### Topic Page: special relativity

**special (theory of) relativity**from

*The Penguin Dictionary of Science*

The area of physics dealing with high speeds (a significant fraction of the ➤speed of light). In special relativity the ➤Galilean transformation (used in classical physics for transforming coordinate systems between moving observers) is replaced by the ➤Lorentz transformation. A crucial feature is that the scale of time is different for observers moving at different speeds, which gives rise to the phenomena of ➤time dilation and ➤length contraction.

**Relativity, Special Theory of**from

*Encyclopedia of Time: Science, Philosophy, Theology, & Culture*

The special theory of relativity (STR) is a fundamental theory of space and time that was formulated and published in 1905 by Albert Einstein (1879-1955). The theory gives up the notion of the absolute and independent character of space and time. Length and time measures change according to an observer's state of motion. Special relativity is based on the assumption that the laws of nature have the same form in any uniformly moving frame of reference (the principle of relativity). As a most popular result of the STR, Einstein derived the famous relation = mc^{2}, which states the equivalence of matter and energy. Over the years the STR has been confirmed in many tests and experiments. Along with the quantum theory, it is considered a foundation pillar of modern physics.

Any quantitative law of mechanics, or rather physics, is expressed with respect to some fixed spatial and temporal frame of reference represented by three spatial coordinates plus a time coordinate. This can be realized in a terrestrial laboratory providing the frame for a measurement. An experiment may be performed equally well in a racing car, cutting a corner at high speed. Quite obviously these two systems are not equivalent: Due to the permanent change of direction (i.e., the velocity vector), pseudo forces appear within the racing car. These prevent, for example, a body, thrown straight upwards by the driver, from following a simple straight trajectory as it would in the stationary laboratory of an institute. The trajectory within the racing car reference does not follow the simple Newtonian law of motion. Among all possible frames of reference, physical laws take the simplest forms in so called inertial systems. These are systems in which bodies that are exposed to no pseudo forces whatsoever, like centrifugal or Coriolis forces, rest or move uniformly; that is, with constant speed and direction. To a good approximation, the earth's surface can be regarded as an inertial system. Although any point on it is moving in a complex way, the appearing pseudo forces are negligible.

Four centuries ago, the Italian naturalist Galileo Galilei (1564-1642) stated that uniform motion is not an inner property of a body but rather a quantity that is to be defined relative to a framework or an observer. Galileo illustrated this by the example of a body resting on a uniformly drifting boat on a silent lake. The question as to whether the concerning body is moving has to be answered in different ways, depending on whether an observer is sitting in the boat or resting on the bank. Both systems, boat and bank, can be considered as inertial systems in an idealized case. Isaac Newton (1642-1727), though sharing Galileo's view in considering relative motion as relevant exclusively, nevertheless assumed the existence of absolute space. He argued, however, on the basis of spiritual rather than physical criteria and admitted the impossibility of proving the existence of absolute space.

After centuries of empirical research, classical electrodynamicsthe theory of electromagnetic fields' and charged particles' dynamicswas brought to its early perfection in the 1860s by the Scottish physicist James Clerk Maxwell (1831-1879). In an accomplishment similar to Newton's reduction of classical mechanics to just three fundamental laws, Maxwell succeeded in expressing the nature and dynamics of electromagnetic fields with a system of just four equations. Following these Maxwell equations, the propagation of light should always occur with the same universally constant velocity. This was, however, in disagreement with Galileo's principle of relativity: The migration velocity of a light beam should certainly have different values depending on whether a resting or a moving observer is watching it. Furthermore, a constant universal speed of light should have the consequence of an absolute space, defined as the frame within which the light velocity adopts the well-known value of 300,000 kilometers per second.

It had long been presumed that, like sound waves, which cannot propagate without substantial material, electromagnetic waves should run within some medium of unknown nature, the so-called ether This picture is clearly accompanied by the concept of an absolute space, namely the ether rest frame. In 1881, for the first time, the physicist Albert A. Michelson (and repeatedly later together with Edward W. Morley) performed a historical experiment to verify or exclude the existence of ether. For this purpose Michelson constructed a set of mirrors placed on two perpendicular tracks, along which he sent two light beams that would finally superpose on a screen. As is well known in the context of waves, one expected to observe some characteristic pattern of interference. The experimental geometry was set up so that one of the beams ran along the earth's orbit around the sun, and the other ran perpendicular to it. This meant the first beam would have to propagate against the ether drift, like a swimmer against the stream, while the other was not affected this way. According to the ether concept, this should have caused the two light beams to have slighdy different velocities. If the whole gadget is turned by 90°, the roles of the beamsas well as their velocitiesmust exchange. This must inevitably lead to visible modification of the interference pattern on the screen. The experiment, conducted with highest accuracy and precision, did not show any such modification, however; the run time of the beams was evidendy not influenced by some ether drift. This important result caused scientists to doubt the existence of ether.

