What is Minkowski Geometry?

Do you like maths?

I used to love playing with numbers in my school years. When the subject of series came up in my university maths classes, I lost my enthusiasm a little bit, but my grades were always good in maths classes.

Mostly, I liked to work with graphs.

Graphs are very useful for understanding the subjects visually.

Let's draw a graph with you today.

I wrote an article on the speed of light the other day, those who follow it know. 

In that article, I wrote what speed is with the formula. 

In fact, we all already know what speed means. There may be speed lovers among us.

However, speed is a relative value that shows how far we have travelled in how much time on a road we are travelling scientifically, that is, with the formula:

Speed=distance/time

In this case, with a little mathematical knowledge, if we know our speed, we can write the formula of the distance we have travelled by taking into account the elapsed time:

Distance = Speed x time

This formula can be used to establish a connection between time and distance if we know our speed. 

Time, i.e. time with a certain coefficient, also means distance!

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It was with this logic that Hermann Minkowski, Einstein's mathematics teacher, realised that we could add the dimension of time (t) to the mathematical coordinates (x), (y) and (z). 

Thus, instead of three dimensions in Newtonian physics, Minkowski realised that the universe could be described in four dimensions.

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As I was saying graphics, we can actually call it geometry, one of the most beautiful branches of mathematics is geometry.

When it comes to geometry, the most well-known geometry theorem is the Pythagorean Theorem. 

This theorem is a theorem found by the famous mathematician known as Pythagoras of Samos, who lived in Ancient Greece between 570 BC and 495 BC.

Briefly, the theorem states that the sum of the squares of the right sides of a right triangle is equal to the square of its hypotenuse.

a2 = b2 + c2

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Albert Einstein is the most intelligent scientist known today.

He revolutionised the world of physics with the theories he developed. 

However, it is not possible for him to develop these theories on his own, he is a person who can analyse the ideas of other scientists who lived before him or in his time, as well as what his teachers taught him.

His knowledge of mathematics is not as good as that of a mathematician. However, in physics, he was sufficient for himself. 

His mathematics teacher Hermann Minkowski (1894-1909) was the person who mathematically formulated Einstein's view that "time, space and motion are not independent of each other", which he mentioned in his theory "Special Relativity", which he put forward in 1905 in Annalen der Physik.

In doing so, he utilises the Pythagorean Theorem. I mean geometry!

In order to symbolise travelling from one point to another at a certain speed, Minkowski puts these two points on a graph, one axis of which is the distance connected with time, i.e. as I wrote above (distance=speed x time), and the other axis of which is only the distance, and puts it into the formula by making a small change in the principle that the sum of the squares of the perpendicular sides in a right triangle is equal to the square of the hypotenuse, which is the concept put forward by Einstein, that time shortens as speed increases.

The resulting formula is 

v= speed,

t= duration,

x=distance 

v2 x T2 = v2 x t2 - x2

Here (T) stands for the time travelling at a certain speed.

The same formula in a three-dimensional space:

v2 x T2 = v2 x t2 - x2 - y2 - z2

can be organised as

This new geometry is called Minkowski geometry.

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Of course, Einstein will then develop the theory of "Special Relativity", which he predicted at constant speeds, and present it to the scientific world in November 1915 at the Prussian Academy of Sciences under the name of the theory of "General Relativity".

However, since his teacher Hermann Minkowski passed away in 1909, Albert Einstein, with the help of mathematician David Hilbert, a friend of his, not only his wife Mileva Maric, who was once a student of his, but also his friend (they divorced Milena in 1919 and then Einstein married his second wife Elsa!), finally established the formulas of General Relativity, and the mass and acceleration, which we now call the bending of space-time and which are included in the formulas, and the force that Newton called gravity, disappeared into history.

Yes, contrary to Newtonian physics, general relativity states that there is no such thing as gravity in the universe. 

Albert Einstein says that the objects move towards each other, which we call gravity, which seems to us to be caused by the gravitational force of matter, because of the bending in space and time.

I do not know how accurately the inverse proportionality of time to the square of velocities and the shortening of distances proportional to the square of velocity, which his teacher Minkowski had proposed based on a simple planar geometry, are formulated in three-dimensional space, but Einstein's General Theory of Relativity, which combines mass, acceleration, time and three-dimensional universe, which David Hilbert was able to solve in a very complicated way using Lorenz transformations, seems to have successfully passed the experimental tests to date.

In 1916, Karl Schwarzschild, an astrophysicist, succeeded in making the first exact and non-zero solution of these nonlinear and complex parabolic differential formulae, and this solution is called the Schwarzschild metric.

Einstein put forward the formulae, but the formulae were so complicated that even he could not come up with a solution from his own formulae.

Hidden within the formulas is that there must be a great gravitational attraction in the galaxy centres, which we call "black holes" today and which absorb all matter and energy around them, including light. 

This secret will be revealed in later years using the Schwarzschild metric. Einstein was unaware of the existence of black holes, but he knew that masses must accumulate at a point due to their formulae.

The term black hole was first coined by American astronomer John Wheeler in 1967, years after Einstein's death in 1955. In 1971, the first physical black hole would be discovered.

Today, black holes can even be photographed. In 2019, the picture taken by the Event Horizon Telescope of the black hole at the centre of the M87 galaxy is the first black hole picture.

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Yes, I started this article by asking if you like maths, but the greyhound has extended to black holes.

Some subjects of mathematics can be very complicated. Sometimes you put forward a formula, but even you cannot solve that formula yourself. Even if you are Albert Einstein, there are things you cannot solve.

However, mathematics is a universal language; what you cannot solve, someone else can solve and you continue on your way.

This was the case for Albert Einstein, first his teacher Minkowski helped him, then his wife Mileva, then his friend David Hilbert.

Karl Schwarzschild was the first to solve the formulae he put forward.

Didn't he know maths?

How could he not? He also had a good knowledge of mathematics, but he was first and foremost a theoretical physicist, a thinker. Why should he waste his precious time in the labyrinths of mathematics? He knew how to get help when he was in trouble.

Yes, I don't bother with maths as much as I used to, it is easy for me to open the calculator on my mobile phone even to add two numbers. I have already forgotten many of the formulas I had memorised. Now we do almost everything with the help of computers and maybe our minds don't work as well as the old people.

Nevertheless, maths is beautiful and I still like to watch videos on maths sometimes.

By the way, my mind is still stuck on the fact that Minkowski solved the space-time connection based on a simple geometry, is the space-time connection really that simple? Is there a different connection in reality? 

We call it dark matter, scientists have been searching for it for years, but they can't find it.

Could it be necessary to revise the Minkowski geometry?

Anyway, the article has gone on too long, let me end this article by saying stay with maths today.

Love and respect to everyone from Moscow