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# Special relativity

(Redirected from Relativistic mechanics)

A simple introduction to this subject is provided in Special relativity for beginners

Special relativity (SR) or the special theory of relativity 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 it is a special case of Einstein's principle of relativity where the effects of gravity can be ignored. Ten years later, Einstein published the theory of general relativity which incorporates gravitation.

 Contents

## Motivation

The principle of relativity was introduced by Galileo. Overturning the old absolutist views of Aristotle, it held that motion, or at 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, which seem like common sense today. Galileo produced five laws of motion.

Next came Newton who inferred from his rotating bucket experiment an "absolute space", that is, an absolute reference frame. Nevertheless he kept the principle of relativity for what can be observed— uniform motion could not detect his absolute space. That concept he used for constructing an improved set of equations containing 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 transverse 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, which moves through it. The idea of an aether seemed to reintroduce the idea of a detectable absolute frame of reference, one that is stationary with respect to the aether.

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

Before special relativity, Hendrik Lorentz and others had already noted that electromagnetic forces differed depending on the position of 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 the speed of light, the resulting laws simplify to the Galilean transformations. He proposed it to be valid for all forces. However, at that point in time he didn't realise the full power of his theory. The theory, today called Lorentz Ether Theory (LET) was criticized, even by Lorentz himself, because of its apparently ad hoc nature. For all practical purposes it is the same theory as SRT, and he taught it as such.

While Lorentz suggested the Lorentz transformation equations, Einstein's contribution was, inter alia, to derive these equations from a more fundamental principle without assuming 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 Fitzgerald derived cleanly from simple geometry and the Pythagorean theorem. The original title for his theory was (translated from German) "On the Electrodynamics of Moving Bodies". Max Planck first 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" (inertial reference systems) 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

Main article: Postulates of special relativity

1. First postulate (principle of relativity)

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.
To state that simply, no property of the universe will change if the observer is in motion. The laws of the universe are the same regardless of inertial frame of reference.

2. Second postulate (invariance of c)

The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, is the same in all directions, and does not depend on the velocity of the object emitting the light. When combined with the First Postulate, this Second Postulate is equivalent to stating that light does not require any medium (such as "aether") in which to propagate.

## Status

Main article: Status of special relativity

Special relativity is only accurate when gravitational effects are negligible or very weak, otherwise it must be replaced by general relativity. At very small scales, such as at the Planck length and below, it is also possible that special relativity breaks down, due to the effects of quantum gravity. However, at macroscopic scales and in the absence of strong gravitational fields, special relativity is now universally accepted by the physics community and experimental results which appear to contradict it are widely believed to be due to unreproducible experimental error.

Because of the freedom one has to select how one defines units of length and time in physics, it is possible to make one of the two postulates of relativity a tautological consequence of the definitions, but one cannot do this for both postulates simultaneously, as when combined they have consequences which are independent of one's choice of definition of length and time.

Special relativity is mathematically self-consistent, and is also compatible with all modern physical theories, most notably quantum field theory, string theory, and general relativity (in the limiting case of negligible gravitational fields). However special relativity is incompatible with several earlier theories, most notably Newtonian mechanics. See Status of special relativity for a more detailed discussion.

A number of experiments have been conducted to test special relativity against rival theories. These include:

## Consequences

Main article: Consequences of Special Relativity

Special relativity 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. (See Lorentz transformation equations)
• Two events that occur simultaneously in different places in one frame of reference may occur at different times in another frame of reference (lack of absolute simultaneity).
• The dimensions (e.g. length) of an object as measured by one observer may differ from the results of measurements of the same object made by another observer. (See Lorentz transformation equations)
• The twin paradox concerns 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).
• The ladder paradox involves a long ladder travelling near the speed of light and being contained within a smaller garage.

## 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 the 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 invariant mass m traveling at velocity v the energy and momentum are given by

$E = \gamma m c^2 \,\!$
$p = \gamma m v \,\!$

where γ (the Lorentz factor) is given by

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

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

$E^2 - (p c)^2 = (m c^2)^2 \,\!$

which is referred to as the relativistic energy-momentum equation. These equations can be more succinctly stated using the four-momentum ${\mathbf p}$ and the four-velocity ${\mathbf v}$ as

${\mathbf p} = m {\mathbf v}$

which can be viewed as a relativistic analogue of Newton's second law.

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

$E \approx m c^2 + \begin{matrix} \frac{1}{2} \end{matrix} m v^2 \,\!$
$p \approx m v \,\!$

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:

$E = m c^2 \,\!$

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. This formula becomes important when one measures the masses of different atomic nuclei. By looking at the difference in masses, one can predict which nuclei have extra stored energy which can be released by nuclear reactions, providing important information which was useful in the development of the nuclear bomb. 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, notably in older textbooks and in some introductory physics courses. 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 m. 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

$M = \gamma m \,\!$

Since γ increases with velocity so does the relativistic mass. This definition is more consistent with (relativistic) length and time and convenient for some purposes. In particular, one can write the equations for energy and momentum as

$E = M c^2 \,\!$
$p = M v \,\!$

which are valid in all reference frames. If the velocity is zero the relativistic mass and the invariant mass become equal.

Neither definition is right or wrong. However, many physicists dislike the concept of relativistic mass because it changes under a Lorentz transformation; they prefer to formulate the special theory of relativity in terms of invariant quantities. The invariant mass is an important quantity in general relativity and quantum field theory. Thus many physicists simply refer to the mass when they actually mean the invariant mass, while they refer to relativistic energy instead of relativistic 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 the diagram to the right 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 the diagram 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.

## The geometry of space-time

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:

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

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:

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

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,

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

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

defined by the equation

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

, or

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

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.

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

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".)

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'.

The geometry of Minkowski space can be depicted using Minkowski diagrams, which are also useful in understanding many of the thought-experiments in special relativity.

## Related topics

People: Arthur Eddington | Albert Einstein | Hendrik Lorentz | Hermann Minkowski | Bernhard Riemann | Henri Poincaré | Alexander MacFarlane | Harry Bateman | 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 | simultaneity | cosmology | Doppler effect | relativistic Euler equations | Aether drag hypothesis
Math: Minkowski space | four-vector | world line | light cone | Lorentz group | Poincaré group | geometry | tensors | split-complex number
Philosophy: actualism | convensionalism | formalism

## References

• Einstein, Albert. The Meaning of Relativity.
• Schutz, Bernard F. A First Course in General Relativity, Cambridge University Press.
• Taylor, Edwin, and Wheeler, John, Spacetime physics, W.H. Freeman and Company, second edition, 1992.
• Wolf, Peter and Gerard, Petit. "Satellite test of Special Relativity using the Global Positioning System," Physics Review A 56 (6), 4405-4409 (1997).
• Will, Clifford M. "Clock synchronization and isotropy of the one-way speed of light," Physics Review D 45, 403-411 (1992).
• Alvager et al., "Test of the Second Postulate of Special Relativity in the GeV region," Physics Letters 12, 260 (1964).

03-10-2013 05:06:04