Have you ever driven on a road that seems like it goes on forever? If you look ahead, you might say you have about 10 km left to go. Another traveler might say the road ahead looks like it’s about 15 km long. If you both measured the road, however, you would agree. Traveling at everyday speeds, the distance you both measure would be the same. You will read in this section, however, that this is not true at relativistic speeds. Close to the speed of light, distances measured are not the same when measured by different observers.

# Proper Length

One thing all observers agree upon is relative speed. Even though clocks measure different elapsed times for the same process, they still agree that relative speed, which is distance divided by elapsed time, is the same. This implies that distance, too, depends on the observer’s relative motion. If two observers see different times, then they must also see different distances for relative speed to be the same to each of them.

The muon discussed in [link] illustrates this concept. To an observer on the Earth, the muon travels at $0.950c$ for $7.05\phantom{\rule{0.25em}{0ex}}\mu \mathrm{s}$ from the time it is produced until it decays. Thus it travels a distance

relative to the Earth. In the muon’s frame of reference, its lifetime is only $2.20\phantom{\rule{0.25em}{0ex}}\mu \mathrm{s}$. It has enough time to travel only

The distance between the same two events (production and decay of a muon) depends on who measures it and how they are moving relative to it.

The Earth-bound observer measures the proper length ${L}_{0}$, because the points at which the muon is produced and decays are stationary relative to the Earth. To the muon, the Earth, air, and clouds are moving, and so the distance $L$ it sees is not the proper length.

# Length Contraction

To develop an equation relating distances measured by different observers, we note that the velocity relative to the Earth-bound observer in our muon example is given by

The time relative to the Earth-bound observer is $\mathrm{\Delta}t$, since the object being timed is moving relative to this observer. The velocity relative to the moving observer is given by

The moving observer travels with the muon and therefore observes the proper time $\mathrm{\Delta}{t}_{0}$. The two velocities are identical; thus,

We know that $\mathrm{\Delta}t=\gamma \mathrm{\Delta}{t}_{0}$. Substituting this equation into the relationship above gives

Substituting for $\gamma $ gives an equation relating the distances measured by different observers.

If we measure the length of anything moving relative to our frame, we find its length $L$ to be smaller than the proper length ${L}_{0}$ that would be measured if the object were stationary. For example, in the muon’s reference frame, the distance between the points where it was produced and where it decayed is shorter. Those points are fixed relative to the Earth but moving relative to the muon. Clouds and other objects are also contracted along the direction of motion in the muon’s reference frame.

Suppose an astronaut, such as the twin discussed in Simultaneity and Time Dilation, travels so fast that $\gamma =\text{30}\text{.}\text{00}$. (a) She travels from the Earth to the nearest star system, Alpha Centauri, 4.300 light years (ly) away as measured by an Earth-bound observer. How far apart are the Earth and Alpha Centauri as measured by the astronaut? (b) In terms of $c$, what is her velocity relative to the Earth? You may neglect the motion of the Earth relative to the Sun. (See [link].)

**Strategy**

First note that a light year (ly) is a convenient unit of distance on an astronomical scale—it is the distance light travels in a year. For part (a), note that the 4.300 ly distance between the Alpha Centauri and the Earth is the proper distance ${L}_{0}$, because it is measured by an Earth-bound observer to whom both stars are (approximately) stationary. To the astronaut, the Earth and the Alpha Centauri are moving by at the same velocity, and so the distance between them is the contracted length $L$. In part (b), we are given $\gamma $, and so we can find $v$ by rearranging the definition of $\gamma $ to express $v$ in terms of $c$.

**Solution for (a)**

- Identify the knowns. ${L}_{0}-\mathrm{4.300\; ly}$; $\gamma =\text{30}\text{.}\text{00}$
- Identify the unknown. $L$
- Choose the appropriate equation. $L=\frac{{L}_{0}}{\gamma}$
- Rearrange the equation to solve for the unknown.
$\begin{array}{lll}L& =& \frac{{L}_{0}}{\gamma}\\ & =& \frac{\mathrm{4.300\; ly}}{\text{30.00}}\\ & =& \text{0.1433 ly}\end{array}$

**Solution for (b)**

- Identify the known. $\gamma =\text{30}\text{.}\text{00}$
- Identify the unknown. $v$ in terms of $c$
- Choose the appropriate equation. $\gamma =\frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}$
- Rearrange the equation to solve for the unknown.
$\begin{array}{lll}\gamma & =& \frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\\ \text{30.00}& =& \frac{1}{\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}}\end{array}$
Squaring both sides of the equation and rearranging terms gives

