PHY197F notes

Einstein, Maxwell, and the speed of light

As we learned last week, astronomers armed with ultrasensitive modern telescopes can detect tiny motions of stars recoiling in response to planetary orbital motion. This is one method used to discover extrasolar planets. The underlying physical principle that helps the most with finding out the speed of moving sources is the Doppler effect for light, known to physicists as redshift/blueshift. This is a consequence of Einstein's Special Relativity.

Using the Doppler effect for light to detect exoplanets

Time dilation

Einstein's fundamental insight was that the speed of light is invariant -- the same in all frames of reference. This deceptively simple looking proposition is amazingly deep, in that it forces us to rethink our conventional, low-speed-based, intuition about how velocities should add and about how every observer should measure the same time. Time has to be relative because of causality: if A causes B, then B cannot also cause A. For example, if a heart attack triggered a car crash, then the car crash cannot have triggered the heart attack.

One of the most amazing things about Einstein's relativity is how he figured out that moving clocks appear to run slow, a relativistic phenomenon known as time dilation. He did it by realizing that sending light pulses between different places in space is the only sensible way to compare, and hence to synchronize, different clocks. After all, light is the most natural thing to use to send signals, because it is the fastest thing in the universe. (You could in principle also use gravitons, because they also go at the speed of light, but they interact only extremely weakly with subatomic particles and so would be much harder to work with. Highly impractical.)

We will now demonstrate how time dilation works in a very simple example which I call Rollerblader Relativity, depicted below. Suppose that you are on rollerblades, at night, and you are wearing a flashing headlight and boots that are as reflective as cateyes. In your frame of reference, a flash at the headlamp will result in your boots lighting up a tiny fraction of a second later. This happens so fast you do not even notice it, but it takes about a nanosecond. The interesting thing is to ask what happens in my frame of reference, which differs from yours because you are moving relative to me (say, to the right). What do I see? The cartoon picture helps explain.

A cartoon figure for explaining time dilation. Snapshots of rollerblader position when (a) headlight flashes and (b) flash of light reaches boot. Image by me using rollerblader image from Google.

What the picture shows is (a) a snapshot of you at the moment the light flash occurs in your headlamp and (b) another snapshot of you at the moment the light hits your boots. In your frame of reference you have not moved horizontally, and the distance the light has to travel between head and boots is about a metre or so. Its path is depicted by the black arrow. But in my frame of reference, things look different. In the Figure I have indicated by the blue arrow what the light's path would look like for me. Since you are moving to the right in my frame of reference, the distance the light has to travel to get from your head to your boot is longer -- a lot longer if you are going at a significant fraction of the speed of light. Now, since the speed of light is the same for both of us, this means that the light takes longer to reach your boot in my frame of reference because it has to go further. I measure a longer time interval between the events than you do. We are both right about what happened, and yet we get different answers for the time interval between events. This is the heart of time dilation: time is not absolute, it is relative. It is only at very low speeds compared to the speed of light that our clocks would agree.

The twin paradox

Einstein invented a number of important ideas in physics. Relativity was the one that made him world-famous. For physicists, a possibly greater discovery was his idea of a Gedankenexperiment. This is just German for thought experiment and is brilliant because it saves physicists huge amounts of time and money. After all, doing an experiment in your brain is quicker and cheaper than doing a real one. And if you think carefully, you can make all the expensive mistakes in your head and not with real equipment.

You may have heard of the Twin Paradox of special relativity. Actually it is not a paradox at all, as I would like to explain with a brief discussion. Let us imagine twin Bart Simpsons, whom we will call homebody-Bart, who stays home on Earth, and astronaut-Bart, who goes off on an adventure at relativistic speed. We are going to ask two questions: (1) Do they age the same? (2) If one twin ages faster, which one? Surprisingly, the answer to the first question is no. The space twin returns younger.

To make things as simple as possible but no simpler, we will suppose that astronaut-Bart rockets away from Earth in a straight line at constant speed, then undergoes brief constant acceleration (rockets firing) to turn around, then comes straight back at the same constant speed as for his outbound journey. Using what we already know, we can realise that, on both straight sections of his trajectory, each twin will think each other twin's clock is running slow. So how come one of them ends up younger when the astronaut comes home?

