SPECIAL THEORY OF RELATIVITY

Prof. J.K. Goswamy

U.I.E.T, Panjab University

Chandigarh

Motion: Basic Terminology

Basics of Motion: Terminology

• Motion is defined as change of position of a body with respect to certain reference.
• The knowledge of position of a body as a function of time defines the trajectory or path of motion. This functional dependence embodies all the information characterizing the motion of body.
• The system with respect to which the motion of a body is discussed is called a frame of reference. Its choice is dictated by convenience of problem.

Particle

• A particle is a system that, for all practical considerations, can be localized to a point.
• During its lifetime, it is specified by constant values of parameters as
• Mass (determining its response to applied force).
• Charge (determining its interaction with electrical forces arising from other charges).
• Spin (intrinsic angular momentum).
• Magnetic dipole moment ……..

Rigid Body

• A body is said to be rigid if the distance of separation between any two of its constituent particles remains constant, irrespective of conditions of rest or motion.

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Clock

• A clock is an entity, which repeats itself regularly, like pendulum or alternating electromagnetic field.
• The time, according to Newtonian mechanics, flows uniformly in an identical manner for all the bodies.

Event

• An event is specified by its location as well as time of its occurrence. Hence an event is known completely if we have information (x, y, z, t) about it.
• Obviously the transformations that relate an event as observed by observers in two reference frames involve time as well as space coordinates.

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Frame of Reference

• Absolute space is an imaginary framework in which the bodies move, which, without any relation to anything external, is always similar and immovable.
• As experiments reveal only relative motion, the absolute space has no physical significance. At best it can locate one body with respect to another.
• A reference frame can be considered as three-dimensional coordinate system fixed in space along with array of clocks synchronized with a master clock.

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Inertial Frames

• Two frames of reference are said to be inertial with respect to one another when they are either at rest or in uniform relative motion.
• A frame of reference is inertial in which a body moves with constant velocity only if there is no net force acting on it.
• The equation of motion of a body takes the simplest form in an inertial frame as it is free from additive terms of fictitious forces arising due to acceleration of frame of reference itself.

Accelerated Frames

• These are the frames, which are in non-uniform motion with respect to each other.
• The common examples of such frames are those under rotation (such as earth) or in acceleration (such a rocket under uplift, upwardly or downwardly accelerating lift).
• Due to acceleration of these frames, fictitious forces such as centrifugal and Coriolis forces arise, which influence the motion of bodies.

Search for Universal Frame of Reference

The Need for Ether

• The wave nature of light suggested that there existed a propagation medium called the ether.
• Ether is assumed to have low density such that the planets move through it without loss of energy. Further it was characterized by nearly zero mass, high elasticity and absolute transparency.
• Ether was proposed as an absolute reference system in which the speed of light was constant and from which other measurements could be made.
• The Michelson-Morley experiment was an attempt to probe the existence of ether.

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The Michelson-Morley Experiment�

Albert Michelson (1852–1931) was the first U.S. citizen to receive the Nobel Prize for Physics (1907). He built an extremely precise interferometer to measure the minute phase difference between two light waves traveling in mutually orthogonal directions.

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The Michelson Interferometer: Working

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• A parallel light from a monochromatic source was allowed to fall on a half silvered mirror, which splits the beam into two components of equal intensity, one of which strikes mirror M₁ after transmission while the other portion strikes mirror M₂ after reflection.
• The beams reflected from mirrors M₁ and M₂ again fall on half silvered mirror and proceed towards viewing screen.
• The plane glass plate, called compensating plate, is placed in the path of beam striking mirror M₁ to make optical path traversed by two beams equal.
• If ether is all pervading, then ether wind must blow in direction opposite to the orbital motion of the earth (v=30km/s).

Earth is moving with respect to the ether (or the ether is moving with respect to the earth), so there should be some directional/season dependent change in the speed of light as observed from the reference frame of the earth.

