Misplaced Pages

Revision as of 20:03, 19 December 2005

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 in his article "On the Electrodynamics of Moving Bodies". It replaced Newtonian notions of space and time and incorporated electromagnetism as represented by Maxwell's equations. The theory is called "special" because it applies the principle of relativity only to the "restricted" or "special" case of inertial reference frames in flat spacetime, where the effects of gravity can be ignored. Ten years later, Einstein published his general theory of relativity (general relativity, "GR") which incorporated these effects.

For History and motivation, see the article: History of special relativity

Postulates

Main article: Postulates of special relativity

1. First postulate (principle of relativity)

The laws of electrodynamics and optics will be valid for all frames of reference in which the laws of mechanics hold good (non-accelerating frames).

In other words: Every physical theory should look the same mathematically to every inertial observer; the laws of physics are independent of the state of inertial motion.

2. Second postulate (invariance of c)

Light is always propagated in empty space with a definite velocity c that is independent of the state of motion of the emitting body; here the velocity of light c is defined as the two-way velocity, determined with a single clock.

In other words: The speed of light in vacuum, commonly denoted c, is the same to all inertial observers, and does not depend on the velocity of the object emitting the light. An observer attempting to measure the speed of light's propagation will get the same answer no matter how the system's components are moving.

Lack of an absolute reference frame

Special Relativity rejects the idea of any observable 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 19 Century, some physicists suggested that the universe was filled with a substance known as "aether" (sometimes written "ether") which transmitted Electromagnetic waves. Aether constituted an absolute reference frame against which speeds could be measured. In other words, the aether was the only fixed or motionless thing in the universe. 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. It was also postulated that light arose from vibrations of the aether. The results of various experiments, including the Michelson-Morley experiment, suggested to some that the Earth was always 'stationary' relative to the Aether — something that is difficult to explain. For many, the most elegant solution was to discard the notion of Aether and an absolute frame, and to adopt Einstein's postulates.

Consequences

Main article: Consequences of Special Relativity

Special relativity leads to different physical predictions than Galilean relativity when relative velocities become comparable to the speed of light. The speed of light is so much larger than anything humans encounter that some of the effects predicted by relativity are initially counter intuitive.

• The time lapse between two events is not invariant from one 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.

The Lorentz transformations of space and time

Full article: Lorentz transformations

Einstein has said that all of the consequences of Special Relativity can be found from examination of the Lorentz transformations.

Relativity theory depends on "reference frames". A reference frame is a point in space at rest, or in uniform motion, from which a position can be measured along 3 spatial axes. In addition, a reference frame has a clock, moving with the reference frame allowing the measurement of the time of events.

An event is an occurrence that can be assigned a single unique time and location in space relative to a reference frame: It is a "point" in space-time. For example, the explosion of a firecracker may be considered to be an "event". We can completely specify an event by its four space-time coordinates: The time of occurrence and its 3-dimensional spatial location from a reference point. Let's call this reference frame ${\displaystyle S}$.

In relativity theory we often want to calculate the position of a point from a different reference point.

Suppose we have a second reference frame ${\displaystyle S'}$, whose spatial axes and clock exactly coincide with that of ${\displaystyle S}$ at time zero, but it is moving at a constant velocity ${\displaystyle v}$ with respect to ${\displaystyle S}$ along the ${\displaystyle x}$ axis.

As we shall see, since there is no absolute reference frame in Relativity, a concept of 'moving' doesn't strictly exist, everything is always moving with respect to some other reference frame. Instead, in relativity theory any two frames ${\displaystyle S}$ and ${\displaystyle S'}$ are said to be comoving.

Let's define the event to have space-time coordinates ${\displaystyle (t,x,y,z)}$ in system ${\displaystyle S}$ and ${\displaystyle (t',x',y',z')}$ in ${\displaystyle S'}$. Then the Lorentz transformation specifies that these coordinates are related in the following way:

${\displaystyle t'=\gamma \left(t-{\frac {vx}{c^{2}}}\right)}$

${\displaystyle x'=\gamma (x-vt)\,}$

${\displaystyle y'=y\,}$

${\displaystyle z'=z\,}$

where ${\displaystyle \gamma \equiv {\frac {1}{\sqrt {1-v^{2}/c^{2}}}}}$ is called the Lorentz factor and ${\displaystyle c}$ is the speed of light in a vacuum.

