GRtensor replacement for Maple

We all know that GRtensor works only with Maple V or older versions of it. Now, Maple have introduced the tensor with somewhat supplements the GRtensor package. Since I just started, I know that it can do the basic tensor calculations. It might also do a little bit more like defining new tensors and taking covariant derivatives. so finding extrinsic curvature etc, should not be a problem.

I have included a Maple command file below which demonstrates the ability of the tensor package to calculate the Ricci scalar from a given metric. Just edit the 2nd and 3rd to input the desired metric in 4D. The rest works fine.

=========================================
> with(tensor);

> gg := array(symmetric, sparse, 1 .. 4, 1 .. 4);

> gg[1, 1] := -f(r); gg[2, 2] := -1/gg[1, 1]; gg[3, 3] := r^2; gg[4, 4] := r^2*sin(theta)^2;

> eval(gg);

> gh := create([-1, -1], eval(gg));

> ginv := invert(gh, ‘detg’);

> coord := [t, r, theta, phi];

> d1g := d1metric(gh, coord);

> cfl1 := Christoffel1(d1g);

> cfl2 := Christoffel2(ginv, cfl1);

> d2g := d2metric(d1g, coord);

> RIEMN := Riemann(ginv, d2g, cfl1);

> displayGR(Riemann, RIEMN);

The Riemann Tensor
non-zero components :
1 d / d \
R1212 = – — |— f(r)|
2 dr \ dr /

> riciten := Ricci(ginv, RIEMN);

> rici := Ricciscalar(ginv, riciten);

[compts = (r^2*(diff(f(r), r, r))+4*(diff(f(r), r))*r-2+2*f(r))/r^2, index_char = []]

A simplistic proof for Fermat’s Last Theorem

The following attempt to prove the Fermat’s Last theorem might be entirely wrong. It is based on a conjecture which I am unable to prove yet. But the actual proof fits in a margin.

Statement of the problem:

Given 3 integers a,b,c ∈ I the relation

is only valid for n=2

Attempted proof :

The Pythagorean triplets (n=2 case) exists as a result of an inverse stereographic projection of rational numbers from real line i.e. onto .
The inverse projection is :
Set . this number would projected onto the coordinates on . These numbers (a,b,c) satisfy the above statement for n=2.

So, we can think that the existence of the triplets for n=2 is because of existence of a closed compact surface having the equation which is the equation of a circle.

Now, can we generalize this statement? Let us make the bold assumption that such triplets can be found for arbitrary “n” if similar projection can be carried out from to a closed surface such that . This statement I cannot prove. Buts intuitively it seems very appealing. The curve has to be closed. If the curve is not closed then the points close to cannot be included. That is to say, large rational numbers would be excluded.

Now, for n=odd the surface is not closed. So, there is no projection at all. But a partial projection can be made for positive rational numbers. But the conjecture is that the surface must be closed. Hence, no triplets for n=odd.

For n=even the projection is possible but we get Pythagorean triplets again. If (a,b,c) are each power n/2 of any integer triplet then becomes true for n=even. However, notice . It can never be because p and q are co-prime. This is true even if p=q. Hence, such triplets do not exist for n=even unless n=2 when a=i^1 i.e. a is an integer which is trivially satisfied.

Hence, such triplets do not exist for .

In conclusion, it looks nice because the proof can be almost written on a margin. But on the other hand the assumption of “existence of higher order triplets due to presence of an inverse projective representation” is crucial and might be wrong.

