January 29, 2018

Time, Gödel & Mahākāl

In the evolutionary ascent of man, the idea of time is perhaps the first concept that differentiates him from his animal past. Unlike a human being, an animal, say a cow, has no -- or very rudimentary -- memories of the past, and certainly no hopes and plans for the future. It lives in the perennial present and is motivated only by the current state of its environment and its own current state of hunger, fear, libido or discomfort. Time is also an enigmatic concept that defies definition. Trapped in a peculiar case of circular logic, where “the snake swallows its tail”, we say that “Time is what is measured by clocks and a clock is what measures time”.  We obviously sense the passage of time but this flow is another mystery because if it indeed flows, like the water of a river, then what exactly are the banks of the river that it flows through? Then again, do we sit still while time flows past us? Or do we move along through stagnant time? There have been questions galore but hardly any convincing answers.

It was Albert Einstein who, in a sense, caught time by its ears and dragged it into the same cage that holds the other three dimensions of length that define what is traditionally known as space. Time in Einstein’s Theory of Relativity is just another dimension that creates a platform for the manifestation of physical phenomena and helps us model the physical world in the language of mathematics. Multiplied by ‘c’ the speed of light, time becomes dimensionally equivalent to a measure of  length and can be treated as such to calculate the Euclidean distance between two events in space-time as opposed to the mere physical distance between two points in pure space. In fact, light -- or rather its speed -- is the magic knot that ties space and time together with measurements along either of these dimensions being affected as the observer travels at a speed close to that of light. Clocks slow down, increasing the time elapsed between events and distances shorten as we increase the speed of the observer.

But just as we thought that we have caught up with time, it gives us the slip and speeds away again. How? Because unlike in the other three dimensions, we cannot stand still in time - we must keep moving, or as we said earlier, time moves past us. But whether we are moving through time or time is moving past us, what is the speed with which time moves? Speed is by definition change of position ( or at a stretch, any other physical quantity like temperature or illumination ) per unit of time, but when it is time that is changing, then the change of time per unit of time is a phrase that is drained of all meaning! Can we extract some meaning from this apparent meaninglessness?

If you are travelling in a train, you cannot make out whether it is moving or not (assuming that there is no sound or vibrations) unless you look out of the window and see the scenery moving “backward”. So if you are moving along in time, you should not be able to determine whether you are moving in time unless you looked out of the “window” and saw something that is not moving through, or with, time. But like a tree that is stuck to the ground and against which you can determine that your train is moving ( though the tree might argue, if it could, that you are stuck and it is moving - and relatively speaking both would be right) is there something that is stuck in time and against which you could measure the quantum of movement that you have made through time?

This is tough. If everything is moving through time at the same “speed” then we cannot determine the “speed” of this movement by looking outside the window. But if looking out of the window does not help, can we look inside? Inside of what? Let us begin by looking into our minds and what do we see there? Memories -- and these are stuck in time, like trees in the landscape outside our speeding train. Memories are artefacts stuck in the past but they are mental objects that lie outside the reach of physical sciences. The corresponding physical objects are events, of which we have memories, and a collection or rather a sequence of events is what gives us our sense of motion or movement through time.

If the relativistic concept of dimension was the first handle that physics had on the elusive nature of time, then the sequence of events is the second handle it could grip it with. This second handle, or perspective, also leads to the key concept of a direction, the Arrow of Time. Unlike a dimension of space in which we can move forward or backward, to and fro, the time dimension is a one way street. You can move from Nagpur to Kanpur and back  to Nagpur again but while you can, and will, move from childhood to old age, you cannot go back to your childhood again. Physics liberates this concept of sequence from its ties to human memory, by formalising it in terms of the Second Law of Thermodynamics that states that entropy, generally understood as the level of disorder in a system, increases with time. Or as Omar Khayyam puts it “The moving finger writes and having writ, moves on; nor all your piety nor wit shall lure it back to cancel half a line, nor all your tears blot out a word of it.”

The fabric of physics now seems all set to wrap up time between the warp of relativistic dimensions and the weft of a thermodynamic direction but even this elegant gift wrap fails to cover this oddity quite completely.  Under the relativistic interpretation, the world is symmetric under time-reversal. This is because the laws of physics must have the same form regardless of any smooth coordinate transformation whatsoever -- and time-reversal is just such a smooth coordinate transformation. We know that when a body moves at high speed through space, a clock attached to it slows down. While moving faster through space it moves slower through time, and so by extension, in principle, when the speed of the body goes past the speed of light -- that mysterious knot that ties up space and time -- time should stop and then go backward. From old age back to childhood? But that is an intolerable state of affairs that is blocked, not only by the Second Law of Thermodynamics but also by our intuitive understanding of time and this is a blemish on the fair face of science.

