Special relativity

Special relativity is a scientific theory about how space and time are related. It was created by the famous scientist Albert Einstein in his 1905 paper titled "On the Electrodynamics of Moving Bodies". The theory has two main ideas.

Albert Einstein around 1905, the year his paper on special relativity was published

The first idea says that the laws of physics work the same way for everyone who is moving at a steady speed. This is called the principle of relativity and was first talked about by Galileo Galilei. The second idea says that the speed of light in a vacuum looks the same to everyone, no matter how fast they or the light source are moving. This is called the principle of light constancy.

Special relativity changed how we think about space and time and how they are linked. It is used in many parts of modern science and technology.

Overview

Relativity is a theory about objects moving very fast, much faster than we usually see. It changes how we think about time. Instead of time moving the same everywhere, it can move differently for each object. For example, a moving clock can tick slower than a clock that is not moving. At normal speeds, we don’t notice this, but it becomes important near the speed of light.

The theory of special relativity is based on two simple ideas. First, the rules of physics work the same whether you are moving or not — like how someone on a train sees things the same whether the train is stopped or moving. Second, the speed of light is always the same, no matter how fast you or the light source are moving. These ideas show how time and distance can look different depending on how fast you are moving, and how mass and energy are connected.

History

Main article: History of special relativity

The idea that the laws of physics work the same for everyone moving at a steady pace was first described by Galileo Galilei in 1632. He used a ship moving smoothly to show that experiments would give the same results whether the ship was moving or not.

Later, James Clerk Maxwell discovered that light always travels at the same speed, no matter how fast you move. This was surprising because it did not fit with Galileo's ideas. Many experiments tried to find the "aether" that light was thought to travel through, but none succeeded.

In 1905, Albert Einstein showed that if we accept that light always moves at the same speed, we must change our ideas about space and time. His theory, called special relativity, works for all situations where there is no gravity or very little gravity.

Terminology

Special relativity is built on some basic ideas from physics. One important idea is speed or velocity, which measures how fast something moves. Another key idea is the speed of light. This is the fastest speed possible and it stays the same no matter where you are or how fast you're moving.

We also think about clocks. In special relativity, every object has its own clock. These clocks can tick at different speeds depending on how the object moves. An event is something that happens at a specific place and time, like a flash of light. Different people watching the same event might see it happen at different times because the information travels at the speed of light.

Other important ideas include spacetime, which combines space and time together, and the spacetime interval. This is a way to measure the distance between events using both space and time. We also use coordinate systems or reference frames to describe where and when things happen, and inertial reference frames are special frames where objects not being pushed in any way move at a steady speed.

Traditional "two postulates" approach to special relativity

Main article: Postulates of special relativity

Albert Einstein developed the theory of special relativity based on two key ideas. First, he believed that the laws of physics should work the same for all observers moving at a steady speed. This is called the principle of relativity. Second, he proposed that the speed of light in a vacuum is always the same, no matter how fast the source of light is moving. This is known as the principle of invariant light speed.

These ideas helped explain how space and time are connected, especially when things move very fast. They were inspired by earlier work in electromagnetism and experiments that showed no evidence of a special substance called the "luminiferous ether" that was once thought to carry light waves.

Principle of relativity

Main article: Principle of relativity

Reference frames help us understand relativity. A reference frame is a way to look at space where the observer is either still or moving at a steady speed. From this frame, we can measure where and when things happen.

In relativity, the main idea is that the rules of physics are the same no matter which steady reference frame you use. This means there is no one special frame that is "right." For example, whether you are sitting still or moving in a car at a constant speed, the laws of physics should look the same to you. This idea helps us understand how space and time are connected.

Lorentz transformation

Main article: Lorentz transformation

The Lorentz transformation is an important idea in special relativity. It shows how space and time change when we look at things from different viewpoints, called reference frames. Albert Einstein used this to explain how the laws of physics work the same, no matter how you move, as long as you move at a steady speed.

Special relativity has two main ideas: the laws of physics are the same everywhere, and the speed of light is constant for everyone. The Lorentz transformation connects these ideas. It shows how space and time must change to keep these rules true. This helps us understand things like time dilation and length contraction in a simple way.

Consequences derived from the Lorentz transformation

See also: Twin paradox and Relativistic mechanics

The rules of special relativity come from something called the Lorentz transformation. These rules show that what we think happens in space and time can change when things move very fast, close to the speed of light. The speed of light is so quick that some of these effects can feel strange at first.

