The Light Clock A Theoretical Proof of Time - Cetin BAL - GSM:+90 05366063183

The Special Theory of Relativity was proposed in 1905 by Albert Einstein (1879-1955). The reason it is "special" is because it is part of, or a "special case" of, the more comprehensive and complex General Theory of Relativity. The latter, General Theory, was proposed by Einstein in 1915.

Space and Time.

In an everyday co-ordinate system, such as a map, you can specify a location using just dimensional distances. For example, to someone looking for buried treasure we could say "go east for 20 miles, north for 5 miles, then dig down 30 feet". We have just specified a three dimensional co-ordinate system. To this, Einstein added another factor, that of time. This still makes sense in our everyday world: "go east for 20 miles, north for 5 miles, dig down 30 feet, and then wait until 3 o'clock when I will meet you to share the treasure!". However, if we go at very high speeds, speeds close to the speed of light, things begin to change in a very strange way. The faster we go the more our clock slows down relative to someone standing still; time, for anything moving, changes! Instead of space and time being separate things they are the same thing, called space-time. In short: "moving clocks run slow".

Einstein's Two Postulates.

The first postulate is seemingly simple and trivial. If I sit and wait an hour in New York, an hour passes. If I sit and wait an hour in Edinburgh then an hour also passes. I am, almost exactly, in the same bit of space (frame of reference), moving around the Sun at the same speed wherever I sit on the Earth. The way time passes in all frames of reference is governed by the same laws.

While the first postulate is pretty much what one might expect the second requires a little more explanation. The speed of light is very close to 300,000 kilometres per second (around 186,300 miles per second). Everyday experience tells us that if a bus is moving north at 30 miles per hour and we are also walking north at 5 miles per hour then the bus is moving away from us at 25 miles per hour.

But what if we move in the same direction as a light beam? Let's say we produce a pulse of light into space by quickly flashing a torch (flashlight) on and off. We then follow the beam of light in a very fast rocket moving at 100,000 km per second. How fast is the light beam moving away from us? Common sense tells us that it is moving away from us 300,000 km minus 100,000 km per second, in other words, the light beam is 200,000 km per second faster than us. Wrong! Remember that the speed of light is always the same regardless of our own speed. From our rocket we would see that the beam of light is still moving away from us at 300,000 km per second! Likewise, if we were moving towards the beam at a very high speed we would still see the light coming at us at 300,000 km per second! This has enormous implications!

Time dilation.

If the speed of light stays the same then what is going on? Something else has to change. That "something" is time.

As odd as it seems time is not constant. More accurately, space-time is not constant. It can be changed, bent and twisted. The faster we go the more time slows down ("moving clocks run slow"). This is only noticeable, normally, at very high speeds such as those approaching the speed of light, 300,000 km per second, which is approximately 7 times around the Earth in a second.

What does all this mean? One of the most dramatic consequences of this is that time itself will run at different speeds for two people moving at different speeds relative to each other (hence "relativity"). Let's have an example. Mary, a 30 year old NASA astronaut, blasts off from Cape Canaveral in her very high speed rocket in the year 2010 on a 10 year mission to a nearby star. After a short time she is travelling at 270,000 km per second, that is, 90% of the speed of light. To Mary everything looks normal in her rocket; the clock seems normal and time passes for her the way it did back on Earth. Her identical twin sister, Susan, is a NASA ground controller for the mission. Ten years pass on Earth before the rocket returns and when it does something is immediately apparent; while the Earth-bound Susan has aged 10 years, her high-flying "twin" sister has only aged 5 years! How can this be? Well, again we are back to "moving clocks run slow". At 90% of the speed of light time slows down to about half of that relative to someone who is stationary. So while 10 years have passed for Susan only 5 years have passed for Mary because her "clock" was running at half the speed of those on Earth. This is called the twin paradox. Remember that while Mary has only aged 5 years she still felt that time was passing normally; this is not a way of living longer! Not only was her clock running slow as far as a ground based observer is concerned, but her time was running slow.

How fast can I go?

For reasons of brevity the first point will not be covered here and the latter two only very briefly.

Under the "rules" of special relativity the mass of any object appears (to a stationary observer) to increase as the speed of the object increases. At 90% of the speed of light mass will approximately double, but at 99% of the speed of light the mass will increases by about 7 fold. As we get closer to the speed of light the mass increases very dramatically until at the speed of light it would be infinite. The more something weighs the more energy is required to move it, as we know from everyday experience. To move something of infinite mass would require infinite energy and this is clearly not possible, hence nothing can travel faster than light. We live in a universe dominated by the Special Theory of Relativity where faster than light travel is not possible. There are, however, ways of breaking out of the constraints of the special theory and travelling at speeds far in excess of that of light, at least for tiny particles. As well as being very unusual and beyond the scope of the special theory, the conditions for this are, as far as is yet known, of no practical consequence.

