Some Scientific Curiosities

This week I thought I would try to be entertaining, to distract myself and others from what has happened in Ukraine. So to start with, how big is a bacterium? As you might guess, it depends on which one, but I bet you didn’t guess the biggest. According to a recent article in Science Magazine (doi: 10.1126/science.ada1620) a bacterium has been discovered that lives in Caribbean mangroves that, while it is a single cell, it is 2 cm long. You can see it (proposed name, Thiomargarita magnifica) with the naked eye.

More than that, think of the difference between prokaryotes (most bacteria and single-cell microbes) and eukaryotes (most everything else that is bigger). Prokaryotes have free-floating DNA while eukaryotes package their DNA nucleus and put various cell functions into separate vesicles and can move molecules between the vesicles. But this bacterium cell includes two membrane sacs, only one of which contains DNA. The other sac contains 73% of the total volume and seems to be filled with water. The genome was nearly three times bigger than those of most bacteria.

Now, from Chemistry World. You go to the Moon or Mars, and you need oxygen to breathe. Where do you get it from? One answer is electrolysis, so do you see any problems, assuming you have water and you have electricity? The answer is that it will be up to 11% less efficient. The reason is the lower gravity. If you try to electrolyse water at zero g, such as in the space station, we knew it was less efficient because the gas bubbles have no net force on them. The force arises through different densities generating a weight difference, and the lighter gas rises, but in zero g, there is no lighter gas – they might have different masses, but they all have no weight. So how do they know this effect will apply on Mars or the Moon? They carried out such experiments on board free-fall flights with the help of the European Space Agency. Of course, these free-fall experiments are somewhat brief as the pilot of the aircraft will have this desire not to fly into the Earth.

The reason the electrolysis is slower is because gas bubble desorption is hindered. Getting the gas off the electrodes occurs because there are density differences, and hence a force, but in zero gravity there is no such force. One possible solution being considered is a shaking electrolyser. Next thing we shall see is requests for funding to build different sorts of electrolysers. They have considered using them in centrifuges to construct models to compute what the lower gravity would do, but an alternative might be to have such a process operating within a centrifuge. It does not need to be a fast spinning centrifuge as all you are trying to do is to generate the equivalent of 1 g, Also, one suggestion is that people on Mars or the Moon might want to spend a reasonable fraction of their time inside one such large centrifuge, to help keep the bone density up.

The final oddity comes from Physics World. As you may be aware, according to Einstein’s relativity, time, or more specifically, clocks, run slower as the gravity increases. Apparently this was once tested by taking a clock up a mountain and comparing it with one kept at the base, and General Relativity was shown to predict the correct result. However, now we have improved clocks. Apparently the best atomic clocks are so stable they would be out by less than a second after running for the age of the universe. This precision is astonishing. In 2018 researchers at the US National Institute for Standards and Technology compared two such clocks and found their precision was about 1 part in ten to the power of eighteen. It permits a rather astonishing outcome: it is possible to detect the tiny frequency difference between the two clocks if one is a centimeter higher than the other one. This will permit “relativistic geodesy”, which could be used to more accurately measure the earth’s shape, and the nature of the interior, as variations in density outcrops would cause minute changes in gravitational potential. Needless to say, there is a catch: they may be very precise but they are not very robust. Taking them outside the lab leads to difficulties, like stopping.

Now they have done better – using strontium atoms, uncertainty to less that 1 part in ten to the power of twenty! They now claim they can test for quantum gravity. We shall see more in the not too distant future.

The M 87 Black Hole

By now, unless you have been living under a flat rock somewhere, you have probably seen an image of a black hole. This image seems to be in just about as many media outlets as possible, so you know what the black hole and its environs look like, right? Not necessarily. But, you say, you have seen a photograph. Well, actually, no you haven’t. That system is so far away that to get the necessary resolution you need to gather light over a very wide array, so the image was obtained from a very large number of radio telescopes and the image was reconstructed by a sequence of mathematical processes. Nevertheless, the black sphere and the ring will represent fairly accurately part of what is there.

