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Special relativity

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The Special relativity (SR) or Special relativity theory (SRT) is the physical theory published in 1905 by Albert Einstein. It replaced Newtonian notions of space and time, and incorporated electromagnetism as represented by Maxwell's equations. The theory is called "special" because the theory does not include a description of gravity. Ten years later, Einstein published the theory of general relativity, which incorporates gravitation.

Table of contents
1 Motivation for the theory of special relativity
2 Postulates of Special Relativity
3 Status of Special Relativity
4 Consequences of Special Relativity
5 Lack of an absolute reference frame
6 Mass, momentum, and energy
7 Simultaneity and Causality
8 The Geometry of Space-time in Special Relativity
9 Tests of postulates of special relativity
10 Related topics
11 External links

Motivation for the theory of special relativity

The principle of relativity was introduced by Galileo. Overturning the old absolute views of Aristotle, it held that motion, or as least uniform motion in a straight line only had meaning relative to something else, and that there was no absolute reference frame by which all things could be measured. Galileo also assumed a set of transformations called the Galilean transformations. These seem like common sense today. Galileo produced five laws of motion. Newton accepted the principle of relativity when constructing an improved set of only three laws of motion.

While these seemed to work well for everyday phenomena involving solid objects, light was still problematic. Newton believed that light was "corpuscular," but later physicists found that a transeverse wave model of light was more useful. Mechanical waves travel in a medium, and so it was assumed for light. This hypothetical medium was called the "luminiferous aether." It seemed to have some conflicting properties, such as being extremely stiff, to account for the high speed of light, while at the same time being insubstantial, so as not to slow down the Earth as it passes through. The idea of an aether seemed to reintroduce the idea of an absolute frame of reference, one that is stationary with respect to the aether.

In the early 19th century, light, electricity, and magnitism began to be understood as aspects of the electromagnetic field. Maxwell's equations showed that accelerating a charge produced electromagnetic radiation which always traveled at the speed of light. The equations showed that the speed of the radiation did not change based upon the speed of the source. This is consistent with analogies to mechanical waves. Presumably, however, the speed of the radiation would change based on the speed of the observer. Physicists tried to use this idea to measure the speed of the Earth with respect to the aether. The most famous attempt was the Michelson-Morley experiment. While these experiments were controversial for some time, a consensus emerged that the speed of light did not vary because of the speed of the observer, and since according to Maxwell's equations it did not vary because of the speed of the source, the speed of light must be invariant for all observers.

Before Special Relativity, Hendrik Lorentz and others had already noted that Electromagnetic forces differed depending on the observer. For example, one observer might see no magnetic field in a particular area while another moving relative to the first does. 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 what appeared at the time to be a reconciliation of Electromagnetics and Newtonian physics by replacing the Galilean transformations. When the velocities involved are much less than speed of light, the resulting laws simplify to the Galilean transformations. 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, Einstein's contribution was, inter alia, to derive these equations from a more fundamental theory, which theory did not require the presence of an aether. Einstein wanted to know what was invariant (the same) for all observers. Under Special Relativity, the seemingly complex transformations of Lorentz and Fitgerald derived cleanly from simple geometry and the Pythagorean theorem. The 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. Comparison 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 relativistic rocket problem. SR can correctly predict the behaviour of accelerating bodies in the presence of a constant or zero gravitational field, or those in a rotating reference frame. It is not capable of accurately describing motion in varying gravitational fields.

Postulates of Special Relativity

  1. First Postulate
The speed of light in vacuum is the same to all inertial observers. This postulate has been verified experimentally.

Second Postulate
Observation of physical phenomena by more than one inertial observer must result in agreement between the observers as to the nature of reality. Or, the nature of the universe must not change for an observer if their inertial state changes.
 
Every physical theory should look the same mathematically to every inertial observer.

Status of Special Relativity

Special relativity is now universally accepted by the physics community. General Relativity is still insufficiently confirmed by experiment to exclude certain alternative theories of gravitation.

Consequences of Special Relativity

Special Relativity has several consequences that struck many people as bizarre, among which are:

Lack of an absolute reference frame

Special Relativity rejects the idea of any absolute ('unique' or 'special') frame of reference; rather physics must look the same to all observers travelling at a constant velocity (inertial frame). This 'principle of relativity' dates back to Galileo, and is incorporated into Newtonian Physics. In the late 19th Century, some physicists suggested that the universe was filled with a substance known as "aether" which transmited Electromagnetic waves. Aether constituted an absolute reference frame against which speeds could be measured. Aether had some wonderful properties: it was sufficiently elastic that it could support electromagnetic waves, those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, including the Michelson-Morley experiment, suggested that the Earth was always 'stationary' relative to the Aether - something that is difficult to explain. The most elegant solution was to discard the notion of Aether and an absolute frame, and to adopt Einstein's postulates.

Mass, momentum, and energy

In addition to modifying notions of space and time, special relativity forces one to reconsider the concepts of mass, momentum, and energy, all of which are important constructs in Newtonian mechanics. Special relativity shows, in fact, that these concepts are all different aspects of the same physical quantity in much the same way that it shows space and time to be interrelated.

