November 17, 2005 Time Capsule

On November 17, 2005, when computers did not work as well as they do today, Bush was serving his second term and I was a youngster of 47, I stumbled across the time capsule portal on the Forbes website and wrote myself a letter from that simpler time.  Here is what it said:

Today is the day before my new Megrez Triplet Fluorite refracting telescope arrives.  I am so looking forward to getting it.  Both our children are doing well in college, our careers are successful (but hectic) and the sun is out on this almost-freezing beautiful November morning.

I just had my "optical cantilever" paper accepted by JOSA B, I corrected the proofs of my new PRA paper on the dipole free SOS expression, and I am anxiously awaiting word from PRL about the "ultimate limit" paper.  In the meantime, I am working on the symmetry paper with David Watkins.  All this while teaching two classes and doing lots of service - not to mention my debate with Bob Olson on religion.  Gotta run to a staff lunch. Oh yeh, and I scored 3 goals and had one assist in my ice hockey game two nights ago!  --Mark

It is interesting to see how some things have changed and others have stayed the same.

I still play hockey and coincidentally scored three goals in my last game.

But, my telescopes have been gathering dust.  I was into telescopes at that time and the Megrez is truly an exceptional instrument.  I enjoyed it for many outings and for a while I stored it on our upper deck, which towers above our neighborhood; well, that is, until I got a frantic call from a neighbor that the tarp was acting as a parachute and the telescope was hovering over our roof.  Given that my memory of those times is fading, perhaps it was a different telescope.  But I can state with emphatic certainty that this beautiful piece of art graces our study today and I still enjoy its elegance.  Some of the photos of the heavens that I took at that time can be found on my website at http://nlosource.com/Astro/Saturn2003-12-06.htm.

The PRL paper was rejected, but I found a good home for it in the still-respectable Physical Review A.  The other papers appeared and have done well, especially the dipole-free paper, which has been used extensively since then to take our work to the next level.  What I didn’t mention was that I was sitting in on a class on general relativity and traveled to Australia for the first time to attend a conference in Sydney.  The famous opera house was truly majestic from a distance, but up close its curved walls of white tiles was dirty as a frat house shower.  Though I spent an afternoon walking around town, visiting the government buildings and the gardens, I spent most of my time at the conference and evenings working on my general relativity homework.

What impresses me most about this window into the past is my efficiency, doing so much with so little time.  Now that I have more time, I spend too much of it trying to get everything just right.  Yesterday, I was telling a student that being on a teaching assistantship might take time away from research, but it makes you more efficient – and I was right.  The lesson from my past self is to take on more and worry less about being perfect.  I also need to remain passionate and engrossed in my activities, like working on our cabin in the wilderness.

All right, here I go…    

How do We Know a Black Hole Lives at the Milky Way's Center?

People often wonder how science can get a handle on out-of-this-world things like the properties of the universe many light years away or the small-scale structure of space-time itself.  In 2005, I presented the distinguished faculty address to give my audience a sense of how this is done.  The title of my talk was, "From Black Holes to the Internet:  How We Use The Scientific Method To Understand the Mysteries of Things Unseen."  It was a very enjoyable hour with lots of wonderful audience participation.  Here I summarize one part of my talk on "how we know" that there is a black hole in the center of our galaxy.

Science often proceeds by taking small steps that together build a general understanding. Here I outline how experiments on the earth along with observations of our universe can lead to an amazing understanding of the natural world.

The outline below shows how simple experiments on the earth's surface can be used to detect a massive black hole at the center of our galaxy, the Milky Way.

