Chapter 2, Part 1: Special Relativity

Inertial frames and Galilean invariance

An inertial reference frame (관성기준틀) is one in which Newton's laws are valid. Why not everywhere? Because \(\vec F = m\vec a\) holds only when the frame itself is not accelerating. In an accelerating frame, objects seem to accelerate with no force acting, and to keep Newton's laws we must invent fictitious (apparent) forces (가상력), such as the inertial force (관성력) and the centrifugal force (원심력). Two frames K and K′ in relative motion along their \(x\) axes are both inertial only when their relative velocity \(\vec v\) is constant.

A reminder from Physics 1. For an object P seen from two observers A and B, the positions satisfy \(\vec r_{PA} = \vec r_{PB} + \vec r_{BA}\). Differentiating with respect to time,

\[ \vec v_{PA} = \vec v_{PB} + \vec v_{BA},\qquad \vec a_{PA} = \vec a_{PB} \quad(\text{when } \vec v_{BA} \text{ is constant}). \]

For example, if a car (P) moves at 100 km/h as seen by a person standing by the road (A), a second car (B) moving the same way at 50 km/h sees it moving at 50 km/h: \(100 = 50 + 50\). If the first car brakes to a stop over a time \(\Delta t\), its velocity changes by \(-100\) km/h for both observers (to 0 for A, to \(-50\) km/h for B), so both measure the same acceleration \(-100\,\text{km/h}/\Delta t\). The same force produces the same acceleration: Newton's laws hold in both frames.

This "classical" invariance is Newton's principle of relativity, or Galilean invariance: it is not possible to determine absolute motion in space by any experiment. In addition, time and space are assumed to be completely separate. The coordinates in K and K′ are related by the Galilean transformation

\[ \begin{gathered} x' = x - vt,\quad y' = y,\quad z' = z,\quad t' = t\\ \text{or}\quad x = x' + vt,\quad y = y',\quad z = z',\quad t = t'. \end{gathered} \]

Newton's laws have the same form under a Galilean transformation, but Maxwell's equations do not: a field that is purely electric in K can appear partly magnetic in K′. Albert Einstein proposed in 1905 that space and time are not separate, and that Newton's laws are only an approximation. This is the special theory of relativity (특수상대성이론); "special" because it deals with inertial frames. The general theory, which includes gravity, comes in Chapter 15.

2.1The apparent need for ether

All known waves other than light need a medium (매질) to propagate: water waves need water, and sound waves need air. Physicists therefore assumed that light, an electromagnetic wave, travels through a hypothetical medium, the luminiferous ether (에테르). Because everyday speeds are tiny compared with the speed of light, it was hard to see any effect of motion through it.

Maxwell's theory gives the speed of light in vacuum as

\[ c = \frac{1}{\sqrt{\mu_0\epsilon_0}} \approx 3\times10^8\ \text{m/s}, \]

which depends only on the electromagnetic properties of the medium. By the classical laws, light moving with velocity \(\vec c\) in K′ should be measured as \(\vec c + \vec v\) in K. But Maxwell's equations do not distinguish between K and K′. Physicists of the late 19th century concluded that there must be one preferred inertial frame, in which the ether is stationary and the speed of light is exactly \(c\).

2.2The Michelson–Morley experiment

The Earth orbits the Sun at about \(v\approx3\times10^4\) m/s \(\approx10^{-4}c\). If the ether exists, the Earth moves through it, and from the Earth's point of view there is an "ether wind" blowing past. Michelson's interferometer splits a light beam at a partially silvered mirror A into two beams that travel along perpendicular arms of lengths \(\ell_1\) (parallel to \(\vec v\)) and \(\ell_2\) (perpendicular), reflect from mirrors, and recombine. Measuring the phase difference between the two beams, as a shift of the interference fringes, measures the difference in their travel times.

