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Special relativity

The special theory of relativity, or SR for short, is the physical theory published in 1905 by Albert Einstein that modified Newtonian physics to incorporate electromagnetism as represented by Maxwell's equations. The theory is called "special" because the theory applies only to the special case of measurements made when both the observer and that which is being observed are not affected by gravity. Ten years later, Einstein published the theory of General Relativity (GR), which is the extension of special relativity to incorporate gravitation.

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Motivation for the theory of special relativity

Before the formulation of special relativity, Hendrik Lorentz and others had already noted that electromagnetics differed from Newtonian physics in that observations by one of some phenomenon can differ from those of a person moving relative to that person at speeds nearing the speed of light. For example, one may observe no magnetic field, yet another observes a magnetic field in the same physical area. Lorentz suggested an aether theory in which objects and observers travelling with respect to a stationary aether underwent a physical shortening (Lorentz-Fitzgerald contraction) and a change in temporal rate (time dilation[?]). This allowed the partial reconciliation of electromagnetics and Newtonian physics. When the velocities involved are much less than speed of light, the resulting laws simplify to Newton's laws. The theory, known as Lorentz Ether Theory (LET) was criticized (even by Lorentz himself) because of its ad hoc nature.

While Lorentz suggested the Lorentz transformation equations as a mathematical description that accurately described the results of measurements, Einstein's contribution was to derive these equations from a more fundamental theory. Einstein wanted to know what was invariant (the same) for all observers. His original title for his theory was (translated from German) "Theory of Invariants". It was Max Planck who suggested the term "relativity" to highlight the notion of transforming the laws of physics between observers moving relative to one another.

Special relativity is usually concerned with the behaviour of objects and observers which remain at rest or are moving at a constant velocity. In this case, the observer is said to be in an inertial frame of reference or simply inertial. Comparision of the position and time of events as recorded by different inertial observers can be done by using the Lorentz transformation equations. A common misstatement about relativity is that SR cannot be used to handle the case of objects and observers who are undergoing acceleration (non-inertial reference frames), but this is incorrect. For an example, see the relativisic rocket[?] problem. SR can correctly predict the behaviour of accelerating bodies as long as the acceleration is not due to gravity, in which case general relativity must be used.

Invariance of the speed of light

SR postulated that the speed of light in vacuum is the same to all inertial observers, and said that every physical theory should be shaped or reshaped so that it is the same mathematically for every observer. This postulate (which comes from Maxwell's equations for electromagnetics) together with the requirement, succesfully reproduces the Lorentz transformation equations, and has several consequences that struck many people as bizarre, among which are:

  • The time lapse between two events is not invariant from observer to another, but is dependent on the relative speeds of the observers' reference frames.
  • Two events that occur simultaneously in different places in one reference frame may occur one after the other in another reference frame (relativity of simultaneity).
  • The dimensions (length, e.g.) of an object as measured by an observer may differ from those by another. This is an obvious difference from Newtonian physics, which showed no such effect.
  • The twin paradox is the "story" of a twin who "ages much more rapidly" than the other. (An even more incredible difference.)

Lack of an absolute reference frame

Another radical consequence is the rejection of the notion of an absolute, unique, frame of reference. Previously it had been believed that the universe traveled through a substance known as "aether" (absolute space), against which speeds could be measured. However, the results of various experiments, culminating in the famous Michelson-Morley experiment, suggested that either the Earth was always stationary (which is absurd), or the notion of an absolute frame of reference was mistaken and must be discarded.

Equivalence of mass and energy

Perhaps most far reaching, it also showed that energy and mass, previously considered separate, were equivalent, and related by the most famous expression from the theory:

<math>E = mc^2</math>

where E is the energy of the body (at rest), m is the mass and c is the speed of light. If the body is moving with speed v relative to the observer, the total energy of the body is:

<math>E = \gamma m c^2 </math>,
where
<math> \gamma = \frac{1}{\sqrt{1 - v^2/c^2}}</math>.

(The term γ occurs frequently in relativity, and comes from the Lorentz transformation equations.) It is worth noting that if v is much less than c this can be written as

<math>E \approx m c^2 + \frac{1}{2}m v^2</math>

which is precisely equal to the "energy of existence", mc2, and the Newtonian kinetic energy, mv2/2. This is just one example of how the two theories coincide when velocities are small.

At very high speeds, the denominator in the energy equation (2) approaches a value of zero as the velocity approaches c. Thus, at the speed of light, the energy would be infinite, which precludes things that have mass from moving at that speed.

