (i.e. quantum mechanics and dark matter/energy). This deficiency in its a priori capability is linked to the fact that general relativity provides no observable mechanism for gravitational transmission. While general relativity does not advocate action at a distance it also does not explain how gravity gets from one place to another. The existence of a graviton particle, has been proposed by field theories to account for this issue, but has currently failed in high-energy situations because it is not subject to renormalization. It remains the case, that for gravitation to be accurately understood, there must exist a specific mechanism for its transmission through space-time.

So while General Relativity remains the most accurate theory of gravitation presently available, it also seems to have some flaw embedded in its original conception. GR proposes that space-time curvature is responsible for gravitational effects, but what exactly is space-time? It should be obvious that there is no such thing as empty or ‘free’ space-time because observations in Quantum Mechanics have shown that any measurable volume cannot be completely devoid of energy. Theoretically, the uncertainty principle shows that absolute zero is impossible to observe and thus requires that a small, but finite amount of energy permeates the entire universe at all events in space-time. This leads to the conclusion that what we define as free space or vacuum must be just as physically tangible as matter, and thus must have its own physical properties that can be directly observed. So the question becomes: what are the physical properties of free space that relate specifically to gravity, and how can they be measured?

A possible answer lies in understanding the fundamental basis for the development of special relativity. The reason Galilean reference frames fail to properly measure the distance between objects in motion is due to the fact that we are inevitably using photons as our measuring device. Additionally, every observation of gravitational phenomenon has been measured via observations of electromagnetic radiation. The strange effects initially described by special relativity occur because electromagnetic radiation always travels at a constant speed regardless of the velocity of the observer. In essence the reason special and general relativity provide such accurate mathematical descriptions is because they fundamentally assume that light provides the basis for measurements in free space. Viewed from a purely empirical perspective, the conclusion to be made from this line of reasoning is that the physical manifestation of space-time must be related in some very basic way to electromagnetic waves, which provide the majority of energy traveling through ‘empty’ space-time.

Here history provides a vital clue as to how this possibility might be explored. The most informative example of theoretical unification is provided by Maxwell’s correction to Ampere’s Law that completes the four equations of electromagnetism. At the heart of this theoretical unification was the physical observation that a changing magnetic field will induce an electric field and vice versa. Electromagnetic induction provided the essential observation of a direct interaction between the two previously separate forces, which provided the extra term needed to complete Ampere’s Law. Because it was possible to study this interaction between fundamental forces and make specific measurements, it was then possible to conclude that the magnetic field could be expressed in terms of the electric field and vice versa. The nature of the fundamental physical forces dictates that if it is possible to observe a direct interaction between two forces then it must also be possible to algebraically interchange them provided some conversion factor. This interchangeability is the basis for unification and thus the theoretical correlation between gravitation and the quantum forces could be found by examining the various observations where the two forces interact.

There are two different phenomena, although they are both simply different aspects of the same quality, which provide observations of direct interactions between light and gravitation. Specifically the gravitational red shift and the lensing effect could work as a guide to understanding the relationship between electromagnetism and gravity. This is of course a more difficult prospect then Maxwell encountered in the unification of electromagnetism because gravity and the electromagnetic force are not comparable in strength. Furthermore, while gravitational red shifting and lensing provide a direct interaction between the two forces there appears to be no basic commonality between the theories that describe gravity and E&M where as electricity and magnetism where both described using the same mathematical language.

The following represents my effort to delve deeper into the phenomenon of the bending and frequency shift observed in light rays traveling through a gravitational field as described in the calculations of general relativity. It is my intent to have learned these methods by the conclusion of this project so that I may continue to pursue this line of research at a higher level. Eventually it will be required that I also learn what this effect looks like from the perspective of quantum electrodynamics, but for now I will focus my efforts on learning the relevant mathematics of general relativity for this paper.

I. The Null Geodesic:

General relativity originally proposed a novel idea as to how to interpret the observations of objects moving thought space-time. The logic of special relativity dictates that if an observer is moving along at a constant velocity, then the reference frame of the observer will remain constant. If a force is applied to the observer the reference frame will change. However, Einstein argued that an observer being attracted by the gravitational field of a massive object will see no change in the local frame of reference. Thus, as previously stated, the theory does not view gravity as a force but instead solves this supposed problem my defining it as an intrinsic curvature of the space-time through which objects travel in relation to one another. From this idea and the equivalence principle, which states that standing in a gravitational field is equivalent to accelerating in a constant direction, the idea of a curved path that appears straight to the observer moving along it was proposed.

