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Line 111: + + + + + + + + + + + + + + + + + + + + + + + + + + + − :Increasingly precise measurements of the precession demonstrate that it conflicts with General Relativity, despite claims of relativists for decades that it predicted the precession accurately in the amount of <math>3{}v^2/c^2</math> revolutions per planet's "year", where <math>v</math> is the planet's average orbital speed.<ref>That is a simple approximation, designed to relate the precession to the planet's speed relative to the speed of light. A more accurate approximation is <math>\frac{3GM}{c^2 a(1-e^2)}</math>, where a is the semi-major axis and e is the eccentricity.</ref> The conflict is greater than the margin of error, and many relativists avoid the discrepancy rather than address it.+ − − Line 213:
Line 238: − ==Experimental and Observational Evidence Confirming Relativity==+ − + − The different effects predicted by special relativity, compared to classical formulations, are extremely tiny. Most relativistic effects are negligible at the speeds of ordinary phenomena observed by humans. The effects only become significant when the speeds involved are a significant fraction of the speed of light, which is <math>3 \times 10^8</math> meters per second—such speeds are called ''relativistic''. (However, it's worth noting that ordinary magnetism can be considered an effect of relativity, dictated by the need for electrostatic theory to be correct under relativity. The speed of light in fact appears in the formulas ([[Maxwell's Equations]]) governing electricity and magnetism, though these equations were developed long before relativity was proposed.)+ − + − Because the effects of relativity are so tiny, scientists have been devising sophisticated and sensitive tests ever since the theory was formulated in 1905.+ − + − The most famous experiment, and the one that is commonly cited in textbooks as the experiment that established the case for relativity<ref>Though relativity did not actually originate from this experiment</ref>, was the [[Michelson-Morley experiment]]. This showed that all observers will obtain the same measured value for the speed of light (3x10<sup>8</sup> meters per second) no matter what their state of motion. This is the first of the two fundamental principles:+ − #''The [[speed of light]] is constant for all observers, regardless of their velocities relative to each other.'' − #''The laws of physics are identical in all reference frames.'' − − A naive "common sense" interpretation of Galilean relativity would require that measurements of the speed of light (or anything else) by different observers would get results that differ by the observers' relative speeds, and hence that principle #1 can't be true. Special relativity fixes this apparent paradox. − − All of special relativity derives for these two principles, plus assumptions of exact conservation of momentum and energy in all cases. − − *At the end of Einstein's original 1905 paper [http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ "Does the Inertia of a Body Depend its Energy Content?"], he speculates on the possibility that the equation <math>E = m c^2</math>, which would normally be very hard to verify, could be verified with the extremely high energies of the newly discovered phenomenon of radioactivity.<ref>This equation is not related to [[quantum mechanics]].</ref> In the 1910s, with the invention of the mass spectrometer, it became possible to measure masses of nuclei accurately. This led to the clearing up of the mystery of atomic masses not being exact integers,and strongly suggested the existence of a "mass defect" (or "packing fraction") consistent with the mass-energy equivalence. In the 1930s, experiments with known nuclear reactions showed a very accurate correlation between the masses of the nuclei involved and the energy released. See [[Quantitative Analysis of Alpha Decay]]. − − *Another prediction of special relativity was time dilation in rapidly moving objects. This effect was most famously verified in the anomalously slow decay of relativistic cosmic muons.<ref>Some have suggested that other explanations are possible for this effect. We are trying to track this down.</ref> Time dilation has since been verified many times, and is routinely taken into account in all high-energy nuclear physics experiments, as in Hadron collision experiments.<ref>Experiments specifically designed to check dilation are rarely conducted any more.</ref> − − [[Image:Cassini-science-289.jpg|right|thumb|The Shapiro effect: A spacecraft signal dipping into a gravity well around the [[Sun]] is delayed slightly.]] − As the 20th century progressed, more tests of general relativity were proposed. − +
More info about precession.
*The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury. There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun. These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century. This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.
*The second "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury. There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun. These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century. This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.
