The Twisted Space-Part 2

So, in my last post I asked some questions for you to ponder upon, if you missed it you can scroll down and check the post Twisted Space Part 1 (you might need it ).
You may be thinking  that we might need some  3-d figure  and then maybe we can see for the discrepancies in its actual measured quantities and theoretically calculated quantities ( such as surface area) like we did for the triangle and the circle. If this was your thought process then you are on the right track. Let's do some analysis to narrow down our search for this 3-d figure. The thing with 3-d curvature is that it will have different values for different orientations i.e you might find a different curvature of space if you measure it in vertical direction and then maybe in horizontal direction. The curvature is said to have components along different orientations, just like a straight line in Cartesian plane having different components along the coordinate axes (obviously depending upon the angle). So it is advised to take a uniform geometry that can give us the average value of curvature in that region of space. That narrows down our search to 1 common 3-d figure- lo and behold........THE SPHERE! ........hey but wait, why are we even doing this, I mean why will  space even  have a curvature in the first place? The answer to this was given by einstien in order to explain gravity.the idea behind this is that any mass placed in space creates a curvature, and with curvature comes a gradient, and like any other phenomenon( like osmosis) a mass would like to go from a higher gradient to a lower gradient, hence by this notion a lighter object (which causes less curvature) would like to move towards an object of greater mass ( resulting in greater curvature).Here is a classic representation of curvature of space visualised in 2-d

                                   
But to give you a true idea of what a curvature in 3-dimensions would look like here is a nice visualisation which shows a concentration of "grids" around a mass, the grid here represents space.

                                 

Pretty interesting, isn't it?With the above facts known, it is obvious to think that a larger mass will have a larger curvature and in turn the deviation from Euclidean geometry will also be more. So this implies that the calculated radius from measuring the exact surface area will be much different from the actual measured radius. To quantify this fact einstien also gave the following formula:
Here r(meas) means the measured radius
M is the mass of the object( due to which there is a curvature)
A is the exact measured surface area
G and c have their usual meaning.

The Twisted Space-Part 1

Almost everyone has come across the  terms like space time curvature,gravitational lensing,black holes, singularities, etc etc. But what do they really mean? Are they just some sophisticated gibberish?

( ha see what I did there.....sophisticated gibberish... Oxymoron( if you didn't get it)) Or do they have some deep meaning that may even alter our perspective about reality. Let's explore !

Let's start with a classical analogy( it's good to have classical analogies as we can easily conceive it, human mind finds it easy to to recreate something that it might have experienced). Have you ever observed the longitudes of earth intersecting with the equator. You might have not observed it but they intersect the equator at right angles and still manage to intersect at the poles. This clearly means that the triangle formed by two  longitudes with the equator have a sum of angles greater than 180 degrees! And in fact if you measure the angle between two longitudes that intersect at right angles at the poles also, you will find that the sum of angles formed by these 2 longitudes and the equator is 270 degrees. Astonishing, isn't it?

But you might say, the lines are not even straight lines, rather they are curves, how can we even expect them to follow Euclidean geometry? Well that's because you are seeing it from a 3 dimensional perspective I.e we know that the surface is spherical (not straight), but what if we put some very intelligent bugs(who know Euclidean geometry) on such a spherical surface who can only perceive in 2 dimensions. When these bugs start drawing straight lines, in the way illustrated above they find that although their triangle looks like any other 2- dimensional figure( in our sense) still it doesn't follow some of the basic rules of geometry( the angle sum property) given by the Euclid of bugs ( of course there might have been a Euclid bug who introduced this geometry in their world......after all the bugs are super intelligent). Similarly the area of a circle won't be pi*r^2 where r is the measured radius of the circle. In fact it will be greater than pi*r^2.

Well that description was for geometries on a sphere, what about something with the opposite curvature like the surface of a saddle? Obviously the effects will be opposite. The angle sum of the triangle will be less than 180 degrees and similarly the area of circle will also be less than pi*r^2.

So far so good. We proved most of euclid's work (which he assumed was always true) wrong...in a few minutes....so yeah we are doing pretty good I guess.

Time to up the ante. What if our own 3-d space is distorted? What if we are the bugs in this experiment now? How are we supposed to measure the curvature of 3-d space, it was pretty easy in 2-d right? 

.....to be continued.

I will give you the opportunity to entertain yourself with these questions, till my next post.

Stay Tuned for part 2.