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LIGO Livingston Observatory NewsLivingston Installation Update
Angular Momentum in General Relativity
National Geographic Photographers Make Safari To LIGO
This month we began installation of detector components at the LIGO Livingston Observatory (LLO). To date, all of the Horizontal Access Module seismic isolation piers have been installed and aligned. In the Laser Vacuum and Equipment Area (LVEA), all of the Beam Splitter Chamber (BSC) piers have also been installed and aligned. The end station BSC piers will be installed in two weeks. Adaptor plates on all of the seismic piers were installed as well. Rich Riesen and Lloyd Demonn have made a tremendous effort (worked like demons?) in the past month to accomplish all this, with the assistance and direction of Dennis Coyne and Joe Giaime.
Figures 1 and 2 at left and right below exhibit views of the seismic isolation piers being installed in the Laser And Vacuum Equipment Area. Sharp-eyed observers discern the form of Dennis Coyne in the foreground of Figure 1.
We have also been cleaning the LVEA to prepare it for clean room operation. We have made multiple passes at cleaning this area. As we learned from the experience of the LIGO Hanford site, this takes persistence and constant vigilance! Present particulate readings are in the neighborhood of 300 0.5 micron particles/cc. Still more cleaning is scheduled.
The interferometer laser table, shown at left, has been installed in its approximate location in the LVEA and baseline RGA scans of the vertex vacuum are in progress. Also, the vacuum bakeout oven has completed commissioning by Allen Sibley (at right) and Kerry Stiff. The background pressures of all amu's above two were less than 10-11 torr! The first bakeout of bellows for the seismic isolation system is now in progress.
Rusyl Wooley, our newest employee at LLO, began work this month after finishing up the Vacuum Equipment work here as a Process Systems International (PSI) employee. Russ has worked for PSI for the last seven years as a design and field construction engineer, installing the Vacuum Equipment at Hanford before coming to Livingston. He has had a busy first month installing the relay racks in the LVEA and end stations and making arrangements to install the detector cabling and cable trays.
(This article was contributed by Anthony Rizzi at the request of Mark Coles. The angular momentum work described here began as my Ph.D. thesis at Princeton University; I would like to acknowledge Demetrios Christodoulou, my thesis advisor, for his invaluable assistance through the use of his personal notes and through insightful conversations.)
General relativity's prediction of the existence of gravity waves includes the idea that they should be able to carry angular momentum if that concept can be defined in a meaningful way. Knowing how to define angular momentum will be useful both in LIGO data interpretation and in implementing match filtering. The interesting problems that surround the existence of angular momentum in general relativity are therefore important to LIGO.
The August 10th 1998 issue of "Physical Review Letters" contains an article with the first definition of angular momentum in general relativity that allows for the exchange of angular momentum. The article's author is Anthony Rizzi, staff scientist at the LIGO Livingston site. "Science" magazine covered the discovery in its October 9th 1998 issue. Until this discovery, one could NOT make the simple statement that an object with angular momentum L emits gravity waves that carry away deltaL thus leaving the object with L-deltaL.
To many this is quite startling. How could it be, they ask, that such a basic quantity as angular momentum was not defined in general relativity. In fact, even conservation of energy and linear momentum are very hard to understand in general relativity. One can go still further, the most basic conservation law in physics, the conservation of energy, probably does not exist, in general, in Einstein's theory; it only exists under specific conditions. It took over 40 years for the definitions to be given for energy and linear momentum in the special conditions. It has taken over 80 years for the same to happen with angular momentum. What could make such definitions so hard? A prerequisite question is: what makes it so easy in everyday physics? Let's explore why Newtonian linear momentum is so easy, and a parallel argument will apply to the more subtle case of angular momentum. Then, we can look at the linear momentum question in general relativity and give its resolution. Again, parallel arguments will apply to angular momentum, but this time only up to a point. This exploration will reveal something about general relativity as well as something about the problem of defining angular momentum in general relativity.
