### special theory of relativity for dummies

Power Engineering | Telecommunication | Electronics | Maxwell equations | Differential Equation### Background

The theory of relativity usually encompasses two interlaced theories developed by Albert Einstein: special relativity and general relativity. Special relativity is applicable to elementary particles and their interactions, describing all their physical phenomena except gravity. General relativity explains the law of gravitation and its affinity to other forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, supplanting a 200-year-old theory of mechanics created primarily by Isaac Newton. It introduced concepts including spacetime as a unified absolute entity, relativity of simultaneity, kinematic and gravitational time dilation, and length contraction. In the field of physics, relativity advanced the science of elementary particles and their fundamental interactions, along with ushering in the nuclear age. With relativity dynamics, cosmology and astrophysics predicted extraordinary astronomical phenomena such as neutron stars, black holes, and gravitational waves

# Special theory of relativity

Consider a statment - " an elephant and an ant are arguing whether human being is the tallest animal in the planet earth." Both are right in their perspectives. The conclusion is that the real size of human being is the same but one regards it as smaller and other regards it as bigger. Quite similar thing happens in case of special relativity. The measures of space and time are different for different observers but the spacetime interval remains the same.

The thoery of relativity has resulted from a combination of the three elements which were called for in a reconstruction of physics: first
, delicate experiment; secondly, logical analysis and thirdly, epistomological(relating to theory of knowledge) consideration. These last played a greater part in the early
stages of the theory than in its finished form, and perhaps this is fortunate, since their scope and validity may be open to question, or at least would
be but for the successes to which they have led. One may say, broadly, that relativity, like other physics, has assumed that when different
observers are doing what is called "observing the same phenomenon" those respects in which their observations differ do not belong to the
phenomenon, but only those respects in which their observations agree. This is a principle which common sense teaches us in early age. A young
child observing a ship sailing away sees the ship continually becoming smaller in size. But later he realizes that the diminution of size
is only apparent . The real size of the ship does not decrease throughout its voyage. In so far as relativity has been inspired by epistemological
consideration, they have been of this common-sense kind, and the apparent paradoxes have resulted from the discovery of unexpected differences
between our observations and those of other hypothetical observers. Relativity like other physics assumes realistic hypotheses, that there are occurences
which different people can observe.

### Ether hypothesis

Albert A. Michelson (1881) tried to measure the relative motion of the Earth and aether (Aether-Wind),
as it was expected in Fresnel's theory, by using an interferometer. He could not determine any relative motion,
so he interpreted the result as a confirmation of the results of Stokes. However, Lorentz (1886) showed Michelson's calculations were
wrong and that he had overestimated the accuracy of the measurement. This, together with the large margin of error,
made the result of Michelson's experiment inconclusive.
In addition, Lorentz showed that Stokes' completely dragged aether led to contradictory consequences, and therefore he advocated
an aether theory similar to Fresnel's. To check Fresnel's theory again, Michelson and Edward W. Morley (1886) performed a repetition of the Fizeau experiment.
Fresnel's dragging coefficient was confirmed very exactly on that occasion, and Michelson was now of the opinion that Fresnel's stationary aether theory was correct.
To elucidate the situation, Michelson and Morley (1887) repeated Michelson's
1881-experiment, and they substantially increased the accuracy of the measurement.
However, this now famous Michelson–Morley experiment again yielded a negative result, i.e.,
no motion of the apparatus through the aether was detected (although the Earth's velocity is 60 km/s different in the northern winter than summer).
So the physicists were confronted with two seemingly contradictory experiments:
the 1886-experiment as an apparent confirmation of Fresnel's stationary aether,
and the 1887-experiment as an apparent confirmation of Stokes' completely dragged aether.
A possible solution to the problem was shown by Woldemar Voigt (1887), who investigated the Doppler effect for waves propagating in an
incompressible elastic medium and deduced transformation relations that left the wave equation in free space unchanged, and
explained the negative result of the Michelson–Morley experiment. The Voigt transformations include the Lorentz factor γ for the y- and
z-coordinates, and a new time variable (0) which later was called "local time". However, Voigt's work was completely ignored by his contemporaries.