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The path difference corresponding to time difference Δt is

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The axes of mirrors A and B were not kept exactly perpendicular so as to get straight fringes in the interference pattern instead of circular ones.

If the path difference d corresponds to shifting of n fringes, then

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Results

• In actual experiment of Michelson and Morley

D=10m and λ=500nm, v = 3 × 10⁴ m/s

were attained. Hence fringe shift expected was:

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• The shift in fringes could be observed only when the apparatus rotated by 90^o. Hence the shift will be twice this value i.e. n=0.4.
• This shift was measurable with the Michelson interferometer. However, surprisingly, no shift was observed.
• The same experiment was repeated in different seasons of the year and at different locations but results remained unaltered.
• Even after several repetitions and refinements with assistance from Edward Morley, the null result got repeated.

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Michelson’s Conclusion

• The ether has no measurable properties, which is equivalent to contradicting the existence of ether. Thus the search for universal frame of reference turned out to be meaningless.
• The light has same speed in free space irrespective of its direction of propagation and motion of the source or observer.

Ether does not seem to exist!

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Possible Explanations of Null Results

Two explanations were proposed for null results:

• Ether Drag hypothesis suggested that the earth somehow dragged the ether along as it rotates on its axis and revolves about the sun. This was contradicted by stellar aberration wherein telescopes had to be tilted to observe starlight due to the earth’s motion. If ether was dragged along, this tilting would not exist.
• Length Contraction hypothesis was proposed independently by both H. A. Lorentz and G. F. FitzGerald which suggested that the length in the direction of the motion was contracted by a factor of thus making the path lengths equal to account for the zero phase shift. This was an adhoc assumption that could not be experimentally tested.

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Relativistic Kinematics

Newtonian Principle of Relativity

If Newton’s laws are valid in one reference frame, then they are also valid in another reference frame moving at a uniform velocity relative to the first system. This is referred to as the Newtonian principle of relativity or Galilean invariance.

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• Same result. Ball rises and ends up in the thrower’s hand. Ball in the air for the same length of time.
• observations appear to be different for ground observer (parabolic trajectory, speed as a function of time) and observer on the truck. However, Newton’s laws remain valid for both of them.

Inertial Frames K and K’

• K is at rest and K’ is moving with velocity along x direction.
• Axes of two frames are parallel.
• K and K’ are said to be inertial coordinate systems (or frames).

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The Galilean Transformation

For a point P

• In system K: P = (x, y, z, t)
• In system K’: P = (x’, y’, z’, t’)

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x

K

P

K’

x’-axis

x-axis

The Galilean Transformations

• K’ is moving with constant relative velocity in the x-direction with respect to K

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• Time (t) for all observers is a fundamental invariant, i.e., the same for all inertial observers

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Inverse Relations for Galilean Transformations

Step 1. Replace with .

Step 2. Replace “primed” quantities with

“unprimed” and “unprimed” with “primed.”

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The speed of light in vacuum has the same value, c=3x10⁸m/s, in all inertial reference frames, regardless of the velocity of the observer or source emitting the light.

All the laws of physics have the same form in all inertial reference frames.

The Solution to Galilean Transformations

Oh my goodness…how can that be right???

Newtonian mechanics worked in all inertial reference frames under Galilean transformations, but does the same hold true for Maxwell’s equations of electromagnetism?

Postulates of Special Theory of Relativity

• The special theory of relativity is concerned with measurements made in inertial frames of reference.
• It involves the analysis of how measurements depend upon the observer as well as upon what is being observed.
• The special theory of relativity ushers in relationships between space and time, mass and energy. These paved way to understanding of vast variety of phenomena observed in the microscopic world.
• The special theory of relativity was postulated with a belief that the Maxwell’s equations must be valid in all inertial frames.

