The Main Foundations in Algebra

1_Vectors, vector spaces, eigenvalues, and eigenvectors – we talked about them in Part 1.

2_Matrices: They are tables of geometric transformations and systems of equations, and they are very concise methods to simplify complex calculations.

3_Linear transformations: In short (because they were explained earlier), they are functions that move vectors from one space to another without changing the basic structure of the vector (such as rotation, stretching, reflection, and projection onto a surface or a line).

4_Determinants: They are numbers associated with a matrix that help to know whether the system can be solved or the transformation is invertible.

5_Rank: It is the number of dimensions in the matrix (in mathematics one can deal with thousands of dimensions). The rank also represents the power of the matrix in holding information: the higher the rank, the more information the matrix carries, and the lower the rank, the less information it carries.

6_There are two types of space:

Null space: It is the set of vectors that, when entered into a matrix, the result equals zero. It is also called the “hidden vectors,” meaning these vectors exist but when transformed the result is zero.

Column space: It is different from the null space but not exactly its opposite. Each column represents a certain vector and real outputs of the matrix appear.
(The null space deals with special inputs, while the column space deals with possible outputs.)

7_High dimensions: As mentioned earlier, algebra does not deal only with ordinary dimensions (such as two, three, or four), but it deals even with very large numbers of dimensions, even if it reaches 1000 dimensions. (It has applications in statistics, artificial intelligence, and others.)

In short, algebra has a very great importance in our life, whether in geometry, physics, or other fields.

Sally
Part 2

Schwarzschild!✨

Schwarzschild Solutions

At first, Albert Einstein’s field equations (the equations of general relativity) were very complicated, but during World War I, on the other hand, the scientist Schwarzschild was trying to find a solution to these equations, and indeed he managed to solve them in the case of a point mass.

The solution was (the spacetime line element):

ds^2=-(1-2GM/(c^2r))*c^2dt^2+(1-2GM/(c^2r))^-1dr^2+r^2dθ^2+r^2sin^2(θ)dφ^2

Where:
M = mass of the central body
G = gravitational constant
c = speed of light
(t,r,\theta,\phi) = coordinates
which are (time, radius, angle, azimuthal angle).

And from it, several things were concluded:

1_ Any body that compresses and crosses the Schwarzschild radius becomes a black hole. This radius is (r_s=2GM/c^2)

2_ Singularity: the point at r=0, where density and curvature become infinite.

3_ Event horizon: r = r_s, beyond which no object can escape, not even light.

4_ Bending of light: (the eclipse experiment proved this).

5_ Escape velocity does not apply beyond the event horizon (because there are no outward trajectories inside a black hole).

6_ Each time we approach objects such as black holes (that cross the Schwarzschild radius), time will slow down relative to another observer (for example, on Earth).

There are other solutions that describe the slowing of time and the stretching of distances (I cannot write them).

Sally

“I find many adults are put off when young children pose scientific questions. Why is the Moon round? … What is a dream? How deep can you dig a hole? When is the world’s birthday? Why do we have toes? … A few more experiences like it, and another child has been lost to science.”

Carl Sagan

Quantum Eraser Experiment

It is a developmental version of the double-slit experiment, which showed that elementary particles such as electrons and photons can behave as both waves and particles.

Steps of the experiment:
1_ A single photon is emitted toward a barrier with two slits.
2_ Behind the slits, there is a screen that records where the photon lands.
3_ When we do not observe which slit the photon passes through, an interference pattern appears — a wave-like behavior.
4_ When we do observe which slit the photon passes through, the interference pattern disappears, and a particle-like pattern appears.

Now, here comes the difference in the experiment:

5_ After the photon passes through, we can remove or erase the information that tells us which slit it went .
6_ When this information is erased — even after the photon has passed — the interference pattern reappears,
as if the universe somehow knows that we no longer know the path!

Result

:

Whether

we

know

or

do

not

know

the

path

changes

the

behavior

of

the photon — even after the event has already occurred.

It’s

as

if

consciousness

of

the

information itself plays a role in determining what actually
happens.

Sally

Problems of Wavefunction Collapse and Scientists’ Interpretations

Firstly, what is the wavefunction?
The wavefunction in quantum mechanics is used to describe the probability of finding particles in a certain position or state.

