A bit of maths and physics

Tag: general relativity

  • The meaning of E=mc²

    The meaning of E=mc²

    Probably the most famous equation on this planet is $E = mc^2$. Energy equals mass times the speed of light squared(beginfootnote)Usually, in mathematics, we leave out the multiplication sign ($\times$).(endfootnote). Usually, the formula is associated with Albert Einstein. This relationship between, energy, mass, and the speed of light, this equation, has a name: the mass-energy equivalence. Perhaps you’ve read or heard people explain that ‘Einstein taught us’, that mass is a form of energy, mass is frozen energy, mass can be converted into energy, and that, in the end, all matter is essentially pure energy.

    The equation is, however, definitely not about all of that. At least, not in this Universe. Here, we will discuss the actual meaning of it. We think it’s time for disposing of some of the unnecessary obscurantism accompanying many popular explanations. We think the involved mathematicians and physicists of yore deserve better.

    Albert Einstein (right) with Dutch physicist Paul Ehrenfest (left) and Ehrenfest’s son in Ehrenfest’s home in Leiden, The Netherlands.

    Standing on the shoulders of giants

    Einstein wasn’t the first to write down this very relationship between mass and energy. There were many others before him who, one way or the other, explored the connection between mass, energy, and velocity. However, most hypothesised that a mechanical mass increase was exclusively due to interactions with electromagnetic fields. They called it electromagnetic self-energy of some kind, giving rise to a form of electromagnetic mass (Miller, 1981; Okun, 1989). Einstein then showed that there was no need for such a concept and was the first to derive the relation correctly.

    However, over the course of a few years, he published several derivations of the equation, none of which were literally written down as $E = mc^2$. You might not recognise them if you saw them. Furthermore, the versions he did write down, weren’t universally true.

    Lastly, the famous equation isn’t the complete version. Usually, people only know the snazzy edition fitting on baseball caps. The full equation is valid in a more universal way and sometimes referred to by contemporary physicists as the ‘correct version’(beginfootnote)cf. https://youtu.be/mkiCPMjpysc(endfootnote). However, this one wasn’t first formulated by Einstein but by Paul Dirac (Eisberg & Resnick, 1974; Miller, 1981). We will get back to that.

    Among others, these incredible minds have all derived and used some version of the famous equation before Einstein.

    Energy is a mathematical idea

    We should remind ourselves that energy isn’t any ‘thing’. As we mentioned in a previous article, it isn’t some invisible, immaterial, fluid-like ‘essence’ which everything is made of, within and behind the façade of the tangible world. Fork, no.

    It has always been just a number, an accounting tool, an important and practical, mathematical concept, first proposed by the 17th century, German scholar Gottfried Wilhelm von Leibniz. How is it a mathematical concept? It’s the number you get when you multiply an object’s mass with its velocity squared(beginfootnote)Which was later calibrated to 1/2 times the mass times velocity squared.(endfootnote). It’s a pragmatic way of keeping the books on these two things.

    Suppose, we have a billiard table with three billiard balls. We ignore any sort of friction. Imagine this isolated system of balls changing internally: the balls constantly collide and bounce off of the edge of the table. He assumed that what doesn’t change, is their mass. What does change, are their velocities. Leibniz then noticed that if you multiply for each ball its mass with its velocity squared, and summed all three products, that sum remained constant, irrespective of how the balls were bouncing in which direction, and how fast, at each point in time (until you changed something to the system by introducing a whack by a cue stick, for example).

    A sketch of a carom billiards table. First panel: three balls on the billiards table have different velocities. Second panel: the three balls have bounced and moved and have, again, different velocities. The sum of the products of the mass of the balls with its velocity squared, of all balls, is constant, however.

    This was the early formulation of what we now fanciful call the law of conservation of energy(beginfootnote)He called this product quantity vis viva, so, not even energy yet. Perhaps he was being poetic. Furthermore, Leibniz also had some fierce competition: his rival Newton had come up with a different quantity, a different way of keeping the books. He stated that the sum of the products of mass and velocity — just velocity, not velocity squared — remained constant. This is the law of conservation of momentum. Later, it was understood that the two laws were complimentary, not contradictory.(endfootnote), which, by the way, is true in a small enough patch of the Universe, such as Earth or the solar system, but in the context of the entire observable Universe, for instance, energy is not conserved. So, indeed, the law of conservation of energy is fundamental enough for us, Earthlings, and our physics experiments, however, contrary to what people usually think, in the grander scheme of the Universe, it’s not(beginfootnote)Courtesy of the genius Emmy Noether (Noether’s theorem) and Albert Einstein (general relativity).(endfootnote).

