7-special theory of relativity

7/31/2019 7-Special Theory of Relativity

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Unit-7

Special theory ofSpecial theory ofRelativityRelativity--II


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Books to consult:

Special relativity

1. Classical Mechanics by Herbert Goldstein2. Classical Dynamics of Particles and Systems by Thornton and Marion

3. Nuclear Physics by Kaplan4. Modern Physics by J. B. Rajam


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1687 1905 Newtonian mechanics ruled the world of physics;

almost always a complete success

Albert Einstein 1905 particles with speeds approaching speed of light,require a completely new form of mechanics relativistic mechanics

Even at lower speeds, Newtonian mechanics is an approximation!

Relativity

All measurements are made relative to some chosen reference system.

Position of a particle, position vector havingcomponents ; relative to some chosen origin and a chosenset of axis

),,( zyx=r),,( zyx

An event at t= 5 s t is 5 s relative to origin t= 0

Theory of relativity the study of the consequences of relativity ofmeasurements

All measurements require a specification of a reference frame


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Einsteins relativity two theories

Special Relativity focuses on un-accelerated frames of reference

General Relativity includes accelerated frames of reference theory of gravitation

Applications of Relativity

General Relativity situations where the predictions differ

appreciably from those of Newtonian gravity intense gravity of Black Holes; of large scale

universeeffect of earths gravity on extremely accurate

time measurements for GPS

Special Relativity in nuclear and particle physics where particlespeeds are near speed of light but gravity isnegligible


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Frames of Reference

Frame of reference a set of coordinatesto measure thingslike position, velocities

In Newtonian mechanics:

Position of a point in space ),,( zyx

An Event like an explosion ),,,( tzyx

Inertial Frame of ReferenceA frame of reference in which Newtons law of inertia holdsA bodynot acted on by an outside force, stays at rest if initially at rest, orcontinues to move at a constant velocity if initially moving.

Non-Inertial Frame of Reference

An accelerating frame of referencee.g. a rotating frame of reference


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Principle of Relativity

The laws of physics take the same form in all frames of referencemoving with constant velocity with respect to one another.

Mathematical meaning: laws of physics expressed in terms ofequations; and the form these equations takein different reference frames moving with

constant velocity with respect to one anothercan be calculated using so-called Galileantransformationin the case ofNewtonianrelativity.

Transformed equations have exactly the same form in all frames ofreference physical laws are the same in all frames of reference.


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Galilean Transformation

Consider two frames SS and

Orientation x || ,x y || ,y z || z

is moving with constant velocity v along xS

Newtonian mechanics assumes a single universal time t= t'

The origins O and O'coincide at t= t'= 0

S fixed to the ground, S fixed to a moving train

Consider an event, like a light bulb B fixed inside the train.

B


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Coordinates ofB in frame S

in frame S

tzyx ,,,

tzyx ,,,

Form figure: OOOBBO =

At time t: VtOO =

therefore

ttzz

yy

txx

==

=

= v

(1)

These four equations are called Galilean transformation.

They give coordinates of any event measured in framein terms of corresponding coords. of the same eventmeasured in frame .

),,,( tzyx),,,( tzyx S

S

Mathematical expression of classical ideas about space and time


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*

x

y

z

x'

y'

z'

B

O

O'

vt

r'

r

Galilean transformation (1) relates thecoordinates measured in two frames instandard configuration

i.e. corresponding axis parallel and relativevelocity along x - axis

General configurationwhere relativevelocity v is in arbitrary direction (figure)

Then the general form of Galilean transformation is:

tt

trr

=

= v(2)

Differentiating first of transformation (2) wrt time:v= rr && (since relative velocity is constant)

Classical velocity addition formulaaccording to ideas of classicalphysics, relative velocities add (or subtract) according to normal rules of

vector algebra


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Another differentiation gives: aa =

acceleration same in both frames

Invariance of Newtons laws under Galilean transformation

Consider an example two inertial frames of reference;

moving relative to with constantvelocity v

two particles connected by a spring of length l

SS and

S S

X-coordinates in S

S

21,xx

21,xx

If is mass of particle at thenfrom Newtons Second Law theequation of motion of the particle is:

1m 1x

)( 2121

2

1 l= xxkdt

xdm (3)

where k is the spring constant

x'

y'y

O'O

v

S S'

m1 m2

llll


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Same pair of masses from the point of view of another frame of

reference moving with a velocityv

relative to , then usingtransformation (1):

S S

txx += v11 and txx += v22

so that v+

= td

xd

dt

dx 11

2

12

2

12

td

xd

dt

xd

=

)( 2121

2

1 l= xxk

tdxdm

substituting above results in equation (3):

and 2121 xxxx =

(5)

since according to Newtonian mechanics, mass of particle is the same inboth frames i.e.

