Now let’s think about our explanations of storms and tides. These involve situations in which Newton’s laws are not the same when we hop into a different frame of reference. In a rotating frame, a rolling ball curves and doesn’t continue in a straight line due to the appearance of the Coriolis Force. We account for this by changing Newton’s second law in the rotating reference frame. It doesn’t look like F = ma any more. It changes into F + Fcor = ma, where Fcor is the Coriolis Force. Similarly, when we think about the origin of the tides, we jumped into the rotating reference frame of the Earth orbiting around the centre of mass of the Earth-Moon system, and saw that F = ma changes into F + Fcen = ma, where Fcen is the Centrifugal Force. In both cases, Newton’s laws do not look the same in the rotating frames because extra ‘fictitious forces’ appear. We can say that Newton’s laws are not invariant when we transform from an inertial reference frame into a rotating reference frame. Incidentally, the Centrifugal Force and the Coriolis Force are always both present, but for the tides the Coriolis Force isn’t important, whilst for cyclones and anti-cyclones, the Centrifugal Force isn’t important.
What do we mean when we speak of symmetries of the laws of Nature, and why do Nobel Prize-winning physicists consider those symmetries so fundamental?
We’ve taken quite a bit of time to discuss these ideas because they are absolutely central to modern physics – and to understanding why Einstein wrote down his theory of relativity.
Einstein was the first to take a very important and, at first sight, rather odd fact seriously. Unlike Newton’s laws, the laws of electricity and magnetism are not invariant under Galilean Transformations. They do not look the same in all inertial frames of reference if you change all the speeds in the way you do for Newton’s laws to account for the shift in perspective. This means that Newton’s laws and the laws of electricity and magnetism are not consistent with each other! This was the situation that Einstein faced in 1905.
This engraving shows a scientist, thought to be James Clerk Maxwell, investigating magnetism, light, and molecular spinning tops.
The reason why the laws of electricity and magnetism cause a problem is a simple one, but a little bit of history is in order first. During the nineteenth century, the exploration of electricity and magnetism was at the cutting edge of physics. The names of many of the scientists are remembered in the language we use to speak about electricity today: André-Marie Ampère gives his name to the Amp, the unit of electric current, and the Volt is named after Alessandro Volta. The greatest experimental breakthrough came during 1831 and 1832 when, in a series of experiments at the Royal Institution and Royal Society in London, Michael Faraday discovered electromagnetic induction, and in doing so invented the electric generator and laid the foundations for the modern world.
During the 1860s, the Scottish physicist James Clerk Maxwell discovered a unified theoretical description of all electrical and magnetic phenomena. Maxwell’s equations are one of the great achievements of the human mind. Einstein later described Maxwell’s work as ‘the most profound and the most fruitful that physics has experienced since the time of Newton.’ The equations are so beautiful that I can’t resist showing them to you. To hell with those who think equations reduce the number of sales of popular science books. Here they are:
The Es and Bs stand for electric and magnetic fields, the basic building blocks of Maxwell’s description of electric and magnetic phenomena. Written in this notation, there are only two other letters in the equations: t stands for time and c stands for the speed of light. This is the key that unlocked the door for Einstein. The speed of light enters Maxwell’s equations as a constant – a fundamental number that does not change. It is one of the axioms – the building blocks of our universe. It is a speed upon which everyone agrees, irrespective of which frame of reference they are in. This is shocking, and looks like a disaster for physics. How can it make any sense that everyone agrees on the speed of light, irrespective of what frame of reference they are in? Recall our example of jumping between different reference frames and observing a rolling ball. All we had to do was add all the speeds together in the intuitive way encoded into the Galilean Transformations and all is well. Maxwell demolishes this idea.
