The Theory of Special Relativity

The Theory of Special Relativity

Learning Objectives • Electromagnetism and Electromagnetic Waves. • Nature of Light. • Do Electromagnetic Waves propagate in a Medium (Luminiferous Ether)? • Galilean transformation in Newtonian physics. • The Michelson-Morley experiment. • Invariance of the speed of light under the Lorentz transformation. • Einstein’s interpretation of the Lorentz transformation.

Electromagnetism • James Clerk Maxwell united the phenomena of electricity and magnetism under the theory of electromagnetism, which is expressed through Maxwell’s equations as published in 1861/1862. James Clerk Maxwell, 1831-1879 Gauss’s Law for Magnetism Single integral with circle: closed loop Double integral with circle: closed surface

Electromagnetism • James Clerk Maxwell united the phenomena of electricity and magnetism under the theory of electromagnetism, which is expressed through Maxwell’s equations as published in 1861/1862. The net outward normal electric flux through any closed surface is proportional to the total electric charge enclosed within that closed surface. James Clerk Maxwell, 1831-1879 Magnetic field has a divergence equal to 0. Or, equivalently, no monopoles. The magnetic field in space around an electric current is proportional to the electric current which serves as its source. Single integral with circle: closed loop Double integral with circle: closed surface The induced electromotive force in any closed circuit is equal to the negative of the time rate of change of the magnetic flux through the circuit.

Electromagnetic Waves • In this course, you do not need to know how to derive or apply Maxwell’s equations. • You only need to know one thing: an accelerating charge produces an electromagnetic wave.

Speed of Electromagnetic Waves • Suppose you have two charges (e.g., two electrons) separated by a distance. • If you move one charge, will the other charge instantaneously feel a change in force from the first charge? e- e-

Speed of Electromagnetic Waves • Suppose you have two charges (e.g., two electrons) separated by a distance. • If you move one charge, will the other charge instantaneously feel a change in force from the first charge? No, change will propagate out from moved charge by the speed of the eletromagnetic wave.

Speed of Electromagnetic Waves • Maxwell’s equations predict the speed of electromagnetic waves to be* • (Although I use the symbol c, which stands for the speed of light, it is important to understand that when Maxwell’s published his equations it had not yet been proven that light comprises electromagnetic waves!) * μ0 is the permeability and ε0 the permittivity of free space (a theoretically perfect vacuum). Permeability/Permittivity is a measure of the amount of resistance encountered when forming an magnetic/electric field in a medium. Vacuum permeability and permittivity are both fundamental constants of nature.

Nature of Light • By mid-1800’s, there had been numerous experiments of increasing accuracy to measure the speed of light. For example, in the 1850’s, the French physicist Léon Foucault measured a value for the speed of light that is only 5% higher than the modern value. • Because the predicted speed of electromagnetic waves appeared to match the measured speed for light, Maxwell postulated (but did not prove) that light comprises electromagnetic waves.

Nature of Light • Foucault’s apparatus involves light reflecting off a rotating mirror towards a stationary mirror. As the rotating mirror will have moved slightly in the time it takes for the light to bounce off the stationary mirror (and return to the rotating mirror), it will thus be deflected away from the original source, by a small angle.

Nature of Light • In a series of experiments beginning in 1865, the German physicist Heinrich Hertz showed that electromagnetic waves have the same properties (e.g., can be transmitted and received, undergoes reflection and refraction, has the same speed) as light, and therefore that light comprises electromagnetic waves.

Luminiferous Ether • Waves that we are familiar with propagate in a medium: - sound waves propagate through air (travelling pressure fluctuations in air) - water waves propagate through water (travelling pressure fluctuations in water) • What kind of medium does electromagnetic waves (light) propagate in?

Luminiferous Ether • In the 1800s, it was thought that light (electromagnetic waves) propagate in luminiferous ether. (Luminiferous = light bearing. Ether = substance which permeates space but does not provide any mechanical resistance.) • Even Maxwell himself thought so, writing that: • If this is the case, then the speed of light should depend on our motion relative to the ether. Which situation then, in this example, would we measure the speed of light to be higher?

Luminiferous Ether • In the 1800s, it was thought that light (electromagnetic waves) propagate in luminiferous ether. (Luminiferous = light bearing. Ether = substance which permeates space but does not provide any mechanical resistance.) • Even Maxwell himself thought so, writing that: • If this is the case, then the speed of light should depend on our motion relative to the ether. • This “commonsense” treatment of space and time (i.e., vectorial addition of velocities) in Newtonian physics is based mathematically on the Galilean transformations.