There are a couple of (equivalent) ways to define momentum and energy in SR. One method uses conservation laws. If these laws are to remain valid in SR they must be true in every possible reference frame. However, if one does some simple thought experiments using the Newtonian definitions of momentum and energy one sees that these quantities are not conserved in SR. One can rescue idea of conservation by making some small modifications to the definitions to account for relativistic velocities. It is these new definitions which are taken as the correct ones for momentum and energy in SR.

Given an object of mass m traveling at velocity v the energy and momentum are given by

where γ is given by
and c is the speed of light. The term γ occurs frequently in relativity, and comes from the Lorentz transformation equations. The energy and momentum can be related through the formula
which is referred to as the relativistic energy-momentum equation.

For velocities much smaller than those of light γ can be approximated using a series expansion and one finds that

Barring the first term in the energy expression (discussed below), these formulas agree exactly with the standard definitions of Newtonian kinetic energy and momentum. This is as it should be, for special relativity must agree with Newtonian mechanics at low velocities.

Looking at the above formulas for energy, one sees that when an object is at rest (v = 0 and γ = 1) there is a non-zero energy remaining:

This energy is referred to as rest energy. The rest energy does not cause any conflict with the Newtonian theory because it is a constant and, as far as kinetic energy is concerned, it is only differences in energy which matter.

Taking this formula at face value, we see that in relativity, mass is simply another form of energy. That is, there may be processes by which mass (in the form of rest energy) may be converted to other forms of energy such as kinetic energy, heat, or light. That these processes do, in fact, occur has been demonstrated vividly in the form of nuclear reactions. The implications of this formula on 20th century life has made it one of the most famous equations in all of science.

On mass

It is often stated that in special relativity the mass of a body increases as its velocity increases. However, this statement depends on one's definition of mass, and in SR there are actually two different notions of mass. The equations above use what is called the invariant mass or rest mass. This mass is an invariant quantity, meaning that it is the same for all inertial observers. In particular, the invariant mass does not increase with velocity.

Another definition of mass is the relativistic mass which is given by

Since γ increases with velocity so does the relativistic mass. This definition is convenient for some purposes. In particular, one can write the equations for energy and momentum as
which are valid in all reference frames. If the velocity is zero the relativistic mass and the rest mass become equal.

Neither definition is right or wrong, it is simply a matter of convenience. It turns out that in applications to general relativity and quantum field theory it is the invariant mass which is more useful. Thus many physicists simply refer to the mass when they actually mean the invariant mass.

Simultaneity and Causality

Special Relativity holds that events that are simultaneous in one frame of reference need not be simultaneous in another frame of reference.

The interval AB in diagram below is 'time-like'. I.e. there is a frame of reference in which event A and event B occur at the same location in space, separated only by occurring at different times. If A precedes B in that frame, then A precedes B in all frames. It is hypothetically possible for matter (or information) to travel from A to B, so there can be a causal relationship (with A the cause and B the effect).

The interval AC in diagram below is 'space-like'. I.e. there is a frame of reference in which event A and event C occur simultaneously, separated only in space. However there are also frames in which A precedes C (as shown) and frames in which C precedes A. Barring some way of traveling faster than light, it is not possible for any matter (or information) to travel from A to C or from C to A. Thus there is no causal connection between A and C.

light cone

The Geometry of Space-time in Special Relativity

SR uses a 'flat' 4 dimensional Minkowski space, usually referred to as space-time. This space, however, is very similar to the standard 3 dimensional Euclidean space, and fortunately by that fact, very easy to work with.

The differential of distance(ds) in cartesian 3D space is defined as:

where are the differentials of the three spatial dimensions. In the geometry of special relativity, a fourth dimension, time, is added, with units of c, so that the equation for the differential of distance becomes:

In many situations it may be convenient to treat time as imaginary (e.g. it may simplify equations), in which case in the above equation is replaced by , and the metric becomes

If we reduce the spatial dimensions to 2, so that we can represent the physics in a 3-D space,
We see that the null geodesics lie along a dual-cone:

image:sr1.jpg

defined by the equation

, or
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.

image:sr3.jpg

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 receiving is X years old.", we are looking down this line of sight: a null geodesic. We are looking at an event 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".)

image:sr1.jpg

The cone in the -t region is the information that the point is 'receiving', while the cone in the +t section is the information that the point is 'sending'.

Tests of postulates of special relativity

Related topics

People: Arthur Eddington | Albert Einstein | Hendrik Lorentz | Hermann Minkowski | Bernhard Riemann | Henri Poincaré | Robert S. Shankland
Relativity: Theory of relativity | principle of relativity | general relativity | frame of reference | inertial frame of reference | Lorentz transformations
Physics: Newtonian Mechanics | spacetime | speed of light | cosmology | Doppler effect
Math: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors
Philosophy: actualism | convensionalism | formalism

External links


General subfields within physics Edit
Classical mechanics | Condensed matter physics | Continuum mechanics | Electromagnetism |
General relativity | Particle physics | Quantum field theory | Quantum mechanics |
Solid state physics | Special relativity | Statistical mechanics | Thermodynamics