Relativity Part II: Galileo versus Maxwell
16/03/17
“You did mechanics at school yes? You remember Newton’s laws of motion?” I asked. Percy nodded and recited, “Law one: a body in a state of rest or constant velocity stays in that state unless acted on by a net force. Law two: the acceleration of a body is equal to the force acting on it divided by its mass. Really law one is just a special case of law two I guess; forces cause accelerations so no force means no change in motion. And law three: forces always come in equal and opposite pairs.”
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“Okay, yeah. So, dry as that may sound, there are a few key lessons to be drawn from it all which are quite counter intuitive to people the first time they learn them. In fact they can be world-view altering if you think about them hard enough. The first is that forces describe accelerations, not velocities. In the Aristotelian view of the world, which is scientist jargon for pre-Newton/Galileo, and also the way I think most people instinctively see the world anyway, the gut notion is that you have to apply a force on something just to keep it moving. Newton tells us that things just keep going the same way forever unless they’re acted on, but that’s not the way it feels to most of us, it feels like you have to keep pushing on an object just to keep it going and that if you leave it alone it’ll just drift to a stop.”
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“Right,” said Percy “but we feel like that because we live in a gravity well surrounded by air and rough surfaces that are always applying resistance and friction and so forth, so we never really see things undergoing anything approaching force free motion.”
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“Exactly. Real objects always slide to a halt when we stop pushing them, not because there’s no force keeping them going per se, but because the environment they’re in is exerting forces on them to slow them down. The force you apply by pushing is really just to counter all those other forces so that there’s no net force and they can just keep moving steadily, as Newton’s laws say they will in free space. But the point to really emphasize here is that if Newton’s laws only describe accelerations, and not velocities, then your velocity has no effect on the way you construct Newton’s laws.”
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“Come again.”
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“In calculus terms, velocity is the first derivative of position with respect to time and acceleration is the second derivative with respect to time.”
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“Sure.”
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“So, if you have some expression that describes your position as a function of time, then you have to differentiate it once to get an expression for velocity, and twice to get the expression for acceleration. Now any terms in that expression that are responsible for describing the initial velocity of the system will be given by a single power of t. Not t squared or t to any other power. Which means they’ll disappear after two differentiations, and won’t have any effect on the expression for acceleration.”
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“Okay.”
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“So accelerations look the same regardless of your velocity. And since Newton tells us that forces concern themselves with accelerations, this means that forces have the same form no matter what the velocity of the frame of reference you are using to describe them. And if laws of physics are all about relating forces to certain configurations of matter, a la gravitation and electromagnetism, then the laws of physics have the same form in any constant-velocity frame of reference.”
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“Yeah okay, I remember this. I’m assuming this is the bit where you talk about the train carriage on super smooth tracks that don’t jostle around at all, and then if you put the blinds down so you can’t see the landscape zipping by outside then there’s no experiment you can do that will tell you how fast the train is moving. Like it could be moving at a million miles an hour and it doesn’t change what you feel inside the train carriage, you can still get up and move around and play catch with a tennis ball and it will all look perfectly normal, as though the train were not moving at all. Right?”
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“Yes. Even with the blinds up you can make the argument that the train carriage is stationary and it’s the landscape that is moving past. The blinds don’t make a difference except psychologically. We’re used to thinking of the landscape as being tied to a kind of absolute background space with respect to which all other motion is defined, but that’s just a psychological crutch. The landscape is just an object with respect to which motion is described in relative terms like any other, as we all acknowledge when we remember that the whole planet and every landscape on it is moving around the sun, and the whole solar system is falling round the centre of the galaxy. Even the jostling when the train goes over bumps, or the vibration of the train’s engine doesn’t change the legitimacy of the argument. You can, with complete physical consistency, describe the tracks as moving beneath the train and say that train is running its engine and jostling along the tracks over all the bumps and so forth, just to resist the movement of the tracks and enable it to stay where it is. Like you on the treadmill at the gym.”