11 mm =

(4)


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then )( 2121

2

1l

=

xxktd

xd

m(6)

from equations (3) and (6), the form of the equation of motion derivedfrom Newtons Second Law is the same in both frames of reference.

the mathematical form of the equations of motion obtained fromNewtons Second Law are the same in all inertial frames of reference.

Conservation of Momentum

If we combine the Second and Third Laws, leading to the law ofconservation of momentum which is:

In the absence of any external forces, the total momentum of a systemis constant.

Thus, in reference frame , the total momentum is:S

constant2211 ==+ Pxmxm && (7)


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Using transformation equations (4):

v)( 212211

2211

mmxmxm

xmxmP

++=

+=

&&

&&

txx += v11 v11111 mxmxm += &&

txx += v22 v22222 mxmxm += &&

Then total momentum in frame :S

or constant)( 21 =+= vmmPP (8)

Therefore, total momentum in both frames is constant.

The general conclusion :

Newtons Laws of motion are identical in all inertial frames of reference.


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Galilean Relativity and Speed of Light

Newtons laws invariant under Galilean transformation same is nottrue for laws of electromagnetism.

Maxwells equations light propagates through vacuum in anydirection with speed:

m/s1031 8

00

==

c

where and are permittivity and permeability of vacuum.0 0

If Maxwells equations hold in frame , then light must travel in anydirection with same speed c as measured in .

S

S

Consider a frame travelling with speed Valong x-axis of .S S


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Imagine a beam of light travelling in the same direction:

In frame speed of light is given by:S cv =

Then in frame using the classical velocity addition formula, the speedof light is given by:SVcv =

If beam of light travelling to left, in :

Vcv +=

S cv =

But in :STherefore, Maxwells equations do not hold in inertial frame .S


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Example

Consider wave equation for light assuming a simple form,

01

2

2

22

2

=

t

E

cx

E(i)

tt

txx

=

= v

in a frame of reference S, with waves moving with speed c.

The Galilean transformation:

Using partial differentiation:

1,0,,1 =

=

=

=

t

t

x

t

t

x

x

xv

Using chain rule:x

t

t

E

x

x

x

E

x

E

+

=

(ii)

Substituting from (ii):

x

E

x

E

=

or

2

2

2

2

x

E

x

E

=

(iii)


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We can also write:

t

t

t

E

t

x

x

E

t

E

+

=

Again substituting from (ii):t

E

x

E

t

E

+

=

v

(iv)

(v)

Differentiating (iv):tt

tE

ttx

tE

xt

E

+

=

2

2

Using (ii):

+

=

t

E

tt

E

xv

t

E

2

2

Using (v):

2

222

2

22

2

2

t

E

tx

E

tx

E

x

E

t

E

+

=

vvv

+

+

+

=

t

E

x

E

tt

E

x

E

xv

t

Evv

2

2


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Substituting from (ii) and (vi) in (i):

021

2

22

2

2

2

22

2

=

+

x

E

tx

E

t

E

cx

Evv

2

22

2

2

2

2

2

2x

E

tx

E

t

E

t

E

+

=

vv (vi)

This is the equation under Galilean transformation for a frame S'movingwith velocity v relative to S.


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Newtons laws hold in all inertial frames Maxwells laws hold only inone frame

What is special about that frame?

Maxwells equations predict that light moves with a certain speed cbut with respect to which frame is this speed measured?

The proposed special frame where Maxwells equations hold and thespeed of light is c frame of the ether.

It was assumed like sound waves, light also need a medium topropagate

medium called ether

Does etherreally exist?

7-special theory of relativity

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