1900: A physics genius wandering around Europe

Chapter 309 Bruce Field Equations! One Solution, One Universe!

Through a pure thought experiment, the disk experiment, Ridgwell proved that the essence of gravity is the curvature of space-time.

Next, he needed to describe the properties of the curvature of space-time.

How exactly is space-time curved?

What is the degree of curvature?

and many more.

All of this requires mathematical knowledge, especially knowledge of geometry.

This is where the most difficult part of general relativity to understand begins.

Mathematics is fatal!

In the previous chapter, Ridgway has demonstrated that if a disk in space rotates, it is no longer in flat space-time.

At this time, the circle's pi is greater than π.

In real history, Einstein was troubled at this point.

As we all know, Einstein's math skills were not very good.

Because at that time, the physics community could only access Euclidean geometry.

This is the flat space-time geometry that we are most familiar with.

Because this geometric form is very consistent with everyday experience.

Many experimental measurements in physics use Euclidean geometry methods.

Therefore, physicists who are not good at mathematics will definitely not specialize in studying other geometries.

So what is Euclidean geometry, and why can't it deal with the curvature of space-time?

Long before Newton, ancient Greek scientists conducted in-depth research on space.

Among them, mathematicians easily believe that space is flat based on their empirical intuition.

That is to say, three-dimensional space is like composed of infinitely long straight lines.

Based on this experience, Euclid, the great ancient Greek mathematician, first defined the concepts of point, line, and plane, and then proposed five axioms.

The so-called axioms are self-evident and are summarized from the universe, like revelation.

First: There is only one straight line connecting any two points.

Second: Any finite straight line can be extended infinitely.

Third: You can make a circle with any point as the center and any length as the radius.

Fourth: All right angles are equal.

Fifth: If two straight lines are intersected by a third straight line, and the sum of the two interior angles on the same side is less than two right angles, then the two straight lines will intersect on that side.

(Or: Through a point outside a straight line, only one straight line can be drawn parallel to the known straight line)

(i.e. parallel lines do not intersect)

Euclid used these five axioms to carry out logically rigorous mathematical deductions, derive 23 theorems, and solve 467 propositions.

This built a stunning geometric edifice, also known as "Euclidean geometry".

Euclid himself is honored as the "Father of Geometry".

Since its creation, Euclidean geometry has dominated the mathematical world for more than two thousand years.

Newton, Descartes and others invented more profound mathematical theories based on it.

For thousands of years, not only mathematicians, but even physicists believed that Euclidean geometry was perfect.

Especially its application in the field of physics is very consistent with the phenomena of the objective real world.

Therefore, physicists firmly believe that space is flat and evenly distributed.

Although the special theory of relativity denies the absoluteness of space, it does not deny that space is flat.

Otherwise, there will be more people criticizing Li Qiwei.

However, in addition to the continuous development of physics, mathematics is also constantly moving forward.

The geniuses and bigwigs in the mathematics world are no weaker than physicists.

There are also super geniuses in the mathematics world who appear only once every hundred or thousand years.

In some ways, it can even be said that mathematicians are "smarter" than physicists.

Of course, what we are referring to here are the top existences in both fields.

Soon, 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 an entirely new system of geometry.

In his theory, Lobachevsky inherited the first four axioms of Euclidean geometry.

But the fifth axiom, he described it this way:
Through a point outside a straight line, at least two straight lines can be drawn parallel to it.

Based on these five axioms, Lobachevsky discovered that he could derive a series of geometric propositions in a logically self-consistent manner.

From this he obtained a new system of geometry.

Later it was called "Lobachevsky geometry".

The difference between Lobachevsky geometry and Euclidean geometry lies in the statement of the fifth axiom.

Later we learned that Lobachevsky geometry actually describes hyperbolic geometry, whose curvature is negative. (The shape of a saddle)
In Lobachevsky 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 when Lobachevsky geometry was published, it caused a huge sensation in the mathematical community.

Instead of being excited, everyone criticized Lobachevsky's theory as heresy and nonsense.

Even Gauss, the absolute king in the field of mathematics, remained silent and did not acknowledge Lobachevsky geometry.

But Gauss's student, Riemann, seriously analyzed Lobachevsky geometry.

He believes that this axiomatic system has great research significance.

Because he perfectly inherited the logical reasoning system of Euclidean geometry.

As long as the fifth axiom of Lobachevsky geometry is recognized, then those incredible conclusions will be correct results under this geometric system.

However, Riemann was not satisfied with this.

Based on Lobachevsky geometry, he developed another geometry, namely spherical geometry.

On the surface of a sphere, no parallel lines can 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 later became "Riemannian geometry".

Both Lobachevsky geometry and Riemann geometry are non-Euclidean geometries. The difference is that the former has negative curvature (the space is concave inward) and the latter has positive curvature (the space is convex outward).

Euclidean geometry has zero curvature, so space is flat.

In 1854, Riemann published his new geometric system.

At that time, like Lobachevsky geometry, almost no one could understand Riemann geometry.

Because it's so counterintuitive.

But at that time, after Einstein learned about Riemannian geometry through Grossmann's recommendation, he was as happy as if he had met his cousin.

Because his theory of space-time curvature is exactly applicable to 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 describing the properties of space and characterizing the geometric structure of space.

