Minkowski Topology

In this passage, we will study space-time, ricci tensor, and minkowski topology.

Minkowski is a well-known German Mathematician who lives in the time of Albert Einstein.  He is pretty friendly with the work of Einstein on explaining space-time curvature. He is a professor of Albert Einstein, in 21 September 1908, he delivered, alongside Einstein works, the meaning of space-time in Special Relativity and General Relativity, on the 80th Assembly of German Natural Scientists and Physician.

"The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

Even if they still use the cartesian diagram to profoundly map relativity, yet the Euclidian did not falter too much from the tensor mathematics and topology.  In short, as a starting, let's imagine two places in space-time, one started in New York City, and say the other one started in Bandung.  Accordingly, let's say the distance is stated in two dimensions, in which started in New York City was 1000 years ago even before the city was built, while the one in Bandung started just several years ago.  According to time reference, these two states were differentiated by the fourth dimension, especially close frame of reference yet quite different as long as quaternion has to comply.  So, the t on the first state equation is say 1000 and the second t let's commute with r is 3.  let's say that the equation is 

S² = x² + c²t² 

and

S² = x² + c²r² 

We can see that the delta S of the component would include two different time measures, in which were 1000 as t and 3 as r.  As how we see it, then S would propagate either as the square root of the first as well as the second equation.  To solve this equation and knows the real dimension of the planar, we would use the equation in differentiation to solve on each slopes and points.

Minkowski Spacetime Cone, Illustration Photograph by Spencer Sutton (fineartamerica.com)

Minkowski define several characterictics to be considered as Einsteinian-Minkowski Topology (Also with the help found in Ricci Tensor Metrices) as, first it is a flat pseudo-Euclidian diagram of space-time, second the space-time interval is described as interval ds² = c²Δt² - Δx² - Δy² - Δz² 

, third it is independent of initial frame of reference that is chosen.

Accordingly as analysis between two points in the diagram we may use Ricci Tensor to make it simpler in breaking-down each equations.  The Ricci tensor is a step to step analysis to study Minkowski Topology and pin point a dimension in space-time, quite different with Dirac Equation in which describes string theories and quantum, Ricci Tensor is more quite much to say "eligible" to be used in calculating Minkowski Topology.

The Ricci tensors as follows:

These elongation in the right hand side is the Reimann Tensor. One perhaps might ask now, where does a calculation the being able to be put in hand in the Topology?

As how we see that Minkowski Euclidian are four basis variable calculation, one may use matrix to calculate such succumbent.

In the diagram, we may simplify each basis and mount it unto an algebraic function, thus then otherwise may make us able to calculate the dimension of the event point in the other dimensional diagrams.

Therefore, Ricci-Reimann Tensor is one of Mathematical Tools in mapping points in four dimensional space-time, disregarding also its use in Black-Holes and Singularities, there would perhaps be a quite interesting foundings if we also use the disclosure of quaternion algebra and Dirac Equation, to bridge in between the examples of Membranes Theories and String Theorems.

References:

- Ricci tensor (planetmath.org)

- Einstein Relatively Easy - The Minkowski metric

- Einstein Relatively Easy - Minkowski's Four-Dimensional Space-Time

- The Dirac Equation (ucsd.edu)

- Drawing Time in Four Dimensional Graphs by Muhammad Sadhra Ali

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