Time in general relativity and cosmology
- Key People:
- Henri Bergson
- Ferdinand Berthoud
News •
In general relativity, which, though less firmly established than the special theory, is intended to explain gravitational phenomena, a more complicated metric of variable curvature is employed, which approximates to the Minkowski metric in empty space far from material bodies. Cosmologists who have based their theories on general relativity have sometimes postulated a finite but unbounded space-time (analogous, in four dimensions, to the surface of a sphere) as far as spacelike directions are concerned, but practically all cosmologists have assumed that space-time is infinite in its timelike directions. Kurt Gödel, a contemporary mathematical logician, however, has proposed solutions to the equations of general relativity whereby timelike world lines can bend back on themselves. Unless one accepts a process philosophy and thinks of the flow of time as going around and around such closed timelike world lines, it is not necessary to think that Gödel’s idea implies eternal recurrence. Events can be arranged in a circle and still occur only once.
The general theory of relativity predicts a time dilatation in a gravitational field, so that, relative to someone outside of the field, clocks (or atomic processes) go slowly. This retardation is a consequence of the curvature of space-time with which the theory identifies the gravitational field. As a very rough analogy, a road may be considered that, after crossing a plain, goes over a mountain. Clearly, one mile as measured on the humpbacked surface of the mountain is less than one mile as measured horizontally. Similarly—if “less” is replaced by “more” because of the negative signs in the expression for the metric of space-time—one second as measured in the curved region of space-time is more than one second as measured in a flat region.
Strange things can happen if the gravitational field is very intense. It has been deduced that so-called black holes in space may occur in places where extraordinarily massive or dense aggregates of matter exist, as in the gravitational collapse of a star. Nothing, not even radiation, can emerge from such a black hole. A critical point is the so-called Schwarzschild radius measured outward from the centre of the collapsed star—a distance, perhaps, of the order of 10 kilometres. Something falling into the hole would take an infinite time to reach this critical radius, according to the space-time frame of reference of a distant observer, but only a finite time in the frame of reference of the falling body itself. From the outside standpoint, the fall has become frozen. But from the point of view of the frame of the falling object, the fall continues to zero radius in a very short time indeed—of the order of only 10 or 100 microseconds. Within the black hole, spacelike and timelike directions change over, so that to escape again from the black hole is impossible for reasons analogous to those that, in ordinary space-time, make it impossible to travel faster than light. (To travel faster than light a body would have to lie—as a four-dimensional object—in a spacelike direction instead of a timelike one.)
As a rough analogy, two country roads may be considered, both of which go at first in a northerly direction. But road A bends round asymptotically toward the east; i.e., it approaches ever closer to a line of latitude. Soon road B crosses this latitude and is thus to the north of all parts of road A. Disregarding Earth’s curvature, it takes infinite space for road A to get as far north as that latitude on road B; i.e., near that latitude an infinite number of “road A northerly units” (say, miles) correspond to a finite number of road B units. Soon road B gets “beyond infinity” in road A units, though it need be only a finite road.
Rather similarly, if a body should fall into a black hole, it would fall for only a finite time, even though it were “beyond infinite” time by external standards. This analogy does not do justice, however, to the real situation in the black hole—the fact that the curvature becomes infinite as the star collapses toward a point. It should, however, help to alleviate the mystery of how a finite time in one reference frame can go “beyond infinity” in another frame.
Most cosmological theories imply that the universe is expanding, with the galaxies receding from one another (as is made plausible by observations of the red shifts of their spectra), and that the universe as it is known originated in a primeval explosion at a date of the order of 15 × 109 years ago. Though this date is often loosely called “the creation of the universe,” there is no reason to deny that the universe (in the philosophical sense of “everything that there is”) existed at an earlier time, even though it may be impossible to know anything of what happened then. (There have been cosmologies, however, that suggest an oscillating universe, with explosion, expansion, contraction, explosion, etc., ad infinitum.) And a fortiori, there is no need to say—as Augustine did in his Confessions as early as the 5th century ce—that time itself was created along with the creation of the universe, though it should not too hastily be assumed that this would lead to absurdity, because common sense could well be misleading at this point.
A British cosmologist, E.A. Milne, however, proposed a theory according to which time in a sense could not extend backward beyond the creation time. According to him, there are two scales of time, “τ time” and “t time.” The former is a time scale within which the laws of mechanics and gravitation are invariant, and the latter is a scale within which those of electromagnetic and atomic phenomena are invariant. According to Milne, τ is proportional to the logarithm of t (taking the zero of t to be the creation time); thus, by τ time the creation is infinitely far in the past. The logarithmic relationship implies that the constant of gravitation G would increase throughout cosmic history. (This increase might have been expected to show up in certain geological data, but apparently the evidence is against it.)
Time in microphysics
Special problems arise in considering time in quantum mechanics and in particle interactions.