FitzGerald (1889) offered another explanation of the negative result of the Michelson–Morley experiment. The theoretical setup for Michelson–Morley experiment was very
simple:

Contrary to Voigt, he speculated that the intermolecular forces are possibly of electrical origin so
that material bodies would contract in the line of motion (length contraction). This was in connection with
the work of Heaviside (1887), who determined that the electrostatic fields in motion were deformed (Heaviside Ellipsoid),
which leads to physically undetermined conditions at the speed of light. However, FitzGerald's idea remained widely
unknown and was not discussed before Oliver Lodge revealed a summary of the idea in 1892. Also Lorentz (1892b)
proposed length contraction independently from FitzGerald in order to explain the Michelson–Morley experiment. For plausibility reasons,
Lorentz referred to the analogy of the contraction of electrostatic fields. However, even Lorentz admitted that that was not a necessary
reason and length-contraction consequently remained an ad hoc hypothesis.

### Relativity of simultaneity

Before the advent of relativity there was no ambiguity about the simultaneity of events in space. Two events which were simultaneous to one observer were assumed to be always simultaneous to any other observer. But relativity changed this viewpoint radically. Two events which appear to be simultaneous to one observer may notbe simultaneous to another observer moving at a constant speed relative to the first observer. Two observer might not agree about the simultaneity of such two events. Both of the observers are in fact right from relativistic viewpoint. The fact is clearly established by Einstein's theory. The problem was the way we were used to see our world. Space and time are not separate entities, which is contrary to what Newton and his predecessors thought. We need four coordinates to represent an event. This compels us to combine space and time to form an absolute entity known as spacetime.

Special theory of relativity is a classical theory that overthrowed many usual conceptions. In 1905 Einstein came with a rather unconventional idea that space and time are not separate and absolute entities. They are part of a more absolute structure known as spacetime. The inclusion of time with space makes geometry not mere geometry but physics : in other words history is combined with geography. The physical world is a set of four coordinates (x,y,z,t). The concept of special relativity is rooted in Maxwell's equation.

Bending of light rays by the sun is explained by theory of relativity.

## Reference frames and relative motion

A reference frame in special theory of relativity is defined to be a system of measurement. Everything in physics is dependent on reference frame. A system of bodies can be regared as a reference frame where usual measurements of length and time are carried out. There is a origin which we keep fixed to carry such measurement. Three orthogona axes represent the three spatial distances carried out in any reference frame while time t is measured by a clock. In special relativity time t has no meaning unless the clock is attached to the reference frame in which its time is measured.A reference frame is in motion with respect to another reference frame. The primed system is in motion relative to the unprimed system with constant velocity v only along the x-axis, from the perspective of an observer stationary in the unprimed system.

Lorentz transformation holds for any two such reference frames which are in a relative velocity. Eientein further realized that all the velocities are relative. There is no such thing as absolute velocity. This idea of relative velocity inspired him much to develop his theory of special relativity. Newton's idea of absolute space and time must be abandoned.

# Lorentz transformation and Minkowski Metric

Lorentz transformation was first introduced by physicist Lorentz but he could not give any physical significance to such rules. It was first interpreted and made intelligible by Einstein. Lorentz transformation shows us how to infer the measures of the space and time appropriate to one reference frame from those appropriate to another reference frame which is moving at constant speed relative to other. That is, there happens a certain event (x, y, z,t) which coordinates are fixed by someone standing on a platform. Lorenzt transformation will enable us to infer how far way the event occurs and when it happens according to someone on a train moving from the platform with velocity v with the assumption that they were at the same place on the platform when the train passes it. Lorentz transformation is an element of a more generalized group O(1,3)[orthogonal group). So it is a matrix of 4X4 elements as the group O(1,3) is the group of all 4X4 matrices. Much can be discussed on group theory. In short, a group is a mathematical structure with a set of elements that through binary relation can produce another element of the same group. Like the group of integers, group of symmetries. Lorentz transformation is also known as Lorentz boost. There are four equations of Lorentz transformation.