Parallel arm (\(A\to C\to A\)): the light moves against the ether wind at speed \(c-v\) and with it at \(c+v\),

\[ t_1 = \frac{\ell_1}{c+v} + \frac{\ell_1}{c-v} = \frac{2c\ell_1}{c^2-v^2} = \frac{2\ell_1}{c}\Big(\frac{1}{1-v^2/c^2}\Big). \]

Perpendicular arm (\(A\to D\to A\)): relative to the stationary ether, the light must travel along a slanted path so that \((ct_2/2)^2 = \ell_2^2 + (vt_2/2)^2\), the same on the way back by symmetry, so

\[ \begin{gathered} t_2 = \frac{2\ell_2}{\sqrt{c^2-v^2}} = \frac{2\ell_2}{c}\frac{1}{\sqrt{1-v^2/c^2}},\\ \Delta t = t_2 - t_1 = \frac2c\Big(\frac{\ell_2}{\sqrt{1-v^2/c^2}} - \frac{\ell_1}{1-v^2/c^2}\Big). \end{gathered} \]

Since the arm lengths are not known to a fraction of a wavelength, Michelson rotated the apparatus by 90°, so that \(\ell_2\) is parallel to \(\vec v\) and \(\ell_1\) perpendicular:

\[ \Delta t' = t_2' - t_1' = \frac2c\Big(\frac{\ell_2}{1-v^2/c^2} - \frac{\ell_1}{\sqrt{1-v^2/c^2}}\Big). \]

The observable is the difference between the time differences, original vs rotated:

\[ \Delta t' - \Delta t = \frac2c\Big(\frac{\ell_1+\ell_2}{1-v^2/c^2} - \frac{\ell_1+\ell_2}{\sqrt{1-v^2/c^2}}\Big). \]

Because \(c\gg v\), use the series expansions \((1-v^2/c^2)^{-1}\approx1+v^2/c^2\) and \((1-v^2/c^2)^{-1/2}\approx1+v^2/2c^2\):

\[ \Delta t' - \Delta t \approx \frac{v^2(\ell_1+\ell_2)}{c^3}. \]

For Michelson's original apparatus, \(\ell_1\approx\ell_2\approx1.2\) m, so \(\Delta t'-\Delta t\approx8\times10^{-17}\) s. That is exceedingly small, but for visible light of wavelength \(6\times10^{-7}\) m the period is \(T = \lambda/c = 2\times10^{-15}\) s, so the effect is 0.04 of a fringe, which is measurable. No shift was observed. The improved 1887 apparatus of Michelson and Morley, with light reflected back and forth to make \(\ell\approx11\) m, should have produced a shift of about 0.4 fringe while detecting shifts as small as 0.005. Still no shift was observed: evidence of no ether.

The Michelson–Morley interferometer, rotated slowly

Arm length \(\ell\) Earth's speed \(v\) Wavelength \(\lambda\)

The apparatus, seen from above

Fringes in the telescope (crosshair fixed)

Predicted fringe shift vs rotation angle

Left: light from the source S is split at the half-silvered mirror A; one beam travels along arm 1 to mirror M₁ and back, the other along arm 2 to M₂ and back (the compensator plate C equalizes the glass each beam passes through), and both recombine into the telescope T. The whole apparatus, floating on mercury, turns slowly while the supposed ether wind blows from a fixed direction. As the apparatus rotates by an angle \(\theta\), the ether theory predicts a fringe shift \(N(\theta) = \dfrac{v^2(\ell_1+\ell_2)}{c^2\lambda}\sin^2\theta\) relative to \(\theta = 0\), reaching its maximum at 90°. The dashed line marks the 1887 sensitivity of 0.005 fringe; the observed shift was zero (magenta). Switching on the Lorentz–FitzGerald contraction makes the prediction zero for every angle as well.

One possible explanation of the null result was the ether drag hypothesis: the Earth somehow drags the ether along with it as it rotates and orbits the Sun. It was rejected by several experiments, including stellar aberration (별의 수차), observed by Bradley in 1728. A telescope moving at speed \(v\) must be tilted by a small angle \(\tan\theta = v/c\) to catch light from a star directly overhead, and as the Earth's velocity changes direction over the year, every star traces a small apparent circle. That effect is actually observed, which would be impossible if the ether moved with the Earth. So, combining the null Michelson–Morley result with stellar aberration: there is no ether. Light and electromagnetic waves need no medium, and there seems to be no single preferred inertial frame in which the speed of light is \(c\).