• The theory of special relativity was formulated on the basis of two postulates:
• Principle of Equivalence: The laws of physics can be expressed in equations having the same form in all inertial frames of reference. This endorses the absence of universal frame of reference.
• Constancy of Speed of Light: The speed of light in free space has same value for all observers, regardless of their state of motion. This implies that speed of light in free space is ultimate.

Re-evaluation of Time

• In Newtonian physics it was assumed that t = t’. Thus with synchronized clocks, events in frames of reference K and K’ can be considered simultaneous.
• Einstein realized that each system must have its own observers with their own clocks and meter sticks. Thus events considered simultaneous in frame of reference K may not remain simultaneous when observed in K’.

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The Problem of Simultaneity�

Observer is at rest and equidistant from events A and B:

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A B

−1 m +1 m

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He sees both flashbulbs go off simultaneously.

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The Problem of Simultaneity�

Another observer is moving to the right with speed v, observes events A and B in different order:

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−1 m 0 +1 m

A B

He sees event B, then A. Hence two events no longer remain simultaneous due to motion of observer.

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We thus observe…

• Two events that are simultaneous in one reference frame (K) are not necessarily simultaneous in another reference frame (K’) moving with respect to the first frame.
• This suggests that each coordinate system must have its own observers with clocks that are synchronized.

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Synchronization of Clocks

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Step 1: Place observers with clocks throughout a given system.

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Step 2: In that system bring all the clocks together at one location.

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Step 3: Compare the clock readings.

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If all of the clocks agree, then they are synchronized.

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Method to Synchronize Clocks

• One way is to have one clock at the origin set to t = 0 and advance each clock by a time (d/c) with d being the distance of the clock from the origin.
• Allow each of these clocks to begin time measurement when a light signal arrives from the origin.

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t = 0

t = d/c t = d/c

d d

The Lorentz Transformations

These are the linear transformations based on postulates of special theory of relativity. These transformations:

• preserve the constancy of the speed of light (c) between inertial observers.
• account for the problem of simultaneity between these observers.

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Lorentz Transformations cont’d…

• If we consider the same situation as taken in developing Galilean transformations, then reasonable relation between x’ and x can be written as:

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k does not depend upon x or t but certainly it depends on v.

• The choice of above relationship is linear so that an event in frame K corresponds to a single event in the frame K’. The inverse relation will be:

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• The factor k must remain same, as there is no difference between direct and inverse relations except the sign of v.
• Since the relative motion between frames K and K’ is along x direction so the transformations y’=y and z’=z must remain preserved.
• The time coordinates t and t’ are not equal, so we can write as:

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• The above three transformation equations (1.1-1.3) relating to x, x’ and t’ satisfy the first postulate of special relativity.

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• To evaluate k, we use second postulate of special relativity. At t=t’=0, the two frames of reference have their origins coincident.
• Suppose a flare is set at t=t’=0 and observers from each frame measure the speed with which light from these flare spreads.
• Both the observers will measure the same speed c. Hence

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• The transformation equations, on substitution of value of k from equation (1.4) are:

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• The inverse Lorentz transformations are:

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The important effects of these transformations are:

• Space and time are now inseparable.
• For all such transformations, the frame velocity is less than c.
• The measurement of time and position depends upon the frame of reference of the observer. As a result two events simultaneous in one frame may not be so in the other frame.
• These transformations reduce to Galilean transformations at low speeds (v<<c).

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Variation of γ with Velocity of Frame

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Lorentz Transformations : Kinematical Consequences

Lorentz transformations result in different observations of length and time as measured by observers at rest or in motion. These are called kinematical consequences which are:

• Length Contraction Lengths in K’ are contracted with respect to the same lengths stationary in K.
• Time Dilation Clocks in K’ run slow with respect to stationary clocks in K.

Time Dilation: Concept of Proper Time

To understand time dilation the idea of proper time must be understood:

• The term proper time (T₀) is the time difference between two events occurring at the same position in a system as measured by a clock placed at that position.

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• Same location

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Beginning and ending of the event occur at different positions.

Not a Proper Time

Time Dilation: Illustration cont’d…

Frank’s clock is at same position in system K when the sparkler is lit in (a) and when it goes out in (b). Hence Frank measures proper time in system K at rest wrt event.

Mary, in the moving system K’, is beside the sparkler at (a). Thus, Mary is at same position as Frank and measures the time in system K’ when the sparkler is lit in (a).

Melinda then moves into the position where and when the sparkler extinguishes at (b). Thus, Melinda, at the new position, measures the time in system K’ when the sparkler goes out in (b).

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According to Mary and Melinda…

• Mary and Melinda measure the two times for the sparkler to be lit and to go out in system K’ as times t’₁ and t’₂ so that by the Lorentz transformation:

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• The Frank records x₂ – x₁ = 0 in K with a proper time T₀ = t₂ – t₁

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Time Dilation: Rigorous Approach

• Time intervals are affected by relative motion.
• Clocks moving with respect to observer appear to tick slower than when they are at rest with respect to an observer.
• If an observer in frame K measures the time interval (t) of an event occurring in the frame K’ (moving w.r.t K at velocity v), then this time interval will be longer than interval (t₀) measured by an observer in frame K’. This effect is called time dilation.

• The observer in frame K measures

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• A moving clock measures a longer time interval between events occurring in a moving frame of reference than does a clock in a stationary frame.

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Why don’t we ordinarily notice time dilation?

Some examples of speeds in m/s

• 464 m/s Earth's rotation at the equator.
• 1000 m/s the speed of a typical rifle bullet
• 1400 m/s the speed of the Space Shuttle when the solid rocket boosters separate.
• 8000 m/s the speed of Space Shuttle just before it enters orbit.
• 11,082 m/s High speed record for manned vehicle
• 29,800 m/s Speed of the Earth in orbit around the Sun
• 299,792,458 m/s the speed of light (about 300,000 km/s)

Length Contraction

To understand length contraction we must have the idea of proper length.

• Let an observer in each system K and K’ have a meter stick at rest in their own system such that each measure the same length at rest.
• The length as measured at rest is called the proper length.

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What Frank and Mary see…

Each observer lays the stick down along his or her on respective x axis, putting the left end at x_ℓ (or x’_ℓ) and the right end at x_r (or x’_r).

• Thus, in system K, Frank measures his stick to be:

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• Similarly, in system K’, Mary measures her stick at rest to be:

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What Frank and Mary measure

• Frank in his rest frame measures the moving length in Mary’s frame moving with velocity v. Thus using the Lorentz transformations, Frank measures the length of the stick in K’ as:

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Both ends of stick must be measured simultaneously, i.e, t_r = t_ℓ

Here Mary’s proper length is

and Frank’s measured length is

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Frank’s measurement

So Frank measures the moving length as L given by

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but since both Mary and Frank in their respective frames measure L’₀ = L₀

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and L₀ > L, i.e. the moving stick shrinks.

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Lorentz Fitzgerald Length Contraction

• Let’s consider a rod lying along x’ axis of K’ frame moving with speed v along x’ direction.
• The coordinates of ends of the rod, measured simultaneously, are found to be x’₁ and x’₂ by an observer in frame K’. This is the frame in which rod is at rest with respect to the observer. The proper length of rod as observed by observer in K’ is:

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• If an observer in frame K measures the coordinates x₁ and x₂ of two ends of rod, then this rod is motion with respect to this observer. The end coordinates in two frames are related as:

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• The length of an object moving with respect to the observer appears to be shorter than when it is at rest. This is called Length contraction.

