Time

The Physics Hypertextbook™
© 1998-2008 by Glenn Elert -- A Work in Progress
All Rights Reserved -- Fair Use Encouraged

prev | up | next

Discussion

introduction

What is time?

What is Time?
source quote
Albert Einstein:

 Zeit ist das, was man an der Uhr abliest.
[Time is what a clock measures.]
Aristotle:

 ὥσπερ οὖν εἰ μὴ ἦν ἕτερον τὸ νῦν ἀλλὰ ταὐτὸ καὶ ἕν, οὐκ ἂν ἦν χρόνος.
[As, if the now had remained the same, time would not have existed.]
Saint Augustine:

 quid est ergo tempus? si nemo ex me quaerat, scio; si quaerenti explicare velim, nescio.
[What, then, is time? If no one ask of me, I know; if I wish to explain to him who asks, I know not.]
Marcus Aurelius:

 Time is like a river made up of the events which happen, and its current is strong; no sooner does anything appear than it is swept away, and another comes in its place, and will be swept away too.
Julian Barbour:

 … I now believe that time does not exist at all, and that motion itself is a pure illusion. What is more, I believe there is quite strong support in physics for this view. I have a vision and I want to tell you about it.
Samuel Beckett:

 Vladimir: That passed the time.
Estragon: It would have passed in any case.
Vladimir: Yes, but not so rapidly.
Henri Bergson:

 Le temps est ce qui empêche que tout soit donné tout d'un coup.
[Time keeps everything from happening all at once.]
Hector Berlioz:

 Time is a great teacher, but unfortunately it kills all its pupils.
Carol Burnett:

 Comedy is tragedy plus time.
Charles W. Chesnutt:

 Time touches all things with destroying hand
Benjamin Disraeli:

 Time is precious, but truth is more precious than time.
Benjamin Franklin:

 You may delay, but time will not.
Merrick Furst:

 The biggest difference between time and space is that you can't reuse time.
Ben Hecht:

 Time is a circus, always packing up and moving away.
Hippocrates:

 Χρόνος ἐστὶν ἐν ᾧ καιρός, καὶ καιρὸς ἐν ᾧ χρόνος οὐ πολύς‧
[Time is that wherein there is opportunity, and opportunity is that wherein there is no great time.]
Aldous Huxley:

 Time, as we know it, is a very recent invention. The modern time-sense is hardly older than the United States. It is a by-product of industrialism — a sort of psychological analogue of synthetic perfumes and aniline dyes.
Time is our tyrant.
To us, the moment 8:17 AM means something — something very important, if it happens to be the starting time of our daily train. To our ancestors, such an odd eccentric instant was without significance — did not even exist. In inventing the locomotive, Watt and Stevenson were part inventors of time.
Franklin P. Jones:

 Time is a versatile performer. It flies, marches on, heals all wounds, runs out, and will tell.
Søren Kierkegaard:

 Life can only be understood backwards, but it must be lived forwards.
Stanislaw Lec:

 People find life entirely too time consuming.
Abraham Lincoln:

 The best thing about the future is that it comes only one day at a time.
Groucho Marx:

 Time flies like an arrow. Fruit flies like a banana.
David B. Norris:

 How you spend your time is more important than how you spend your money. Money mistakes can be corrected, but time is gone forever.
Ovid:

 Time is the devourer of all things.
Theodore Roethke:

 Time marks us while we are marking time.
William Shakespeare

 But thought's the slave of life, and life time's fool;
And time, that takes survey of all the world,
Must have a stop.
Bob Talbert:

 Time neither subtracts nor divides, but adds at such a pace it seems like multiplication.
Henry David Thoreau:

 As if we could kill time without injuring eternity!
H.G. Wells:

 Clearly… any real body must have extension in four directions: it must have length, breadth, thickness, and duration…. There are really four dimensions, three which we call the three planes of space, and a fourth, time. There is, however, a tendency to draw an unreal distinction between the former three dimensions and the latter, because it happens that our consciousness moves intermittently in one direction along the latter from the beginning to the end of our lives.
Stephen Wright:

 Everywhere is in walking distance if you have the time.
Chinese Proverb:

 An inch of time cannot be bought with an inch of gold.
unknown:

 The speed of time is one second per second.
unknown:

 Time heals all wounds — except deadly ones.
Tom Stoppard:

 Eternity is a terrible thought. I mean, where's it going to end?