Lorentz and FitzGerald proposed instead that length is contracted by the factor \(\sqrt{1-v^2/c^2}\) in the direction of motion: \(\ell_1\) for \(\Delta t\), and \(\ell_2\) for \(\Delta t'\). Then

\[ \begin{gathered} \Delta t = \frac2c\Big(\frac{\ell_2}{\sqrt{1-v^2/c^2}} - \frac{\ell_1\sqrt{1-v^2/c^2}}{1-v^2/c^2}\Big) = \frac2c\frac{\ell_2-\ell_1}{\sqrt{1-v^2/c^2}},\\ \Delta t' = \frac2c\frac{\ell_2-\ell_1}{\sqrt{1-v^2/c^2}}. \end{gathered} \]

So \(\Delta t = \Delta t'\), and rotating the apparatus produces no shift. (Neither time difference is zero unless \(\ell_1 = \ell_2\); what matters is that they are equal.) But why should lengths contract? And the hypothesis is hard to test, because any measuring device would presumably be shortened by the same factor. The puzzle remained until Einstein (1879–1955) presented his explanation.

2.3Einstein's postulates

Einstein believed that Maxwell's equations must be valid in all inertial frames, and he based special relativity on two postulates:

1. The principle of relativity. The laws of physics, including electromagnetism, are the same in all inertial systems. There is no way to detect absolute motion, and no preferred inertial system exists.

2. The constancy of the speed of light. Observers in all inertial systems measure the same value for the speed of light in a vacuum.

An inertial frame is now one in which all the laws of physics are valid. Previously it was assumed that \(t = t'\). Einstein insisted instead that each system must have its own observers with their own clocks and metersticks, and that an event in a given system is specified by both its space and time coordinates.

The relativity of simultaneity. Two light flashes go off at \(x = \pm1\) m in system K, and Frank, standing at \(x=0\) in K, sees them simultaneously. Mary, in system K′ moving to the right with speed \(v\), is beside Frank when the flashes go off. Because the speed of light is the same for Mary (postulate 2), the flash on the right reaches her before the flash on the left: for her the flashes are not simultaneous.

Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K′) moving with respect to the first.

Two flashes, two observers

Mary's speed \(v/c\) Time \(t\) in K

Viewed in K, where the flashes at \(\pm1\) m both go off at \(t=0\) (time in nanoseconds; light covers 1 m in 3.34 ns). Frank (blue, on the ground of K) receives both pulses at the same moment. Mary (magenta, riding the platform of K′ above him and passing him at \(t = 0\)) meets the right-hand pulse at \(t = (1\,\text{m})/(c+v)\) and the left-hand one at \((1\,\text{m})/(c-v)\). The Lorentz transformation (§2.4) shows that in K′ the right-hand flash really did go off first, by \(\Delta t' = 2\gamma v(1\,\text{m})/c^2\).

"Information" can travel only as fast as the finite speed of light. How, then, do we synchronize clocks? Clocks synchronized at one spot at rest would have to be moved to their positions, and moving them might affect the synchronization. This is exactly the question of \(t\) vs \(t'\).

2.4The Lorentz transformation

We need the correct transformation, the Lorentz transformation, which makes the laws of physics invariant between inertial frames. Let a light flash go off at the common origin when \(t = t' = 0\), with K′ moving along the \(x\) and \(x'\) axes at speed \(v\) (no motion along \(y\) and \(z\)). By postulate 2, the speed of light is \(c\) in both systems, so the wavefronts observed in both must be spheres centered on each system's own origin:

\[ x^2+y^2+z^2 = c^2t^2,\qquad x'^2+y'^2+z'^2 = c^2t'^2. \]

This is inconsistent with the Galilean transformation: substituting \(x' = x-vt\), \(t'=t\) gives \(x^2+y^2+z^2 - 2xvt + v^2t^2 = c^2t^2\), with extra terms. The Lorentz transformation equations are

\[ \begin{gathered} x' = \frac{x-vt}{\sqrt{1-v^2/c^2}},\qquad t' = \frac{t-(vx/c^2)}{\sqrt{1-v^2/c^2}},\\ y' = y,\qquad z' = z. \end{gathered} \]