    For the purpose of this post, however, this is irrelevant. What is important to note, is that since then, with their propensity to invent intricate systems of categorisation, humans have made distinctions between various forms of energy. Think of potential energy (something’s high up and can fall down or wound up and unwind rapidly), thermal energy (something’s hot), chemical energy (something’s ‘charged’), and kinetic energy (something moves). All of these, including their mother-concept ‘energy’, are human constructs. Thinking of any of these nouns as referring to physically separate, physical, tangible things is, ironically, a category mistake.

    Energy isn’t an ephemeral and/or ethereal substance. It’s a mathematical measure for the product of mass or momentum, and speed. In this hastily taken photograph by an eyewitness, it’s not beams of pure energy that you’re seeing, even though this does appeal more to our imagination of what ‘pure energy’ is supposed to be. They are particle rays (orange-coloured bundles of radially polarised protons), emitted by portable particle accelerators on their backs, aimed at a ghost. The Ghostbusters, as they call themselves, stated that ghosts are negatively charged energy in the form of slime-like ectoplasm. So, even here — fictional or not — when there’s something strange in your neighbourhood, it’s never ‘pure energy’. (© Sony Pictures Home Entertainment)

    It’s not all about that mass

    Sometimes, people misinterpreted Albert Einstein. In the old days, again, being the talented labellers that they are, humans split up the term ‘mass’ into rest mass and relativistic mass. Rest mass is the mass when the object is at rest. Relativistic mass is the mass when the object is in motion. If then the object would start to move faster and faster, then this particular mass would become larger and larger, because, you know, that’s what he said.

    Well, no. He wrote in his third 1905 paper (1905a, p. 920), Zur Elektrodynamik bewegter Körper, an equation for the kinetic energy of an electron, which went as follows:

    While it may not look like it, you could say that this was his first expression for the relationship between (kinetic) energy, mass, and speed of light squared(beginfootnote)Incidentally, Max Abraham had published Walter Kaufman’s work just before Einstein, showing the same equation for kinetic energy. Einstein probably wasn’t aware of this (Miller, 1981).(endfootnote). It does require a little translation, but it’s easy. Ignore the part in the middle, focus on the letter $W$, and the part behind the last equals sign. You should know that in modern notation, kinetic energy $W=E_k$, rest mass $mu=m_0$, and speed of light $V=c$. So, what it says is:

    \begin{equation} E_k = \frac{m_0c^2}{\sqrt{1-\dfrac{v^2}{c^2}}} – m_0c^2. \end{equation}

    So, this is slowly starting to resemble our familiar $E=mc^2$. It doesn’t state, however that mass increases. It only says that when the speed of the object $v$ approaches the speed of light $c$, the result of this whole equation is infinity – infinite energy.

    As the standing interpretation is that mass equals energy (mass-energy equivalence), people nevertheless concluded that if the energy of a moving object becomes infinite at the speed of light, that an object’s mass becomes infinite, or relativistic mass, to be precise (in the minds of the old folk).

    This, however, is something of the past – well, technically. As soon as 1940, the great Lev Landau and Evgeny Lifshitz ignored the distinction between rest mass and relativistic mass in their book The Classical Theory of Fields. The legendary John A. Wheeler and Edwin F. Taylor also brought an up-to-date Spacetime Physics to the reading table. Unfortunately, many textbooks today still mention archaic notions, terms, and notation.

    Contemporary professional physicists don’t speak of relativistic mass anymore. In special relativity, Einstein showed that observations and measurements depend on one’s frame of reference. By definition, there are at least a couple of things that do not depend on the motion of observers. Besides the spacetime interval, the laws of physics, and the speed of light, this turns out to be rest mass.

    An object’s rest mass is invariant, i.e. it doesn’t vary or change, regardless of the motion of the observer relative to the object (Taylor & Wheeler, 1992, p. 211). And, as you can see in Einstein’s equation for the kinetic energy, only rest mass is used. No mentioning of relativistic mass whatsoever. In fact, Einstein himself wrote (as cited in Okun, 1989, p. 32):

    ‘It is not good to introduce the concept of mass $M = m/(\sqrt{1-v^2/c^2})$ of a body for which no clear definition can be given. It is better to introduce no other mass concept than the ‘rest mass’ $m$. Instead of introducing $M$ it is better to mention the expression for momentum and energy of a body in motion.’

    Of course, $M$ is what humans would later call ‘relativistic mass’, which means that they did it anyway, against Einstein’s wishes.

