hello everybody i am rajdeep chatterjee from
the department of physics iit roorkee and
i shall be talking on the special theory of
relativity
well we plan for the next series of lectures
is something like this ah i shall talk on
how relativity arose while reconciling the
laws of mechanics and electrodynamics but
to be more precise reconciling the transformation
laws of mechanics and electrodynamics in that
context i will be talking of galilean and
lorentz transformations moving over to be
all important postulates of special relativity
okay
and then i shall of course go over to the
consequences which are quite interesting we
will be talking of length contraction time
dilatation mass energy equivalence and this
is one thing perhaps many of you are quite
familiar with e = mc square let us try to
see we will try to see at a certain point
of time how it all arose and in explaining
all these things what i shall do is that as
and when necessary we will talk of certain
problems
we will try to do some problems so as to illustrate
the principles involved okay okay so let us
start at the beginning mechanics
this f = ma this perhaps is the most famous
equation in mechanics if i may say so you
all know that if the ah force of is applied
on a particle of mass m it is going to accelerate
with acceleration e you can even tell me that
this is actually newtons second law of motion
okay but what is assumed here apart from of
course in this particular case that we treat
m that is the mass of a particle that is that
is always constant okay
well if you have any can argue that if you
have a variable mass you can talk of force
as rate of change of momentum but let us stick
to this form of newtons second law for the
motion okay yeah but what is important here
is that we always assume here that somehow
we have a frame of reference ah where we are
able to measure this acceleration okay now
frame of reference is actually a very fancy
name for for a simple coordinate system
i mean coordinate system you know simplest
one of course it is the ah cartesian coordinate
system i mean if you see where the walls of
of this room meet i mean the room you are
in if it meets with the floor and then you
see the axis of the coordinate system so we
have quite familiar with the word coordinate
system so that is a fancy name the frame of
reference okay
now another thing is we sometimes hear of
this word inertial frame so what is an inertial
frame well a very solid definition is an inertial
frame is one in which newtons laws of motion
are valid okay you know newtons laws of motion
are valid you know inertial frames so what
does this explain where explain a person explain
and tell an engineer or a scientist where
to look for this inertial frame we need to
be a little bit more precise than this idealistic
definition and in doing that in defining that
we can say that it is a frame whose coordinate
axis are fixed relative to the to the average
position of a fixed are fixed star in space
of course
or it is that frame is moving with an uniform
linear velocity that is a constant velocity
relative to the star and of course there should
not be acceleration of this star otherwise
the definition is not valid well once i have
said this and you realize that any frame the
earth itself the earth itself is evolving
and it is there is day and night there is
this rotation due to which you have of course
they are night and and and revolution around
the sun that is the that gives the change
of season
so there is some amount of acceleration yes
ok physically any coordinate system attached
to its surface is known inertia but for many
purposes this acceleration is slight and then
by many purposes i course do not mean for
all purposes if this acceleration can be considered
slight here this particular frame that is
the frame on earth can be considered inertia
a little better would be a non rotating frame
with with a origin fixed at the earths center
and access appointed towards the fixed star
so that is we can say it is approximately
interaction well even if you have this frame
to be fixed at the center of the sun for example
that will be more inertia than compared to
the frame i just described ok but having said
ah this we should be clear that non inertial
frames are actually quite common in in mechanics
i mean must have heard of corioli forces okay
so these are not inertial forces ok so let
us try to have a ah visual explanation of
what we have been trying to say in more definitive
terms ok
so here we have frame s ok so this is a three
dimensional cartesian the system a right handed
cartesian system ah i have not written the
z-axis here but you can all guess and you
can all you can all figure out that z-axis
here actually points outside the screen okay
so this is frame s and then we have another
frame s prime let us say and then this frame
s prime is moving with a constant velocity
or let us say uniform velocity v along the
common x x prime axis okay
now the coordinates here are the x prime y
prime and z prime and here is that prime again
points outside the screen now ah at the beginning
at the very beginning ah general let us we
have two observers who are let us say at the
origin so both these frames and at the beginning
on both these flames coincide that is s and
s prime they have a common origin at the beginning
at time t equal to and t and t prime = 0
so i will talk of this chord at some time
a little bit later so how to do that i mean
at simpler and you start with two observers
