Chapter 2: Special theory of relativityPart 2: velocities, spacetime, momentum, and energy

2.6Addition of velocities

A spaceship moves at \(v = 0.60c\) relative to a space station (system K). Mary, its commander (system K′), orders protons fired toward an asteroid at \(u' = 0.99c\) relative to the ship. According to Newtonian mechanics the protons move at \(u = v + u' = 1.59c\) relative to the station, more than \(c\), in disagreement with the Lorentz transformation. Instead, take the Lorentz transformation of differentials,

and with \(u_x = dx/dt\), \(u_x' = dx'/dt'\), and so on,

the Lorentz velocity transformations. Even though \(y = y'\) and \(z = z'\), the perpendicular components change: \(u_y\ne u_y'\) and \(u_z\ne u_z'\). The inverse transformations follow by swapping primes and \(v\to-v\):

For the protons, \(u_x = (0.99c + 0.60c)/(1 + 0.60\times0.99) = 0.997c\lt c\). The Lorentz transformation does not allow a material object to have a speed greater than \(c\); only massless particles such as light can travel at \(c\). Indeed, if \(u_x = c\), then \(u_x' = (c-v)/(1-v/c) = c\): light moves at \(c\) in every frame. (A spot of light can sweep across a distant surface faster than \(c\), for example a rapidly rotating floodlight sweeping across the Moon, but no energy or information travels along the surface at that speed; the light itself only travels radially, at \(c\).)

Example 2.4. Suppose instead Mary fires the protons perpendicular to the ship's motion, \(u_x' = 0\), \(u_y' = 0.99c\). With \(\gamma = 1/\sqrt{1-0.60^2} = 1.25\), an observer on the station finds \(u_x = 0.600c\), \(u_y = 0.99c/1.25 = 0.792c\), and a speed \(u = \sqrt{u_x^2+u_y^2} = 0.994c\), again below \(c\).

Example 2.5: Fizeau's experiment. In 1851 Fizeau measured the speed of light in water flowing at speed \(v\) and found \(u = u' + (1-1/n^2)v\), where \(u' = c/n\) is the speed of light in still water and \(n\) the index of refraction. Fresnel had predicted this partial "ether drag" coefficient \(1-1/n^2\), and it was taken as support for the ether. Relativistic velocity addition explains it without any ether: since \(v\ll c\), expanding \((1+x)^{-1}\approx1-x\),

2.7Experimental verification

Muon decay

Cosmic rays entering the Earth's atmosphere from outer space produce \(\pi\)-mesons (pions), which decay into muons, which in turn decay into other particles according to the radioactive decay law

Classically, muons traveling at \(0.98c\) cover a 2000 m path in \(6.8\times10^{-6}\) s, so of 1000 muons at the top of a mountain only about 45 should reach sea level. Experimentally about 542 do. Time dilation explains this: with \(\gamma = 5.0\), the muons' own clocks record only \(t' = 6.8\times10^{-6}\ \text{s}/5.0\), which gives about 538 survivors. Equivalently, in the muons' frame the 2000 m height is length contracted to about 400 m.

Atomic clocks and other tests

In 1971 Hafele and Keating flew cesium atomic clocks eastward and westward around the Earth on commercial airliners and compared them with clocks kept at the U.S. Naval Observatory. Both special-relativistic time dilation and the gravitational effect of general relativity contribute. The predicted and observed time differences were \(-40\pm23\) ns vs \(-59\pm10\) ns eastward, and \(275\pm21\) ns vs \(273\pm7\) ns westward: moving clocks really do run differently.

Velocity addition was tested directly at CERN (Alväger and colleagues, 1964). Neutral pions moving at \(\beta\approx0.99975\) decay into two \(\gamma\) rays. Galilean addition would give the forward \(\gamma\) rays a speed \(0.99975c + c = 1.99975c\); relativistic addition gives \(u = (c + 0.99975c)/(1 + 0.99975) = c\). Timing the \(\gamma\) rays between detectors about 30 m apart agreed with the relativistic prediction, not the Galilean one.

Quantum electrodynamics (QED), which includes special relativity, has been tested to one part in \(10^{12}\). Lorentz symmetry, the requirement that the laws of physics be the same for all observers, is the foundation of relativistic quantum field theory. Recent experiments have found no violation, to one part in \(10^{32}\) for electrons, \(10^{31}\) for neutrons, and \(10^{27}\) for protons, though some theories of quantum gravity suggest violations at length scales around \(10^{-35}\) m.

