## Theory of relativity

The theory of relativity was formulated by Albert Einstein in 1905. He showed that the primary quantities in mechanics, such as space, time and mass are relative or variant. According to classical Newtonian mechanics, space, time and mass are absolute quantities and they are independent of relative motion of the body and observer or two frames of reference. Some Definitions about the theory of relativity Event : Any happening or occurrence which is taking place in particular space at particular time is called an event. For example:

1) A collision between two bodies

2) Ticking of a clock gives a series of events

3) An event be described in four coordinated e.g. (x, y, z, t), three coordinates are of space and fourth is of time.

### Observer-

a) A person or his measuring instruments or equipment utilized to observe or measure the event is called an observer.

b) An observer draws his conclusion about the event on the basis of his observation.

c) An observer draws is supposed to possess all kind of accurate instruments, apparatus or equipments, necessary to measure an event.

### Frame of reference

It is a system which asset of coordinates is attached to describe an event. Frames of reference are of two types:

1) Internal frame of reference

a) It is that frame of reference in which Newton’s low of inertia and all other fundamental laws of mechanics hold good.

b) For example, train moving with a constant velocity will act as an inertial frame of reference.

c) It is in respect of an inertial frame of reference i.e. all inertial frame of reference are equivalent.

2) Non-internal frame of reference

a) It is that frame of reference in which Newton’s low of motion and other fundamental laws of mechanics do not hold good.

b) For example, train moving with a variable velocity will act as a non-inertial frame of reference.

c) The fundamental laws of physics have different meanings in different non-inertial frame of reference.

d) Since earth is having orbital motion and spinning motion due to which its velocity is not uniform, Hence earth is a non-inertial frame of reference. Note. If observations are to be taken for a smaller interval of time on events, then during that time, the velocity of earth will be constant, and then earth is taken as inertial frame of reference

### Galilean transformations

1) These are the set of equations which describe how an event in one inertial frame of reference appears to an observer in another inertial frame when the two frames of reference are in relative motion with a relative speed very small as compared to the velocity of light.

2)

- Let an inertial frame of reference S’ be moving with relative velocity u w.r.t. inertial frame of reference S along the positive direction of X-axis and an event occurs at point P. If (x, y, z, t) are the space-time coordinates as observed by observer in system S and (x’, y’, z’, t’) are the space time coordinates as observed by observer in system S’, then according to Galilean transformations.

*x’ = x – vt * ………………..( equ. A)

y’ = y

z’ = z

And t’ = t

or x = x’ – vt’ ………………..( equ. B)

y = y’

z = z’

And t = t’

If the point P is displaced slightly along the XX’ axis, then

*Δx’=Δx-vΔt and Δt’=Δt * ……(iii)

*∆x’/∆t’= ( ∆x/∆t) – u or u _{x}’ = u_{x}-v * ……(iv)

Hence ux’ is the velocity of P w.r.t. frame S’ and u

_{x}is the velocity of P w.r.t. frame S.

As motion is confined only along XX’ axis so u

_{y}’ = u

_{y}and u

_{z}’ = u

_{z}.

Thus according to Galilean transformations, velocity is a variant quality. That is why, momentum (=mv), K.E (=1/2 mv

^{2}) are also variant quantities according to Galilean transformations.

From (iv), a

_{x}’ = a

_{x}

a

_{y}’ = a

_{y}and a

_{z}’ = a

_{z}.

It means, the acceleration is invariant (constant) under Galilean transformation.

Further as, force = mass × acceleration, hence force is also invariant in Galilean transformation.

It means Newton’s second low of motion remains unchanged under Galilean transformation.

It can be show at work ( = force × distance ); torque ( = forc × perpendicular distance ), potential energy ( = mgh) are all invariant under Galilean transformation.

### Principle of Newtonian relativity

It states that, the law of Physics has same meanings in all inertial frames of reference. It means, there is no preference for any particular inertial frame of reference.

### Michelson- Morley’s experiment

1) This experiment was designed and used by Michelson and Morley to detect the presence of hypothetical medium ether, to be used as an absolute frame of reference.

2) Theoretical fringe shift in Michelson : Morley’s experiment, when the apparatus is rotated through 90˚, should be.**Δn = 2lv ^{2}/λc^{2}**. Where l = distance of each mirror, M1 and M2 from half silvered glass plate P.v = velocity of apparatus through ether (i.e. velocity of earth through ether). C = velocity of light, λ = wavelength of light used.

