Minkowsky’s Myth

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 20^th 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.