The reason is that astronaut-Bart does something distinctive: he accelerates, which is necessary for him to change from his original speed and heading in order to turn around. If he did not accelerate, he would keep going for ever and ever and never come back to Earth. This acceleration is what makes the astronaut different from the homebody in the eyes of Einstein's general theory of relativity.

It is possible to figure out the time dilation for the constant-deceleration part of the astronaut's trajectory using only special relativity and calculus. What you find is that the deceleration only adds to the time dilation effect; it does not reverse it. Also, you find that

This feature of Relativity is difficult to explain intuitively: we do not build up the mathematical technology to explain it to physics majors until their fourth year of undergrad. Suffice it to say that it always happens this way: accelerate using rockets, and you age less quickly. Give it a few years and someone will try to sell you a ride in a sub-orbital rocket ship as an anti-ageing solution. The problem for any budding entrepreneurs is that you would never go fast enough powered by human rockets for it to be humanly noticeable.

Theoretical predictions of Relativity have been tested in particle physicist laboratory experiments and found to work perfectly so far. So this business is real life, not just a theory!

Maxwell's unification of Electricity with Magnetism

James Clerk Maxwell was a genius. He managed in the 1860s to unify electricity and magnetism together theoretically, in a way that (accidentally!) incorporated Special Relativity. He did this decades before Einstein formulated Special Relativity.

Aspects of the unity of electricity and magnetism were noticed by a number of important physicists before Maxwell put the puzzle pieces together and wrote down his beautiful equations for electromagnetic radiation, which we list here for artistic reasons,

\begin{align} {\vec{\nabla}} \cdot {\vec{B}} &=0 & {\vec{\nabla}} \times {\vec{E}} + {\frac{1}{c}}{\frac{\partial{\vec{B}}}{\partial t}} &= 0 \notag\\ {\vec{\nabla}} \cdot {\vec{E}} &=\rho & {\vec{\nabla}} \times {\vec{B}} - {\frac{1}{c}}{\frac{\partial{\vec{E}}}{\partial t}} &= {\vec{J}} \,. \end{align}

These encapsulate discoveries of previous workers. Gauss had found that electric fields ${\vec{E}}$ are sourced by charges $\rho$. Oersted and Ampere had found that magnetic fields ${\vec{B}}$ are sourced by currents ${\vec{J}}$. Faraday had found that a time-varying magnetic field ${\vec{B}}$ induces a spatial gradient in the electric field ${\vec{E}}$. Maxwell found that a time-varying electric field ${\vec{E}}$ induces a spatial gradient in the magnetic field ${\vec{B}}$. The executive summary is that Maxwell's equations show electric and magnetic fields to be inherently intertwined.

A powerful way of seeing for yourself how explicitly electricity and magnetism are interconnected is to watch a compass needle during an electrical storm: it swings around. Bolts of lightning involve a lot of moving charged particles, creating electrical currents, which in turn create magnetic fields. Your compass needle can measure the magnetic field of the Earth, and it is sensitive enough to detect changing magnetic fields in electrical storms as well. Try it sometime! Just take the usual safety precautions during an electrical storm: do not talk on a landline phone or stand/sit near a conductor -- like an oven, sink, or metal toilet.

Maxwell's equations also have the feature that electromagnetic fields (${\vec{E}}$ and ${\vec{B}}$) propagate at the speed of light. The subatomic particles that make up big fat classical electromagnetic fields described by Maxwell's equations are photons.

Electromagnetic Spectrum

You may have heard of gamma rays, X-rays, ultraviolet (UV), visible light, infrared (IR), RADAR, microwave, radio, wi-fi, and television. Every one of these types of electromagnetic radiation is made up of photons. A photon is a quantum -- a lump -- of the electromagnetic field. The difference between the different types of electromagnetic radiation is simply that the frequency or wavelength is different. There is a simple relationship between wavelength $\lambda$, frequency $f$, and wave speed $c$,

\begin{equation} f = {\frac{c}{\lambda}} \,. \end{equation}

Wavelength is measured in metres. Frequency is measured in Hertz (Hz), which is equivalent to inverse seconds (1/s). For example, if my electromagnetic wave has a frequency of 99.1MHz which is CBC Radio One, that means it waves 99,100,000 times per second. The spectrum diagram illustrated below is organized by wavelength.

