Timing: Part 10 - The Influence Of Relativity

To see why perfect synchronization can never be achieved, we have to look at relativity, which is the physics of co-ordinate systems.

Relativity is forever associated with Albert Einstein, who moved forward from the work of Lorenz and Poincaré. Nevertheless there is a difference between working something out and being able to explain it, and the best explanations of relativity are due to Richard Feynmann, who remains an inspiration.

It was René Descartes who thought that space had three mutually orthogonal axes, traditionally called x, y and z and that the position of any points with respect to a first point could be expressed by combinations of those three variables. In a 3D printer, the first point, the frame of reference, is the bed of the machine.

But what about inter-galactic space, in which our, and other, solar systems move? Where is x = y = z = 0? It doesn't exist, or looked at another way, there is an infinite number of places it could be. Without an origin, there is no such thing as an absolute position. Without an absolute position reference, all practical frames of reference are relative, hence the term relativity, and there is no way to tell if the arbitrary reference one has chosen is moving linearly.

Frames Of Reference
It is of course possible to tell if a frame of reference is rotating, as that requires stuff to accelerate and the necessary forces can be detected. Foucault's pendulum is a good example of that. If a frame of reference is moving in a straight line, there is no way to detect the motion. Isaac Newton said as much, and has to be acknowledged as a pioneer of relativity.

The Michelson Morley experiment confirmed Newton's suspicion. The Earth covers almost a million miles a day in its orbit round the sun, which is pretty fast by human standards, yet the motion couldn't be detected. The implications are far reaching. One could pick any frame of reference moving in a straight line and every experiment that could be performed, every machine that could be made would work as if the frame of reference were static. That's not too difficult to swallow, but the one that is hard is the case where two experimenters are travelling at vastly different speeds in two space ships and each one of them emits some light.

If a fighter plane fired a gun, the velocity of the plane would be added to the muzzle velocity in accordance with Newton's Laws, but with light that doesn't happen. Relative to the point of origin, the light has the speed c, but when the light reaches the other space ship, it will still have the speed c. Two things follow from that. The first is that Newton's laws are not telling the whole story and the second is that the three Cartesian co-ordinates are not independent of time. Instead we have a four-dimensional co-ordinate system known as spacetime.

Conspiring Against Us
At the time of the Michelson Morley experiment these findings could not be explained. No experiment could determine absolute speed. Poincaré was of the view that nature seemed to be in a conspiracy to defeat man's efforts.

It is interesting to consider what might have happened if that debate had been attempted today. The "fact checkers" on Poincaré's social network would have concluded that he was a conspiracy theorist who was posting misleading information and his account would have been blocked.

It was Lorenz who worked out how to calculate speed when seen from a different frame of reference. The necessary equations are called a Lorenz Transformation. It follows from Lorenz that neither space ship can claim to have a static frame of reference. It is not known which one is moving or if both are. It further follows that nothing can go faster than the speed of light.

In one sense these things still cannot be explained. Science is empirical, based on observation, and so it documents how things are. Science can say what is quite well, but we should not expect science to say why.

The whole story is fascinating, because it involves time. Speed is distance moved through time. The whole basis of relativity is that frames of reference that are moving with respect to one another do not share the same time or distance frames. A pair of events that are synchronous in one frame of reference will not be when seen from another. These characteristics place limits on the accuracy of synchronization systems. The phenomenon is called failure of simultaneity at a distance.

GPS Accuracy
Just as airflow changes its behavior as the speed of sound is approached and effects are seen that are absent at low speeds, relativistic effects are negligible in everyday life, which is why they weren't detected sooner. They only become relevant as speeds approach the speed of light, or when exceptional accuracy is required, as in GPS, for example. Relativistic effects are to the speed of light what compressibility is to the speed of sound, with the major difference that the speed of sound can be exceeded.

Imagine a surfer balanced on top of a wave and moving at the same speed. To the surfer, the wave isn't moving. Imagine an infinitely small microphone moving at the speed of sound. It doesn't detect any sound travelling in the same direction; time has been halted. Finally a thought experiment: Imagine travelling at the speed of light and looking at some light going the same way. The frequency of the light will be zero. To the light itself, time has no meaning. At the speed of light, time stops.

Light is a form of energy, and if it is emitted from a moving object, the motion alters the energy. Light energy is the product of the frequency and Planck's constant whereas c is the product of frequency and wavelength. To stay consistent with those equations, light seen in a different frame of reference to the one in which it was emitted will have a different combination of frequency and wavelength, which is observed as a Doppler shift.

Nothing having a rest mass can reach the speed of light, but as any object is accelerated to relativistic speed, from a frame of reference assumed fixed, several things happen. Time with respect to the moving object appears normal, but appears to have slowed down in comparison with time at the fixed reference. The dimensions of the moving object have reduced and its mass has increased.

Approaching Light Speed
According to Newton, if a force is applied to a mass, it accelerates. If the force is sustained, there is no limit to the speed that can be reached. Yet nothing can exceed the speed of light, so how does that work? Again the Lorenz Transformations reveal that as speed gets higher, so does the mass of a moving object, so that the energy put in causes an increase in momentum without much of an increase in speed. As c is approached, the mass goes to infinity and so does the energy needed to increase the speed.

It then follows that mass and energy are essentially the same thing, as Einstein's famous equation shows.

If it is accepted that c cannot be exceeded and that linear motion cannot be determined, then these strange things must follow. It follows that time must slow down for a body subject to acceleration. In 1905, Einstein didn't have access to atomic clocks. Once sufficiently accurate clocks had been developed, it was possible to perform an experiment. Two identical clocks were involved, which kept pace with one another. Then one was flown around the world on a series of airliners, while the other one stayed home.

Sure enough, the one that went round the world was behind the one that stayed home when it returned. All of the accelerations it had experienced on the journey had slowed it. That summarizes the whole story of relativity. All of the phenomena that were predicted by relativity were eventually measured and found to agree with the predictions.

Satellite Predictions
When it became possible to place satellites in Earth orbit, more of Einstein's predictions were found to be true. The acceleration due to gravity on the surface of the Earth is not what is experienced by satellite, so the speed of a clock in orbit differs from the speed of the same clock on Earth not just because of the speed but also because of the difference in acceleration.

It doesn't make a whole lot of difference to a transatlantic phone call bounced off a satellite, but in the case of GPS it does make a difference. GPS works by computing the time it takes for signals to get from the satellites to the receiver, and a one nanosecond error corresponds to a position error of about a foot. The errors due to relativistic effects are significantly greater than that.

The accuracy delivered by GPS is only possible because the clocks driving the transmitters in the satellites are programmable and can be adjusted from the ground so that relativistic effects can be compensated.

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