Mathematics Awareness Month Theme Essay

From Black Holes to Dark Energy: Cosmology in the 21st Century

Cosmology in the 20th century was almost in its entirety the outgrowth of Einstein's foundational paper in 1915 on general relativity. Two years later he presented his first model of the universe based on general relativity together with Riemann's notion of the three-sphere.

Side-by-side with the theoretical advances, observational astronomy led to great leaps in astrophysics, as the life cycles of stars were discovered and elaborated, the existence of other galaxies outside our own was confirmed, and the expansion of the universe was demonstrated to the satisfaction of nearly all. Einstein was able to use general relativity on the one hand to explain earlier observations, such as the amount of precession of the planet Mercury, and on the other hand to make new predictions for the observers to confirm or refute. The first and most widely heralded of those was the prediction of the bending of light as it passed close to a large mass such as the sun. Others, such as gravitational red-shift, gravitational lensing, and "frame-dragging" around a rotating body, were confirmed one by one over the course of the century. Still others, like the existence of gravity waves, remain a high priority for 21 st century experimentalists.

The Nobel Prize-winning work of Russell Hulse and Joseph Taylor stemmed from their discovery in 1974 of a pulsar whose "pulses" varied in a regular fashion, leading them to conclude that it had an invisible companion, the pair forming a familiar binary system each one circling the other (or actually, their common center of gravity) in an approximately elliptical orbit. In this case, the pair consisted of two bodies each as massive as the sun, but compressed into a tight ball whose diameter was the size of a small town, and each completed its orbit around the other in about eight hours. Under such extreme conditions, the relativistic effects would be considerable. One of those effects would be the production of gravity waves, and Einstein's equations predicted that those waves would radiate energy in a way that cause the two bodies to gradually get closer, which would in turn speed up the rate that they completed each orbit by a very precise amount. After observing the variations in the pulse rate over a period of four years, Hulse and Taylor were able to show that the speed-up was indeed taking place at the rate predicted, to within less than one percent deviation. That provided the first experimental evidence for the existence of gravity waves.

That evidence, however, was indirect. In fact, the strength of the predicted waves in the case of the binary pulsars was far too weak for any hope of direct detection on earth. However, the same general principles would apply to a binary pair of black holes, and there the calculations indicated that the strength of the waves could be just within the limits of possible detectability with suitably crafted apparatus. Attempts to do so had already begun in the late 1950s with Joseph Weber. At that time, not only was the reality of gravity waves in doubt, but the existence of black holes was generally greeted with skepticism.

The idea of black holes (although not the name) arose very soon after Einstein formulated general relativity. Karl Schwarzschild, despite the fact that he was in the German army stationed in Russia , and that it was in the midst of World War I, read Einstein's paper, and almost immediately was able to solve Einstein's equations for the case of the gravitational field surrounding a (non-rotating) spherically symmetric body. A few weeks later he was able to solve the equations and describe the space-time curvature in the interior of the body. One of the consequences of the Schwarzschild solution seemed to be that a sufficiently massive body compressed within a sufficiently small radius (where "sufficiently" was made precise by the Schwarzschild equations) would have the property that no radiation or matter could ever escape. Oddly, a very similar conclusion was reached by purely Newtonian methods in 1783 by John Michell in England , and became widely known through Laplace 's famous 5-volume Le Système du Monde. In both cases, however, the question remained whether it was possible for a real-world physical body to exist within those parameters. The first theoretical evidence was adduced in a 1939 paper by Robert Oppenheimer and Hartland Snyder, who calculated the spacetime geometry around an imploding massive star, under certain simplifying assumptions, and concluded that the star would eventually become invisible.

As for the reality of black holes, it was hard for the experts, much less the general public, to decide whether they represented science or science fiction. Many leaders in the field, from Einstein to John Wheeler had serious doubts. It was not until the advent of X-ray astronomy that the balance was tilted in favor of science. Since X-rays from outer space do not penetrate our protective atmosphere, this research developed hand-in-hand with rocket science. The big discovery was the existence of a powerful X-ray source in the constellation Cygnus, designated Cyg X-1. This discovery was made in a rocket flight in 1964. The first X-ray satellite, Uhuru , was launched in 1970, while its successor, Einstein , launched in 1978, was an X-ray telescope that was able to make X-ray images as sharp as an optical telescope. Gradually, the scientific community became convinced that Cyg X-1 was indeed a real-life black hole whose physical characteristics corresponded closely to those predicted by the theory. Evidence has accumulated for other X-ray sources arising from the vicinity of black holes, as well as black holes in the center of quasars and large galaxies, such as our own.

One tool that has become increasingly more important in the study of black holes as in the rest of astronomy and cosmology has been computer simulations. By 2001, such simulations were able to predict the nature of the gravitational waves that we might be able to detect from the collision and merging of two black holes, and to display the results in dramatic images. The reality of black holes, and in particular, their role in the production of gravity waves, is now widely enough accepted that large amounts of money are being invested in experimental devices, such as the LIGO project, to detect associated gravity waves.

Given the extent to which theoretical predictions about black holes appear to be confirmed by observations, why the continued hesitancy about their wholehearted acceptance? One answer is that each of the predictions is based on certain simplifying assumptions and continuing unknowns. For example, the early models were for a spherically symmetric non-rotating body, whereas physical reality almost certainly corresponds to rotating bodies with concomitant bulges at the equator. But most importantly, what was missing in the early studies, and is conspicuously absent in the above discussion, is the central role of quantum effects. That, however, would lead us far afield, and can be found in many of the references given below. Instead, we indicate briefly two further subjects of particular mathematical interest.

