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Space and Time Beyond Einstein
by Abhay Ashtekar and Jerzy Lewandowski

Einstein's theory of general relativity is widely regarded as an intellectual triumph of twentieth century Science. Conceptually, it displays Francis Bacon's "strangeness in proportion" that characterizes the most sublime of human creations. Mathematically, it is beautiful and, observationally, it has withstood the most stringent tests ever performed. In this theory, Einstein wove the gravitational field into the very fabric of space and time. The gravitational force now stands apart from all other forces because it is a manifestation of curved space-time geometry. Matter, through its gravity, tells space-time how to bend and curved space-time, in turn, tells matter how to move.

This radical insight led to startling conclusions. Einstein predicted the existence of gravitational waves; ripples in space-time geometry which travel with the speed of light. Their existence has been verified by the orbits of dense binary stars observed by Joseph Taylor, Alexander Wolszczan and others. General relativity also predicted that the universe began with a big-bang some 15 billion years ago, relics of which have been observed in careful measurements of the cosmic microwave background. And it predicts the existence of black holes which are now believed to reside in the centers of most galaxies, often serving as powerful engines for the most energetic phenomena observed in the universe.

Yet there is a consensus among physicists that, at a fundamental level, general relativity is incomplete. For, it totally ignores quantum physics which dominates all atomic and subatomic phenomena. The world of general relativity is classical, marked by geometric precision, continuity and certainty while the quantum world is discrete, probabilistic and 'jumpy.' Since matter, which bends space-time, definitely exhibits these quantum characteristics, so should the curvature of space-time. This suggests that Einstein's space-time continuum must be an approximation. The sheet of this newspaper in front of you appears to be continuous to a human eye but we know that, when looked at under an electron microscope, it would reveal a discrete atomic structure. The situation, then, must be analogous with space-time geometry itself. If so, what are the building blocks - the atoms - of space-time geometry? What are their properties? How can we fuse the pristine, geometric world of Einstein with quantum physics, without robbing it of its soul? These are extremely difficult questions. Indeed, Einstein himself had suggested that the continuum picture is only an approximation. However, the approximation will break down only at the tiniest of scales - 10-33cm - called the Planck length. This is some twenty orders of magnitude smaller than the radius of a proton and we have no experimental means to directly observe these effects.

Yet, over the past decade very significant progress has been made through theoretical considerations. This work originated at Syracuse and Penn State Universities in the US. Soon, a dozen or so groups world-wide - including one at the University of Warsaw - joined and made key contributions. Through this systematic effort, a quantum theory of geometry has now emerged which offers language to formulate the desired generalization of Einstein's theory. Now, the fundamental 'excitations' of geometry are 1-dimensional, rather like a polymer. The continuum we are all used to is only an approximation. Perhaps the simplest way to visualize these ideas is to look at a piece of a fabric. For all practical purposes, it represents a 2-dimensional continuum; yet it is really woven by 1-dimensional threads. The same is true of the fabric of space-time. It is only because the 'quantum threads' which weave this fabric are tightly woven in the region of the universe we inhabit that we perceive a continuum. Upon intersection with a surface, each thread, or polymer excitation, endows it with a tiny, 'Planck quantum' of area of about 10-66cm2. So, an area of 100cm2 has about 1068 such intersections; because the number is so huge, the intersections are very closely spaced and we have the illusion of a continuum. The rigorous mathematics of quantum geometry predicts that lengths, areas and volumes are 'quantized' in a very specific way and enables one to calculate their 'spectra,' i.e., allowed, discrete values. These results have been used to resolve some long standing puzzles. We will conclude with two examples.

The first comes from the big-bang. General relativity predicts that the gravitational field as well as matter become infinite at the big bang; physics stops. However, there is a general consensus that these are unphysical results and quantum effects must intervene very near the big-bang and remove these infinities. This expectation is borne out in quantum geometry. Classical space-time "dissolves" very near the big-bang, when the radius of the universe is less than about 10-29cm but physics does not stop. Space-time curvature can become very large, about 1077 times that at the horizon of a solar mass black hole but it does not become infinite. The quantum state of the universe is perfectly well-defined and one can even evolve it 'backwards in time.' One can study 'initial conditions' at the big-bang and hope to analyze their ramifications for structure formation in the early universe.

The second comes from black holes. In the beginning of the twentieth century, we learnt that matter and energy are fundamentally the same; one can be converted into another. General relativity taught us that geometry is on the same physical footing as matter. So, it is natural to ask: Can the two be converted into one another? In 1974, Stephen Hawking showed black holes do radiate quantum mechanically, thereby shrinking in area. This is a strong hint that the geometrical area of a black hole horizon can be converted into matter. However, Hawking's calculation assumed a classical space-time as in general relativity. Only matter was treated quantum mechanically; there were no quanta of area. With quantum geometry, we can re-examine the situation. Now, it is literally true that the black hole horizon acquires its area through its intersections with polymer geometry. In the Hawking process, quanta of area are converted to quanta of matter. Thus, Einstein's vision on the physical nature of geometry is now realized at the quantum level. This transmutation of geometry to matter is the "Einsteinian alchemy!"

From the Polish publication RZECZPOSPOLITA (April 2002)

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