The electromagnetic spectrum

For light, there is no medium. Light can travel perfectly well in a vacuum. The speed of light $c$ describes the speed of light in vacuum (empty space with no particles in it). When light moves through a medium, like the glass in a window pane, the photons interact with atoms in the material, and this delays them a little, effectively making them move slower through the medium. This slowing is quantified by something called the refractive index.

So - how do photons help us discover things about the universe?

Blueshift/Redshift

Hydrogen gas is by far the most common gas in our universe. It was created during the Big Bang. So astronomers have to be good at studying hydrogen and the interactions it has with light. The technical term for this is studying the absorption and emission spectra - which physicists do by looking at the kind of EM radiation emitted by the H atoms over whole frequency bands.

The fascinating thing that you find if you study the spectrum of any stable atom is that it has spectral lines. Photons are emitted from atoms only with very specific frequencies, and photons are absorbed by atoms only with very specific frequencies. The reason is that the energy of atomic electrons around the nucleus is quantized, and so if you want to shift from one allowed energy level to another you have to do it in quantum jumps. (We will discuss this more thoroughly when we get to Quantum Mechanics later this semester.) Spectral lines for atoms are the same regardless of whether the atoms are in Montreal, on Mars, or in the Andromeda galaxy. Photons are emitted by the atom at the same frequency in their rest frame. But what happens when the hydrogen gas is moving? Time gets dilated, so we know something must happen to the photons.

What happens is broadly similar but not identical to the story of the Doppler effect for sound. Light sources moving towards a detector will have their frequency upshifted by a combination of the motion and time dilation; wave crests they emit get squished together because of the movement. Conversely, sources moving away from our detector will have their frequency downshifted by a combination of the motion and time dilation: wave crests they emit get stretched out because of the movement. This is illustrated schematically below.

Spectral absorption lines, redshifted or blueshifted

The upshifting of light by motion of the source is called blueshift and it is a direct result of special relativity. Similarly, the downshifting of light by motion of the source is called redshift. The amount of blueshift or redshift is precisely related to the velocity of the source relative to the detector. So, by looking at how far spectral lines get shifted from their rest-frame counterparts, we can work backwards and figure out how fast the source was moving.

(For anyone interested in the math, the formula is $f' =f \sqrt{{\frac{1\pm v/c}{1\mp v/c}}}$, where $f'$ is the Doppler shifted frequency, $f$ is the original frequency, $v$ is the speed of the source relative to the detector and $c$ is the speed of light. You use the upper sign for blueshift and the lower sign for redshift.)

Photons and the relationship between energy and frequency

Since photons go at the speed of light, which is the same speed no matter what, you might think motion of photons would be boring. But you would be wrong. Photons can go in any direction (heading), and they can also have any energy or any frequency. Each photon is an indivisible lump (quantum) of electromagnetic field. It is characterised by its frequency, or equivalently by its energy. There is a very important relationship between photon energy E and frequency f. It is embodied in the equation

\begin{equation} E=hf \,. \end{equation}

The constant of proportionality $h$ is known as Planck's constant, and in SI units it has the approximate value of $6.626 \times 10^{-34}$Js. In other words, it is tiny in human-sized units.

To show you how this equation works, let us pick an explicit example. We know that if we tune to 99.1MHz on our FM radio dial, we get CBC Radio One. So how much energy does one single CBC Radio One photon possess? To find out, we multiply the frequency by Planck's constant. We get approximately $E = 6.6 \times 10^{-26}$J. By comparison, the amount of energy emitted by a 100W lightbulb in one second is 100J. So when you are listening to CBC Radio One your radio receiver is getting lots and lots of photons. You are bathed in them. But they do not damage you one little bit. We will explain the fundamental reason why low-frequency electromagnetic waves are not dangerous to humans when we get to Quantum Mechanics.

Energy, Momentum and Mass

One of the most famous equations in all of physics is $E=mc^2$. Actually, this equation is only one part of the true equation relating energy $E$, momentum ${\vec{p}}$ and mass $m$. Energy and momentum can vary - and they typically do differ in different frames of reference - but the mass is an invariant: it stays fixed. The constant in the equation c is again the speed of light.

\begin{equation} E^2 = |{\vec{p}}|^2c^2 + (mc^2)^2 \,. \end{equation}

This equation, known to particle physicists as the mass shell relation, holds for all objects. You can use it to handle special cases, of course.