Lots of fancy words for what could be stated more simply, or can it? Simple concepts are often the most difficult to explain. Many times, there's one notion that we carry around with us and another, more technical and specific, that's used in physics. What time is it? Pretty simple question, no? Saturday, 04-Jul-2009 11:32:00 GMT. Now tell me again what time is. Is my definition any good? Well, of course it is. Otherwise I wouldn't have written it down. Time in the physical sense is always relative and never absolute. That is to say, we always consider time as the interval between two events and not as some cumulative measure of this abstract thing we talk about when when ask, "What time is it?" When did time begin? In the year 1 CE? Hardly. That's just a cultural fixed point. Do you think Socrates walked around thinking, "I can't believe it's 535 BCE already. I can't stop writing 536 BCE on my checks"? When the aliens land on Mount Shasta they'll have a different chronometer reading in their spaceships, using different units and a different zero point. Furthermore, why is it Saturday, 04-Jul-2009 11:32:00 GMT in London and Saturday, 04-Jul-2009 04:32:00 PDT in Chicago? This idea of local time is not usually what we mean by time in physics.

Have you ever heard people say "Daylight Saving Time is ending so we'll gain an hour Saturday night"? Well let me tell you right now that nobody ever gained an hour when Daylight Saving Time ended, not literally anyway. If you were in the hospital, nearing death at 1:59 AM on the last Saturday of October in the United States would you have lived an extra hour if you died two minutes later at 1:01 AM? Would you say you went back in time? No way. There's time -- in the sense of how long it takes for something to happen -- and then there's time -- in the sense of "What time is it?". Time in the former sense, as used in physics, is measured with a stopwatch or an interval timer, while time in the latter sense is determined with a wristwatch or clock. In the course of your study of physics you'll have to understand the difference between the two meanings of this one word.

Time is a measure of the interval between two events. Time is also a cultural construct whereby an event can be associated with a series of numbers. When the quantity called time appears in a physical equation, it is always referring to the measure of the interval between two events.

natural units

year month day

The length of a year depends on your choice of calendars or how you define a year.

How long is a year?
type in days in seconds notes
standard 365 31,536,000  
leap 366 31,622,400  
julian 365.25 31,557,600 adopted by Julius Caesar in 46 BCE
gregorian 365.2425 31,556,952 adopted by Pope Gregory XIII in 1582
tropical 365.242189352… 31,556,925.16… period between vernal equinoxes
sidereal 365.256363… 31,558,149.76… orbital period measured relative to the stars
anomalistic 365.259635… 31,558,432.46… orbital period measured between perihelia
eclipse 346.620075… 29,947,974.48… period of lunar node passages

cultural units

hour minute second

week

decade century millennium eon

calendars

blah

time zones

Greenwich Mean Time (GMT)

international atomic time (tai)

The SI unit of time is the second [s].


[magnify]

The hyperfine transition is the basis of International Atomic Time (TAI). By definition, the outermost electron in an ordinary cesium 133 atom cycles through this transition 9,192,631,770 times in one second.

International Atomic Time (abbreviated TAI after the French Temps Atomique International) began at midnight GMT on the first day of 1958 and has continued advancing forward at the rate of one second every 9,192,631,770 periods of the hyperfine transition in 133Cs. TAI is maintained by the Bureau International des Poids et Mesures (BIPM) in Paris, which periodically averages the time kept by various atomic clocks around the world. The BIPM then disseminates correction factors needed to synchronize these clocks with the master clock in the Observatoire de Paris.

coordinated universal time (utc)

Coordinated Universal Time (abbreviated UTC) is the basis of legal time throughout the world. All local civil times differ from UTC by either a whole number of hours or an odd number of half hours, but never by any other amount. One second of Coordinated Universal Time is the same as one second of International Atomic Time, but UTC and TAI are slightly out of step. TAI marches forward uniformly, while UTC is adjusted from time to time to keep it synchronized with the earth's rotation.