Newtonian physics assumes that the laws of Euclidean geometry apply in space and that the time variable is totally separable from the space variables. Euclidean space is completely symmetrical under translation. In an empty Euclidean space there is no absolute origin. To see this, imagine--when you imagine (i.e. make a mental image) you will naturally default to Euclidean space--that you are in an empty space; you will note that no point has any distinction from another.
Now, Newtonian mechanics is not just Euclidean geometry; one can imagine many types of laws that might apply, for example, in place of Newton's third law: "F = m a." Hence, one can think of the underlying space as being, in some sense, independent of the laws of Newtonian mechanics. The laws of Newtonian mechanics do in fact also respect the translational symmetry. This means there is nothing in those laws or in the underlying space that will distinguish one point in space from another. Hence, linear momentum, given no outside forces, cannot change from its initial state; it is conserved. One obtains, for the case of Newtonian mechanics, the familiar statement that every object will continue in its state of rest or uniform motion unless acted on by an outside force--which is the crucial concept of inertia which dates back to John Buridan in the Middle Ages.
One can recast this qualitative outline in more mathematical terms. The equations of motion and the definition of momentum (comes from the formalism) can be obtained by Hamilton's principle. Hamilton's principle requires the difference between the kinetic energy and the potential energy "averaged" over a period of time be minimized (or in rare cases possibly maximized). The "averaged difference" is called the action. In detail, Hamilton's principle requires, in an abstract calculus known as calculus of variations, that the first derivative of the action be zero. Setting this derivative to zero gives the equations of motion; F = m a in the Newtonian case. Recall that Newtonian physics with no forces is obtained when one has no potential energy; so one is extremizing the kinetic energy only (i.e. T-V=1/2 m v2). Note that the underlying Euclidean space is hidden in what mathematical entities are defined and how; for example, the dot product between two velocity vectors, which gives the square of the magnitude of velocity = v2, implicitly utilizes the Euclidean space metric. In Hamilton's formalism, the conservation of linear momentum results from the action having translational symmetry (i.e. not changing mathematical form when changing origin).
Now we are ready to address the two major issues in general relativity. First, in general relativity one does not, in general, have translational symmetry as discussed above because the underlying space is curved in an unrestrained fashion. Second, gravity takes on a different status in general relativity; it is no longer treated like any other force. The notion of inertial frames is completely reassessed. The change in status of the gravitational field implies that momentum associated with the gravitational field must be treated differently from that of other fields. To see how this comes about we must understand inertial frames more completely. This takes us back to Newtonian physics.
In Newtonian physics, one introduces the concept of global inertial frames in which time flows uniformly, independent of space. Given one inertial frame, one can transfer to another frame by moving at a uniform speed with respect to the first frame, and the laws of physics will remain unchanged. Newton's way of looking at mechanics is obviously consistent and powerful, but it contains an oddity associated with the concept of mass. When an object is pulled by the gravity of a planet (or any other object), the mass that enters the equation for the planet's pull on the object (the object's gravitational mass) and the object's resistance to pull (its inertial mass) are exactly the same; this is the weak equivalence principle. In simple terms, it means that, given a certain experimental accuracy, a man in free fall will see an inertial frame with that accuracy as long as he confines himself to a small enough region, that is, as long as he looks locally. In an arbitrary gravity field, far from the man in free fall, his frame will not see things behave inertially. Noting the importance of the equivalence principle is one of Einstein's key insights.
Instead of assuming the equality to be a cosmic coincidence, Einstein takes it to be a principle. In this view, gravity is considered not a force but rather that which establishes a system of inertial frames. There is no global inertial frame and therefore, in general, no universal time. Locally things look inertial (i.e. they look special relativistic), but globally it's a different ball game. When someone is doing an experiment, he usually thinks locally, and considers himself to be in a frame fixed to the earth; it is then easy for him to think of gravity as a force in an earth-fixed inertial frame (neglecting rotation). This view is usually perfectly satisfactory for earth bound experiments, so the experimenter can easily neglect to consider the previous line of argument and thereby miss the uniqueness of gravity.