Invariant interval

Figure 4–1. The three events (A, B, C) are simultaneous in the reference frame of some observer O. In a reference frame moving at v = 0.3c, as measured by O, the events occur in the order C, B, A. In a reference frame moving at v = −0.5c with respect to O, the events occur in the order A, B, C. The white lines, the lines of simultaneity, move from the past to the future in the respective frames (green coordinate axes), highlighting events residing on them. They are the locus of all events occurring at the same time in the respective frame. The gray area is the light cone with respect to the origin of all considered frames.

In simple ideas about motion, space and time are separate ideas. But in special relativity, space and time are connected. This creates something called an invariant interval, written as Δs². It combines changes in time and space: Δs² = c²Δt² − (Δx² + Δy² + Δz²). This interval looks the same no matter how you move.

There are three important cases:

Δs² > 0: The events happen far apart in time but close in space, called timelike separated. This means we can find a view where the events happen at the same time.
Δs² = 0: The events are lightlike separated. This means the speed between them is exactly the speed of light, and this is true for everyone.

Relativity of simultaneity

Two things that happen at the same time in one place might not happen at the same time in another place. This happens because space and time are linked in special relativity.

Time dilation

Length contraction

Lorentz transformation of velocities

Causality and prohibition of motion faster than light

Special relativity tells us that causes must happen before effects. If something could move faster than light, it might seem to go backward in time, which would cause big problems. This is why nothing can travel faster than light. Only light and information that travels at or slower than the speed of light follow this rule.

Item	Measured by the stay-at-home	Fig 4-4	Measured by the traveler	Fig 4-4
Total time of trip	T = 2 L v {\displaystyle T={\frac {2L}{v}}}	10 yr	T ′ = 2 L γ v {\displaystyle T'={\frac {2L}{\gamma v}}}	8 yr
Total number of pulses sent	f T = 2 f L v {\displaystyle fT={\frac {2fL}{v}}}	10	f T ′ = 2 f L γ v {\displaystyle fT'={\frac {2fL}{\gamma v}}}	8
Time when traveler's turnaround is detected	t 1 = L v + L c {\displaystyle t_{1}={\frac {L}{v}}+{\frac {L}{c}}}	8 yr	t 1 ′ = L γ v {\displaystyle t_{1}'={\frac {L}{\gamma v}}}	4 yr
Number of pulses received at initial f ′ {\displaystyle f'} rate	f ′ t 1 {\displaystyle f't_{1}} = f L v ( 1 + β ) ( 1 − β 1 + β ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1+\beta )\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}} = f L v ( 1 − β 2 ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}	4	f ′ t 1 ′ {\displaystyle f't_{1}'} = f L v ( 1 − β 2 ) 1 / 2 ( 1 − β 1 + β ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1-\beta }{1+\beta }}\right)^{1/2}} = f L v ( 1 − β ) {\displaystyle ={\frac {fL}{v}}(1-\beta )}	2
Time for remainder of trip	t 2 = L v − L c {\displaystyle t_{2}={\frac {L}{v}}-{\frac {L}{c}}}	2 yr	t 2 ′ = L γ v {\displaystyle t_{2}'={\frac {L}{\gamma v}}}	4 yr
Number of signals received at final f ″ {\displaystyle f''} rate	f ″ t 2 {\displaystyle f''t_{2}} = f L v ( 1 − β ) ( 1 + β 1 − β ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1-\beta )\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}} = f L v ( 1 − β 2 ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}}	4	f ″ t 2 ′ {\displaystyle f''t_{2}'} = f L v ( 1 − β 2 ) 1 / 2 ( 1 + β 1 − β ) 1 / 2 {\displaystyle ={\frac {fL}{v}}(1-\beta ^{2})^{1/2}\left({\frac {1+\beta }{1-\beta }}\right)^{1/2}} = f L v ( 1 + β ) {\displaystyle ={\frac {fL}{v}}(1+\beta )}	8
Total number of received pulses	2 f L v ( 1 − β 2 ) 1 / 2 {\displaystyle {\frac {2fL}{v}}(1-\beta ^{2})^{1/2}} = 2 f L γ v {\displaystyle ={\frac {2fL}{\gamma v}}}	8	2 f L v {\displaystyle {\frac {2fL}{v}}}	10
Twin's calculation as to how much the other twin should have aged	T ′ = 2 L γ v {\displaystyle T'={\frac {2L}{\gamma v}}}	8 yr	T = 2 L v {\displaystyle T={\frac {2L}{v}}}	10 yr