    Today, however – well, at least since 1940 – professional physicists speak only of mass. We tossed out relativistic mass as, with Einstein, it’s ‘not good’. Furthermore, the adjective ‘rest’ in rest mass is redundant. Mass is about an object at rest. If it’s in motion, we speak of the product of some proportion of mass and velocity: either momentum or energy. An object’s mass does not grow by its motion.

    Resistance is crucial 

    So, what is mass then? Well, just to be clear, mass isn’t weight: the same amount of mass has different weight on different planets. We were taught this in high school. A spring scale measures weight, not mass. A balance with calibrated counter-weights – *cough* masses – is your best option during interplanetary travels. These masses should have been calibrated to the definition of a kilogram according to the International Bureau of Weights and Measures.

    Mass isn’t matter either. Mass and matter are different categories. The first is a property, the second is a ‘thing’. Elementary ‘particles’, things(beginfootnote)We use quotations marks in ‘particles’ because, while it’s easier to use that term, we acknowledge it’s actually quantum field oscillators we should be talking about or, even prettier, wave functions.(endfootnote), such as the electron, have a certain amount of mass. An electron gains its mass through interacting with the Higgs field, the existence of which was proved in 2012 at CERN. And the amount of elementary particles does correlate with the amount of mass. However, the mass of an object isn’t defined by merely the amount of elementary particles: it’s also the motion of gluons inside protons, the motion of electrons, atoms, molecules – the kinetic, thermal, and chemical energy contained within the (resting) object.

    Einstein wrote this in his fourth paper of 1905, Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig? (1905b, p. 641):

    If a body releases the energy L in the form of radiation, its mass decreases by $L/V^2$,

    where, in modern notation, $L=E$, and $V=c$. Also, the energy he’s talking about is not the kinetic energy (of a body in motion) but the internal energy (of a body at rest), such as thermal energy. And yes, this does mean that mass increases or decreases depending on the object’s internal energy.

    Two structurally identical balls of steel have different mass if one ball is hotter (more mass) than the other due to their diverging thermal energy content. Two structurally identical mobile phones have different mass if one is charged (more mass) and the other is out of juice (electrochemical energy). Note, we are talking about the whole object being at rest in its reference frame.

    As soon as either object starts radiating light or heat, they lose mass. The fraction $E/c^2$, however, is very small because the speed of light is very high. And so, the extra mass gained or lost is so small that this may be a reason for people confusing mass with the amount of matter. It’s almost the same. It’s the amount of matter plus something more, its internal energy.

    Mass could be best described by the resistance to acceleration – a ratio between the force needed to accelerate it to the extent it’s accelerating. Mass is inertial mass, an object’s inertia (as Einstein put it in the title of his 1905a paper).

    The complete equation

    If you’re not into equations any longer, skip this section. If you want to know the real equation, let’s go.

    Einstein published several derivations. One of the more familiar was the following for the total energy:

    \begin{equation} E_T = \frac{m_0c^2}{\sqrt{1-\dfrac{v^2}{c^2}}}. \end{equation}

    If you just to happen to be fluent in algebra, then you could see how we obtain $E=mc^2$. If not, not to worry. If an object is at rest, is has no speed, so $v=0$. If you would fill in that number in the equation, the denominator of the big fraction becomes the value 1. And anything divided by 1 equals exactly that same anything. So, that means that what you get is $E=m_0c^2$.

    Of course, since, nowadays, there is only one mass, which is $m$, since ‘rest’ is redundant, we should really leave out the subscript 0. This also means that the famous equation $E=mc^2$ is only applicable if the object isn’t moving in our (inertial) reference frame. Moreover, it’s not applicable to phenomena without mass either, such as a photon. Hence,

    $E=mc^2$ is not universally true. Only in an inertial reference frame, where the object isn’t moving, and only in the case of ‘particles’ with mass, does this equation hold, so, this equation isn’t valid for photons and gluons.

    This following equation, however, does hold for massless as well as massive particles, and while Einstein laid the groundwork, the genius Paul Dirac was to write this down for the first time in 1928 (Eisberg & Resnick, 1974; Miller, 1981), albeit in a slightly more technical fashion than presented here. The following equation handles all objects, including light:

    \begin{equation} E^2 = m^2c^4 + p^2c^2. \end{equation}

    The letter $p$ is the momentum. Suppose, we want to calculate the energy of a photon. Since the photon has no mass ($m=0$), this equation becomes $E = pc$, which is, indeed, the correct relation between energy and a photon. If you would try to use $E = mc^2$ to calculate the energy of a photon, you would get a silly answer.

