Physics › Turning points › The consequences of special relativity
The consequences of special relativity
Grant Einstein his two postulates and the furniture of common sense starts to move: clocks in flight run slow, lengths shrink along the direction of travel, and mass grows with speed until no push can gain another metre per second. Each claim has been measured, and each measurement agreed.
Builds on The Michelson-Morley experiment and Mass-energy and binding energy.
IN THIS TOPIC
- Use the time dilation equation with proper time, and cite muon decay as its evidence.
- Use the length contraction equation with proper length.
- Describe how mass and kinetic energy vary with speed, and Bertozzi's direct test.
WHAT YOU PROBABLY THINK
With a strong enough accelerator you could push an electron past the speed of light.
Time dilation
Hold both postulates at once and something has to give, and what gives is simultaneity's little luxuries: universal time first. The proper time t0 between two events is the time measured in the inertial frame where both events happen at the same place, a clock riding along with them. Any observer moving at speed v relative to that clock measures a longer time:
The root is smaller than one, so t is bigger than t0: a moving clock runs slow as judged from the frame it moves through. At everyday speeds the correction is invisibly tiny; near c it is enormous.
WORKED EXAMPLE
A muon's stretched lifetime
A muon at rest lives about t0 = 2.2 μs before decaying. Find its lifetime as measured by a laboratory it passes at 0.99c.
t = t0/√(1 − v2/c2) = 2.2/√(1 − 0.992) = 2.2/0.141 = 16 μs.
The muon's own clock still reads 2.2 μs at decay; the laboratory watches that clock run seven times slow. Proper time belongs to the muon, because both events, birth and decay, happen at the muon.
The muons that should not arrive
Nature runs this experiment constantly. Cosmic rays striking the upper atmosphere create muons about 10 km up, travelling near c. One undilated lifetime at 0.996c covers 660 m, and the 10 km descent lasts fifteen lifetimes: decay being exponential, barely two muons in ten million should survive it. Detectors on the ground disagree: around a quarter of them arrive.
Time dilation balances the ledger exactly. At 0.996c the factor is 11.2, the laboratory-frame lifetime stretches to 25 μs, the descent lasts only 1.4 dilated lifetimes, and the predicted survival is 26 per cent, matching what detectors count. Muon showers are time dilation measured daily, with the whole sky for apparatus.
Length contraction
The same root reaches the metre rule. The proper length l0 of an object is its length measured in its own rest frame; an observer it passes at speed v measures it shorter along the direction of motion:
Nothing is crushed: lengths, like durations, are simply relations between frames. The muon insists on it. In the muon's rest frame its lifetime really is 2.2 μs; what saves the muon is that the onrushing atmosphere is length-contracted from 10 km to under 900 m, which at 0.996c takes 3.0 μs, the same 1.4 lifetimes the ground observer counted. Two frames, two stories, one identical survival rate: relativity's books always balance.
YOUR TURN
The travelling metre
A metre rule flies past at 0.80c, aligned with its motion. Find its measured length, and its measured length if instead it flies at the same speed aligned across its motion, before opening the working.
Show the working
Along the motion: l = 1.00 × √(1 − 0.802) = 1.00 × 0.60 = 0.60 m.
Across the motion: 1.00 m, unchanged. Contraction acts only along the direction of travel, a detail exam questions love to probe.
Mass, energy, and Bertozzi's electrons
The last casualty is mass. Push on an object near light speed and the work done buys ever less speed, behaving as if the inertia itself grows: the mass of an object moving at speed v rises as m0/√(1 − v2/c2), where m0 is its rest mass, and the equivalence of mass and energy prices it all:
As v approaches c the denominator collapses toward zero, so the mass, and with it the kinetic energy, grows without limit. That is what stops any accelerator, however strong, pushing an electron past c, and why lesson one's check question broke: eV = ½mv2 keeps pretending the work becomes speed, when past a few tenths of c it increasingly becomes mass-energy instead. An electron's rest energy m0c2 is 8.2 × 10−14 J, about 0.51 MeV, and every joule an accelerator adds beyond that buys mostly inertia.
In 1964 William Bertozzi tested this as directly as possible: accelerate electrons through a known pd, so the kinetic energy is known from eV, and time their flight over 8.4 m to measure the speed outright. Classically v2 should climb in proportion to the energy forever; his electrons instead saturated just below c, their measured speed flattening while the energy, delivered as heat to the target, kept rising exactly as supplied. Kinetic energy varies with speed the relativistic way, by direct measurement.
TRY IT UNSEEN
An impossible speed, corrected
An electron is given 2.0 MeV of kinetic energy. Show that classical physics predicts an impossible speed, and find the true speed (electron rest energy 0.51 MeV).
Show the working
Classically ½m0v2 = 2.0 MeV gives v = 8.4 × 108 m s−1: 2.8 times the speed of light, forbidden.
Relativistically the total energy is E = 2.0 + 0.51 = 2.51 MeV, so E/m0c2 = 4.9, and v/c = √(1 − 1/4.92) = 0.98: v = 2.9 × 108 m s−1, just under c, exactly where Bertozzi's electrons sat.
THE EXAM BIT
- Choose the proper quantity first, and say why: proper time is measured in the frame where the two events happen at the same place; proper length in the object's rest frame. Most lost marks are proper-quantity errors.
- Sanity-check every answer's direction: moving clocks read less time (t0 is the smallest), moving lengths measure shorter (l0 is the longest).
- The muon answer needs numbers in a chain: lifetime dilated by the factor, descent time compared with it, survival transformed from negligible to substantial, matching detection.
- Sketching mass or kinetic energy against speed: start at the rest value, stay near it to a few tenths of c, then rise steeply toward a vertical asymptote at c, never touching it.
- Bertozzi scores on the method's directness: kinetic energy known from the accelerating pd (and checked calorimetrically), speed measured by time of flight, and the measured v2 flattening below c.
CHECK YOURSELF
A spacecraft passes Earth at 0.60c. Its on-board clock records 100 s between two ticks, and its proper length is 50 m. Find the time between those ticks and the craft's length as measured from Earth, and state which measurement each observer would call "correct".
Show a hint
The root is √(1 − 0.36) = 0.80; decide which quantity dilates and which contracts.
Show the answer
Time: the ticks happen at the ship, so 100 s is proper time, and Earth measures t = 100/0.80 = 125 s: the moving clock runs slow.
Length: 50 m is proper length, and Earth measures l = 50 × 0.80 = 40 m, contracted along the motion.
Both are correct. Each observer's measurements are right in their own inertial frame; relativity's content is precisely that these quantities are frame-dependent, with the proper values belonging to the frame riding with the clock and the rule.
Moving clocks run slow and moving lengths shrink, by the same root, and the muons prove it.
Mass grows with speed and E = mc² prices it: c is a wall no energy can breach.
WORKBOOK
Printable practice for this topic: original exam-style questions with room to work, and a fully worked answer book. Free to use; please do not redistribute or sell.
CHECK YOUR PROGRESS
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- Use the time dilation equation with proper time, and cite muon decay as its evidence.
- Use the length contraction equation with proper length.
- Describe how mass and kinetic energy vary with speed, and Bertozzi's direct test.
Open the full revision checklist to track your progress across the whole unit.
No animated video for this topic yet; these notes stand alone. InkPhysics on YouTube.