As we can see the ${\displaystyle y}$ and ${\displaystyle z}$ coordinates are unaffected, but the ${\displaystyle x}$ and ${\displaystyle t}$ axes are mixed up by the transformation. In fact they are a form of rotation.

Lorentz Contraction and Time Dilation

Multiplying through we get:

${\displaystyle t'=\gamma t-\gamma {\frac {vx}{c^{2}}}}$

${\displaystyle x'=\gamma x-\gamma vt\,}$

Examination of the first term of the equation for ${\displaystyle x'}$ in the Lorentz transformation shows that all positions ${\displaystyle x}$ in one frame are multiplied by gamma, a number greater than one, to calculate the spatial interval in the second comoving frame. This may be correctly interpreted as a physical contraction of any object from full sized, and at rest in one frame, to the second frame in which it is moving. This is termed Lorentz Contraction.

Similarly, in the equation for time ${\displaystyle t'}$, ${\displaystyle t}$ is multiplied by gamma in the second comoving frame. This may be interpreted as time proceeding more slowly when an object is moving relative to another frame of reference. This is termed Time Dilation.

It might be expected that since one frame seems contracted, from the contracted frame of reference, the other would seem expanded; and similar effects with time. However since the Lorentz equations are symmetrical with respect to opposite relative speed, each frame sees the other as equally contracted and equally time dilated.

These effects are not merely appearances; the time in the different frame of references essentially do travel at different rates to each other and the lengths of objects really are physically changed whilst in relative motion.

See also the twin paradox.

Simultaneity

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

Simultaneity can be seen by considering the second term of the expanded Lorentz equation for ${\displaystyle t'}$. Here as the velocity ${\displaystyle v}$ varies two events move forwards or backwards in time relative to each other if they are physically separated in space. This can be observed in Diagram 1; some events may be observed moving from the past to the future and back again as acceleration between reference frames occurs and time passes.

Lack of simultaneity implies that, for example, the two ends of a rod in a comoving frame actually are not equally old — so for example, a cast radioactive rod would be older and have lower activity at the trailing edge than the leading edge. Indeed, lack of simultaneity explains why Lorentz Contraction occurs — the rod is partially tilted in time as it accelerates giving a foreshortening in the spatial dimension.

Causality

In diagram 2 the interval AB 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 direct causal connection between A and C. However, many points in spacetime would be in the light cone of both C and A and can be causally related to either or both of these events, and similarly both C and A could have been caused by an earlier event.

Since the set of points of spacetime that is in any events light cone is completely independent of reference frame, then causality is absolutely assured.

Addition of velocities

If the observer in ${\displaystyle S}$ sees an object moving along the ${\displaystyle x}$ axis at velocity ${\displaystyle w}$ then the observer in the ${\displaystyle S'}$ system will see the object moving with velocity ${\displaystyle w'}$ where

${\displaystyle w'={\frac {w-v}{1-wv/c^{2}}}}$.

This equation can be derived from the space and time transformations above. Notice that if the object is moving at the speed of light in the ${\displaystyle S}$ system (i.e. ${\displaystyle w=c}$), then it will also be moving at the speed of light in the ${\displaystyle S'}$ system. Also, if both ${\displaystyle w}$ and ${\displaystyle v}$ are small with respect to the speed of light, we will recover the intuitive Galilean transformation of velocities: ${\displaystyle w'=w-v}$.