    So, $E = mc^2$ isn’t even a universal equation because it doesn’t fly for massless ‘particles’: photons and gluons. The equation first written down by Paul Dirac does, however. And it still fits on a T-shirt. 

    Often, though, it’s written as

    \begin{equation} E^2 = (pc)^2 + (mc^2)^2, \end{equation}

    which makes it possibly even snazzier as it shows a beautiful Pythagorean relationship triangle.

    Paul Dirac

    The meaning of E = mc²

    All well and good, but, technicalities aside, what does it mean?

    What it means is that energy is mass, proportioned by a factor of $c^2$.

    What it also means is that an object’s mass is a measure of its total intrinsic energy (potential, thermal, chemical, electrical, even kinetic, if parts inside the object have motion) proportioned by a factor of $1/c^2$.

    $E = mc^2$ should actually be written $E_0 = mc^2$ as it’s about the energy of an object at rest and the subscript 0 usually denotes something at rest.

    However, it isn’t the full equation.

    What it doesn’t mean is that energy is matter, and, conversely, it doesn’t also mean that matter is energy. Mass isn’t matter. This is a category mistake.

    It also doesn’t mean that mass can be converted into energy or vice versa. For one, mass cannot be converted as it isn’t a ‘thing’. Secondly, energy isn’t a ‘thing’ either. Mass is a property, a measurable property. Energy is also a property, a calculable property, which can be done by measuring mass.

    Imagine an object had the following properties: size, colour, hardness, and energy. Suppose, the equation would have said $E =$ hardness $\times c^2$. Perhaps it’s more clear now that this doesn’t mean that hardness gets converted into energy. What it means, is that you have a mathematical way of calculating one measure in terms of the other measure. The only thing that’s being converted here, is a number, a quantity.

    Matter is a different beast. It’s a clump of things: ‘particles’. An object is a clump of matter and matter interactions. As CERN show on a daily basis, matter in motion can be converted into a thousand other things in motion. If people insist on talking about things getting converted, then they could talk about converting particle A with motion $a$ and particle B with motion $b$ into particles C, D, E, F, G, $\dots$ with motions $c,d,e,f,g,\dots$

    So, next time someone thinks they should explain to you that $E = mc^2$ means that mass can be converted into energy or that no object can gain the speed of light because its mass would become infinite, you can just reply with, ‘Nah, mate, mass is an invariant property of an object, calculable through the complete equation, you know, $E^2 = (pc)^2 + (mc^2)^2$. Although you would have to solve for $m$ and merely use the pseudo-Euclidean norm for momentum, not the whole four-vector, but that shouldn’t be a problem – it makes it easier.’ Then pause, and add, ‘In Minkowski space, obviously.’(beginfootnote)Minkowsi space is like Euclidean space but in four dimensions. This might be a good time to add the footnote that there is an even more fundamental equation, which is Einstein’s field equation of general relativity (Carroll, 2014), but that’s something to discuss at a later point in time.(endfootnote)

    Also, pure energy = pure nonsense. If Leibniz were somehow able to hear this, he would cackle and turn over in his grave – if he could muster the energy for it. To be fair, physicists use the word energy all the time, all over the place. It’s short, sweet, and simple to use on a daily basis, which is fine, just as long as we’re all in agreement about what we mean.

    Energy isn’t fundamental to motion, it’s motion(beginfootnote)Of quantum fields as described by wavefunctions(endfootnote) and interactions giving rise to the construct of energy.

    However, if you do bump into a floating blob of pure energy down the long narrow hall upstairs of your rich aunt’s mansion, then, well, yes, that would most certainly be something strange.

    References

    Carroll, S. (2014) Spacetime and Geometry: Pearson New International Edition : an Introduction to General Relativity. 1st. Pearson.

    Einstein, A. (1905a) ‘Zur Elektrodynamik bewegter Körper’, Annalen der Physik, 322(10), pp. 891-921.

    Einstein, A. (1905b) ‘Ist die Trägheit eines Körpers von seinem Energieinhalt abhängig?’, Annalen der Physik, 323(13), pp. 639-641.

    Eisberg, R. M. and Resnick, R. (1974) Quantum physics of atoms, molecules, solids, nuclei, and particles. New York: Wiley.

    Miller, A. I. (1981) Albert Einstein’s special theory of relativity : emergence (1905) and early interpretation (1905-1911). Reading, Mass ;: Addison-Wesley.

    Okun, L. B. (1989) ‘The Concept of Mass’, Physics Today, 42(6), pp. 31-36.