are there and they look at the watches and
say that okay fine so our watches agree and
this is the time we set as t = our t prime
=0 and then s prime frame starts moving with
uniform velocity v with respect along the
common x x prime axis so the only line if
we have a point p which has coordinates x
y z and then its measured at a certain time
t remember block is moving
i mean time clock is working it is moving
i mean that time is flowing ah at a certain
time t so at that same instant let us say
observer in s prime measures the coordinate
as x prime y prime and z prime okay so how
are they related well you can say that it
is simple actually so how are they related
x prime is related by this relation x prime
= x - v of t v and t ah y = y prime and z
equal to z prime okay and it is important
that both these time coordinates agree okay
so so that is we this point is measured at
the same instant of time okay so they all
started with synchronized watches so t prime
= t here okay now this transformation ah we
call this transformation as galilean transformation
okay let us delve a little bit further
how are the velocities if you measure the
velocities in in both these frames how are
they related well that actually will be given
by the galilean velocity addition formula
so let us see ah how it how it can be derived
we have this we have this equation x prime
= x - vt ah see that now you differentiate
x prime with respect to the time in its own
frame so that is dx prime dt prime and on
the right hand side all you got to have dx
by dt prime - v dt dt prime okay and now you
realize that on the right hand side you have
a dx by dt prime now x is the coordinate in
the s frame but t prime that is the time that
is measured in the primed frame that is s
prime frame
so you need so if you need velocities so you
need coordinates of the same frame okay so
a coordinate and the time of course in the
same frame i should say so ah the third step
clarifies how to do that so you have dx dt
and then you take this dt dt prime - v of
dt dt prime okay so then you realize that
dx prime dt prime that is the u prime that
is the velocity velocity that is being measured
in the primed frame okay
now you realize since t = t prime in galilean
transformation so dt by dt prime that is equal
to 1 so ah you realize and then dx dt that
is u and so dt by dt prime that is 1 so - v
of – v so you have u prime = u – v now
i mean there is a subtraction sign here so
ah do not be too bothered about that when
i use the word addition formula because you
can very well write ah the velocity which
is in the s frame in terms of the s prime
frame by saying that u = u prime + v
and what is v by the moment by the way so
it is just the velocity with which the s prime
frame is moving uniform that is uniform velocity
with which the s prime frame is moving with
respect to the s frame along the common x
prime axis ok so that goes for the velocity
velocity addition formula what about the acceleration
mm-hmm
well we start with what we have obtained for
the velocity so that is u prime = u – v
so if you differentiate this once again so
you are going to have du prime by dt prime
is du dt of course you know how to get this
now so this is du by dt you know how it would
be so if du by dt frame take du by dt prime
and then you have dt prime dt by dt prime
that is equal to 1 so the acceleration is
going to be the same in both frames
so these frames are moving with uniform velocity
s with respect to one another so what we have
is acceleration is being unaffected by if
you have frames which are moving with uniform
relative velocities okay now on top of that
if you consider that mass is unaffected by
motion of reference frames ah you come to
and we come to a very interesting conclusion
we see that the form of newtons second law
is valid well actually newtons second law
is valid in both these frames in both these
inertial frames okay
so so what does this mean well this means
that by doing experiments entirely in one
of these frames you cannot distinguish it
from the other okay so if you are doing extreme
experiments entirely in one of the frames
so and this frame is moving with a uniform
velocity v with respect to another frame you
will you cannot distinguish this particular
frame from any other inertial frame okay so
by by mechanical experiments alone that is
what im going to say here
so what you can ask so so what happens is
that since in newtons laws of motion are being
valid are valid in these two frames so our
equations of motion okay we charge right from
from them and consequently the conservation
laws so going to have the conservation laws
same in all these inertial frames okay so
if you do your mechanics in one of one of
these frames and you derive a conservation
law can be rest assured that another inertial
frame it is going to be the same it is going
to be valid okay
so ah we could say that the laws of mechanics
are being invariant in in all inertial frames
okay this is an this is an important conclusion
so next we move over to what is going to happen
in electrodynamics okay
so what is the electro dynamics so you see
it is an interesting thing so here ah you
have what does it give you it gives you that
if you have a changing electric field you
going to have a magnetic field and then if
you have a changing magnetic field you are
going to have an electric field okay now we
ask this question that is electrodynamics
or the laws of electrodynamics invariant under
galilean transformation
