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Relativity: Time Dilation and Length Contraction
Modern Physics
Relativity says that time and length are not absolute: a fast-moving clock ticks more slowly and a fast-moving object is shorter, as measured by someone at rest.
What relativity says
Special relativity rests on two postulates: the laws of physics are the same in all inertial frames, and the speed of light in a vacuum, c = 3.0 x 10^8 m/s, is the same for all observers, however they move. From these two simple statements, strange consequences follow. Time dilation means a moving clock ticks more slowly than a clock at rest, as seen by a stationary observer. Length contraction means a moving object is shorter along its direction of motion. The effects are described by the Lorentz factor gamma, which is close to 1 at everyday speeds and grows without limit as speed approaches c.
Evidence for relativity
Time dilation and length contraction are not illusions; they have been measured directly. Muons created high in the atmosphere live longer, as measured on Earth, than they do at rest, because they travel near the speed of light and their internal clocks run slow in our frame, letting them reach the ground. Atomic clocks flown on aircraft tick measurably slower than identical clocks left on the ground, and the GPS system must correct for both special and general relativistic effects, or its positions would drift by kilometres each day. The faster an object moves, the harder it becomes to accelerate it further, because its energy grows with gamma.
Mass, energy and gravity
The most famous result is E = mc^2: mass and energy are two forms of the same thing, and a mass m at rest has energy mc^2. This explains why the Sun shines: in fusion, a small amount of mass is lost and appears as a large amount of energy, because c^2 is enormous. The same equation governs nuclear power and nuclear weapons, positron emission in medical PET scans, and the annihilation of particles in accelerators. General relativity extends the idea to gravity, describing it as the curvature of spacetime, and it predicts the bending of light by the Sun and the existence of black holes and gravitational waves.
Worked example: time dilation and length contraction at 0.8c
A spacecraft travels at v = 0.80 c relative to Earth. First find gamma: gamma = 1 / sqrt(1 - v^2/c^2) = 1 / sqrt(1 - 0.64) = 1 / sqrt(0.36) = 1 / 0.60 = 1.67. Time dilation: a clock on the ship ticks 10 s of proper time, delta t0 = 10 s. On Earth the time is delta t = gamma x delta t0 = 1.67 x 10 = 16.7 s. Length contraction: a 100 m ship, measured on Earth, is L = L0 / gamma = 100 / 1.67 = 60 m. So Earth observers see the ship's clock run slow and its length shortened along its motion.
Common mistakes
- Applying time dilation to everyday speeds: at ordinary speeds gamma is so close to 1 that the effect is far too small to notice.
- Saying the moving clock is really slow in an absolute sense: each observer sees the other's clock run slow; the effect is symmetric between inertial frames.
- Thinking E = mc^2 means mass turns into energy only in bombs: it applies to every mass, and a small mass holds an enormous amount of energy.
Practice questions
- A clock ticks 8.0 s in its own frame while moving at 0.60 c. How much time passes for a stationary observer?
Show answer
gamma = 1/sqrt(1 - 0.36) = 1/sqrt(0.64) = 1/0.80 = 1.25. delta t = 1.25 x 8.0 = 10 s.
- At 0.60 c, what length does a stationary observer measure for a 50 m object?
Show answer
L = 50/1.25 = 40 m.
- What is the rest energy of 1.0 g of mass?
Show answer
E = 0.0010 x (3.0 x 10^8)^2 = 0.0010 x 9.0 x 10^16 = 9.0 x 10^13 J.
Frequently asked questions
Can anything travel faster than light?
No. The theory says no object with mass can reach or exceed c, because its energy would become infinite.
If each observer sees the other's clock run slow, who is right?
Both are correct in their own frames; there is no single absolute time, only time measured in a given frame.