A person throws an arrow which travels from A to B. The person who throws the arrow will see that the arrow starts from A and reaches B at latter time. Another person is moving relative to this person whose space coordinates are marked with x` axis. What will he observe?

He will observer that event A happens before event B. Another person who is moving in another direction may judge event B happens before A. His coordinate axes are represented by (x``, t``). This is the consequences of Lorentz transformation. The speed in all these cases should be close to that of light. Otherwise relativistic effect will be neglegible.

Second equation of Friedman comes from the other component field equations. General Lorentz transformation group is :

Where r is a position vector : **r** = ix + jy + kz in three dimension. beta (β) = v/c and the rotation matrix is R and lorentz boost is our original Lorentz transformation. If we multiply
rotation martix R with Lorentz boost L we get general Lorentz transformation. The rotation is possible in three independent direction of space and a boost is just a
translation ( moving the frame in some direction : the direction of velocity vector). When velocity v; is very less than c , Lorenz transformation turns into Galillian transformation. Maxwell equations are invariant under Lorenz transformation but not under Gallilian transformation.

Putting Lorentz boost in matrix form again :

Lorentz transformations can be thought of hyperbolic rotations of spacetime coordinates in xt, yt and zt cartesian plane of 4D Minkowski space ( x, y,z, t).
This can be thought as hyperbolic analog of circular rotation. In case of circular rotation we shift all points along a circle by some angle. In hyperbolic rotation we shift all points along a hyperbola.

Hyberbolic angle ζ = w which is the measure of velocity by the relation

For every value of the velocity v there will be two corresponding hyperbolas defined by the two equations x(`)^2 -c^2t(`)^2 = x^2 and c^2t^2 - x^2 = c^2(t`)^2 .
Then points on the hyperbola (xcoshw , xsinhw) will shift along one of those hyperbole as we change parameter w. For each rotation , all the points of ( x, ct) frame can be mapped to {x(`), ct(`)} frame by the relation :

The angle should be thought as imaginary as we can not physically picture time dimension. Ct axis can be thought as the imaginary axis.

These four equations relate a set of coordinates (x,y, z, t) to another set of coordinates (z`,y`,z`,t`), corresponding to two arbitrary reference frames. These are the equations from which we can come to the conclusion about time dilatation, length contraction, mass and energy equivalence relation. The equations for the time dilatation, length contraction and relativistic mass is :

There is no universal or absolute time. The measured time (T0) in a reference frame will be less than time (T) measured in other reference frame moving at constant speed relative to the former one. Two observers in two such different reference frames will conclude that their clocks are ticking slowly relative to each other. Moving clocks runs slowly or gives different time measure. If you move close to the speed of light , the effect of time dilation will be large. When the velocity is very less than that of light, the time measured in both frame can be taken as the same. As a result Newtonian physics takes over. The same is true for length measurement and relativistic mass. They all alter by a factor (Y) called Lorentz factor.

## Time dilation

An example can make it clear. Am observer on the rocket will see that the light pulse bounce from the ceiling of the rocket vertically. That is, the light signal will go from the source to the mirror vertically. There is no time dilation for this observer.But according to an observer outside the spaceship the light will not move vertically. By the time it reaches the mirror the spaceship will move some distance forward. And by the time it reaches the source again it will move some distance forward. Here the observer's clock will run slowly as the speed of light needs to remain the same.

A moving clock will be slowed down or click more slowly than a stationary clock. The clock's length will be shortented along the direction of its motion. And last of

all the clock's mass will increase which can not be visualized in the picture given above.