remember laws of mechanics they were invariant
under galilean transformation so we asked
another branch of physics electrodynamics
are the laws they are invariant under galilean
transformation well for that let us see what
those laws are okay ah with the basic laws
they are the encapsulated all in maxwells
equations
well divergence of e that that that is equal
to rho by epsilon 0 and e as you know is the
electric field a rho is the charge density
and epsilon epsilon 0 that is the permittivity
of free space and then the curl of e that
is - del b del t b is the magnetic field and
then the divergence of v that is zero the
curl of v that is mu0 j mu0 that is the permeability
of free space and j that is the current density
that plus of course mu0 epsilon 0 del e del
t
now do not be bothered too much about this
mathematical details so what do they stand
for i mean i mean that is what i have written
on the right hand side obvious equations the
first one is actually gausss law well so it
is a very important law it actually allows
you to calculate the electric field if you
have symmetries in the in the problem symmetry
the charge distribution that is okay
the second one curl of e that is equal to
- del v del t that is actually faradays law
and i am surely aware of it because had this
law not been there i mean did not have motors
you electric motors that is you must have
heard of faraday is important experiment in
which you had this he was moving us a magnet
within a solenoid and then he detected an
emf within the leads of the solenoid okay
then the third thing i mean a third equation
that i have written here and should not be
talked as a third law that is divergence of
b = zero which which well does not have a
law it does not have an name as such that
it is physical implication is that there are
no magnetic monopoles so like that like you
have a you have a positive and negative charges
you do not have charges in a magnetic poles
in isolation okaydo not have magnetic monopoles
the curl of b that is mu0 j + mu0 epsilon
0 del u del t that is actually a curl of b
= mu0 j that is actually amperes law okay
and then added to that is maxwells correction
well that maxwell or well maxwells corrections
are actually quite important you sre going
to see later on because it had rather quite
rather very interesting implications in showing
that these the maxwells equations well by
the way so what you see here is of all e and
b are coupled
these are coupled partial differential equations
so well this correction is very important
because he was able to show that well he put
he introduced this concept of displacement
current and then was able to show that these
equations when written in terms of only e
or only b could be framed in terms of the
wave equation the more of that a little later
okay okay
so our maxwells equations invariant under
galilean transformations under the transformations
mechanics is invariant on so what were these
transformations once again so so you have
an s prime frame moving with an uniform velocity
v along the common x x prime axis okay again
z and that prime this axis are moving are
actually out of the screen okay they are pointing
out outside the screen and this transformations
x prime = x – vt y prime = y z prime = z
and t prime = t
so our maxwells equation invariant under this
the answer is no okay they are invariant under
a different transformation maxwells equation
are not invariant on the galilean transformation
but maxwells equations are invariant under
lorentz transformation okay so what is that
so again we have the frames s prime moving
the uniform relative velocity v along the
common x x prime axis but here we need x prime
to be given by not only x - vt divided by
root over of 1 - v square by c square okay
and of course here a y prime = y z prime is
equal to remember we are moving along common
xx prime axis
what is interesting here is that see that
the times are not matching in galilean transformation
we had t is equal to t prime but here t prime
= t - vx by c square divided by root over
of 1 - v square by c square ok so this v is
actually the velocity with which the frame
s prime is moving with respective s frame
ok what is this see here well if you have
guessed that is the speed of light but all
of a sudden how come this speed of light is
there so remember this is the transformation
under which maxwells equations are invariant
ok
so do we see do we see c and that is the velocity
of light im sorry the speed of light in vacuum
ah explicitly in maxwells equation i mean
on the left hand side i have written that
once again just for your convenience well
it is not present explicitly ah so we asked
this question so where is this coming from
okay so is c ingrain somewhere within maxwells
equation itself ok for that what you have
to do as i said is that maxwells equation
these are posh coupled partial differential
equations
now if you uncouple them ok we have if there
is a price to pay you see that you have a
second order equation then ok
so you have del square e = mu 0 epsilon 0
del 2 lt square of e and similarly for the
magnetic field also you have the laplacian
i should say del square b or plus en of p
that is mu 0 epsilon 0 is del2 by del t square
okay now this has an uncanny resemblance with
the wave equation you know waves water waves
sound waves so it is wave equation here so
the laplacian of f that is equal to 1 by v
square of del 2 f del t square okay
so t is the time here and what is v v is the
