Difference between revisions of "Theory of relativity"
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'''Relativity''' refers to two closely-related theories in [[physics]], and to [[postulate]]s that led to the first [[theory]]. Special relativity (SR) is a theory which describes the laws of motion for non-accelerating bodies traveling at a significant fraction of the [[speed of light]]. As speeds approach zero, Special Relativity tends towards equivalence with [[Newton's Laws of Motion]]. Special Relativity was developed by [[Hendrik Lorentz]], [[Henri Poincaré]], Hermann Minkowski<ref>"German mathematician who developed the geometrical theory of numbers and who made numerous contributions to number theory, mathematical physics, and the theory of relativity." [http://www.britannica.com/eb/article-9052860/Hermann-Minkowski Hermann Minkowski -- Britannica Online Encyclopedia]</ref><ref>[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Minkowski.html Hermann Minkowski, Biography]</ref>, and [[Albert Einstein]]. | '''Relativity''' refers to two closely-related theories in [[physics]], and to [[postulate]]s that led to the first [[theory]]. Special relativity (SR) is a theory which describes the laws of motion for non-accelerating bodies traveling at a significant fraction of the [[speed of light]]. As speeds approach zero, Special Relativity tends towards equivalence with [[Newton's Laws of Motion]]. Special Relativity was developed by [[Hendrik Lorentz]], [[Henri Poincaré]], Hermann Minkowski<ref>"German mathematician who developed the geometrical theory of numbers and who made numerous contributions to number theory, mathematical physics, and the theory of relativity." [http://www.britannica.com/eb/article-9052860/Hermann-Minkowski Hermann Minkowski -- Britannica Online Encyclopedia]</ref><ref>[http://www-groups.dcs.st-and.ac.uk/~history/Biographies/Minkowski.html Hermann Minkowski, Biography]</ref>, and [[Albert Einstein]]. | ||
Revision as of 04:17, 9 November 2008
Relativity refers to two closely-related theories in physics, and to postulates that led to the first theory. Special relativity (SR) is a theory which describes the laws of motion for non-accelerating bodies traveling at a significant fraction of the speed of light. As speeds approach zero, Special Relativity tends towards equivalence with Newton's Laws of Motion. Special Relativity was developed by Hendrik Lorentz, Henri Poincaré, Hermann Minkowski^{[1]}^{[2]}, and Albert Einstein.
General Relativity (GR) is a theory which explains the laws of motion as viewed from accelerating reference frames and includes a geometric explanation for gravity. This theory was developed by David Hilbert and Albert Einstein as an extension of the postulates of Special Relativity.^{[3]} A dramatic but later discredited claim by Sir Arthur Eddington of experimental proof of General Relativity in 1919 made Einstein a household name.
Contents
Special Relativity
Special Relativity is usually explained in terms of two assumptions (postulates):
- The speed of light is constant for all (inertial) observers, regardless of their velocities relative to each other.
- The laws of physics are identical in all inertial reference frames.
In layman's terms, these two assumptions can be restated as:
- It is impossible ever to transmit information faster than the speed of light.^{[4]}
- The laws of physics are identical, without any variation, in every location throughout the universe.
- The laws of physics are identical, without any variation, no matter how fast something is traveling (in the absence of acceleration).
Special Relativity (SR) was initially developed by Henri Poincaré and Hendrik Lorentz, working on problems in electrodynamics and the Michelson-Morley experiment, which had not found any sign of luminiferous aether, which was believed to be the substance which carried electromagnetic waves. Special relativity alters Isaac Newton's laws of motion by assuming that the speed of light will be the same for all observers, despite their relative velocities and the source of the light. (Therefore, if A sends a beam of light to B, and both measure the speed, it will be the same for both, no matter what the relative velocity of A and B. In Newtonian/Galilean mechanics, If A sends a physical object at a particular velocity towards B, and nothing slows it, the velocity of the object relative to B depends on the velocities of the object and of B relative to A.)
At low speeds (relative to light-speed), the Einstein-Lorentz relativity equations are equivalent to Newton's equations. The famous equation attributed to Einstein, E=mc^{2}, describes the relationship between energy and the rest mass of a body.