A geodesic is basically that idea. It is a geometrical line drawn though space-time to represent the path and the various dimensions of any particular particle one may wish to calculate. When measured in any specific local Lorentzian frame of reference, the path is indistinguishable from a straight line and uniformly parameterized. A geodesic in space-time can either be time-like where the path behaves like a massive particle, space-like where the path cannot be connected by any physical object, or null where the path can only be connected via a massless object such as a photon. A time-like geodesic is usually the curve that scientists are most interested in, as it is used to represent the path of any massive particle, be it a molecule or a black hole, through space-time in terms of a specified affine parameter. The purpose of the affine parameter is to provide measurable intervals on the path of the particle moving along the geodesic. For a time-like geodesic this parameter is typically the proper time of the particle, however this is not the only possible affine parameter, and in general, the proper time can not be used in this capacity for either of the other possible categories of geodesics.

There are several possible ways to go about calculating the equations of a geodesic:

In peticular, the Schwarzschild solution to the Einstein field equations can be written as follows.

This metric is actually only a solution for vacuum field equations, as it does not adequately describe the gravitational field within a massive object. To avoid complication, the Euler-Lagrange equations will be used for this purpose as it provides a reasonably intuitive approach to the necessary calculations. The Euler Lagrange equation is a method for optimizing dynamic systems and thus can be used to calculate the shortest distance between two points in a curved geometrical manifold. The point of deriving these equations is to then pick off the Christoffel relations, which specify the local curvature of the geodesic equation. Plugging the Schwarzschild metric into this equation as L,

we can then solve for each of the four polar coordinates. The solution for is as follows

Pulling off the Christoffel symbols for this solution we obtain:

Due to spherical symmetry of the coordinates, there is no loss of generality if we assume that q = p/2. This restricts the generality of the calculations to specify only particles moving in the equatorial plane of the massive object. From this it follows that the integrals of the remaining two equations should yield constants. These two integration constants provide the relationship between coordinate

time and proper time and an expression of the conservation of angular momentum. We can now make use of the definition of proper time from the Schwarzschild metric to produce:

This equation is obtained from the line element of the Schwarzschild space-time, where t is held constant. The right hand side of this equation comes out as c² because the line element describes the path of a particle moving slower then the speed of light. To find the equation that defines the path of a photon the right hand side of equation 13 will be zero giving:

Through a series of substitutions to the equation for the time-like line element, we arrive at a general equation for the energy of the specified particles.

For massive particles traveling slower then c where E = c²(k²-1)/h², and

For massless particles moving at the speed of light, where F = c²k²/h². The last term of these represent the relativistic correction to the Newtonian orbital equations for particles moving though curved space-time and can be used to calculate the effects of the gravitational field.

II. Spectral Frequency Shift and Gravitational Lensing:

This section discusses the equations describing the spectral frequency shift and the gravitational lensing effects in the language of general relativity. The spectral shift plays a very important role in cosmology as it defines the gravitational field lines for light and thus the contours of space-time curvature. The current observational data suggests that the universe is expanding at an accelerating rate and it is measurements in this shift that provides the main evidence. Using the null geodesic we define our affine parameter to be u, where u_e is the emission event and u_r is the reception event along the geodesic. From the equation presented in the previous section we can calculate that the change in coordinate time between events should come out to be T_r1 – T_e1 = T_r2 – T_e2 which is equivalent to T_r2 – T_r1 = T_e2 – T_e1 which again is the same as ∆T_r = ∆T_e so the coordinate time difference at the point of emission equals that at the point of reception. If we instead look at the proper time interval of an observer at each event, as stated the respective coordinate time intervals are equal. Thus, through a bit of manipulation we arrive at the equation:

By defining the proper time in terms of frequency where n is the number of cycles measured, this quickly yields the equation for the gravitational effect on frequency.

It is interesting to note that this same equation can be obtained through an analysis of the change in potential energy at varying radii from the massive object. The basic thing to understand about this effect is that the frequency as measured by observers at different distances from a massive object and an emitter will change, but observers at the same distance from the massive object as an emitter will see no change in the frequency from that emitter. Even in free fall, an observer traveling with the source will see no difference in the behavior of that source from that of a reference frame devoid of any massive objects. This is important because if we assumed that light always traveled in straight lines then an observer accelerating with a source would see the photon stream curving away from them. This does not turn out to be what happens. An observer far away from the reference frame of the massive object will see the path thought which the light is traveling curve towards the massive object. This is the gravitational lensing effect and can also be described in terms of the null geodesic.

