Jacobi’s principle and space axes

[AntonioLao]

Carl Gustav Jacob Jacobi (1804-1851), please see a short biography at the following link
http://www-groups.dcs.st-and.ac.uk/~...ns/Jacobi.html was the first mathematician to study the intimate relation between dynamics and the geometry of curved spaces. Hence, a proper starting point for our investigation would be his principle of 1845.
In the spacelike region of the light cone, we apply Jacobi’s principle to determine the orientation of one space axis by minimizing the action integral in the configuration space.

[math]\bar{A}=\int^{\tau_2}_{\tau_1}\sqrt{2(E-V)}\bar{ds}[/math]

This expression is identical to the one given by Cornelius Lanczos in his book entitled ‘The Variational Principles of Mechanics’, 4th Edition, page 135, Dover, New York, 1986. If we restrict ourselves to the case of a single spacetime event, the line element [math]\bar{ds}[/math] becomes identical with the line element of ordinary 3D space in arbitrary curvilinear coordinates. The principle then bears a significant resemblance with Fermat’s principle of least time in optics which determines the optical path by minimizing the integral

[math]I=\int^{\tau_2}_{\tau_1} n \bar{ds} [/math]

Where n is the refractive index. A workable hypothesis is the connection between time axis and space axis and the wave-particle nature of the photon and the speed of light in vacuum.

[Guille]

When the space and time axis get together is where there is vacuum space, speed of light, and light cone, right?

[AntonioLao]

When the space and time axis get together

space axis and time axis can never get together. They are always orthogonal, perpendicular, 90 degrees apart. Our own space-time axes always subtended at a given angle to the universal space-time axis of the universe. This angle gives our average human life-span. If our space-time axis is at 45 degrees then we live forever. However, for photons, their space-time axes are all at nearly 45 degrees to the universal space-time axis, I think that is why photon are stable meaning they do not aged, do not grow old even they can lose energy. Energy can be explained as the interactions of photon's positive and negative space axes.

[Guille]

I mean when the space and time axis of an object get together. If it's 45 degrees in the time exis, and 45 in the space, then the both lines reach.

[AntonioLao]

I mean when the space and time axis of an object get together. if it's 45 degrees in the time exis, and 45 in the space, then the both lines reach.

each of us has our own individual time-space axis. This axis is then referenced to the time-space axis of the universe. If our time axis aligned with the time axis of the universe, we are dead. But this also implies that our space axis also aligned with the space axis of the universe since time/space axes are always perpendicular to each other. At 45 degrees, our time axis aligned with the time axis of light and space axis aligned with the space axis of light. All photons have their time axes at 45 degrees from the time axis of the universe, and their space axes also at 45 degrees from the space axis of the universe.

[Guille]

each of us has our own individual time-space axis. This axis is then referenced to the time-space axis of the universe. If our time axis aligned with the time axis of the universe, we are dead. But this also implies that our space axis also aligned with the space axis of the universe since time/space axes are always perpendicular to each other. At 45 degrees, our time axis aligned with the time axis of light and space axis aligned with the space axis of light. All photons have their time axes at 45 degrees from the time axis of the universe, and their space axes also at 45 degrees from the space axis of the universe.

What states that they are perpendicular to each other? I believe they aren't, I believe thay have nothing to do with each other in their axis.

[AntonioLao]

What states that they are perpendicular to each other?

The principle of orthogonality. Complementarity of the trigonometric functions of sine and cosine. The orthogonality of eigenvectors in eigenvalue problems. The perpendicular planes of electric and magnetic field of electromagnetic radiation. If the electric and magnetic field vectors are not perpendicular then the Poynting vector of radiation is null meaning no propagation is possible.