Special relativity

Special relativity (or the special theory of relativity) is a theory (Well supported idea) in physics that was developed and explained by Albert Einstein in 1905. It applies to all physical phenomena, so long as gravitation is not significant. Special relativity applies to "flat spacetime", which is space without gravity.

Einstein knew that some weaknesses had been discovered in older models. For example, older models said light moved in a luminiferous aether. Various tiny effects were expected if this theory were true. Gradually it seemed these predictions were not going to work out.

Eventually, in 1905, Einstein drew the conclusion that the concepts of space and time needed a fundamental rethink. His idea was special relativity, which brought together a new principle (Law), the constancy of the speed of light (the speed of light is always the same), and the previously established principle of relativity (physics is the same for everyone)

Galileo had already established the principle of relativity, which said that physical events must look the same to all observers, and no observer has the "right" way to look at the things studied by physics. For example, the Earth is moving very fast around the Sun, but we do not notice it because we are moving with the Earth at the same speed; therefore, from our point of view, the Earth is at rest. However, Galileo's math could not explain some things, such as the speed of light. According to him, the measured speed of light in an empty vacuum should be different for different speeds of the observer in comparison with its source. However, the Michelson-Morley experiment showed that this is not true, at least in a vacuum. Einstein's theory of special relativity explained this among other things.

Basics of special relativity

Imagine that you are moving towards something that is also moving to you. If you measure that thing's speed, it will look like like it's moving faster than if you were not moving towards them. Now imagine the opposite, you are moving away from something that is moving toward you. If you measure its speed again, it will look like it's moving more slowly. This is the idea of "relative speed"—the speed of the object compared to you.

Before Albert Einstein, scientists were trying to measure the "relative speed" of light. They were doing this by measuring the speed of star light reaching the Earth. They expected that if the Earth was moving toward a star, the light from that star should seem faster than if the Earth was moving away from that star. However, they noticed that no matter who performed the experiments, where the experiments were performed, or what star light was used, the measured speed of light in a vacuum was always the same.^[1]

Einstein said this happens because there is something unexpected about length and duration, or how long something lasts. He thought that as Earth moves through space, all measurable durations change a little bit. A clock used to measure a duration will be wrong by exactly the amount so that the speed of light remains the same. Imagining a "light clock" allows us to better understand this remarkable fact for the case of a single light wave.

Einstein also said that as Earth moves through space, all measurable lengths change (but just by a little). A thing measuring both length and time will give a length and time so that the speed of light remains the same.

The most difficult thing to understand is that events that appear to be simultaneous (Happening at the same time) in one frame may not be simultaneous in another. This has many effects that are not easy to understand. Since the length of an object is the distance from head to tail in one particular moment, it means that if two observers disagree about what events are simultaneous then this will change (sometimes by a lot) what they think the measurements of the length of objects are. Furthermore, if a line of clocks look the synchronised/the same to an observer who is standing still and then appear to become out of sync to that same observer after they accelerate to a certain velocity, this means that during that acceleration, the clocks ran at different speeds. Some may even turn run backwards. This kind of thinking leads to general relativity.

Other scientists before Einstein had written about light seeming to go the same speed no matter how it was observed. What made Einstein's theory so note worthy is that it considers the measurement of the speed of light to be constant by definition, in other words it is a basic law of how the Universe works. This has big implications because speed-related measurements, length and duration, must change in order to accommodate this.

The Lorentz transformations

Einstein's theory here was inspired by some Mathematical works called the "Lorentz transformations". These are equations (a way scientists calculate something). This particular equation tell us how we can calculate the way time and space changes (in the context of special relativity), for example: when two people are moving at constant velocities relative to each other, without acceleration.

To describe these transformations (The way it changes), we use the Cartesian coordinate system to show where and when the events take place. Every observer can describe an event using four "coordinates": (x, y, z, t). The letters x, y, z tell us where the thing is, and t is for the time.

The spatial position (where something is) is described as "from" an origin point, a thing happening at (3, 3, 3) is 3 units away (in each direction) from the "origin" point which is (0, 0, 0). The time marker t likewise tells us when something happened, usually scientists use minutes or seconds so 3 could be 3 minutes from the "origin" or start.

The time of an event is when that thing occurred or happened. Not when someone first noticed or observed it. If someone observed an event and wanted to calculate when it actually happened, we subtract the time it took for the signal (such as light) to travel from the event to the observer from the observed time. In other words, record when the thing was noticed and then subtract how long it takes for the thing to travel from the point to the observer.

${\displaystyle t_{e}=t_{o}-d_{o}/c}$

where:

${\displaystyle t_{e}}$ is time of event

${\displaystyle d_{o}}$ is the distance from the observer to the event

and ${\displaystyle c}$ is speed of light.