Imagine someone holding a torch. Light streams out of the torch at the speed of light: 299,792,458 metres per second. Now imagine someone else looks at the situation from a different inertial frame of reference, flying towards the torch at half the speed of light. We might expect that we will be able to describe everything in either frame as long as we add the speeds, in accord with the Galilean Transformations. The person flying towards the torch would conclude that the light whizzes past them at 450,000,000 metres per second – which is c + ½ c. Maxwell’s equations demand that this is not the case. They say that both observers measure the speed of light to be precisely equal to 299,792,458 metres per second. The speed of light doesn’t change, irrespective of how you look at it. It is a constant – a fundamental property of Nature.
If this sounds weird, it is. I have no way of explaining why, other than to say that our universe is constructed like this. Maxwell’s equations are correct. The statement that the speed of light is a constant in all inertial frames of reference is on the same footing as the principle of inertia. It is because it is.
Einstein’s brilliance – let us call it genius – was to take Maxwell’s equations at face value and insist that when we hop between inertial frames of reference we keep the speed of light the same. We are not allowed to add velocities in the way that we have been doing; it is simply wrong. The Galilean Transformations are wrong, and therefore Newton’s laws, which possess the symmetry represented by the Galilean Transformations, are also wrong.
Maxwell’s equations are one of the great achievements of the human mind. Einstein later described Maxwell’s work as ‘the most profound and the most fruitful that physics has experienced since the time of Newton.’
The High Redshift Galaxy Cluster has been a source of some confusion for astronomers. It is one of a number of objects termed ‘giant galactic blobs’.
The High Redshift Galaxy Cluster has been a source of some confusion for astronomers. It is one of a number of objects termed ‘giant galactic blobs’.
Somewhere in spacetime
We can now bring all these ideas together. Einstein rebuilt physics from the ground up by insisting on two axioms, which are known as Einstein’s postulates. The first is one with which we are very familiar indeed.
The laws of physics are the same in all inertial frames of reference.
The second postulate is the one that comes from taking Maxwell’s equations at face value:
The speed of light in a vacuum is the same in all inertial frames of reference.
If we were writing a physics textbook, we’d now proceed to derive all the consequences of these two postulates, and in the process discover treasures such as E = mc2 – the statement that mass and energy are interchangeable. This isn’t a textbook. Here, we want to explore a very particular consequence of Einstein’s two postulates: the idea that space and time are not what they seem.
Claude Monet’s Poppy Field (Giverny). This painting reveals a moment in time – an event – which can help us explore Einstein’s theory that space and time are not what they seem.
Let’s return to the beginning; the moment at which Monet sat down in a field of poppies just outside Argenteuil and, lungs filled with the scents of a late-spring afternoon, dabbed a delicate spot of red paint onto his canvas. The position in space and time of the dab of red paint is known in the language of relativity as an event. Because we live in three-dimensional space, we need three numbers to describe the position of the painted poppy on the canvas. These numbers could be the latitude and longitude of the easel in the poppy field and the height of the canvas above sea level. These three numbers specify where the event happened. We also need a time and date to specify when it happened; noon on 26 May 1873. An event in space and time has four coo
rdinates; three to specify its position in space, and one to specify its position in time.
As the light fades, Monet slips the half-finished canvas under his arm, walks back to his room in the village and closes his door. The click of the lock marks another event, with a different latitude, longitude and height above sea level and a different time; by his watch.
Now consider another event. As the light fades, Monet slips the half-finished canvas under his arm, walks back to his room in the village and closes his door. The click of the lock marks another event, with a different latitude, longitude and height above sea level and a different time, by his watch. It’s now 8pm on 26 May 1873.
Let’s imagine that Monet decided to measure the distance between his easel and his door and found it to be precisely 2 kilometres, and that they are at the same height above sea level. This is the distance in space between the two events. The difference in time is 8 hours, by Monet’s watch.