Relativity is essential for massive or fast-moving bodies; for electromagnetism; for light and other radiation; for quantum field theory; for spin; and for nuclear energy. Particles at low mass and low speed can be accurately approximated by classical mechanics (such as Isaac Newton's laws of motion). At the two extremes, modeling the behavior of electrons requires that relativistic effects be taken into account (the chemically significant phenomenon of electron spin arises from relativity), and the course light passing through a region containing many massive bodies such as galaxies will be distorted (classical mechanics, in which light travels in straight lines, does not predict this). These are both experimentally confirmed (electron spin was known before relativity arose, and telescopic observations confirm that galactic clusters distort the paths of the light passing through them).
General Relativity
General Relativity is a theory of gravity that is compatible with Special Relativity. The theory was inspired by a thought experiment developed by Einstein involving two elevators. The first elevator is stationary on the Earth, while the other is being pulled through space at a constant acceleration of g. Einstein realized that any physical experiment carried out in the elevators would give the same result. This realization is known as the equivalence principle and it states that accelerating frames of reference and gravitational fields are indistinguishable. General Relativity is the theory of gravity that incorporates Special Relativity and the equivalence principle.
General Relativity is a mathematical extension of Special Relativity. GR views space-time as a 4-dimensional manifold, which looks locally like Minkowski space, and which acquires curvature due to the presence of massive bodies. Thus, near massive bodies, the geometry of space-time differs to a large degree from Euclidean geometry: for example, the sum of the angles in a triangle is not exactly 180 degrees. Just as in classical physics, objects travel along geodesics in the absence of external forces. Importantly though, near a massive body, geodesics are no longer straight lines. It is this phenomenon of objects traveling along geodesics in a curved spacetime that accounts for gravity.
The GR field equations are
where G_{uv} is the Einstein curvature tensor, and T_{uv} is the stress-energy tensor, G_{uv} and T_{uv} are both rank 2 symmetric tensors. The GR field equations is a system of partial differential equations that relates the curvature of space to the mass occupying the space.
General relativity provides one explanation for the seemingly anomalous precession of Mercury's perihelion. There are other explanations based in Newtonian gravity, such as factoring in the pull of the other planets on Mercury's orbit. One Newtonian explanation requires a slight alternation to the precise inverse-square relation of Newtonian gravity to distance, which is disfavored by mathematicians due to its inelegance in integrating.
British Historian Paul Johnson declares the turning point in 20th century to have been when fellow Briton Sir Arthur Eddington, an esteemed English astronomer, ventured out on a boat off Africa in 1919 with a local Army unit to observe the bending of starlight around the sun during a total eclipse. Upon his return to England declared that his observations proven the theory of relativity. In fact recent analysis of Eddington's work revealed that he was biased in selecting his data, and that overall his data were inconclusive about the theory of relativity. The prediction was later confirmed by more rigorous experiments, such as those performed by the Hubble Space Telescope ^{[5]}^{[6]}^{[7]}. Lorentz has this to say on the discrepancies between the empirical eclipse data and Einstein's predictions.
- It indeed seems that the discrepancies may be ascribed to faults in observations, which supposition is supported by the fact that the observations at Prince's Island, which, it is true, did not turn out quite as well as those mentioned above, gave the result, of 1.64, somewhat lower than Einstein's figure.^{[8]}
The prediction that light is bent by gravity is predicted both by Newtonian physics and relativity, but relativity predicts a larger deflection.
Special relativity is the limiting case of general relativity where all gravitational fields are weak. Alternatively, special relativity is the limiting case of general relativity when all reference frames are inertial (non-accelerating and without gravity).
Consequences of the Theories
Time dilation
One important consequence of SR's postulates is that an observer in one reference frame will observe a clock in another frame to be "ticking" more slowly than in the observer's own frame. This can be proven mathematically using basic geometry, if the postulates are physically true without exception.
The length of an event , as seen by a (relative) stationary observer observing an event is given by:
Where
- is the "proper time" or the length of the event in the observed frame of reference.
- is the relative velocity between the reference frames.
- is the speed of light (3x10^{8} ms^{-1}).
Evidence for time dilation was discovered by studying muon decay. Muons are subatomic particles with a very short halflife (1.53 microseconds at rest) and a very fast speed (0.994c). By putting muon detectors at the top (D_{1}) and bottom (D_{2}) of a mountain with a separation of 1900m, scientists could measure accurately the proportion of muons reaching the second detector in comparison to the first. The proportion found was different to the proportion that was calculated without taking into account relativistic effects.
