Theory of relativity

Unit -1

THEORY OF RELATIVITY

Topic followed are:-

1.Space, Time & Motion

2. Frame of Reference

3.Galileo’s principle of relativity

4.Michelson-Morley experiment     ok

5.Special theory of relativity

6.Transformation of space and time

7.Time dilation

8.Dopplers Effect   ok

9. Length Contraction     ok

10.Twin Paradox

11. Relativistic Mass

12.Variation of mass with velocity

13.Kinetic energy

14.Equivalence of mass and energy

15. Relation between energy and momentum.

The Theory of Relativity, proposed by the Jewish physicist Albert Einstein (1879-1955) in the early part of the 20th century. The theory of relativity, or simply relativity, encompasses two theories of Albert Einstein: special relativity and general relativity.^[1] However, the word "relativity" is sometimes used in reference to Galilean invariance.

1.Space, Time & Motion

To understand the relativity let us have some examples

Space: Let us imagine a river flowing and two persons (A and B) are standing on the banks of the river facing  towards each other. Now the person A will say that house is on the right side while the person B will say that house is on the left side. But actually of house is fixed. This indicates that position is relative.

Time : Suppose , if we are in India then we will say that  now its 3:00 pm but if we are in America then at the same instant we will say that time is 7:00 pm. Both time are different. So, we cannot answer the question without reffereng the place.Hence, time is relative.

Motion: Suppose a person A is moving with speed of 60 km/hr(A is observer) and car B is moving with the speed 20 km/hr in the same direction while the car C is moving with speed 30 km/hr in opposite direction.Now for the observer A, the speed of A is 60 km/hr, the speed of car B is(60-20=40 km/hr)

40 km/hr and the speed of car C is (60 + 30=90 km/hr)90km/hr.So, we can say that motion is relative.

       Thus, the state of rest or the state of motion of a body can only be defined relative to some other object. This  theory which deals with the relativity of motion and rest is called the theory of relativity.

• The resultant theory agrees with experiment better than classical mechanics, e.g. in the Michelson-Morley experiment that supports postulate 2, but also has many surprising consequences. Some of these are:
•  Relativity of simultaneity: Two events, simultaneous for one observer, may not be simultaneous for another observer if the observers are in relative motion.
• Time dilation: Moving clocks are measured to tick more slowly than an observer's "stationary" clock.
• Length contraction: Objects are measured to be shortened in the direction that they are moving with respect to the observer.
• Mass–energy equivalence: E = mc², energy and mass are equivalent and transmutable.
• Maximum speed is finite: No physical object or message or field line can travel faster than light.

The defining feature of special relativity is the replacement of the Galilean transformations of classical mechanics by the Lorentz transformations. (See Maxwell's equations of electromagnetism and introduction to special relativity).

1.  Frame of Reference

The frame relative to which the positionor the motionof a body is specified is called the frame of reference.

 A system of co-ordinate axes which defines the position of a particle in two or three dimensional space is called a frame of reference. The simplest frame of reference is the cartesion system of co-ordinates in which the position of particle is specified by its three co-ordinates x,y,z along the three perpendicular axes.

                  In Newtonian physics and special relativity, an inertial frame of reference (or Galilean reference frame) is a frame of reference in which Newton's first law of motion applies: an object moves at a constant velocity unless acted on by an external force. All inertial frames are in a state of constant, rectilinear motion with respect to one another; they are not accelerating (in the sense of proper acceleration that would be detected by an accelerometer). Measurements in one inertial frame can be converted to measurements in another by a simple transformation (the Galilean transformation in Newtonian physics and the Lorentz transformation in special relativity). In general relativity, an inertial reference frame is only an approximation that applies in a region that is small enough for the curvature of space to be negligible. ^[1][2]

3.Galileo’s principle of relativity

     Galilean invariance or Galilean relativity is a principle of relativity which states that the fundamental laws of physics are the same in all inertial frames.

Specifically, the term Galilean invariance today usually refers to this principle as applied to Newtonian mechanics, that is, Newton's laws hold in all inertial frames. In this context it is sometimes called Newtonian relativity.The equation that relates the co-ordinates of the two frames of refrences are called the transformation equation.
Among the axioms from Newton's theory are:

1. There exists an absolute space, in which Newton's laws are true. An inertial frame is a reference frame in relative uniform motion to absolute space.
2. All inertial frames share a universal time.

