Prerequisites : Frames Of Reference and Co-ordinate Systems
In
1905, Albert Einstein stunned the scientific community through his epoch-making
five publications on Brownian motion, photoelectric effect and special
relativity theory and changed the way the science viewed the nature forever. Historians
called this year as annus Mirabilis –
the miracle year. Though he won a Nobel Prize for his work on photoelectric
effect, he is most popularly known for his theory of relativity which
completely modified our understanding of the space and time at a very
fundamental level.
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| Albert Einstein |
The
theory of special relativity is based on two seemingly simple postulates,
however, the results that follow from these are startling and in direct
contrast with our general views (inspired by our daily experience) on terms
like mass, energy, time, motion, etc. For example, according to special
relativity
(a)
There
is a limit to which an object can be accelerated i.e. there is a maximal
velocity which nothing can exceed.
(b)
Mass
of an object is not a constant but changes with the velocity of the object.
(c)
E
= mc^2 i.e. Mass and energy are inter-convertible, contrary to our Newtonian
way of looking at these as disparate quantities.
(d)
Observed
dimensions of an object and time depend on relative motion (length contraction
and time dilation).
These
are just a few out of many. All these follow from the following two postulates
:
(a)
All
the laws of physics have the same form in all the inertial frames of reference,
regardless of their position or velocity. This is known as the principle of
equivalence.
(b)
The
speed of light (299792458 m/s) remains the same in all inertial frames of
reference (irrespective of their velocities). Thus, speed of light is a
universal constant.
These
two postulates form the basis of special relativity from which all of the above
results (along with those not stated here) can be derived through mathematical
reasoning. The first postulate is nothing but the principle of equivalence
which holds in the realm of Newtonian mechanics also and I have already
addressed this in one of the previous posts. The second postulate is more
intriguing. At a first glance these two may seem two disparate statements,
however, they are intrinsically related and I am going to bring out this
relation here by means of deductive reasoning.
To
begin with, let us first take the following example to understand what we mean
by velocity of light being a universal constant.
Suppose
that you are on a car moving with a velocity of 40 km/hr (~11 m/s) and a friend
of yours is on a Ferrari moving with a velocity of 80 km/hr (~22 m/s) in the
same direction. So you would see your friend getting ahead of you with at the
rate of 40 km/hr (~11 m/s). This is how
we perceive (relative) motion and this is also how the classical (Newtonian)
mechanics also perceives motion. But something remarkable happens when we
observe light. Suppose that a person standing on the road with a torch flashes
light at some instant of time t. For
this person the velocity of light will be some number c (actually it is
299792458 m/s). If now you observe this flash while you are moving at 40 km/hr
(~11 m/s), the velocity of light you would observe now won’t be (299792458 – 11) m/s as you might guess from
your classical intuition, but it is observed to be 299792458 m/s. That is, no
matter how fast you are moving relative to the source of light, the light
always appears to be moving at the same rate (299792458 m/s). Therefore, we say
the velocity of light is a universal constant.
Now
let us resume with our task of relating above two statements. We begin make an
assumption.
Infinite velocities do not exist in nature.
We
can make this assumption based on our observations that nothing in nature has
ever been encountered to travel with infinite velocity. In other words, we have
never observed anything (an information or an object) to travel instantaneously
between two different positions in space.
Now
let us consider an inertial frame, call it A.
The above assumption implies that no object or signal in this frame can travel
with infinite velocity. Therefore, the set of all the possible velocities in
this frame must be bounded from above and below. The lower limit (infimum) is obviously equivalent to 0
m/s i.e. a body at rest. Let the minimum upper limit (also known as the supremum or maximal velocity) of this set be M0. This means the velocity v of any object or signal in this frame satisfies
0 ≤ v ≤ M0 or it can only
take a value within the set [0, M0]. The value of such a maximal velocity is characteristic of an inertial frame.
Consider
another inertial frame (call it B)
moving relative to A with a velocity vB. Our assumption of
non-existence of infinite velocity must also hold in this inertial frame as
both the frames are equivalent. Hence, following the same argument as in case
of A, we say that there must exists a
maximum velocity limit (supremum of the set of velocities) for this inertial
frame. Let us call this maximum limit as M1.
As
there is infinitude of inertial frames there will be infinite number of these
maximal velocity limits M0,
M1, M2,... Now let us assume that M0 ≠ M1.
Since values of maximum limit of velocities are characteristic of inertial frames and are different for the inertial frames A and B,
this implies that these maximal velocities are some function of a property which is
different for these two inertial frames. If such a property existed then it
would be possible to distinguish these two inertial frames on the basis of
different values of this property. Thus the statement M0 ≠ M1
is equivalent to saying that it is possible to distinguish the two inertial
frames A and B. This is in direct contradiction with the principle of
equivalence. Hence, in order for the principle of equivalence to hold under the
assumption that infinite velocities are not possible, we must have M0 = M1. Since we had arbitrarily chosen the inertial frames A and B, our results apply to all inertial frames. Therefore, we can
conclude that the value of the maximal velocity in all inertial frames is the
same i.e.
M0 = M1 = M2
= ... = M
Thus
an object or a signal moving with maximal velocity M moves at the same rate in all inertial frames and has some
peculiar properties. It experimentally turns out that light (electromagnetic
radiation) also exhibits these properties. Hence, we say the velocity of light c is equal to the maximal velocity M and remains constant in all the
inertial frames of reference. When we do the math it turns out that no massive
object (object having mass) can move
with the velocity equal to or greater than c
and any signal (or information) can at most travel with the velocity c.
When
we further extend our arguments through mathematics we arrive at all those
fascinating results that I had mentioned earlier viz. E = mc^2, length contraction, time dilation, etc. It is
important here to realise that the reason behind constancy of the velocity of
light is very fundamental and is highly unlikely to be wrong (or at least very
difficult to prove wrong). In order to do so you would have to discover
something moving instantaneously between two different positions in space. This
is why physicists are always skeptic about experiments indicating/claiming the
discovery of something moving faster than light.
Important Remarks
We
started with two simple assumptions: principle
of equivalence and that nothing can
travel at infinite velocity and through deductive reasoning we could arrive
at a result which, at first, seemed completely unrelated. This is the kind of simple
but powerful reasoning which physicists use quiet often to derive as much
information as possible from a small set of assumptions before resorting to
maths. Such techniques are of critical importance while tackling a new problem
or developing a new theory.
Credit for the picture :
http://ffden-2.phys.uaf.edu/212_fall2009.web/Ralph_Sinnok/RS_seventhPage.html
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