So, in general relativity, we have recast inertial frames to be the local area around the path of an object in free fall; what, you may say, does this say about our problem: momentum in general relativity? Well, it means that locally (if there is no spin) one does not even know that the gravity field exists! How could we define the local momentum of something that we cannot express locally?
Finally, we are in a position to solve the two major problems with defining momentum. The first problem, the lack of symmetry, is resolved by choosing asymptotically flat space-times and looking only that far from all masses. Far from all masses, the space-time gets flat (Euclidean) and one recovers the translational symmetry. Specifically, we look at "null infinity"; the place and time far from all mass where gravity waves eventually reach. By looking here, one can obtain a definition that allows for the exchange of angular momentum through gravity wave emission. The second problem is resolved by integrating over a sphere encompassing all the mass; this enables us to pick up the effects of the gravitational field because the integration samples globally rather than locally. The definition, thereby, becomes a non-local definition. The sphere must be at null infinity because, as stated, it needs to be there to regain the translational symmetry. The same solutions work with angular momentum with a major caveat. In the above argument, translational symmetry is replaced with rotational symmetry, but when it comes to integrating over a surface, we run into an ambiguity about what surfaces to use; this problem is called the super-translation ambiguity.
This brief introduction gives a flavor of the sort of problems that arise due to the subtleties of general relativity. Anyone interested in learning a little more about the angular momentum definition should see the layman's treatment written by Matt Visser in the October 9, 1998 issue of "Science" magazine.
Last month, February 7 through 9, photographers from the "National Geographic" Magazine visited the LIGO Livingston Observatory to take pictures for a story on cosmology which will appear in the October 1999 issue. According to Bill Douthitt, Special Projects Editor at National Geographic, the October issue will feature several science stories in a special section dealing with the very large, such as cosmology, and the very small, such as microbiology. The section is in turn part of the "millennium series" National Geographic has been producing this year and last. The February issue, which looks at bio-diversity, is the latest installment in this series, of which Mr. Douthitt is the manager. Other themes have included exploration, the physical earth, and population. The cosmology story, which falls under "the universe" heading, will look at the current understanding of the universe and the Big Bang model, and work in progress to test this understanding through efforts such as LIGO.
We took advantage of having National Geographic's photographers at our site to strengthen our connection with the local community. We invited the student year book staff from Doyle High School, along with their principal, Bill Spears, to LIGO for a pizza lunch (shown in Figure 1 at left) while Joe McNally, the lead photographer for this story, and his assistant, Nina Sabo, talked to the students about photojournalism and photographic techniques. Joe also brought a wonderful series of slides from previous assignments which he showed the Doyle High students.
Joe McNally is considered to be one of the best photographers working for the National Geographic, and is widely regarded as one of the most accomplished and versatile editorial photographers in the United States. He has photographed cover stories for "Life," "Time," "Newsweek," "Sports Illustrated," "Fortune," and "New York" Magazine. Joe described for the students his recent experiences with the Russian space program and with NASA, as well as his work photographing the preparations for the shuttle mission that flew John Glenn, to be featured in an upcoming National Geographic article. The students showed their greatest interest when Joe described some of his encounters with movie stars, and in particular his date with Michelle Pfeiffer (in fact we were all interested in that particular event).
At left, Joe McNally entertains students from Doyle High School with some interesting anecdotes drawn from his career. At right, Joe hovers above LIGO Vacuum Equipment in search of the perfect shot.
We all enjoyed having Joe and Nina here and appreciate the time they took to talk with the Doyle High Students. The two had some very creative ideas for photographing LIGO. One was some 5:30 am shots of the dawn light viewed along the arms of the interferometer through an early morning mist. Another was of the sunset along the west arm of the interferometer. We look forward eagerly to seeing LIGO in "National Geographic" this fall.