1.  Establish Newton's Theory of Gravity by measuring forces between hanging masses here on earth.
2. Use Newton's Theory of Gravity to determine the mass of the earth and check if it is consistent with what we know about the earth's composition and size.
3. Predict the orbital shape and period of the moon based on the earth's mass. It is found that the calculated period is consistent with the measured one and the shape of the orbit is accurately predicted (ellipse with the earth at the focus). This evidence suggests that gravity acts on lunar distances.
4. From the orbits of the planets, the mass of the sun is determined. Every planetary orbit gives the same solar mass (with the sun at the focus of every ellipse), so gravity appears to work even on these larger scales. Furthermore, the density of the sun that is determined from its mass is consistent with what we know of the sun's composition from independent spectroscopic measurements.
5. The orbital properties of all bodies in the solar system obey Newton's Theory of Gravity with impeccable precision. This includes comets, asteroids, moons, satellites, and space ships.
6. Mercury's orbit is found to deviate ever so slightly from Newton's predictions. This irks physicists until Einstein formulates the General Theory of Relativity in 1918 which fully accounts for the small deviation. It turns out that Newton's Theory of Gravity is a special case of General Relativity, which predicts the possibility of the existence of black holes.
7. Telescopes that view infrared light are able to penetrate the dust that obscures the center of the Milky Way to visible light to see stars at our galaxy's center.
8. Astronomers measure the orbits of these stars over more than a decade. As predicted by Newton, the orbits of the stars are perfect ellipses. The foci of these ellipses all coincide with an invisible object.
9. Using the orbital data, the calculated mass of the dark object is almost 4 million solar masses.
10. Some of the stars get very close to the dark object, so an upper limit of the object's size is determined from the distance of closest approach.
11. The dark object's mass and density fall in the range predicted by general relativity for a black hole.

The infrared observations of stellar orbits as described above do not prove the existence of a galactic black hole; but provide strong evidence. There are other independent measurements that all point to a black hole (see below). When the pieces of the puzzle are assembled, the picture that emerges is one of a huge black hole at the galactic center.

Incidentally, such massive black holes are found at the centers of other galaxies and even globular clusters. Since evidence of smaller black holes are routinely "observed" in binary star systems, it becomes clear that black holes are out there.

This journey illustrates something fundamental about nature, and about the breadth of physical laws. Richard Feynman said it best, "Nature uses only the longest threads to weave her patterns, so that each small piece of her fabric reveals the organization of the entire tapestry."

A youtube video shows the motions of the stars around the central black hole that comes from direct telescopic observations.  The data covers over a decade of observations. A more dramatic version can be seen here.

Scientists are slow to accept a new idea or theory unless there are multiple pieces of supporting evidence.

The case for black holes is bolstered by X-ray observations of the hot gas surrounding the galactic black hole at the heart of our Milky Way. The observed X-ray spectrum can be used to determine the gas temperature using the same principles that betray the temperature of glowing orange embers in our fireplaces.

If the gas is to remain stationary, the inward gravitational tug of the black hole must be balanced by the outward pressure of the gas. The calculation is simple enough for a high school physics student, requiring only Newton's Universal Law of Gravity (see below) and the gas laws.

This simple calculation leads to a black hole mass of about 3.4 billion suns, in agreement with the observations of stellar orbits.

Newton's Universal Law of Gravity

Newton's universal law of gravity states that the force of attraction between two objects is proportional to the product of the two masses and inversely proportional to the square of the distance between their centers. The constant of proportionality is G.

G can be determined be measuring the forces between masses that are measured with a torsion balance. This is a common experiment done in most physics departments by students, and even in high schools.

 

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Kepler's Laws (1571 - 1630) Kepler's laws follow from Newton's theory of gravity.   Kepler's Laws state that: Planets travel in elliptical orbits with the sun at one of the foci (shown above). Equal areas are swept in equal times (see diagram below). The time it takes to complete one orbit is proportional to d3/2 All objects in our solar system are observed to obey Kepler's Laws, and therefore confirm Newton's more general theory. Einstein's Theory of General Relativity is the most general theory that makes small corrections to the orbit of Mercury and is required to make the GPS system work. But don't believe the authorities. Anyone can observe the motion of Jupiter's moons to determine the mass of the gas giant. I took the photo below of Jupiter and three of its moons.

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 Cutting Through the Dust

Raleigh found that the degree of scattering is proportional to the inverse of the fourth power of the wavelength of light. Blue light has a shorter wavelength than red light, so is more strongly scattered. Rayleigh scattering explains why the sky is blue.

