# Theory of relativity/draft

Relativity refers to two closely-related theories in physics. Special relativity (SR) is a theory that describes the laws of motion for non-accelerating bodies traveling at a significant fraction of the speed of light. At speeds approaching zero, Special Relativity is identical to Newton's Laws of Motion. Special Relativity was developed by Hendrik Lorentz, Henri Poincaré, 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.[1]

## Galilean Relativity

In the 17th century, Galileo Galilei stated what became known as the special principle of relativity:

The laws of physics are identical in all inertial (non-accelerating) reference frames.

The principle is represented mathematically by the Galilean transformations, which relate positions and velocities in one inertial frame to those in another.

Isaac Newton later formulated his three laws of motion, which held that space and time are absolute.

## Special Relativity

Special Relativity, introduced by Albert Einstein in 1905, is based on two postulates:

1. The laws of physics are identical in all inertial reference frames.
2. The speed of light is identical for all observers, regardless of their velocities relative to each other.

The first postulate is simply the special principle of relativity, from which the theory gets its name. The second is contradictory to the expectations of Newtonian mechanics. When these postulates are used as a basis for predicting the behavior of matter and energy, the results are contrary to everyday human experience:

• Relativity of Simultaneity: Simultaneous events in one reference frame are not simultaneous in another.
• Relativity of Length: Objects in a moving frame are shortened along the direction of motion, a phenomenon known as length contraction.
• Relativity of Time: Clocks in a moving frame tick slower than clocks in a stationary frame, a phenomenon known as time dilation.
• Mass-Energy Equivalence: Mass and energy are different forms of the same entity, as demonstrated by the famous equation E=mc2.

The groundwork for Special Relativity was begun by Hendrik Lorentz and Henri Poincaré as part of an effort to discover transformations under which Maxwell's Equations are invariant. These equations, known as the Lorentz transformations, became the mathematical framework for Special Relativity after Einstein recognized their broader applicability.

At low speeds (relative to light-speed), the Lorentz transformations are equivalent to the Galilean transformations. The famous equation attributed to Einstein, E=mc2, 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 path of light passing through a region containing many massive bodies such as galaxies will be distorted (classical mechanics does not predict the same degree of distortion). 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 mathematical extension of Special Relativity. GR proposes that space-time is curved by massive bodies, so that near any massive body, the sum of the angles in a triangle is not exactly 180 degrees.

The GR field equations are

$G_{uv} = 8\pi\, T_{uv}$

where Guv is the Einstein curvature tensor, and Tuv is the stress-energy tensor, Guv and Tuv 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 [2] [3] [4]. 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.[5]

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).

## Time dilation

Light-cone diagram

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 t, as seen by a (relative) stationary observer observing an event is given by:

$t = \frac{t_{0}} {\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Where

t0 is the "proper time" or the length of the event in the observed frame of reference.
v is the relative velocity between the reference frames.
c is the speed of light (3x108 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 (D1) and bottom (D2) 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 D1 to D2 as measured by a stationary observer is:

$t = \frac{s}{v} = \frac{1900}{0.994\times(3\times10^{8})} = 6.37\mu\textrm{s}$

The fraction of muons arriving at D2 in comparison to D1 was 0.732. (Given by $\frac{N}{N_0} = 0.732$)

Since (from the equation for exponential decay) $\frac{N}{N_{0}} = e^{-\lambda t_{0}}$ then

$t_{0} = \frac {ln(0.732)}{ln (0.2)} \times 1.53\times 10^{-6} = 0.689\mu\textrm{s}$

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:

$t = \frac{t_{0}}{\sqrt{1 - \frac{v^{2}}{c^{2}}}} = \frac{0.689 \times{10^{-6}}}{\sqrt{1 - \frac{0.994^{2}}{1^{2}}}} = 6.3\times 10^{-6}\textrm{s}$

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, l, of an object as seen by a (relative) stationary observer is given by:

$l = l_{0} \sqrt{1- \frac{v^{2}}{c^{2}}}$

Where

l0 is the "proper length" or the length of the object in the observed frame of reference.
v is the relative velocity between the reference frames.
c is the speed of light (3x108 ms-1).

## Mass increase

We also see that as a body moves with increasing velocity its mass also increases.

The mass, m, of an object as detected by a (relative) stationary observer is given by:

$m = \frac{m_{0}} {\sqrt{1 - \frac{v^{2}}{c^{2}}}}$

Where

m0 is the "rest mass" or the mass of the object when it is at rest.
v is the relative velocity of the object.
c is the speed of light (3x108 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 F=γma where γ varies with velocity as mass m remains constant. Force F is a vector and thus can handle the directional aspect of the relativistic effects better than the concept of relativistic mass can.

## References

1. "[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
2. http://www.spaceimages.com/gravlen.html
3. http://csep10.phys.utk.edu/astr162/lect/galaxies/lensing.html
4. http://www.iam.ubc.ca/~newbury/lenses/glgallery.html
5. http://ia331314.us.archive.org/2/items/theeinsteintheor11335gut/11335-h/11335-h.htm