For many centuries wise men have
followed the golden rule: To every impossible question there is an inaudible
answer. However, in today’s world when
electronics has augmented our perceptive powers to the extent that sounds and
images can be shared and exchanged with those who are worlds apart, the silence
must be broken. One such poser is Minkowsky’s Point Event introduced in
early twentieth century.
It is said that Minkowsky, using the principles of
special relativity and the four dimensional geometry of Rieman was able, in
1908, to present to the world a new and unique concept of four dimensional
space time continuum which is represented mathematically by the second order
differential equation ds2 = dx2 + dy2 + dz2
+ c2dt2. The square of time has no practical significance
by itself, but multiplied by the square of velocity it represents an area like
the other terms of the equation. Similarly, in the Cartesian coordinate system
x.y, y.z and z.x represent area but x.x means nothing. In terms of 20th
century physical science this means that the minute displacement represented by
ds is not completely described by the three coordinate dimensions dx, dy and
dz, as in classical Euclidean geometry; but in addition by the time dimension
dt also, forming the fourth coordinate of relativistic geometry. The quantity
ds is renamed as a "point event" and is not exactly a distance in
space, but an element in the four dimensional Minkowsky space-time continuum
which is `simultaneously finite and boundless.' However, in this continuum,
matter is able to assert itself as "the pressure of matter distorts the
curvature of the four dimensional space-time continuum which is the physical
universe."
Minkowsky’s equation can be rewritten as ds2/dt2=dx2/dt2+dy2/dt2+dz2/dt2+c2
in which the left hand term is an acceleration equal to the sum of
accelerations in each of the three coordinate directions plus the velocity of
light squared. A body with such acceleration would exceed the velocity of light
in no time and in the case of a body at rest or moving constant speed we get 0=c2 which is absurd.. The constant c2 in the equation also indicates that the
basic relationship might be a third order differential equation. You could
multiply both sides with m to get
mds2/dt2= mdx2/dt2+mdy2/dt2+mdz2/dt2+mc2
in which the
left side is a force but the last term
on right hand side would be energy. Moreover in classical mathematical analysis
higher powers of infinitesimal quantities are ignored as zero, which if applied
would make it 0=c2. We can
also write ds=√(dx2 + dy2 + dz2 + c2dt2)
which means that the displacement ds is equal to root sum of squares of displacements in
three coordinate directions plus the distance covered by light in time dt.
It is mind-boggling as to how the displacement of any or every particle in
space can be related to the
displacement of a photon in equal time anywhere in vacuum. However, the
mathematical geniuses of that era were able to utilize it for greater
achievements.
One can only wonder how the self contradicting expression of
“simultaneously finite and boundless” was accepted by the scientific community
of that time. Such a thing is possible only if you stand in a hall of mirrors
and insist that each of the innumerable images you are seeing is a real human
being. Perhaps this was the beginning of
the century of illusions which I shall discuss in future.
