Special relativity

Special relativity (or the special theory of relativity) is a theory 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 Minkowski space, or "flat spacetime" (phenomena which are not influenced by gravitation).

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

Eventually, Einstein (1905) drew the conclusion that the concepts of space and time needed a fundamental revision. The result was special relativity theory, which brought together a new principle "the constancy of the speed of light" and the previously established "principle of relativity".

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 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 not for all cases. Einstein's theory of special relativity explained this among other things.

Basics of special relativity

Suppose that you are moving toward something that is moving toward you. If you measure its speed, it will seem to be moving faster than if you were not moving. Now suppose you are moving away from something that is moving toward you. If you measure its speed again, it will seem to be moving more slowly. This is the idea of "relative speed"—the speed of the object relative 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.

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 very slightly. Any clock used to measure a duration will be wrong by exactly the right 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.

Also, Einstein said that as Earth moves through space, all measurable lengths change (ever so slightly). Any device measuring length will give a length off by exactly the right amount so that the speed of light remains the same.

The most difficult thing to understand is that events that appear to be simultaneous in one frame may not be simultaneous in another. This has many effects that are not easy to perceive or understand. Since the length of an object is the distance from head to tail at one simultaneous moment, it follows that if two observers disagree about what events are simultaneous then this will affect (sometimes dramatically) their measurements of the length of objects. Furthermore, if a line of clocks appear synchronized to a stationary observer and appear to be out of sync to that same observer after accelerating to a certain velocity then it follows that during the acceleration the clocks ran at different speeds. Some may even run backwards. This line of reasoning 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 revolutionary is that it considers the measurement of the speed of light to be constant by definition, in other words it is a law of nature. This has the remarkable implications that speed-related measurements, length and duration, change in order to accommodate this.

The Lorentz transformations

The mathematical foundation of special relativity lies in the Lorentz transformations. These equations describe how measurements of space and time change for two observers moving at constant velocities relative to each other, without acceleration.

To define these transformations, we use a Cartesian coordinate system to represent the position and time of events. Each observer can describe an event using four coordinates: (x, y, z, t)—where x, y, z specify the location in space, and t represents the time.

The spatial position is defined relative to an origin point (0, 0, 0), so an event at (3, 3, 3) lies three units away in each spatial direction. The time coordinate t is measured relative to a chosen zero point in time, using consistent units like seconds or years.

Let there be an observer K, who assigns time coordinates t to events and describes their positions using spatial coordinates x, y, z. This establishes the reference frame of observer K, which serves as our initial point of view.

We define the time of an event as the actual moment it occurred, not simply when it was observed. To determine this, we subtract the time it took for the signal (such as light) to travel from the event to the observer from the observed time:

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

where:

$t_{e}$ is time of event

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

and $c$ is speed of light.

This 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 reach 30 miles). So we have:

$t=d/c$

This framework defines how any observer measures time: as the moment an event actually occurred, based on light-travel time correction. With this established, consider a second observer K′ with the following properties:

• K′ moves at a constant velocity v along the x-axis of observer K.
• K′ uses its own spatial coordinates: x′, y′, z′.
• The x′-axis remains coincident with the x-axis of K, while the y′-axis and z′-axis remain parallel to the corresponding axes of K at all times. This setup implies that if K′ describes an event as occurring at (3, 1, 2), the x-coordinate (3) corresponds to the same position in space as K would describe, due to the shared x-axis. However, the y′ = 1 and z′ = 2 values are only parallel to K’s y and z axes, not identical in position. We also define that K and K′ 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

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

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

$y'=y$, and

$z'=z$.

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:

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

By applying the inverse Lorentz transformation, 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:    $\Delta x'=x'_{2}-x'_{1}\ ,\ \Delta t'=t'_{2}-t'_{1}\ .$

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

we get

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

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

If we take differentials instead of taking differences, we get

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

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

Mass, energy and momentum

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

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

and

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

A frequently made error (also in some books) is to rewrite this equation using a "relativistic mass" (in the direction of motion) of $m_{r}={\frac {m}{\sqrt {1-{\frac {v^{2}}{c^{2}}}}}}$. The reason why this is incorrect is that light, for example, has no mass, but has energy. If we use this formula, the photon (particle of light) has a mass, which is according to experiments incorrect.

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

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

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

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

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).

To have James Clerk Maxwell's equations be consistent with both astronomical observations^[1] and Newtonian physics,^[2] Maxwell proposed in 1877 that light travels through an ether which is everywhere in the universe.

In 1887, the famous Michelson-Morley experiment tried to detect the "ether wind" generated by the movement of the Earth.^[3] The persistent null results of this experiment puzzled physicists, and called the ether theory into question.

In 1895, Lorentz and Fitzgerald noted that the null result of the Michelson-Morley experiment could be explained by the ether wind contracting the experiment in the direction of motion of the ether. This effect is called the Lorentz contraction, and (without ether) is a consequence of special relativity.

In 1899, Lorentz first published the Lorentz equations. Although this was not the first time they had been published, this was the first time that they were used as an explanation of the Michelson-Morley's null result, since the Lorentz contraction is a result of them.

In 1900, Poincaré gave a famous speech in which he considered the possibility that some "new physics" was needed to explain the Michelson-Morley experiment.

In 1904, Lorentz showed that electrical and magnetic fields can be modified into each other through the Lorentz transformations.

In 1905, Einstein published his article introducing special relativity, "On the Electrodynamics of Moving Bodies", in Annalen der Physik. In this article, he presented the postulates of relativity, derived the Lorentz transformations from them, and (unaware of Lorentz's 1904 article) also showed how the Lorentz Transformations affect electric and magnetic fields.

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

In 1908, Max Planck endorsed 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 for the Nobel prize in physics due to their pioneering work on relativity. Unfortunately, relativity was so controversial then, and remained controversial for such a long time that a Nobel prize was never awarded for it.

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 $E=pc$. (In Newtonian physics, this is expected to be $E={\begin{matrix}{\frac {1}{2}}\end{matrix}}pc$.)
• 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.

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 ether wind should have been present 6 months later, but none was observed.

Related pages

• General relativity

References

• W. Rindler, Introduction to Special Relativity, 2nd edition, Oxford Science Publications, 1991, ISBN 0-19-853952-5.
• Web article on the history of special relativity Archived 2013-12-09 at the Wayback Machine
• Relativity Calculator - Learn Special Relativity Mathematics Archived 2008-11-08 at the Wayback Machine The mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.

Other websites

• http://math.ucr.edu/home/baez/physics/Relativity/SR/rocket.html Archived 2015-10-13 at the Wayback Machine