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 m₀ traveling at velocity u the energy and momentum are given by

${\displaystyle E=\gamma m_{0}c^{2}\,\!}$

${\displaystyle p=\gamma m_{0}u\,\!}$

where γ (the Lorentz factor) is given by

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

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

${\displaystyle E^{2}-(pc)^{2}=(m_{0}c^{2})^{2}\,\!}$

which is referred to as the relativistic energy-momentum equation. These equations can be more succinctly stated using the four-momentum ${\displaystyle P^{a}}$ and the four-velocity ${\displaystyle U^{a}}$ as

${\displaystyle P^{a}=m_{0}U^{a}}$

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

${\displaystyle E\approx m_{0}c^{2}+{\begin{matrix}{\frac {1}{2}}\end{matrix}}m_{0}u^{2}\,\!}$

${\displaystyle p\approx m_{0}u\,\!}$

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 (u = 0 and γ = 1) there is a non-zero energy remaining:

${\displaystyle E=m_{0}c^{2}\,\!}$

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 are meaningful.

Taking this formula at face value, we see that in relativity, mass is simply another form of energy. In 1927 Einstein remarked about special relativity:

Under this theory mass is not an unalterable magnitude, but a magnitude dependent on (and, indeed, identical with) the amount 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 have made it one of the most famous equations in all of science.

Relativistic mass

Introductory physics courses and some older textbooks on special relativity sometimes define a relativistic mass which increases as the velocity of a body increases. According to the geometric interpretation of special relativity, this is often depreciated and the term 'mass' is reserved to mean 'rest mass' and is thus independent of the inertial frame, i.e., invariant.

Using the relativistic mass definition, the mass of an object may vary depending on the observer's inertial frame in the same way that other properties such as its length may do so. Defining such a quantity may sometimes be useful in that doing so simplifies a calculation by restricting it to a specific frame. For example, consider a body with an invariant mass m₀ moving at some velocity relative to an observer's reference system. That observer defines the relativistic mass of that body as:

${\displaystyle m=\gamma m_{0}\!}$

"Relativistic mass" should not be confused with the "longitudinal" and "transverse mass" definitions that were used around 1900 and that were based on an inconsistent application of the laws of Newton: those used F=ma for a variable mass, while relativistic mass corresponds to Newton's dynamic mass in which p=mv and F=dp/dt.

Note also that the body does not actually become more massive in its proper frame, since the relativistic mass is only different for an observer in a different frame. The only mass that is frame independent is the invariant mass. When using the relativistic mass, the used reference frame should be specified if it isn't already obvious or implied. It also goes almost without saying that the increase in relativistic mass does not come from an increased number of atoms in the object. Instead, each atom, indeed each subatomic particle increases its relativistic mass as the object accelerates.

Physics textbooks sometimes use the relativistic mass as it allows the students to utilize their knowledge of Newtonian physics to gain some intuitive grasp of relativity in their frame of choice (usually their own!). "Relativistic mass" is also consistent with the concepts "time dilation" and "length contraction".

The geometry of space-time

SR uses a 'flat' 4-dimensional Minkowski space, which is an example of a 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:

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}}$

where ${\displaystyle (dx_{1},dx_{2},dx_{3})}$ 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:

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}$

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

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}+c^{2}(dt')^{2}}$

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

${\displaystyle ds^{2}=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}$

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

defined by the equation

${\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}-c^{2}dt^{2}}$

or

${\displaystyle dx_{1}^{2}+dx_{2}^{2}=c^{2}dt^{2}}$

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×(±time).

${\displaystyle ds^{2}=0=dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}-c^{2}dt^{2}}$

${\displaystyle dx_{1}^{2}+dx_{2}^{2}+dx_{3}^{2}=c^{2}dt^{2}}$

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 ${\displaystyle d={\sqrt {x_{1}^{2}+x_{2}^{2}+x_{3}^{2}}}}$ 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.

Relativity and unifying Electromagnetism

The magnetic field generated by a moving charge disappears and becomes a purely electrostatic field in a comoving frame of reference. As electric and magnetic fields are reference frame dependent and thus intertwined, one speaks of electromagnetic fields. Special relativity provides the transformation rules for how an electromagnetic field in one inertial frame appears in another inertial frame.

Status

Main article: Status of special relativity

Special relativity is accurate only 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 few key experiments can be mentioned that led to special relativity:

• The Trouton-Noble experiment showed that the torque on a capacitor is independent on position and inertial reference frame — such experiments led to the first postulate
• The famous Michelson-Morley experiment demonstrated the directional invariance of the two-way speed of light — "the speed of light" as defined in the second postulate.