Nevertheless most scientists at that time were not willing to give up the idea of absolute space and time. The Dutch mathematician and physicist Hendrik A. Lorentz (1853-1928) conceived a mathematical gimmick to reconcile the Michelson-Morley result with the ether concept. He suggested any object moving with respect to the ether would contract by a tiny amount along the direction of motion. For this he postulated some mechanism of electromagnetic nature, acting on an atomic scale, but not yet determined in greater detail. Furthermore he invented a purely mathematically motivated new time coordinate, the proper time, as well depending on the reference system's velocity against the ether. With his Lorentz transformation rules it was perfectly possible to describe any physical scenario from any inertial system's perspective and still preserve the principle of relativity for electrodynamics. The Maxwell equations are, purportedly, invariant against the Lorentz transformation.

In the first years of the 20th century, Albert Einstein, then a “3rd class expert” in the Swiss patent office, started to address the physical and philosophical problem of space and time. Presumably without knowing about the Michelson-Morley result, he was convinced that absolute concepts of space and time would not be maintainable. He further postulated a universal version of Galileo's (mechanical) principle of relativity. According to Einstein, not only the rules of mechanics but rather any law of nature should keep the same form in any inertial system. This theory again assumes the universal constancy of the speed of light independent of the observer. He published his epoch-making work “Zur Elektrodynamik bewegter Körper” (“On Electrodynamics in Moving Bodies”) on September 25,1905. Therein he interpreted in a revolutionary manner Lorentz's mathematical preparatory works in terms of really varying spatial and temporal measures, depending on the observer's state of motion. Distances should reduce with increasing velocity, just as do time intervals. The speed of light acts as a universal limit: Neither matter nor information can propagate faster than light.

It is possible to demonstrate the relativistic concept of time in a vivid manner using the term simultaneity. As a thought experiment, we consider a train passing over a track with considerably high velocity. Let the train be open along its complete extension, so that the light of a bulb at the middle of the train can reach both the front (F) and the back (B) inside. There in each case a photocell is located, activating a signal when a light beam is detected. Let the train have the length L. We denote the signal event at the front/back end by S_{F} and S_{B} respectively. At some time t_{0} a passenger switches on the light, located exactly at the halfway point M (middle) between the detectors at F and B. The observer inside the train and in motion along with the train is called the train observer. He will observe that the light beams approach both of the train endings F and B with identical speed and that the signals S_{F} and
S_{B} are activated at the identical time t_{1} = t_{0} + ½ L/c (where c is the speed of light). Thus the events S_{F} and S_{B} appear simultaneous to the train observer. We call the time interval Δ = t_{1}-t_{0}= ½ L/c the runtime of the light beams in the train inertial system.

Let us now take the perspective of a person resting on the track (the track observer) and repeat the simple experiment. Both observers synchronize their clocks at the time t_{0}, when the light bulb in the train is switched on. Further we choose the time t_{0} so that the bulb at M is just passing the track observer exactly at time t_{1} = t_{0} + Δt (that is, the time when the signals are activated in the train inertial system). After the light bulb is turned on, the light beams in each direction are propagating with identical (light) speed c. During the time interval Δt, the beam approaching the front end covers the distance cΔt towards F During the time change Δt, the train (and so F) is moving on by a distance vΔt, where v is the speed of the train. Thus the front signal S_{F} will not yet be activated at t_{1} = t_{0} + Δt, because F, moving forward at the speed of the train, tried to “escape” from the light beam moving forward behind it. On the other hand, the back end signal S_{B} will move a little toward the light beam traveling backward during the same time and will thus be activated a little earlier than t_{1}. From the track observer's view, this obviously means that the signal event S_{B}occurs earlier than S_{F}while it was simultaneous for the train observer. Nevertheless both inertial systemstrain and trackare equitable in the physical sense. The concept of simultaneity is in fact dependent on the observer.

In a similar way it is possible to demonstrate a consequence of the Lorentz transformations, according to which the clock of a moving observer is running more slowly than a resting clock. For this purpose we construct a light clock in the following way. On both the ceiling and the floor of a train car we fix mirrors that face each other, their reflecting surfaces adjusted horizontally face to face. At time t_{0} we send out a very short light signal (say one photon) from the floor mirror to the ceiling mirror; this signal will then be reflected up and down between the mirrors for an arbitrary time. We denote the train velocity by v and the distance between the mirrors by d. The situation is trivial in the train inertial system. The photon starting at t_{0}reaches the upper mirror at t_{0} + d/c, is reflected, arrives at the lower mirror at t_{0} + 3d/c, gets to the ceiling mirror again at t_{0} + 3d/c, and so on. We call the time interval between the two reflections δt.