The Galilean Transformations • The Galilean transformations treats space and time separately and as absolutes. • Consider a laboratory where there are meter sticks and synchronized clocks everywhere so that the position and time of any event that occurs in this laboratory can be measured at the location of that event; i.e., the purely geometrical effect of time delay inherent for light to propagate from a given location to an observer can be safely ignored. • Consider two inertial reference frames, S and S´, constituting two such laboratories where the frame S´ is moving in the positive x-direction with constant velocity u. The clocks in the two reference frames are started when the origins of the coordinate systems, O and O´, coincide at time t = t´ = 0.

The Galilean Transformations • Consider, for simplicity, a stationary object (apple) located at position (x, y, z) in the S frame. After a time t has elapsed, this same object will be located at a position as measured in the S´ frame of at a time in the S/ frame of Obviously, the same transformations apply for a moving object.

The Galilean Transformations • How does the velocity v of an object as measured in the S frame relate to the velocity v´ of the same object in the S´ frame? • We simply take the time derivatives of Eqs. 4.1-4.3 to get or in vector form

The Galilean Transformations • How does the acceleration a of an object as measured in the S frame relate to the acceleration a´ of the same object in the S´ frame? • We simply take the time derivatives of Eq. 4.5 (keeping in mind that u is constant) to get • Thus, the same acceleration is measured in both reference frames. • In summary, the Galilean transformations are simply the “commonsense” treatment of space and time in Newtonian physics.

Does Light Propagate in a Luminifereous Ether? • In 1887, Albert A. Michelson and Edward W. Morley began an experiment (which they gradually improved over time) to determine whether light propagates in a luminifereous ether. Luminifereous ether Albert A.Michelson, 1852-1931 Motion of Earth through ether Edward W.Morley, 1838-1923

Does Light Propagate in a Luminifereous Ether? • To understand their experiment, consider the following: Suppose we arrange a competition between two swimmers in a river that is 100-m wide and flowing at 3 m/s. Both swimmers are equally strong and swim at a constant speed of 5 m/s. Swimmer 1 is asked to swim 100 m upstream and back. Swimmer 2 is asked to swim directly across the river and back to the same point. Who wins? Swimmer 2 100 m Swimmer 1 100 m

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2m/s, taking 50s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4m/s directly across. Swimmer 2 therefore takes 25s to swim 100 m directly across to the other band, and another 25s to swim 100 m directly back. Total time = 25 + 25 = 50 s. θ

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4m/s directly across. Swimmer 2 therefore takes 25s to swim 100 m directly across to the other band, and another 25s to swim 100 m directly back. Total time = 25 + 25 = 50 s. θ

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5 s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4m/s directly across. Swimmer 2 therefore takes 25s to swim 100 m directly across to the other band, and another 25s to swim 100 m directly back. Total time = 25 + 25 = 50 s. θ

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5 s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4 m/s directly across. Swimmer 2 therefore takes 25 s to swim 100 m directly across to the other band, and another 25 s to swim 100 m directly back. Total time = 25 + 25 = 50 s. θ

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5 s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4 m/s directly across. Swimmer 2 therefore takes 25 s to swim 100 m directly across to the other band, and another 25 s to swim 100 m directly back. Total time = 25 + 25 = 50 s. • Cross-stream swimmer wins! • If we stage the competition at an even wider river, would their time difference be smaller or larger? θ

Does Light Propagate in a Luminifereous Ether? • Swimmer 1 swims upstream at a net speed of 5-3=2 m/s, taking 50 s to swim 100 m. Swimmer 1 then swims downstream at a net speed of 5+3 = 8 m/s, taking 12.5 s to swim another 100 m. Total time = 50 + 12.5 = 62.5 s. • In 1 s, swimmer 2 swims 5 m at an angle θ to the bank and is swept 3 m downstream, and therefore experiences a net speed of 4 m/s directly across. Swimmer 2 therefore takes 25 s to swim 100 m directly across to the other band, and another 25 s to swim 100 m directly back. Total time = 25 + 25 = 50 s. • Cross-stream swimmer wins! • If we stage the competition at an even wider river, would their time difference be smaller or larger? Larger. θ

The Michelson-Morley Experiment • Light from a source partially passes through a half-silvered mirror, is reflected back, and reflected by the half-silvered mirror to observer. • Light from the same source is partially reflected by the half-silvered mirror, is reflected back, and passes through the half-silvered mirror to observer. The original Michelson-Morley experimental setup Half-silvered mirror Light source Direction of Earth’s motion through ether