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Percy was biting the rim of his glass pensively. He swirled the remaining contents around a few times and then said “Okay. But this is all Newtonian stuff. Where does Einstein come in? All the stuff about the speed of light, and time not being absolute and E=MC2 and what not?”
“We’re getting to that” I replied. “There are multiple levels to the series of ideas that now come under the broad moniker ‘relativity’. What we’ve described so far is the first one, what is sometimes called Galilean relativity. Galileo described it in terms of butterflies flying around in a ship’s cabin. Imagine you have a ship moving steadily in a particular direction. Now you’re sitting in the cabin, and you’ve just released a jar of butterflies and you’re watching them fly randomly around the cabin. What do you see? Do you see them tending to drift towards the back of the cabin as the ship moves beneath them? Do they have to compensate for the motion of the ship by putting in extra effort to fly towards the bow and less effort for the stern? Of course not. They fly around randomly in all directions with equal ease. They were on board the ship as it left the doc and accelerated up to speed and Newton’s laws tells us that they need no extra help to stay moving at that speed. They might need to make some concessions to the air moving past them (or if you prefer, the air that they are moving through) if they were up on deck, but this is just another complication that distorts our instincts about these processes in conventional situations. In fact it’s no different to how you have to put in extra effort, i.e. brace yourself, to remain stationary if you’re in a strong wind. If you can understand this then you understand, in essence, the way the stage of physical knowledge was set at the end of the seventeenth century. Motion is relative according to all the laws of physics. There’s no absolute background space relative to which you define one system as definitely moving and another as definitely stationary. Actually there’s no absolute space in two senses. The first, no absolute meaning to position, no special point zero out there in space. In other words no privileged coordinate system in which the laws of physics take on a uniquely elegant or simple formulation. Because the laws of physics don’t change at all with location. This is related to what we were talking about earlier with the derivatives, but in an even more elementary way. Forces define accelerations, not positions or velocities. If you have an expression for position in terms of time you have to differentiate twice to get the form for acceleration and by that point the terms defining initial velocity and positions have vanished. Position disappears with the first differentiation, in fact, so even if forces cared about velocities they would still be position invariant. So yeah, to summarise; physics doesn’t care about position, there’s no absolute meaning to it, the position of an object is just a relative term that refers to its distance from, and orientation to, other objects. Secondly, physics doesn’t care about velocity either. It seems there’s no absolute meaning to it, no absolute reference frame with respect to which all other velocities can be defined once and for all. Velocity is just a relative term that describes the way the relative position of two objects, with respect to each other, is changing with time. This deals a further identity blow to space because it means that even if you choose an arbitrary coordinate system and assign a value to every point in space at any given moment, there is no absolute meaning to the identity of those points and a later (or indeed earlier) moment in time. It depends on your reference frame, what one frame describes as the same point just at a later time, another reference frame will think is a different point in space altogether. Say I’m on a train going at sixty miles an hour and I tap the centre of the table in front of me. Then, a minute later, I tap it again. In the frame of the train those two taps happened in the same place. In the frame of the surrounding landscape, they happened a mile apart.”
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Percy had taken out a biro and was drawing something on a napkin. “What’s that?” I asked.
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“A space-time diagram. At least the way my A-level teacher depicted them.”
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“Sure.”
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“I’m just trying draw roughly what you’re saying.” He showed me the paper.
“So the horizontal axis represents a dimension of space,” he said “and the vertical axis is time. Now in the left diagram we’re treating me as stationary and you as moving, so my arrow points straight up because I’m not moving with time, and yours is slanted because you are.”
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“Yup.”
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“And because we’re in different reference frames we draw the time axis differently. The vertical and slanted lines respectively represent what each of us thinks is a point of constant position across time. And because the laws of physics make no preference for one reference frame over any other, both accounts are equally legitimate. In fact if we asked you to draw what was going on, you would probably depict the situation the way the diagram on the right does. Both depictions represent the same physics.”