Example 1: consider the rest frame of a massive particle. In this frame of reference, the velocity is zero, so therefore the momentum is also zero: ${\vec{p}}=\vec{0}$. The equation then tell us that $E=mc^2$. This is called the rest energy and is the amount of energy locked up in a particle just because it has mass. When this energy is liberated it can be ferocious -- e.g. in atom-bomb or H-bomb nuclear weapons.

Example 2: consider massless particles (like photons, gluons, and gravitons). For massless particles, $m=0$. Therefore, the mass shell relation tells us that $E = |\vec{p}| c$. So for massless particles, the energy and the momentum are always directly proportional.

Energy can come from (a) rest energy, (b) kinetic energy, a.k.a. energy of motion, and (c) potential energy, arising from forces like gravity. Typical particles moving around will have all three.

The Equivalence Principle

As mentioned earlier, Maxwell actually incorporated Special Relativity into his equations for electromagnetism a full forty years before Einstein developed it. This was not the only clue that Einstein had pointing towards Relativity. He looked at a myriad of clues. One of the things that he thought about really hard was what it would be like to chase a photon. He thought about acceleration, as well as velocity. He thought about freefall, and about gravity. For years and years and years. His conclusion from a lot of really hard mental effort and calculations in well-organized notebooks? That acceleration feels exactly like a gravitational pull.

You can check this equivalence principle by bringing a bathroom scale into an elevator. As you accelerate up to cruising speed, you temporarily weigh more! You did not get more massive, but the acceleration of the elevator added to the acceleration due to gravity that you already feel every day, and made the scale read a higher number. Your weight increased but your mass did not. Similarly, when the elevator decelerates at the top, your weight temporarily reads a lower number on the bathroom scale. Einstein codified this into a principle which he called the Equivalence Principle: gravity feels indistinguishable from acceleration.

The Equivalence Principle states that a gravitational pull is indistinguishable in a local experiment from an acceleration

This Equivalence Principle was part of what led Einstein to formulate General Relativity (GR), an amazing theoretical construct which incorporates Special Relativity, gravity, and the Equivalence Principle. In GR, gravity becomes the fabric of spacetime. As Greene puts it, The agent of gravity, according to Einstein, is the fabric of the cosmos.. Spacetime is a geometrical concept, and uses a lot of pretty mathematics known as Riemannian geometry. The Einstein equations themselves, offered here for artistic reasons only, look like this:-

\begin{equation} R_{\mu\nu}-{\frac{1}{2}}g_{\mu\nu}R = {\frac{8\pi G_N}{c^4}} T_{\mu\nu} \,. \end{equation}

The left hand side of this equation talks about the curvature of spacetime, i.e. how much the fabric of spacetime is warped. The right hand side of the equation talks about the energy and pressure of the matter in the spacetime. The proportionality constant is a pure number, which involves Newton's gravitational constant $G_N$, and the speed of light $c$. The Einstein equations describe relativistic gravity. In more poetic words, matter tells spacetime how to curve, while spacetime tells matter how to move..

A cartoonized version of a black hole spacetime

Einstein's picture of gravity corrects Newton's picture when speeds are a significant fraction of the speed of light and when gravity is strong. It reduces back to Newton's theory when gravity is weak and all speeds are low everywhere.

GPS

The one place where GR makes a direct difference to your everyday life is with GPS: the Global Positioning System. GPS works by having satellites in low Earth orbit above us, which have extraordinarily good timekeepers on board. They send radio signals down to Earth, which GPS receivers can pick up passively. If you can get a fix on at least three satellites, you can triangulate your position. Neato!

So what does GPS do that requires relativity? Well, the satellites that are sending the timing signals down to your GPS device are moving (a) at a measurable fraction of the speed of light, and (b) are higher up Earth's gravitational field than we are. This means that the photons in the signals experience both special relativistic time dilation and gravitational time dilation. If the chips in our GPS devices did not take special and general relativity into account when computing our position, we would be off by kilometres per day. Clearly that would not be optimal on a canoeing trip!

A GPS receiver