The most practical implication of this theory is that it puts an upper limit to the laws (see Law of nature) of Classical Mechanics and gravity formed by Isaac Newton at the speed of light. Nothing carrying mass or information can move faster than this speed. As an object's velocity approaches the speed of light, the amount of energy required to accelerate it approaches infinity, making it impossible to reach the speed of light. Only particles with no mass, such as photons, can actually achieve this speed (and in fact they must always travel at this speed in all frames of reference), which is approximately 300,000 kilometers per second or 186,300 miles per second.

The name "tachyon" has been used for hypothetical particles which would move faster than the speed of light, but to date evidence of the actual existence of tachyons has not been produced.

Special relativity also holds that the concept of simultaneity is relative to the observer: If matter can travel along a path in spacetime without changing velocity, the theory calls this path a 'time-like interval', since an observer following this path would feel no motion and would thus travel only in 'time' according to his frame of reference. Similarly, a 'space-like interval' means a straight path in space-time along which neither light nor any slower-than-light signal could travel. Events along a space-like interval cannot influence one another by transmitting light or matter, and would appear simultaneous to an observer in the right frame of reference. To observers in different frames of reference, event A could seem to come before event B or vice-versa; this does not apply to events separated by time-like intervals.

Special relativity is now universally accepted by the physics community, unlike General Relativity which is still insufficiently confirmed by experiment to exclude certain alternative theories of gravitation. However, there are a handful of people opposed to relativity on various grounds and who have proposed various alternatives, mainly Aether theories.

The Theory

SR uses tensors or four-vectors to define a non-cartesian space. This space, however, is very similiar, and fortunately by that fact, very easy to work with. The differential of distance(ds) in cartesian space is defined as:

<math>ds^2=dx_1^2+dx_2^2+dx_3^2</math>
where <math>(dx_1,dx_2,dx_3)</math> are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, except it is treated as an imaginary quantity with units of c, so that the equation for the differential of distance becomes:
<math>ds^2=dx_1^2+dx_2^2+dx_3^2-c^2dt^2</math>

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space,

<math>ds^2=dx_1^2+dx_2^2-c^2dt^2</math>
We see that the null[?] geodesics lie along a dual-cone:

defined by the equation

<math>ds^2=0=dx_1^2+dx_2^2-c^2dt^2</math>
, or
<math>dx_1^2+dx_2^2=c^2dt^2</math>
Which is the equation of a circle with r=c*dt. If we extend this to three spatial dimensions, the null geodesics are continuous concentric spheres, with radius = distance = c*(+ or -)time.

<math>ds^2=0=dx_1^2+dx_2^2+dx_3^2-c^2dt^2</math>
<math>dx_1^2+dx_2^2+dx_3^2=c^2dt^2</math>

This null dual-cone represents the "line of sight" of a point in space. That is, when we look at the stars and say "The light from that star which I am recieving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event <math>d=\sqrt{x_1^2+x_2^2+x_3^2}</math> meters away and d/c seconds in the past. For this reason the null dual cone is also known as the 'light cone'. (The point in the lower left of the picture below represents the star, the origin represents the observer, and the line represents the null geodesic "line of sight".)

Geometrically, all "points" along the null dual-cone represent the same point in space-time( because the distance between them is zero). This can be thought of as 'the window of combustion' of forces. ("Connection is when two motions, once thought to be mutually exclusive, meet in a single moment." -James Morrison) It is where events in space-time intersect; how space interacts with itself. It is how a point "sees" the rest of the universe and is "seen" by it. The cone in the -t region is the information that the point is 'recieving', while the cone in the +t section is the information that the point is 'sending'. In this way, we can envision a space of null dual-cones:

and recall the concept of cellular automata, applying it in a spatially and temporally continuous fashion. This also holds for points in uniform translatory motion to eachother, a.k.a. inertial frames:

This means that the geometry of the universe remains the same regardless of the velocity(<math>\partial x/\partial t</math>) (inertia) of the observer. Let us recall Newton's law of motion: "An object in motion tends to stay in motion; an object at rest tends to stay at rest." - the law of conservation of kinetic energy.

However, the geometry does not remain constant when there is acceleration (<math>\partial ^2x / \partial t^2</math>) involved, as this implies an application of force (F=ma), and consequently a change in energy, which brings us to general relativity, in which the intrinsic curvature of space is directly proportional to the energy density at that point.

Modifications of special relativity

In the early 21st century a number of modified versions of special relativity have been postulated. One of the most notable of these is doubly-special relativity, where a characteristic length is added to the list of invariant quantities.

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