FitzGerald–Lorentz contraction, whereby a sphere, moving rapidly with speed v, is regarded as being Xattened in its direction of motion, by a factor γ = (1 - v^2/c^2)^(-1/2), Imagine that the sphere passes horizontally overhead at a speed approaching that of light. It is easy to imagine that this Xattening ought surely to be perceivable to an observer standing at rest on the ground. By the relativity principle, the effect should be identical with what the observer perceives if it is the observer who moves with speed v in the opposite direction and the sphere remains at rest. But to an observer at rest viewing a sphere at rest, the sphere is certainly perceived as something with a circular outline. This would seem to contradict the ‘perceived circles go to perceived circles’ assertion of the preceding paragraph. In fact, there is no contradiction, because this FitzGerald–Lorentz ‘flattening effect’ is, in fact, not directly observable. This follows by detailed consideration of the path lengths of the light that appears to be coming to an observer, with respect to whom the sphere is in motion. The light which appears to come from the rear of the sphere reaches the observer from a more distant point than that which appears to be coming from the sphere’s front

# Relativistic energy and momentum relation

The equation of relativistic energy is applicable to any particles moving with velocity less than the velocity of light or massless particle such as photon. In case of light the rest mass m(0) = 0; and light has only relativistic energy or momentum even though light particle (photon) does not have rest mass in usual sense. There is no reference frame moving at the speed of light. But we can bring photon under the scope of the relativistic energy equation to calculate it's relativistic energy or momentum. I guess we have no other reason than to regard the fact that rest mass of photon is zero is a true hypothese. Even though for individual photon there is no rest frame , a group of photons moving at different direction can have a center of mass , which can be treated as a reference frame. The derivation of energy and mass equivalence is simple :

The above equation needs some explanation. We start with proper mass which is denoted by m(nought). It is the mass that is measured in the object's own frame. As the object moves its relative mass increases. The ratio of this mass and rest mass is exactly the inverse of Lorentz factor. The relativistic mass is a function of velocity. If we expand the Lorentz factor by Binomial theorem , higher order terms in v can be neglected for small velocity. We finally arrive at expression which contains kinetic energy of the object, which is added with other terms. This means these terms must be some form of energy themselves. So we have now expression of total energy E = m c(squared) which is the sum of object's rest energy added the kinetic energy. If the velocity is zero the object's total energy is the rest energy. This suggests energy can exist in time dimension. Relativity dictates we always have an energy at rest. The above relation also suggests that when the velocity is very less than that of light , the change of mass is the same as change of kinetic energy (letting c=1 ). The kinetic energy is added as extra mass to the object. So classical notion of energy is only the approximation when the velocity is very less than that of light. Actual energy should be conserved according to measured mass. The energy related to time dimension also necessitates us to add one extra component to the usual momentum vector. 4-momentum is the momentum which has both time and spacial components. The momentum in time direction is the same kind as measured mass in relativity. There is a multiplicative constant c which can be taken as 1 for convenience. This results from the velocity of the object in time coordinate.

The usual momentum has three components (p1, p2, p3). An extra component corresponding to proper velecity(cdt/dt) is added to make generalized four-momentum vector. Magnitude
of the four-momentum is calculated using dot product of the momentum vector with itself. The time component suggest we have a momentum of mc, which also implies we are always moving
at the speed of light in time dimension.
**Proof of kinetic energy from relativity principle**
We can use the usual definition of work.

This also proves the mass and energy equivalence.
That means we need to have a generalized distance formula of Pythagorus , which must include distance in time. This distance should be called interval between two events in space-time.

The Minkowski metric is the small space-time interval between two neighboring events. It can be written in differential form .

It is an invariant quantity. the space and time measurement can be different for different observer but the total spacetime interval must be same for all. That is the assumption Einstein made to make his two postulates consistent with each other. The speed of light must remain the same no matter which reference frame it is measured from.

View this video for more explanation:

The figure above describes the four momentum of a particle. A light cone is depicted . The world line of the particle falls inside the cone.