The earth is not an effective timekeeper. For most of the last two hundred years the mean solar day has been slightly longer than the 86,400 s currently defined by the International System. Universal Time UT, or more specifically UT1, is in effect the mean solar time. It is continuous (i.e. there are no leap seconds) but has a variable rate because of the Earth's non-uniform rotation period. It is needed for computing the sidereal time, an essential part of pointing a telescope at a celestial source. The quantity UT1-UTC, which typically changes by 1 or 2 ms per year, can only be obtained by observation, though seasonal trends are known and the IERS listings are able to predict some way into the future with adequate accuracy for pointing telescopes.


[magnify]


[magnify]

When UT1 lags too far behind UTC, a leap second is inserted at the end of the day before January 1 or July 1 as appropriate. When this happens, 23:59:59 is followed by the unusual time of 23:59:60 before turning over to 00:00:00 and starting the next day. In the unlikely event that UT1 were to lead UTC (that is, if the earth's rate of rotation were to increase) the provision exists for the insertion of a negative leap second. Were this to ever occur, 23:59:58 of one day would be followed by 00:00:00 of the next, skipping 23:59:59 altogether. In any case the absolute difference between UTC and UT1 must never exceed 0.9 s.

The decision on when to insert a leap second is made by the International Earth Rotation Service (IERS) in Frankfurt based on observations of the Earth's orientation in space.


[magnify]

Adding leap seconds to Coordinated Universal Time (UTC) keeps it in line with the rotation of the earth as measured by Universal Time (UT1).

Leap Seconds and Cumulative Adjustments to UTC
year month offset   year month offset   year month offset   year month offset
1970 n/a* n/a*   1980 January 19   1990 January 25   2000   32
1971 n/a* n/a*   1981 July 20   1991 January 26   2001   32
1972 July 11   1982 July 21   1992 July 27   2002   32
1973 January 12   1983 July 22   1993 July 28   2003   32
1974 January 13   1984   22   1994 July 29   2004   32
1975 January 14   1985 July 23   1995   29   2005   32
1976 January 15   1986   23   1996 January 30   2006 January 33
1977 January 16   1987   23   1997 July 31   2007 ? ?
1978 January 17   1988 January 24   1998   31   2008 ? ?
1979 January 18   1989   24   1999 January 32   2009 ? ?
* Adjustments from 1961 to 1971 follow a different, more complicated protocol and were omitted.

Coordinated Universal Time, CUT -- Temps universel coordonné, TUC

Are leap seconds even necessary?

What would be so wrong with that? Does it really matter what number we assign to a position of the sun in the sky?

In 1582 Pope Gregory XIII managed to extract ten days from the calendar. On 4 October 1582 the Catholic world went to sleep. When they woke up it was 15 October 1582. By 1752 the protestant nation of England and her American colonies also accepted the change. (They needed to add 11 days to catch up.) In 1873 Japan made the switch. (They needed 12 days.) Then Russia in 1917 and China in 1949. (13 days.) The Greek Orthodox Church is possibly the only European agency that has not accepted this change (although the nation of Greece made the switch in 1923).

I think the big thing is that everyone agrees what time (or day) it is. Not that the time is any particular number. Time is a social construct, remember.

network time protocol (ntp)

Network Time Protocol (NTP) is a 64 bit binary counter that advances in synchrony with International Atomic Time (TAI). The first 32 bits (binary digits) count the seconds and the second 32 bits count the fractions of a second.