2.8The twin paradox (쌍둥이 역설)

Twins Frank ("fixed") and Mary ("moving"): Mary travels at \(0.8c\) to a star 8 light-years away and returns. According to Frank, Mary's biological clock ticks more slowly than his, so she will return younger than he (compare the father and daughter in the movie Interstellar). The paradox: Mary can claim that Frank is moving rapidly with respect to her, so that when she returns Frank should be the younger twin.

The correct answer is that Mary returns as the younger twin. According to Frank, Mary's trip takes \((8\ \text{ly}/0.80c)\times2 = 20\) years. Mary's clock ticks slowly by the factor \(\sqrt{1-0.8^2} = 0.6\), so she ages \(10\ \text{y}\times0.6\times2 = 12\) years (equivalently, she sees the distance contracted to \(8\sqrt{1-0.8^2} = 4.8\) ly, covered in 6 years each way). The important fact: Frank's clock is in an inertial system during the entire trip, but Mary's is not. Mary feels the acceleration when she slows down to turn around; she leaves her original inertial system and returns in a completely different one. The situation is not symmetric. The table in the next demo, following A. P. French's analysis, has each twin send signals at a fixed rate by their own clock and count the signals received, using spacetime (§2.9) and the Doppler effect (§2.10).

2.9Spacetime (시공간)

It is convenient to present events on a spacetime diagram (Minkowski diagram), with \(x\) horizontal and \(ct\) vertical, so that both axes have dimensions of length; this is the origin of the term "fourth dimension" for time. The path of an object is its worldline. A light signal has slope \(c/c = 1\) (45°), and a spaceship moving at constant velocity \(v\) has a straight worldline of slope \(c/v>1\). Clocks at \(x_1\) and \(x_2\) can be synchronized by a light signal sent from the point halfway between; if the clocks are in a moving system K′, their worldlines are slanted and the signals do not arrive simultaneously in K.

The light cone divides spacetime. Only events inside the past light cone can affect the present, and the present can affect only events inside the future light cone; events "elsewhere" cannot affect the present, because of the limit \(v\le c\).

Invariant quantities have the same value in all inertial frames. Consider \(s^2 = x^2 - (ct)^2\). Using the Lorentz transformation,

so \(s^2\) is invariant under the Lorentz transformation; it extends to \(s^2 = x^2+y^2+z^2-(ct)^2\). For two events, the spacetime interval

is the same in any inertial frame, and it classifies the pair:

1. \(\Delta s^2 = 0\) (\(\Delta x^2 = c^2\Delta t^2\)): the events can be connected only by a light signal, a lightlike separation.
2. \(\Delta s^2 > 0\) (\(\Delta x^2 > c^2\Delta t^2\)): no signal can travel fast enough to connect the events; they are not causally connected, a spacelike separation.
3. \(\Delta s^2\lt 0\) (\(\Delta x^2\lt c^2\Delta t^2\)): the events can be causally connected, a timelike separation. There is an inertial frame, moving at less than \(c\), in which the two events occur at the same place but at different times, and they can never occur simultaneously.

Example 2.7 draws the twin trip on a spacetime diagram. Mary leaves Earth at \((x, ct) = (0,0)\) and returns at \(x=0\), \(ct = 20\) ly; her worldlines have slopes \(c/0.8c = 1.25\) outbound and \(-1.25\) on return, Frank's is the vertical line \(x=0\), and annual light signals have slopes \(\pm1\). The demo reproduces it for any speed and distance.

Four-vectors. A 3-vector \(\vec R = (x,y,z)\) has an invariant space interval \(d\ell^2 = dx^2+dy^2+dz^2\) under rotations. In the four-dimensional space with components \((x, y, z, ict)\), the corresponding interval \(ds^2 = dx^2+dy^2+dz^2-c^2dt^2 = ds'^2\) is invariant under the Lorentz transformation, and can be positive, zero, or negative. The four-vector formalism makes form-invariant equations easier to construct, at the cost of more mathematics (matrices, tensors, spinors) and differing conventions; the textbook does not use it.

2.10The Doppler effect

The "classical" Doppler effect is the increase in frequency of sound as a source approaches a receiver and the decrease as it recedes; it also occurs when the source is fixed and the receiver moves, and the size of the change depends on which is moving. That does not violate the principle of relativity, because sound has a special frame, that of its medium (air, water, a steel plate). For light there is no such medium, so only the relative motion of source and receiver can matter.