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“Exactly. What you’ve just drawn is called a Galilean transformation which is a conversion from description under one reference frame to another, in pre-Einsteinian physics.”
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“Pre-Einsteinian physics?”
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“Yeah. Physics as it was thought to be from the 1600s until 1905. It turns out, in fact, that things are a little more complicated than that, although this way of looking at things is still an excellent approximation for situations in which the velocities involved are significantly slower than the speed of light.”
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“Which is to say most everyday situations that we have to think about?”
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“Yes. But now it’s time to move beyond those situations and start discussing the speed of light.”
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“What is the significance of the speed of light?”
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“Well, we’ll start in the 1860s. A Scottish physicist called James Clerk Maxwell completed what is usually regarded as one of the greatest achievements of unification in physical theory to date. It had been known for a long time that there was some sort of connection between the phenomena of electricity and magnetism, but Maxwell was the first person to comprehensibly form a single theory that accounted for all of the phenomena normally regarded as either electrical or magnetic in terms of a single phenomenon. Which we now call electromagnetism.”
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“Yeah I think I’ve heard this story too. Didn’t he reduce it all to four equations?”
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“Yes, basically. We don’t need to go through the details here but essentially yes, one equation that describes electric fields in terms of charge density, one that basically says there are no magnetic monopoles and two more that describe and extend the processes of electromagnetic induction documented by Faraday half a century earlier. But the point is that as he was describing electromagnetic fields and relating them to charged objects he showed that as configurations of charged systems change their arrangements, and the distribution of the field changes accordingly, the influence of that effect is not instantaneous but propagates outward at a finite speed.”
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“The speed of light? Because it is light?”
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“Right, yes, that’s what he realised. Light and radio and X-rays and microwaves and ultraviolet are all waves, of various frequencies, in the electromagnetic field, travelling at this fixed speed, about three hundred thousand kilometres per second, determined by two constants that appear in his equations. Here is the problem though. Relative to what is this fixed speed defined? The equations just sort of prescribe this fixed speed of propagation that does not depend of the speed of the source that emits it. Sound waves are a bit like that, they have a fixed speed that basically does not depend on their source, but with them it’s well defined; their speed is relative to the material medium that carries them. But light doesn’t have such a medium”
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“Isn’t the medium of light the electromagnetic field?”
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“That might sound like a natural answer but what would that even mean? It’s quite hard to turn an abstraction like that into anything concrete. How do you define the reference frame of the electromagnetic field? How do you know if you’re at rest or moving with respect to it? We can only interact with it by observing the behaviours of the charged systems that couple to it and, by proxy, to each other. How do we distinguish its velocity from theirs?”
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“Um, well, you know how fast the theory tells you light is supposed to propagate, which you said is irrespective of the emitting systems. So you measure the speed of light, see how much your measured value deviates from the expected one and that tells you how fast you are moving with respect to the field.”
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“Ok. Which is essentially what they tried to do. They described it a little differently, our terminology bears all the hallmarks of historic retrospection, but they proposed that there was some kind of space-filling medium, which they called the ‘luminiferous aether’, that mediates the effects of electromagnetic disturbances. And they tried to do the experiment you just proposed, two guys called Michelson and Morley did an experiment in 1887 that essentially involves comparing the measured velocity of light in different directions. If you think about it, assuming the Earth is moving through ‘aether’, then from the perspective of the Earth we’re sitting in a kind of ‘aether wind’. So light that is moving in the same direction as the Earth (or against the ‘aether wind’) will have a reduced relative velocity and light moving in the opposite direction will seem to be moving faster.”
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“And light that is travelling perpendicular to the ‘aether wind’ is unaffected?”