The first 32 bits divide NTP into eras lasting 136 years, since …

232 s = 4,294,967,296 s ≈ 136 years

or more precisely …

232 s = 4,294,967,296 s ≈ 136 years, 36 days, 6 hours, 28 minutes, 16 seconds

but don't quote me on the number of days. The years shift between normal years with 365 days and leap years with 366 days in a roughly four year cycle. The exceptions that made me write "roughly" make this an exact number that is a pain to determine. My guess is that for most eras there'd be 136 years and 36 days plus the fractions in this cycle and for exceptional eras (like those containing the years 2400 or 2800) there'd be 136 years and 37 days plus the fractions. But all this is idle speculation. NTP is probably the longest continuously running protocol on the Internet, but it would be astounding if it lasted four hundred years.

The first era of NTP began 1 January 1900 at midnight Greenwich Mean Time (GMT) — the predecessor of Coordinated Universal Time (UTC). No NTP clock literally began ticking at this time, of course. The necessary technology hadn't been invented. (Digital computers first appeared in the late 1930s and the first atomic clock was built in 1949.) It's just a convenient place to put the zero for us humans who are used to years, days, hours and minutes. When UTC went into effect at 00:00:00 on 1 January 1972 the NTP time was 2,272,060,800; the first 32 bits of which looks like this …

10000111011011001110010110000000

or with all 64 bits, like this …

1000011101101100111001011000000000000000000000000000000000000000

The second 32 bits allows for a precision of about one quarter of a nanosecond since …

2−32 s = 2.328 × 10−10 s ≈ ¼ ns

This is a ridiculous level of precision given that the maximum speed at which any signal can propagate through a network is the speed of light. To take advantage of the last bit in the 64 bit NTP code, the time server would have to be closer than the distance traveled by a beam of light in 2−32 s.

s = ct = (299,792,458 m/s)(2−32 s) = 0.06980 m ≈ 7 cm

My favorite NTP server is time.nist.gov at the Time and Frequency Division of the National Institute of Standards and Technology in Boulder, Colorado. This is roughly 2900 km from my desktop computer in New York City. The signal that I get is delayed by at least …

t = s ∕ c = (2,900,000 m) ∕ (299,792,458 m/s) = 0.009673 s ≈ 10 ms

I could get increased accuracy by switching to time-a.nist.gov located in Gaithersburg, Maryland; which is only 380 km away. This signal would have a maximum delay of only …

t = s ∕ c = (380,000 m) ∕ (299,792,458 m/s) = 0.001268 s ≈ 1 ms

This increase in accuracy is meaningless given the way signals propagate over the Internet. Data (like email, web pages, streaming video, and time signals) are broken up into packets that are free to take whatever path the network allows. Unlike a classical telephone network where information is sent along a single dedicated channel, the packets of an Internet message need not follow the same path. Each one is free to negotiate its own journey. The path each packet follows is determined by the architecture of the network and the instantaneous traffic at every node and through every link. Packets can even get lost or "dropped" along the way. Only after they've all arrived are the packets reassembled into the original message.

The electronic act of chopping up a message into packets and reassembling them back into a message takes time, as does the negotiation that each packet makes at the junctions or nodes in the network. The overall result is that the actual transmission time is several times the speed-of-light time. Typical ping times across the United States are on the order of 10 to 100 ms. For NTP this means a deviation from TAI of around 10 to 100 ms in practice. Erring on the side of caution, most software companies only claim accuracy to within a second.

Some Events in NTP
event common era ntp era ntp time
julian day number zero begins 1 January 4713 BCE −49 1,795,583,104
common era begins 1 January 1   −14 202,934,144
gregorian calendar begins 15 October 1582   −3 2,874,597,888
ntp begins 1 January 1900   0 0
unix life begins 1 January 1970   0 2,208,988,800
utc begins 1 January 1972   0 2,272,060,800
last second of second millennium 31 December 1999   0 3,155,673,599
first second of third millennium 1 January 2000   0 3,155,673,600
ntp era change at 06:28:16 UTC 7 February 2036   1 0
fourth millennium begins 1 January 3000   8 352,930,432
Adapted from David Mills, University of Delaware

 

Summary

Problems

practice

  1. To the nearest order of magnitude, determine the number of seconds that pass or have passed …
    1. in the life of an average human
    2. in all of recorded history
    3. since modern humans first appeared
    4. since the dinosaurs went extinct
    5. since the earth was formed
    6. since the beginning of time

    Since all of these times are normally stated in years, we should start by determining the number of seconds in a year.

