#clocksynchronization

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“You don’t understand something until you can explain it”   (AvdV 2017)

Reference Frames

If you want to describe a physical system in quantitative terms it is generally useful to use a reference system so that you can talk about the location of things and the timing of events.  Set up a set of axes, and a point of origin.  But if you want to make your physics easy to understand, you had better take some care about your choice of reference frame.  Descriptions that are simple in one frame can be quite complex in another.

The purest reference frame is free from the effects of gravity, accelerations, rotations and rotational accelerations. This is called an “inertial reference frame.” Nearly every conceivable situation in the Universe has some degree of gravitation, acceleration or rotation.  So an inertial reference frame is something of an idealized simplification and it is almost non-existent in practice.  Most of the reference systems that we use in practice are non-inertial to some degree. (To reduce typing maybe we should called them “ertial” frames!) 

If you want to describe anything that involves movement or change, you need to add a time dimension as well.  But this not so easy.

Sir Isaac Newton found it convenient to describe his observations and experiments as if there was an absolute reference frame for space and time. But Einstein showed that all measurements of space, time and speed are relative to each other and can vary according to how they are observed. The most problematic dimension is time.

Time

We may as well clarify a few things before confusion sets in. There are two types of answers to the question “What is the time?”  You might be asking for the locally agreed reading for a clock.  But this is a momentary thing.  You might also be asking an official for the result of a race. But in this case you are asking for the difference between the start and the end times. A duration in fact. The language and notation used in physics can be a bit careless about the difference.  This adds to the confusion.

Then there is the problem that one observer’s calculations of a duration might differ substantially from that of another observer. Add to this the fact that clocks run slower in a stronger gravitational field, no matter what method they used for measuring durations. But we are getting ahead of ourselves.

We are accustomed to thinking that time is well defined, reliable, accurate and easily agreed between different observers. And for day to day affairs this is fair enough. But for understanding physics we have to put this aside. Time as we commonly perceive it is an illusion.

Time is just a sequence of events as perceived by observers. Events happen. And sequences of events happen. And if the events are regular and we can agree on them as some sort of standard of time, then that is a bonus.  But do not expect observers in different circumstances to agree with each other. They will not even be able to agree that the same two events happened simultaneously or not simultaneously. Time will turn out to be a relative concept that depends on the circumstances of the observers. It is not at all absolute entity, or even particularly well behaved.

For early man, time was just how high the sun was in the sky, and time durations were just the passing of the days, seasons,solar cycles, lunar cycles, and the period of rule of various kings. It was not a precise concept, and time keeping was not standardized or synchronized.

For the ancient Egyptians, daytime was divided into twelve period ruled by the sun gods (Ra, Sol, Horus etc.)  Horus gave his name to words like “hours” and “horizon”. Night was divided into twelve periods ruled by Set or Seth. Hence the word “sun-set”. This approach was refined by the Sumerians and Babylonians. Sundials were developed but noon varies from place to place and the behaviour pattern of the shadow cast by the gnomon is complicated by the eccentricity of the earth’s orbit. So they were rarely very precise.

Rudimentary clocks were developed based on repetitive set of events - the drip of a water clock, the time for a candle to burn etc. Pendulum clocks and clocks with springs and escapements were invented.  Timekeeping became more precise. Clocks were installed in churches and public buildings.  These were only approximately synchronized with each other.

As time keeping became more consistent, reliable and standardized, it became possible to make meaningful statements about when an event happened or about the time separation between two different events. Clocks started to be synchronized across wide geographical areas or even whole kingdoms, generally with 12 midday (noon) set approximately to coincide with the highest part of the sun’s apparent movement through the sky. 

Synchronizing clocks is not always easy.  Consider two clocks some set distance apart. By repetitive signalling it is possible for observers at clock A to be satisfied that their clock is running at the same rate as clock B and vice versa.  That is a useful start. Synchronizing the clocks is a bit more difficult. Any signalling necessarily involves delays in getting a signal from one place to the other. There are a variety of ways to synchronize two clocks, and then to extend this to cover a network of clocks:  

Get a very good clock and synchronize it directly with clock A. Then transport that clock very slowly to Clock B and adjust Clock B accordingly.  Repeat in the reverse direction just to make sure.

Get each clock to signal the other and to return the signal immediately upon detection.  Each clock will detect a time delay in seeing their signal when it comes back again.  Divide this duration by two and use that interval to fine tune the synchronization process.  Then send signals to each other at precisely the same time on each clock (on the hour for instance) and record when that signal is received by the other.  If each clock notices the same time lag then all well and good.  However there is a possibility that the time lag is different in one direction from the other.  For example: suppose that a pilot station is trying to synchronize with an offshore lighthouse and is using a foghorn for signalling.  If the air is still this might work reasonably well.  However, if there is a constant onshore breeze the lighthouse to shore signal might travel faster than the shore to lighthouse signal.  The pilot station and the lighthouse will have to recognize the problem and do some sums to refine their adjustment process.

Once you have worked out a way to synchronize two clocks, check that you can extend this to yet more clocks.  Then check that if you have A synchronized to B and B synchronized to C then C is still synchronous with A.  

Suppose that the observers at Clock A and Clock B are now given telescopes so that they can see the other clock.  Will they see a time the same as their clock?  No – they will see a time reading that is less than theirs.  The difference will be the duration required for that light signal to reach their telescope.

This is a long discussion for a simple point, but it will turn out that in some situations two observers will be unable to agree on whether two events are simultaneous or not.

So What is the Speed of Light?

If you ask this question today a pedant could point to an international convention that took effect in the 1970’s that the speed of light in a vacuum will henceforth be regarded as being exactly 299,792,458 meters per second.  Call this number c.