Quantum mechanical aspects of time
In quantum mechanics it is usual to represent measurable quantities by operators in an abstract many-dimensional (often infinite-dimensional) so-called Hilbert space. Nevertheless, this space is an abstract mathematical tool for calculating the evolution in time of the energy levels of systems—and this evolution occurs in ordinary space-time. For example, in the formula AH − HA = iℏ(dA/dt), in which i is Square root of√−1 and ℏ is 1/2π times Planck’s constant, h, the A and H are operators, but the t is a perfectly ordinary time variable. There may be something unusual, however, about the concept of the time at which quantum mechanical events occur, because according to the Copenhagen interpretation of quantum mechanics the state of a microsystem is relative to an experimental arrangement. Thus, energy and time are conjugate: no experimental arrangement can determine both simultaneously, for the energy is relative to one experimental arrangement, and the time is relative to another. (Thus, a more relational sense of “time” is suggested.) The states of the experimental arrangement cannot be merely relative to other experimental arrangements, on pain of infinite regress, and so these have to be described by classical physics. (This parasitism on classical physics is a possible weakness in quantum mechanics over which there is much controversy.)
The relation between time uncertainty and energy uncertainty, in which their product is equal to or greater than h/4π, ΔEΔt ⋜ h/4π, has led to estimates of the theoretical minimum measurable span of time, which comes to something of the order of 10−24 second and hence to speculations that time may be made up of discrete intervals (chronons). These suggestions are open to a very serious objection—viz., that the mathematics of quantum mechanics makes use of continuous space and time (for example, it contains differential equations). It is not easy to see how it could possibly be recast so as to postulate only a discrete space-time (or even a merely dense one). For a set of instants to be dense, there must be an instant between any two instants. For it to be a continuum, however, something more is required—viz., that every set of instants earlier (later) than any given one should have an upper (lower) bound. It is continuity that enables modern mathematics to surmount the paradox of extension framed by the Pre-Socratic Eleatic Zeno—a paradox comprising the question of how a finite interval can be made up of dimensionless points or instants.
Time in particle interactions
Until recently it was thought that the fundamental laws of nature are time symmetrical. It is true that the second law of thermodynamics, according to which randomness always increases, is time asymmetrical, but this law is not strictly true (for example, the phenomenon of Brownian motion contravenes it), and it is now regarded as a statistical derivative of the fundamental laws together with certain boundary conditions. The fundamental laws of physics were long thought also to be charge symmetrical (for example, an antiproton together with a positron behave like a proton and electron) and to be symmetrical with respect to parity (reflection in space, as in a mirror). The experimental evidence now suggests that all three symmetries are not quite exact but that the laws of nature are symmetrical if all three reflections are combined: charge, parity, and time reflections forming what can be called (after the initials of the three parameters) a CPT mirror. The time asymmetry was shown in certain abstruse experiments concerning the decay of K mesons that have a short time decay into two pions and a long time decay into three pions.
Time in molar physics
The above-mentioned violations of temporal symmetry in the fundamental laws of nature are such out-of-the-way ones, however, that it seems unlikely that they are responsible for the gross violations of temporal symmetry that are apparent in the visible world. An obvious asymmetry is that there are traces of the past (footprints, fossils, tape recordings, memories) and not of the future. There are mixing processes but no comparable unmixing process: milk and tea easily combine to give a whitish brown liquid, but it requires ingenuity and energy and complicated apparatus to separate the two liquids. A cold saucepan of water on a hot brick will soon become a tepid saucepan on a tepid brick, but the heat energy of the tepid saucepan never goes into the tepid brick to produce a cold saucepan and a hot brick. Even though the laws of nature are assumed to be time symmetrical, it is possible to explain these asymmetries by means of suitable assumptions about boundary conditions. Much discussion of this problem has stemmed from the work of Ludwig Boltzmann, an Austrian physicist, who showed that the concept of the thermodynamic quantity entropy could be reduced to that of randomness or disorder. Among 20th-century philosophers in this tradition may be mentioned Hans Reichenbach, a German-U.S. Positivist, Adolf Grünbaum, a U.S. philosopher, and Olivier Costa de Beauregard, a French philosopher-physicist. There have also been many relevant papers of high mathematical sophistication scattered through the literature of mathematical physics. Reichenbach (and Grünbaum, who improved on Reichenbach in some respects) explained a trace as being a branch system—i.e., a relatively isolated system, the entropy of which is less than would be expected if one compared it with that of the surrounding region. For example, a footprint on the beach has sand particles compressed together below a volume containing air only, instead of being quite evenly (randomly) spread over the volume occupied by the compressed and empty parts.
Another striking temporal asymmetry on the macro level—viz., that spherical waves are often observed being emitted from a source but never contracting to a sink—was stressed by Sir Karl Popper, a 20th-century Austrian and British philosopher of science. By considering radiation as having a particle aspect (i.e., as consisting of photons), Costa de Beauregard has argued that this “principle of retarded waves” can be reduced to the statistical Boltzmann principle of increasing entropy and so is not really different from the previously discussed asymmetry. These considerations also provide some justification for the common-sense idea that the cause–effect relation is a temporally unidirectional one, even though the laws of nature themselves allow for retrodiction no less than for prediction.
A third striking asymmetry on the macro level is that of the apparent mutual recession of the galaxies, which can plausibly be deduced from the red shifts observed in their spectra. It is still not clear whether or how far this asymmetry can be reduced to the two asymmetries already discussed, though interesting suggestions have been made.
The statistical considerations that explain temporal asymmetry apply only to large assemblages of particles. Hence, any device that records time intervals will have to be macroscopic and to make use somewhere of statistically irreversible processes. Even if one were to count the swings of a frictionless pendulum, this counting would require memory traces in the brain, which would function as a temporally irreversible recording device.