    1 year = (60 s/min)·(60 min/day)·(24 h/day)·(365.25 day/year) = 3.156 × 107 s
     
    1. Life expectancy varies from country to country. According to the United Nations Statistics Division, Japan had the longest life expectancy at 82 years and Zambia the shortest at 32 years (2000-2005). Women typically live longer than men. In the United States, the difference is roughly six years. To answer this part of the question, we'll use the global mean of 66 years for both genders (2003).
       
      t = (66 years)·(3.156 × 107 s/year) = 2.1 × 109 s ~ 109 s
       
    2. Writing was independently invented by humans in at least three and possibly five different places …
       
      location years before present
      mesopotamia 5300
      china 3500
      mesoamerica 2500
      egypt? 5100
      indus river valley? 4600

      Let's go with the oldest of these, Mesopotamia, where cuneiform was invented more than 50 centuries ago.
       
      t = (5300 years)·(3.156 × 107 s/year) = 1.7 × 1011 s ~ 1011 s
       
    3. The oldest fossils of anatomically modern humans date back about 195,000 years, which agrees with molecular evidence for the age of Homo sapiens. Genetic diversity and the rate at which mutations occur point to a common ancestor who lived in Africa some 200,000 years ago. Let's use the nice round number.
       
      t = (200,000 years)·(3.156 × 107 s/year) = 6.3 × 1012 s ~ 1013 s
       
    4. This number is quite well known by the general population, largely since dinosaurs and their legendary extinction have worked their way into popular culture. All of the dinosaurs and about 50% of the other species inhabiting the earth disappeared 65 million years ago at a point in geological history known as the Cretaceous-Tertiary extinction event. This was most certainly caused by some cataclysmic natural catastrophe, most likely a collision between the earth and an asteroid or comet.
       
      t = (65 × 106 years)·(3.156 × 107 s/year) = 2.1 × 1015 s ~ 1015 s
       
    5. The oldest known terrestrial rocks are 4.4 billion years old as determined by radioisotopic dating. The oldest known meteorites are a bit older -- 4.6 billion years. Since all the bodies of the solar system are assumed to have formed at the same time, this second number is usually quoted as the age of the earth. The earth is a geologically dynamic body. Its surface is constantly being ground down by erosion, consumed at subduction zones, and regurgitated from volcanoes. As a result, none of the surface rocks are in a form that would enable them to be used for dating the earth as a whole. Meteorites are small bits of material left over from the birth of the solar system that did not manage to coalesce into a body of any significant size. They do not share the same complicated history as the earth's crust and are in near pristine condition (at least, from a radioisotopic point of view that is). The age of the earth is therefore the age of the oldest known meteorites.
       
      t = (4.6 × 109 years)·(3.156 × 107 s/year) = 1.5 × 1017 s ~ 1017 s
       
    6. The universe as we know it began when a microscopic bubble of space and time appeared in the background foam of quantum uncertainty and then suddenly and mysteriously found itself rapidly increasing in size. It is estimated that the diameter of the universe increased 1050 times in a mere 10−34 s. This is when the period of inflation ended -- that is, the period when space was being pushed outward by some as yet unknown form of dark energy -- but the momentum of the ordinary matter and energy was enough to keep space expanding for a long time afterwards. In fact, it's still expanding now at the relatively slow rate of one part in 1018 every second or about the diameter of an atom every ten years. Stepping back in time again, the very early universe was too dense and hot for light to propagate any significant distance and space was basically opaque. Approximately 380,000 years after the initial period of inflation, the universe had expanded and cooled enough so that light was able to live an existence independent of matter. The universe is about 1000 times bigger now than it was then and the wavelength of this radiation has shifted from highly dangerous gamma rays to relatively innocuous microwaves. Studies of the variations in the temperature of this cosmic microwave background radiation have yielded the most accurate measurement of the age of the universe to date. The universe began in a big bang explosion some 13.7 ± 0.2 billion years ago.
       