Based on this concept, data such as geodesics in Riemannian geometry (the line with the shortest distance between two points in Riemannian geometry) can be calculated.

The curvature can be calculated based on the geodesic, and the curvature is the trajectory of matter in space.

This is also the path that light takes.

At this point, the mathematical model of the space-time structure of general relativity can be constructed.

And now, Li Qiwei's mathematical level is much better than Einstein's original level.

For future physics doctoral students, mathematics is also a compulsory course.

Riemann geometry is even more famous. He studied it a lot in his previous life, and now it can finally come in handy.

Now that we have the mathematical means to deal with the curvature of spacetime, the next step is simple: to study the degree to which different substances bend space.

For example, the density, mass, energy, etc. of matter, and the curvature of space-time caused by it.

Click, click, click!
Li Qiwei worked on the paper for a full half hour.

He finally wrote out an equation.

This is the famous gravitational field equation, also called the Einstein field equation.

But now, it has to be renamed "Bruce Field Equation".

The equation looks like this:
The equation on the left represents the curvature of space-time, and the equation on the right represents the distribution of matter.

The literal version of this formula is: matter tells space-time how to curve, and space-time tells matter how to move.

This equation may seem simple, but it is actually very complicated. (See comments)

This is a second-order nonlinear partial differential equation with ten unknowns.

The sentence punctuation is: second order, nonlinear, (partial) differential, equation.

Don’t worry, we will analyze it bit by bit so that you can understand what is so 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

A differential equation is an equation based on an ordinary equation that contains unknown functions and their derivatives.

For example, assuming 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, and this equation is called a differential equation. (u′ must exist in the equation, but u can be absent)

If a differential equation has only one derivative with respect to one independent variable, it is called an ordinary differential equation.

For example, the above equation has only one independent variable x, and only one derivative of the independent variable x, u′, so 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, called 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 type of differential equation is called a partial differential equation.

[Second level]

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 derived again.

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 and x, y is a straight line, it means linearity.

If it is not a straight line, it means nonlinearity.

At this point, the concept of the Bruce field equation, a second-order nonlinear partial differential equation, is understood.

It can be seen that finding the exact solution to this equation is an extremely complicated task.

There is no trick, only brute force solution.

That is to take all variables into consideration.

Such as mass, energy, density, space, time and so on.

Therefore, without the supercomputers of later generations, it is conceivable how difficult it would be to solve this equation by hand.

Even with the help of computers, it is not easy to solve.

Even the simplest movement between two celestial bodies.

If we consider the properties of general relativity, then until later times, there was no way to simulate its precise space-time relationship.

In real history, the exact solution given by Schwarzschild was actually the simplest one, taking the least variables into consideration.

He assumed that there was only one point of matter in the entire universe.

Although the Bruce field equations cannot be solved exactly, they can be approximately solved through mathematical means.

For example, the famous problem of the precession of Mercury's perihelion was answered using an approximate solution, thus giving a perfect explanation.

The Bruce field equations have very rich connotations.

Each exact solution to this equation represents a different universe.

And it is a universe that is constantly evolving from the past to the future.

Because there is a parameter called time t in the field equation, the equation will change continuously with time.

This also represents that the universe is constantly moving and changing.

The possibility of going back to the past that is often mentioned in later generations actually refers to a specific solution to the field equation.

Solving the Bruce equation is a specialized subject.

The relationship between all time, space and matter in the universe is encompassed by this equation.

call!
Li Qiwei exhaled heavily.

At this point, the content of general relativity has been completed.

However, the paper is not over yet.

Because many incredible conclusions can be derived based on this field equation.

On the day of publication, Li Qiwei will attach all these conclusions to the paper as his predictions.

He put all the prophecies for later generations together, and one can imagine the shock it brought to everyone.

However, the wild nature of general relativity makes it extremely difficult to prove it.

In real history, in the early days, there were three most important pieces of evidence in chronological order.

The first one is the problem of the precession of Mercury's perihelion, which can be perfectly explained using 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 should take into account various factors such as the rotation of the sun.

It's entirely possible that this is also causing Mercury's odd behavior.

At least you can't prove the conjecture wrong.

Therefore, the first proof is slightly weaker.

The second one is the famous bending of starlight.

That is, Eddington proved through the total solar eclipse experiment that the path of light will bend after passing through the sun.

This evidence strongly proved the correctness of general relativity and elevated the theory to a divine position.

The third is the phenomenon of gravitational redshift.

According to the deduction of general relativity, the wavelength of light will become longer after leaving the gravitational field. (It is a bit complicated and will not be explained in detail for now)
Therefore, the position of light on the spectrum moves closer to red light, which is called redshift.

This inference was not proven until around 1950 by a very, very sophisticated experiment.

Li Qiwei looked at the first draft of the paper in his hand, feeling deeply moved.

The special theory of relativity unifies time and space, and time and space are originally one.

The general theory of relativity unifies the interaction between space, time and matter.

The approximation of the special theory of relativity is Newton's three laws of mechanics.

The approximation of general relativity leads to the law of universal gravitation.

Ridgwell's theory of relativity thoroughly incorporated Newtonian mechanics.

(End of this chapter)