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The development of relativity is essentially a series of simplifications, because each simplification lays the foundation for the next. Without these simplifications few people would be able to understand relativity. In fact, without these simplifications, it is uncertain whether relativity could have been discovered.

The following is a brief overview of the simplifications made by successive masters of relativity, which gives us a rough idea of how complicated relativity could have been. In the process, two threads eventually converge to form a clever simplification.

Riemann

Riemann was a key figure in one of the main threads, proposing differential geometry formulas, which are a series of formulas describing curved spaces, defined on the basis of Riemann degrees. One of the most famous is the Riemann curvature tensor formula, which describes the relationship between the curvature of space and the curvature of curves. But in Riemann's day, these formulas were a bunch of unsimplified polynomials. Coordinate transformations, in particular, require a series of complex algebraic operations with no obvious pattern to follow. Similarly, Maxwell's equations consist of four elegant and concise formulas, but occupy pages of algebraic computation in Maxwell's time.

If relativity develops only in this form, we are likely to remain stuck in special relativity and unable to advance beyond Lorentz transformations.

Ricky

Ricci-Curbastro was the leader of the next step in simplification in this thread, developing tensor calculus, a mathematical method based on coordinate transformations. The symbols of tensor calculus represent arrays of numbers that are transformed in a simple, well-defined way between different coordinate systems and observers. This approach greatly simplifies the complexity of describing spatial curvature and physical phenomena.

Tensor calculus is powerful in that it combines shear and compressive stresses on different axes into a single stress tensor and unifies the corresponding strain conditions. This method can express the relationship between all these tensors in one symbol and can make correct transformation between different coordinate systems and observers.

Ricci also discovered a simple way to describe curvature in multidimensional space, the Ricci curvature tensor, which is important for the study of general relativity.

Einstein

At the same time, in another clue, Einstein introduced two great simplifications to what we now call "special relativity."

Michelson-Morley experiments show that no matter how the speed of light is measured, the result is always the same. Physicists have struggled to explain why the laws of physics known at the time seemed to conspire to conceal the variations in time and space that affected the scale of measuring equipment. Einstein simplified the whole discussion by assuming that the constancy of the speed of light is an inherent property of the universe, as a fundamental law of nature. This proved to be the only explanation for the strange variations in time and space.

The constancy of the speed of light prevents one from finding an absolute reference to the speed of space, the speed associated with the ubiquitous presence known as the ether. Einstein shelved the problem by assuming that there is no frame of reference in which relative velocity is the only meaningful velocity.

the Minkowski

Minkowski combined these two simplified ideas, combining time and space so that tensor calculus could be applied to the resulting spacetime. This is a big logical leap, because in order for tensor calculus to make sense, Minkowski must introduce a strange negative sign into the definition of distance in four-dimensional spacetime. The result is the following simplification:

Although Einstein had shown that variations in time and space with variations in relative motion were consistent with the assumption that the speed of light was constant, Minkowski was able to generalize this result to all forces. Thus, for example, the proof of the existence of a magnetic field can be deduced from this assumption, and the final simplification is that Maxwell's equations are reduced to only one tensor equation.

Einstein's assumption that all motion is relative leads to seemingly paradoxical, such as the twin paradox, because two observers observing the same object/event, with different relative velocities, seem to have contradictory observations. Minkowski's tensor analysis simplifies these apparent inconsistencies, because the same tensor analysis can explain why an object looks different from different perspectives, and can also explain the extra differences due to different relative velocities.

Minkowski discovered that the three-dimensional stress tensor discovered by Richie, when extended to four-dimensional spacetime, becomes a tensor, which includes mass, energy and momentum as well as stress. This discovery is a fundamental simplification of gravity research, because in Newtonian gravity, which is so successful in most applications of the solar system, the source term is mass. Thus, in the development of general relativity, Minkowski's energy-momentum-stress tensor is a four-dimensional extension of mass.

As a result of this simplification, the field equations of general relativity have a simple form:

Among them,

T is Minkowski energy-momentum-stress tensor;

g is called the "metric tensor" and is a simplification of the general formula for distances in curved spaces discovered by Riemann by Ricci;

R is the Ricci curvature tensor mentioned above, and the Ricci curvature tensor is an algebraic differential function of g;

Lambda is a cosmological constant, sometimes included and sometimes not included in history. It simplifies the discussion of some cosmological problems;

G is Newton's gravitational constant;

c is the speed of light in a vacuum.

Minkowski's "flat" four-dimensional spacetime is the boundary condition of this equation.

Without these simplifications, Einstein's research might have taken decades or more, not five years.

This article comes from Weixin Official Accounts: Laohu Science (ID: LaohuSci), Author: I am Lao Hu

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