Relativity
For three centuries physics rested on a quiet assumption: that space and time form a fixed stage, the same for everyone, on which events simply play out. In 1905 Einstein showed that assumption is wrong. Starting from just two postulates — that the laws of physics look the same to every uniformly moving observer, and that light travels at the same speed c for all of them — he was forced to a startling conclusion: moving clocks run slow, moving objects contract, and two events one observer calls simultaneous another does not. Space and time are not separate and absolute but woven together into a single spacetime whose measurements depend on who is doing the measuring. This chapter follows that logic from the postulates to its famous consequences — time dilation, length contraction, the Lorentz transformation, the relativistic velocity and Doppler laws — and ends at \(E = mc^2\), the equivalence of mass and energy that lights the stars.
- Einstein's two postulates: the laws of physics are the same in every inertial frame, and light moves at the same speed \(c = 299\,792\,458\,\mathrm{m/s}\) for all observers — an ultimate speed nothing with mass can reach.
- Simultaneity is relative: two observers in relative motion generally disagree about whether two separated events happen "at the same time."
- Time dilation: a moving clock runs slow. With the proper time \(\Delta t_0\) (measured where the events coincide), any other frame measures \(\Delta t = \gamma\,\Delta t_0\), where \(\beta = v/c\) and the Lorentz factor is \(\gamma = \dfrac{1}{\sqrt{1-\beta^{2}}}\).
- Length contraction: an object of proper length \(L_0\) is measured shorter along its direction of motion, \(L = \dfrac{L_0}{\gamma}\).
- The Lorentz transformation ties space and time together, \(x' = \gamma(x - vt)\) and \(t' = \gamma\!\left(t - \dfrac{vx}{c^{2}}\right)\), and yields the velocity addition rule \(u = \dfrac{u' + v}{1 + u'v/c^{2}}\), so speeds never exceed \(c\).
- Mass is energy: relativistic momentum is \(p = \gamma m v\), the rest energy is \(E_0 = mc^{2}\), the total energy is \(E = \gamma mc^{2} = mc^{2} + K\), and they obey \(E^{2} = (pc)^{2} + (mc^{2})^{2}\).
What Is Physics?
One principal subject of physics is relativity: the study of events — things that happen — and how their measured positions and times transform between reference frames in relative motion. By 1905 moving reference frames were routine; what was not routine was Einstein's special theory of relativity, which begins from two innocent-looking postulates and ends by overturning common sense. The word special signals that the theory treats only inertial reference frames — frames in which Newton's laws hold, with no gravitational acceleration. (The harder case of accelerating frames is the province of general relativity.)
Einstein's central discovery is that space and time are entangled: the time between two events depends on how far apart they happen, and that entanglement differs for observers moving relative to each other. Time does not tick off on some master clock that governs the universe; its rate is adjustable by relative motion. This is not philosophy. Every engineer working with the GPS satellites must correct for relativity — both special and general — because clocks aboard the satellites run at a different rate from clocks on the ground, and without those corrections the system would drift into uselessness within a day. The mathematics here is light; the difficulty is being scrupulous about who measures what, and how.
The Postulates
The entire theory rests on two statements, both exhaustively tested with no exception ever found.
Galileo had already assumed the laws of mechanics were frame-independent; Einstein extended that to all physics, including electromagnetism and optics. The first postulate does not claim every measured quantity is the same for all observers — most are not. It is the laws relating those measurements that agree. The second postulate is the radical one: light's speed does not depend on the motion of its source or its observer.
Measuring an Event
An event is something that happens — a bulb flashing, two particles colliding, a clock hand sweeping past a mark — and it carries three space coordinates and one time coordinate, collectively its spacetime coordinates. An event does not "belong" to any frame; anyone, in any inertial frame, may assign coordinates to it, and observers in relative motion will in general assign different ones.
A practical snag lurks here: light from a distant event reaches you late, so you cannot simply read a clock at your own location and call that the event's time. The fix is to imagine an array of measuring rods filling the frame, with a tiny synchronized clock at every intersection. Each observer reads the position off the nearest rods and the time off the nearest clock, eliminating any worry about signal travel time. The clocks are synchronized with light: standing at the origin, an observer flashes a pulse at \(t = 0\), and a helper a distance \(r\) away sets the local clock to \(t = r/c\) the instant the pulse arrives.