# Relativistic Doppler shift

Doppler effect in sound is the relative change of pitch(relative frequency) according to observer who is moving away or towards sound source. If the source is moving towards the source, the sound will be harsh and high-pitched. If the observer moves away from the source the sound will be higher pitched. The similar effect can be found in case of light. The light's wave length can be stretched or squeezed according to the observer who is moving towards or away from the light source. The speed of light must remain the same. So we hope there will be a change in both its wavelength and frequency so that their product remains the same.

The parameter Z is defined as the the difference of measured wave lengths both at rest and moving.

As the clock slows down for moving observer, it can be shown that the relative(observed) frequency of light wave will decrease by Lorentz factor multiplied by an additional factor of (1 - v/c). When we substitute the value of the Lorentz factor in expression for frequency we get the observed wave length. So the amount of Doppler shift can be calculated easily from the first equation. Similar effect can be oserved in case of gravity. Light coming from strong gravitational field seems to be red shifted. The observed light will appear blue as seen by a distant observer.

So what can be the meaning of relativity in very simple terms ?

**
Take two watches or clock which are identical in all respect. Keep one watch on the ground and move the other with a velocity close to light. The moving watch when returned
to ground after a somewhat long journey will not give the same time as the other on the ground. Time will be slower for the moving watch. In other words, you leave the earth
to travel to outer space with a velocity very close to that of light. When you return to the earth after some years your friend will be older than you.
That is the summary of theory of special relativity.
**

### Notes and additional comments

A moving coil inside an magnetic field will develop a potential difference across it. If the coil is stationary and a magnet moves through it, a voltage will also
develop across it. What matters is the relative velocity between the magnet and the coil. This fact really worried Einstein and he thought if there is a relative
velocity between two observers, results of their observations will differ. They will not agree on certain aspects of the phenomena they observe.

Two postulates of theory of special relatvity are

1) The laws of the nature are the same in all reference frames. The natural world allows no “privileged” frames of reference. As long as an object is moving in
a straight line at a constant speed (that is, with no acceleration), the laws of physics are the same for everyone. It’s a bit like when you look out a train
window and see an adjacent train appear to move — but is it moving, or are you? It can be hard to tell. Einstein recognized that if the motion is perfectly uniform,
it's literally impossible to tell — and identified this as a central principle of physics.

2) light travels at an unvarying speed of 186,000 miles a second. No matter how fast an observer is moving or how fast a light-emitting object is moving,
a measurement of the speed of light always yields the same result.

There seems to be an apparent contradiction between these two postulates.
Somebody is standing on a train station and
at the same time a train crosses the station with a velocity 200000 m/s.
Now if a light signal goes from the station in the direction of the moving train,
the person in the train should see the light moving at 2800000000 m/s.
And the person on the station should see the same light going at 300000000 m/s. How can the velocity of light be the same
according to second postulate? Maxwell 's equations predicted that light speed is the same for all observers no
matter how they are moving. Einstein raised this fact to the second postulate. Thus the two postulates become
inconsistent with each other. In order to avoid this contradiction Einstein claimed space and time must be changing for different
observers. This way two postulates can be consistent with each other.
Space and time measurement are purely subjective to each observer. Space and time is really entangled. They are not separate entities.
is not run by a grand master clock. There is no single cosmic time.

That is to say, if a body is moving at a velocity v with respect to a first body and another third body is moving at velocity v with respect to the first body , then the velocity of the third body with respect to the second body will be found according to the formula given above.

The most remarkable feature of theory of relativity , from philosophical standpoint, is already apparent in special theory of relativity. That is the feature,
already spoken about, is the merging of space and time. But theory of special relativity is only approximation , which does not hold true in the neighborhood of
matter. But it is a necessary step toward understanding General theory of relativity. Special theory of relativity does not necessarily abandon all the common sense
notions
of space and time , which General theory compels us to abandon.

Time-like interval is the interval between two events in which material particle can be present. When two events happen to a person,
those two events still can have time -like interval whic a the time measure by a clock in his hand. These two events , still have physical significance.
Thus psychological time is unaffected by relativity. assuming that everything that is concerned with psychology happens, from physical point of
view, in the body of the person whose mental events are being considered. This analysis needs further explanation which
will be done in proper time.