This equation (how scientists calculate something) is correct because distance, divided by speed gives the time it takes to go that distance at that speed (e.g. 30 miles divided by 10 mph: give us 3 hours, because if you go at 10 mph for 3 hours, you go 30 miles). So we have:

${\displaystyle t=d/c}$

This equation is useful because we can use it to measure time. If we know the distance (how far something is) and how fast it travels (Velocity) we can work out exactly what time it occurred. Consider this example:

• H moves at a constant velocity (v) along the x-axis of observer K.
• H has their own spatial coordinates: x, y, z (Which will be listed as H(x), H(y) etc)
• H's x-axis remains coincident (remains the same) with the x-axis of K, while H's y axis: H(y), and H(z) remain parallel, they remain completely separate, to the corresponding axes of K at all times. This would imply (suggest), if H describes an event as occurring at (3, 1, 2), the x-coordinate (3) is the same point in space that K would describe, due to the shared x-axis. However, the H(y) = 1 and H(z) = 2 values are parallel to K’s y and z axes, they are not at the same point. We can also say that K and H share the same origin in space and time at the moment t = t′ = 0. This means that both observers agree on at least one event: the origin event at (0, 0, 0, 0) in both frames.

The Lorentz Transformations then are

${\displaystyle t'=(t-vx/c^{2})/{\sqrt {1-v^{2}/c^{2}}}}$

${\displaystyle x'=(x-vt)/{\sqrt {1-v^{2}/c^{2}}}}$

${\displaystyle y'=y}$

${\displaystyle z'=z}$.

An event in spacetime is described by the coordinates (t,x,y,z) in reference frame S, and by (t′,x′,y′,z′) in another frame S′, which is moving at a constant velocity v relative to S, specifically along the x-axis.

.

Solving the above four transformation equations for the unprimed coordinates yields the inverse Lorentz transformation:

${\displaystyle {\begin{aligned}t&=\gamma (t'+vx'/c^{2})\\x&=\gamma (x'+vt')\\y&=y'\\z&=z'.\end{aligned}}}$

By applying the Lorentz transformation but backwards (use -v in place of v(, we find that the unprimed frame (S) appears to move at velocity v′ = −v when observed from the primed frame (S′). This reciprocity confirms the symmetry of relative motion between inertial frames.

Importantly, the x-axis is not uniquely special. The Lorentz transformation can be generalized to motion along any direction in space. The key distinction lies in components:

• Parallel to the direction of motion (e.g., x): affected by time dilation and length contraction, scaled by the Lorentz factor γ.
• Perpendicular to the direction of motion (e.g., y and z): remain unchanged.

For full generalization, the Lorentz transformation extends to arbitrary directions via vector decomposition into parallel and perpendicular components relative to the motion vector.

A physical quantity that remains unchanged under Lorentz transformations is called a Lorentz scalar. Examples include the spacetime interval and the rest mass of a particle.

Writing the Lorentz transformation and its inverse in terms of coordinate differences, where one event has coordinates (x₁, t₁) and (x′₁, t′₁), another event has coordinates (x₂, t₂) and (x′₂, t′₂), and the differences are defined as

Eq. 1:    ${\displaystyle \Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .}$

Eq. 2:    ${\displaystyle \Delta x=x_{2}-x_{1}\ ,\ \ \Delta t=t_{2}-t_{1}\ .}$

we get

Eq. 3:    ${\displaystyle \Delta x'=\gamma \ (\Delta x-v\,\Delta t)\ ,\ \ }$ ${\displaystyle \Delta t'=\gamma \ \left(\Delta t-v\ \Delta x/c^{2}\right)\ .}$

Eq. 4:    ${\displaystyle \Delta x=\gamma \ (\Delta x'+v\,\Delta t')\ ,\ }$ ${\displaystyle \Delta t=\gamma \ \left(\Delta t'+v\ \Delta x'/c^{2}\right)\ .}$

If we take differentials instead of taking differences, we get

Eq. 5:    ${\displaystyle dx'=\gamma \ (dx-v\,dt)\ ,\ \ }$ ${\displaystyle dt'=\gamma \ \left(dt-v\ dx/c^{2}\right)\ .}$

Eq. 6:    ${\displaystyle dx=\gamma \ (dx'+v\,dt')\ ,\ }$ ${\displaystyle dt=\gamma \ \left(dt'+v\ dx'/c^{2}\right)\ .}$^[2]

Mass, energy and momentum

In special relativity, the momentum ${\displaystyle p}$ and the total energy ${\displaystyle E}$ of an object as a function of its mass ${\displaystyle m}$ are

${\displaystyle p={\frac {mv}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$

and

${\displaystyle E={\frac {mc^{2}}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$.

A very common mistake (also in some books) is to rewrite this using a "relativistic mass" (in the direction of motion) of ${\displaystyle m_{r}={\frac {m}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}}$. The reason why this is not right is that light, for example, has no mass, but has energy. If we use the incorrect equation here, the photon (particle of light) has a mass, which is incorrect according to experiments.

