# Division by zero and Black Holes

Discussion in 'Mathematics' started by AVDBA, Sep 21, 2019.

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1. ### AVDBANew commenter

First image of a black hole, using Event Horizon Telescope observations of
the center of galaxy M87. The Event Horizon Telescope Collaboration.
Source: https://eventhorizontelescope.org

Pythagoras explicitly started to make the concept of distance in space a common notion. One writes two points on a Cartesian plane and the distance between the two points is obtained by using the Pythagoras Theorem. This is so, because Pythagoras Theorem is valid for a flat plane. For analytically working with these points, i.e., for using numbers, these points must be labeled, named. This necessity requires the use of coordinates, which are labels for these points. Using Cartesian coordinates x and y, one writes a point P - having the coordinates P(x, y) = (x, y). Using polar coordinates r and a, the same point P has coordinates P(r, a) = (rcos(a), rsin(a)). In our 3-dimensional quotidian space, a point P(x, y, z) = (x, y, z) is written in Cartesian Coordinates, and the same point P(r, a, b) = (rsin(b)cos(a), rsin(b)sin(a), rcos(b)) is written in spherical coordinates. A point at the north pole of the r-sphere has
r = r, b = 0, and a myriad of possible values for a - whereas the same point labeled by Cartesian
coordinates is represented by 3 well defined values (labels), namely by P(x, y, z) = (0, 0, r). The point cannot be represented by well defined values in spherical coordinates at the pole. There is nothing special with the pole. The spherical labeling system is not a good system for labeling the totality of points on a sphere.

The Theory of Gravitation developed by Albert Einstein, the General Theory of Relativity, brings the concept of spacetime defining its structure by means of the intricacy with matter and energy - since matter and energy defines the geometric shape of spacetime, which is a space that needs 4 labels for its points (the spatial positions x, y, and z, as well as time t - with no formal separation of this structure, the reason we have written spacetime instead of space-time). As well as described above to the more geometrically simple case of our 3-dimensional quotidian space, one also needs to label the points of spacetime, now with the need of 4 coordinates. At choosing a set of coordinates for labeling a point, it can be the case that one encounters limitation of labeling the spacetime at some location(s), as we have similarly described for the case of spherical coordinates. When one defines a given spherical distribution of mass and performs the calculations according to the General Theory of Relativity, the spacetime arena is described by a rule that defines the distance between 2 points belonging this arena, but, now, the rule is not the same that is given by the Pythagoras Theorem. A term of this rule has the factor 1/(1 - C/r). This is problematic, since, when r = C, one deals with a division by zero. C defines the location of the so-called event horizon of a Black Hole. This occurred by virtue of a limitation of labeling the points of spacetime by using this set of coordinates when labeling such points pertaining to the event horizon. This is a problem with the chosen coordinates, not with the spacetime. By using different coordinates, the points pertaining to the event horizon are labeled fine, similarly to the previous example where the pole was well represented by the Cartesian labels instead of spherical labels. Labels and coordinates are the same thing. Different labeling systematization, i.e., how the labels are being implemented, mean different systems of coordinate. Some systems cannot deal with the entire space, they have limitation for labeling. So, nothing special has to occur with something crossing the event horizon - this is just a region that is defined by points that are forbidden to be labeled by that coordinates that were forbidden to label these points by virtue of generating a division by zero. A different system of coordinates, a different labeling systematization that does not numerically complain to the duty of describing the points of the event horizon by means of numbering them does not detect any problem with this region. But, when there is a mathematical problem with a point of the spacetime that persists even when different labels are used, there exists a serious condition being reached. This occurs for the case of a spherical distribution at r = 0. In this case, this location is called a Genuine Singularity. Again, the problem is by virtue of division by zero. But now, the problem is due to the spacetime itself, not with a particular system of labels, since the problem occurs with any labeling systematization.

A division by zero may also mean serious conditions in the physical world, as the one one encounters within a Black Hole.

Is he doing this for a bet?

3. ### DMNew commenter

TES will be pleased. This is the most activity this forum has had for about five years,

Indeed, I can remember when Maths used to be quite an active forum. Now TES seems to be mainly used to argue about Brexit or post pictures of cats on Personal.

Someone needs to tell Getling that they disagree with his choice of best calculator. That'll wake things up a bit.

blue451 likes this.

I rather miss Getling's posts. He could be obnoxious but at least you were left in no doubt as to his opinion.

7. ### DMNew commenter

Is he still around? I'd forgotten about him. He was everywhere at one point.