## Relativistic mechanics

With the advent of special relativity entirely new kind of mechanics had been formulated. It is called relativistic mechanics. Whenever the velocity of an object approaches that of light relativistic effects are taken into account. The definition of force is now :Now we can formulate the Lagrangian of relativistic mechanics as sum of kinetic and potential energy:

### Why velocity cannot be infinity?

The mass becomes infinite at the velocity of light.The force is proportional to the mass. So we need an infinite amount of push to move such a body to reach the velocity of light. Thus it is impossible to achieve the speed of light.

## Special relativity in the presence of gravity

The equations of special relativity are altered in the presence of gravity. So in the presence of a gravitational potential φ the energy mass equivalence relation becomes :Here g(00) is the first component or time-time component of metric tensor g(uv). It is a misconcept to think that the velocity of particle can not be greater than that of light. When we say that the velocity of light in context of relativity we refer to the usual velocity of light in vacuum. But the velocity of light can be less than that in some medium. It is due to refractive index of the medium. In Cherenkov radiation this phenomena of faster than light can be observed. The light forms a cone when a particle travels in the medium faster than it. It happens when a charged particle passes through a dielectric medium with speed greater than the phase velocity of light.

But velocity of light in vacuum plays the part of infinity in relativity. No material particle can reach this speed limit.

### Light Cone

Light one is a set of events traced out by light rays. Here is an example of how light cone is constructed. The interval between anytwo events on the light cone is always zero. This is the limiting condition between time-like and space-like interval. The parts of all light rays comming out of point p (point representing present moment) form the future light cone whereas the parts of all the light rays that implode at point P after coming from backward form past light cone. Material particle always falls inside the light cone.

The general view of physicists is that time itself started at a specific point about 13.8 billion years ago with the Big Bang, when the entire universe suddenly spreaded out from an infinitely hot, infinitely dense singularity, a point where the laws of physics as we understand them simply go haywire. This can be regarded the “birth” of the universe, and the beginning of time as we know it. Before the Big Bang, there just was no space or time, and you cannot go further back in time than the Big Bang, in much the same way as you cannot go any further north than the North Pole of the earth. As a theoretical physicist Stephen Hawking notes in his 1988 book A Brief History of Time, even if time did not begin with the Big Bang, and there was another time frame before it, no information is available to us from that previous time-frame, and any events that occurred then would have no effect on our present time-frame. Any putative events from before the Big Bang can therefore be considered effectively meaningless (or at least the province of philosophical speculation, not physics).

## Invaraints and their representation

In Euclidean geometry distance between two points is an invariant quantity. If you tanslate the system in any direction the spatial relationship between points do no change. The transformation preserve distance and angles.The distance , after any change is the system , will always be the same. Einstein's reasoning was similar. The space-time interval should be the same in all reference frames.

### Introduction to general theory of relativity

What does it mean that g(00) is not unity? Events are labelled by four coordinates (e.g. ct; r; µ; Á). It doesn't necessarily mean that t is the time measured by a clock at a speci¯c location. We know this already, since if the clock is moving, there will be Doppler-type effects. The proper time is the time elapsed on a clock comoving with the object in question, and depends on its location, and its motion. We can get the proper time ¿ by using ds2 = c2d¿2 = g¹ºdx¹dxº: -- (36)So for a stationary clock (whose spatial coordinates are fixed, so dxi = 0; i = 1; 2; 3),

ds2 = g(00)dt^2 ---(37)

so that dt` = g(00)dt. In the weak-field case, where g(00) = (1 + φ/2) ---(38)

and we see that t coincides with t` only if φ = 0. So t is the time elapsed on a stationary clock at infinity, if we adopt the common convention that t` = 0 at innity. The time elapsed on a stationary clock in a potential t` is not t - it runs at a different rate. We also see that clocks run slow in a potential well (' < 0).