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

• Kaufman's experiment — electron deflection in exact accordance with Lorentz-Einstein prediction
• Hamar experiment — no "ether flow obstruction"
• Kennedy-Thorndike experiment — time dilation in accordance with Lorentz transformations
• Rossi-Hall experiment — relativistic effects on a fast-moving particle's half-life
• Experiments to test emitter theory demonstrated that the speed of light is independent of the speed of the emitter.

In addition, particle accelerators run almost every day somewhere in the world, and routinely accelerate and measure the properties of particles moving at near lightspeed. Many effects seen in particle accelerators are highly consistent with relativity theory and are deeply inconsistent with the earlier Newtonian mechanics.

See also

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

External links

• The Hogg Notes on Special Relativity A good introduction to special relativity at the undergraduate level.
• Beneath the Foundations of Spacetime Special relativity can be derived with moving rulers in such a way that the astonishing connection between space and time can be clearly understood.
• Relativity calculator Geometric calculations of relativistic problems such as the addition of velocities. Note that it is Java-based and can take several minutes to load using a 56k modem.
• Relativity in its Historical Context The discovery of special relativity was inevitable, given the momentous discoveries that preceded it.
• Nothing but Relativity There are many ways to derive the Lorentz transformation without invoking Einstein's constancy of light postulate. The path preferred in this paper restates a simple, established approach.
• Reflections on Relativity A complete online book on relativity with an extensive bibliography.
• Special Relativity Lecture Notes is a standard introduction to special relativity containing illustrative explanations based on drawings and spacetime diagrams from Virginia Polytechnic Institute and State University.
• Brane World Mach Principles and the Michelson-Morley experiment
• Petites expériences de pensée : five interesting thought experiments about special relativity quoted in the French Language Misplaced Pages.
• Special relativity theory made intuitive A new approach to explain the theoretical meaning of Special Relativity from an intuitive geometrical viewpoint
• Special Relativity Stanford University, Helen Quinn, 2003
• Relativity: the Special and General Theory at Project Gutenberg, by Albert Einstein
• Special Relativity This is chapter two of Christoph Schiller's 1000 page walk through the whole of physics, from classical mechanics to relativity, electrodynamics, thermodynamics, quantum theory, nuclear physics and unification. 61 pages.
• "Why Einstein may have got it wrong" by David Adam, The Guardian, April 11, 2005.
• Through Einstein's Eyes The Australian National University. Relativistic visual effects explained with movies and images.
• Greg Egan's Foundations.
• Short Essay Explaining Special Relativity by S. Abbas Raza of 3 Quarks Daily
• Einstein Light An award-winning, non-technical introduction (film clips and demonstrations) supported by dozens of pages of further explanations and animations, at levels with or without mathematics.
• Enlightening Ideas a humoristic animation about the special relativity for the general public, Yannick Mahé, 2005
• On the abuse and use of relativistic mass October 21, 2005

References

Textbooks

• Einstein, Albert. "Relativity — The Special and the General Theory", Routledge Classics 1993 ISBN 0-415-25538-4
• Tipler, Paul; Llewellyn, Ralph (2002). Modern Physics (4th ed.). W. H. Freeman Company. ISBN 0716743450
• Schutz, Bernard F. A First Course in General Relativity, Cambridge University Press. ISBN 0521277035
• Taylor, Edwin, and Wheeler, John (1992). Spacetime physics (2nd ed.). W.H. Freeman and Company. ISBN 0716723271
• Einstein, Albert (1996). The Meaning of Relativity. Fine Communications. ISBN 1567311369
• Geroch, Robert (1981). General Relativity From A to B. University of Chicago Press. ISBN 0226288641

Journal articles

• On the Electrodynamics of Moving Bodies, A. Einstein, Annalen der Physik, 17:891, June 30, 1905 (in English translation)
• 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).

Template:Link FA Template:Link FA Template:Link FA

Template:Link FATemplate:Link FA

• Special relativity