    Taylor, E. F. and Wheeler, J. A. (1992) Spacetime physics : introduction to special relativity. 2nd ed. edn. New York: W.H. Freeman.

    Featured image: NASA’s Solar Dynamics Observatory captured this image of an X2.0-class solar flare bursting off the lower right side of the sun on Oct. 27, 2014. The image shows a blend of extreme ultraviolet light with wavelengths of 131 and 171 Angstroms. Credit: NASA/SDO. Retrieved 30 Aug 2019, from https://www.nasa.gov/content/goddard/sun-release-x20-class-flare-on-oct-27-2014

    Einstein and Ehrenfest. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 21 Aug 2019, from https://quest.eb.com/search/132_1510430/1/132_1510430/cite

    Oliver Heaviside (1850-1925) – Science and Society Museum/ Universal Images Group. Oliver Heaviside, English physicist, c 1900.. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_541915/1/102_541915/cite

    Hendrik Lorentz (1853-1928) – Science and Society Museum/ Universal Images Group. Hendrik Antoon Lorentz, Dutch physicist, c 1920.. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_523268/1/102_523268/cite

    Henri Poincaré (1954-1912) – akg-images / Universal Images Group. Henri Poincare / Photo c. 1890. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 3 Sep 2019, from https://quest.eb.com/search/109_171035/1/109_171035/cite

    Joseph J. Thomson (1856-1940) – Science and Society Museum/ Universal Images Group. Sir Joseph J. Thomson, English physicist, late 19th century/early 20th century.. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/102_547694/1/102_547694/cite. Cropped by @kjrunia.

    George Frederick Charles Searl FRS(1864-1954) – Royal Society. As printed in Thomson, G. (1955) ‘George Frederick Charles Searle. 1864-1954’, Biographical Memoirs of Fellows of the Royal Society,1, p. 247. Cropped by @kjrunia.

    Wilhelm Wien (1864-1928) – NATIONAL LIBRARY OF CONGRESS / SCIENCE PHOTO LIBRARY / Universal Images Group. Wilhelm Wien, German physicist. [Photography]. Encyclopædia Britannica ImageQuest. Retrieved 1 Sep 2019, from https://quest.eb.com/search/132_1255736/1/132_1255736/cite

    Max Abraham (1875-1922) – Niedersächsische Staats- und Universitätsbibliothek, Göttingen. Max Abraham around 1905. Public domain. Slightly cropped by @kjrunia.

    Albert Einstein, Swiss-German physicist. [Photograph]. Encyclopædia Britannica ImageQuest. Retrieved 22 Aug 2019, from https://quest.eb.com/search/132_1510416/1/132_1510416/cite

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  • Einstein’s special relativity in under 6.999 minutes for people on the move

    Einstein’s special relativity in under 6.999 minutes for people on the move

    In 1905, Albert Einstein published an article on moving bodies and electrodynamics. He noticed that Newton’s mechanics of moving bodies weren’t compatible with Maxwell’s equations of electromagnetism. In this article, he reconciled the two by modifying the first. These ideas and mathematical derivations became what we now know as Einstein’s special (theory of) relativity. In this post, we will describe some of its important bits. We start with two fundamental propositions. For the geeks, we will end with answering why special relativity is special.

    Einstein’s postulates

    His first postulate is basically that, whether or not you’re standing on a moving object, such as a ship, a train, or in your car, the same laws of physics apply. If you’re standing on a platform at the train station and you throw a ball in the air, the laws of physics ensure your ball comes down again. If you’re standing in a moving train, that ball still comes down again because the same laws of physics apply. The fact that you’re in motion doesn’t change anything to the rules of the Universe.

    Einstein’s second postulate is basically that, by extension of the first one, the speed of light is the same for everyone. He recognised that Maxwell and colleagues correctly describe light as an electromagnetic disturbance propagating according to the electromagnetic laws of physics. He also recognised that the velocity of the light source plays no role in Maxwell’s equations. And so, if the first postulate is correct, then irrespective of the velocity of an observer relative to the light source, light travels at the same speed $c$ ($c$ is about 300 000 000 m/s) and nothing can go faster.

    An image of the interior of an underground train. Passengers are sitting across each other.
    Whether you’re on the train or standing on the platform, the laws of physics are the same. The propagation of light is a law of electrodynamics not involving the velocity of its source. Hence, its velocity is the same on the train as on the platform, irrespective of its source or the train’s velocity.

    Intuition works mostly, just not really

    Suppose, you’re standing on a train station’s platform. A train passes at a speed of 30 metre per second. From inside the train, our friend throws a tennis ball out the window but in the direction of where the train is headed, at 2 metre per second, right at you. You catch it. At what speed does the ball hit your hand?