No radiation can escape from a black hole so the black bit in the middle is fair, however the image as presented gives no idea of its size. Its radius is about 19 billion km, which is a little under five times the distance from the sun to Pluto. This is really a monster. Ever wondered what happens to photons that are emitted at right angles to the gravitational field? Well, at 28 billion km or thereabouts they go into orbit around the black hole and would do that for an infinite time unless they get absorbed by dust falling in. The bright stuff you see is outside the rotating photons, and is travelling clockwise at about half light speed.

The light is obviously not orange and the signals were received as radio waves, but when emitted they would be extremely high energy photons. We see them as radio waves because they have lost that much energy climbing out of the black hole’s gravitational field. One way of looking at this is to think of light as a wave. The more energy the light has, the greater the frequency of wave crests passing by. As the energy lowers due to gravity lowering the energy of the light, the wave gets “stretched” and the number of crests passing by lowers. At the edge of the black hole the wave is so stretched it takes an infinite amount of time for a second crest to appear, which means no light can escape. Just outside the event horizon the gravity is not quite strong enough to stop it, but a gamma ray wave might take 100,000 of our years for the next crest to pass when it gets to us. The wave is moving but it is so red-shifted we could not see it. Further away from the event horizon the light is a little less stretched, so we see it as radio waves, which is what we were looking at in this image, even if it still started as gamma rays or Xrays.

It has amused me to see the hagiolatry bestowed upon Einstein regarding this image. One quote: “Albert Einstein’s towering genius is on display yet again.” As a comment, I am NOT trying to run down Einstein, but let us be consistent here. You may note Newton also predicted a mass at which light could not escape. In Newtonian mechanics the energy of the light would be given by E =mc^2/2, while the gravitational potential energy would be GMm/R. This permits us to calculate a radius where light cannot escape as R = 2GM/c^2, which happens to be exactly the same as the Schwarzchild radius from General Relativity.

Then we see statements such as “General relativity describes gravity as a consequence of the warping of space-time.” Yes, but that implies something that should not be there. General Relativity is a geometric theory, and describes the dynamics of particles in geometric terms. The phrase “as a consequence of” should be replaced with “in terms of”. The use of “consequence” implies cause, and this leads to statements involving cosmic fabric being bent, and you get images of something like a trampoline sheet, which is at best misleading. Here is another quote that annoys me: “Massive objects create a sort of dent or well in the cosmic fabric, which passing bodies fall into because they’re following curved contours (not as a result of some mysterious force at a distance, which had been the prevailing view before Einstein came along)”. No! Both theories are done a great disservice. Einstein gave a geometric description of how bodies move, but there is no physical cause, and it has the same problem, only deeper, than the Newtonian description had, because you must then ask, how does one piece of space-time know exactly how much to distort? Meanwhile, Newton gave a description of the dynamics of particles essentially in terms of calculus. Whereas Einstein describes effects in terms of a number of tensors, which most people do not understand, Newton invented the term “force”.

Now you will often see the argument that light is bent around the sun and that “proves” General Relativity is correct. Actually, Newtonian physics predicted  the same effect, but general Relativity bends it twice as much as predicted by Newtonian physics, so yes, in that sense General relativity is correct if the bend is correctly found to be twice that of Newton. You will then see statements along the lines this proves the bent path is “due to the warping of spacetime”. That is, of course, nonsense. The reason is that in Einstein’s relativity E = mc^2, which is twice that of the Newtonian energy, as you can see from the above. The reason for the difference appears to be the cosmic speed limit of light speed, which Newton may or may not have considered, but had no reason to go further. Why do I say Newton might have considered it? Because as a postulate, the fundamental nature of the speed of light goes all the way back to Empedocles. Of course, he did not make much of it.

Finally, I saw one statement that “the circular nature of the black hole again confirms the correctness of Einstein’s theory of General Relativity. Actually, Aristotle provided one of the first recorded reasons why gravity leads to a sphere. Newton would certainly have predicted a basic sphere, and of course the algorithm used to make the image would not have led to any other result unless there were something really dramatically non spherical. The above is not intended to downplay Einstein, but I am not a fan of the hagiolatry that accompanies him either.