$\text{900}\text{.}0=\frac{1}{1-\frac{{v}^{2}}{{c}^{2}}}$so that

$1-\frac{{v}^{2}}{{c}^{2}}=\frac{1}{\text{900}\text{.}0}$and

$\frac{{v}^{2}}{{c}^{2}}=1-\frac{1}{\text{900}\text{.}0}=0\text{.}\text{99888}\text{.}\text{.}\text{.}.$Taking the square root, we find

$\frac{v}{c}=0\text{.}\text{99944},$which is rearranged to produce a value for the velocity

$\text{v=}\phantom{\rule{0.25em}{0ex}}0\text{.}\text{9994}c.$

**Discussion**

First, remember that you should not round off calculations until the final result is obtained, or you could get erroneous results. This is especially true for special relativity calculations, where the differences might only be revealed after several decimal places. The relativistic effect is large here ($\text{\gamma =}\text{30}\text{.}\text{00}$), and we see that $v$ is approaching (not equaling) the speed of light. Since the distance as measured by the astronaut is so much smaller, the astronaut can travel it in much less time in her frame.

People could be sent very large distances (thousands or even millions of light years) and age only a few years on the way if they traveled at extremely high velocities. But, like emigrants of centuries past, they would leave the Earth they know forever. Even if they returned, thousands to millions of years would have passed on the Earth, obliterating most of what now exists. There is also a more serious practical obstacle to traveling at such velocities; immensely greater energies than classical physics predicts would be needed to achieve such high velocities. This will be discussed in Relatavistic Energy.

Why don’t we notice length contraction in everyday life? The distance to the grocery shop does not seem to depend on whether we are moving or not. Examining the equation $L={L}_{0}\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}$, we see that at low velocities ($v\text{<<}c$) the lengths are nearly equal, the classical expectation. But length contraction is real, if not commonly experienced. For example, a charged particle, like an electron, traveling at relativistic velocity has electric field lines that are compressed along the direction of motion as seen by a stationary observer. (See [link].) As the electron passes a detector, such as a coil of wire, its field interacts much more briefly, an effect observed at particle accelerators such as the 3 km long Stanford Linear Accelerator (SLAC). In fact, to an electron traveling down the beam pipe at SLAC, the accelerator and the Earth are all moving by and are length contracted. The relativistic effect is so great than the accelerator is only 0.5 m long to the electron. It is actually easier to get the electron beam down the pipe, since the beam does not have to be as precisely aimed to get down a short pipe as it would down one 3 km long. This, again, is an experimental verification of the Special Theory of Relativity.

# Summary

- All observers agree upon relative speed.
- Distance depends on an observer’s motion. Proper length ${L}_{0}$ is the distance between two points measured by an observer who is at rest relative to both of the points. Earth-bound observers measure proper length when measuring the distance between two points that are stationary relative to the Earth.
- Length contraction $L$ is the shortening of the measured length of an object moving relative to the observer’s frame:${\text{L=L}}_{0}\sqrt{1-\frac{{v}^{2}}{{c}^{2}}}=\frac{{L}_{0}}{\gamma}.$

# Conceptual Questions

To whom does an object seem greater in length, an observer moving with the object or an observer moving relative to the object? Which observer measures the object’s proper length?

Relativistic effects such as time dilation and length contraction are present for cars and airplanes. Why do these effects seem strange to us?

Suppose an astronaut is moving relative to the Earth at a significant fraction of the speed of light. (a) Does he observe the rate of his clocks to have slowed? (b) What change in the rate of Earth-bound clocks does he see? (c) Does his ship seem to him to shorten? (d) What about the distance between stars that lie on lines parallel to his motion? (e) Do he and an Earth-bound observer agree on his velocity relative to the Earth?

# Problems & Exercises

A spaceship, 200 m long as seen on board, moves by the Earth at $0\text{.}\text{970}c$. What is its length as measured by an Earth-bound observer?

48.6 m

How fast would a 6.0 m-long sports car have to be going past you in order for it to appear only 5.5 m long?

(a) How far does the muon in [link] travel according to the Earth-bound observer? (b) How far does it travel as viewed by an observer moving with it? Base your calculation on its velocity relative to the Earth and the time it lives (proper time). (c) Verify that these two distances are related through length contraction $\text{\gamma =}3\text{.}\text{20}$.