Using the equation for exponential decay, they could use this proportion to calculate the time taken for the muons to decay, relative to the muon. Then, using the time dilation equation they could then work out the dilated time. The dilated time showed a good correlation with the time it took the muons to reach the second sensor, thereby supporting the existence of time dilation.
The time taken for a muon to travel from D_{1} to D_{2} as measured by a stationary observer is:
The fraction of muons arriving at D_{2} in comparison to D_{1} was 0.732. (Given by )
Since (from the equation for exponential decay) then
This gives the time for the proportion of decay to occur for an observer who is stationary, relative to the muon.
Putting this into the time dilation equation gives:
This is in good agreement with the value calculated above, thereby providing evidence to support time dilation.
Time Dilation and Creation Science
For a more detailed treatment, see Starlight problem.
Prevailing theories among creation scientists such as physicists Dr. Russell Humphreys and Dr. John Hartnett are time dilation explains why the earth is only 6,000 years old even though cosmological data (background radiation, supernovae, etc.) set a much older age for the universe.
Length contraction
When two inertial reference frames move past each other in a straight line with constant relative velocity, an observer in one reference frame would observe a metre rule in the other frame to be shorter.
The length, , of an object as seen by a (relative) stationary observer is given by:
Where
- is the "proper length" or the length of the object in the observed frame of reference.
- is the relative velocity between the reference frames.
- is the speed of light (3x10^{8} ms^{-1}).
Mass increase
We also see that as a body moves with increasing velocity its mass also increases.
The mass, , of an object as detected by a (relative) stationary observer is given by:
Where
- is the "rest mass" or the mass of the object when it is at rest.
- is the relative velocity of the object.
- is the speed of light (3x10^{8} ms^{-1}).
Since speed is relative, it follows that two observers in different inertial reference frames may disagree on the mass and kinetic energy of a body. Since all inertial reference frames are treated on an equal footing, it follows that mass and energy are interchangeable.
There is a logical difficulty, however, to an increase in relativistic mass. Such increase would only exist in the direction of motion, and the rest mass would remain intact with respect to a force applied in a direction orthogonal to velocity. But mass is not a vector, and the notion of the mass of an object having different values depending on the direction of an applied force is unacceptable. Accordingly, most physicists today avoid Einstein's original reliance on relativistic mass and his suggestion that mass increases. Instead, most physicists today teach that
where is the momentum defined by , is the standard Lorentz factor, and is the proper time. Force F defined this way is a vector and thus can handle the directional aspect of the relativistic effects better than the concept of relativistic mass can.
Evidence for Relativity
In 1972, scientists flew extremely accurate clocks around the world in both directions on commercial airlines, and were directly able to observe the relativistic "twin paradox" the eastbound clock gained 273 ns and the westbound clock lost 59 ns, matching the predictions of general relativity to within experimental accuracy ^{[9]}^{[10]}^{[11]}
Predictions of relativity have historically been used to make the Global Positioning System (GPS) function properly. A 1996 article says:
- "The Operational Control System (OCS) of the Global Positioning System (GPS) does not include the rigorous transformations between coordinate systems that Einstein's general theory of relativity would seem to require - transformations to and from the individual space vehicles (SVs), the Monitor Stations (MSs), and the users on the surface of the rotating earth, and the geocentric Earth Centered Inertial System (ECI) in which the SV orbits are calculated. There is a very good reason for the omission: the effects of relativity, where they are different from the effects predicted by classical mechanics and electromagnetic theory, are too small to matter - less than one centimeter, for users on or near the earth."^{[12]}^{[13]}
This article, which was published in 1996, goes on to propose relativistic corrections that might be used to design more accurate GPS systems. Clocks on board GPS satellites require adjustments to their clock frequencies if they are to be synchronized with those on the surface of the Earth.
Tom Van Flandern, an astronomer hired to work on GPS in the late 1990s, concluded that "[t]he GPS programmers don't need relativity." He was quoted as saying that the GPS programmers "have basically blown off Einstein."^{[14]} Asynchronization can be easily addressed through communications between the satellites and ground stations, so it is unclear why any theory would be needed for GPS. But other obscure physicists having no connection with GPS design claim that Van Flandern is wrong about GPS, and insist that relativity provides the best explanation for its timing adjustments.^{[15]}
Some internet articles claim that GPS timing differences confirm the Theory of Relativity or its Lorentzian counterpart (which uses a preferred frame of reference). GPS clocks run slower in the weaker gravitation field of the satellites than on ground stations on Earth, with the effects predicted by general relativity far outweighing the effects predicted by special relativity. However, the articles claiming that the slower GPS satellite clocks confirm relativity do not address the effect, if any, of the weaker gravitational force under Newton's theory on the GPS satellite clocks.
