We always see this equation, but what is it? What does it state, and why is it important in physics?
Before this equation appeared, physics relied on classical physics (Newton’s laws), which stated that mass is constant, space and time are absolute (unchanging under any circumstance), and energy 1/2mv^2 is given by
However, with the advent of special relativity, these concepts changed, introducing more precise ideas that explained many phenomena Newton’s laws could not, the most important being the constancy of the speed of light. Among the concepts that changed are:
Spacetime (time and space) is relative.
Energy, motion, and mass are two sides of the same coin (the same thing).
Why did Einstein change these laws?
Because he observed that objects moving near the speed of light cannot have their kinetic energy correctly described by classical laws. Also, the mass of an object increases as its speed increases, meaning its energy increases.
Here, Einstein concluded that even stationary objects carry enormous energy, which he called “mass energy.”
An example in the universe:
When two atoms, such as helium and hydrogen, fuse in a reaction like those in stars, the mass after the reaction is less than the mass before. This shows that the lost mass has been converted into thermal energy. In other words, when mass loses energy, it also loses some of its mass, and vice versa.
In short: “mass = rest energy.”
Energy is also related to spacetime, meaning time and space determine the form of interaction between mass and energy (thermal, radiative,...).
As Einstein famously said:
“Spacetime tells energy how to move, and energy tells spacetime how to curve.”
Now the question arises: We can produce energy from mass, but can we convert energy into mass?
The answer is yes. This is what the principle of “pair production” states, where energy transforms into electrons and positrons.
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The Development of the Theory of Relativity: From Special to General Relativity
Albert Einstein arrived at the general theory of relativity through the concept of accelerated coordinates and their associated laws.
But what are accelerated coordinates?
Let us go back in time to the emergence of special relativity. This theory dealt only with inertial frames (where bodies move with constant velocity).
But in our reality, we do not deal only with inertial frames; we also deal with accelerated bodies (such as cars, rockets, and other objects with increasing velocity).
Therefore, the Rindler transformations appeared mathematically to describe accelerated coordinates. They revealed several results for the accelerated observer, which are:
Event Horizon (Rindler Horizon):
The accelerated observer does not see the light cone in the same way as the inertial observer. This means that there are events the accelerated observer cannot see and will never be able to see, no matter how long they wait or how hard they try (in other words, part of the light cone is hidden).
Accessible Region (Rindler Wedge):
These transformations also showed that the path of an accelerated body is not straight but curved. This makes the observer see only a portion of spacetime, which is the accessible region.
After the appearance of these transformations, and also with the equivalence principle, which states that:
“There is no local difference between uniform acceleration and the effect of gravity”
Albert Einstein concluded the following:
Since gravity is the acceleration of bodies toward the center of, for example, the Earth, then gravity is a geometric curvature in the fabric of spacetime.
And here he arrived at the general theory of relativity.
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Imagination will often carry us to worlds that never were. But without it, we go nowhere. Skepticism enables us to distinguish fancy from fact, to test our speculations. The Cosmos is rich beyond measure—in elegant facts, in exquisite interrelationships, in the subtle machinery of awe
-Carl Sagan
Cosmos
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The Principle of Equivalence
The Principle of Equivalence was established in 1907 by Albert Einstein and is considered one of the most fundamental pillars of General Relativity.
Its core idea states that:
“It is impossible to distinguish between the effects of gravity and the effects of acceleration.”
For example, if you were aboard a spaceship in outer space with no windows, and this spaceship had the same acceleration as Earth’s gravitational acceleration (9.8 m/s²), you would not be able to tell whether you were on Earth or in space.
The principle of equivalence is divided into several types:
1. The Weak Equivalence Principle (WEP):
All objects fall with the same acceleration in a gravitational field, regardless of their mass or composition.
Examples of this include Galileo’s experiment and the Apollo 15 experiment by astronaut David Scott. He repeated Galileo’s test under better conditions by dropping a feather and a hammer from the same height. On Earth, the feather falls slower due to air resistance. However, when he repeated the experiment on the Moon—where there is no atmosphere—both objects hit the surface at the same time, as shown on camera.
2. The Local Equivalence Principle:
This is the same example given above: if you are in a spaceship with no windows accelerating at 9.8 m/s², you cannot tell whether you are experiencing Earth’s gravity or acceleration in space.
3. The Strong Equivalence Principle (SEP):
This states that all the laws of physics hold true in the same way, whether in a gravitational field or in an accelerating frame of reference.
One of the most important consequences of this principle concerns light and gravity. For example, if a light beam is emitted on Earth, it bends when passing through a gravitational field. The same bending would occur if the light were in an accelerating frame (or even more if the acceleration were greater).
This led Einstein to predic
t the bending of starlight, which was later confirmed during the 1919 solar eclipse experiment.
This principle paved the way for General Relativity:
Einstein showed that gravity is not a force but rather the result of the curvature of spacetime. Objects move along geodesics in this curved spacetime, rather than being “pulled” as classical physics suggested.
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Hi
In principle, what you are going to read now is a personal confusion.
We all know that black holes, when they die through Hawking radiation and the other theories that modified this idea (such as Leonard Susskind’s theory) and the spreading of information into space, their spacetime returns to its natural state.
But, if we look at the matter from another perspective, such as wormholes connected to black holes,
if these wormholes are stable (supported by exotic matter), they will remain even after the disappearance of the black hole.
More clearly: the spacetime of the black hole (the wormhole) will remain even after the black hole disappears.
Therefore, this spacetime (since it is a curved spacetime) will generate gravity! but without energy or mass!
Only from the exotic matter.
In principle, what you are going to read now is a personal confusion.
We all know that black holes, when they die through Hawking radiation and the other theories that modified this idea (such as Leonard Susskind’s theory) and the spreading of information into space, their spacetime returns to its natural state.