With \(\beta = v/c\) and the relativistic factor

\[ \gamma = \frac{1}{\sqrt{1-\beta^2}} = \frac{1}{\sqrt{1-v^2/c^2}}\ \ge 1\quad(\gamma=1 \text{ when } v=0), \]

\[ \begin{gathered} x' = \gamma(x-\beta ct),\quad y'=y,\quad z'=z,\quad t' = \gamma(t-\beta x/c),\\ \begin{pmatrix}x'\\ t'\end{pmatrix} = \begin{pmatrix}\gamma & -\gamma\beta c\\ -\gamma\beta/c & \gamma\end{pmatrix}\begin{pmatrix}x\\ t\end{pmatrix}, \end{gathered} \]

a linear transformation. Check that it keeps the light sphere a sphere:

\[ \begin{aligned} x'^2+y'^2+z'^2 &= \gamma^2x^2 - 2\beta\gamma^2cxt + \gamma^2\beta^2c^2t^2 + y^2+z^2\\ &= (\gamma^2-1)x^2 - 2\beta\gamma^2cxt + \gamma^2\beta^2c^2t^2 + c^2t^2\\ &= \gamma^2\beta^2x^2 - 2\beta\gamma^2cxt + \gamma^2c^2t^2\\ &= c^2\gamma^2\big(t - \beta x/c\big)^2 = c^2t'^2. \end{aligned} \]

(The second line uses \(y^2+z^2 = c^2t^2 - x^2\); the third uses \(\gamma^2-1 = \gamma^2\beta^2\) and \(\gamma^2\beta^2+1 = \gamma^2\).) Both systems have a spherical wavefront centered on their own origin under the Lorentz transformation. Note that \(t'\ne t\): each system must have its own clocks and metersticks, for example a lattice of synchronized clocks and uniform measuring sticks placed throughout the system.

A "proof". The textbook argues from the simplest linear transformation, \(x' = \gamma(x-vt)\) and \(x = \gamma'(x'+vt')\). Postulate 1 requires \(\gamma = \gamma'\), and \(\gamma\) must be close to 1 for \(v\ll c\). Following the wavefront of the light pulse along the \(x\) axes, \(x = ct\) and \(x' = ct'\):

\[ \begin{gathered} ct = \gamma(ct'+vt') \Rightarrow t = \gamma\Big(1+\frac vc\Big)t',\\ ct' = \gamma(ct-vt) \Rightarrow t' = \gamma\Big(1-\frac vc\Big)t, \end{gathered} \]

so \(t' = \gamma^2(1-v/c)(1+v/c)\,t'\), which gives \(\gamma = 1/\sqrt{1-v^2/c^2}\). Then using \(t = x/c\) for the light pulse, \(t' = \gamma(t-vx/c^2)\). The full set and its inverse (swap primes and \(v\to-v\)):

\[ \begin{aligned} x' &= \frac{x-vt}{\sqrt{1-v^2/c^2}}, & y'&=y, & z'&=z, & t' &= \frac{t-(vx/c^2)}{\sqrt{1-v^2/c^2}},\\ x &= \frac{x'+vt'}{\sqrt{1-v^2/c^2}}, & y&=y', & z&=z', & t &= \frac{t'+(vx'/c^2)}{\sqrt{1-v^2/c^2}}, \end{aligned} \]

which reduce to the Galilean transformation when \(v\ll c\). Even at the Earth's orbital speed of 30 km/s, \(\gamma\approx1.000000005\); at \(v = 0.1c\), \(\gamma\approx1.005\). The linear equations ensure that a single event in one system is a single event in another. But space and time are not separate: to express \(x\) in K′ we need both \(x'\) and \(t'\). And the Lorentz transformation does not allow a speed greater than \(c\), since \(\gamma\) would become imaginary; later in this chapter we show that no object of nonzero mass can travel faster than \(c\).

Galilean or Lorentz? Follow the light

\(\beta = v/c\) Time \(t'\) in K′

Light wavefront seen in K′ at time \(t'\)

The relativistic factor \(\gamma(\beta)\)

Transform an event from K: \(x\) (light-seconds) \(t\) (s)

Left: a flash at the origin at \(t=t'=0\). The Lorentz transformation (solid) gives a circle of radius \(ct'\) centered on K′'s own origin: light moves at \(c\) in every direction, as postulate 2 demands. The Galilean transformation (dashed) puts the center at \(x' = -vt\), so light would move at \(c+v\) backward and \(c-v\) forward. Right: \(\gamma\) stays very close to 1 until \(v\) is a sizable fraction of \(c\), then grows without bound. The event calculator uses units with \(c = 1\) light-second per second.