The Relativity of Simultaneity
Suppose Sam sees two separated events happen at the same instant. Will Sally, gliding past him at constant velocity, also call them simultaneous? In general, no.
Picture two ships and a red flare and a blue flare striking their ends. If the expanding light fronts reach Sam (at his ship's midpoint) together, he rightly declares the flares simultaneous. But Sally, moving toward the red flare's source and away from the blue, meets the red light first and rightly declares red happened before blue. There is one light front from each event, traveling at \(c\) in both frames, exactly as the second postulate demands — and that single fact forces the disagreement. When relative speeds are tiny compared with \(c\), the discrepancy is unmeasurably small, which is why this never troubles everyday life.
The Relativity of Time
Let Sally, riding a train, fire a light pulse straight up to a mirror a distance \(D\) above and catch it on return. For her the emission and the return happen at the same place, so she reads the interval off one clock: \(\Delta t_0 = 2D/c\). Sam, on the platform, sees the train carry the apparatus sideways during the flight, so for him the light travels a longer, slanted path of length \(2L\) between two events at different places, requiring two synchronized clocks. Because light travels at the same \(c\) for Sam, a longer path means a longer time. Eliminating the geometry gives the central result.
This is not an illusion. Muons created high in the atmosphere with a rest-frame lifetime of \(2.2\,\mu\mathrm{s}\) reach the ground in far greater numbers than classical physics allows, because in our frame their fast motion stretches that lifetime by the factor \(\gamma\). Hafele and Keating flew atomic clocks around the world and saw the predicted offsets; today any transported atomic clock is corrected for its motion.
The Relativity of Length
To measure a moving rod's length you must mark its two ends simultaneously in your frame — and since simultaneity is relative, length must be relative too. The length measured in the object's own rest frame is its proper length \(L_0\); any frame moving along that length measures less.
The Lorentz Transformation
Time dilation and length contraction are special cases of a single set of equations relating the coordinates one observer (frame \(S\)) assigns to an event and those another (frame \(S'\), moving at speed \(v\) along their common \(x\)-axis) assigns to the same event. At low speeds the old Galilean rules \(x' = x - vt,\ t' = t\) seemed obvious — they quietly assumed time runs the same for everyone. Einstein replaced them.
Some Consequences of the Lorentz Equations
The difference equations reproduce, in a few lines, everything argued earlier from the postulates.
The Relativity of Velocities
If a particle moves at velocity \(u'\) along the \(x'\)-axis of frame \(S'\), which itself moves at \(v\) relative to \(S\), what velocity \(u\) does \(S\) measure? Dividing the Lorentz difference equations gives the answer — and it is not simple addition.
Doppler Effect for Light
The Doppler effect for sound depends on the motion of source and detector through the air. Light needs no medium, so its Doppler shift depends only on the relative velocity of source and detector. Let \(f_0\) be the proper frequency (measured in the source's rest frame).
A New Look at Momentum
If we keep the classical definition \(\vec p = m\vec v\), total momentum is not conserved across inertial frames once speeds approach \(c\). Rather than abandon conservation of momentum, we refine the definition: measure the displacement \(\Delta x\) in the observer's frame but divide by the proper time \(\Delta t_0\) measured by a clock riding with the particle. Using \(\Delta t = \gamma\,\Delta t_0\) then gives the relativistic momentum.
A New Look at Energy
Einstein's deepest consequence is that mass is a form of energy, so conservation of energy becomes conservation of mass–energy. In chemical reactions the mass change is far too small to weigh; in nuclear reactions, where the energy released is roughly a million times larger, the mass change is easily measured.
If the products are lighter than the reactants, \(\Delta M < 0\), \(Q > 0\), and energy is released — the case for the fusion of hydrogen in the Sun, which lightens the products and pours the difference out as sunlight. If the products are heavier, \(Q < 0\) and energy must be supplied. Mass and energy are two currencies for the same thing, exchangeable at the rate \(c^{2}\).
Putting It to Work
Problem. Your starship cruises at \(v = 0.9990c\). You travel outward for \(10.0\,\mathrm{y}\) by your own clock, then turn and return in another \(10.0\,\mathrm{y}\). How much time passes on Earth? (Neglect the turnaround.)