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Addition of Velocities

Taking differentials of the inverse Lorentz transformation, velocities may be calculated:

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Defining velocities as:

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The inverse velocity transformations for u_x is:

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The Direct Lorentz Velocity Transformations

In addition to the previous relations, the Lorentz velocity transformations for u’_x, u’_y , and u’_z can be obtained by switching primed and unprimed and changing v to –v:

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Experimental Verification: Meson Decay

• The mesons are the unstable particles, which are produced on bombardment of atmospheric molecules by cosmic rays.
• These are produced at considerable altitudes and travel with a speed of 0.998c. These particles have a lifetime of 2μs.
• These particles, according to simple calculations, must traverse a distance before their decay, which is given as:

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• However these particles are found to reach the surface of earth, even though are produced at much higher altitudes than 600m.

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• This paradox can be resolved by using the results of special theory of relativity.
• If we study of propagation of meson in its own frame, the meson lifetime will remain unaffected while the distance of ground from position of production will suffer shortening due to length contraction.

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• Hence the mesons will be able to reach the ground despite its short lifetime.

• Let’s observe the propagation of meson from the frame of reference of an observer on the ground. In this frame the altitude remains y₀ but the lifetime of the meson in this frame will be extended to a value

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which is almost 16 times than when it is at rest.

• The distance traveled with speed 0.998c will be again 9500m.

Meson paradox illustrates length contraction and time dilation effects.

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Experimental Verification: Atomic Clock Measurement

• Two airplanes took off (at different times) from Washington, D.C., where the U.S. Naval observatory is located.
• The airplanes traveled east and west around earth as it rotates.
• Atomic clocks on the airplanes were compared with similar clocks kept at the observatory to show that the moving clocks in the airplanes ran slower.

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Experimental Verification: Twin Paradox

• The Set-up: Twins Mary and Frank at age of 30 decide on two career paths: Mary decides to become an astronaut and to leave on a trip 8 light years (ly) from the earth at a great speed and to return; Frank decides to reside on the earth.
• The Problem: Upon Mary’s return, Frank reasons that her clocks measuring her age must ran slow. As such, she will return younger. However, Mary claims that it is Frank who is moving and consequently his clocks must run slow.
• The Paradox: Who is younger upon Mary’s return?

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The Resolution

• Frank’s clock is in an inertial system during the entire trip; however, Mary’s clock is not. As long as Mary is traveling at constant speed away from Frank, both of them can argue that the other twin is aging less rapidly.
• When Mary slows down to turn around, she leaves her original inertial system and eventually returns in a completely different inertial system.
• Mary’s claim is no longer valid, because she does not remain in the same inertial system. There is no doubt as to who is in the inertial system as Frank feels no acceleration during Mary’s entire trip, but Mary does.

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The Doppler Effect

• The Doppler effect in sound relates to increase in frequency as a source approaches a receiver and a decreased frequency as source recedes. Also, the same change in sound frequency is observed when the source is fixed and the receiver is moving.
• The change in frequency of the sound wave depends on whether the source or receiver is moving.
• Doppler effect in sound seems to violate the principle of relativity, until we realize that there is a special frame of reference for propagation of sound waves. Sound wave propagation depends on media such as air, water or a steel plate.

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Recall the Doppler Effect in Light

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The Relativistic Longitudinal Doppler Effect

Consider a source of light (for example, a star) and a receiver (an astronomer) approaching one another with a relative velocity v.

• Consider the receiver in system K and the light source in system K’ moving toward the receiver with velocity v.
• The source emits n waves during the time interval T.
• As speed of light is always c and the source is moving with velocity v, the total distance between the front and rear of the wave train transmitted during the time interval T is:

Length of wave train = cT − vT

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Because there are n waves, the wavelength and frequency are given by

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Source and Receiver Approaching

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Source and Receiver Receding

Relativistic Transverse Doppler Effect

Relativistic Dynamics

Relativity of Mass

• Let’s consider an elastic collision between two particles A and B and analyze the observations made by observers in frame S and another frame S’ (moving with velocity v along x direction w.r.t frame S).
• The properties of A and B are identical when determined in reference frames in which they are at rest.
• Before collision, particle A has been at rest in frame S while particle B is at rest in frame S’.