### Are there parallel universes?

Astrophysical data suggests space-time might be "flat," rather than curved,
and thus that it goes on forever. If so, then the region we can see (which we think of as "the universe") is just one patch in an
infinitely large "quilted multiverse." At the same time, the laws of quantum mechanics dictate that there are only a finite number of
possible particle configurations within each cosmic patch (10^10^122 distinct possibilities). So, with an infinite number of cosmic
patches, the particle arrangements within them are forced to repeat — infinitely many times over. This means there are infinitely
many parallel universes: cosmic patches exactly the same as ours (containing someone exactly like you), as well as patches that
differ by just one particle's position, patches that differ by two particles' positions, and so on down to patches that are
totally different from ours.

Is there something wrong with that logic, or is its bizarre outcome true? And if it is true, how might we ever
detect the presence of parallel universes? Check out this excellent perspective from 2015 that looks into what
"infinite universes" would mean.

### Why is there more matter than antimatter?

The question of why there is so much more matter than its oppositely-charged and oppositely-spinning twin, antimatter, is actually a question of why anything exists at all. One assumes the universe would treat matter and antimatter symmetrically, and thus that, at the moment of the Big Bang, equal amounts of matter and antimatter should have been produced. But if that had happened, there would have been a total annihilation of both: Protons would have canceled with antiprotons, electrons with anti-electrons (positrons), neutrons with antineutrons, and so on, leaving behind a monotonous sea of photons in a matterless expanse. For some reason, there was excess matter that didn't get annihilated, and here we are. For this, there is no accepted explanation. The most detailed test to date of the differences between matter and antimatter, announced in August 2015, confirm they are mirror images of each other, providing exactly zero new paths toward understanding the mystery of why matter is far more common.## Summary of special relativity

All the equations are put into one package:## Parallel transport

Parallel transport is the displace of a vector such that the covariant derivative vanishes. It is hard to explain without some mathematics of general theory of relativity but the main concept is as follows:## Hyperbolic geometry in Minkowski Geometry

In Euclidean geometry, the locus of points that are a Wxed distance a from a Wxed point O is a sphere. In E4, of course, this is a 3-sphere S3. What happens in M? There are now two situations to consider, depending upon whether we take a to be a (say positive) real number or (in effect) purely imaginary (where I am adopting my preferred + - - - signature; otherwise the roles would be reversed);Now our 'sphere' consists of two pieces, one of which is 'bowl-shaped', H +, lying within the future light cone, and the other, H +, 'hill-shaped', lying within the past light cone. We shall concentrate on H - (the space H - being similar) What is the intrinsic metric on H +? It certainly inherits a metric, induced on it from its embedding inM. (The lengths of a curve in H +, for example, is deWned simply by considering it as a curve inM.) In fact, for this case, the dl^2 (with signature +---) is the better measure, since the directions along H + are spacelike. We can make a good guess as to H þ’s metric, because it is essentially just a 'sphere' of some sort, but with a 'sign flip'. What can that be? Recall Johann Lambert's considerations, in 1786, on the possibility of constructing a geometry in which Euclid’s 5th postulate would be violated. He considered that a /sphere' of imaginary radius would provide such a geometry, provided that such a thing actually makes consistent sense. In fact, our construction of H +, as just given, provides just such a space -a model of hyperbolic geometry—but now it is 3-dimensional. To get Lambert’s non-Euclidean plane (the hyperbolic plane), all we need to do is dispense with one of the spatial dimensions in what has been described above. In each case the 'hyperbolic straight lines' (geodesics) are simply intersections of H + with 2-planes through O

In Minkowski geometry rules are somewhat different. In Euclidean geometry sum of lengths of two sides of a triangle is always greater than the length of the third. But in Minkowski or relativity geometry it is the opposite. That is to say, the sum of lengths of two sides is always less than the third. If ABC is any Euclidean triangle

AB + BC => CA

with equality holding only in the degenerate case when A, B, and C are all collinear. In Lorentzian geometry, we only get a consistent triangle inequality when the sides are all timelike, and now we must be careful to order things appropriately so that AB, BC, and AC are all directed into the future. Our inequality is now reversed: AB + BC <= AC,