Galilean relativity can be shown as follows. Consider two inertial frames S and S' . A physical event in S will have position coordinates r = (x, y, z) and time t; similarly for S' . By the second axiom above, one can synchronize the clock in the two frames and assume t = t' . Suppose S' is in relative uniform motion to S with velocity v. Consider a point object whose position is given by r = r(t) in S. We see that

The velocity of the particle is given by the time derivative of the position:

Another differentiation gives the acceleration in the two frames:

It is this simple but crucial result that implies Galilean relativity. Assuming that mass is invariant in all inertial frames, the above equation shows Newton's laws of mechanics, if valid in one frame, must hold for all frames. But it is assumed to hold in absolute space, therefore Galilean relativity holds.

 Newton's theory versus special relativity

A comparison can be made between Newtonian relativity and special relativity.
Some of the assumptions and properties of Newton's theory are:

1. The existence of infinitely many inertial frames. Each frame is of infinite size (covers the entire universe). Any two frames are in relative uniform motion. (The relativistic nature of mechanics derived above shows that the absolute space assumption is not necessary.)
2. The inertial frames move in all possible relative uniform motion.
3. There is a universal, or absolute, time.
4. Two inertial frames are related by a Galilean transformation.
5. In all inertial frames, Newton's laws, and gravity, hold.

In comparison, the corresponding statements from special relativity are:

1. Same as the Newtonian assumption.
2. Rather than allowing all relative uniform motion, the relative velocity between two inertial frames is bounded above by the speed of light.
3. Instead of universal time, each inertial frame has its own time.
4. The Galilean transformations are replaced by Lorentz transformations.
5. In all inertial frames, all laws of physics are the same (this leads to the invariance of the speed of light).

4.Michelson-Morley experiment

The Michelson–Morley experiment was performed in 1887 by Albert Michelson and Edward Morley. Its results are generally considered to be the first strong evidence against the theory of a luminiferous aether.

Physics theories of the late 19th century postulated that, just as water waves must have a medium to move across (water), and audible sound waves require a medium to move through (such as air or water), so also light waves require a medium, the "luminiferous aether". Because light can travel through a vacuum, it was assumed that the vacuum must contain the medium of light. Because the speed of light is so great, designing an experiment to detect the presence and properties of this aether took considerable ingenuity.
Earth travels a tremendous distance in its orbit around the sun, at a speed of around 30 km/s or over 108,000 km per hour. The sun itself is travelling about the Galactic Center at even greater speeds, and there are other motions at higher levels of the structure of the universe. Since the Earth is in motion, it was expected that the flow of aether across the Earth should produce a detectable "aether wind". Although it would be possible, in theory, for the Earth's motion to match that of the aether at one moment in time, it was not possible for the Earth to remain at rest with respect to the aether at all times, because of the variation in both the direction and the speed of the motion.
At any given point on the Earth's surface, the magnitude and direction of the wind would vary with time of day and season. By analysing the return speed of light in different directions at various different times, it was thought to be possible to measure the motion of the Earth relative to the aether.
The expected difference in the measured speed of light was quite small, given that the velocity of the earth in its orbit around the sun was about one hundredth of one percent of the speed of light.
Experiment
To detect relative motion between the body(earth) and a hypothetical medium(ether).
A schematic diagram of the Michelson–Morley experiment is shown in fig ……  A collimated beam of light from the source S is incident upon a half silvered glass plate P placed at 45⁰ to the beam. It split into two beam 1 (reflected) and 2 (refracted).These beams travel right angles to each other.They incident normally on mirrors M₁ and M₂ and reflected back to  P. the two beams returned to P are directed towards a telescope T and interference takes place. The interference fringes are observed in the telescope.
    Let the mirror M₁ and M₂ be at the same distance L from the glass time to return to P. Let us assume the earth with a velocity v, therefore the apparatus is moving in the space with velocity v.
 Suppose c is the velocity of the light through the ether. The beam 2 moving towards M₂ has a velocity (c-v) relative to the apparatus on the outgoing journey and (c+v) on the return journey. If T₂ is the total time by this beam to go from P to M₂ and back, then
                                            T₂=L/(c-v) + L/(c+v) = 2Lc/c²-v² = 2L/c/(1-v²/c²)     ------------(1)
The beam 1 that travels towards mirror M₁ will strike the mirror at A’ but only after P has moved to new position P’.
Let a be shifted to a’ in time T’.
From ∆PA’ R we have (PA)²=(A’R)² + (PR)²
                                         d²=L^{2 +}  v²d²/c²            or  d= 1/√1-v²/c²
Therefore, the time for the round trip PA’P is T₁ = 2d/c =  2L/c /√1- v²/c².     --------------(2)
The time difference between two paths due to the motin of the instrument relative to the ether is given by:
   ∆T = T₂ - T₁ = 2L/c   [ 1/1 – v²/c²  - 1/√1- v²/c²]