The effects of relativity have been observed with the frequency shift of the signal being sent back to Earth several times as various spacecraft have dipped into the gravity wells around massive objects such as the sun (see image at right)^{[16]} or Saturn^{[17]}. A satellite called Gravity Probe B was put in orbit about the Earth to examine the effects of frame dragging and geodetic warping of space^{[18]}^{[19]}, but the results were inconclusive. Note, however, that Newtonian mechanics also predicts deflection of light by gravity, and in the initial theory of relativity it predicted the same amount of deflection.^{[20]} Adjustments to the theory of relativity resulted in a prediction of a greater deflection of light than that predicated by Newtonian mechanics, though it is debatable how much deflection Newtonian mechanics should predict.
None of the NASA spacecraft incorporates predictions of relativity into their own timing mechanisms, as Newtonian mechanics is adequate even for probes sent deep into space.^{[21]}
A decade of observation of the pulsar pair PSR B1913+16 detected a decline in its orbital period, which was attributed to a loss in energy by the system. It is impossible to measure the masses of the pulsars, their accelerations relative to the observers, or other fundamental parameters. Professors Joseph Taylor and Russell Hulse, who discovered the binary pulsar, found that physical values could be assigned to the pulsars to make the observed decline in orbital period consistent with the Theory of General Relativity, and for this they were awarded the 1993 Nobel Prize for Physics, which is the only award ever given by the Nobel committee for the Theory of Relativity.^{[22]} In 2004, Professor Taylor utilized a correction to the derivative of the orbital period to fit subsequent data better to the theory.
The perihelion of Mercury's orbit precesses at a measurable rate, but even after after accounting for gravitational perturbations caused all other planets in the solar system, Newton's theory (assuming a precise inverse-square relationship for distance) predicts a rate of precession that differs from the measured rate by approximately 43 arcseconds per century. General relativity was developed in part to provide an estimate for this rate of precession that better matches observations.^{[23]} ^{[24]} ^{[25]}
General relativity predicts twice as much bending in light as it passes near massive objects than Newton's theory predicts,^{[26]} a phenomenon known as gravitational lensing. A large number of instances of gravitational lensing have been observed, and it is now a standard astronomical tool.^{[27]} ^{[28]} ^{[29]}
Ostensible Paradoxes
The Theory of Relativity implies that physical constants like the speed of light have remained constant. But at least one study suggests that physical constants, and possibly even the speed of light, have changed as the universe has aged.^{[30]}
"For the first time, scientists have experimentally demonstrated that sound pulses can travel at velocities faster than the speed of light, c. William Robertson's team from Middle Tennessee State University also showed that the group velocity of sound waves can become infinite, and even negative. ... Although such results may at first appear to violate special relativity (Einstein's law that no material object can exceed the speed of light), the actual significance of these experiments is a little different. These types of superluminal phenomena, Robertson et al. explain, violate neither causality nor special relativity, nor do they enable information to travel faster than c. In fact, theoretical work had predicted that the superluminal speed of the group velocity of sound waves should exist. 'The key to understanding this seeming paradox is that no wave energy exceeded the speed of light,' said Robertson."^{[31]}
"A team of researchers from the Ecole Polytechnique Fédérale de Lausanne (EPFL) has successfully demonstrated, for the first time, that it is possible to control the speed of light – both slowing it down and speeding it up – in an optical fiber, using off-the-shelf instrumentation in normal environmental conditions. Their results, to be published in the August 22 issue of Applied Physics Letters, could have implications that range from optical computing to the fiber-optic telecommunications industry."^{[32]} Both slowing down and speeding up of light within a substance other than a vacuum is made possible, because the light travels through the material, and that material affects the speed of light, i.e. a photon hits an electron, which then exits and emits a slightly lower energy photon out in the direction that the original photon was traveling, thus maintaining conservation of momentum. No matter how transparent an object may appear, it radically impacts the speed of the light traveling through it, as demonstrated by the refractive production of a rainbow by a crystal, which Newton himself discovered.
