Chapter 309 Bruce Field Equation! One solution to one universe!
Li Qiwei proved that the essence of gravity is the curvature of space and time through a pure thought experiment, the disk experiment.
Following this, he needed to describe the nature of the curvature of space and time.
How is space and time curved?
What is the degree of bending?
etc.
And these require mathematical knowledge, especially geometry.
From this point on, it is also the most difficult part to understand the general theory of relativity.
Mathematics kills people!
In the previous chapter, Ridgway has demonstrated that if a disk in space rotates, it will no longer be in flat space-time.
At this time, the pi of the circle is greater than π.
In real history, Einstein ran into trouble at this point.
As we all know, Einstein’s mathematical skills were not very good.
Because the physics community at that time almost only had access to Euclidean geometry.
That is the straight space-time geometry that we are most familiar with.
Because this geometric form is very consistent with daily experience.
Many experimental measurements in physics use the method of Euclidean geometry.
Therefore, physicists who are not good at mathematics will definitely not study other geometries specifically.
So what is Euclidean geometry, and why can’t it deal with the curvature of space-time.
Long before Newton, scientists in ancient Greece had conducted in-depth research on space.
Among them, mathematicians easily believe that space is flat based on empirical intuition.
That is, the three-dimensional space is like an infinitely long straight line.
Based on this experience, the great ancient Greek mathematician Euclid first defined the concepts of points, lines, and planes, and then proposed five major axioms.
The so-called axioms are self-evident and are summarized from the universe, just like the apocalypse.
First: There is and is only one straight line connecting any two points.
Second: Any finite straight line can be extended infinitely.
Third: With any point as the center and any length as the radius, a circle can be drawn.
Fourth: All right angles are equal.
Fifth: Two straight lines are intercepted by a third straight line. If the sum of two interior angles on the same side is less than two right angles, the two straight lines will intersect on that side.
(or: passing a point outside the straight line, only a straight line parallel to the known straight line can be drawn)
(That is, parallel lines do not intersect)
Using these five axioms, Euclid conducted logically rigorous mathematical deductions, derived 23 theorems, and solved 467 propositions.
This created a shocking geometric building, also known as "Euclidean geometry".
Euclide himself is revered as the "Father of Geometry".
Euclidean geometry has dominated the mathematical world for more than two thousand years since its creation.
Newton, Descartes and others all invented more and more profound mathematical theories based on it.
For thousands of years, not only mathematicians, but also physicists have believed that Euclidean geometry is perfect.
Especially its application in the field of physics is very consistent with the phenomena of the objective real world.
Therefore, physicists are convinced that space is flat and evenly distributed.
Although the special theory of relativity denies the absolute nature of space, it does not deny that space is flat.
Otherwise, there will be more people criticizing Ridgway.
However, in addition to physics that is constantly moving forward, mathematics is also constantly moving forward.
Geniuses and big names in mathematics are no weaker than physicists.
There are also super geniuses in the world of mathematics who are rare to see in a century or a thousand years.
Even from a certain perspective, mathematicians can be considered "smarter" than physicists.
Of course, this refers to the top existences in both fields.
Soon, the Russian mathematician Lobachevsky discovered that things were not that simple.
There is a problem with the fifth axiom of Euclidean geometry!
In 1826, he published a completely new geometric system.
In Lobachevsky's theory, he inherited the first four axioms of Euclidean geometry.
But the fifth axiom, he described it like this:
Through a point outside the straight line, at least two straight lines can be drawn parallel to it.
Based on these five axioms, Lobachevsky discovered that a series of geometric propositions can be derived logically and self-consistently.
From this he obtained a new geometric system.
Later it was called "Roche geometry".
The difference between Roche geometry and Euclidean geometry lies in the expression of the fifth axiom.
Later we learned that Roche geometry actually describes hyperbolic geometry, and its curvature is negative. (The shape of a saddle)
In Roche geometry, the sum of the interior angles of a triangle is no longer equal to 180°, but less than 180°.
