relativity5

B. Tech Physics

Chapter 1

Theory of Relativity

Lecture 1.5

Question:

Consider a spacecraft moving at speed v₁ = 0.5c relative to earth fires a projectile at speed v₂ = 0.5c in the same direction.

What is the velocity of the projectile relative to an observer who is at rest on earth ?

Is it v₁ + v₂ = c or something else ?

Addition of the velocities:

Consider two frame of references S and S’ such

that frame S’ is moving with uniform velocity v w.r.t. S. Consider a body moving through distance dx in frame s in time dt. Let in frame S’ this distance and corresponding time be dx’ and dt’ respectively.

Velocity of body in frame S is

---(1)

In frame S’ the velocity is

According to Lorentz transformations,

---(3)

Differentiating above Eq. we have

Put the values from Eq. (4) in Eq. (2),

---(5)

Similarly,

---(6)

Eq. (5) and Eq. (6) gives relativistic laws of velocity addition. In classical mechanics it was just u = u’+v.

Question: Consider a spacecraft moving at speed v₁ = 0.5c relative to earth fires a projectile at speed v₂ = 0.5c in the same direction. What is the velocity of the projectile relative to an observer who is at rest on earth ?

Ans:

To find the answer of above problem we shall use the Eq. (6) where

v₁ = u’ = 0.5c, v₂ = 0.5c

And then the velocity of the projectile relative to earth = 0.8c.

Note that velocity of the projectile is not equal to = v₁+v₂= c

Exercise:

(1) Discuss the case when we have non-relativistic case.

(2) Discuss the case when we have a photon in one frame

moving with velocity c. What will be the velocity of photon in other frame?

Exercise: Suppose the components of the velocity of some object are

in S frame and

in frame S’

Prove the following relations

Variation of mass with velocity:

The mass of a body moving with very high velocity v (near to velocity of light) relative to an observer is greater as compared to when it is at rest.

Derivation:

Let us consider two frames S and S’ such that the Frame S’ is moving with uniform velocity v w.r.t S. To study the variation

of mass with velocity we consider the collisions of two masses m₁

and m₂ in frame S’ and then observe this collision from frame S.

Let in frame S’, the masses m₁ and m₂ are moving with velocity

Now suppose after collision the two masses combine in one single and comes to rest in frame S’.

The velocity of two objects in two frames are related by velocity Addition relation,

---(1)

The velocity of combined mass (m₁+m₂) will be v as observed from S.

Now applying the law of conservation of momentum

---(2)

From last Eq. we have

---(4)

Now from eq. (1) we write

Last Eq. Further lead to following Eq.

---(5)

Similarly, we have

Using (5) and (6) in Eq. (4) we get

---(7)

We can further write above Eq. as

= ---(8)

Since the right and left sides of the above eq. are independent of each other, so we can equate them to a constant

= = m₀ ---(9)

From Eq. (9) we write,

and

If we take m₁ = m and u₁ = v, then we write in general

---(9)

Above Eq. Give the relativistic relation of mass with velocity.

Relativistic momentum :

We know the relativistic mass is given as,

---(1)

The momentum of the particle is written as

---(2)

In fig m is actually m₀ (rest mass). Note that in the at small speed non-relativistic momentum and relativistic momentum matches whereas at high speed there are deviations in the relativistic

Relativistic force : 2nd law in relativistic form:

However when then we have γ much less then one and we get the classical formula.

Einstein mass energy relation: The relation E = mc2 is known as Einstein mass-energy relation. It says that the mass can be converted into energy and vice-versa.

Proof: The work done, W on an object by a constant force F because

of which the object moves through distance s in direction of force is given by,

W = Fs ---(1)

If no other force acts on the object and object starts from the rest then the whole work done converts into K.E. i.e. KE = Fs.

In general when force is not constant then the K.E. is,

In the non-relativistic case the Kinetic energy of an object of mass m and speed v is (mv 2/2).

To get the relativistic form of the K.E. We shall use the relativistic Form of 2nd law of motion i.e.

---(3) Integrating by part

---(4)

Above eq. Says that the kinetic energy of an object is equal to the Increase in its mass due to its relativistic motion multiplied

by the square of the speed of the light. Above eq. can be written as,

---(5)

If we call mc2 as the total energy of the object then we see that when

the object is at rest its KE = 0 and it still possess an energy equal to

Thus we total energy of the particle

---(6)

where, E₀ = m₀c2 is rest energy

If object is moving then the total energy is

E = mc2 = ---(7)

Above eq. (7) gives us the famous Einstein mass-energy relationship

Above Einstein mass-energy relation ship says that the mass can be created or destroyed but when this happens an equivalent amount Of energy simultaneously vanishes or comes into being and

Vice-versa.

Relativistic energy-momentum relationship:

Consider an object of relativistic mass m and moving with velocity v. Its momentum is

---(1)

The relativistic mass is given by

---(2)

Last eq. further can be written as

---(3)

Also we know Einstein mass-energy relation is

---(4) Using (3) in (4) we have

So we have Above Eq. Is the relativistic energy momentum relationship

---(6)

Concept of massless particles:

The momentum of the particle is defined by

And energy is

E =

Now if m₀ = 0 and v<c, then E = p = 0.

However, when m₀ = 0 and v = c, than E = 0/0 and p = 0/0, which mean the energy and momentum are indeterminate and can possess any values. Thus a mass-less particle with finite value of E and p can exist provided they move with the speed of light.

Example : Photons are mass-less particle which move with speed c

and have rest mass zero.

The total energy of such particles (using and m₀ = 0),

Energy, E = pc

Exercise:

 Write the expression for relativistic kinetic energy and show