Home Page Overview Site Map Index Appendix Illustration About Contact Update FAQ

Relativity, Cosmology, and Time

A Brief History of Time
Arrows of Time
Asymmetry of Time
Measuring Time
Mathematical Time
Symmetry Breaking of Time
Quantization of Time
Imaginary Time
End of Time
Two Times
Thermal Time
Reborn of Time

A Brief History of Time

For 10 billion years, the universe has been in existence without bothering with the definition of time. It started about 3.5 billion years ago when unicellular organisms took up residence on Earth. They had to adjust their activities according to the daily and yearly cycles. Since then all living beings including human come equipped with biological clocks within to adopt to these rhythms. For thousands of years, protohumans probably had only dim notions of time: past, present, and future. Beginning around 2500 BCE, systemic definitions of time were developed in the form of
Sundail calendars. The Egyptian pioneers first divided a day into 24 units. Other calendars were linked to religion and the need to predict days of ritual significance, such as the summer solstice. All calendars had to resolve the incommensurate cycles of days, lunations and solar years, usually by intercalating extra days or months at regular intervals. The Julian calendar was established at 46 BCE. The first mean of measuring daily time was probably the Egyptian sundail, dating from about 1500 BCE (Figure 11a). It was soon followed by the water clock or clepsydra and the sandglass or hourglass, in which time is measured by change in level of flowing water or sand. The earliest mechanical clocks containing movable parts were built about 700 years ago. It had no minute hand. In the beginning,

Figure 11a Sundail [view large image]

time is cyclic, but it becomes linear when the cycles are added up together turning into history.

Newton When Newton published the three natural laws in 1686, time is no longer confined to record the daily and yearly rhythms. It had become a mathematical entity - a parameter to keep track of motions in a fixed, infinite, unmoving space. Einstein changed this with his relativity theories, and once wrote, "Newton, forgive me." In the new theories, time is treated almost on the same footing as the spatial dimensions with some essential differences. Recently, theory in quantum gravity considers time and space to be discrete at Planck scale with a minimum temporal interval and spatial size of about 10 -43 sec and 10-33 cm respectively. At this scale, they are useless as framework for the motion of other objects. It is suggested that time and space are the active participants in the dynamics of this world.

Figure 11b Newton [view large image]

Representation of Space, Time While space and time can emerge in mathematical formula or graph in symbol or curve (Figure 11c,a), the five senses of human fail to perceive space and time directly. We recognize space through its representation by some features such as a house, a hill, or a galaxy, ... (Figure 11c,b) and similarly, the representation of time has to be a sequence of events (Figure 11c,c). Scholars over the eon have lamented about the difficulty of defining time. A lot of confusion could be mistaken representation of time as the "TIME" itself. Thus, instead of trying to define time or provide an answer to the philosophical question of "What is time?", some of the characteristics and/or interpretations of time-representation are listed in the followings :

Figure 11c Representations of Space, Time [view large image]


Arrows of Time

Arrows of Time The arrow of time often invokes an image of time flying into one direction. In reality, time is ticking at its own pace; only the underlying representation is moving in that direction as shown by the entropy evolution in Figure 11e below. In this instance, the universe has so many different configurations (in terms of position and velocity) for the objects to occupy; there is very little chance that they will return to the more confined pattern once the holding force is removed. The situation is quite different for a closed system of only one or two particles, it is more likely for them to return to the previous configuration. Thus, the arrow is applicable mostly to an emsemble of many objects.

Figure 11d Arrows of Time [view large image]

  • Thermodynamic Arrow of Time - The laws of physics do not care about the direction of time, i.e., the mathematical formulations are invariant if the direction of time is reversed. Yet there is a big difference between the forward and backward directions of real time in ordinary life. We have no difficulty to tell the sequence of events in Figure 11c,d, because it is highly improbable for the crumbling dust to return to the structured building in the reversed direction. It is usually explained by the second law of thermodynamics, which states that in the macroscopic world there is a tendency for a closed system moving toward greater disorder. This is the thermodynamic arrow of time in Figure 11d.
  • Figure 11e Thermodynamic Arrow of Time [view large image]

    As shown in Figure 11e, ultimately it is the ever increasing entropy that causes the seemingly flow of time in one direction.

    Cosmic Expansion
  • Cosmological Arrow of Time - Since volume expansion tend to increase the entropy of a system (Figure 11e), the cosmic expansion would be another indicator for the direction of time (Figure 11f). But what would happen if and when the universe stops expanding and begins to contract? According to Hawking, he used to believe that disorder would decrease when the universe recollapse. Now he realizes that the no boundary condition (see imaginary time) implies that disorder would in fact continue to increase during the contraction. He attributes the association between the thermodynamic and cosmolgical arrows to the no boundary condition instead of volume expansion.
  • Figure 11f Cosmic Expansion

    Figure 11f illustrates one more time that it is the ever increasing cosmic expansion that causes the apparent flow of time in one direction.

    Psychological Time
  • Psychological Arrow of Time (Subjective Time) - The psychological perception of time is affected by such things as medications, time of day, level of happiness, external stimuli, and even the temperature. Einstein once remarked, "When you spend two hours with a nice girl, you think it's only a minute. But when you sit on a hot stove for a minute, you think it's two hours." This statement is exemplified by Einstein and friend (Figure 11g) in a 1985 movie called "Insignificance". Thus, the time line experienced by the conscious brain is often quite different from the "objective" time line of events occurring in the real world. Beside the circadian clock, which controls activities in 24-hour cycle, and the millisecond timing, which is involved in fine motor motion; it is found that there is a region of the brain called the striaturm (a cluster of nuclei that includes the putamen and the caudate nucleus), which is used to perceive the passage of time in the seconds-to-hours range. As neurons in the brain regions go about their business, coordinating movement, attention, memory and so on, they produce waves of electrical excitation that are detected by the striatum and integrated into an estimate of how much time has passed -
  • Figure 11g Einstein and Friend [view large image]

    producing the subjective time. It is suggested that such subjective time can be manipulated by brain chemistry, in particular the dopamine (one kind of neurotransmitters that controls arousal levels). It is known that patients with
    disorders in its secretion, such as Parkinson's disease, Huntington's or Schizophrenia, also suffer disturbances in their perception of time. It turns out this is because their neurochemistry - specifically their dopamine level - somehow alters the speed of their subjective internal clock. Schizophrenics have too much dopamine in the brain, their clock is so fast that it feels like the whole world is crazy. Stimulants such as cocaine, caffeine and nicotine make time passing faster, while sedatives such as Valium and cannabis slow it down.


    Asymmetry of Time

    Time Machine The asymmetry between past and future has two aspects. On the one hand, we know events happened in the past by memory recall, history books, fossil records, and astronomical observations etc., but we know nothing about the future. The direction of time in which we remember the past and not the future is referred to as the psychological arrow of time (Figure 11g). On the other hand, we can only move forward into the future and can never go backward to the past physically. This kind of asymmetry is related to the principle of cause and effect (causality), which is an important concept in physical theories. For example, the notion that events can be ordered into causes and effects is necessary to prevent contradictions such as the grandfather paradox, which asks what happens if a time-traveller kills his own grandfather before he ever meets his grandmother. Within special relativity, causality can be preserved by forbidding information from traveling faster than the speed of light - the worldline is not allowed to loop back to the past. It is strongly suspected that general relativity also preserves causality and forbids agents from changing the past, despite the possibility of developing a closed time-like curve. In term of the huge number of possible configurations, the asymmetry can be explained by the very low probability for reproduction of the past events; by the same token we cannot predict what will be the next configurations.

    Figure 11h Time Machine [view large image]

    Regardless, the idea of using time machine to visit the past is still a hot topic in science fictions. Figure 11h shows a fictitious time machine.


    Measuring Time

    Time is often equated to the change of a variable dR. Mathematically, it can be expressed as dR/dt = constant rate of change, where dt denotes the change in time. One revolution of the Earth around the Sun defines a year. One complete rotation of Earth defines a day. The moving hands of clocks define hour, minute, and second. In these cases, the variable R is the angle of rotation. In short, the units of time are often defined by cyclic motions and their subdivisions. The accuracy in clocks has improved over the last 700 years until today (Figure 12a), an atomic clock known as NIST-7 is accurate to 10-9 second per day (Figure 13).
    Clock Accuracy
    Quartz Clock
    Atomic Clock Modern quartz clocks use the piezo-electric properties of the quartz crystal, which vibrates at a specific frequency when placed in an alternating electric current circuit. The induced "crystal current" (the oscillator) is amplified and used to operate an LED display or electrically actuated hands (see Figure 12b). The circuit advances the time display by 1 second for every 9,998,876,995 oscillations (Hz).

    Atomic clocks use the frequency of atomic emission to regulate a quartz crystal clock (see Figure 13). One second is now defined as 9,192,631,770 oscillation (the frequency of the atomic line emission) associated with the microwave from the hyperfine transition of cesium atoms. All types of design depend on the tuning of the microwave cavity to find the frequency,

    Figure 12 a, b Clocks [view large image a, b]

    Figure 13 Atomic Clock
    [view large image] [more image]

    which induces the maximum number of transitions between the hyperfine states. This is used as reference to measure error in the output frequency
    (to the clock). Any deviation of the output frequency (originated within the quartz oscillator) against this standard will be corrected by the Servo Control(see "more image" under Figure 13). The process is similar to the adjustment of the hand manually in the conventional wall clock by comparing it to a more accurate time piece.

    Atomic Clock Next Generation In the next generation atomic clock, the microwave cavity is replaced by the frequency comb, which generates a train of million laser pulses each one with a different frequency (in optical range). The frequency comb is used to probe a lattice of atoms cooled to microKelvin temperature. The sweep produced a profile (the bell curve in Figure 14a). Due correction is made according to the difference between the frequencies at the peak (corresponding to the most intense signal induced by one of the comb frequencies) and the actual output to the clock. It is claimed that this new model is ten times

    Figure 14a Atomic Clock, Next Generation [view large image]

    more accurate than the conventional atomic clock. One of the problems is synchronization, which is more difficult to achieve than the microwave signals from the older model.


    Mathematical Time

    The crucial difference between Classical Mechanics and Special Relativity is the appearance of the velocity of light c in two inertial frames moving with constant velocity V relative to each other (Figures 14b and 14c). In classical mechanics time ticks at the same pace in the two reference frames while the velocity of light appears to be different, i.e., it has a value c with respect to one frame and a different value c' = c - V with respect to the other one. Such view has been proved to be wrong by experiments, which show that the velocity of light is the same in all inertial frames. The novel concept reveals many unusual phenomena such as shorten of length and dilation of time as V c. They are the consequence of trying to reconcile the following two equations :

    Classical Mechanics Special Relativity x2 + y2 + z2 = c2 t2 or
    x2 + y2 + z2 - c2 t2 = 0     (for an observer in the S frame) ---------- (1)

    x'2 + y'2 + z'2 = c2 t'2 or
    x'2 + y'2 + z'2 - c2 t'2 = 0     (for an observer in the S' frame) ---------- (2)

    Figure 14b Classical Mechanics [view large image]

    Figure 14c Special Relativity [view large image]

    Thus, the simple Galilean transformation between inertial frames :
    x' = x - Vt, is replaced by the Lorentz transformations (Figures 14b, 14c):

    x' = [x - (V/c)ct] / (1 - V2/c2)1/2,    y' = y,    z' = z,    ct' = [ct - (V/c)x] / (1 - V2/c2)1/2 ---------- (3)
    Black Hole In general relativity, time or actually i ct attains an equal footing with the spatial coordinates and denoted as x4 = i ct. Space and time are merged together called space-time. For example, the space-time interval for a particle is expressed in the form : -ds2 = gdxdx where and are the indices running from 1 to 4. In particular, the space-time interval for a point mass M in spherical coordinates is :
    -ds2 = -(1 - 2GM/c2r) c2dt2 + dr2 / (1 - 2GM/c2r) + r2 ( d2+sin2 d2) ---------- (4)
    in which g44 = (1 - 2GM/c2r), g11 = 1/(1 - 2GM/c2r), g22 = r2, g33 = r2 sin2.
    For the case when r = 2GM/c2, g44 = 0, while g11 . This is the Schwarzschild radius or event horizon

    Figure 14e Black Hole
    [view large image]

    of a black hole (Figure 14e) for which it is often dubbed as space-time warp. In fact, space-time behaves as usual, only the metric tensor g11 becomes infinity and nobody know what's beyond the horizon.

    Proper Time On the other hand, time dilation (Figure 15,c) may be a kind of time warp from the view point of the proper time , which is defined as the time moving together with the observer and hence the spatial derivatives vanish in this particular choice of frame, i.e., ds2 = c2d2 (Figure 15,a); while for another observer ds2 = c2dt2[1-(V/c)2]. Since the space-time interval ds is an invariant under the Lorentz transformation between different inertia frames, the proper time d = dt[1-(V/c)2]1/2, which shows that proper time runs slower according to an outside observer and hence the twin paradox.
    In general relativity, the proper time would be the one associated with the free-fall frame where the supporting force F becomes zero (Figure 15,b). Thus, ds2 = c2d2 ; while for a rest frame ds2 = g44c2dt2 . It follows that d = (g44)1/2dt. For the case of running around a black hole horizon (g44)1/2 ~ (1 - 2GM/c2r)1/2, the proper time runs very slow according to an outside observer not plunging into the black hole.

    Figure 15 Proper Time
    [view large image]

    The proper time in cosmological expansion is called comving time d = R(t)dt, where R(t) is the scale factor. The corresponding space-time interval is : -ds2 = -c2d2 + R(t)2 [dr2 + r2 (d2 + sin2 d2)].
    As R(t) evolves from 0 to 1 at the moment of Big Bang to present, the proper time again shows its peculiar property near the singularity as R(t) 0, i.e., it almost stopped ticking at the beginning of the universe according to an external observer, i.e., the spectral line is red shifted to infinity. BTW, this is the beginning of time according to the "Standard Cosmology". The latest data from observation of the CMBR yield an age of 13.8 billion years. Some cosmological models assert that there is no beginning of time (see "Cyclic Universe").

    In quantum field theory, ct plays the same role as in special relativity, i.e., it is treated the same way as x, y, z. Indeed, the theory as to be invariant under the Lorentz transformation. The requirement is relaxed in quantum mechanics as its formulation is non-relativistic. Anyway, in both cases, time is the conjugate variable of the energy E, meaning they can be transformed into each other by Fourier transform. This is evident in the uncertainty principle E t > , and in the commutative relation (Et - tE) = i. The uncertainty principle implies that the energy of the system is stationary (the eigen-energy) if t ; on the other hand virtual particle with certain mass-energy E can pop up to its brief
    Quantum Time existence in finite t allowed by the uncertainty. The eigenvalue solutions in Quantum Mechanics are very important in understanding the atomic and molecular structures, while the virtual particles would alter the dynamic of particle interactions (Figure 16a). The commutative relation requires E to be an operator in the form i d/dt.

    Figure 16a Quantum Time
    [view large image]

    Most theoretical formulations are symmetrical under the time reversal t -t operation (designated by T). However, the observable universe does not show such symmetry, primarily due to the "second law of thermodynamics".

    Time Reversal
      T violation can occur in three levels as shown below (from intrinsic to man-made) :

    1. Theory on the weak interaction.
    2. Second law of thermodynamics.
    3. Quantum non-invasive measurements.

    Figure 16b Time Reversal, Symmetry and Violation [view large image]

    Figure 16b shows the time reversal invariance in physical law and quantum process with T represents the operation. It also portrays a sequence of time irreversible implosion.


    Thermal Time

    Thermal Time Another attempt to do away with time asserts that it is all an illusion. The thermal time hypothesis (Figure 20) suggests that a statistical effect gives rise to the "appearance" of time. Similar to temperature, which is the average of the momentum of each molecule, the same applies to the thermal time but including many more constituents such as space (which is expanding on the average). It predicts that the ratio of the observer's proper time to the thermal time is the surrounding temperature. A toy model has been constructed successfully from the CMBR data to explain the cosmic expansion as described by standard cosmology. It can also reproduce the temperature of a black hole associated with the Hawking radiation. The idea follows the rework of quantum mechanics without time. The evolution in time is replaced by the variation of correlations between things as mentioned above. Instead of "collapsing" the wave function of an electron, both the electron and the measuring device are described by a single wave function, and a single measurement of the entire set-up causes the collapse.

    Figure 20 Thermal Time
    [view large image]

    It is anticipated that combining quantum mechanics with general relativity would become less daunting when it is rewritten in time-free form. The loop quantum theory is an example adopting such idea.

    So much for getting rid of time, now let's celebrate "Reborn of Time".

    Go to Next Section
     or to Top of Page to Select
     or to Main Menu