In 1953, a young differential geometer, Eugenio Calabi, made a study of complex manifolds, and was led to conjecture that under very general conditions there should be a metric on each manifold of a particularly symmetric nature. This Calabi conjecture was a subject of great interest, and was finally proved in 1977 by Shing-Tung Yau. Although of considerable mathematical interest, the Calabi-Yau manifolds, as they came to be known, had no obvious connection to cosmology until the advent of string theory introduced a whole new dimension - or more precisely, set of dimensions - into play. What the theory required, was that in addition to our familiar four-dimensional space-time, there would be six additional "curled-up" space dimensions. Furthermore, the equations of string theory imply that this six-dimensional component must have a very particular structure, and in 1984 it was proved that the Calabi-Yau manifolds have precisely the structure needed.

The gift of mathematics to physics provided by Calabi-Yau manifolds was amply repaid when physicists discovered what was termed "mirror symmetry" between pairs of geometrically distinct but physically linked pairs of Calabi-Yau manifolds. Using this link, Philip Candelas and his collaborators were able to suggest precise numerical answers to a problem in algebraic geometry that had seemed far beyond the capabilities of any known method to provide: the number of rational curves of given degree on a fifth-degree algebraic hypersurface in projective four-space. Those numbers that algebraic geometers were able to calculate directly confirmed the predictions arising from physics.

The other circle of ideas involve what is known as "curvature flow." The simplest example consists of starting with a smooth closed curve in the plane, and defining a "flow" by moving the curve in a direction orthogonal to itself at each point, and at a speed proportional to the curvature at the point. Intuition suggests that the curve should become progressively more circular. In 1986, Michael Gage and Richard Hamilton were able to prove the result, starting from an arbitrary convex curve and normalizing the flow to fix the area enclosed. In a rather different situation, Hamilton was led to define and study a "Ricci flow" on an arbitrary Riemannian manifold, in which the rate of change of the metric tensor is proportional to the Ricci tensor. After some rather spectacular successes in which Hamilton was able to use his method to prove that under certain assumptions such a flow tended toward a constant curvature metric, Grigori Perelman announced in 2003 that he used extensions of the method to give complete proofs of Poincaré's conjecture and the Thurston "Geometrization Conjecture." Perelman's proof is still under review by the mathematical community before being fully endorsed by the experts. Possible cosmological implications relate to characterizing shapes of three-dimensional manifolds that may constitute the universe as it evolves in time.

In another direction, the curvature flow for curves was generalized to higher-dimensional hypersurfaces, leading to a proof of the "Riemannian Penrose inequality," first by Huisken and Ilmanen in 1997 for a single black hole, and then for multiple black holes in 1999 by Hubert Bray. Roger Penrose was led to the inequality in 1973 by a physical argument about the nature of black holes.

It need hardly be said that the number of topics touched upon here represents a minuscule portion of the activity in recent decades in astrophysics, cosmology, and related parts of mathematics. In some cases, theory has led the way, suggesting observations that might be made and what to look for in those observations. In others, the results of the observations have forced theorists to rethink some of their fundamental assumptions. One of the most striking examples along those lines was the discovery in 1998, in the course of examining a certain class of supernovae, that the expansion of the universe, rather than slowing down under the restraining force of gravity, appeared to be speeding up as a result of some mysterious, hitherto undreamed of force, dubbed "dark energy." One immediate thought was that this was due to Einstein's notorious "cosmological constant." But even if it worked mathematically, that would be no more of a physical explanation than when Einstein originally inserted it into his equation for what turned out to be the wrong reason: His equations seemed to imply that the universe was expanding or contracting, rather than static in time, and this was just before the realization that the universe actually was expanding. The search for a satisfactory explanation of dark energy is sure to occupy a central place in the mathematics of cosmology for some time to come.

General References

  1. Marcia Bartusiak, Einstein's Unfinished Symphony: Listening to the Sounds of Space-Time , Berkley Publishing 2003.
  2. Brian Greene, The Elegant Universe: superstrings, hidden dimensions, and the quest for the ultimate theory , New York W.W. Norton 1999.
  3. Roger Penrose, The Road to Reality: A Complete Guide to the Laws of the Universe , Knopf 2005.
  4. Saul Perlmutter, "Supernovae, Dark Energy, and the Accelerating Universe," Physics Today, April 2003, 53-60.
  5. Joseph Silk, The Big Bang , (Revised and Updated Edition) New York , Freeman 1989
  6. George Smoot and Keay Davidson, Wrinkles in Time , New York , William Morrow 1993.
  7. Kip S. Thorne, Black Holes and Time Warps: Einstein's Outrageous Legacy , New York , W.W. Norton 1994.
  8. Edward Witten, "Reflections on the Fate of Spacetime," Physics Today, April 1996, 24-30.

Technical References

  1. Hubert L. Bray, "Black Holes, Geometric Flows, and the Penrose Inequality in General Relativity," Notice of the AMS 49 (2002), 1372-1381.
  2. Ignazio Ciufolini and John A. Wheeler, Gravitation and Inertia , Princeton University Press 1995.
  3. David A. Cox and Sheldon Katz, Mirror Symmetry and Algebraic Geometry , AMS 2000.
  4. Charles W. Misner, Kip S. Thorne, and John A. Wheeler, Gravitation , New York , W.H. Freeman 1973.
  5. Barrett O'Neill, Semi-Riemannian Geometry, with Applications to Relativity , New York , Academic Press 1983.
  6. P.J.E. Peebles, Principles of Physical Cosmology , Princeton University Press 1993.

Mathematics Awareness Month is sponsored each year by the Joint Policy Board for Mathematics to recognize the importance of mathematics through written materials and an accompanying poster that highlight mathematical developments and applications in a particular area.
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