3) Experimental result of Michelson-Morley experiment

a) No fringe shift could be detected through experiment was repeated day and night, in different seasons and at different heights.

b) Thus the motion of earth through ether could not be detected.

c) It means Michelson and Morley got negative result through their experiment.

4) Outcome of Michelson-Morley experiment Einstein in 1905 accounted the negative result of Michelson-Morley experiment by starting that the motion through either in a meaningless concept, only relative motion to a frame of reference has a physical significance. He then started that the velocity of light is invariant quantity and it does not change due to relative motion of the two frames of reference or of source and observer. Using this idea, Einstein developed a special theory of relativity.

### Postulates of Special theory of relativity

1) The law of physics has the same meaning in all inertial frames of reference, moving with constant velocity relative to one another.

2) The speed of light in a free space (vacuum0 is always a constant It is independent of relative motion of inertial system, the source and the observer.

### Consequences of special theory of Relativity

1) **Length contraction**

a) According to theory of relativity, length is no longer absolute.

b) It can be shown that L = L_{0} √(1-v2/c2) Where L0 = true length of the body , when there is no relative between body and observer. L = observed length of the body, when in motion with a velocity v w.r.t. observer.

c) If a rod of length L_{0} is moving with velocity of light c, it will appear to be a point to an observer at rest.

d) A circular body while moving with high speed will appear to be an elliptical body with semi minor axis in the direction of motion of the circle.

2) **Time dilation**

a) According to theory of relativity, time is relative i.e. time travel between given two events will be different in different frames of reference which are in relative motion.

b) It can be shown that, t = t_{0} /√(1-v2/c2). Where t_{0} actual time interval between two events are measured by an observer in frame S’. t = observed time interval between the same two events are measured by an observer in frame S.

c) As v is comparable to c, t >t_{0}. It means observed time interval is greater than the true time interval between two events. This shows that to an observer in frame S, the clock of frame S’ seems to be running slow, as compared to his own clock.

d) If v=c, then t = 8. It means a clock moving with a speed of light appears to be completely stopped to a stationary observer.

3) **Relative addition of velocity**

a) From theory of relativity, it can be shown that the observed velocity of a body from frame S (i.e. u’) when it is moving with velocity u in frame S’ which is in relative motion with velocity v w.r.t. frame S is u’ = (u-v)/(1-uv/c^{2}).

b) It can be shown that, U = (u’-v)/(1-u’v/c^{2}).

c) If u’ = c, then u = (c+v)/ (1+ cv/c^{2})=c

d) It means the velocity of light in vacuum is constant and is independent of the frame of reference.

4) **Relativity of mass**

a) According to relativity mass of a body is also not invariant.

b) It can be shown that m= m_{2}/√(1-v^{2}/c^{2}) Where m_{0} = rest mass of the body i.e. mass of the body when it is at rest relative to an observer. m = effective mass of the body when it is moving with a velocity v w.r.t. observer.

c) If v increases, then m> m_{0} i.e. mass of the body increases with its velocity.

d) If v = c, then m = ∞. It is due to this reason we say that the maximum velocity of a material body cannot be greater than the velocity of light.

e) If v<< c, then v/c <<1 and m≈m_{0}. It means for small speed, mass of a body remains practically constant.

5) **Mass of Energy relation**

a) It is one of the most important outcomes of special theory of relativity.

b) The mass-energy relation is the universal equivalence between mass and energy.

c) The mass-energy relation is given by E = mc^{2} Where m = effective mass of the body, E = total energy associated with the body, c = velocity of light.

d) From mass-energy relation, it is clear that if certain mass disappears, an equivalent amount of energy is produced.

e) The kinetic energy(E_{k}) of a moving body with velocity v is given by E_{k} = mc^{2} – m_{0}c^{2} = m_{0}v^{2}[ (1/√(1-v^{2}/c^{2})) -1].

f) The relativistic relation between total energy E and momentum p is E = √(m^{2}c^{4}+p^{2}c^{2})

g) Fact supporting mass-energy equivalence are

i) binding energy of nucleus

ii) energy produced in atom bomb or hydrogen bomb

iii) Pair production etc.