Is it real?

This is all well and good, but what evidence is there? After all it's just a "theory"! Well, there have been very many experiments carried out on special relativity and all of them have shown it to be correct. These range from experiments involving sub-atomic particles in high speed accelerators to the slight, but expected, different clock rates of some space exploration vehicles (such as the Voyager inter-planetary probes) as compared with those on Earth.

One of the "proofs" is a much simpler experiment that was first carried out in 1971 and has been repeated many times since then. Atomic clocks have been carried on aircraft making long flights such as over the Atlantic ocean. An aircraft, even the fastest, goes at a tiny fraction of the speed of light, but at any speed "moving clocks run slow". With an atomic clock on the ground and one in the aircraft it is possible to measure the tiny differences in time produced by moving the clock. At the speed an aircraft travels these differences are measured in millionths of a second, but they are real and in exact agreement with what special relativity says they should be. If Albert Einstein could have seen these experiments he would be quietly and politely pleased, but not surprised.

The Light Clock A Theoretical Proof of Time

A Theoretical Proof.

In other pages in this series we have seen that there is direct and measurable evidence for time dilation. One example from the many available, that of atomic clocks carried on aircraft, has shown that moving clocks do indeed run more slowly than stationary ones, just as predicted by Einstein.

The distinction between scientific "proof" and "evidence" is a complex one and will be dealt with in another page. For the moment though it is sufficient to say that scientific proof is only available in models, and never in reality. The primary laboratory of the theoretical physicist is his or her own mind and so it is there that any initial experiments must take place. These experiments are often called "thought experiments". Of course, as Einstein was happy to admit, the only real way to test a theory is by carrying out experiments in reality. However, the germ of any physical experiment must start in the mind, and this page looks at one of the best examples of a such a thought experiment; namely that of the theoretical light clock.

A Light Clock.

Clocks exist in many forms. Among the many types of clocks that have been made there are:

Most clocks measure how many times a repetitive action is carried out. For example, in a quartz watch the quartz crystal usually vibrates at 32,768 times a second. These vibrations are counted by electronic circuits. After 32,768 "ticks" have been counted a second is added to the watch's display.

We can also use light to make a clock, at least in theory. To do this we need to bounce a pulse of light between two mirrors that are a known distance apart. Light travels at 186,300 miles per second, so if we separate the mirrors by a distance of 93,150 miles (i.e. half 186,300) each individual mirror will be struck by the pulse of light once a second. In other words, the round trip from one mirror to the other and back again will take the light pulse one second. We now have a clock:

There are a number of practical problems with such a clock. Probably the most obvious one is the separation distance of the mirrors, but in reality we could put them very close together. The large separation used here is just to demonstrate the principle and make the mathematics easy. In reality the mirrors would absorb some of the light each time they were struck by the pulse and after a time the light pulse would dissipate completely. Also, the fact that we can see the light at all means that at least some of it is being scattered thereby further weakening the pulse. None of this really matters however, because we are dealing with a theoretical proof and not an experimental one.

As an aside, there are systems that use the principle of a light clock in order to perform important tasks. Radar is probably the best known example. In a radar system pulses of "light" (i.e. electromagnetic radiation) are beamed out at very close to the speed of light. If the beam hits an object some of it will be reflected back (as if from a mirror) and can then be detected by the radar receiver equipment. The time taken between the beam being emitted and re-absorbed can be used to calculate the distance of the reflecting object, usually an aircraft.

A Moving Light Clock and Pythagoras.

There is nothing really that extraordinary about a stationary light clock. In fact there is nothing really that extraordinary about a moving light clock if we are on the same moving platform as it. Imagine being on a rocket moving at half the speed of light and that on this rocket we have a light clock. As we travel through space we can see the clock ticking away quite happily and there wouldn't be anything odd about it (okay, we have to use our imagination here because the clock, as we have seen, would be either enormous or so small that we can't actually see the pulses, but we must remember that this is a thought experiment!).

Now let's imagine that we are being watched by an external, and stationary, observer. We whiz past the observer holding the light clock to the window. Will we both see the light clock doing the same thing? No! To us on our rocket the pulses of light just go up and down the way we would expect them to, but to the observer they will follow a different path, one that maps out a series of triangles. The diagram below shows the track of the light pulse as it moves past the observer:

At first this may not seem so strange. After all we could do the same experiment with anything that went up and down in a transparent box, but this is light and light has some very strange properties. From the other pages in this series we know that light has a constant speed. This is where things start to become interesting!