2.5Time dilation and length contraction

The Lorentz transformation has consequences for time and length measurements made by observers in different inertial frames.

Time dilation (시간 팽창)

Frank lights a sparkler at position \(x_1\) in system K, and it burns out at the same place. The time it stays lit, measured by his clock at rest, is the proper time (고유시간) \(T_0 = t_2 - t_1\): the time interval between two events occurring at the same position in a system, as measured by a clock at rest in that system (the subscript 0 marks proper quantities). In the moving system K′ the two events (lit and burnt out) do not occur at the same place, so two observers with synchronized clocks are needed: Mary, beside the sparkler when it is lit, and Melinda, beside it when it goes out. Three clocks in all perform the measurement: \((x_1,t_1)\to(x_1,t_2)\) in K (Frank), and \((x_1',t_1')\) (Mary) and \((x_2',t_2')\) (Melinda) in K′. From the Lorentz transformation, with \(x_2 - x_1 = 0\),

\[ T' = t_2'-t_1' = \frac{(t_2-t_1) - \frac{v}{c^2}(x_2-x_1)}{\sqrt{1-v^2/c^2}} = \frac{T_0}{\sqrt{1-v^2/c^2}}, \]

\[ T' = \gamma T_0 \ \ge\ T_0. \]

The time interval measured in the moving system K′ is greater than the proper time measured in K, where the sparkler is at rest: time dilation, a direct result of Einstein's two postulates. It occurs because simultaneity is relative: the events do not occur at the same place and time coordinates in the two systems. In short, moving clocks run slow by the factor \(\gamma^{-1}\), and this applies to any clock: an hourglass, a heartbeat, an atom. The proper time is always the smallest possible time interval between two events.

Important: each person claims that the clock in the other (moving) system is running slow. If Mary had a sparkler at rest in K′, Frank would measure a longer interval on his clocks, because the sparkler would be moving with respect to his system (Example 2.1: \(T = \gamma T_0'\)). There is no contradiction, because the two measurements compare different pairs of clocks.

A gedanken ("thought") experiment. Mary in K′ flashes a light along her \(y'\) axis toward a mirror at height \(L\) and measures the proper time for its return, \(T_0' = 2L/c\). Frank in K sees the light travel along a slanted path while K′ moves a distance \(vT\), so by postulate 2

\[ \begin{gathered} T = \frac{\text{distance}}{\text{speed}} = \frac{2\sqrt{(vT/2)^2+L^2}}{c},\qquad L = \frac{cT_0'}{2},\\ \Rightarrow\ c^2T^2 = v^2T^2 + c^2T_0'^2\ \Rightarrow\ T = \frac{T_0'}{\sqrt{1-v^2/c^2}} = \gamma T_0', \end{gathered} \]

consistent with the earlier result.

Length contraction (길이 수축)

A meterstick at rest in K′ has proper length (고유길이) \(L_0' = x_r' - x_\ell' = L_0\). Frank, at rest in K, measures the length of the moving stick. The key is to measure both ends simultaneously in K, at \(t = t_r = t_\ell\) (these are not simultaneous in K′: \(t_r'\ne t_\ell'\) in general). Then

\[ L_0 = x_r'-x_\ell' = \frac{(x_r-x_\ell) - v(t_r-t_\ell)}{\sqrt{1-v^2/c^2}} = \frac{L}{\sqrt{1-v^2/c^2}} = \gamma L, \]

\[ L = L_0\sqrt{1-v^2/c^2} = \frac{L_0}{\gamma}\ \le\ L_0, \]

the length or space contraction (Lorentz–FitzGerald contraction). It occurs only along the direction of motion.

