Download RELATIVITY

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LIMITATIONS OF NEWTON’S LAWS OF MOTION
EINSTEIN SPECIAL THEORY OF RELATIVITY
SOME IMPORTANT DEFINITIONS
FRAME OF REFERENCE
INERTIAL FRAME OF REFERENCE
NON INERTIAL FRAME OF REFERENCE
GALILEAN TRANSFORMATION
In classical physics,space time and mass are
regarded as absolute .This means that
i) length of an object is independent of condition
under which it is measured such as position and
motion of object or the observer.
ii) The time interval between two events has the
same value for all observers.
iii) The mass of a body is constant and is
independent of its motion relative to an observer.
 Newtonian mechanics fails to explain the motion
of particles which are moving with high speeds
close to velocity of light.
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All the limitations of Newtonian mechanics
led Einstein to modify the concept of space,
time and mass. He introduced the new
modified laws in order to get consistent
results in all situations.
According to Einstein, absolute space,time
and mass have no meaning .Space and time
are relative.
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PARTICLE:- A particle is a
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EVENT:-An event is defined
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OBSERVER:-An observer is
very small quantity of matter
having practically no linear
dimensions,but only a
position. Measure of the
quantity of the matter is mass
of the particle.
as a simple or complicated
occurrence taking place in
space at a given time .For
example collision between
two cars on the road,
lightning in the sky etc.
a person or an equipment
meant to observe or measure
events.
FRAME OF REFERENCE
A frame of reference is a
system of co-ordinate
axes which specify the
position of a particle or
an event in two or three
dimensional space.
The simplest and the
most commonly used
frame of reference is the
Cartesian system of coordinates with observer
at the origin. It is not
essential that the position
of observer should
coincide with that of
origin .However, it is
convenient to do so.
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Inertial Frame of reference, is that frame in
which Newton’s laws of motion hold good ,
i.e.
it moves with a constant velocity when there
is no net force acting on it.
Thus an inertial frame of reference is a nonaccelerating frame of reference.
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Let, S be an inertial
frame of reference
with co-ordinate
axes OX,OY and OZ
and the origin O.
S’ is another frame
of reference with
origin O’ and coordinate axes
OX’,OY’ and OZ’.
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Let the two frames have their origin coincident at a
certain time ,say ,t=0.
After time t ,S’ moves to a position shown in fig. so that
displacement of origin O’ is
R=vt
Let r and r’ represent position vectors of a particle P
with reference to frames S and S’ respectively.
then r=R+r’
Or
r’=r-R=r-vt
dr’/dt=dr/dt-v
And d2r’/dt2=d2r/dt2 (the velocity v being constant)
d2r/dt 2 and d2r/dt2 are the accelerations of the particle
in frames of reference S and S’ respectively. Thus
acceleration of the particle as measured in two frames
of reference is always same i.e.,if the particle is at rest
in inertial frame S, it would appear to be at rest in
another inertial frame S’ also.
Thus inertial frame of reference are non-accelerating
frames.
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When a frame of reference is accelerated
relative to an inertial frame,the form of basic
physical laws such as Newton’s 2nd law ,
becomes completely different.
Such relatively accelerated frames of
reference are known as non inertial frames of
reference.
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Consider a non inertial frame of reference S’ moving with
an acceleration ao relative to any stationary frame s.then all
the particles which are stationary w.r.t. frame S ,have
acceleration –ao in frame S’ .
Now if a particle of mass m moves with acceleration ai in
frame S ,the apparent acceleration of the particle as
observed in S’=ai-ao Hence the apparent force acting on
the particle in frame S’ is given by
F=ma=m(ai-ao)=mai-mao
F=mai+Fo
When mai=0 , F=Fo
The force Fo=-mao is called fictitious or pseudo force.
Thus a force which doesn’t really exist but appears only
due to relative acceleration of the frame of reference is
called fictitious force.
So a non inertial frame is either a frame having uniform
linear acceleration or a uniformly rotating frame.
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Galilean transformation is a set of equations,
which represents how the co-ordinates of an
event, which occurs in space at any time,in
inertial frame of reference are related to the
co-ordinates of same event in another frame
of reference, moving with constant velocity
relative to the former frame.
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S is an inertial frame of
reference with origin at O. Let
it be at rest.
S’ is another inertial frame of
reference with origin O’. It is
moving with constant velocity
v along the positive direction
of X or X’ axes .
Let time be measured from the
instant when origin O’ just
coincides with origin O.
Let an event occur in space at
a point P at any instant. The
co-ordinates of P observed by
an observer in S are x,y,z,t .
Similarly co-ordinates of same
event as observed by an
observer in S’ are x’,y’,z’,t’.
In time t , S’ covers the distance OO’ =vt
along positive direction of x - axes i.e.
O’A=OA-OO’
But O’A=x’,OA=x and OO’=vt
So,
x’=x-vt
As there is no relative motion of S’ along Y
and Z axes therefore
y’=y and z’=z
But time is absolute in classical relativity so
t’=t
Hence Galilean transformation are given by
x’=x-vt
y’=y
z’=z
t’=t