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The Birth in Minkowski Space

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How is the curvature in general relativity working

It would be better to read the initial part of this article,if you haven’t read An Introduction to General Relativity


In the 4 dimensional Minkowski space,the shortest distance between two points are always straight lines,just like Euclidean. But the square of their values could be possibly negative,zero or positive.


It was calculated that if a massive particle or energy field,while travelling in this space,will try to choose the minimum distant path,for which Minkowski have already derived an equation. But Einstein was not satisfied. He felt something missing.


He already had found that,light would curve its path in strong gravitational fields. So he set out to make some changes in the Minkowski Formulations. He used Tensors,in with Riemann geometrical aspects. He derived a new equation,made of tensors,in which the minimum distance between two points in a gravitationally strong space was found to be curved. Now the problem was solved. He could now compare why objects and Light curves before gravity. They are just following the shape of Space.

How the minimum distance between two points could ever be curved?

Consider two points on the surface of a sphere,it would be a curve,never a straight line If you want to get a straight line,shortest distance between two points on the surface of a sphere,you would have to drill through the sphere. These curved lines or curved paths are called as Geodesics

Captains’ of the Ship use such paths,while travelling through the sea,even though found to be straight in the ocean,they are actually curved.


You could even consider the distance between two points on the surface of a cylinder,they too would be curved,excluding the parallel axial paths.


Through this,findings Einstein modified his Principle of Equivalence and stated that the curvature in space is equivalent to the presence of matter or conversely,presence of matter curves the space. This is the curvature which everyone hears,when ever they talks about or hears about General Relativity.


J.A.Wheeler
 have stated this in a beautiful phrase “Spacetime tells matter how to move,while mass tells Spacetime how to curve”


To be more technical,Einstein used energy and momentum inside a single tensor form,namely the Energy-Momentum Tensor;which relates it to the curvature of spacetime. Which also,indirectly stated that,this curvature in spacetime,represented by the points(events) also must be a tensor,which is dependent on the coordinates chosen. However,Einstein formulated the general equation connecting the matter and the space around it. He unified the Space and Matter : The birth of General relativity,in Minkowski Space


It took some more years for Karl Schwarzchild to find the solution to Einstein’s equation,that too only for the simplest symmetrically easier spherical model.


From that,he was able to state the existence of points,in space,at particular situations. Such points,arising from the solution created big consequences in the general theory at first.


In a point,the curvature was found to be infinity. That was against logic. Infinite curvature means infinitely curving to maximum,which means a fall. Infinite curvature is a fall to a pit or a hole found near. This hole is found in space. If anything comes near by,instead of going through a curved path,the object just falls. This infinite curvature,hole doesn’t seem to allow anything to climb back,because of the properties of space and mass.


When ever we are approaching a curve,consider we are in car; we will soon be in the normal path,soon after the curve has finished. But this infinite curve would never finish,and we would never be in the normal route. We will be curving and curving alone forever. Sometimes,this could be a bad interpretation of the space,however we don’t know it yet. 


This hole,as always invisible to every known radiations (which means,this hole makes every known radiation to fall into it,and never reflect to make us see),we named it as Black ,the absence of light : A Black Hole.


However this was what we get when we approach this spacetime properties through Classical View. Approaching General Relativity through Quantum Physics,one could find that these blackholes fitting to certain criteria would emit certain radiations. This was proved by Stephen Hawking,hence named after him : Hawking radiation. We haven’t observed this yet,hope we will,in future.


Its not late to read the basics,even if you know it already An Introduction to General Relativity

Complete Article on LetsPhysics

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