Another gedanken experiment. Mary flashes a light along her \(x'\) axis toward a mirror at the end of a stick of proper length \(L_0\); the round trip takes \(\Delta t' = 2L_0/c\) in K′. Frank sees the stick moving, so the light needs \(ct_1 = vt_1 + L\) to catch the mirror, \(t_1 = L/(c-v)\), and \(c(t_2-t_1) = L - v(t_2-t_1)\) to come back, \(t_2 - t_1 = L/(c+v)\):

\[ \Delta t = t_2 = \frac{L}{c-v} + \frac{L}{c+v} = \frac{2cL}{c^2-v^2} = \frac{2L}{c(1-v^2/c^2)}, \]

as in the Michelson–Morley experiment. Time dilation requires \(\Delta t = \gamma\Delta t'\), i.e. \(\frac{2L}{c(1-v^2/c^2)} = \frac{2L_0}{c\sqrt{1-v^2/c^2}}\), so again \(L = L_0\sqrt{1-v^2/c^2} = L_0/\gamma\) (Problem 18).

Two light clocks on one spaceship

\(v/c\)

Mary's ship carries two identical light clocks of proper length \(L_0\): one across the direction of motion (Mary's \(y'\) axis) and one along it (her \(x'\) axis). Both tick every \(T_0 = 2L_0/c\) for Mary. Seen from K, the vertical clock ticks every \(\gamma T_0\). The horizontal clock ticks every \(\frac{2L}{c(1-v^2/c^2)}\), which matches only if its arm is contracted to \(L = L_0/\gamma\). Turn the contraction off and the two clocks drift apart. Mary, carrying both, would then see her own clocks disagree, which would let her detect her absolute motion, contradicting the principle of relativity. This is the Michelson–Morley experiment again.

Example 2.2: a trip to Alpha Centauri

In the year 2150, antimatter-fueled spaceships carry provisions for 16 years of travel time. How fast must a ship go to make the round trip to a star 4.30 light-years away (one light-year, the distance light travels in a year, is \(9.46\times10^{15}\) m)? The astronauts' 16 years are the proper time \(T_0'\), since the trip begins and ends at the ship's location in K′; ignoring acceleration and turnaround, the time measured on Earth is \(T = 2L/v\), so

\[ \begin{gathered} T = \frac{2L}{v} = \frac{T_0'}{\sqrt{1-v^2/c^2}}\\ \Rightarrow\quad \frac{2(4.30\ \text{ly})(9.46\times10^{15}\ \text{m/ly})}{v} = \frac{16\ \text{y}}{\sqrt{1-v^2/c^2}}, \end{gathered} \]

which gives \(v = 0.473c = 1.42\times10^8\) m/s. The astronauts age 16 years, but their friends on Earth age \(\gamma T_0' = 18.2\) years. Can that really be true? We return to the question in §2.8 (the twin paradox).

Plan your own trip

Distance \(L\) (light-years) Provisions \(T_0'\) (years)

Solving \(2L/v = \gamma T_0'\) for \(v\): \(\beta = \big[1 + (cT_0'/2L)^2\big]^{-1/2}\). The plot shows the trip duration on Earth (\(2L/v\)) and aboard the ship (\(2L/\gamma v\)) against speed, for your distance. The ship's time can be made as short as you like by going fast enough; the Earth's time can never be less than \(2L/c\).

What does a moving object look like?

Length contraction is a statement about where the parts of an object are at the same time in K. A camera, or an eye, records light that arrives at the same time, and light from farther parts of the object left earlier. Computer simulations (Scott and Viner, 1965) show that boxes moving at \(0.8c\) photographed this way are not simply squashed: horizontal lines are length-contracted, vertical lines become curves, and the objects appear slightly rotated.

Contracted vs photographed

\(v/c\)

At rest

Positions at one instant in K (contracted)

What a camera records

Four wireframe boxes moving to the right, viewed by a pinhole camera at rest whose image plane faces them, with the nearest faces 5 units away. All three views share one scale, but each is centered on its own image, since light-travel delay also shifts the photographed boxes to the left. Middle: every point placed where it is at a single instant of K time, so the boxes are simply contracted along \(x\). Right: every point placed where it was when it emitted the light that reaches the camera at that instant, accounting for the light-travel time. Edges along the line of sight now bend and the boxes look rotated; the effect is purely optical, layered on top of the real contraction.

Chapter 2 continues in Part 2.

Chapter 2: Special theory of relativityPart 1: from the ether to the Lorentz transformation