(a) 1.387 km = 1.39 km

(b) 0.433 km

(c) $\begin{array}{}L=\frac{{L}_{0}}{\gamma}=\frac{1.387\times {10}^{3}\phantom{\rule{0.25em}{0ex}}\text{m}}{3.20}\\ & =& \mathrm{433.4\; m}=\text{0.433 km}\end{array}$

Thus, the distances in parts (a) and (b) are related when $\gamma =\text{3.20}$.

(a) How long would the muon in [link] have lived as observed on the Earth if its velocity was $0\text{.}\text{0500}c$? (b) How far would it have traveled as observed on the Earth? (c) What distance is this in the muon’s frame?

(a) How long does it take the astronaut in [link] to travel 4.30 ly at $0\text{.99944}c$ (as measured by the Earth-bound observer)? (b) How long does it take according to the astronaut? (c) Verify that these two times are related through time dilation with $\text{\gamma =}\text{30}\text{.}\text{00}$ as given.

(a) 4.303 y (to four digits to show any effect)

(b) 0.1434 y

(c) $\text{\Delta t}={\text{\gamma \Delta t}}_{0}\Rightarrow \gamma =\frac{\text{\Delta t}}{{\text{\Delta t}}_{0}}=\frac{4\text{.}\text{303 y}}{0\text{.}\text{1434 y}}=\text{30}\text{.}0$

Thus, the two times are related when $\text{\gamma =}\text{30}\text{.}\text{00}$.

(a) How fast would an athlete need to be running for a 100-m race to look 100 yd long? (b) Is the answer consistent with the fact that relativistic effects are difficult to observe in ordinary circumstances? Explain.

**Unreasonable Results**

(a) Find the value of $\gamma $ for the following situation. An astronaut measures the length of her spaceship to be 25.0 m, while an Earth-bound observer measures it to be 100 m. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

(a) 0.250

(b) $\gamma $ must be ≥1

(c) The Earth-bound observer must measure a shorter length, so it is unreasonable to assume a longer length.

**Unreasonable Results**

A spaceship is heading directly toward the Earth at a velocity of $0\text{.}\text{800}c$. The astronaut on board claims that he can send a canister toward the Earth at $1\text{.}\text{20}c$ relative to the Earth. (a) Calculate the velocity the canister must have relative to the spaceship. (b) What is unreasonable about this result? (c) Which assumptions are unreasonable or inconsistent?

- College Physics
- Preface
- Introduction: The Nature of Science and Physics
- Kinematics
- Introduction to One-Dimensional Kinematics
- Displacement
- Vectors, Scalars, and Coordinate Systems
- Time, Velocity, and Speed
- Acceleration
- Motion Equations for Constant Acceleration in One Dimension
- Problem-Solving Basics for One-Dimensional Kinematics
- Falling Objects
- Graphical Analysis of One-Dimensional Motion

- Two-Dimensional Kinematics
- Dynamics: Force and Newton's Laws of Motion
- Introduction to Dynamics: Newton’s Laws of Motion
- Development of Force Concept
- Newton’s First Law of Motion: Inertia
- Newton’s Second Law of Motion: Concept of a System
- Newton’s Third Law of Motion: Symmetry in Forces
- Normal, Tension, and Other Examples of Forces
- Problem-Solving Strategies
- Further Applications of Newton’s Laws of Motion
- Extended Topic: The Four Basic Forces—An Introduction

- Further Applications of Newton's Laws: Friction, Drag, and Elasticity
- Uniform Circular Motion and Gravitation
- Work, Energy, and Energy Resources
- Linear Momentum and Collisions
- Statics and Torque
- Rotational Motion and Angular Momentum
- Introduction to Rotational Motion and Angular Momentum
- Angular Acceleration
- Kinematics of Rotational Motion
- Dynamics of Rotational Motion: Rotational Inertia
- Rotational Kinetic Energy: Work and Energy Revisited
- Angular Momentum and Its Conservation
- Collisions of Extended Bodies in Two Dimensions
- Gyroscopic Effects: Vector Aspects of Angular Momentum

- Fluid Statics
- Fluid Dynamics and Its Biological and Medical Applications
- Introduction to Fluid Dynamics and Its Biological and Medical Applications
- Flow Rate and Its Relation to Velocity
- Bernoulli’s Equation
- The Most General Applications of Bernoulli’s Equation
- Viscosity and Laminar Flow; Poiseuille’s Law
- The Onset of Turbulence
- Motion of an Object in a Viscous Fluid
- Molecular Transport Phenomena: Diffusion, Osmosis, and Related Processes