    Intuitively, you might say, that’s 30 + 2 = 32 metre per second. This way of calculating is very useful, most of the time. Instinctively, you would simply add the train’s speed and the throwing speed together. This is how Isaac Newton(beginfootnote)And Galileo Galilei before him as this is the so-called Galilean transformation.(endfootnote) would want you to do it. And, mostly, he’s not wrong. Except, well, he is kinda.

    Let’s wonder what would happen if our friend didn’t throw a ball, but, instead, switched on a pocket torch. The train still passes at 30 m/s. Light, however, flies out the torch at a speed of about 300 000 000 m/s. You lift up your hand. It ‘catches’ the light. At what speed does it hit your hand?

    You might say, that’s 30 + 300 000 000 = 300 000 030 m/s. But no. That’s wrong. Remember Einstein’s second postulate? Irrespective of the motion of the observer, light always travels at 300 000 000 m/s and nothing goes faster, full stop. So, by our simple addition, we would have invented a way for light to go faster than light! We would be Nobel Prize winners, surely. Except, it doesn’t, and we’re not.

    Something’s gotta give

    So, if the speed of light is the same to our friend, on the moving train, as it is to us, standing on the platform (do read this again and realise how bonkers this is), then how the Helheim does the Universe achieve this? After he did some relatively simple mathematics – which a student in secondary education can do – Einstein realised that something was up with metres and seconds. He proved that what is a metre to us isn’t a metre to our friend and vice versa. Furthermore, what is a second to us isn’t a second to our friend either.

    Basically, since we, on the platform, measure light to be going at 300 000 000 metre per second, and so does our friend on the train, well, that means that our understanding of what metres and seconds are is wrong.

    The point

    Here’s what’s happening. If two ‘things’, people, ‘objects’, or reference frames as physicists call them, move with respect to each other, weird things happen to space and time. Yes, the actual space and time. They are weird. We thought they were just there. Static. Always and everywhere the same. Two unchanging entities. Well, they’re not.

    In the Dutch town of Leiden, the project Leiden Wall Formulas have scattered physics equations throughout the town centre. This is the Lorentz contraction, aptly placed alongside a train track: it describes how space (in this case, a one-dimensional length) is contracted, according to Einstein’s special relativity. Click here for Google Street View.

    Suppose, we are on the train this time. We are in motion relative to our friend on the platform. Our friend then observes that space along our direction of motion becomes smaller, it contracts. They will actually measure our train to be shorter as compared to when it was standing still relative to our friend. They will also observe that our time is being stretched, i.e. our clock slows down. If one second goes by on their clock, they see only 0.9999999995… seconds have past on ours(beginfootnote)This number is a metaphor. The real time difference is too small for my calculator to show.(endfootnote).

    And to us, being on the train, it’s our friend who is travelling (backwards!) relative to us, in fact, the whole world is travelling relative to us, backwards. So, indeed, we measure the world to be shorter in the opposite direction of our motion as well as their time being slowed down.

    Twin paradox

    Now, you might say, hold on: if both of us see each other’s clock slow down, then surely, there is no difference between our clocks. However, when we ride back to our friend, stop at the platform, and compare our clock to our friend’s clock, we do see that our clock is behind. This is the so-called twin paradox. If both can state the same thing, how do their clocks still differ in the end, causing one half of the twin (on the train) to be younger than the half who stayed behind (on the platform)?

    This is due to the switching of reference frames: we were on a moving frame (the train), switched to a moving frame in the opposite direction (returning to our friend), and, lastly, switched to the platform’s frame, standing next to our friend, to compare clocks, while our friend never switched – he stayed on the platform(beginfootnote)Contrary to many popular and even introductory physics texts, this has less to do with acceleration, even though this does plays a role – without it, in the real world, one wouldn’t be able to switch reference frames. However, mathematically, acceleration isn’t necessary for solving the so-called twin paradox, switching frames is.(endfootnote). So, the situation isn’t symmetric.

    Length contraction and time dilation. They’re not illusory, they’re real. Many experiments showed that space contracts and time dilates for things in motion as observed by things with a different motion.

    Newton vs Einstein

    To summarise in a slightly more mathematical way – Newton taught us that one platform metre equals one train metre and that the same is true for seconds:

    1 platform metre = 1 train metre,
    1 platform second = 1 train second.