## Symmetry groups of Minkowski space

The group of symmetries of E4 (i.e. its group of Euclidean motions) is 10-dimensional, since (i) the symmetry group for which the origin is fixed is the 6-dimensional rotation group O(4) (because n(n - 1)/2 = 6 when n = 4; and (ii) there is a 4-dimensional symmetry group of translations of the origin. When we complexify E4 to CE4, we get a 10-complex- dimensional group (clearly, because if we write out any of the real Euclidean motions of E4 as an algebraic formula in terms of the coordinates, all we have to do is allow all the quantities appearing in the formula (coordinates and coefficients) to become complex rather than real, and we get a corresponding complex motion of CE4. Since the first preserves the metric, so will the second. Moreover, all continous motions of CE4 to itself which preserve the complexified metric Cg are of this nature.

The symmetry group of a geometric object is the group of all the transformations under which the object studied remains invariant and a group endowed with law of composition.

## Rapidity

Let us go back to Hyperbolic geometry in Minkowski space M. Minkowski first developed a kind of geometry where the laws of special relativity apply. This is known as Minkowski geometry. This geometry is not the same as Euclidean geometry as we have already shown. There is a close connection between this geometry and conformal geometry of Beltrami. Actually the latter can be developed using the former.The construction for hyperbolic geometry as the 'pseudosphere' H + can be directly related to Beltrami's conformal and projective representations that were described (in the 2-dimensional case) . In Figure below, I have illustrated the way that both of these can be obtained directly from H þ, explicitly depicting the 2-dimensional case of pseudospheres in Minkowski 3-space M3 (with coordinates t, x, y). Taking H + to have equation t^2 - x^2 - y^2 = 1, we obtain Beltrami’s 'Klein' (i.e. projective) representation by projecting it from the origin (0, 0, 0) to the plane t = 1, and we obtain Beltrami's 'Poincare´' (i.e. conformal) representation by projecting from the 'south pole' ( - 1, 0, 0) to the 'equatorial plane' t = 0 (i.e. 'stereographic projection';

In Minkowski 3-space M3, the hyperbolic 2-geometry of H + (given by t2 - x2 - y2 = 1) directly relates to Beltrami's conformal and projective representations. Beltrami’s projective ('Klein') model is obtained by projecting H+ from the origin (0,0,0) to the interior of the unit circle in the plane t = 1. Beltrami's conformal (Poincare´) model is obtained by projecting H+ from (-1,0,0) to the interior of the unit circle in t = 0. The analogous construction works also for hyperbolic 3-geometry in M.

## Derivation of Lorentz transformation

First thing is to notice that like Galilian transformation Lorentz transformation is linear. The linearity of transformation is due to the fact that the relative velocity V is a constant vector. Consider two reference frames O and O` which are in relative motion. An event p is measured from O and O`. The wavefront from P will spread from P and the velocity of light is constant. So for the point Pr = ct and r` = ct` where r and r` are the distance of P from the origins of O and O` respectively. The equation of a sphere in frame O is given

Similarly for the wavefront in frame O` is given by

The reference frame is moving along x axis so

Again the position x` must vary linearly with t and x

x` = γx + σt

For the origin of O` x` and x is given by

The value of γ is to be determined . γ is a constant and must reduce to 1 for v<< c (velocity of light) . The above two equations gives the relationship between t and t` also

No replacing x`, y`, z` and t` in the equation of sphere of O` frame

and therefore

Which impies

Comparing the coefficients of t^2 in this equation with the those of t^2 in the equation of spherical wavefront in O frame c^2γ^2 - v^2γ^2 = c^2 [ ^ is the exponential ] which upon rearranging we get

Thus we get the final form of Lorentz transformations as

## Triangular law of vector addition

The addition of two vectors gives the resultant of those vectors. The resultant vector is found by the trianglular law of vector addition. It has the following mathematical properties.You can provide your comment and response on special relativity below: