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  Kinematic Consequences of Lorentz Transformation, Physics tutorial

Length contraction:

Length of any object in the moving frame will appear foreshortened in direction of motion, or contracted. Amount of contraction can be computed from Lorentz transformation. Length is maximum in frame in which object is at rest.

The length of the object which is at rest with respect to the observer in the - frame is simply difference between spatial coordinates of its end-points.

Length L0 = x1 - x2. L0 is known as rest length, as body is at rest relative to observer in -frame.

1617_Length contraction.jpg

To the observer O' in S'- frame, which is in relative motion to - frame at the relativistic velocity v along common positive -direction, coordinates the ends of rod are estimated as x'1 and x'2. Now measurements in two frames are related through Lorentz transformation as

x1 = γ(x'1 + vt') and x2 = γ(x'2 + vt')

L is the length of the object as measured by O'

L = L0√(1 - β2) = L0√(1 - (v2/c2))

If L is less than one, as Lorentz factor is √(1 - (v2/c2) is less than one. As Lorentz factor i.e. √(1 - (v2/c2) decreases as v increases. Therefore, to observer in motion, object seems to be shortened in direction of motion. The rod seems shortened only if direction of motion is parallel to its length. This shortening of length is known as Lorentz - Fitzgerald contraction. This effect is reciprocal. In other words,

L0 = L√1 - β2 = L√(1 - (v2/c2)

Therefore, objects seem to be longest when they are at rest relative to observer. When it is in motion, it seems to be contracted in direction of motion by the factor √1 -β2.

Though, as y = y' and z = z' are perpendicular to motion, apparent dimensions of object remain unchanged in the direction perpendicular to motion.

Time Dilation:

When it is said that something is dilated, it means that it is enlarged. With that in mind, let us consider problem of measurement of time interval of two events as estimated in two reference frames that are in relative motion at relativistic velocity.

Assume we have clock at fixed point x' in S'- frame that is in relative motion to S-frame at constant velocity v along common positive x-axis. This clock is utilized to evaluate interval between two events that happen at same position x' at different instants of time t'1 and t'2. Time interval between the events is t'2 - t'1 = Δt'

Relationship between times of occurrence of events in two frames is given in terms of Lorentz transformation as

t1 = γ(t'1 + β(x'/c)) and t2 = γ(t'2 + β(x'/c)) t2 - t1 = γ(t'2 - t'1)

Δt = Δt'/(√1 - β2)

t2 - t1 = (t'2 - t'1)/(√1 - (v2/c2))

This equation explains the fact that as v increases, γ and thus Δt increases. Time estimated by observer O in -frame is longer. This is called as time dilation. Now, observer O' in S'-frame is at rest relative to clock and occurrences of events at same position. Time of the occurrence of events estimated by him (Δt') is called as proper time.

Velocity Addition:

Assume particle is the photon of light emitted in S'- frame in x'- direction. Then, as estimated by O in S-frame, velocity of particle will be c + v that clearly contradicts postulate of special relativity that is speed of light in free space is independent of speed of its source.

1430_Velocity addition.jpg

Now compute components of velocity vector V of particle as Vx = dx/dt, Vy = dy/dt, Vz = dz/dt and V'x = dx'/dt', V'y = dy'/dt', V'z = dz'/dt' respectively.

To properly transform velocity of particle from one frame to another, utilize Lorentz coordinate transformation.

x' = γ(x - vt) = (x - vt)/(√1 - β2)

y' = y

z' = z and t' = γ(t - (βx/c)) = (t - (vx/c2))/(√1 - β2) where β = v/c and γ = 1/(√1 - β2)

Thus

V'x = (Vx - v)/(1 - (v/c2)Vx), V'y = [dy/(dt - (vdx/c2)/(√1 - β2)] = Vy(√1 - β2)/(1 - v/c2Vx)

V'z = Vz(√1 - β2)/(1 - (v/c2)Vx)

These Equations are relativistic velocity transformation equations. Corresponding inverse velocity transformation equation are achieved if simply replace v by -v in above set of equations, that is

Vx = (V'x + v)/(1 + (v/c2)V'x)

Vy = V'y(√1 - β2)/(1 + (v/c2)V'x)

Vz = V'z(√1 - β2)/(1 + (v/c2)V'x)

Twin Paradox:

The story is that one of the pair of twins leaves on the high speed space journey during which he travels at the large fraction of the speed of light whereas other remains on Earth. Due to time dilation, time is running more slowly in spacecraft as seen by earthbound twin and traveling twin will find that earthbound twin will be older upon return from journey.

The clear implication is that traveling twin would indeed be younger, but scenario is difficult by fact that traveling twin should be accelerated up to traveling speed, turned around, and decelerated again upon return to Earth. Accelerations are outside realm of special relativity and need general relativity.

In spite of experimental difficulties, experiment on the commercial airline confirms existence of the time difference between ground observers and reference frame moving with respect to them.

Relativistic Doppler Effect:

As the car or airplane approaches you, pitch (frequency) of its sound increases and as it moves away pitch of its sound decreases. So we can say that, for mechanical waves (waves which need elastic material media for propagation), when source of wave and observer are in relative motion with respect to material medium in which wave propagates, frequency of waves observed is different from frequency of source. The relationship between frequencies observed by observer and emitted by source is provided by Doppler equation as:

f ' = f(v - v0)/(v - vs)or f' = f[(1 - v0/v)/(1 - vs/v)]

Where v is velocity of wave relative to medium, vs velocity of source relative to medium v0 and velocity of observer relative to medium. The source and observer are supposed to move along same straight line. In more general case where source and observer move at theangle θ relative to each other, it could be shown that frequencies are related by equation

f' = f(1 - v0s/vcosθ)/(1 - vs/v)

Where vos = v0 - vs is relative velocity between observer and source.

If vos < 0 i.e. negative, then observer is approaching source and if vos > 0, i.e., then observer is moving away from source.

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