Is the Earth’s Core Younger than the Crust?

There was a rather interesting announcement recently: three Danes calculated that the centre of the earth is 2.5 years younger than the crust ( U I Uggerhøj et al. The young centre of the Earth, European Journal of Physics (2016). DOI: 10.1088/0143-0807/37/3/035602 ). The concept is that from general relativity, the gravitational field of earth warps the fabric of space-time, thus slowing down time. This asserts that space-time is something more than a calculating aid and it brings up a certain logic problem. First, what is time and how do we measure it? The usual answer to the question or measurement is that we use a clock, and a clock is anything that has a change over a predictable period of time, as determined by some reference clock. One entity that can be used as a clock is radioactive decay and according to general relativity, that clock at the core would be 2.5 years younger than a clock on the surface; another is the orbit of the Earth around the star, and here the core has carried out precisely the same number of orbits as the crust. Where this becomes relevant is that according to relativity all clocks must behave the same way towards velocity, otherwise you could take your rocket ship and by comparing two different types of clocks you could measure your absolute velocity. So, does that mean gravitational time dilation is conceptually different from velocity time dilation? I believe this matters because it brings into question exactly what is space-time?

The above does not mean that time dilation does not occur. It is unambiguous. Thus we know that the muon travelling at relativistic velocities has its lifetime extended relative to a stationary one. If we assume that the process of decay is unaffected by the velocity, then the passage of time has to have slowed down. But that raises the question, is the assumption valid? An analogy might be, suppose I have a clock that is powered by a battery, and as the voltage drops, the clock slows. I would argue this is because the lower voltage is inadequate to keep the mechanism going at its previous rate, and not that time itself has slowed down.

Now, consider the mechanism of muon decay. If apparent mass increases according to velocity, why should not the rate of decay of a muon slow down, after all, it has accumulated more mass/energy so it is not the same entity? Is the accretion of mass equivalent to the change of gravitational potential?

Perhaps what relativity tells us is the rate at which clocks move indicates their altering the scale of the passage of time, rather than time itself slowing down. By that, I mean that when a clock hand completes one period, we say an hour has passed, but at relativistic speeds it might say that γ hours have passed per clock period, where γ = 1/√(1 – v2/c2). In terms of gravitational fields, it is not that time slows down, but rather clocks do, together with the rate of physical processes affected by the gravitational field.

That suggests we take our concept over to inertial motion. If a body travels near the velocity of light, then our equations tell us that time appears to dilate, but has time really slowed, or is it the process that leads to the decay that has slowed? Does it matter? In my opinion, yes, because it is through understanding that we are more likely to make progress into new areas.

The reason it is asserted that it is time itself that slows down comes from the principle of relativity, first (as far as I can tell) loosely stated by Galileo, used as the basis of his first law by Newton, and perhaps more clearly stated by Poincaré: the laws of physical phenomena must be the same for a fixed observer as for an observer who has uniform translational motion relative to him, so that we have not, nor can we possibly have, any means of discerning whether or not we are carried along in such motion. When added to the requirement from Maxwell that the velocity of light is a constant, we end up with Einstein’s relativity.

The question is, is the principle correct? It has to be in Galilean relativity, as it is the basis of Newtonian dynamics. If velocities are added vectorially, there is no option. But does it translate over into Einstein’s dynamics?

My argument is that it does not. There is an external fixed background, and that is the cosmic microwave background. The microwave energy comes almost uniformly from all directions, and through the Doppler shift one can detect an absolute velocity relative to it. (The accuracy of such determinations at present is not exactly high, but that is beside the point.) Very specifically, at 1977 our solar system was travelling with respect to this black body radiation at 390 +60 km/s in the direction 11.0+ 0.6 h right ascension and 6o +10 o declination. (Smoot et al., 1977, Phys Rev Lett. 39, 898 – 901). So we DO have the means of discerning whether or not we are carried along with such motion.