Since v is very small compared to c , therefore
     1/ 1- v²/c²  = (1- v²/c²)^-1 = 1+ v²/c²
      1/√1- v²/c² = (1- v²/c²)^-1/2 = 1 + v²/2c²
   ∆T=2L/c   [1 + v²/c² -1- v²/c²] = Lv²/c³                                                                            ---------------------(3)
If the apparatus is turned through 90^o , the role path 1 and 2 is interchanged. In this case, the time T₂’ and T₁’ required to trvel the paths PA’P’, then
                              T₂’ = 2L/c /√1- v²/c²  and  T₁’ = 2L/c /1- v²/c².
              ∆T’= T₂’ - T₁’ =2L/c [1/√1- v²/c² - 1/1 – v²/c²  ]
            ∆T’ = - L v²/c³                                                                             --------------------------(4)
            ∆T_(total) =∆T - ∆T’ =     L v²/c³- (-L v²/c³)
              ∆T_(total) = 2L v²/c³                                                                  --------------------------(5)
The path difference introduced between the components of beam 1 and 2 will be, therefore
                             δ = c(∆T_(total)) c x 2L v²/c³  =  2L v²/c²               ---(6)
The number of fringes passing through a reference mark will be
                            N = Path difference/ λ = =  2L v²/c²λ          -----(7)
Where  λ is the wavelength of light used.
In Michelson mMreley’s experiment
         L=11, λ=6000 A^o and v= 3x 10⁴ m/s.
N   =     2 x 11 x (3 x 10⁴  m/sec)²      =   0.37
            (3 x 10⁸  m/sec)² x 6 x 10^-7
There must be a shift of fringes by 0.37 times the fringe width when the apparatus turned by 90^o.
Explanation of the negative result :

(i)                 The result showed that the speed of light is same for all observers which is not true for waves that need a material medium in which to occur ( such as sound waves and water waves), this is called principle of constancy of speed of light.

(ii)               Lorentz and Fitgerald gave an explanation of the  negative result of a moving body is altered due to its motion through stationary ether. They showed that the length of path in the direction of ether flow will be shortened by L √1- v²/ c²  instead of L. Replacing L by L √1- v²/ c²  in eq(i), we  have

                                                  T2 =2 L √1- v²/ c²   =      2L                           -------------------(8)
                                                c (1- v²/ c^{2 )}         c√1- v²/ c²
                                     Hence,   T₂=T₁

(iii)             The moving earth drags the ether with it. Hence there is no relative motion between the two so that no shift is observed.

    Einstein’s theory of the Michelson-Morley Experiment

 Einstein proposed that the basic concepts that preceded Michelson-Morley experiment were wrong .Einstein argued that it is meaningless to talk of the motion through ether. He also proposed that the speed of light is an absolute constant. It is same for all observers irrespective to their state of motion. He said that only motion relative to a frame of reference has physical significance. While discussing and motion the frame of reference must be specified, which may be road, the earth’s surface, the sun, the center of our galaxy etc.

   If we were isolated in the universe, then it could not be determined whether we are in motion or not. This is the reason why it is impossible to perform any experiment for detecting earth’s motion through ether.

                                    

              

Special relativity is based on two postulates which are contradictory in classical mechanics:

1. The laws of physics are the same for all observers in uniform motion relative to one another (principle of relativity),
2. The speed of light in a vacuum is the same for all observers, regardless of their relative motion or of the motion of the source of the light.

Michelson had a solution to the problem of how to construct a device sufficiently accurate to detect aether  flow.

Transformation of space and time( Lorentz transformation)

  On the basis of Einstein postulate of special theory of relativity, Lorentz gave transformation equation which  replaced Galilean transformation and also explain the Michelson Morley experiment result.

            Let us consider two frames of reference S and S’.Let frame S’ move with velocity v along positive X- direction with respect to frames S. The axis of two frames coincide at t=0. Let one observer be at O and the other be at O’ and both observers observe an event at the same time.

If the event occur at an instant t at point(x,y,z) as observed by O and at instant t’ at point(x’,y’,z’) observed by O’ when the event  is observe by the observer is at O, then

Distance of P from  O is

            :               r² = x² + y² + z²

Hence     x² + y² + z²= c²t²,

       [ velocity = distance/ Time, c = r/t  =>