But, if we look at the matter from another perspective, such as wormholes connected to black holes,
if these wormholes are stable (supported by exotic matter), they will remain even after the disappearance of the black hole.
More clearly: the spacetime of the black hole (the wormhole) will remain even after the black hole disappears.
Therefore, this spacetime (since it is a curved spacetime) will generate gravity! but without energy or mass!
Only from the exotic matter.
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For clarification, “exotic matter” is a type of energy, but unlike ordinary energy. It is often described as possessing negative energy.
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English science
Hi In principle, what you are going to read now is a personal confusion. We all know that black holes, when they die through Hawking radiation and the other theories that modified this idea (such as Leonard Susskind’s theory) and the spreading of information…
Oh….
It turns out that the two scientists, "Morris and Thorne", talked about this topic 🗿 and in much more depth — mathematically as well 😂
It turns out that the two scientists, "Morris and Thorne", talked about this topic 🗿 and in much more depth — mathematically as well 😂
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We will never see any living creature in space!
It is impossible to observe any living being outside the Earth, not because Earth is the only place for life—quite the opposite.
This happens because what we see is the past of other planets, or rather “the past of the universe,” so we cannot ever see any living creature beyond Earth.
And if there were living beings elsewhere, they also would not be able to observe us or see us in any way, because what they would see is Earth’s past, not its present.
(Unless they had sent spacecraft to us millions or even billions of years ago, depending on the distance.)
But before such a time, was Earth even in existence? And were there any other creatures on Earth at all?
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Quantum Consciousness
It is a theory that mixes neuroscience and physics, developed by physicist Roger Penrose and neuroscientist Stuart Hameroff (who is an anesthesiology specialist).
Its main principle is:
Human consciousness does not arise from classical interactions between neurons,
but is the result of quantum processes occurring inside the microtubules within neurons.
“Perception is linked to fundamental physical processes in spacetime itself.”
To explain:
In quantum mechanics, particles like electrons can exist in multiple states simultaneously, until they are observed or measured, at which point they collapse into a single state. (This is called quantum superposition.)
Penrose’s view was as follows:
He rejected the idea that human consciousness causes the collapse of superposition, and added that the collapse does not happen because of consciousness or observation but occurs due to the structure of spacetime.
This is because, in a superposition state, a particle exists in two different places, which causes two curvatures in different locations in spacetime. And since spacetime cannot sustain this “split between two contradictory forms” for a long period, it leads to the collapse of superposition “as if spacetime itself chooses one of the states.”
Then Hameroff connected this to human consciousness:
Penrose previously suggested that these collapses might be related to human consciousness.
Hameroff was the one who proposed this hypothesis in more depth, stating that human consciousness arises from quantum collapses within the brain, specifically inside the microtubules.
Sally
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Coordination between Free Motion and General Relativity
Free motion is fundamentally the motion of bodies unless acted upon by external forces.
In classical physics (Newtonian physics), free motion is described by Newton’s first law (the law of inertia): a body at rest remains at rest, and a body in motion continues to move at a constant speed along a straight line. With gravity, free motion also follows the second law of acceleration.
In Newton’s framework, gravity was considered a force, so the laws of motion applied to it. Consequently, objects in free fall under Earth’s gravity were not considered to be in free motion, as they were subjected to a force.
Einstein’s perspective differs fundamentally. Gravity arises from the curvature of spacetime, rather than as a conventional force. Therefore, according to Einstein, objects in free fall are indeed in a state of free motion, contrary to Newton’s concept.
Einstein also emphasized that objects in free fall move along geodesic paths in spacetime. A geodesic path represents the natural, shortest path in spacetime.
Regarding the perception of weight: in Newtonian physics, weight results from the gravitational force acting on a body. In Einstein’s view, weight arises from resisting free motion.
For example, if you were in an elevator and its cable snapped, you would experience free fall — not because gravity is absent, but because you are moving along a geodesic in spacetime. Conversely, if you stand still on the ground, your body naturally wants to continue in free fall, but the Earth pushes upward against you (a normal reaction force).
In other words:
According to Newton, weight is the pull of gravity dragging you downward.
According to Einstein, weight is the push of the Earth’s surface preventing you from moving along a geodesic path.
Sally
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Linear Algebra
Linear algebra is one of the fundamental branches of mathematics, focusing on linear transformations of vectors as mathematical entities that have both magnitude and direction, as well as vector spaces, since they contain vectors.
Magnitude is the length of the vector.
Direction is the orientation of the vector in space.
(Magnitude can represent quantities such as speed or force.)
(Direction can represent the movement of objects, for example, the direction of the wind.)
There are several operations on vectors:
Addition: When two vectors are added, their direction and magnitude may differ, resulting in a new vector.
Scalar Multiplication: A vector can be multiplied by a number, changing its length (increasing or decreasing) and potentially its direction.
Linear Transformations:
Rotation: For example, rotating a vector 60° around a point, though some transformations (such as projection) may partially change its length or direction.
Scaling: Adding one vector to another increases the length but preserves the direction.
Projection onto a line or plane: In other words, it preserves the fundamental structure of the vector.
After that, matrices are used, as they help simplify algebraic operations. For example, if there are multiple equations, the results and numbers can be organized into matrices to facilitate finding solutions and values, acting like a compact computational tool.
Various methods can then be applied to matrices, such as matrix inversion or Gaussian elimination, allowing solutions to be found even for thousands of equations with thousands of unknowns.
There are also eigenvalues and eigenvectors. Here, the length of a vector may change (increase or decrease) while preserving its direction. These are called eigenvectors, while eigenvalues are the numbers that scale the vector’s length.
Sally
Part 1
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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.
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.
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.)
Sally
Part 2
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
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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
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