Solution. Your \(10.0\,\mathrm{y}\) for each leg is a proper time \(\Delta t_0\) (start and end occur where you are), so Earth measures the dilated \(\Delta t = \gamma\,\Delta t_0\).
Both legs are alike, so the round trip takes \(2 \times 224 = 448\,\mathrm{y}\) of Earth time while you age only \(20\,\mathrm{y}\). You cannot visit the past, but high-speed travel is a genuine one-way ticket into the future.
Problem. A positive kaon has a proper (rest-frame) lifetime of \(0.1237\,\mu\mathrm{s}\). Moving at \(0.990c\) through the lab, how far does it travel before decaying, (a) by classical physics and (b) by relativity?
Solution. Classically, distance is just speed times the rest lifetime. Relativistically, the lab-frame lifetime is dilated, \(\Delta t = \gamma\,\Delta t_0\), with \(\gamma = 1/\sqrt{1 - 0.990^{2}} \approx 7.09\).
The relativistic distance is about seven times the classical one — and matches experiment. The kaon's clock runs slow in the lab frame, so it survives long enough to cross roughly \(260\,\mathrm{m}\), not \(37\,\mathrm{m}\). Such tests became routine in particle labs long ago.
Problem. A starship moving at \(0.980c\) detects a microwave burst at one base and, \(\Delta t = 1.10\,\mathrm{s}\) later, an explosion at another, \(\Delta x = 4.00\times10^{8}\,\mathrm{m}\) away in the ship frame. In the planet–moon frame, what are \(\Delta x'\) and \(\Delta t'\), and could the burst have caused the explosion?
Solution. With \(\gamma = 1/\sqrt{1 - 0.980^{2}} \approx 5.0252\), transform the pair of events.
The sign flips: in the planet–moon frame the explosion occurs before the burst. Checking the speed information would need to travel between them gives about \(3.6\times10^{8}\,\mathrm{m/s} > c\), which is impossible — so neither event could have caused the other. They are causally unrelated, and only that is why their order may legitimately reverse.
Problem. A "\(2.53\,\mathrm{MeV}\) electron" has kinetic energy \(K = 2.53\,\mathrm{MeV}\). The electron's rest energy is \(mc^{2} = 0.511\,\mathrm{MeV}\). Find (a) its total energy and (b) its momentum in \(\mathrm{MeV}/c\).
Solution. Total energy is rest plus kinetic; momentum comes from the energy–momentum relation.
So \(p \approx 3.00\,\mathrm{MeV}/c\). Note \(c\) here is the symbol for the speed of light, used as part of the momentum unit, not a separate factor — a convention worth getting comfortable with in particle physics.
Chapter Summary
Laws of physics are the same in all inertial frames; light moves at the same \(c\) for all. \(c\) is an ultimate speed no massive particle can reach.
Observers in relative motion generally disagree on whether two separated events are simultaneous. It is relative, not absolute.
\(\Delta t = \gamma\,\Delta t_0\) with proper time \(\Delta t_0\), \(\beta = v/c\), \(\gamma = 1/\sqrt{1-\beta^{2}}\). Moving clocks run slow.
\(L = L_0/\gamma\) along the motion only, with proper length \(L_0\).
\(x' = \gamma(x - vt)\), \(t' = \gamma(t - vx/c^{2})\), \(y'=y\), \(z'=z\); space and time entangled.
\(u = \dfrac{u' + v}{1 + u'v/c^{2}}\); combining sub-light speeds never exceeds \(c\).
Separating: \(f = f_0\sqrt{\dfrac{1-\beta}{1+\beta}}\); transverse: \(f = f_0\sqrt{1-\beta^{2}}\) (time dilation).
\(p = \gamma mv\), \(E = \gamma mc^{2} = mc^{2}+K\), \(K = mc^{2}(\gamma-1)\), \(E^{2} = (pc)^{2}+(mc^{2})^{2}\).
Problems
Use the exact value \(c = 299\,792\,458\,\mathrm{m/s}\) where precision matters. Identify the proper quantity first: the proper time is read on a single clock present at both events, and the proper length is measured in the object's rest frame. Keep distance and time in the same frame whenever you compute a speed, and never mix one event's coordinates with another's when applying the Lorentz equations.