      t = (13.7 × 109 years)(3.156 × 107 s/year) = 4.3 × 1017 s ~ 1018 s
       
      Note that even though the mantissa (the decimal portion of the number that comes before the exponential part) is less than five this number rounds up to the next higher power of ten. That's because the rule for rounding to the nearest exponent is different from the rule for rounding to the nearest decimal. We use five for decimals because it divides a decimal interval in half. For example, 1.5 is midway between 1 and 2, while 25,000 is midway between 20,000 and 30,000. If we apply similar reasoning to exponents we find that it's 3 that divides an exponential interval -- or, more precisely, 100.5 = √10 = 3.16228. For example, 101.5 ≈ 30 is midway between 101 = 10 and 102 = 100, while 10−2.5 ≈ 0.003 is midway between 10−2 = 0.01 and 10−3 = 0.001. Since 4.3 x1017 s is greater than 3.16228 x1017 s (or 1017.5 s) it rounds up to 1018 s.

    To summarize …

    event       duration
    one year ≈ 107 s ≈ ten million seconds
    one human lifetime ≈ 109 s ≈ one hundred years
    all of written history ≈ 1011 s ≈ one hundred lifetimes
    entirety of human existence ≈ 1013 s ≈ one hundred times older than all of written history
    extinction of the dinosaurs ≈ 1015 s ≈ one hundred times older than the human race
    age of the earth ≈ 1017 s ≈ one hundred times older than the extinction of the dinosaurs
    age of the universe ≈ 1018 s ≈ three times older than the earth


  2. Write something.
    • Answer it.
  3. Write something.
    • Answer it.
  4. Write something completely different.
    • Answer it.

conceptual

  1. For each of the following questions, decide whether the question is asking for the time interval between two events or the time at which an event occurred.
    1. How long have you been waiting for me?
    2. How old are you?
    3. What time is it?
    4. When did that happen?
    5. When were you born?

numerical

  1. London is usually five hours ahead of New York. Say it takes the Concorde two hours to fly from New York to London.
    1. What time is it when you arrive in London if you depart New York at 9:00 PM on the outgoing trip?
    2. How many hours have you lost? What will you do to make up for them?
    3. What time is it when you arrive in New York if you depart London at 9:00 PM on the return trip?
    4. How many hours have you gained? What will you do with them?
  2. The Network Time Protocol (NTP) is widely used to synchronize computers and other electronic devices connected to the Internet. When your computer connects to a time server it is sent a 128 digit binary number. The first 64 bits are the number of whole seconds since the NTP era began. (The remaining 64 bits are the decimal portion.) David Mills, the creator of NTP, has reportedly said that the "64 bit second value is enough to provide unambiguous time representation until the universe goes dim." Verify this claim by the following set of calculations.
    1. How many years is 264 seconds?
    2. Find a value for the currently accepted age of the universe. (This has been relatively relatively easy since 2003 when several methods converged on a reliable number. Be sure to state your source.)
    3. How many more "ages of the universe" can we cycle through before NTP would need another binary digit?
    4. Find a claim for the expected lifetime of the universe. (This is slightly more difficult than part b. Work in this field is still highly speculative and values will be stated with very little confidence. Be sure to state your source.)
    5. According to the value you found in part d, will NTP outlive the universe as we know it?
      1. If your answer to part e is "yes", give an approximate value for the time of the end of the universe in 64 bit binary seconds.
      2. If your answer to part e is "no", state the approximate fraction of the universe's expected life span that NTP will cover.

Resources

prev | up | next

Another quality webpage by

Glenn Elert		 eglobe logo home | contact

bent | chaos | eworld | facts | physics