But what is a meter and what is second I hear you ask.  Good point.

A second has been defined to be a set number of oscillations in the radiation emitted in hyperfine transitions within Cesium 133 atoms held in certain conditions.

A meter has been defined to be the length that light travels in 1/c seconds.

So the official definition of a meter hangs off the official definition of a second plus the official definition of the speed of light.

I’m not sure whether the defined value for the speed of light means “as it is here on Earth” or whether it has been adjusted for the Earth’s gravitational field.

In a later essay I will discuss the fact that a gravitational field affects time duration and hence the measurement of lengths and speeds.  It turns out that clocks on earth run more slowly than the same clocks when they are free falling around the Earth in a satellite orbit.

I don’t know if the official definition of the speed of light uses a slow earthbound atomic clock.  If so then the official definition of a meter also uses a slow clock and hence is arguably a bit short.  And so you could say that the speed of light is exactly 299,792,458 ‘short meters’ per ‘short second’.  Right answer but not so simple as you might think.

I will talk about the problems of synchronizing laboratory based clocks with clocks in orbiting satellites used for Global Positioning Systems in another essay.  If you can’t wait then just Google it and read the Wikipedia article.

Has the speed of light always been the same?  Earthling officials can officiate all they like about the speed of light.  If we are talking about events that happened billions of years ago in galaxies that are billions of kilometers away then we might excuse such galaxies if they have not yet heard about our earthling pronouncements.  So keep an open mind.

See a later blog essay for a discussion about the experiments on light around the end of the 19th century that were involved in the lead up to Einstein’s new way of looking at the world – the Theory of Special Relativity.

The Speed Of Light

The speed of light is a lot more complicated than you might think. You can’t just time a photon over a 100 metre dash using a stopwatch because you would need to allow for the time it takes for light to reach you from each end of your measurement scale. And even if you did allow for such a delay it is still not simple.

Speed is distance divided by duration. But you have to ask yourself – is the speed constant, is the distance calculation reliable and is the time duration measurement reliable? Where light is concerned, almost none of this can be taken for granted.

The fact that light has a speed at all was not clear until the Danish astronomer Olaus Roemer demonstrated it in 1676. Roemer noted that one of Jupiter’s moons (Io) passed behind Jupiter very regularly and this created a celestial clock. However, the period varied by several minutes depending on the changing distance between Jupiter and Earth. This suggested that light from Jupiter took a finite time to reach Earth and that the variable distance between Earth and Jupiter was causing the variation in timing. By careful calculations Roemer came up with an estimate of the speed of light that was about 83% of the modern calculation. To cynics who argued against him Roemer replied that the next eclipse would be 11 minutes late. And so it was.

Fizeau performed a calculation based on a rapidly rotating toothed wheel (later replaced with a rotating mirror) and a distant mirror. For light to get out through one gap and back in through the next it would have to travel at a certain speed. Fizeau obtained a reasonable estimate for the speed of light near the surface of the Earth as measured over a two-way journey.

A problem with this experiment is that it is done on Earth. There are three issues with this. Firstly it is done in air and not in a vacuum. Secondly it is done in the Erath’s gravitational field and this turns out to matter slightly. Thirdly, the Earth is rotating on its axis and is in orbit around the Sun that in turn is in orbit around the centre of the Milky Way galaxy. Fizeau’s experiment and others carried out on Earth are inevitably done in a non-inertial reference frame.

Nevertheless, at the start of the 20th Century Albert Einstein was prepared to assert as a foundation of his Theory of Special Relativity that the speed of light as measured in a vacuum within an inertial reference frame and a synchronized set of clocks is always the same value and is independent of the speed of the observer or the orientation of the inertial reference frame.

I repeat:  Whenever observers measure the (two-way) speed of light in vacuum and in an inertial reference frame, they always get exactly the same answer. It does not matter if the source of the light is moving towards the observers or away from the observers. It does not matter if the observers are moving towards the source of the light or away from it. It does not matter how the whole measuring apparatus is oriented in space.

This is very strange!  If light is some sort of particle fired out by its sources, you would expect the speed of the source to add to speed of the emitted particle. But it does not.

If light is some sort of wave travelling at a well defined speed in some sort of medium, then you would expect the speed of the observer relative to that medium to add to the speed of the detected light. But it does not.

All (two-way) measurements in an inertial reference frame and a vacuum find that the speed of light is the same constant answer. The speed is always the same, no matter which way the reference system is moving through the universe, as long as it is not rotating or accelerating or in a gravitational field.

But note the importance of clock synchronization in all of this. The world of Special Relativity is so far removed from everyday experience that every part of Einstein’s logic has to be examined very carefully. It is not meaningful to try to describe the one way speed of light without being crystal clear about what this means exactly. In fact is not sensible to talk about the one way speed of any moving object unless you are able to synchronize the clocks at the start and end of the test run.  Please refer to the Wikipedia article on Einstein synchronization. Paraphrasing some of this…  

Einstein synchronisation can be applied consistently if and only if the Von Laue-Weyl's round-trip condition holds. ….  Von Laue and Weyl's round-trip condition is this: The time needed by a light beam to traverse a closed path of length L is L/c, where L is the length of the path and c is a constant independent of the path.

The Einstein synchronisation looks this natural only in inertial frames. One can easily forget that it is only a convention. In rotating frames, even in special relativity, the non-transitivity of Einstein synchronisation diminishes its usefulness. If clock 1 and clock 2 are not synchronised directly, but by using a chain of intermediate clocks, the synchronisation depends on the path chosen. 

Synchronisation around the circumference of a rotating disk gives a non vanishing time difference that depends on the direction used. This is important in the Sagnac effect and the Ehrenfest paradox.