- The mean lifetime of muons at rest is \(2.2000\,\mu\mathrm{s}\). Cosmic-ray muons observed from Earth have a measured mean lifetime of \(16.000\,\mu\mathrm{s}\). To five significant figures, find the speed parameter \(\beta\) of these muons relative to Earth.
- To eight significant figures, find the speed parameter \(\beta\) when the Lorentz factor \(\gamma\) equals (a) \(1.0100000\), (b) \(10.000000\), and (c) \(100.00000\).
- An unstable particle leaves a track \(1.05\,\mathrm{mm}\) long in a detector before decaying; its speed relative to the detector was \(0.992c\). What is its proper lifetime — how long would it have lasted at rest in the detector?
- A spaceship of rest length \(130\,\mathrm{m}\) passes a timing station at \(0.740c\). (a) What length does the station measure? (b) What time interval does the station clock record between the passage of the ship's front and back ends?
- A rod's rest length is \(1.70\,\mathrm{m}\) and it moves along the \(x\)-axis at \(0.630c\). What is its measured length in that frame?
- A spaceship is measured to be exactly half its rest length. (a) To three significant figures, what is its speed parameter \(\beta\)? (b) By what factor do its clocks run slow relative to the observer's?
- A traveler moves at \(0.9900c\) toward the star Vega, \(26.00\,\mathrm{ly}\) away. How much time elapses on Earth clocks when the traveler reaches Vega, and how much does the traveler age over the same trip (measured in her frame)?
- Observer \(S\) reports an event on the \(x\)-axis at \(x = 3.00\times10^{8}\,\mathrm{m}\), \(t = 2.50\,\mathrm{s}\). Frame \(S'\) moves in the positive \(x\)-direction at \(0.400c\) with origins coinciding at \(t = t' = 0\). Find the event's \(x'\) and \(t'\).
- An experimenter triggers two flashes simultaneously, one at the origin and one at \(x = 30.0\,\mathrm{km}\). An observer moving at \(0.250c\) in the \(+x\) direction also views them. (a) What time interval does she measure between the flashes? (b) Which does she say came first?
- A particle moves along the \(x'\)-axis of frame \(S'\) at \(0.40c\), while \(S'\) moves at \(0.60c\) relative to \(S\). What is the particle's velocity in frame \(S\)?
- Galaxy A recedes from us at \(0.35c\); Galaxy B recedes at the same speed in exactly the opposite direction. What recessional speed (as a multiple of \(c\)) would an observer on Galaxy A find for (a) our galaxy and (b) Galaxy B?
- Assuming the low-speed shift law holds, how fast would you have to approach a red traffic light (\(620\,\mathrm{nm}\)) for it to appear green (\(540\,\mathrm{nm}\))?
- A spaceship recedes from Earth at \(0.900c\) and transmits at \(100\,\mathrm{MHz}\) in its own frame. To what frequency must Earth receivers be tuned?
- Certain spectral lines from a galaxy in Virgo are \(0.4\%\) longer than the same lines from an Earth source. (a) What is the galaxy's radial speed? (b) Is it approaching or receding?
- How much work must be done to increase an electron's speed from rest to (a) \(0.500c\), (b) \(0.990c\), and (c) \(0.9990c\)?
- What momentum (in \(\mathrm{MeV}/c\)) does an electron have when its kinetic energy is \(2.00\,\mathrm{MeV}\)? (Take the electron rest energy as \(0.511\,\mathrm{MeV}\).)
- What must the momentum of a particle of mass \(m\) be, in terms of \(mc\), so that its total energy is \(3.00\) times its rest energy?
- For the reaction \(p + {}^{19}\mathrm{F} \to \alpha + {}^{16}\mathrm{O}\) with \(m(p) = 1.007825\,\mathrm{u}\), \(m(\alpha) = 4.002603\,\mathrm{u}\), \(m({}^{19}\mathrm{F}) = 18.998405\,\mathrm{u}\), \(m({}^{16}\mathrm{O}) = 15.994915\,\mathrm{u}\), calculate the \(Q\) of the reaction.