Let a source (star) at rest in K′ approach a receiver (astronomer) in K at speed \(v\). During a time \(T\) in K the source emits \(n\) waves, and the front of the wave train moves \(cT\) while the source moves \(vT\), so the length of the wave train is \(cT - vT\):

In its rest frame K′ the source emits \(n = f_0T_0'\) waves during the proper time \(T_0' = T/\gamma\) (time dilation), so

valid also when the source is fixed and the receiver approaches it. For source and receiver receding, replace \(v\to-v\): \(f = \sqrt{(1-\beta)/(1+\beta)}\,f_0\). If the light is emitted at an angle \(\theta'\) to the \(x'\) axis in K′, and makes the angle \(\theta\) with the \(x\) axis in K,

For \(\theta' = 0\) and \(180°\) this recovers the longitudinal formulas. When the light reaches the receiver at right angles to the source's motion in K (\(\theta = 90°\)), \(f = f_0/\gamma\): the transverse Doppler effect, a purely relativistic effect due directly to time dilation. The same relations explain stellar aberration.

Example 2.8: the twins again. Frank's annual signals reach Mary on the outbound trip, with \(\beta = -0.8\) (receding), at \(f = \sqrt{0.2/1.8}\,f_0 = f_0/3\): one every 3 years, so during her 6-year outbound trip she receives only 2. On the return (\(\beta = +0.8\)), \(f = 3f_0\), so she receives 18 more in 6 years: 20 in all, and she concludes Frank has aged 20 years. Mary's signals reach Frank at \(f_0/3\) until he sees her turnaround, 18 years into the trip (10 years for her to reach the star plus 8 for the light to return), by which time he has received 6. The remaining 6 arrive at \(3f_0\) during the last 2 years. Frank receives 12 in all and concludes Mary has aged 12 years, in agreement with the spacetime diagram.

Applications of the Doppler effect include astronomy (radial velocities, the rotation of stars and planets: Venus's retrograde rotation was found by comparing light reflected from its two sides), radar (speed guns, weather radar), the broadening of spectral lines, and laser cooling, where atoms moving toward a laser see it Doppler shifted into resonance, absorb its momentum, and slow down to microkelvin temperatures (Nobel Prize 1997: Chu, Cohen-Tannoudji, Phillips).

2.11Relativistic momentum

Newton's second law \(\vec F = d\vec p/dt\) is invariant under a Galilean transformation: with \(x' = x - vt\), \(p' = m\,dx'/dt' = p - mv\) and \(F' = dp'/dt' = dp/dt = F\), since \(m\) and \(v\) are constant. It is not invariant under a Lorentz transformation. Worse, by Newton's law a particle already moving very fast could be accelerated past \(c\), in conflict with the Lorentz transformation. Newton's second law must be modified at high speeds.

Apply the fundamental law of linear momentum conservation to a collision. Frank (system K) throws a ball along \(+y\) with speed \(u_0\); Mary (system K′, moving at \(v\) along \(x\)) throws an identical ball along \(-y'\) with speed \(u_0\) in her frame. The balls collide perfectly elastically, and each twin catches their own ball as it rebounds. Frank's ball returns along \(-y\) with speed \(u_0\), so with the classical momentum \(\vec p = m\vec u\), as observed by Frank,

Mary's ball has \(u_{Mx}' = 0\) and \(u_{My}' = -u_0\), so by the velocity transformation Frank measures \(u_{Mx} = v\) and \(u_{My} = -u_0/\gamma = -u_0\sqrt{1-v^2/c^2}\). Before the collision \(p_{My} = -mu_0\sqrt{1-v^2/c^2}\); after it \(u_{My}' = +u_0\), so \(p_{My} = +mu_0\sqrt{1-v^2/c^2}\). Hence

Linear momentum is not conserved if we use the classical form. Assuming momentum conservation is fundamental, modify the momentum instead: try \(\vec p = \Gamma(u)m\vec u\), with \(\Gamma\) depending on the speed \(u = |\vec u|\) of the particle (not the relative speed \(v\) of the two frames). Momentum is conserved in the collision if \(\Gamma(u) = 1/\sqrt{1-u^2/c^2}\) (Example 2.9). Using the proper time \(\tau\),

the relativistic momentum, which reduces to the classical momentum for \(u\ll c\). (Some physicists call \(m\) the "rest mass" \(m_0\) and write \(m = \gamma m_0\), so that \(\vec p = m\vec u\) keeps its form and the mass seems to increase at high speed. Most prefer to keep mass as an invariant, intrinsic property of an object, and so do we: "mass" means the rest mass \(m\), and \(\vec p = \gamma m\vec u\).)