Newton, and everyone else before Einstein came along, would agree with the common-sense notion that any observer who decided to measure the distance between Monet’s easel and door and the time between the dab of paint and the click of the lock would be in complete agreement with Monet, assuming that their rulers and watches were accurate and synchronised. Einstein discovered that, if he imposed his two postulates, this is not the case. Different observers do not agree on the spatial distance and temporal difference between events. Let’s be specific. Imagine that an enterprising French lady with access to a futuristic aircraft was flying past Monet on 26 May 1873 at half the speed of light. She would measure the time difference between Monet’s dot on the canvas and the click of his lock to be 9 hours and 14 minutes and the distance between the easel in the poppy field and his door to be 1.73 kilometres. This discrepancy has nothing to do with the way time and distance are measured, or the measuring devices used. Furthermore, neither Monet nor the aviator is wrong; each is absolutely entitled to claim that their measurements are correct. Rather, Einstein discovered that in reality there is no such thing as absolute time and no such thing as absolute space. Let’s repeat this, because it’s very odd. From the point of view of the aviator, Monet’s time passes more slowly than hers, which means that Monet ages more slowly than she does, and Monet really does walk 1.73 kilometres on his way home. The converse is also true. If Monet glanced up and saw the aircraft fly by, he would see the aviator’s clock ticking more slowly than his, and he would conclude that she was ageing more slowly than he. He would also conclude that her aircraft is 0.866 times shorter than it appears to her. Arguably he wouldn’t have continued to paint a poppy field had this really happened, but the point is that this is not theoretical; the effect is real. Nature really is constructed this way. The slowing down of moving clocks is known as time dilation, and the shrinking of moving objects is called Lorentz Contraction.
If you are comfortable with a bit of mathematics, you’ll find a derivation of the result that Monet’s clock runs slow as viewed by the aviator, and by how much, on here. The conclusion follows directly from Einstein’s two postulates, and the argument is quite simple and requires no mathematics beyond Pythagoras’s theorem. If you’re happy to take our word for it without reference to here, then accept it and read on!
The reason why Einstein’s theory predicts that distances in space and intervals of time are not the same in different frames of reference are his two postulates – the requirement that the laws of Nature take the same form in all inertial frames of reference and that the speed of light is constant in all inertial frames of reference. These two postulates imply that moving clocks run slow, as proved on here. In more precise language, Einstein had to replace the Galilean Transformations, which tell us how to switch between different inertial frames, with a new set of equations called the Lorentz Transformations. Lorentz Transformations leave the speed of light the same, as required by the second postulate, but there is an apparently terrible price to pay: Distances in space and intervals in time do change under Lorentz Transformations: moving rulers shrink and moving clocks run slow!
So where does this leave us? We’ve discovered that space and time are not as they seem, if we accept that the speed of light must remain constant for all observers. There is no such thing as absolute space, because observers moving at different speeds relative to each other disagree on the distance between events. The comfortable picture of the Universe as a big box, where every star, planet and galaxy has a well-defined place, cannot be right, because the distances between the stars, planets and galaxies cannot be defined in a unique way. Similarly, there is no such thing as absolute time, because it is not possible to define the time between events in a unique way.
This is fun, and strange, but it also presents a serious problem for physics. The problem lies in Einstein’s first postulate: the laws of physics are the same in all inertial frames of reference. The laws of physics are the tools that we use to predict the outcome of real-world experiments; they are descriptions of Nature. If they are to be the same in all inertial frames, then it follows that they should be constructed out of quantities that are the same in all inertial frames. But the laws of physics we learn at school concern distances measured by rulers and times measured on clocks. Think about Newton’s second law of motion, F = ma, which describes how fast an object of mass, m, accelerates in response to a force, F. Acceleration is measured in metres per second squared – a quantity that involves changes in distance over some time interval. But since we’ve discovered that distance intervals and time intervals are not the same in all reference frames, it follows that Newton’s laws are not the same either! This looks like a disaster.
It isn’t, fortunately, because Einstein found a way out. He discovered that, whilst the distance in space between two events and the difference in time between two events each change, there is a quantity that does not change if we switch perspective between inertial frames: the distance in space and time, taken together in a very special way.
