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Special relativity

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Special relativity

From Wikipedia, the free encyclopedia.

The special theory of relativity (SR) 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, or GR for short, which is the extension of special relativity to incorporate gravitation.

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. 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 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 inertial observer. This postulate (which comes from Maxwell's equations for electromagnetics) together with the requirement, successfully 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.
The twin paradox is the "story" of a twin who flies off in a spaceship travelling near the speed of light. When he returns he discovers that his twin has aged much more rapidly than he has (or he aged more slowly).
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 (e.g. length) of an object as measured by an observer may differ from those by another.
The mass of a particle increases as it's velocity increases. This led to the famous equation E = mc². See below.

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 suggested that the universe was filled with a substance known as "aether" (absolute space), against which speeds could be measured. Aether had some wonderful properties: it was sufficiently elastic that it could support electromagnetc waves, those waves could interact with matter, yet it offered no resistance to bodies passing through it. The results of various experiments, culminating in the famous Michelson-Morley experiment, suggested that either the Earth was always stationary, 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:

E = mc²

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:

E = γmc²,

where

$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$ .

(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

$E \approx m c^2 + \frac{1}{2}m v^2$

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 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.

Simultaneity

Special relativity also holds that the concept of simultaneity is relative to the observer: A 'time-like interval' has end-points separated by a path along which it is possible for a hypothetical matter or light to travel. A 'space-like interval' has end-points separated by a path in space-time along which neither light nor any slower-than-light signal could travel. No information can pass between points separated by a space-like interval. 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.

Status of Special Relativity

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. One alternative theory is doubly-special relativity, where a characteristic length is added to the list of invariant quantities.

The Geometry of Space-time in Special Relativity

SR uses a 'flat' 4 dimensional 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:

ds^2=dx_1^2+dx_2^2+dx_3^2

where (dx₁,dx₂,dx₃) 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:

ds^2=dx_1^2+dx_2^2+dx_3^2-c^2dt^2

In many situations it may be convenient to treat time as imaginary (e.g. it may simplify equations), in which case t in the above equation is replaced by i.t', and the metric becomes

ds^2=dx_1^2+dx_2^2+dx_3^2+c^2(dt')^2

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

ds^2=dx_1^2+dx_2^2-c^2dt^2

We see that the null geodesics lie along a dual-cone:

image:sr1.jpg

defined by the equation

ds^2=0=dx_1^2+dx_2^2-c^2dt^2

, or

dx_1^2+dx_2^2=c^2dt^2

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

ds^2=0=dx_1^2+dx_2^2+dx_3^2-c^2dt^2

dx_1^2+dx_2^2+dx_3^2=c^2dt^2

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 $d=\sqrt{x_1^2+x_2^2+x_3^2}$ 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'. We can envision a space of null dual-cones:

image:sr2.jpg

and recall the concept of cellular automata, applying it in a spatially and temporally continuous fashion.

Tests of postulates of special relativity

Michelson-Morley experiment - ether drift
Hamar experiment - obstruction of ether flow
Trouton-Noble experiment - torque on a capacitor
Kennedy-Thorndike experiment - time contraction
Forms of the emission theory experiment