If we can measure an absolute velocity, it follows there is an absolute time, and as I have noted before, we can always measure when we are by determining the age of the Universe. Therefore I am reasonably confident in saying that the core of the Earth has aged at precisely the same rate as the crust once the Earth formed, and since there has not been complete mixing, it is more likely the core is older, as it on average would have accreted first. One the other hand, isotope decay there should have been held back by about two and a half years.

What is the cause of the Lorentz contraction?

Do you ever ask yourself, what is the cause of . . . ? I think you cannot expect to make significant advances on what you are familiar with unless you can answer such questions. However, what I consider to be one of the more serious problems of modern physics is the belief that if you have an equation that accounts for observation, then the problem is solved. For only too many, physics IS mathematics, and everything about us is determined by mathematics. That is not a new thought; the Pythagoreans and devotees of Plato held this belief. That is why there were five elements (earth, air, fire, water, ether): there are five and only five “Platonic solids” in solid geometry. We all know how well that turned out. Now the mathematics are more complicated, but we get the same outcome: belief.

What has inspired this are my thoughts on Special Relativity. The question is, physically, what is the primary cause of what is happening, and what are the consequences? What we see mathematically is that to make correct calculations when motion approaches the velocity of light, we must alter the value of certain variables by multiplying or dividing by what is called the Lorentz contraction term. Thus if a rod is moving at a speed v approaching the speed of light c and is aligned in the direction of motion, then lengths in this direction will appear to be shorter by a factor of √(1 – v2/c2). The question then is, what is going on?

It is easy to derive this in terms of length contraction if we believe the speed of light is constant. If it is, following Feynman in “Six Not-so easy Pieces” then we can build a simple apparatus with two equal arms at right angles to each other. If we put mirrors on the end, and send light signals each way, and if the result must not be able to be used to measure the velocity of the instrument if it is moving without acceleration, then with a bit of algebra you find this only makes sense if the length in the direction of travel has contracted by the factor √(1 – v2/c2). That time must dilate by a factor of 1/√(1 – v2/c2) arises because if it did not, again you could work out the absolute velocity of your apparatus. The length contraction arises because light going there and back at right angles to the direction of motion has to travel along the hypotenuse of a right-angled triangle, which is perforce longer than the two arms. However, as Feynman notes, you can also get all of special relativity if you assume the mass increases by the “rest mass” being multiplied by 1/√(1 – v2/c2).

The length contraction is derivable, nevertheless it raises another interesting question: how does the length “know” to contract? The length of the space ship is possibly understandable, but the problem is it is also the length in whatever direction you are travelling. Alternatively, we could ask why does the time dilate? While the velocity of light must be constant, that does not explain why, because it does not relate the effect to anything external. As an example, in my cat paradox, if you are in a space ship raveling at almost light speed towards Epsilon Eridani, and if you fix the frame of reference as your ship, then it appears Epsilon Eridani is hurtling towards you, and you are stationary. Now, if you put v into the equations, it is Epsilon Eridani that suffers time dilation. Of course you can get around this, but my argument is every time you try, you do not use the space ship’s frame of reference.

Elsewhere, Patrice Ayme has argued that given time dilation, you also get a corresponding mass increase for a physical reason rather than a mathematical one ( ). If I have this right, the argument is that if time dilates, then collision of the object with “force carriers” is much less frequent and hence inertial mass is, or appears, higher. The logic of mass increase, however, also implies time dilation by a reverse logic.

So what is fundamental? The constancy of the speed of light? Or does that arise from the length/time dilation? Which is the chicken and which is the egg? One interesting fact is that you have to choose one to be fundamental, and the rest follows. You could start with the mass augmentation, and everything else follows from the usual relationships. The constancy of the speed of light is an attractive place to choose for its being fundamental because that arises essentially from Maxwell’s electromagnetic theory, namely c = 1/√(εμ), where ε and μ are the permittivity and permeability of space. If you choose that, then the equations of special relativity follow from Feynman’s argument about length contraction.

