Sunday, 25 May 2008

Faster than Light?

Probably the best known consequence of Einstein’s Special Theory of Relativity is that nothing can travel faster than light. Science fiction writers have tried to get round the problem in various ways, most notably the warp drive of Star Trek fame but the reality was best summed up by the late Arthur C. Clarke in the notes accompanying his 1986 novel The Songs of Distant Earth when he said that “ Warp Six will ever get you from one episode to another in time for next week’s episode. The great Producer in the Sky did not arrange his programme planning that way”.

But why can’t we go faster than light? For instance, what would happen if we were travelling at the speed of light, then went a bit faster? The answer turns out to be that only light can travel at the speed of light; anything else – no matter how fast it tries to go – can never quite reach much less exceed the speed of light.

This was the conclusion that Einstein came to when he considered relative velocities – that is to say the speed and direction of travel of one object relative to another. Hence – the Theory of Relativity. Before we come to this, however, let us look at the speed of light itself.

To paraphrase the late Douglas Adams, light is fast. Mind-bogglingly fast. Light is so fast that it was for long time believed that its speed must be infinite – switch on a sufficiently bright light on, say, Mars, and it would become instantly visible from Earth. That this was not the case was demonstrated in an ingenious fashion by the Danish astronomer Ole Christensen Roemer. Roemer’s demonstration arose out of Galileo’s attempt to solve the vexed “longitude problem” of determining the longitude of a ship at sea. All attempts had been confounded by the lack of any means of establishing the time of day with any degree of accuracy once a ship was out of sight of land. Galileo hit on the idea of timing the eclipses of Jupiter’s four principle moons. The moons regularly disappear behind the giant planet in a completely predictable fashion: thus the Jovian system could in effect be used as a clock.

Unfortunately the method proved impractical due to the difficulties of making the required observations from a ship at sea. Nevertheless in the 1660s and 1670s several astronomers began making observations with a view to compiling an ephemeris for the use of ships’ captains. These included Roemer and his French colleague Jean-Felix Picard at Hven, near Copenhagen, and Giovanni Cassini, who was observing from Paris. Cassini had noticed discrepancies in his observations - when Jupiter was closer to Earth the eclipses would occur earlier than expected: conversely, when it was further away they would be late. This he correctly attributed to light having a finite speed: the further away the Jovian system was, the longer the light from the eclipsed moon would take to reach Earth, and the later the eclipse would appear to be.

Cassini however did not pursue the matter until 1672, when Roemer went to Paris and began working as his assistant. Roemer made further observations and eventually determined the discrepancy amounted to between ten and eleven minutes (the actual value is just over eight minutes). Cassini published the results in 1675 in a short paper and Roemer, using data collected by Picard and himself, published a more detailed treatment a year later. Using currently accepted values for the distances of Earth and Jupiter from the Sun, which were not known with great accuracy in Roemer’s day, a calculation of the speed of light using Roemer’s observations yields a value of 227,000 km per second for the speed of light, rather less than the currently-accepted value of 299,792.5 km per second. However Roemer himself never attempted to calculate an actual value.

Despite this strong evidence, it would be another 50 years that it became generally accepted that light has a finite speed. In the 172os, the third Astronomer Royal, James Bradley, discovered and then explained a phenomenon now known as stellar aberration. Bradley and his friend and colleague Samuel Molyneux were attempting the measure the so-called parallax of the star Gamma Draconis. Parallax is the apparent motion of an object when seen from two viewpoints against a distant background. If, for example, I look at a lighthouse visible against a background of stars from my hotel window on a fine night, then decide to go down to the beach for a midnight swim, and there see the lighthouse from a different angle, it will appear to have moved against the background stars. Two things will determine the apparent movement (it has of course in reality remained stationary); the distance between my hotel room and the beach; and the distance between myself and the lighthouse (which we assume to be large in comparison, say three kilometres versus a couple of hundred metres. We don’t know how far away the lighthouse is, but we do know the distance between the hotel room and the beach, and we can measure the apparent movement – or parallax – of the lighthouse. Armed with this information, simple trigonometry can be used to calculate the distance to the lighthouse.

Bradley and Molyneux were attempting to measure the distance from the solar system to Gamma Draconis using a similar method, based the fact that as the Earth goes round the Sun, the star would be viewed from a slightly different angle and so should show a parallax against more distant objects. Because the distance from the Earth to the Sun (the so-called astronomical unit) is tiny in relation to the distance to even the nearer stars, the parallax shown will generally be tiny and very difficult to measure. Bradley and Molyneux were unsuccessful in their attempts to determine the distance of Gamma Draconis by this method (it is actually too far away for the parallax method to be practical), but they did pick up an unexplained “wobble” in its position.

Molineux died in 1728, but Bradley carried on the work on his own and explained the results in terms of the speed of the Earth as it moves round the Sun, and the finite speed of light. The speed of light is not infinite compared with the Earth’s orbital velocity and when the two are combined the result is a small apparent displacement of distant celestial objects from their true positions. After the passage of a year, objects returns to their original positions. This is the wobble Bradley and Molyneux observed for Gamma Draconis. The maximum displacement of an object by this effect is small – just 22.47 seconds of an arc, but it was within the resolving capacity of the instruments of that era. Bradley’s observations yield a value of 298,000 km per second for the speed of light, much closer to the accepted value.