It can be said that Roche's geometry caused a huge sensation in the mathematical community when it was published.
Instead of being excited, everyone criticized Lobachevsky's theory as fallacies and nonsense.
Even Gauss, the absolute king in the field of mathematics, remained silent on this and did not acknowledge Roche's geometry.
But Riemann, a student of Gauss, seriously analyzed Roche geometry.
He feels that this axiom system is of great research significance.
Because he perfectly inherited the logical reasoning system of Euclidean geometry.
As long as the fifth axiom of Roche's geometry is recognized, then those incredible conclusions will be the correct results under this geometric system.
However, Riemann was not satisfied with this.
Based on Roche's geometry, he developed another geometry, namely spherical geometry.
On the surface of a sphere, parallel lines cannot be drawn through a point outside the straight line.
And the sum of the interior angles of a triangle on a sphere is greater than 180°.
This is what later became "Riemannian geometry".
Roche geometry and Riemannian geometry are both non-Euclidean geometries. The difference is that the former has negative curvature (space is concave inward) and the latter has positive curvature (space is convex outward).
Euclidean geometry has zero curvature, so space is flat.
Riemann published his new geometric system in 1854.
At that time, like Roche geometry, almost no one could understand Riemannian geometry.
Because it's so counterintuitive.
But after Einstein learned about Riemannian geometry on Grossman's recommendation, he was as happy as meeting his cousin.
Because his space-time curvature theory happens to be suitable for Riemannian geometry.
Now that his theory had a solid mathematical foundation, Einstein used the metric tensor invented by Riemann to study the curvature of space-time.
The so-called metric tensor can be roughly understood as it describes the properties of space and characterizes the geometric structure of space.
Based on this concept, data such as the geodesic in Riemannian geometry (the line with the shortest distance between two points in Riemannian geometry) can be calculated.
Curvature can be calculated based on geodesics, which is the trajectory of matter in space.
Light also takes this path.
At this point, the mathematical model of the space-time structure of general relativity can begin to be constructed.
Now, Li Qiwei’s mathematical level is much better than that of Einstein.
For future doctoral students in physics, mathematics is also a required course.
Riemannian geometry is even more famous. He studied it a lot in his previous life, and now he can finally put it to use.
Now, there is a mathematical treatment for the curvature of space-time. The next step is simple, and that is to study how different materials bend space.
For example, the density, mass, energy, etc. of matter cause the curvature of space-time.
Crack!
Li Qiwei worked on the paper for half an hour.
An equation was finally written out by him.
This is the famous gravitational field equation, also called Einstein's field equation.
It’s just that now, it has to be renamed [Bruce Field Equation].
This equation looks like this:
The formula on the left represents the curvature of space-time, and the formula on the right represents the distribution of matter.
The text version of this formula is: matter tells space-time how to bend, and space-time tells matter how to move.
This equation seems simple, but it is actually very complicated. (See comment area)
This is a second-order nonlinear partial differential equation containing ten unknown quantities.
The sentence fragments are: second order, nonlinear, (partial) differential, equation.
Don't worry, we will analyze it bit by bit to let you understand what is difficult about the equation.
[Equation]
First of all, everyone knows what an equation is.
x+1=2.
This is the most common and simple equation.
[Partial Differential]
The differential equation is an equation based on ordinary equations with unknown functions and their derivatives in the formula.
For example, assuming that u is a function of x, it can be expressed as u=f(x), and u′ is the derivative of u with respect to x.
Then x＋u＋u′=1, this equation is called a differential equation. (u′ must be present in the equation, but u may not be present)
If there is only one derivative of the independent variable in the differential equation, it is called an ordinary differential equation.
For example, the above formula has only one independent variable, x, and only u′, the derivative of the independent variable x. It is an ordinary differential equation.
And if u is not only a function of x, it is also a function of y, then u=f(x, y).
u′(x) is the derivative of u with respect to x, which is called a partial derivative;
Similarly, u′(y) is the derivative of u with respect to y.
