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starting to merge
The prediction that light is bent by gravity is predicted both by Newtonian physics and relativity, but relativity predicts a larger deflection.
The prediction that light is bent by gravity is predicted both by Newtonian physics and relativity, but relativity predicts a larger deflection.
==Special relativity==
Special relativity is the limiting case of General relativity where all gravitational fields are weak. It is based on two postulates; one, that the laws of physics are identical to all [[inertial observers]], and two, that the [[speed of light]] ''in vacuo'' is a universal constant.
===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.
The length of an event <math>t</math>, as seen by a (relative) stationary observer observing an event is given by:
<math> t = \frac{t_{0}} {\sqrt{1 - \frac{v^{2}}{c^{2}}}}</math>
Where
:<math>t_0</math> is the "proper time" or the length of the event in the observed frame of reference.
:<math>v</math> is the relative velocity between the reference frames.
:<math>c</math> is the speed of light (3x10<sup>8</sup> ms<sup>-1</sup>).
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<sub>1</sub>) and bottom (D<sub>2</sub>) 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 proving the theory.
The time taken for a muon to travel from D<sub>1</sub> to D<sub>2</sub> as measured by a stationary observer is:
<math> t = \frac{s}{v} = \frac{1900}{0.994\times(3\times10^{8})} = 6.37\mu\textrm{s} </math>
The fraction of muons arriving at D<sub>2</sub> in comparison to D<sub>1</sub> was 0.732. (Given by <math> \frac{N}{N_0} = 0.732 </math>)
Since (from the equation for exponential decay) <math> \frac{N}{N_{0}} = e^{-\lambda t_{0}} </math> then
<math> t_{0} = \frac {ln(0.732)}{ln (0.2)} \times 1.53\times 10^{-6} = 0.689\mu\textrm{s}</math>
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:
<math> t = \frac{t_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{0.689 \times{10^{-6}}}{\sqrt{1 - \frac{0.994^{2}}{1^{2}}}} = 6.3\times 10^{-6}\textrm{s}</math>
This is in good agreement with the value calculated above, thereby providing evidence to support time dilation.
===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, <math>l</math>, of an object as seen by a (relative) stationary observer is given by:
<math> l = l_{0} \sqrt{1- \frac{v^{2}}{c^{2}}}</math>
Where
:<math>l_0</math> is the "proper length" or the length of the object in the observed frame of reference.
:<math>v</math> is the relative velocity between the reference frames.
:<math>c</math> is the speed of light (3x10<sup>8</sup> ms<sup>-1</sup>).
===Mass increase===
We also see that as a body moves with increasing velocity its [[mass]] also increases.
The mass, <math>m</math>, of an object as detected by a (relative) stationary observer is given by:
<math> m = \frac{m_{0}} {\sqrt{1 - \frac{v^{2}}{c^{2}}}}</math>
Where
:<math>m_0</math> is the "rest mass" or the mass of the object when it is at rest.
:<math>v</math> is the relative velocity of the object.
:<math>c</math> is the speed of light (3x10<sup>8</sup> ms<sup>-1</sup>).
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.
==Evidence for Relativity==
==Evidence for Relativity==