"A pair of German physicists claim to have broken the speed of light - an achievement that would undermine our entire understanding of space and time. ... Dr Nimtz told New Scientist magazine: 'For the time being, this is the only violation of special relativity that I know of.'"^{[33]}
Pending research
Today some physicists are working on hypothesizing how general relativity might have related to the other three forces of nature during the first fraction of a second of the Big Bang. Two of the more commonly studied attempts are string theory and loop quantum gravity, but they have failed to produce any evidence that science mandates a science must have, and both typically take large amounts of work to even conform to what scientists believe. Critics increasingly point out that string theory and loop quantum gravity are largely untestable and unfalsifiable, and thus potentially unscientific under the principles of science advanced by Karl Popper.^{[34]}
Relativity continues to be tested and some physics professors remain skeptical of the theory, such as University of Maryland physics professor Carroll Alley, who served as the principle physicist on the Apollo lunar project.^{[35]}
Political aspects of relativity
There has been little recognition by the Nobel Prize committee of either theory of relativity, and particularly scant recognition of the Theory of General Relativity.
There is a correlation between enthusiasm for the theory of relativity and political views. For example, Democratic presidential candidate Barack Obama helped publish an article by liberal law professor Laurence Tribe to apply the relativistic concept of "curvature of space" to promote a broad legal right to abortion.^{[36]} As of June 2008, over 170 law review articles have cited this liberal application of the theory of relativity to legal arguments.^{[37]} Applications of the theory of relativity to change morality have also been common.^{[38]} Moreover, there is an unmistakable effort to censor or ostracize criticism of relativity.^{[39]}
Physicist Robert Dicke of Princeton University was a prominent critic^{[40]} of the theory of relativity, and Dicke's alternative "has enjoyed a renaissance in connection with theories of higher dimensional space-time."^{[41]} Despite being one of the most accomplished physicists in the 20th century, Dicke was never awarded a Nobel Prize.
Government Support for Relativistic research
The Theory of Relativity enjoys a disproportionate share of federal funding of physics research today,^{[42]}. In at least one case that research has been unsuccessful. The $365 million dollar LIGO project has failed to detect the gravity waves predicted by relativity.^{[43]}
References
- ↑ "German mathematician who developed the geometrical theory of numbers and who made numerous contributions to number theory, mathematical physics, and the theory of relativity." Hermann Minkowski -- Britannica Online Encyclopedia
- ↑ Hermann Minkowski, Biography
- ↑ "[T]he German mathematician David Hilbert submitted an article containing the correct field equations for general relativity five days before Einstein."Nobel Prize historical account
- ↑ This assumption is commonly restated in this manner. For example, a discussion of hypothetical tachyons talks "about using tachyons to transmit information faster than the speed of light, in violation of Special Relativity."[1] However, there is some question whether the Theory of Special Relativity really restricts faster-than-light communication of information.
- ↑ Hubble Gravitational Lens Photo
- ↑ Gravitational Lensing
- ↑ [2]
- ↑ Lorentz, H.A. The Einstein Theory of Relativity
- ↑ Hafele-Keating Experiment
- ↑ As described in What is the experimental basis of Special Relativity?, a personal web page, which cites Haefele and Keating (1972), Science Vol. 177 pp 166-170 as its source
- ↑ Sullivan, Walter (1972), "Relativity Theory Awaits Affirmation", September 23, 1972, p. 61. Note: Article refers to a different experiment, which Sullivan discusses, saying that if successful it would be "the second time within a year" that relativity had been confirmed, then proceeds to discuss Hafele[sic] and Keating's experiment as the first.
- ↑ http://tycho.usno.navy.mil/ptti/1996/Vol%2028_16.pdf
- ↑ Some do claim that relativity is "vital" to GPS even though GPS developed independently of theoretical predictions and theoreticians disagree about how the relativistic effects for GPS should be calculated. See id. See also [3]
- ↑ http://archive.salon.com/people/feature/2000/07/06/einstein/index.html See also [4], where Van Flandern discusses how relativistic corrections might improve GPS accuracy.
- ↑ Ibid.