Furthermore, there is fairly clear evidence that the time dilation effect is real. Muons are generated by cosmic ray events in the upper atmosphere and they travel at relativistic velocities. Muons have a very short lifetime and if they were constrained to that lifetime without time dilation they could only travel so far before they decayed, but we know from observation that they can travel so much further. They can also travel so much further in particle accelerators, and this is only explicable in terms of their “clocks” that govern their decay rate are going slow, in accordance with relativity. Of course, their clocks might go slow because of mass augmentation.

So, what is the issue? For me, if length and time contraction are fundamental, in the case of my previous posts on a space ship going to Epsilon Eridani, all the space between the ship and Epsilon Eridani has to “know” to contract, otherwise a light beam from the ship to a mirror somewhere near Epsilon Eridani travelling at the same velocity as the ship would be a means of determining the ship’s velocity, which is allegedly forbidden. (One can always find the velocity by reflecting light from something fixed about Epsilon Eridani; the reason for having the mirror travelling at the same velocity is so there is nothing external to the frame of reference.) On the other hand, if the mass enhancement is the real cause, then the effect is a consequence of the energy poured into the entity during acceleration. Thus in one case, the effect is due to space “knowing what is coming”, the effect is non-local and only applies to the travelling object. By that, I mean if the fast moving ship overtakes a snail, the space on the path somehow distinguishes between them. (Mathematically, the distinguishing is trivial, but physically?) Why does the space light years away respond differently and correctly to two objects at the same time? If it is mass that is enhanced, the effect is due to the entity “recording what has been done to it” and the effect is local to it. By that I mean its clock will slow, and the distance will appear to contract. I know what I prefer, and as usual with me, it is not the same as what everyone else seems to think.

Revisiting the Relativistic Cat Paradox

I have put forward my relativistic cat paradox both in a previous post here, but also elsewhere, and I received various comments. Accordingly, I thought I should resurrect it, and address some of the points made that are not recorded in the original post. At the risk of being repetitive, the essence of the paradox, which is a variant of the twin paradox, is as follows:

To test the premise that all inertial reference frames are of equal standing, Fred and my five-year-old cat Horatio participate in a thought experiment. Horatio is put into a cat friendly space ship (SS1), I put myself into SS2, and the two ships accelerate rapidly then coast as close as possible to light speed in the direction of Epsilon Eridani, which is about 10.5 light years away, leaving observer Fred behind. The ships loop around the back of Epsilon eridani, then returns to Earth, landing where we took off. Fred and I ach calculate what should happen to Horatio, bearing in mind a cat seldom lives more than 20 years, then we open the hatch to SS1, and the question then is, which calculations are correct?

First, the issue is not, can we agree what we shall see before we open the hatch? Of course we can! The question is, can we agree on what we shall see when we view a third object from two inertial frames of reference without assigning a special preference for one, because Einstein argued that all inertial frames of reference are equally valid. Actually, that comes straight from Galilean relativity, but in that there is no effect on time or space from velocity.

Reviewing the situation, both parties agree there are three phases of acceleration, and two of coasting. I left acceleration out of the original post because all observers agree on the rate of acceleration. (That is an oddity, if you like, of all relativity, including Galilean.) Acceleration will cause time dilation, but since all parties agree on the rate of acceleration, both Fred and I agree what that will be. Accordingly we focus on the inertial stages. Here, time is dilated through a factor of γ where γ = 1/√(1 – v2/c2). Accordingly, Fred sees Horatio sailing through space where v ≈ c, and hence concludes Horatio should hardly age. Fred predicts Horatio will be alive. I, however, see v = 0 in my frame of reference, and Horatio ages about 11 years on reaching Epsilon Eridani, and another 11 years on the way back. (If I look forward, from the Doppler effect on the spectra I see Epsilon Eridani hurtling towards me at near light speed, but that only means that to me, anyone in that star system is not aging.) Accordingly, I predict Horatio will be dead.