• At some instant A is thrown in +y direction with speed V_A while B is thrown in –y’ direction at speed V’_B such that V_A = V’_B.
• The behaviour of particle A in frame S is exactly the same as that of particle B in frame S’.
• When the two particles collide, A rebounds in –y direction at speed V_A while B rebounds in +y’ direction with speed V’_B.
• If the particles are thrown from positions, which are distance Y apart, then collision occurs at y= Y/2 in frame S while at y’ = Y/2 in frame S’.

• The round trip time as measured in frame S is

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• Similarly the round trip time measured in the frame S’ is

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• The conservation of momentum in this collision as observed in the frame S demands that

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• In frame S the speed of particle B is expressed as:

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• Inserting the velocity of B in the requirement of conservation of momentum in the frame S gives:

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• The difference in mass of two particles reveals that like space and time, the measurements of mass also depends upon relative motion between observer and whatever he is observing.
• Mass of a body in motion with respect to observer is larger than when it is at rest.

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• The relativistic increase in mass is about 0.5% when body moves at speed 0.1c while becomes nearly 100% when it is moving with speed ~ 0.9c.
• The atomic particles such as electrons, protons etc move with such high relativistic speeds.
• The effect of relativistic increase in mass was depicted in experiment, where e/m ratio was measured for different speeds of electron. It was observed that this ratio decreases with velocity of electron. This was manifested as resulting due to relativistic increase in mass.

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Relativistic Momentum

• Mass of a body in motion with respect to observer is larger than when it is at rest.
• The momentum of body will now be redefined as:

• Now the momentum of a body suffers increase not only due to increase in velocity but also consequent relativistic increase in mass.

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Relativistic Energy

• Due to the new idea of relativistic mass, the concepts of work and energy also get redefined.
• The kinetic energy (T) possessed by the body is equal to the work done to bring it to a state of motion (with velocity v) from the state of rest. Hence

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• According to Newton’s second law, the applied force can change momentum of body, which may be manifested as due to change in velocity or mass. Hence we write force as:

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• Inserting the expression of force in expression for work, we have:

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• Integrating by parts, we get:

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• The quantities mc² and m_oc² are called total energy and rest mass energy of the body respectively.
• If the body is at rest then it is the relation defining the mass-energy equivalence.
• Increase in kinetic energy results due to increase in the mass of the body as a consequence of motion.
• Other forms of energy can also manifest themselves as increase in mass.

At v<<c, we can write as:

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• This is the expected result in the realm v/c<<1, where results of Newtonian mechanics hold true.
• The relativistic and classical kinetic energies diverge considerably for v/c >0.6.

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Total Energy and Rest Energy

We can write total energy (E) in the Newtonian approximation as:

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The term m₀c² is called the rest energy and is denoted by E₀. Thus total energy of the particle is the sum of the kinetic energy and rest mass energy.

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Energy-Momentum Relation

The relativistic momentum of a body is expressed as:

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• It is useful relation between total energy and momentum of particle.
• The quantities (E² – p²c²) and m are invariant quantities.

Conservation of Mass-Energy

• The conservation of energy and mass no longer remain separate laws. Instead the quantity called total mass-energy is always conserved.
• According to law of conservation of mass-energy, the mass can be created or destroyed if and only if equivalent amount of energy gets vanished or is created.
• This is commonly observed in processes involving atoms, molecules and nuclei.

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• The transformation of small amount of mass into energy during a process results in increase in stability of the final system formed.
• Combination of atoms to form a molecule, the binding energy liberated results from conversion of mass into energy.
• However transformation of energy into mass usually takes place during fragmentation of the system caused by increase in its total energy.
• Disintegration of a nucleus requires extra energy which is supplied by an external agency and further gets converted into mass.

