- +1-530-264-8006
- info@tutorsglobe.com

18,76,764

Questions

Asked

21,311

Experts

9,67,568

Questions

Answered

Start Excelling in your courses, Ask an Expert and get answers for your homework and assignments!!

Submit Assignment2015 © Tutors Globe. All rights reserved.

## 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 L

_{0}= x_{1}- x_{2}. L_{0}is known as rest length, as body is at rest relative to observer in -frame.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 asx

_{1}= γ(x'_{1 }+ vt') and x_{2}= γ(x'_{2}+ vt')L is the length of the object as measured by O'

L = L

_{0}√(1 - β^{2}) = L_{0}√(1 - (v^{2}/c^{2}))If L is less than one, as Lorentz factor is √(1 - (v

^{2}/c^{2}) is less than one. As Lorentz factor i.e. √(1 - (v^{2}/c^{2}) 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,L

_{0}= L√1 - β^{2}= L√(1 - (v^{2}/c^{2})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 -β

^{2}.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

t

_{1 }= γ(t'_{1}+ β(x'/c)) and t_{2}= γ(t'_{2}+ β(x'/c)) t_{2}- t_{1}= γ(t'_{2}- t'_{1})Δt = Δt'/(√1 - β

^{2})t

_{2}- t_{1 }= (t'_{2}- t'_{1})/(√1 - (v^{2}/c^{2}))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.

Now compute components of velocity vector V of particle as V

_{x}= dx/dt, V_{y}= dy/dt, V_{z}= 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 - β

^{2})y' = y

z' = z and t' = γ(t - (βx/c)) = (t - (vx/c

^{2}))/(√1 - β^{2}) where β = v/c and γ = 1/(√1 - β^{2})Thus

V'

_{x}= (V_{x}- v)/(1 - (v/c^{2})V_{x}), V'_{y}= [dy/(dt - (vdx/c^{2})/(√1 - β^{2})] = V_{y}(√1 - β^{2})/(1 - v/c^{2}V_{x})V'z = V

_{z}(√1 - β^{2})/(1 - (v/c^{2})V_{x})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

V

_{x}= (V'_{x}+ v)/(1 + (v/c^{2})V'_{x})V

_{y}= V'_{y}(√1 - β^{2})/(1 + (v/c^{2})V'_{x})V

_{z}= V'_{z}(√1 - β^{2})/(1 + (v/c^{2})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 - v

_{0})/(v - v_{s})or f' = f[(1 - v_{0}/v)/(1 - v_{s}/v)]Where v is velocity of wave relative to medium, v

_{s}velocity of source relative to medium v_{0}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 equationf' = f(1 - v

_{0s}/vcosθ)/(1 - v_{s}/v)Where v

_{os}= v_{0 }- v_{s}is relative velocity between observer and source.If v

_{os}< 0 i.e. negative, then observer is approaching source and if v_{os}> 0, i.e., then observer is moving away from source.Tutorsglobe: A way to secure high grade in your curriculum (Online Tutoring)Expand your confidence, grow study skills and improve your grades.

Since 2009, Tutorsglobe has proactively helped millions of students to get better grades in school, college or university and score well in competitive tests with live, one-on-one online tutoring.

Using an advanced developed tutoring system providing little or no wait time, the students are connected on-demand with a tutor at www.tutorsglobe.com. Students work one-on-one, in real-time with a tutor, communicating and studying using a virtual whiteboard technology. Scientific and mathematical notation, symbols, geometric figures, graphing and freehand drawing can be rendered quickly and easily in the advanced whiteboard.

Free to know our price and packages for online physics tutoring. Chat with us or submit request at info@tutorsglobe.com