As we have seen the light moves in such a way, as viewed by the external observer, that it traces out a series of triangles. We know the mirrors are separated by the distance that light travels in half a second (i.e. 93,150 miles) and that the spaceship is travelling at half the speed of light, i.e. covering the same distance in the same time. We have the opposite and adjacent measurements of a right-angled triangle and all we need is a little help from Pythagoras to work out the length of the hypotenuse:

The mathematics are correct but the actual result is wrong! If it was correct it would mean that the pulse of light was travelling a total distance of 2 times 131,734 miles = 263,468 miles every second. It is a pulse of light however and can't travel faster than 186,300 miles in a single second, nothing can!

What's going on?

The external observer knows that the distance tracked out by the pulse of light in a single second can't be more than 186,300 miles. He also knows that the speed of light is constant. If the speed of light can't change is there anything else that can? Einstein pondered this problem and came to a breathtaking conclusion; if the speed of light is constant it must be space and time that change.

Einstein realised that what the external observer would really see would be a light clock that appears to be slowed down. The clock has to behave like this otherwise it would break the universal speed limit. The "ticks" of the clock would now appear to be slower as viewed by the external observer than as viewed by the person on the rocket. The rocket is moving at 50% of the speed of light so according to the external observer the time the pulse takes to get to the top mirror and back again would be about 1.1 seconds. As viewed from an external stationary position the rocket and everything on it would be running in slow motion. Time on the rocket has, according to an external observer, slowed down by about 10%. To the person on the rocket however, time would still seem to be passing normally.

Not only that but the spaceship and all its contents, including the light clock, would appear squashed in the direction of motion according to the external observer, but normal for the person on the spaceship. Again, this is a consequence of the speed of light being constant and so forcing space (more accurately space-time) to shrink. If the space traveller takes a ruler and measures something the results will appear normal because the ruler has shrunk as well.

The two observers would be experiencing space and time in different ways relative to each other. Note that in each individual's frame of reference everything seems normal; they would both feel time passing normally and the laws of physics would still be the same (as the first postulate states). It is only when observing each other's frame of reference that they notice anything strange. As the speed of light is approached these effects become even more apparent:

Moving Clocks.

We have seen that due to the constant speed of light a moving light clock will appear to run slowly according to an external observer. It was stated at the start of the page that the light clock was a thought experiment and so it is. However, not only is the light clock experiment expected to work in reality but every clock that has ever been observed at high speeds slows down in just the way that the special theory of relativity says it should. It is not just light clocks that run slowly at high speeds, all clocks, including our own body clocks, slow down at high speeds. Time for anything moving changes.

Time Dilation Worked Examples

The Time Dilation Equation.

Of all the major advances in physics from about 1900 onwards special relativity is the only one that can be understood in its entirety without recourse to mathematics beyond that of high school level. However, like all physics special relativity has at its base a precise set of mathematical formulas from which predictions can be made and tested against experimental results. It will come as no surprise then that time dilation has a precise mathematical formula. This is it:

Example 1: Solving the Equation as a Factor of 1.

The effects of time dilation don't become really noticeable until very high speeds are reached so for this worked example I will use a speed of 90% of that of light, that is 270,000 km per second (the speed of light is very close to 300,000 km per second). The first thing we must do is to write down the equation:

We now need to "plug in the numbers". Because V²/c² is a ratio we can either use the exact values or just the percentages of each value. It is easier to do the latter. Because we are only interested in the dilated time factor we can set the stationary time to be 1. Note that we can drop the percentage symbols and that c is equal to 100% of the speed of light. Plugging in the numbers we get:

We can now begin to solve the equation. The first thing we can do is remove the leading number 1 (anything multiplied by 1 is itself), then square the last two terms:

We can now reduce the equation by carrying out the division:

We then carry out the subtraction:

And finally we take the square root (and round the answer to a workable value):

This result means that at 90% of the speed of light local time has slowed down to 43.6% of that relative to an external observer. Put another way, if a rocket is sent out into space on a 10 (Earth) year mission at 90% of the speed of light the rocket and its occupants will have only aged by about four and a half years when they return, while everyone and everything on Earth will have aged 10 years.

Example 2: Solving the Equation as a Measure of Time.

The first example solved the equation as a factor of one. This example puts a real time scale into the equation. In this example we will look at how time changes over 10 years travelling at a speed of 50% of that of light.

Instead of the step-by-step approach of the last example I will just carry out the equation:

So for a rocket travelling at 50% of the speed of light 8.66 years will pass in the same time as 10 years pass for a stationary observer. Note that while the unit of time used was years it could have been any other unit of time, such as seconds or millennia, as long as the result is in the same units as the units used in the equation.

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