Special relativity

Special relativity is a scientific theory that explains the relationship between space and time. It was introduced by the famous scientist Albert Einstein in his 1905 paper titled "On the Electrodynamics of Moving Bodies". The theory is based on two main ideas, or postulates.

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

The first postulate says that the laws of physics work the same way for all observers who are not accelerating, meaning they are moving at a constant speed. This idea is called the principle of relativity and was first suggested by Galileo Galilei. The second postulate states that the speed of light in a vacuum is the same for everyone, no matter how fast they or the light source are moving. This is known as the principle of light constancy.

Special relativity changed our understanding of space, time, and how they are connected. It has important applications in many areas of modern physics and technology.

Overview

Relativity is a theory that describes objects moving at very high speeds, much faster than we usually experience. It changes our idea that time flows the same everywhere. Instead, time can flow differently for each object. For example, a moving clock runs slower than a stationary one. At everyday speeds, we don’t notice this, but near the speed of light, it becomes very important.

The theory of special relativity is based on just two simple ideas. First, the laws of physics work the same whether you are moving or not — like how someone on a train sees things happening the same way 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 two ideas lead to many interesting results, like how time and distance can seem different depending on how fast you are moving, and how mass and energy are related.

History

Main article: History of special relativity

The idea that the laws of physics work the same for all observers 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 at rest. This idea was later used in Newtonian mechanics.

Later, James Clerk Maxwell discovered that light always travels at the same speed, no matter how fast you move. This was strange because it didn't fit with Galileo's ideas. Many experiments tried to find this special "aether" that light was thought to travel through, but none succeeded. Finally, 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, which is the fastest speed possible and 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, and these clocks can tick at different speeds depending on how they move. 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, 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 are important in understanding relativity. A reference frame is a way of looking at space where the observer is not moving or is moving at a steady speed. From this frame, we can measure where and when things happen.

In relativity, the big idea is that the rules of physics work the same no matter which steady reference frame you use. This means there isn't 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 a key idea in special relativity. It shows how space and time coordinates change when moving between different viewpoints, called reference frames. Albert Einstein used this to explain how the laws of physics stay 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 work the same everywhere, and the speed of light is constant for all observers. The Lorentz transformation connects these ideas, showing exactly how space and time must change to keep these rules true. This helps us understand phenomena like time dilation and length contraction in a simple way.

Consequences derived from the Lorentz transformation

See also: Twin paradox and Relativistic mechanics

The consequences of special relativity come from the Lorentz transformation equations. These transformations show that physical predictions differ from Newtonian mechanics, especially when speeds approach the speed of light. The speed of light is so fast that some effects of relativity seem surprising 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. But in special relativity, space and time are linked. This creates something called an invariant interval, written as Δs². It combines changes in time and space: Δs² = c²Δt² − (Δx² + Δy² + Δz²). This interval stays the same no matter how you move.

There are three important cases:

Δs² > 0: The events are separated more in time than space, called timelike separated. This means it's possible to 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 in every view.

Relativity of simultaneity

Two events that happen at the same time in one view might not happen at the same time in another view. 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 shows that causes must come before effects. If something could move faster than light, it might seem to go backward in time, causing problems. This is why nothing can travel faster than light. Only light and information limited by light speed 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 discovered that light travels slower in water than in air. This helped support some theories about how light behaves and challenged others. One famous experiment was done by Hippolyte Fizeau, who wanted to know how fast light would travel if the water was moving. He set up a way to send light beams through flowing water in opposite directions and then recombine them. This showed that the moving water did change the speed of the light, but not as much as some theories predicted at the time.

Relativistic aberration of light

Main articles: Aberration of light and Light-time correction

Because light travels at a constant speed, the way we see things can change if they are moving. If something is moving toward or away from us, or even sideways, the light from it takes time to reach us, which 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 at very high speeds, the color of the light changes. If it moves toward us, the light appears bluer (higher energy), and if it moves away, it appears redder (lower energy). 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 predicts there will be. Depending on the exact setup, the light can appear either slightly bluer or redder due to the effects 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 dealt strictly with kinematics, the study of motion without considering the forces that cause it. This section looks at mass, force, energy, and similar ideas, needing 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 everyday physics, energy and momentum are separate, but in relativity, they form a four-vector. This connects energy and momentum in a special way. For an object at rest, its energy-momentum four-vector has one part for energy and three parts for momentum, which are zero. By changing viewpoints with a Lorentz transformation, the energy and momentum change, showing that mass and energy are related.

Einstein's 1905 demonstration of E = mc²

In one of his 1905 papers, Einstein showed how mass and energy are equivalent. He used ideas like the Doppler shift for light, and the laws that energy and momentum are conserved. His work led to the famous equation E = mc², though some details were debated 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 expect. If a spaceship could speed up constantly, after a year it would nearly reach the speed of light. Over time, people on Earth would age much faster than travelers on the spaceship, allowing trips to very distant places in a shorter time for the travelers.

Elastic collisions

When particles crash into each other, scientists study what happens to learn about the tiny building blocks of nature. In special relativity, mass isn't 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, where points on the circle help us understand angles. In special relativity, we use a similar idea but with 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, which are like the regular trigonometric functions but work with hyperbolas. This makes adding speeds much simpler than using the regular formulas. When we use rapidity, many equations in special relativity become straightforward and easy to work with.

Minkowski spacetime

Main article: Minkowski space

Special relativity can be understood using a special kind of geometry called Minkowski spacetime. This spacetime is similar to ordinary 3D space, but it has an important difference because of how time is treated. In regular space, distances are measured using simple formulas involving the coordinates (x, y, z). In Minkowski spacetime, we add time as a fourth dimension, creating a 4D space where the distance between two points includes both space and time.

This idea helps explain how space and time are linked in special relativity. One key feature is that the "distance" between two events in spacetime stays the same no matter how you move — this is similar to how distances on a map stay the same even when you look at them from different angles. This property makes special relativity work consistently across all moving viewpoints, or "frames of reference."

The concept of "4-vectors" — mathematical objects with four components representing space and time together — is very useful in describing motion and forces in this spacetime. These tools help scientists write physical laws that work the same way no matter how fast you're moving.

Acceleration

Further information: Acceleration (special relativity)

Special relativity can also handle situations where objects are speeding up or moving in ways that change direction. Some people think special relativity only works for objects moving at steady speeds, but that's not true. When we talk about gravity, we need a different theory called general relativity.

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

When dealing with objects that are speeding up, we have 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 a few 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 discovered how waves travel. They found that the speed at which electric and magnetic fields move means that charged particles, like those in magnets or static electricity, must behave in certain ways. This idea of moving charges is part of what led to the development of special relativity.

Special relativity helps us understand how electric and magnetic fields change depending on how fast you are moving. Sometimes, what looks like a magnetic field to one person might look like an electric field to someone moving at a different speed. This shows that electric and magnetic fields are closely connected. Special relativity gives us the rules for how these fields appear different to observers 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 working on how to bring together general relativity and quantum mechanics into one theory.

In 1928, a scientist named Paul Dirac created an important equation that works with both special relativity and quantum mechanics. This equation helped explain the behavior of electrons and even predicted the existence of a particle called the positron.

Status

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

Special relativity is a theory that explains the relationship between space and time. It works very well when gravity is weak, but in strong gravity, we use a different theory called general relativity. Special relativity matches what we see in experiments to a very high degree of accuracy.

Many experiments support special relativity. For example, particle accelerators show that particles moving close to the speed of light behave as the theory predicts. These experiments confirm that the rules of special relativity are necessary for understanding the world at high speeds.