:Increasingly precise measurements of the precession demonstrate that it conflicts with General Relativity, despite claims of relativists for decades that it predicted the precession accurately in the amount of <math>3{}v^2/c^2</math> revolutions per planet's "year", where <math>v</math> is the planet's average orbital speed.<ref>That is a simple approximation, designed to relate the precession to the planet's speed relative to the speed of light. A more accurate approximation is <math>\frac{3GM}{c^2 a(1-e^2)}</math>, where a is the semi-major axis and e is the eccentricity.</ref> The conflict is greater than the margin of error, and many relativists avoid the discrepancy rather than address it.
==Experimental and Observational Evidence Confirming Relativity==
The different effects predicted by special relativity, compared to classical formulations, are extremely tiny. Most relativistic effects are negligible at the speeds of ordinary phenomena observed by humans. The effects only become significant when the speeds involved are a significant fraction of the speed of light, which is <math>3 \times 10^8</math> meters per second—such speeds are called ''relativistic''. (However, it's worth noting that ordinary magnetism can be considered an effect of relativity, dictated by the need for electrostatic theory to be correct under relativity. The speed of light in fact appears in the formulas ([[Maxwell's Equations]]) governing electricity and magnetism, though these equations were developed long before relativity was proposed.)
Because the effects of relativity are so tiny, scientists have been devising sophisticated and sensitive tests ever since the theory was formulated in 1905.
The most famous experiment, and the one that is commonly cited in textbooks as the experiment that established the case for relativity<ref>Though relativity did not actually originate from this experiment</ref>, was the [[Michelson-Morley experiment]]. This showed that all observers will obtain the same measured value for the speed of light (3x10<sup>8</sup> meters per second) no matter what their state of motion. This is the first of the two fundamental principles:
#''The [[speed of light]] is constant for all observers, regardless of their velocities relative to each other.''
#''The laws of physics are identical in all reference frames.''
(The second is just a restatement of Galilean relativity, that is, the "common sense" that had been accepted for centuries.)
A naive "common sense" interpretation of Galilean relativity would require that measurements of the speed of light (or anything else) by different observers would get results that differ by the observers' relative speeds, and hence that principle #1 can't be true. Special relativity fixes this apparent paradox.
All of special relativity derives for these two principles, plus assumptions of exact conservation of momentum and energy in all cases.
*At the end of Einstein's original 1905 paper [http://www.fourmilab.ch/etexts/einstein/E_mc2/www/ "Does the Inertia of a Body Depend its Energy Content?"], he speculates on the possibility that the equation <math>E = m c^2</math>, which would normally be very hard to verify, could be verified with the extremely high energies of the newly discovered phenomenon of radioactivity.<ref>This equation is not related to [[quantum mechanics]].</ref> In the 1910s, with the invention of the mass spectrometer, it became possible to measure masses of nuclei accurately. This led to the clearing up of the mystery of atomic masses not being exact integers,and strongly suggested the existence of a "mass defect" (or "packing fraction") consistent with the mass-energy equivalence. In the 1930s, experiments with known nuclear reactions showed a very accurate correlation between the masses of the nuclei involved and the energy released. See [[Quantitative Analysis of Alpha Decay]].
*Another prediction of special relativity was time dilation in rapidly moving objects. This effect was most famously verified in the anomalously slow decay of relativistic cosmic muons.<ref>Some have suggested that other explanations are possible for this effect. We are trying to track this down.</ref> Time dilation has since been verified many times, and is routinely taken into account in all high-energy nuclear physics experiments, as in Hadron collision experiments.<ref>Experiments specifically designed to check dilation are rarely conducted any more.</ref>
[[Image:Cassini-science-289.jpg|right|thumb|The Shapiro effect: A spacecraft signal dipping into a gravity well around the [[Sun]] is delayed slightly.]]
As the 20th century progressed, tests of general relativity were proposed.