| u | = u = d x / d t . {\displaystyle \mathbf {|u|} =u=dx/dt\,.}

| u ′ | = u ′ = d x ′ / d t ′ . {\displaystyle \mathbf {|u'|} =u'=dx'/dt'\,.}

u ′ = d x ′ d t ′ = γ ( d x − v d t ) γ ( d t − v d x c 2 ) = d x d t − v 1 − v c 2 d x d t = u − v 1 − u v c 2 . {\displaystyle u'={\frac {dx'}{dt'}}={\frac {\gamma (dx-v\,dt)}{\gamma \left(dt-{\dfrac {v\,dx}{c^{2}}}\right)}}={\frac {{\dfrac {dx}{dt}}-v}{1-{\dfrac {v}{c^{2}}}\,{\dfrac {dx}{dt}}}}={\frac {u-v}{1-{\dfrac {uv}{c^{2}}}}}.}

u = u ′ + v 1 + u ′ v / c 2 . {\displaystyle u={\frac {u'+v}{1+u'v/c^{2}}}.}

u = ( u 1 , u 2 , u 3 ) = ( d x / d t , d y / d t , d z / d t ) . {\displaystyle \mathbf {u} =(u_{1},\ u_{2},\ u_{3})=(dx/dt,\ dy/dt,\ dz/dt)\ .}

u ′ = ( u 1 ′ , u 2 ′ , u 3 ′ ) = ( d x ′ / d t ′ , d y ′ / d t ′ , d z ′ / d t ′ ) . {\displaystyle \mathbf {u'} =(u_{1}',\ u_{2}',\ u_{3}')=(dx'/dt',\ dy'/dt',\ dz'/dt')\ .}

u 1 ′ = u 1 − v 1 − u 1 v / c 2 , u 2 ′ = u 2 γ ( 1 − u 1 v / c 2 ) , u 3 ′ = u 3 γ ( 1 − u 1 v / c 2 ) . {\displaystyle u_{1}'={\frac {u_{1}-v}{1-u_{1}v/c^{2}}}\ ,\qquad u_{2}'={\frac {u_{2}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ ,\qquad u_{3}'={\frac {u_{3}}{\gamma \left(1-u_{1}v/c^{2}\right)}}\ .}

u 1 = u 1 ′ + v 1 + u 1 ′ v / c 2 , u 2 = u 2 ′ γ ( 1 + u 1 ′ v / c 2 ) , u 3 = u 3 ′ γ ( 1 + u 1 ′ v / c 2 ) . {\displaystyle u_{1}={\frac {u_{1}'+v}{1+u_{1}'v/c^{2}}}\ ,\qquad u_{2}={\frac {u_{2}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ ,\qquad u_{3}={\frac {u_{3}'}{\gamma \left(1+u_{1}'v/c^{2}\right)}}\ .}

Optical effects

Main article: Fizeau experiment

In the 1800s, scientists learned that light travels slower in water than in air. This helped them understand how light behaves. A famous experiment was done by Hippolyte Fizeau. He wanted to know how fast light would travel if the water was moving. He sent light beams through flowing water in opposite directions and then brought them back together. This showed that moving water changes the speed of light, but not as much as some theories at the time thought.

Relativistic aberration of light

Main articles: Aberration of light and Light-time correction

Because light always travels at the same speed, the way we see things can change if they are moving. If something moves toward or away from us, or even sideways, the light from it takes time to reach us. This can make the object look like it’s in a different place than it really is. This effect is called aberration. Early scientists tried to measure this but got confusing results. Later, Einstein’s theory of special relativity helped explain these observations better.

Relativistic Doppler effect

Main article: Relativistic Doppler effect

Normally, if a sound source moves toward you, the sound seems higher in pitch, and lower if it moves away. This is called the Doppler effect. With light, something similar happens, but it’s more complicated because of special relativity. When a light source moves toward or away from us very fast, the color of the light changes. If it moves toward us, the light looks bluer. If it moves away, it looks redder. This is called the relativistic Doppler effect.

Transverse Doppler effect

The transverse Doppler effect is a special case where the light source moves sideways relative to us, with no forward or backward motion. Classical physics would say there should be no change in color, but special relativity says there will be. Depending on how it’s set up, the light can look slightly bluer or redder because of time dilation—a key idea in relativity.

Measurement versus visual appearance

Main article: Terrell rotation

Sometimes what we see isn’t exactly what’s there! When objects move very fast, the light from different parts of the object takes different times to reach our eyes. This can make a fast-moving object look stretched, squished, or even rotated, even though its actual shape hasn’t changed. This strange visual effect is called Terrell rotation. For example, a fast-moving sphere might look like a flattened disk, even though it’s perfectly round.