- Temperature, Kinetic Theory, and the Gas Laws
- Heat and Heat Transfer Methods
- Thermodynamics
- Introduction to Thermodynamics
- The First Law of Thermodynamics
- The First Law of Thermodynamics and Some Simple Processes
- Introduction to the Second Law of Thermodynamics: Heat Engines and Their Efficiency
- Carnot’s Perfect Heat Engine: The Second Law of Thermodynamics Restated
- Applications of Thermodynamics: Heat Pumps and Refrigerators
- Entropy and the Second Law of Thermodynamics: Disorder and the Unavailability of Energy
- Statistical Interpretation of Entropy and the Second Law of Thermodynamics: The Underlying Explanation

- Oscillatory Motion and Waves
- Introduction to Oscillatory Motion and Waves
- Hooke’s Law: Stress and Strain Revisited
- Period and Frequency in Oscillations
- Simple Harmonic Motion: A Special Periodic Motion
- The Simple Pendulum
- Energy and the Simple Harmonic Oscillator
- Uniform Circular Motion and Simple Harmonic Motion
- Damped Harmonic Motion
- Forced Oscillations and Resonance
- Waves
- Superposition and Interference
- Energy in Waves: Intensity

- Physics of Hearing
- Electric Charge and Electric Field
- Introduction to Electric Charge and Electric Field
- Static Electricity and Charge: Conservation of Charge
- Conductors and Insulators
- Coulomb’s Law
- Electric Field: Concept of a Field Revisited
- Electric Field Lines: Multiple Charges
- Electric Forces in Biology
- Conductors and Electric Fields in Static Equilibrium
- Applications of Electrostatics

- Electric Potential and Electric Field
- Electric Current, Resistance, and Ohm's Law
- Circuits, Bioelectricity, and DC Instruments
- Magnetism
- Introduction to Magnetism
- Magnets
- Ferromagnets and Electromagnets
- Magnetic Fields and Magnetic Field Lines
- Magnetic Field Strength: Force on a Moving Charge in a Magnetic Field
- Force on a Moving Charge in a Magnetic Field: Examples and Applications
- The Hall Effect
- Magnetic Force on a Current-Carrying Conductor
- Torque on a Current Loop: Motors and Meters
- Magnetic Fields Produced by Currents: Ampere’s Law
- Magnetic Force between Two Parallel Conductors
- More Applications of Magnetism

- Electromagnetic Induction, AC Circuits, and Electrical Technologies
- Introduction to Electromagnetic Induction, AC Circuits and Electrical Technologies
- Induced Emf and Magnetic Flux
- Faraday’s Law of Induction: Lenz’s Law
- Motional Emf
- Eddy Currents and Magnetic Damping
- Electric Generators
- Back Emf
- Transformers
- Electrical Safety: Systems and Devices
- Inductance
- RL Circuits
- Reactance, Inductive and Capacitive
- RLC Series AC Circuits

- Electromagnetic Waves
- Geometric Optics
- Vision and Optical Instruments
- Wave Optics
- Introduction to Wave Optics
- The Wave Aspect of Light: Interference
- Huygens's Principle: Diffraction
- Young’s Double Slit Experiment
- Multiple Slit Diffraction
- Single Slit Diffraction
- Limits of Resolution: The Rayleigh Criterion
- Thin Film Interference
- Polarization
- *Extended Topic* Microscopy Enhanced by the Wave Characteristics of Light

- Special Relativity
- Introduction to Quantum Physics
- Atomic Physics
- Introduction to Atomic Physics
- Discovery of the Atom
- Discovery of the Parts of the Atom: Electrons and Nuclei
- Bohr’s Theory of the Hydrogen Atom
- X Rays: Atomic Origins and Applications
- Applications of Atomic Excitations and De-Excitations
- The Wave Nature of Matter Causes Quantization
- Patterns in Spectra Reveal More Quantization
- Quantum Numbers and Rules
- The Pauli Exclusion Principle

- Radioactivity and Nuclear Physics
- Medical Applications of Nuclear Physics
- Particle Physics
- Frontiers of Physics
- Atomic Masses
- Selected Radioactive Isotopes
- Useful Information
- Glossary of Key Symbols and Notation