Although attempts go back to the early 17th Century, it was not until 1849 that Earthbound techniques were first successfully used to measure the speed of light. In that year, the French physicist A.H.L. Fizeau used an apparatus consisting of a beam of light that was directed to mirror several kilometres away. The beam passed through a rotating cogwheel which was spun at a rate whereby the beam would pass through one gap on the way out and another on the way back. If the rate of rotation was slightly higher or lower, the beam would be blocked by a tooth on its return. Based on the rate of rotation of the wheel and the distance to the mirror, the speed of light could be calculated. Fizeau obtained a figure of 313,000 km per second. Leon Foucault refined the method, substituting a rotating mirror for the cogwheel and obtained a better estimate of 298,000 km per second. Further refinements by the Americans A.A. Michelson and Simon Newcomb eventually yielded a figure of 299,860 +/- 60 km per second.

In 1887 Michelson collaborated with E.W. Morley in one of the most important experiments in the history of science at the Case Western Reserve University in Cleveland, Ohio.

In the late 19th Century it was widely believed that just as sound waves require a medium such as air through which to propagate, so light waves must require some form of medium, referred to as the luminiferous aether, in order to propagate. Because light could propagate through a vacuum, it was assumed that the vacuum of space must be permeated with this substance. Because waves propagate through a medium at a constant speed, the speed of light should be fixed in relation to the aether. Also, because the Earth is in motion around the Sun, its movement though the aether should manifest itself in variations in the speed of light as measured indifferent directions (think of the aether as a “wind” blowing against the Earth; the speed of light should vary depending on whether it is measured in a direction facing directly into the wind or away from it).

Michelson and Morley devised an apparatus known as an interferometer, in which a single beam of light is split by a half-silvered mirror and later recombined by mirrors after the two portions have been sent in different directions along two arms of equal length, 90 degrees apart. By using a coherent light source (one in which all the waves were in phase), any difference in travel times caused by the motion against the aether should show up as interference patterns, where waves either reinforce each other (constructive interference) or cancel out (destructive interference). It would then be possible to calculate the variations in the speed of light and from this, the speed and direction of the Earth’s movement through the aether.

But surprisingly, absolutely no difference was detected. The speed of light was the same regardless of what direction it was measured in. The experiment was repeated at different times of the year to rule out the possibility that Earth might happen to be stationary with respect to the aether at a certain time of the year: if so, it could not be six months later when it would be moving in the opposite direction. But still no differences were found. The speed of light remained resolutely immutable.

The paradox implied here can be illustrated by considering an express train passing through a station at night. Suppose a woman in the train gets up from her seat to go to the refreshment car at the head of the train. She walks past her fellow passengers at a brisk 5 km per hour. The train passes through the station at 100 km per hour. A man standing on the platform sees the woman on the train. He concludes she is passing at 100 + 5 = 105 km per hour, i.e. the sum of the train’s speed plus her own speed along the carriage.

All very straightforward, but now suppose that the man on the platform (who is in fact none other than that Einsteinian folk-hero, the stationary observer) measures the speed of light coming from the train and that from the platform lighting. Applying the same logic as before, one would expect the former to be approaching at 100 km per hour faster. This is of course an infinitesimal difference, but one that is well within the range of a present-day undergraduate physics lab.

But that isn’t what happens if the conclusions of the Michelson-Morley experiment are accepted. The speed of light, regardless of how fast the source is travelling in relation to the observer, is always the same old 299,792.5 km per second. This even applies to the light from distant galaxies, moving away from us at significant fractions of the speed of light.

In 1905 Einstein proposed what is now known as the Special Theory of Relativity in one of his four so-called Annus Mirabilis papers, published in the scientific journal Annalen der Physik. Einstein postulated that all frames of reference were equivalent: thus it was equally valid for the man on the station platform to claim that the woman on the train was moving at 105 km per hour; for other passengers to claim that the woman was moving at 5 km per hour; and for the woman herself to claim that she was stationary, but the other passengers were moving at 5 km per hour and the man on the platform was moving at 105 km per hour! Einstein claimed that there was no physics experiment one could perform that would distinguish between the three frames of reference. He also claimed that this included measuring the speed of light, which would always come out at 299,792.5 km per second.

One of the consequences of this is that velocities are not additive. The 100 + 5 = 105 km per hour calculation above is not, strictly speaking, correct. The actual speed is ever so slightly less than 105 km per hour. The difference is negligible at such low speeds, but becomes ever more significant as the speed of light is approached. Thus if the train was moving at 95% of the speed of light and the woman was moving down the carriage at 15% of the speed of light, the man on the platform would not measure the woman’s speed at 95% + 15% = 110% of light speed but a speed given by the formula:

S = (V + U) / (1 + (V/C) * (U/C) where V = speed of train, U = speed of woman and C = speed of light.

With V = 0.95, U = 0.15 and c = 1, S comes out at (1.1) / (1 + 0.95 * 0.15) = .96, i.e. still less than light speed.

The correctness of this formula implies that supraluminal speed is impossible since even if U and V are arbitrarily close to C, S will still come out at less than C. Also, if U and V are small in relation to C (as in the first example), the approximation S = U + V holds to all intents and purposes.

That velocities are not additive have further consequences. Physicists define momentum as the product of mass and velocity and while there is an upward limit on velocity, there is no upward limit on momentum. This implies that as an object approaches the speed of light, its mass increases. The effect has been verified in the laboratory – it is been shown that so-called relativistic particles in an accelerator are more massive than their stationary counterparts. At lightspeed a particle would have infinite mass, thus to reach the speed of light would require an infinite amount of energy – another way of saying that faster than light speeds are impossible.

Finally the energy required to produce the mass increase can be calculated. There is a direct equivalence between mass and energy, which extends to an object at rest, the energy equivalent of which is the rest mass multiplied by the square of the speed of light.

The relationship is usually expressed by that most familiar of equations, E = mc squared.

© Christopher Seddon 2008

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