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“Not quite. Think about it. Let’s say it’s midday so the ‘aether wind’ is blowing roughly from west to east as the Earth moves around the sun (we’ll ignore the Earth’s rotation since it contributes a much smaller effect). Remember that the speed of light is defined relative to the ‘aether’, not the Earth. If a ray of light moves straight from north to south, in the reference frame of the Earth, then that means it does have some east-west motion relative to the ‘aether’. So it has already ‘used up’ some of its speed fighting against the ‘aether’ and has less available to travel from north to south with. Another way to look at it is, light in the ‘aether wind’ has a tendency to drift sideways. That doesn’t mean that if you shine a beam of light in one direction it will curve round as it travels, because we’re not accelerating through the ‘aether’ here, just moving with a constant velocity. But it does mean that if you had, say, a laser that would shoot a beam of light straight forward out in space, it would fire at a slight diagonal if you tried to shoot it from north to south here on Earth, or if you had a spherical light source that emits light in all directions, and should emit a spherical distribution of light in free space, it will produce a skewed, stretched distribution in this experiment. So from the ‘aether’s’ perspective any light ray that does travel straight from north to south in the Earth’s frame must have cancelled its drift tendency with an east-west component of velocity, giving it a smaller one left with which to travel north to south. But anyway, as it turns out, all of this is moot because the actual experiment did not reveal anything of the sort. The measured light did not exhibit any directional preference, it travelled with equal speed in all directions.”
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“So the Earth is not moving with respect to the ‘aether’?”
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“So it would seem. Which is a pretty problematic proposition. This was two and a half centuries post-Galileo so nobody was parochial enough to suggest that ‘aether’ was somehow centred on the Earth, moving and rotating with it, at least not universally. There was some suggestion that perhaps the ‘aether’ is not perfectly uniform but gets ‘dragged’ around a bit by the objects in it so that the ‘aether wind’ would be locally reduced because it would be sort of moving with the Earth within the planet’s local vicinity. But that seems a bit vague and evasive given how ad hoc a proposition it was to begin with. So it seems that we’re back where we were. Maxwell’s equations prescribe a speed for light, but we have no clear notion of how do define the reference frame for that motion.”
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“What’s the solution then?”
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“Remarkably simple actually. If you’re willing to drop a few, previously quite foundational, assumptions. The speed of light given by Maxwell’s equations is a law of nature and by all accounts there is nothing in the theory that specifies a medium with a well-defined velocity with respect to which that speed can be specified. And all of our previous descriptions of physical laws make it pretty apparent that they should not depend on reference frame. So the answer to which the theory seems to be pushing us is that light travels at the speed given by Maxwell’s laws in every reference frame.”
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“But that’s impossible. If there’s a ray of light travelling away from me at three hundred thousand kilometres a second, and you’re travelling away from me in the same direction at one hundred thousand kilometres per second, then the light is gaining two hundred thousand kilometres on you every second. So in your reference frame it’s travelling at two hundred thousand kilometres per second. It doesn’t make sense otherwise.”
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“It can. With one fundamental modification to your assumptions. One that didn’t come easily to anyone in the nineteenth century, and still doesn’t to most people today, at least not initially. We know that observers in relative motion have different measures of the distance between two events that happen at different times. We saw that with the example where I was tapping on a table in a train carriage.”
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“Yeah.”
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“And speed is distance per time. So how can we have the same measure of a system’s speed, but different measures of covered distance? Different values for distance, same value for distance per time.”
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“We would each have to have a correspondingly different value for time? Ah, I see. Time dilation.”
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“Exactly. Time runs slower for moving observers. The textbook example of this is to imagine a highly impractical clock comprised of two mirrors, one suspended above the other (let’s say by about one and a half metres), and to have a single particle of light (a photon) bouncing back and forth between to two. So this light particle will do about a hundred million round trips up and down per second. Presumably it loses a bit of energy per bounce or nobody would be able to detect the bouncing and use it to tell the time, but we’ll pretend that amount is really tiny and we just have amazingly sensitive detection equipment.”
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“Doesn’t sound very realistic.”
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“No. Gedankenexperiments never are.”
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“What never are?”