In special relativity, an object's mass, total energy and momentum are related by the equation

${\displaystyle E^{2}=p^{2}c^{2}+m^{2}c^{4}}$.

For an object at rest, ${\displaystyle p=0}$ so the above equation simplifies to ${\displaystyle E=mc^{2}}$. Hence, a massive object at rest still has energy. We call this rest energy and denote it by ${\displaystyle E_{0}}$:^[3]

${\displaystyle E_{0}=mc^{2}}$.

History

The need for special relativity arose from Maxwell's equations of electromagnetism, which were published in 1865. It was later found that they call for electromagnetic waves (such as light) to move at a constant speed (i.e., the speed of light).

In order for James Clerk Maxwell's equations to be consistent (to be right, in light on new findings) with new astronomical observations^[1] and Newtonian physics,^[2] Maxwell made the idea (in 1877) that light travels through an ether which is everywhere in the universe.

In 1887, the famous Michelson-Morley experiment tried to detect this "ether wind" which according to Maxwell's ether theory was generated by the movement of the Earth.^[3] The experiment consistently generated a null result, meaning the experiment contradicted what the scientist thought they would find, which in this case was ether. The fact they could not find any "ether wind", confused many physicists at the time, and many of them questioned whether ether existed at all

However, In 1895, Lorentz and Fitzgerald said that the null result of the previously mentioned Michelson-Morley experiment could be explained by the "ether wind contracting the experiment in the direction of motion of the ether". We now know that this effect is called the Lorentz contraction, and (without ether) is a consequence of special relativity.

In 1899, Lorentz first published his "Lorentz equations". Although this was not in fact the first time they had been published, it was the first time that they were used to suggest why Michelson-Morley's experiment had a null result, since the Lorentz contraction is a result of them.

In 1900, the French scientist Poincaré gave a famous speech in which he debated the possibility that some "new physics" was needed to explain the Michelson-Morley experiment, in other words that something unknown needed to be discovered to explain the results of Morley's experiment

In 1904, Lorentz showed that electrical and magnetic fields can be modified (made into) one another through the previously mentioned Lorentz transformations.

In 1905, Einstein published his article introducing his special relativity idea, it was called "On the Electrodynamics of Moving Bodies", in the German scientific journal Annalen der Physik. In the article, he wrote about simple relativity and it's fundamental ideas, and in the process made the Lorentz transformations from them, he also (completely unaware that Lorentz had already discovered this the year earlier) showed how the Lorentz Transformations affect electric and magnetic fields.

Later in 1905, Einstein published another article presenting E = mc².

In 1908, Max Planck agreed with Einstein's theory and named it "relativity". In that same year, Hermann Minkowski gave a famous speech on Space and Time in which he showed that relativity is self-consistent and further developed the theory. These events forced the physics community to take relativity seriously. Relativity came to be more and more accepted after that.

In 1912, Einstein and Lorentz were nominated (suggested) for the Nobel prize in physics due to their excellent work on relativity. Unfortunately, relativity was very controversial at the time, and for a long time after, many people did not like Einstein's theory. This meant he did not win the prize for it (although he would later win it, for unrelated work in 1921).

Experimental confirmations

• The Michelson-Morley experiment, which failed to detect any difference in the speed of light based on the direction of the light's movement.
• Fizeau's experiment, in which the index of refraction for light in moving water cannot be made to be less than 1. The observed results are explained by the relativistic rule for adding velocities.
• The energy and momentum of light obey the equation ${\displaystyle E=pc}$. (In Newtonian physics, this is expected to be ${\displaystyle E={\begin{matrix}{\frac {1}{2}}\end{matrix}}pc}$.)
• The transverse doppler effect, which is where the light emitted by a quickly moving object is red-shifted due to time dilation.
• The presence of muons created in the upper atmosphere at the surface of Earth. The issue is that it takes much longer than the half-life of the muons to get down to Earth surface even at nearly the speed of light. Their presence can be seen as either being due to time dilation (in our view) or length contraction of the distance to the earth surface (in the muon's view).
• Special relativity plays a great role in making and function of particle accelerators. Subatomic particle accelerates with speed of light. The apparent change in the particles could be observed such as apparent masses of particles. This leads to formation of new and heavier particles.^[4]

Notes

• ^[1] Observations of binary stars show that light takes the same amount of time to reach the Earth over the same distance for both stars in such systems. If the speed of light was constant with respect to its source, the light from the approaching star would arrive sooner than the light from the receding star. This would cause binary stars to appear to move in ways that violate Kepler's Laws, but this is not seen.
• ^[2] The second postulate of special relativity (that the speed of light is the same for all observers) contradicts Newtonian physics.
• ^[3] Since the Earth is constantly being accelerated as it orbits the Sun, the initial null result was not a concern. However, that did mean that a strong aether wind should have been present 6 months later, but none was observed.

Related pages

• General relativity