velocity of the wave now ah you see in these
two sets of equation if you compare ah maxwells
equation of b and e with the wave equation
what you are going to see is that this term
mu 0 epsilon 0 can be compared with 1 by v
square okay so which means that if it if it
i mean since it resembles the wave equation
ah mu 0 epsilon 0 somehow has some sort of
relation with velocity okay
it is actually you are going to see that 1
by mu 0 epsilon 0 is that does indeed turn
out to be and 1 by root over of mu0 epsilon
0 does indeed have the dimension of velocity
it is actually 3 to 10 to power 8 meter per
second okay so on that value later on when
you put in values okay but also from physical
principles in hindsight you can also check
that mu 0 epsilon 0 should have the dimensions
of 1 by velocity square well how to do that
well well check any one of any one of maxwells
equations in e or b
check the first one the laplacian of e = mu0
epsilon 0 del2 e del t square now this laplacian
of e laplacian how does it look like del 2
del x square + del 2 del y square + del 2
del z square that kind of a thing so it has
a dimension 1 by length squared okay so on
the left hand side i mean for the moment look
at these operators that is then that is more
important now because e and e so the ene they
have the same dimension
so what we need to do is to balance the dimensions
of rest of the operators and rest of the things
here but on the right hand side will be concerned
with mu 0 epsilon 0 del 2 del t squared okay
now you have a t squared in a denominator
here so which means that that is time squared
okay so on the left hand squared you have
on the left hand side you have 1 by length
square and then on the right hand side you
have 1 by time square okay
so what should be then the dimension of mu0
epsilon 0 so that you have this entire thing
mu 0 epsilon 0 del 2 del t square to have
the dimension of length square okay well it
has to have then the dimension of 1 by velocity
squared ok so then in hindsight we can actually
we actually can figure out that mu0 epsilon
0 should have the dimension of 1 by velocity
square that similarly has the same thing the
same conclusion you realize from the second
equation laplacian of v is mu 0 epsilon 0
del to be del t square ok
now on this value 3 into 10 to power 8 meter
per second okay and you might have already
guessed that this number is actually the speed
of light okay so you see speed of light is
actually ingrained within maxwells equation
itself and then del 2 laplacian of e is actually
equal to a plus sign of the electric field
and the laplacian of the magnetic field you
see that it is 1 by c square and then here
on top you have del v del t square and for
the magnetic field l to be del t square okay
now since it follows the pattern since it
follows the form of the wave equation okay
so maxwell concluded that then light must
be an electromagnetic wave okay now this had
a profound significance because light electromagnetic
wave and you see that i have written wave
in in italics
because in those days the bend in the 19th
century actually people thought that waves
actually require a material medium to propagate
why was that the the reason that okay you
have water waves which water to propagate
you have sound waves you need medium air we
need air or even sound waves can travel through
another material for example to a metal but
in any case you need a medium to propagate
so so the reason that perhaps they also light
also should require a medium to propagate
so and then they just name this medium as
ether or actually used to call it the luminiferous
ether okay
and then further reason that as light can
travel through vacuum then vacuum must contain
this medium of light which is ether so vacuum
is full of ether a okay that is the medium
of light okay now like every assertion in
physics and if you even if you make a theory
it has to be proved it has to be validated
by experiments and so that is the challenge
that confronted physicists at in those days
in the late 19th century is to detect ether
and its properties okay
so they were thinking of a possible experiment
in which to measure the speed of light in
different inertia frames okay and and to see
if these speeds were different in in these
different systems okay
now in case they were different ah they will
look for evidence of a ah special frame where
that is the ether frame and that that is going
to be a preferential frame where the speed
of light is seated that c 3 into 10 to power
8 meter per second that is the speed of light
in vacuum okay so they were they were looking
for an ether frame and this experiment remember
was to be done on earth so sitting or not
they were supposed to detect ether
now consider the fact that earth is in motion
okay so if an experiment has been done on
earth and then earth is in motion so you should
be able to detect an ether wind in a sense
quote unquote an ether wind okay and then
the magnitude and direction of ether of this
heat oven would vary with season and of course
the time of the day because of rotation of
the earth okay
so the point was to the suggested experiment
was to measure the return speed of light okay
so going and coming back okay since ether
was always gas in an ether frame in in different
seasons and in various times of the day okay
why because if earth is moving relative to
the ether frame the return speed of light
would be different and this difference could
be detected then and that would be a test