Then x+y+u′(x)+u′(y)=1, this equation contains two or more derivatives.
This kind of differential equation is called a partial differential equation.
[Second order]
The order refers to the order of the derivative. For example, u′ is the first-order derivative, and u″ is the second-order derivative, that is, the derivative is then derived.
< br>x＋y＋u′(x)+u′(y)+u″(x)+u″(y)=1.
This equation is a second-order partial differential equation.
[Nonlinear]
Linear and nonlinear are easier to understand.
If the functional relationship between u, x, and y is a straight line, it means linearity.
If it is a non-linear line, it means non-linearity.
At this point, the concept of the Bruce field equation, a second-order nonlinear partial differential equation, is understood.
It can be seen that if you want to find the exact solution of this equation, it is a very complicated matter.
Without any skills, we can only solve it violently.
That is, taking all variables into consideration.
Such as mass, energy, density, space, time, etc.
So, without the supercomputers of later generations, it is understandably difficult to tear this equation apart by hand.
Even with the help of computers, it is not easy to solve.
Even the simplest movement between two celestial bodies.
If you consider the properties of general relativity, there will be no way to simulate its precise space-time relationship until later generations.
In real history, the exact solution given by Schwarzschild was actually the simplest one, taking into account the fewest variables.
He postulated that there is only one particle in the entire universe.
Although the Bruce field equation cannot be solved exactly, it can be solved approximately through mathematical means.
For example, the famous problem of Mercury's perihelion precession is answered using approximate solutions, thus providing a perfect explanation.
The Bruce field equation has too rich connotations.
Each exact solution to this equation represents a different universe.
And it is the kind of universe that is constantly evolving from the past to the future.
Because there is a parameter of time t in the field equation, the equation will continue to change with time.
This also represents that the universe is constantly moving and changing.
Later generations often talk about the possibility of going back to the past, which actually refers to a specific solution to the field equation.
The solution to the Bruce equation is a specialized subject.
All the relationships between space, time and matter in the universe are encompassed by this equation.
call!
Li Qiwei exhaled heavily.
At this point, the contents of the general theory of relativity are all completed.
However, the paper is not over yet.
Because many incredible conclusions can be derived based on this field equation.
And these conclusions, Li Qiwei will all be attached to the paper as his prediction on the day it is published.
He put all the prophecies of future generations together, and you can imagine the shock to everyone.
However, the wildness of general relativity makes it very difficult to prove it.
In real history, in the early stage, in chronological order, there are three most important pieces of evidence.
The first one is the problem of Mercury's perihelion precession, which can be perfectly explained by the Bruce field equation.
But there is a drawback to this proof.
That is, if other people insist on using the law of universal gravitation to calculate, they will take into account various factors such as the rotation of the sun.
It's entirely possible that it could also be responsible for Mercury's odd behavior.
At least you can't prove this conjecture wrong.
Thus, the first proof is slightly weaker.
The second one is the famous starlight bending.
That is, through the total solar eclipse experiment, Eddington proved that the path of light will bend after passing through the sun.
This evidence strongly proves the correctness of general relativity and elevates the theory to god status.
The third one is the phenomenon of gravitational red shift.
According to the derivation of general relativity, the wavelength of light will become longer after leaving the gravitational field. (It’s a bit more complicated, so I won’t go into details yet)
Therefore, the position of light on the spectrum is closer to the direction of red light, which is called red shift.
This inference would not be proven by a very, very subtle experiment until around 1950.
Li Qiwei looked at the first draft of the paper in his hand and was filled with emotion.
Special relativity unifies time and space, and space and time are inherently integrated.
The general theory of relativity unifies the interaction between space-time and matter.
The approximation of special relativity is Newton's three laws of mechanics.
The approximation of general relativity will lead to the law of universal gravitation.
Litchwick’s theory of relativity completely incorporated Newtonian mechanics into it.
(End of this chapter)