    Einstein, however, taught us that:

    1 platform metre $\equiv \dfrac{1\text{ train metre}}{\gamma}$,
    1 platform second $\equiv \dfrac{1\text{ train second}}{\gamma}$.

    This $\gamma$ (Greek letter gamma) is crucial. It’s a factor necessary to make sure that the speed of light doesn’t get any faster than the speed of light, even if it’s on-board a moving train. This factor is called the Lorentz factor(beginfootnote)Named after Hendrik Lorentz. The expression for his Lorentz factor is as follows: \[ \gamma = \dfrac{1}{\sqrt{1-\dfrac{v^2}{c^2}}}, \] where $v$ is the speed of the other relative to us and $c$ is the speed of light.(endfootnote).

    We never notice these things though. Usually, our speeds are way to slow for the effects of special relativity to be noticeable. Except for things that do go fast. Without Einstein’s special relativity, GPS satellites wouldn’t work(beginfootnote)Of course, we also need Einstein’s General Relativity for that, but that’s for another time.(endfootnote). Research institutes such as CERN and Fermilab also need to take special relativity in account for the high-energy, fast-flying particles.

    So, our intuition (Newton) is mostly just fine though not precisely right.

    What’s so special about special relativity

    This is a somewhat technical question, requiring a somewhat technical answer, our apologies. Contrary to what is being said in many popular science texts, special relativity isn’t really about constant speeds as opposed to general relativity dealing with acceleration. In fact, special relativity is able to deal with acceleration. It’s just that it works fine as long as we’re assuming things exist in so-called Euclidean space, adhering to Euclidean geometry(beginfootnote)Named after Euclid.(endfootnote). However, after ample deliberation, Einstein concluded through his general relativity that we’re not living in Euclidean-geometric reality at all. We’re living in a Riemannian manifold(beginfootnote)Named after Bernhard Riemann.(endfootnote). In short, Euclidean space is a special case of the more general Riemannian manifold(beginfootnote)If we’re being precise, Hermann Minkowski developed a modified version of Euclidean space which we now call Minkowski space, while Riemannian manifold should actually be named pseudo-Riemannian manifold but many physicists simply call this Riemannian manifold anyway, perhaps because they’re not mathematicians.(endfootnote). Hence, we have special relativity as opposed to general relativity.

    Featured image by StockSnap from Pixabay, modified by @kjrunia (added equations).

  • What is a spacetime interval?

    What is a spacetime interval?

    Einstein and collaborators taught us that space and time are not fixed quantities. They can stretch and contract. They vary. There is one thing, though, that does not vary. It is the invariance of the spacetime interval.

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    Spatial interval

    Suppose, a photon is emitted from origin $O$ and travels to point $F$ as depicted in Figure 1. Let us write down the expression for its distance-squared, $d(O,F)^2$, in terms of the other distances using the good old Pythagorean theorem:

    $ d(O,F)^2 = d(O,A)^2 + d(A,B)^2 + d(B,F)^2. $

    Figure 1 A photon travels from O to F in a three-dimensional space

    We can also write the previous expression in terms of their distance from $O$. We then write the following:

    \begin{align}
    F &= (x, y, z),\quad O = (0,0,0), \\
    d(O,F)^2 &= (x-0)^2 + (y-0)^2 + (z-0)^2, \\
    \therefore d(O,F)^2 &= (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2. \label{eq:distance O-F}
    \end{align}

    As the axes of the space in Figure 1 are spatial and Euclidean, $d(O,F)^2$ is called a spatial or Euclidean interval. It can also be thought of as a rectangular cuboid represented by its space diagonal $d(O,F)$, tracing out a region of 3D space.

    In the real world, to make sure we meet at the correct place, we could, for instance, give the following coordinates: 1 Einstein Drive, 2nd floor. Think of Einstein Drive as some place along the the $x$-axis (next to $x$-axis places like Battle Road, Mercer Road), number 1 as some place along the $y$-axis, and 2nd floor as some place along the $z$-axis.

    What we still need, though, is an extra bit of information: when do we meet?

    Time interval

    Suppose, Figure 2 shows a timed series of our photon on its way to point $F$ and beyond. It demonstrates that we live in a world where we do not just need three spatial coordinates, but also a time coordinate. It is only logical to not just tell the people you are supposed to meet, where in space you will be, but also when in time you will be there.