*One important "classical" test of general relativity was the advance of the perihelion of the orbit of Mercury. There are many complex effects contributing to this, including gravitational perturbations from other planets and the effect of the oblateness of the Sun. These are hard to calculate accurately, but, by 1900 it was known quite accurately that there was an "anomalous" precession, that is, a precession beyond all other known effects, of 43 arc seconds per century. This is a very tiny effect, but astronomical measurements were sufficiently accurate by that time to show it clearly.
:This created quite a problem—physicists by then were accustomed to having their theories check out very accurately. One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2. He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>. By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year. Whatever value is chosen for <math>\delta\,</math>, it gives the same precession, per revolution, for all orbiting bodies, but gravitational effects from other planets diminish that effect the further the planet is from the sun.
:This created quite a problem—physicists by then were accustomed to having their theories check out very accurately. One proposal that was made, by Simon Newcomb and Asaph Hall, was that the exponent of the radius in the gravitational formula wasn't exactly 2. He showed that, by choosing an exponent of <math>2+\delta</math>, the precession, as a fraction of a full orbit per planet's year, is <math>\delta/2</math>. By setting <math>\delta</math> to .000000157, that is, an exponent of 2.000000157, Newcomb was able to get a precession of .000000078 revolutions per Mercury year, or 43 arcseconds per Earth year. Whatever value is chosen for <math>\delta\,</math>, it gives the same precession, per revolution, for all orbiting bodies, but gravitational effects from other planets diminish that effect the further the planet is from the sun.
:A good approximation for the precession under general relativity is <math>\frac{3GM}{c^2 a(1-e^2)}</math> revolutions per planet's "year", where a is the semi-major axis and e is the eccentricity. A simpler but less accurate one is <math>3{}v^2/c^2</math>, where <math>v</math> is the planet's average orbital speed.
:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.
:While Newcomb's theory, and general relativity, don't lead to closed-form solutions, both theories can be solved numerically to as much precision as one desires.
:The following table shows some approximate parameters for the planets. Note that Mercury has the smallest orbit, and the fastest speed. Precession of planets other than Mercury is extremely hard to measure, but measurements of the actual anomalous precessions are in good agreement.<ref>http://www.mathpages.com/rr/s6-02/6-02.htm</ref>
:The following table shows some approximate parameters for the planets. Note that Mercury has the smallest orbit, the fastest speed, and the highest gravitational pull. Precession of planets other than Mercury is extremely hard to measure, but measurements of the actual anomalous precessions are in good agreement .<ref>http://www.mathpages.com/rr/s6-02/6-02.htm</ref>
{| class="wikitable"
{| class="wikitable"
|}
|}
:Considering only the ''anomalous'' precession, that is, the precession that remains after all known other factors (other planets and asteroids, solar oblateness) have been accounted for, and using very accurate calculations rather than the approximations given above, general relativity predicts 42.98 ±0.04 arcseconds per century. Some observed values, as of 2008, are:
::: 43.11 ± 0.21 (Shapiro et al., 1976)
::: 42.92 ± 0.20 (Anderson et al., 1987)
::: 42.94 ± 0.20 (Anderson et al., 1991)
::: 43.13 ± 0.14 (Anderson et al., 1992)
::: [Source: [http://arxiv.org/PS_cache/astro-ph/pdf/9804/9804258v1.pdf Pijpers 2008]]
:These error bars, and that of the general relativity prediction, all overlap.
(The second is just a restatement of Galilean relativity, that is, the "common sense" that had been accepted for centuries.)
*One was the ''Shapiro effect'', involving time delay in radio signals passing through the gravity well of the Sun or a planet. Various spacecraft have confirmed this.
*Another is the ''Shapiro effect'', involving time delay in radio signals passing through the gravity well of the Sun or a planet. Various spacecraft have confirmed this.
*Another is ''gravitational time dilation''. This is an effect separate from the time dilation of special relativity. It was tested by the Pound-Rebka experiment in 1959.
*Another is ''gravitational time dilation''. This is an effect separate from the time dilation of special relativity. It was tested by the Pound-Rebka experiment in 1959.