Yellow fog lights work on the same principle. When white light is filtered to remove the blue light, what remains is green and red, which appears yellow. The longer wavelengths pass further through the fog and the scattered glare from the blue light is eliminated, making it easier to see.

The infrared range of the spectrum is made from light of even longer wavelengths, allowing telescopes to see through the muck. Special detectors are used to image the light. The image below is of the center of the Milky Way, taken by researchers at Max-Planck Intitut fur extraterrestrische Physik.

Another semester and another new class - general relativity and thermodynamics

When I was an undergraduate at the University of Pennsylvania, I took a class in general relativity (GR). It was taught by the eccentric Professor Jeffrey Cohen, not to be confused with the equally eccentric Professor Michael Cohen.

Michael Cohen had instilled in me a deep appreciation for truly understanding physics. Just as Michael Cohen felt that he would never attain the depth of understanding commanded by his adviser, THE Richard Feynman , I too feel that I will never approach the physical intuition of Michael Cohen. It is fortunate that the singularities that we call great physicists are born with abilities far superior to their contemporaries.

After more than three decades have passed, I recall little from my undergraduate flirtation with GR. However, some of the mathematical formalism of differential forms has taunted me for much of my career. I recall Jeffrey Cohen mentioning a paper on the topic of the properties of a black hole in some complex geometry that took forty pages of derivations in an article that appeared in The Physical Review. Using the trickery of differential forms, he was able to solve the problem in just a few steps.

The trick was to formulate the problem in a coordinate independent way, then to project the results into the coordinate system that reflected the symmetry of the problem. In contrast, the Physical Review paper used the inelegant brute-force approach of picking the coordinate system up front, and then by necessity painstakingly plodding through all the messy mathematics.

Given the complexity of the problems that we work on as a matter of daily routine in our research, I am always looking for simplifying tools. In teaching various classes, my intention is to sneak in a little bit of differential forms to wet the appetites of my acolytes and to teach my old brain some new tricks. Furthermore, the geometric interpretation of the mathematics adds a deeper layer of understanding.

In the upcoming spring semester, I am teaching graduate statistical mechanics for my first time. As usual, preparing for a new course if filled with grand excitement. You can imagine my elation when I realized that a homework assignment in the textbook could be done with ease using differential geometry. Since then, it has been difficult for me to think about anything else.

The problem is a simple one that normally requires a bit of math. The student is to show that the 6N-dimensional volume element in phase space for an N-particle system is invariant under a canonical transformation. To put this into simple English, the problem seeks to show that a transformation of coordinates does not change the nature of the results. Be reformulating the problem so that the volume element is represented as a wedge product of what are called one-forms, the volume element is shown to be the same when the so-called Poisson bracket yields unity -- the requirement of a canonical transformation. Thus, the problem is solved without the need for messy mathematics.

This realization makes me feel like a kid at Christmas. Ironically, tonight is Christmas Eve, the focal point of my family's celebration. My father has made what may be his last trip to Pullman from Philadelphia. He is 94 and still lives on his own, drives a car, and prepares meals for senior citizens at the Ukrainian Cultural Center in Fox Chase, Pennsylvania. Though still vigorous, his body betrays the telltale signs of wear and tear due to old age. Both of my children are home for the holidays, and all the fragrance from the traditional Ukrainian foods simmering on the stove and in the oven permeate the house. As I write this post, my wife is busily making last-minute preparations.

It is fortunate for me that my family values my passion for physics, and allows me to occasionally be a recluse. Just a few minutes ago, my wife called out a query about my whereabouts. I simply answered, "I am excited about something." Though she undoubtedly had some mundane duty for me to perform, she immediately signaled her understanding of my state of mind, and left me alone. I am truly fortunate to be living with someone who shares in my passions.

The intensity and meaningfulness of spirituality that I derive from physics far exceeds all others, including the times in my distant past when I had embraced religion. As my family turns in for the night, I continue to sit at my desk, full of excitement in my new-found understanding, and looking forward to sharing this understanding with my family and my students. It is a truly privileged life that allows me to rekindles the child-like wonder of Christmas on a daily basis.