Dynamics

Section § Consequences derived from the Lorentz transformation talked about kinematics, which is the study of motion without looking at forces. This section looks at mass, force, energy, and similar ideas, and it needs more than just the Lorentz transformation.

Equivalence of mass and energy

Main article: Mass–energy equivalence

Mass–energy equivalence is a result of special relativity. In normal physics, energy and momentum are separate, but in relativity, they are connected as a four-vector. This connection shows that mass and energy are related. For an object that is not moving, its energy-momentum four-vector has one part for energy and three parts for momentum, which are zero. When we change how we look at it using a Lorentz transformation, the energy and momentum change, showing the link between mass and energy.

Einstein's 1905 demonstration of E = mc²

In one of his 1905 papers, Einstein showed how mass and energy are the same. He used ideas like the Doppler shift for light, and the rules that energy and momentum are saved. His work led to the famous equation E = mc², even though some details were talked about later.

How far can you travel from the Earth?

Because nothing can go faster than light, you might think we can't travel very far from Earth. But because of time dilation, a spaceship could go much farther than we think. If a spaceship could keep speeding up, after a year it would almost reach the speed of light. Over time, people on Earth would get older much faster than travelers on the spaceship, which means trips to very far places could happen in less time for the travelers.

Elastic collisions

When tiny parts crash into each other, scientists study what happens to learn about the small building blocks of nature. In special relativity, mass is not separate from energy, which changes how we understand these crashes compared to older physics ideas.

( H 0 − E 0 ) − ( H 1 − E 1 ) = L ( 1 1 − v 2 / c 2 − 1 ) {\displaystyle \quad \quad (H_{0}-E_{0})-(H_{1}-E_{1})=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}

6-1

K 0 − K 1 = L ( 1 1 − v 2 / c 2 − 1 ) {\displaystyle \quad \quad K_{0}-K_{1}=L\left({\frac {1}{\sqrt {1-v^{2}/c^{2}}}}-1\right)}

6-2

K 0 − K 1 = 1 2 L c 2 v 2 {\displaystyle \quad \quad K_{0}-K_{1}={\frac {1}{2}}{\frac {L}{c^{2}}}v^{2}}

6-3

p → = γ m v → and E = γ m c 2 {\displaystyle \quad \quad {\vec {p}}=\gamma m{\vec {v}}\quad {\text{and}}\quad E=\gamma mc^{2}}

6-4

γ 1 m v 1 → + 0 = γ 2 m v 2 → + γ 3 m v 3 → {\displaystyle \quad \quad \gamma _{1}m{\vec {v_{1}}}+0=\gamma _{2}m{\vec {v_{2}}}+\gamma _{3}m{\vec {v_{3}}}}

6-5

γ 1 m c 2 + m c 2 = γ 2 m c 2 + γ 3 m c 2 {\displaystyle \quad \quad \gamma _{1}mc^{2}+mc^{2}=\gamma _{2}mc^{2}+\gamma _{3}mc^{2}}

6-6

β 1 γ 1 = β 2 γ 2 cos ⁡ θ + β 3 γ 3 cos ⁡ ϕ {\displaystyle \quad \quad \beta _{1}\gamma _{1}=\beta _{2}\gamma _{2}\cos {\theta }+\beta _{3}\gamma _{3}\cos {\phi }}

6-7

β 2 γ 2 sin ⁡ θ = β 3 γ 3 sin ⁡ ϕ {\displaystyle \quad \quad \beta _{2}\gamma _{2}\sin {\theta }=\beta _{3}\gamma _{3}\sin {\phi }}

6-8

γ 1 + 1 = γ 2 + γ 3 {\displaystyle \quad \quad \gamma _{1}+1=\gamma _{2}+\gamma _{3}}

6-9

β 2 = β 1 sin ⁡ ϕ { β 1 2 sin 2 ⁡ ϕ + sin 2 ⁡ ( ϕ + θ ) / γ 1 2 } 1 / 2 {\displaystyle \quad \quad \beta _{2}={\frac {\beta _{1}\sin {\phi }}{\{\beta _{1}^{2}\sin ^{2}{\phi }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}

6-10

β 3 = β 1 sin ⁡ θ { β 1 2 sin 2 ⁡ θ + sin 2 ⁡ ( ϕ + θ ) / γ 1 2 } 1 / 2 {\displaystyle \quad \quad \beta _{3}={\frac {\beta _{1}\sin {\theta }}{\{\beta _{1}^{2}\sin ^{2}{\theta }+\sin ^{2}(\phi +\theta )/\gamma _{1}^{2}\}^{1/2}}}}