This is similar to the twin paradox, for which there are various explanations. The twin paradox is often explained by the twin on Earth sending signals to the ship, and applying something similar to the Doppler effect. Such signals are irrelevant; whatever happens should be irrespective of any signalling. Some other people considered the time dilation was due to acceleration, and we all agree on what has accelerated. Thus following acceleration, there is time dilation caused by “enhanced mass”, because mass is enhanced by γ. We could also say the traveller knows he has accelerated, as has Horatio, and therefore he knows he is the one sustaining the time dilation. Another comment was that the distance has shortened, so the amount of time evolved must be less. That is all very correct, but there is a flaw in the argument.: these explanation still ultimately depend on γ, which in turn depend on the value of v put into the term. If you are coasting, or you are observing Horatio, in your frame of reference, v = 0. What has happened is that these explanations, while they have the virtue of each party getting the same result, which also is most likely to be correct, they only get it because the traveller is still using the stationary observer’s frame of reference. Following the acceleration, they know they are moving at near light speed, therefore they get the same answer by putting in the same value of v but that means there is only one frame of reverence: Fred’s. This avoids the problem of the paradox, but only by deleting the cause of the paradox and ignoring the point the paradox was addressing.

A true theory should always give the correct answer, irrespective of who is calculating, but in this case that can only happen if both use the same value of v because v is the only variable in the equation that is used to calculate the effect. If you do not accept v as a variable, then you reject the fundamental theory. But v depends on the frame of reference, as a velocity represents the change of distance between the object and a reference point. Accordingly, you cannot get a constant value of v unless both parties use the same frame of reference.

If you accept a preferred frame of reference, you have to explain why it is preferred. In the above, the preferred frame of reference was where the journey started and ended, and that is logical because then the time dilation is proportional to the kinetic energy imparted by the applied force. An alternative is that there is indeed a preferred background frame of reference. If so, we could not discern this as yet because the Earth would be moving too slowly with respect to the background to make any discernible difference. But either way, in my opinion you cannot have all frames of reference being equally valid.

A relativistic cat paradox.

When travelling near the speed of light, time goes more slowly, by a factor of γ, where

γ = 1/√(1 – v2/c2)

Here, c is the velocity of light, which is constant for all observers, while v is the velocity. The problem with the velocity is that it depends on the frame of reference used, which in turn depends on the motion of the observer, and according to Einstein, there is no preferred frame of reference. That means any frame of reference is as good as another, and apparently Einstein illustrated the principle of relativity once by remarking at the end of a train journey, “Ha, the Zurich railway station is approaching, and will soon stop outside the train.”

Consider this problem. I have a friend Fred and a five-year old cat Horatio, and these two have agreed to participate in a thought experiment to test Einstein’s argument that there is no preferred frame of reference. Horatio is put into a cat friendly space ship (SS1), I put myself into SS2, and the two ships travel as close as possible to light speed in the direction of Epsilon eridani, leaving observer Fred behind. The ships loop around the back of Epsilon eridani, then head back to Earth, landing where we took off. Fred and I open the hatch to SS1, and the question then is, what do we see? Before opening the hatch, we can use the time dilation equation to make our prediction, but we get different answers.

From Fred’s point of view, the two space ships have been in flight for twenty-two years, say, but they sustain the relativistic time dilation effect because v ≈ c, and time should almost stop. Accordingly, following Einstein’s equations, Horatio will leap out, a little older than when he entered SS1. However, from my point of view, once underway, I look out the window of SS2, and see SS1 stationary beside me, and Epsilon eridani hurtling towards me at just under the speed of light. However it does not reach me for a bit under 11 years, and the same thing happens on the way back, except Earth is now hurtling towards me at almost light speed instead of receding. Accordingly, Horatio should have experienced a bit under 22 years of travel, but since cats do not live longer than about 18 years, and given his first five years were over before he started, I expect to see a long dead residue of Horatio.

Two adjacent observers must see the same thing. What do they see when they open the hatch of SS1, and why?

The paradox goes away if there is a preferred frame of reference, and the velocity both use in the equation is the velocity with respect to that frame of reference. Note that one can argue that there is a preferred frame of reference in the cosmic microwave background, and motion of our solar system relative to that has been found to be approximately 390km/sec. (Smoot et al., 1977 Phys. Rev. Lett. 39: 898).