An essay on time : Saugata Chatterjee

Time is something we experience every day. Curiosity about the nature of time dates pre-civilization times, more than 30,00 years ago, when men drew the phases of moon and counted days. But serious thoughts about nature of time was begun only much later, in Europe, by Socrates, who plunged the idea deep into the philosophical school of Greece and taken forward by Plato and in India. where people pondered over true nature of creation. The Greco-Roman concept of time was practical, involving measurement of time. The Indians took it many more leagues further by describing time as infinite (Lord Vishnu floating eternally and creating universe out of nothingness). The concept of infinity might have also entered Vedic culture in the context of creation and the role of time, but infinity might be used from a different perspective. The endless cycles of time divided into 4 major “Yugas” clearly depicts that ancient Hindus conceived time as endless but the only way they could incorporate the notion of infinite time is to introduce cyclicity. This way they could avoid dealing with the disturbing notion of infinity. However soon they realized that the concept of infinity was inevitable and allowed for it in the creation cycle.
But still, starting from Zeno’s granular nature of time to Vedic liquidity, the true nature of time remains elusive. One thing is certain, the measurement of time requires subdivisibility while measurement of motion requires continuity. So in a way the Greeks were concerned with measuring or the practicalities of time while the Vedic culture was more interested in its esoteric properties. I would think only about howto measure time.
Time could be assigned properties which are mathematical and properties which are purely physical (mechanical). For example, the rate of change of velocity with time is a measure of force experienced. By reciprocity this means perception of time change is induced by existence of forces. As long as there are forces in nature of fundamental or mechanical in origin, time must change.
But General Relativity runs contrary to this argument. It says as long as no force acts on a unaccelerated observer it would continue to move along geodesics which are labelled (parametrized) by its own proper time. A geodesic observer is one who moves along a path of maximum proper time. From the point of view of the geometry nothing special is claimed but a naturalness condition that the manifold be allowed to have a metric or rather the spacetime be a metric space. And any path in a manifold could be parametrized but geodesics are special because it allows an invariant notion of distance (or rather a scalar product) on that manifold which results in the choice of an unique metric. Time is defined as the affine parameter labelling the path of a geodesic observer. So, here is a paradox. GR says time could very well exist in absence of force or anything. Not only that, even if nothing is happening, Time still evolves. The solution to this apparent paradox is the question : How can someone measure the change in time if he has nothing to compare to, no hands of clock changing orientation, no moon going around Earth? He can’t !! So the bottomline is that Time keeps changing but to observe that change some other associated change is necessary in a closed configuration.
This brings us to a philosophical question.Can time be perceived without fundamental forces of nature?Relativity makes Time a coordinate and says it exists on its own. But to measure Time an observer must carry a clock. What does a clock do? It changes its configuration in a cyclic manner. What is the reason for this periodic change? How is it brought about? The change could only be induced by external (gravity) or internal (flip of ammonia; EM force) forces acting on it.
So my conjecture is that Time exists in its own accord but its existence could only be perceived by the accompanied existence of natural forces or some agent which induces a change in the state of a system. Some skeptics could ask that what if someone is locked in a featureless room for days, would that mean that time is not changing? First of all this is a ill-posed situation because the person would age, feel hunger, his biological clock in his brain induce him to sleep etc. which would tell him time is progressing. But place a stone(something which cannot feel hunger etc.) in absolute space, before creation, everywhere there is total vacuum. Now, how would that stone tell that time is evolving?
Many people with a religious inclination might be willing to believe that whatever is discussed is a copy from Book of Genesis or from Vedanta, but I am not here to prove the existence of divine entities. Instead it boils down to plain facts, that forces are a necessary ingredient in the study of Time.
CP violating forces would give the famous “Arrow of time” – a direction in which direction Time must flow. In fact any irreversible phenomenon gives a direction to Time. But since all these irreversible processes have associated increase entropy, it implies that increasing entropy is a tell tale sign for increase in Time.
But we have to ask the question- why entropy always increases or why all physical processes are irreversible? The answer is simple- so that time could increase. If a broken glass could mend itself, then time might run backwards for some time and we would not even notice it.
Irreversibility is natures way of telling that Time must increase. But why is the description purely thermodynamic- the state of system goes to the maximum density of states in the phase space?
Reason might be that the forces(gauge fields) decide on measurement of time and matter fields, which take values in the state space or ensemble decide how is Time going to evolve- backwards or forwards. And since gauge fields and chiral matter fields are dual description to the same Lie algebra, it must imply that measurement of Time and its evolution are dual to each other or 2 faces of the same coin.

(to be continued ….)