If we call the distance between Monet’s easel and door Δx and the time difference between the dab of red paint and the click of the lock Δt, then the ‘distance’ Δs2 = c2Δt2 - Δx2 does not change. Both the aviator and Monet agree on Δs, even though they disagree on Δt and Δx. The quantity Δs is known as the distance in spacetime between the two events. The speed of light, c, has entered the equation in a rather subtle way, multiplying the time difference Δt. Why? One thing we can say immediately is that some speed or other had to be there to make the definition of the distance in spacetime sensible. Let’s say we chose to measure time differences Δt in seconds and distances between events Δx in metres. We can’t simply subtract something in seconds from something in metres – that’s like subtracting five apples from ten oranges. But if we multiply the time difference in seconds by a speed, which is measured in metres divided by seconds, then we get the object c Δt, which is measured in metres, and we can happily go ahead and subtract Δx from it. That argument doesn’t inform us what value c should take, but it does tell us that it has to be some speed or other.
The world’s only supersonic passenger airliner, the Concorde, is shown here in the maintenance hangar at Heathrow Airport. The swept-back delta wings are designed to maximise lift generation during take-off and to minimise drag at high speeds. The wing design also provides sufficient stability that horizontal stabilisers are not needed on the tail, unlike conventional passenger aircraft. Concorde made its first commercial flight in 1976 and retired from service in 2003.
In an undergraduate lecture course on physics, we would now proceed to consider how energy and momentum are treated in special relativity and show that this special speed can be interpreted as the speed of massless particles. Coincidently, as far as we know, photons happen to be massless and therefore travel at the special speed c – and this is why we call it the speed of light.
The fact that distances between events in spacetime are agreed upon by everyone suggests that we should rebuild our laws of Nature out of qu
antities like Δs, and this is precisely what Einstein did, replacing Newton’s laws and quantities familiar to physicists such as energy and momentum with spacetime versions.
The fact that distances between events in spacetime are agreed upon by everyone suggests that we should rebuild our laws of Nature out of quantities like Δs, and this is precisely what Einstein did, replacing Newton’s laws and quantities familiar to physicists such as energy and momentum with spacetime versions. This is where E = mc2 comes from. I think it’s quite satisfying that Nature is constructed in this way. Events, after all, form the narrative of our lives. We don’t separate our memories into separate spatial and temporal components. I remember a perfect summer’s day in August 1972 when a yellow sun lifted the scents of the lawn and Doppler-shifted bees drowned out the hum of the town. We set up a paddling pool in my parents’ garden and played in the water so long we chafed our thighs. I remember this as an event, not a moment with a separate latitude, longitude and time stamp.
There is a vivid way of visualising these ideas known as a spacetime diagram. In order to draw it, we can represent the position of events in space along the horizontal axis and the position of events in time on the vertical axis, as shown in the diagram opposite. We’ve neglected two spatial dimensions here for clarity because they don’t matter for our argument, and it’s hard to draw a four-dimensional diagram on a piece of paper. Let’s draw my life as a spacetime diagram. It’s important to define precisely what frame of reference we’re in when we draw a spacetime diagram. In this case, Oldham Royal Infirmary, where I was born on 3 March 1968, will be our frame of reference (which we’ll assume to be an inertial frame). This means that we set up a grid of Oldham rulers and Oldham stopwatches at rest relative to OldhamRoyal Infirmary. We agree to zero the Oldham stopwatches at the moment of my birth, and because I was born inside Oldham Royal Infirmary, the co-ordinates of my birth event are x = 0, ct = 0, where x is the distance from Oldham Royal Infirmary and t is the time as measured by the Oldham stopwatch at position x = 0. We’ll label this event ‘3 March 1968’, and it sits at the origin of the spacetime diagram.
Forces of Nature Page 12