2.12Relativistic energy

Modify Newton's second law to use the relativistic momentum,

The work done by \(\vec F\) moving a particle from position 1 to 2 along a path is \(W_{12} = \int_1^2\vec F\cdot d\vec s = K_2 - K_1\). Let the particle start from rest; then with \(d\vec s = \vec u\,dt\) and \(m\) invariant,

integrating by parts and substituting \(x = (u/c)^2\) in the remaining integral. The relativistic kinetic energy is

For \(u\ll c\), \(\gamma\approx1+u^2/2c^2\) and \(K\approx\tfrac12mu^2\), the classical kinetic energy. As \(u\to c\), \(K\to\infty\): since infinite energy is not available, no object with mass can reach the speed of light (Conceptual Example 2.10). Experiments accelerating electrons to many MeV confirm that their speed approaches \(c\) but never reaches it (Bertozzi, 1964).

Rewrite \(K = (\gamma - 1)mc^2\) as \(\gamma mc^2 = K + mc^2\). Even with zero velocity, and thus no kinetic energy, a particle has the rest energy \(E_0 = mc^2\) through its mass. The total energy is

expressing the equivalence of mass and energy and the conservation of mass-energy. For example, two masses \(m\) colliding at speed \(v\) and sticking together, with total energy \(E = 2mc^2 + 2K\), form a single body of mass \(M = E/c^2 > 2m\); the fractional mass increase is \(f_r = (M-2m)/2m = K/mc^2\). For \(m = 0.1\) kg and \(v = 10\) m/s, \(f_r\approx6\times10^{-16}\): far too small to notice in everyday life.

Energy and momentum. From \(p = \gamma mu\), with \(\beta^2 = 1 - 1/\gamma^2\),

\(E^2 - p^2c^2 = m^2c^4\), and \(m\), are invariant quantities. For massless particles such as photons, \(m = 0\) gives \(E = pc\). Since also \(E = \gamma mc^2\) would require \(\gamma\to\infty\) with \(m\to0\), i.e. \(u = c\): massless particles must travel at the speed of light.

2.13–2.14Computations, and electromagnetism and relativity

Section 2.13 introduces units convenient for modern physics, such as electron volts and MeV/\(c^2\) for masses, and binding energies of atoms and nuclei; read the textbook if you are interested.

Einstein wrote that what led him more or less directly to special relativity was the conviction that the electromagnetic force on a body moving in a magnetic field is nothing but an electric field. Maxwell's equations and the Lorentz force law are invariant in different inertial frames; here we show qualitatively that the magnetic force one observer sees is simply an electric force according to an observer in another frame. Electric fields come from charges, and magnetic fields from moving charges: the difference is relative motion.

Place a positive charge \(q_0\) outside a neutral wire in which electrons move at speed \(v\), with \(q_0\) also moving at \(v\). In the frame of the wire, the positive ions are at rest and the moving electrons have the same density, so there is no electric field: the moving electrons produce a magnetic field, which exerts the force \(q_0\vec v\times\vec B\) on the moving charge. In the frame of \(q_0\) (and the electrons), \(q_0\) is at rest and feels no magnetic force. But now the positive ions move, and their spacing is length contracted, while the electrons, now at rest, are more widely spaced than before. The wire has a net positive charge, and the resulting electric field repels \(q_0\). In the Lorentz force \(\vec F = q_0(\vec E + \vec v\times\vec B)\), the two observers simply split the same physics differently between \(\vec E\) and \(\vec B\): electric and magnetic fields are relative to the frame in which they are observed.

This idea extends to two conducting wires with currents (Problem 86) and underlies the definition of the ampere, and with it of electric charge. We have come full circle: the laws of electromagnetism expressed by Maxwell's equations are invariant in all inertial systems; only their interpretation as electric and magnetic fields is relative.

This completes Chapter 2. Chapter 3 turns to the experiments that led to quantum physics.

Further reading

• S. T. Thornton, A. Rex, and C. E. Hood, Modern Physics for Scientists and Engineers, 5th ed. (Cengage Learning), Chapter 2.
• A. P. French, Special Relativity (Norton, 1968).
• J. C. Hafele and R. E. Keating, "Around-the-world atomic clocks," Science 177, 166 and 168 (1972).
• T. Alväger, F. J. M. Farley, J. Kjellman, and I. Wallin, "Test of the second postulate of special relativity in the GeV region," Phys. Lett. 12, 260 (1964).
• W. Bertozzi, "Speed and kinetic energy of relativistic electrons," Am. J. Phys. 32, 551 (1964).
• E. F. Taylor and J. A. Wheeler, Spacetime Physics, 2nd ed. (Freeman, 1992).