Using the energy equation derived for the null geodesic and turning the mass down to zero we get the equation:

an obvious solution for the path of the photon would be u = u₀sinf where u = 1/r and u₀² = F. Assuming that the photon is approaching from a direction f = 0, then in a massless system the photon would continue on a straight tragectory leaving at an angle f = p. The point at which the photon comes closes to the chosen origin would then be r₀ and upon introducing a massive object this situation can be expected to change around this radius.

Introducing mass to the previous equation we get:

Assuming that our values for r are much larger then m we can limit the approximation to the first power, thus viewing the extra term as a relativistic correction to the equation for flat space-time. Now because du/df = 0 when u = u₀ the value of the constant F = u₀²(1-u₀2m), then substitution will produce:

We can assume that this equation should have a solution close to that of the photon moving through flat space-time: u = u₀sinf+2mn, where n is a function of f. Again substitution gives:

Rewriting this equation and integrating will then produce the function represented by n:

From our original assumption we can then show that A is a constant of integration and will equal –u₀² and the path of the first order equation for the photon will be:

Using a small angle approximation we can assume that the photon will now be deflected in the direction p+a. Setting u = 0 and f = p+a and using the approximations sin(p+a) » -a and cos(p+a) » -1, the previous equation becomes:

so then a = mu₀ which gives the solution for the deflection of light by a massive body:

The variation in direction and frequency of light passing by a massive object seem as though they should be two different pieces of the same basic action. Thereby a solution that incorporates both the spectral shift and gravitational lensing should be obtainable, and necessarily related in that the equation would take the form:

when the change in radial distance = 0 and the form:

when the path of the photon is perpendicular to the surface of a spherical mass or f = π/2. This equation would then effectively describe the complete interaction of light with a gravitational field in terms of current experimental observations. Despite several weeks of work on this problem I have not managed to produce a term that would completely satisfy the conditions mentioned above. Although I have produced several possible equations, I have not yet found a solution that flows naturally from the previous arguments. I feel that I am on the correct track however and simply need more time to work through the problem.

Conclusion:

In conclusion to the work I have done this semester, I have successfully learned how to calculate a simple form of the null geodesic curvature for a massive object. Thus I have accomplished what I had set out to and the results of this are two fold. First, it allows me to pursue the connection between the spectral shift and gravitational lensing to a more complete description, specifically in terms of null geodesic mathematics. The second is that it provides a direction for my work to further understand the nature of the mathematics used to describe Electromagnetism. The tensor formulation for electromagnetism provides a rich field to explore and my brief delve into it has enriched my understanding of its utility.

Also it is clear that for further development of this idea it will be required that I either learn to, or possibly invent, a satisfactory way to describe gravitational lensing and the gravitational frequency shift in the language of quantum field theory, specifically quantum electrodynamics. This assumes the undertaking of a great deal more theoretical work, which I do not necessarily plan to pursue professionally, in favor of more pressing problems on the forefront of experimental particle physics. However, it is my strong belief that the resolution of this problem will be found in a more profound understanding of angular momentum in modern quantum theory, and I do plan to pursue this direction at some point in my future research and education.

Hello everyone! If you are reading this it is probably safe to say that you are a close friend or relative. I just want to let you know that though emma and I are perhaps a bit farther away then usual we send you our love. I wrote this not long after we finally made it to California. I will post more stories and pictures from our trip and new home in the days and weeks to come. Enjoy...

After much consideration and contemplation about the future, Emma and I have moved to Santa Cruz, CA. At the suggestion of Ben I've decided to try to blog about our trip and how things progress as we settle in to our new home for family and friends. Because the job scene is pretty tight around here I've decided to put ads on this page to perhaps generate a small amount of capitol while we search for work.
I'm going to start by recapping our cross country trip (which was fantastic). Something that I had never done and had been intensely jealous of all my friends who had already made similar voyages. This is all that I knew though before we left. I had no idea how to plan such a trip and was kind of at a loss for how to go about it.
Luckily I have Emma, a master organizer of the first degree. She sat down on the computer and in an hour had a trip schedule mapped out for us on our calender. All that was left to do was call people and let them know we would be coming through. The first people on our way out west were Erik who lives in Bloomington, IN where he goes to grad school, Mr. and Ms. Tipping who live in Kansas City, and Seth, who had just moved to Denver a few weeks earlier. It felt like it was a mark of good fortune that everyone we knew lived no more then 10 hours from each other.
I could say a lot about before we actually got on the road. There was a lot of packing and finishing up of projects in Oneonta and then Allentown, but we kept to our schedule and left for Bloomington on November 23.

We left a little bit later then we had planned because even though we got up on time we realized that the car was completely full and we still had our overnight bags to stow, so we spent another two hours repacking the car.

We left Allentown at approximately 9:30 and began our drive to Erik and the University of Indiana.