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“Thought experiments. They get a German prefix in academic parlance in honour of Ernst Mach, who was a big influence on Einstein. The point of them is to try to figure out the implications of a theory for our view of the world in the starkest terms possible. They normally involve constructing highly abstract and unrealistic scenarios, not because theorists think these situations are plausible, but for precisely the opposite reason. Anyone who has ever tried to conduct even the simplest physical experiment in the real world knows how difficult it is to discern the real principles at work because the real world in so damn complex. So many hidden and chaotic forces obscure the underlying principles at work, so much so that, for example, it took until the seventeenth century for something as concrete and Newton’s laws to become apparent. The point of a thought experiment is to run a kind of simulation in the mind, highly simplified so that the implications of the basic underlying laws can be seen more directly. If the situation in question involves principles that are quite counter intuitive then this methodology can help us to avoid the biases of common sense, because common sense often involves notions that are not at all as basic or fundamental as we think, but really heuristics that apply to a very limited class of situations and involve quite complex and high level principles, even if we have evolved to find them obvious because of their prevalence in everyday life. An example of this could be the pre-Newtonian/Galilean instinct that moving systems naturally drift to a stop without any forces on them, for which we have to construct frictionless environments with very simple-shaped objects to unambiguously illustrate the real dynamics at work. Or, even more primitively, the idea that there is a special direction in space (downwards) along which all objects naturally tend because of their weight. Most moderns are pretty used to it but historically even that idea required a bit of refinement before people could accept the idea that the world is a globe because it requires appreciating that the special ‘downwards’ direction is radial, not Cartesian, meaning one person’s up is another’s down and someone else’s is a combination of the two and so forth.”
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“Yeah ok, I see. Take us back to the light clock, with the two mirrors.”
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“Right, right. So we have a photon bouncing back and forth between two mirrors, a hundred million times per second. But now we’ll set up the same contraption on a moving train carriage. Let’s say I’m on board the train with the clock and you’re standing on the platform watching the carriage go by. What are you going to see?”
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“Well, the photon is going to keep moving at the speed of light. Except now it’s not just moving up and down, it’s got to have some horizontal velocity to keep pace with the train. If its overall speed is still the same speed of light, and it has a horizontal velocity component, then that means that its vertical velocity component is a bit smaller. So it takes longer to complete each bounce. So the clock will tick slower.”
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“Yup. A moving light clock will run slower. But, if you think about it, won’t we have violated the Galilean principle that there is no physical experiment that can distinguish two frames in relative motion? Won’t I, on the train, now know for sure that I am in fact the one in motion, because my light clock is running slower?”
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“Um…no?”
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“And why not?”
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“...Because your time will be running commensurately slower. Not just the tick-tocking of your light clock, but the speed of everything on board the carriage. The speed at which you move and breath and think, everything. Everything that defines the passage of time for you. You’ll be operating in slow motion relative to me. So by your standards, you’ll still think the clock is operating at the correct speed.”
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“Precisely. Time for moving observers really does run slower. It has to for all observers to measure the same value for the speed of light. In real life, of course, trains do not move anywhere near fast enough for these effects to be detectable by most of the relatively crude measuring equipment that we use in everyday life. But it is real. Experiments have been done with very precise atomic clocks that can measure to the nanosecond, in which one is loaded on board an aircraft and another is left in the laboratory back on the ground. The two clocks begin the experiment in perfect synchronization. Then one is flown around the world at high speed and when it returns the two are compared and it always turns out the clock that flew around the world has ‘experienced’ slightly less time than the one that stayed behind, by exactly the amount predicted by the theory. And in particle physics experiments certain particles that normally decay after a predictable amount of time can have their ‘lifetimes’ extended by accelerating them to high speeds.”
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“And if I had a spaceship that could travel at super high speeds, arbitrarily close to the speed of light, then time on board could be slowed right down so that I might only experience hours on board while back on Earth years passed by.”
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“Exactly. The galaxy might be a hundred thousand years across, but if you could travel close enough to the speed of light, you could slow the passage of time on board the spacecraft right down, enabling you to get across within your lifetime, even though a hundred thousand years would elapse back on Earth.”