for the presence of heat remember that devising
such an experiment was indeed very difficult
okay so but but there were smart people they
were wherever as they were michelson and morley
who in the later part of 19th century
they device then interesting instrument they
devised actually devised an interferometer
which goes by their name
so that had a light source okay so light is
emitted from the source it comes and hits
the semi silvered better that is actually
a beam splitter okay so then you have mirrors
on two sides perpendicular and parallel to
this light source and then and detector on
the other side like it is shown here so what
happens is that light comes and hits this
semi silvered mirror it splits into two parts
okay goes to the mirrors is reflected back
okay
so that you see the science a little bit different
symbols for this reflected rays and for this
reflected ray then re combines and goes to
the detector and there will be constructive
contract constructive and destructive interferences
due to which there will be a fringe pattern
at this detector okay now remember this experiment
is being done on earth okay now as earth is
moving in the ether frame okay
and so if this flow of ether is parallel to
one of the one of these beam directions let
us say parallel to going from if the direction
of ether is from light source towards the
mirror on your right hand side and then what
will happen is that the returned speed of
light will be different from the returned
speed on the perpendicular to the ether flow
why because if you spiral to the flow of ether
and then once it goes parallel it is towards
it is flowing with you know in the direction
of ether
but when it is been reflected black its opposite
to the flow of it okay so there there is going
to be a difference in time of the return speed
of of the return of light in both this axis
and what you are going to have is that this
difference is going to cause a shift in the
fringe pattern at the detector okay
so the expected result was that there would
be a friend shift at the detector which would
confirm the presence of ether okay but surprise
surprise the actual result was that although
this was done on a different season different
times of the day no discernible friendship
was observed i mean you could have argued
that maybe a more sophisticated instrument
or later on they could have rechecked
it was checked even by other people and also
by by more sophisticated equipments and there
was no evidence of this ether frame well jokingly
of course sometimes people call and this is
the most famous field experiment okay so there
was no ether
now towards the end of the 19th century on
the other hand albert einstein was also very
concerned and he was also concerned on a different
thing he was concerned that the laws of classical
mechanics and electrodynamics we are not following
the same transformation laws they were following
galilean and they all in transformation laws
okay so this was quite troublesome to him
he being a theoretician so he is not that
does it mean that an inertial system which
is actually indistinguishable by mechanical
experiments remember we saw earlier that with
the help of mechanical experiments you are
not able to distinguish between inertial systems
different inertial systems because newtons
law is going to be valid in each one of them
in the same form ok so does it mean that okay
i mean with mechanical experiments it is not
being possible
but by other means by by other electromagnetic
means maybe optical methods can you then distinguish
between inertial systems that to einstein
was a very why something because here you
have then different branches of physics following
different transformation laws okay now he
reason that this need not be so this that
there is there is somehow there is a there
is a problem somewhere
so he figured out that actually it is the
lorentz transformations which are more general
than the galilean transformations we will
put the words more general in italics yes
so explain that a little bit more later on
and he talked of the need to modify a mechanics
the laws of mechanics accordingly so that
electrodynamics and mechanics follow the same
transformation laws okay now to do this einstein
had to make two important assumptions okay
they are actually the postulates of special
relativity so the first postulate that is
the principle of relativity so which tells
us that the laws of physics are going to be
the same in all inertial frames okay so there
is no that there should not be any preferred
inertial frame okay there is no preferred
inertial and no preferred inertial frame axis
okay and then the second postulate which says
which talks of the constancy of the speed
of light
the second assumption postulate and the speed
of light in free space it has the same value
c in all inertial frames okay now with these
two postulates einstein started his calculations
and let us go let us check a little bit more
on let us let us take this idea a little bit
more on the second postulate
so here we have the two frames s and s prime
moving with velocities going with a velocity
v with respect this prime frame is moving
with a velocity v along the common x x prime
axis then and then of course at t = t prime
they started so the it coincide and we considered
a ray of light starting from a common origin
and reaching point p okay and then let us
measure the distance op and o prime p in both
these frames
so what would an observer in s frame measure