Next we analyze the light clock from the view of an external observer, resting on the track again, whose clock is synchronized with the clock of the train observer at t_{0}. Because the mirrors move along with velocity v, the movement of the photon also has a forward velocity component, as it remains between the mirrors. If it continued to move up and down with light speed c, the photon would obtain a total velocity
c' = √(c^{2}+v^{2}) that is, the vector sum of the horizontal and the upright components of its movement), in clear contradiction to the universal constancy of the speed of light. The upright velocity component must be lower than c by a small amount in order to not exceed a total velocity of c. Necessarily, due to the slowed perpendicular velocity, the time intervals At' between two reflections increase. As Δt' > Δt, the light clock is “ticking” more slowly in the track observer's frame.

The existence of time as a relative quantity is confirmed today by a multitude of experiments. Well known are the measurements with the help of atomic clocks, covering long distances in airplanes. When they return to the place from which they departed, these clocks show a delay by a tiny fraction of a second compared with the clocks resting at the airport. Another proof for this so-called time dilation is the fact that certain elementary particles, muons, created in high layers of the atmosphere by cosmic rays, reach the earth's surface at a much higher rate than one might expect according to their tiny average lives. In the relativ-istic framework, this result is interpreted in terms of a considerably slowed run of the muons' “clocks.” Thus the muons obtain significantly longer average lifetimes and can easily reach the ground instead of decaying halfway to the earth's surface.

Analogous to the effect on time intervals, spatial measures are also subject to relativistic effects at high velocities. As recognized by Lorentz prior to Einstein, length scales in a moving frame of reference are contracting from a resting observer's perspective, an effect called the Lorentz contraction. Whereas Lorentz considered this as a real electromagnetic effect on an atomic scale and a result of motion against the ether, in Einstein's interpretation it was generated purely by the change of reference frame. The above-mentioned argument, whereby muons reach the earth's ground due to their enlarged lifetimes, is quite obviously useless from the muons' own perspective: In their view, rather, a clock resting on the earth's surface is running more slowly than their own! Nevertheless the muons do reach the earth. The reason is that from their perspective, the thickness of the atmospherethe distance they have to pass down to the earthis greatly shortened due to the Lorentz contraction. This explains again conclusively the high rate at which muons are detected on the ground.

Effects of relativistic nature lack any relevance for our daily lives. This is simply due to the enormously high speed of light compared with any human experience. Relativistic effects of noticeable degrees only occur in frames of reference moving against each other at at least 10% of the speed of light. Muons from high up in the atmosphere arrive at more than 99% of this speed. Our lives would be considerably different if the speed of light were much lower, perhaps 200 kilometers per hour or so. Then a motorcyclist would confront relativistic effects, and a fast car driver could travel the entire length of US Route 66 and back while aging only a few minutes.

In the framework of special relativity, space and time lose the absolute character that has been considered self-evident over millennia. The German physicist Hermann Minkowski (1864-1909), who was devoted to a stringent mathematical formulation during his last years, suggested that we should understand space and time henceforward as components of a four-dimensional spacetime. In the common three-dimensional position, spacethe distance d between two pointsis calculated employing the appropriate x-, y- and Z-coordinates as

In the four-dimensional Minkowski space, the spatiotemporal “distance” between two events (i.e., between points in spacetime) is determined in an analogical manner using the formula

The light speed c in the first summand of the right-hand side guarantees the time coordinate to obtain the dimension of a length and can thus be directly compared with the spatial components. As is of special interest, because it is invariant against Lorentz transformations. This means that the spatiotemporal distance (As)_{A} between two events measured by some observer A is the same as
(As)_{B}observed from a moving system B.

The day following the submission of his work “On Electrodynamics of Moving Bodies,” Einstein sent a follow-up paper to the journal Annalen der Physik titled “Ist die Masse eines Körpers ein Maβ für dessen Energieinhalt?” (“Is the Mass of a Body a Measure for its Energy Content?”) Therein he discusses the consequences of momentum (i.e., the product of mass and speed of a body) in a theory of relativity. In this context Einstein shows in a simple mathematical derivation that the mechanical energy E of a body and its mass m are always connected via the relation E = mc^{2}. Mass and energy are thus nothing but distinct realizations of the same physical quantity. This plays an important role in physics, for example, in the study of nuclear bonds. A helium nucleus, for example, has a mass lower than the sum of its constituents' masses (two protons plus two neutrons). The mass difference corresponds to the amount of energy needed to dismantle the nucleus to its parts. Equally, the Einstein formula explains the destructive power of nuclear weapons, which are based on the release of the very same binding energy.

Cosmogony, Einstein, Albert, Galilei, Galileo, Light, Speed of, Newton, Isaac, Quantum Mechanics, Relativity, General Theory of, Space, Space and Time, Spacetime, Curvature of, Spacetime Continuum, Time, Relativity of

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