And the Measurement Principle?
In quantum mechanics, a particle before being observed or measured is in a state of quantum superposition — meaning the particle exists in several states at once, until it is observed or measured. Then, all probabilities disappear (the wavefunction collapses), and only one possibility remains.

Here lies the problem:

Where did all those other probabilities go?

Many scientists have offered interpretations (such as Copenhagen, Hugh Everett, and Bohm) as well as the decoherence interpretation.

For example, the Copenhagen interpretation explains that there is no real state before measurement, only probabilities. (This is the most famous interpretation.)

Hugh Everett, whose interpretation is called the Many-Words Interpretation, proposed that there is no collapse at all — instead, when measurement occurs, the universe splits into multiple branches, and each possibility goes into a different universe. We simply live in one of these branches.

David Bohm suggested that the particle already has a definite position, and the wavefunction is only a guiding wave. Therefore, there is no real collapse, just hidden evolution.

There are also the consciousness-based interpretations, proposed by von Neumann and Wigner, suggesting that human consciousness itself causes the collapse upon observation.
This is considered a philosophical interpretation.

Sally

Juan Martin Maldacena!✨

Feynman’s famous quote:

“If you think you understand quantum mechanics, you don’t understand quantum mechanics.”

What does it mean? And does it literally imply that no one can understand quantum mechanics?
The answer is certainly no.

It’s not meant literally that no one can understand quantum mechanics, but rather it was an illustration of the nature of quantum mechanics — a clarification of its probabilistic essence and theories.
He presented two possibilities for understanding quantum mechanics, or two “states” of awareness:
(If you think you understand, then you don’t understand).

The probability of your understanding quantum mechanics also implies the probability that you do not understand it.

For example, in one of the quantum theories — quantum superposition — in Schrödinger’s cat thought experiment, before the wave function collapses, the cat is both alive and dead.
Here, the cat falls into the realm of probability, which is the core of quantum mechanics.

As another example, John Wheeler once said in relativity:

“Spacetime tells matter how to move; matter tells spacetime how to curve.”

This, too, was meant to explain relativity but in a simplified statement — just like Feynman’s quote above.

Sally

Gravitational Waves

They are disturbances that occur in the fabric of spacetime as a result of asymmetric acceleration of the distribution of mass and energy, as predicted by general relativity. They appear most clearly during the motion of extremely massive objects or their collisions with one another.

For example, if we have a piece of fabric and we throw a ball or any other particle that carries mass onto it, we will see that a curvature forms in the region where the mass falls, but the remaining parts of the fabric will experience ripples as a result of the large motion that occurred in a specific region of the fabric.

This example illustrates the waves in a simplified way. However, to explain them in a more scientific manner: all bodies or regions that contain (mass, density, energy) will create curvature in the fabric of spacetime. When an object with a very large mass moves, these waves will appear, or during supernova explosions and others.

(This does not mean that bodies or regions with small mass do not generate gravitational waves. Waves are produced by all bodies, but the smaller the mass, the weaker the gravitational waves. This means that they do have waves, but they are short-lived and cannot be detected.)

Now comes the question: how do we know about the existence of gravitational waves? Or how do we know if a movement of massive objects has occurred or if a supernova has taken place?
Their existence is detected through specialized instruments, but how do we predict their existence or describe them theoretically?
Here comes the role of mathematics, specifically “tensors,” which we previously discussed in this research:
https://t.me/Science_40/23

Sally

Young’s double-slit experiment

It is an experiment that proves whether light behaves as a particle or a wave

The experiment is:
Placing a barrier with two narrow slits

And shining light on them (a laser, because it is monochromatic)

And placing a screen behind the barrier to observe the result through it

If light behaves as a particle, it will appear as only two bright lines

But the result showed an interference pattern (bright and dark bands, alternating)

And this indicates that light behaves as a wave, because waves undergo constructive interference or destructive interference

The physical explanation of this case is that each slit becomes a secondary wave source, and the two waves superpose and propagate, and according to the path of the two waves it is determined whether constructive or destructive interference occurs

(In the twentieth century this experiment was repeated but with electrons)

Sally