Relativity tries to do plastic, or cosmetic surgery, to erase this scar by stating that as the spatial speed of a body reaches the speed of light, its mass would become infinite and so it would take an infinite force, and infinite energy, to accelerate it past that magic figure and back into the temporal past. Hence while it may be theoretically possible to travel back into the past, it will never be practically possible.

But even in the realm of massless objects, like photons of light, that travel, obviously, at the speed of light, we run into the problem of establishing the state of simultaneity. To determine the sequence of events along the arrow of time, we need to map these events to corresponding simultaneous ticks of a universal clock. But since relativity puts an upper limit on the speed at which information can be transmitted through light beams, there is no guaranteed way to determine if two events that are separated in space are, or were, simultaneous in time! The blemish on the face becomes a scar!

Enter Gödel, to deliver the coup de grâce.

Who is Gödel? Gödel was an Austrian mathematician and logician, and close friend and confidant of Einstein during his last days at Princeton. In his youth, Gödel had driven a stake through the heart of mathematics by using mathematics to prove that mathematics was incomplete in the sense that there will exist facts that are true but not provable by mathematics. This was the Theorem of Incompleteness, where, as his biographer Paul Yourgrau puts it, “Gödel had used the letter of a false doctrine to demolish its spirit”. Gödel’s Theorem is very well known and respected in the mathematical community but what is not known is the way he demolished the concept of time as it is understood by the Theory of Relativity! Gödel’s genius lay in demonstrating that time as it is captured by the laws of physics cannot be the same as the time that is intuitively understood as flowing through a sequence of ordered events. He did this by showing that under certain circumstances -- and admittedly these are extreme circumstances, comparable in extremity associated with black holes -- relativistic time can bend back on itself. Just as by going around a spherical world surface you can be back where you started without turning around, you could also come, or go, back to the past if the universe was expanding and spinning at the same time.  Time travel is possible, but only if we agree that this scientific time is not the intuitive time that we generally understand as time. Physics claims to have a caught a bird and put it in a relativistic cage only to realise that what it has caught is only a clay image of the bird. The real bird -- time as it is intuitively understood -- is still flying free in the sky, unfettered by the logical bars that define the cage of science.

Einstein admitted that Gödel was correct in his analysis but defended the relevance of his own work by stating that Gödel’s requirements were unlikely to be met with in the real world. But then many of Einstein’s own reservations about the real world, like the possibility of black holes and the expanding universe, were later overturned by physical evidence. For more detailed explanation of how Gödel demolishes time, the reader can look up Yourgrau’s book, “The Forgotten Legacy of Gödel and Einstein”.

But if time cannot be talked about logically and scientifically, how should one go about trying to describe or articulate the idea? Or in that case, is it something not worth discussing at all?

Ludwig Wittgenstein, one of the most influential philosophers of the western world in the 20th century, had originally stated in his Tractatus Logico-Philosophicus that  “Whereof one cannot speak, thereof one must be silent.” But with the passage of time, this stark utterance has been beautifully re-interpreted by Doxiadis in his famous graphic novel Logicomix as “The things that cannot be talked about logically are the only ones that are truly important”. This, in a sense, is also echoed in the Sanatana Dharma where the principle of  neti neti (नेति नेति) or “not this, not this” --  an expression of something inexpressible -- has been used to arrive at an approximate description of the Atman. Then again we have Gödel, who was a great fan of fairy tales, stating that  “Only fables present the world as it should be and as if it had meaning.”

So our search for the meaning of time must now explore the legends, fables and the divinity associated with time. In the Indic worldview, the world is viewed as moving cyclically with the Kālchakra and rotating recursively through a sequence of solar-years, yugas, mānavantars, kalpas and Brahma varshas  whose cumulative span is longer than the scientifically dated age of the universe.

Our universe obviously exists in space and in time. But for space to exist it must first acquire and demonstrate the quality of persistence and persistence by its very definition is a property that can only be manifested in the flow of time. Hence time must precede space -- it cannot be at the same level of significance as just another dimension of a hybrid space-time. This makes time the most fundamental component,  the primordial matrix of the manifest universe.

The iconic imagery of Kāli and Mahākāl --that which is time or even beyond time,  is an excellent expression of this perspective.

This article originally appeared in Swarajya, the magazine that reads India right.

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