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Four or Minkowski Space

Concept of Space-time

• When describing events in relativity, it is convenient to represent events on a space-time diagram.
• In this diagram one spatial coordinate x, to specify position, is used and instead of time t, ct is used as the other coordinate so that both coordinates will have dimensions of length.
• Space-time diagrams were first used by H. Minkowski in 1908 and are often called Minkowski diagrams.
• Paths in Minkowski space-time are called worldlines.

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Particular Worldlines

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The Light Cone

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Space-time Interval

• All observers measure same speed of light and hence must observe spherical wave fronts.

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• If we consider two events, we can determine the quantity Δs² between them which remains invariant in any inertial frame. There are three possibilities for the invariant quantity Δs²
• Light-like (Δs² = 0: Δx² = c² Δt²) where two events can be connected only by a light signal.
• Space-Like (Δs² > 0: Δx² > c² Δt²) where no signal can connect the two events.
• Time-Like (Δs² < 0: Δx² < c² Δt²) where two events can be causally connected.

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Four Vectors

• The results of special theory of relativity can be expressed in an elegant manner if we use a four dimensional space called Four Space or Minkowski Space to specify the coordinates of events.
• The quantities represented in such a space are called four vectors, which are invariant under Lorentz transformations.
• The usage of four vectors gives an advantage that laws of physics become more generalized to handle the phenomena in the microscopic world.

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Common Four Vectors

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• If the four vector is specified as (A₁, A₂, A₃, A₄), then the generalized Lorentz transformations are expressed as:

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Electromagnetism and Relativity

• Einstein was convinced that magnetic fields appeared as electric fields when observed in another inertial frame. This conclusion is the key to electromagnetism and relativity.
• Maxwell’s equations can describe electromagnetism in any inertial frame formed the basis of Lorentz transformations.
• Maxwell’s assertion that electromagnetic waves propagate at the speed of light and Einstein’s postulate that the speed of light is invariant in all inertial frames are intimately connected.

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Force Between Current Carrying Conducting Wires

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Let’s consider two ideal conductors containing equal number of positive and negative charges at rest which are equally spaced.

As conductors are electrically neutral, there is no force between them.

Let currents I₁ and I₂ flow, in the same direction, in the conductors 1 and 2 respectively. The negative charges flow towards right in both the conductors.

• Consider a frame of reference in which electrons of conductor 1 are at rest and observe the charges in conductor 2.
• The negative charges in other conductor 2 will also appear to be at rest and equally spaced.
• The positive charges in conductor 2 will appear to be moving towards left with velocity 2v. The spacing between them will suffer length contraction.
• The conductor 2 appears to be positively charged and will attract negative charges in the conductor 1.

• Consider a frame of reference in which positive charges of conductor 1 are at rest and observe the charges in conductor 2.
• The positive charges in other conductor 2 will appear to be at rest and equally spaced.
• The negative charges in conductor 2 will appear to be moving towards right with velocity 2v. The spacing between these charges will suffer length contraction.
• The conductor 2 appears to be negatively charged and will attract positive charges in the conductor 1.

• Similarly arguments can be extended by considering rest frame of positive and negative charges in conductor 2.
• Hence all charges in each conductor will experience force of attraction from opposite charges in other conductor leading to net force of attraction between two current carrying conductors.
• However both the conductors being electrically neutral, the mutual attraction is attributed to magnetic interaction between the currents.
• Hence a conductor which is electrically neutral in one frame of reference may not be so in the other inertial frame.

Influence on Modern Physics

• Today, fundamental physics is formulated in the language of Relativistic Quantum Field Theory.
• This subject combines the postulates of Special Relativity with those of Quantum Mechanics.
• The Standard Model of particle physics, in principle, explains every interaction in nature not involving gravity.
• The Standard Model has been subjected to extremely sophisticated precision tests. Each of these, among other things, is a test of Special Relativity!

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True lessons from Einstein’s life and work:

• Think clearly.
• Follow your intuition.
• Do not be discouraged by others.
• Work hard.
• Learn all you can – but use only what you need.
• And above all, have a goal that you care about.

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