6-11

cos ⁡ ( ϕ + θ ) = ( γ 1 − 1 ) sin ⁡ θ cos ⁡ θ { ( γ 1 + 1 ) 2 sin 2 ⁡ θ + 4 cos 2 ⁡ θ } 1 / 2 {\displaystyle \quad \quad \cos {(\phi +\theta )}={\frac {(\gamma _{1}-1)\sin {\theta }\cos {\theta }}{\{(\gamma _{1}+1)^{2}\sin ^{2}\theta +4\cos ^{2}\theta \}^{1/2}}}}

6-12

cos ⁡ θ = β 1 { 2 / γ 1 + 3 β 1 2 − 2 } 1 / 2 {\displaystyle \quad \quad \cos {\theta }={\frac {\beta _{1}}{\{2/\gamma _{1}+3\beta _{1}^{2}-2\}^{1/2}}}}

6-13

Rapidity

Main article: Rapidity

Special relativity uses a concept called "rapidity" to make some calculations easier. Imagine you are looking at a diagram with a unit circle. In special relativity, we use something similar called a unit hyperbola instead of a circle. This helps us understand how things move at very high speeds.

Rapidity uses special math called hyperbolic functions. These are like the regular math we use for angles, but they work with hyperbolas. This makes adding speeds much simpler. When we use rapidity, many equations in special relativity become easier to solve.

Minkowski spacetime

Main article: Minkowski space

Special relativity uses a special kind of geometry called Minkowski spacetime. This spacetime is like regular 3D space, but with a big difference because of time. In regular space, we measure distances using simple formulas with coordinates (x, y, z). In Minkowski spacetime, we add time as a fourth dimension. This makes a 4D space where the distance between two points includes both space and time.

This idea shows how space and time are connected in special relativity. One important feature is that the "distance" between two events in spacetime stays the same, no matter how you move. This is like how distances on a map stay the same even when you look from different angles. This helps special relativity work the same in all moving viewpoints, or "frames of reference."

The idea of "4-vectors" — math tools with four parts that show space and time together — is very useful. These tools help scientists describe motion and forces in this spacetime. They help write physical laws that work the same, no matter how fast you are moving.

Acceleration

Further information: Acceleration (special relativity)

Special relativity can describe objects that are speeding up or changing direction. Some people think special relativity only works for objects moving at steady speeds, but this is not true. When we talk about gravity, we need a different theory called general relativity.

Figure 7–4. Dewan–Beran–Bell spaceship paradox

When objects speed up, we need to be careful. In special relativity, all speeds are relative, but changes in speed are absolute. General relativity, which deals with gravity, treats all kinds of motion as relative. To make this work, general relativity uses the idea of curved space-time.

In this part, we look at examples where objects are speeding up and see how special relativity helps us understand what happens. One famous example is called the Dewan–Beran–Bell spaceship paradox, which shows how things can seem confusing if we don’t think about space and time in the right way.

Main article: Bell's spaceship paradox

Main articles: Event horizon § Apparent horizon of an accelerated particle, and Rindler coordinates

Relativity and unifying electromagnetism

Main articles: Classical electromagnetism and special relativity and Covariant formulation of classical electromagnetism

In classical electromagnetism, scientists learned how waves move. They found that the speed of electric and magnetic fields affects how charged particles behave, like those in magnets or static electricity. This idea helped create special relativity.

Special relativity shows us how electric and magnetic fields change when you move at different speeds. Sometimes, what looks like a magnetic field to one person might look like an electric field to someone moving faster. This shows that electric and magnetic fields are linked. Special relativity explains the rules for how these fields look different to people moving at different speeds.

Theories of relativity and quantum mechanics

Special relativity can be combined with quantum mechanics to form relativistic quantum mechanics and quantum electrodynamics. Scientists are still trying to find a way to bring general relativity and quantum mechanics together into one theory.

In 1928, a scientist named Paul Dirac made an important equation. This equation works with both special relativity and quantum mechanics. It helped explain how electrons behave and even predicted a new particle called the positron.

Status

Main articles: Tests of special relativity and Criticism of the theory of relativity

Special relativity is a theory about space and time. It helps us understand how they are connected. It works best when there is not much gravity. When gravity is strong, we use a different theory called general relativity. Special relativity has been tested many times and it matches what we see in experiments very well.

Many tests support special relativity. For example, in particle accelerators, particles that move close to the speed of light act just as the theory says they should. These tests show that special relativity helps us understand the world when things move very fast.