The Michelson-Morley Experiment • Light from a source partially passes through a half-silvered mirror, is reflected back, and reflected by the half-silvered mirror to observer. • Light from the same source is partially reflected by the half-silvered mirror, is reflected back, and passes through the half-silvered mirror to observer. • Imagine that you align the blue arm to be in the direction through which Earth is moving through the ether. For light to take equal times to travel along the two arms and hence interfere constructively to produce maximum light intensity at the screen, you would need to make the green arm longer than the blue arm. (Remember: cross-stream swimmer wins!) Half-silvered mirror Light source Direction of Earth’s motion through ether

The Michelson-Morley Experiment • Light from a source partially passes through a half-silvered mirror, is reflected back, and reflected by the half-silvered mirror to observer. • Light from the same source is partially reflected by the half-silvered mirror, is reflected back, and passes through the half-silvered mirror to observer. • Now rotate the entire experimental setup by 90. The green arm is now aligned with the direction through which Earth is moving through the ether. Light takes longer to travel along the green (longer) arm than the blue (shorter) arm, and so the two light rays do not (in general) arrive at the screen in phase and do not interfere constructively. • In practice, do not know which direction Earth is moving through ether, so have to repeat experiment at various orientations. Also, on a given day, Earth may happen to moving at the same speed and direction as the ether, so have to repeat the experiment over different months. Light source Half-silvered mirror Direction of Earth’s motion through ether

The Michelson-Morley Experiment • Michelson and Morley used multiple mirrors to increase the arm lengths so as to attain a measurable time delay in light travel time (if indeed light propagates in luminiferous ether) between the two arms. (Recall that if we stage the competition between the two swimmers in a wider river, their time difference would be correspondingly larger.) The original Michelson-Morley experimental setup

The Michelson-Morley Experiment • In practice, see fringes at the screen because the two light rays are not exactly parallel and the wavefront not exactly planar. This is an advantage as it is easier to see changes in the fringe pattern rather than changes in brightness at the screen. • On rotating the equipment, expect the fringe pattern to change (i.e., fringes move radially in our out). • Despite repeating the experiment in different orientations and at different times, Michelson and Morley could not detect any changes, thus demonstrating that light does not propagate in a luminiferous ether (and obey the Galilean transformations)! Half-silvered mirror Light source

How to Interpret the Michelson-Morley Experiment? • The Dutch physicist HendrikAntoon Lorentz, along with George FitzGerald, Joseph Larmor, and Woldemar Voigt, wished to preserve the idea that light propagates in a luminiferous ether. In that case, the Galilean transformation had to be abandoned, otherwise light would have different speeds depending on our motion relation to the ether. • They proposed a mathematical transformation – which Poincaré in 1905 named the Lorentz transformations – under which Maxwell’s equations were invariant when transformed from the ether to a moving reference frame. Thus, under the Lorentz transformation, electromagnetic waves – and therefore light – have the same speed in all reference frames even though light propagates in a luminiferous ether. • Lorentz and his collaborators understood that this transformation predicted the effects of length contraction and time dilation, and indeed used these effects to explain the results of the Michelson-Morley experiment under the notion that light propagates in a luminiferous ether.

How to Interpret the Michelson-Morley Experiment? • Lorentz proposed that all matter comprises electrical charges in the empty space of ether, held together by electric and magnetic forces. When matter moves with respect to the ether, moving electrical charges create magnetic fields (according to Maxwell’s equations) such that the electrical charges settle into a new configuration with the body suffering contraction along the direction of motion. • In other words, Lorentz believed that space is an absolute, and that objects contract along the direction of motion. Half-silvered mirror Light source Matter at rest in ether Matter moving to right with respect to ether Direction of Earth’s motion through ether

How to Interpret the Michelson-Morley Experiment? • Instead of matter contracting in such a way so that we measure the same speed for light as we move through the ether, Albert Einstein discarded the notion that light propagates in a luminifereous ether and postulated that light travels at the same speed in all reference frames. • In other words, Einstein discarded the idea that space is an absolute and that objects contract along the direction of motion. Instead, space is not an absolute, and has dimensions that depend on our relative motions. • Einstein assumed that all other postulates in Newtonian physics remained valid. Thus, his postulate that light travels at the same speed in all reference frames was the only difference compared with, but marked a crucial break from, Newtonian physics. • Based on this single postulate (preserving all other postulates in Newtonian physics), Einstein derived the transformation for space (length) and time as measured from different reference frames. As we shall see, not surprisingly, the transformations Einstein derived were identical to the Lorentz transformations, but with an entirely different interpretation.

The Theory of Special Relativity