optical signal tcp protocol
This is the embarrassing problem of using signal to align clock [cover face]
No matter how you do it, there is always a problem that cannot be avoided: (ct)^2 = x^2 + y^2 + z^2.
So, light speed determines the measured system error . Since
is a systematic error, it can be considered that the four-dimensional interval between two points is more suitable as their characteristics than their three-dimensional distance and time .
So, the four-dimensional interval between two s infinitely close to is:
ds^2 = (cdt)^2 - dx^2 - dy^2 - dz^2.
This is Special Relativity Theory .
In short, Einstein believes that time and distance that is not affected by signal speed, so he needs to use and their squared difference to measure the motion of objects.
2, the transformation of curve coordinates,
Einstein also believes that in bending space , cannot give real rectangular coordinate system :
People think rectangular coordinate system is actually a curve coordinate system, but it has not been processed according to curve coordinates before.
Coordinate system on the ellipsoid
So, Einstein needs to use a set of mathematical of curve coordinates to calculate physical problems.
This set of mathematics was invented by Riemann : Riemann geometry .
curve coordinate system requirements are much lower than those of the rectangular coordinate system:
A, the curve coordinates of different points are 's unique ,
, which must be met, because the coordinates are . In order to distinguish different points of , it would be meaningless if the coordinates are not unique.
B, when converting points between different coordinate systems, corresponds to one by one.
That is to say, point A in coordinate system 1 is converted to point A' in coordinate system 2, and the conversion relationship is inverse function :
C, arc length or is expressed by the coordinate quadratic term ,
in the rectangular coordinate system, because Pythagorean theorem , the square of the arc length of is exactly the sum of squares of the coordinates :
ds^2 = dx^2 + dy^2 + dz^2.
curve coordinate system, Pythagorean theorem has become like this:
ds^2 = A dx^2 + B dy^2 + C dz^2 + D dxdy + E dxdz + F dydz.
where A, B, C, D, E, F are all functions of coordinates, abbreviated as:
where i and j are the numbers of coordinates, and the same numbers above and below represent summing: counting from 0, i,j = 0, 1, 2, 3.
3, is located at different points in , how to compare?
is the same as the Cartesian coordinate system. Translate one of the s to the other point , and then compare it at the same location at .
vector translation
Whether it is a rectangular coordinate system or a curve coordinate system, this translation must ensure that scalar is the unchanging of .
vector inner product is scalar :
Therefore, the translation of inner product is unchanged.
If these two points are infinitely close to , then the differential of the inner product of is unchanged:
According to this point, Riemann derived a set of derivative rules in the curve coordinate system , called Riemann Contact .
vector translation along the curve
Riemann also derived the relationship between Riemann contact and Riemann standard .
degree is the coefficient matrix in the arc length formula and the quadratic term of the coordinate differential.
can solve the metrics, and then you can translate the vector along the curve .
speed is also a vector. The curve formed by translating it along the tangent line of space-time is to measure ground line .
geodesics can be regarded as "straight line" in curved space-time.
4, gravity and curvature,
Because the motion law in gravitational field has nothing to do with the mass of an object , so Einstein regards the gravitational field as a kind of geometric effect .
Newton regard gravity as a force , so gravity will produce acceleration:
mg = GMm/r^2, and further obtain g = GM/r^2, which is the gravitational acceleration at the distance from the sun r.
However, Einstein did not allow gravity to change the acceleration, but instead changed the curvature of space-time in .
Spatial curvature changes, the metric changes, and the geodesics changes.
The object moves along the geodesic , and the motion pattern of the object changes.
That is to say, Einstein turns the problem of Newton's second law into the problem of Newton's first law .
If Newton's second law is used, then the gravitational field changes the acceleration of the object.
If Newton's first law is used, then the gravitational field changes the curvature of space.
Think about it, Einstein was born 200 years later than Newton . If he didn't change his perspective, how could he publish a paper? [covering his face]
Since Einstein asked gravity to change the curvature, how did he calculate curvature ?
is actually still translated through vector !
In a straight space-time, when vector translates a circle back to the initial position, the direction is that does not change.
Translation of the rectangular coordinate system
However, in the curved space-time, when vector translates one circle back to the initial position, the direction changes .
Based on this point, you can calculate the curvature of space , which is called Gaussian curvature .
Because of Gauss , the quarrel that lasted 2000 Euclidean fifth public issue. Anything related to curvature is basically related to gauss.
The translation of the curve coordinate system is one circle
Integrate the space-time curvature and the energy-momentum of the object together. According to the variation method of Euler-Lagrangian , find the minimum value of of this integral to obtain Einstein's gravitational field equation :
Let = 0 in the brackets be the legendary Einstein's gravitational field equation .
This formula comes from "Landau's Field Theory" .
g is the curve meter of Jacobian determinant , because gravitational potential energy is only 0 at infinity, so g 0.
When integrating according to Gaussian theorem , it is the coefficient in front of the space-time coordinates. The coefficient of the rectangular coordinate of
is 1, so the double integral on the high number is, but the polar coordinate is, and the latter r is the value of the Jacobian determinant.
Energy corresponds to time , momentum corresponds to space , and is written together in 4-dimensional coordinate .