Simple relativity

During the summer break, I got involved in the issue of whether time was relative, but before I can discuss that, I need to be sure readers understand what relativity is. Most would consider relativity to be essentially mathematical. Not really. The principle of relativity is quite simple, and goes back a long way. In Il Dialogo, Galileo pointed out that if you were below decks in a boat, you have no idea how fast it was going, nor for that matter, in what direction. You could get up on deck and work out how fast you were going relative to the water, but there is no absolute velocity, for if you can see land, you may have a different velocity if there is a tidal flow or current. Then, of course, the earth is rotating, orbiting the sun, the sun is orbiting the galactic centre, and the galaxy is also moving relative to other galaxies. The point is, unless there is a fundamental reference there is no absolute velocity, but only a velocity relative to something else, and that depends on your perspective. As Einstein once remarked when on a train, “The Zurich Railway Station is approaching, and will shortly stop outside the train.” Bizarre though that may sound, that encompasses relativity.

The simplest way to look at this is to answer the question, “Where are you?” There are two probable answers. One is “Here!” Not very helpful when half the population answer the same way, in which case “here” is a different place for different people. The second answer is to give an address, or coordinates. The means you are defining your position as being at some distance from something else. Velocities represent the rate of change of position, and are vectors, which means they have magnitude and direction. Coming and going have quite different effects. Think of standing in the middle of a road and there is a car on it. However, when direction is properly taken into account, velocities are additive, at least in Galilean relativity. Suppose we have two fleets of ships heading to each other. Each is entitled to consider itself as motionless in its own frame of reference, with the other fleet approaching at a velocity that is the sum of the vectors in a third frame of reference.

James Clerk Maxwell gave physics a huge problem by writing his equations of electromagnetism in the form of a wave equation, when he found the velocity of his wave was more or less equal to the known speed of light. Accordingly, he stated that light was an electromagnetic wave that travelled at velocity c. The problem was, relative to what? His equation equated the velocity to constants that were properties of space itself. Still, if the waves moved through something, namely aether, they would have a velocity relative to the aether. When Michelson and Morley carried out an experiment to measure this, they found nothing. (Actually, they found a very small velocity, but that was put down to experimental error because it did not reflect the earth’s movement properly.) For Einstein, the velocity of light was constant to any observer, and there was no aether, nor any absolute motion. Making sense of this involves mathematics that are a little more complicated than those of Newtonian physics, and now we have a problem as to what it means. The interpretation most people accept was proposed by George Fitzgerald and Hendrik Lorentz, and involved space contraction in the direction of motion. The basis of this can be imagined by considering two space ships flying parallel to each other, and going in a fixed direction. Suppose one sends a signal to the other that is reflected. The principle of relativity is from the space-ships’ point of view, the other ship is stationary, but from an external observer, the signal does not go directly to the other ship, but rather travels along the hypotenuse of a right-angles triangle, which now requires Pythagoras’ theorem to untangle the maths. Complicated?

There are some seemingly absurd results obtained from relativity, but it should be noted that these arise from what different observers, each travelling at near light speed, interpret an event they see. The complication is each sees the light coming to them at the same velocity, and this leads to some more complicated maths. Strange though it may seem, the equations always give correct agreement with observation, and there is little doubt the equations are correct. The question then is, are the observations and equations being properly interpreted? Generally speaking, the maths have taken relativity quite some distance, using a concept called space-time, and in that, time is always relative as well. It would generally be thought to be near impossible to solve anything of significance in General Relativity without the use of space-time, so it must be right, surely? In my Gaius Claudius Scaevola trilogy I make use of the time dilation effect. To fix a problem in the 23rd century, a small party of Romans have to be abducted by aliens in the first century. They travel extremely close to the speed of light, and when they return, they arrive at the right time, having burned through 2,200 years, and have aged a few years. Obviously, I believe time is relative too, don’t I? More next week.