- ↑ Saturn-Bound Spacecraft Tests Einstein's Theory
- ↑ Encounter with Saturn confirms relativity theory
- ↑ NASA Gravity Probe B mission page
- ↑ Gravity Probe B project page
- ↑ http://www.mathpages.com/rr/s6-03/6-03.htm
- ↑ There is no reported reliance on relativity by any space probe.
- ↑ http://nobelprize.org/nobel_prizes/physics/laureates/1993/press.html
- ↑ http://physics.ucr.edu/~wudka/Physics7/Notes_www/node98.html#SECTION032121000000000000000
- ↑ http://www.alberteinstein.info/gallery/pdf/CP6Doc30_English_pp146-200.pdf
- ↑ http://farside.ph.utexas.edu/teaching/336k/lectures/node117.html
- ↑ http://www.mathpages.com/rr/s6-03/6-03.htm
- ↑ http://imagine.gsfc.nasa.gov/docs/features/news/grav_lens.html
- ↑ http://astro.berkeley.edu/~jcohn/lens.html
- ↑ http://www.iam.ubc.ca/~newbury/lenses/glgallery.html
- ↑ James Glanz and Dennis Overbye, "Cosmic Laws Like Speed of Light Might Be Changing, a Study Finds," August 15, 2001.[5]
- ↑ http://www.physorg.com/news88249076.html
- ↑ http://www.scienceblog.com/light.html
- ↑ http://www.telegraph.co.uk/earth/main.jhtml?xml=/earth/2007/08/16/scispeed116.xml
- ↑ See, for example, Not Even Wrong, by Peter Woit
- ↑ http://science.nasa.gov/headlines/y2004/21jul_llr.htm
- ↑ Tribe and Obama argued that the Constitution should be interpreted to establish a right to federally funded abortion and that, more generally, Roe v. Wade does not go far enough. They insisted that a relativistic "curvature of space" could achieve this result by expanding application of the Constitution based on its impact on personal choice. "The Roe v. Wade opinion ignored the way in which laws regulating pregnant women may shape the entire pattern of relationships among men, women, and children. It conceptualized abortion not in terms of the intensely public question of the subordination of women to men through the exploitation of pregnancy, but in terms of the purportedly private question of how women might make intimately personal decisions about their bodies and their lives. That vision described a part of the truth, but only what might be called the Newtonian part. ... [A] change in the surrounding legal setting can constitute state action that most threatens the sphere of personal choice. And it is a 'curved space' perspective on how law operates that leads one to focus less on the visible lines of legal force and more on how those lines are bent and directed by the law's geometry." Laurence H. Tribe, The Curvature of Constitutional Space: What Lawyers Can Learn from Modern Physics, 103 Harv. L. Rev. 1, 16-17 (1989).
- ↑ Search conducted by User:Aschlafly in the LEXIS database "US Law Reviews and Journals, Combined," conducted June 1, 2008.
- ↑ "Mistakenly, in the minds of many, the theory of relativity became relativism."[6]
- ↑ Although the liberally biased Wikipedia contains lengthy criticisms of the subjects of many entries, and even though publications like The Economist recognize the lack of scientific satisfaction in the theory (see, e.g., "Weighing the Universe," The Economist (Jan. 25, 2007)), Wikipedia's entry on Theory of Relativity omits one word of criticism.
- ↑ http://www.time.com/time/magazine/article/0,9171,943324,00.html
- ↑ "Initially a popular alternative to General Relativity, the Brans-Dicke theory lost favor as it became clear that omega must be very large-an artificial requirement in some views. Nevertheless, the theory has remained a paradigm for the introduction of scalar fields into gravitational theory, and as such has enjoyed a renaissance in connection with theories of higher dimensional space-time."[7]
- ↑ The Democratic Congress insisted on the $250 million LIGO project despite substantial criticism by scientists that it was wasting scarce research dollars. John Travis, "LIGO: a $ 250 million gamble; Laser Interferometer Gravitational-Wave Observatory; includes related article," Science p. 612 (Apr. 30, 1993). "Adding to the acrimony is LIGO's $ 250 million price tag, which some hold responsible for NSF's recent funding woes." Id.
- ↑ http://www.npr.org/programs/atc/features/2002/sept/gravitywaves/index.html
External Links
- The Einstein Theory of Relativity, by H.A. Lorentz.
- Relativity Calculator - Learn Special Relativity Mathematics The mathematics of special relativity presented in as simple and comprehensive manner possible within philosophical and historical contexts.