    Our photon $P$ flies through $F$ at $t=3$. This entails that the time coordinate of the event that the photon reaches $F$ is
    $ t_{F}=3. $

    Assuming that at $t=0$, photon $P$ is at the origin,

    $ t_{O}=0, $
    then we can write for the temporal interval between the photon leaving $O$ and reaching $F$:

    $ \Delta t_{OF} = t_{F} – t_{O} = 3 – 0 = 3. $

    Figure 2 A photon travels from O to F in a three-dimensional space over a period of time

    Time to distance unit conversion

    As the previous two sections showed, we need four coordinates to describe an event, for instance, the event where photon $P$ reaches $F$. The three spatial distances are measured in a unit of distance, usually, metres. The one temporal distance is not a distance in the traditional sense and is usually expressed in seconds. This makes it difficult to make sensible comparisons.

    To convert the time-units to distance-units, we multiply by a constant of nature, the speed of light $c$, which, by Einstein’s second postulate [1], happens to be invariant: no matter which frame of reference you choose, the speed of light is constant. For a longer description of this conversion, read section 4.3 of Deriving the Lorentz transformations from a rotation of frames of reference about their origin with real time Wick-rotated to imaginary time. We conclude that our time interval becomes a temporal distance:

    $ \Delta t \mapsto c\Delta t. $

    Spacetime interval

    In Figure 3, we left out the spatial $z$-axis and replaced it with the temporal $ct$-axis (which is thus time expressed in distance-units) in order to make a comprehensible drawing on a flat surface. In reality, of course, the photon still moves in the $z$-direction as well. (We have thus not yet been successful to draw a four-dimensional object on a flat surface.) Mind the unit vector diagram top right and the points of distances $\Delta x$, $\Delta y$, and $c\Delta t$. Then think, really hard, of an added fourth spatial distance $\Delta z$, somewhere.

    To calculate the spatial distance $d(O,F)$ for our photon, we look again at Equation \eqref{eq:distance O-F}:

    $ d(O,F)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2.\quad\eqref{eq:distance O-F} $

    Since we know that speed, in general, is calculated through $v = \Delta x / \Delta t$, where $x$ is the travelled distance in one direction, and $v = c$ for our photon, we can write for the travelled distance of our photon from $O$ to $F$:

    \begin{align} d(O,F) &= v\Delta t, \\ d(O,F)^2 &= (v\Delta t)^2, \\ \therefore d(O,F)^2 &= (c\Delta t)^2. \end{align}

    This is becoming interesting, since $(c\Delta t)^2$ is also (the square of) the temporal distance. If we substitute Equation \eqref{eq:distance O-F} into this last equation, we get

    $ (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 = (c\Delta t)^2. $

    If we rearrange this a little bit, we get

    \begin{equation}
    – (c\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 = 0. \label{eq:spacetime-homogeneity}
    \end{equation}

    While this may seem nice and simple, the question we should be asking is, what is zero? If we know the answer to that, we know the answer to what all the terms are on the left-hand side of the equals sign.

    In physics and mathematics, whenever something equals zero, something special is going on: it may entail a certain system in a certain configuration that is stable, static even, it may point to constant motion, an energy well, an attractor, a root, a conservation law, homogeneity, or a minimum or maximum of some kind.

    In general, it means that there is a certain kind of symmetry at play, which in turn means that something is conserved. There are beautiful, deep insights to be made as Emmy Noether showed us [2], and her genius deserves nothing less than an entire series of articles on their own.

    However, for now, let us conclude that independent of which coordinate system one uses, rendering different values for $\Delta x$, $\Delta y$, $\Delta z$, and even $\Delta t$, as we have come to learn from the Lorentz transformations, the sum of all these variances remains invariant. The zero points to the fact that irrespective of its four moving parts—no matter what frame of reference you prefer—the resultant is a constant, i.e. invariant.

    The quantity on the left-hand side has a name; it is called the spacetime interval and is denoted by $(\Delta s)^2$. The $s$ stands for ‘separation’. It is about the separation between events. If we had used the word distance, it might have had inadvertently referred too much to a spatial distance, hence, we use separation, $(\Delta s)^2$. And so, the spacetime interval is usually written:

    \begin{equation}
    (\Delta s)^2 = – (c\Delta t)^2 + (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2.
    \end{equation}

    The signs before the terms may have been flipped in some texts, but important to note is that, while time has been made comparable to space unit-wise by multiplication by $c$, you can still see that time has a special place in the interval of the fabric of the cosmos.

    Figure 3 A spacetime diagram with two spatial dimensions and one temporal dimension.

    Featured image: Klaus P. Rausch

    References
    1. A. Einstein, Zur Elektrodynamik bewegter Körper, Annalen der Physik 322(1905), no. 10, 891—921.
    2. E. Noether, Invariante Variationsprobleme, Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse 1918(1918), 235—257.