op as and what an observer in s prime frame
measure o prime ps okay so the distance wise
that would be op that would be x square +
y square + z square so that is equal to c
square t square the member c is the speed
of light and then o prime p that is x prime
square + y prime square +z prime square that
is equal to c square t prime square
now i notice of course so get into the second
postulate we have again the speed of light
to be the same in both these frames okay
now you are assured that x square + y square
+ z square now you would you subtract out
c square t square okay it is going to give
you 0 and a similar thing where going to happen
if you subtract out c square and t prime square
from the primed from x prime square + y prime
square + z prime square that is going to that
that that two is going to give you 0 so now
the same thing is going to happen if you go
to another frame moving with certain other
velocity v prime let us say or v double prime
let us say ok
so the distance there could be x double prime
square + y double prime square + z double
prime square and if the observer there has
measured time t prime - c square t prime squared
remember that this speed of light is taken
the same in all inertial frames here but we
point out that the quantity x square + y square
+ z square - c square t square is an invariant
quantity ok so that is the thing that is not
changing
now for this invariant quantity so which of
these transformations galilean or lorentz
preserves this invariance ok the answer is
and you can actually check this out you can
put x prime = x - vt x prime is y prime = y
is that z prime = z t prime = t and check
if this invariance is preserved you are going
to see that it is not so it is only the lorentz
transformation which is going to preserve
this invariance okay
now let us see what that is so well we have
we have been introduced to lorentz transformation
before well i have been talking about the
laws of electrodynamics so but let us write
down once again
so that is x prime = x - vt by root over of
1 - v square by c square y prime = y z prime
= z and t prime = t - vx by c square root
over 1 - v square by c square of course i
mean if this looks a little bit more complicated
sometimes people actually write it a more
compact form by taking this ratio v by c as
beta and then writing gamma as 1 - bi by 1
by root over of 1 - v square by c square which
is the same as 1 by root over of 1 - beta
square
so lorentz transformations can be very concisely
written in terms of in this fashion written
in this white box x prime = gamma times x
- beta ct ok so why beta c because you see
beta is equal to v by c ok and here we had
x – vt so we had to write beta c here okay
so the ys and zs are the same here but it
is very interesting to write the time coordinates
so if you multiply that by c so it has the
dimension of length again
so ct prime = gamma of ct - beta x have you
noticed one thing it is that notice the thing
for this x prime the transformation equation
for the x prime and the ct prime okay see
that x prime and the last you have beta times
ct ok but when you have ct prime you have
in the last you have beta times x okay and
then so you see it is it looks very symmetrical
so when have the when you have the length
coordinate you have the time coordinate
and when you have the time coordinate you
have the length coordinate now this is something
new this is something new to us this is something
which is not natural to us i mean we are quite
used to galilean transformation in our real
life okay but here what you see is that in
this time coordinate you have some you have
you have the length coordinate as well okay
so naturally this is going to have consequences
and we are going to check all these things
in subsequent lectures okay
just one more thing we we were actually talking
of all we were actually always talking of
what is the quantity in the s prime frame
in terms of quantities in the s frame so we
can also talk of the opposite thing so what
is the how can you say what what are the quantities
in s frame in terms of quantities in s prime
frame for that it is very easy to concern
it is it is it is the same situation if you
consider s frame to be moving with a velocity
-v with respect to the s prime frame
in that case you can simply write down x = gamma
times x prime + beta ct prime of course y
and y prime are the same z and z prime of
the same and then ct = gamma times ct prime
+ beta x prime now what i have done here as
i said was to express the quantities in in
in s frame so you want to understand calculate
quantities in x frame and in the s frame in
terms of quantities in the primed frame okay
so like we can take a break here and in the
next talk we are going to focus on again on
the postulates of relativity then the consequences
of the lorentz transformations where we are
going to carry our discussions a little more
okay so to summarize what we have been doing
today we have been looking at the laws of
mechanics and electrodynamics and we saw that
actually they were not a transformation laws
of mechanics and electrodynamics
they are not the same there were two galilean
and lawrence because einstein was so concerned
and then he he he showed that you take if
he took he actually showed that the lorentz
transformations were actually the more general
transformation laws and if you take that you
want to you need to change mechanics okay
so these are certain things that we will be
considering in our future talk’s okay thank
you
