square momentum

[AntonioLao]

The square of momentum denoted by ��² becomes important for relativistic dynamics where and when its directional invariance properties emerged as the velocity component approaches lightspeed. It appears within the physical definition of 4-vectors. This is an extension of the idea of vectors, which has so far considered to have only three spatial components, but now to include a time component. That is to say there will be vectors with four components, three of which are like the components of an ordinary vector; the fourth component is defined as the time component. If these 4-vectors are translated in a uniform linear motion then the time component is translated as a scalar quantity and these 4-vectors are then equivalent to the transformation of quaternion. On the other hand, if the 4-vectors undergo rotational Lorentz transformation such that the space-time interval remains a relativistic invariance, and which eventually provides a general covariance then the time component becomes the energy and the three space components become the three components of linear momentum.

Since relativistic geometry is now given as a Lorentzian hyperbolic non-Euclidean geometry, the square of the mass as the hypotenuse is given, not by the sum of the square of energy and momentum, but by their difference: ��²��²��²=��²-��²��² and if ��=1 then ��²=��²-��². Solving for square of momentum gives ��²=��²-��². This alternative formulation clearly suggests the scalar property of the square momentum, which is plausible only by constructing its corresponding quadratic form given by ��²=Q(2)=(��₂-��₁)²=��₁₁|��₁|²��������₁₁+ ��₂₂|��₂|²��������₂₂-��₁₂|��₁||��₂|��������₁₂-��₂₁|��₂||��₁|��������₂₁ where |��₂|=|��₁|=��, ��������₁₁= ��������₂₂=0° or 180°, ��������₁₂= ��������₂₁=90° or 270°, and ��₁₁=��₂₂=½ such that ��²=½|��₁|²+½|��₂|² and orthogonality of ��₂ and ��₁ implies (��₂-��₁)²=(��₁-��₂)²=(��₁+��₂)² wherever and whenever expanded as their respective quadratic forms.

[Bogie]

I always find your descriptions interesting. But to deal with momentum and geodesics plotted out by moving points in three D space and each with an event time attached seems so "mathematical" . Where are you taking us?

[AntonioLao]

Where are you taking us?

Where and when the directional property of a vector quantity like linear momentum disappears. The directional property of gravity was made to disappear by Einstein's general theory of relativity such that gravity is replaced by the metric tensor of the space-time curvature. However, the metric tensor of GR is just a special case of the general covariance of the quadratic form of space-time with 4 variables (4D). But GR cannot predict the existence of graviton until the 11 dimensions of superstring theories as an extension of Kaluza-Klein 5-dimensional quadratic form. All these require velocity configurations approaching lightspeed.
However, 11D minus 3D minus 3D minus 3D is equal to a 2D configuration that is a quadratic form with only two variables.

[Bogie]

Where and when the directional property of a vector quantity like linear momentum disappears. The directional property of gravity was made to disappear by Einstein's general theory of relativity such that gravity is replaced by the metric tensor of the space-time curvature. However, the metric tensor of GR is just a special case of the general covariance of the quadratic form of space-time with 4 variables (4D). But GR cannot predict the existence of graviton until the 11 dimensions of superstring theories as an extension of Kaluza-Klein 5-dimensional quadratic form. All these require velocity configurations approaching lightspeed.
However, 11D minus 3D minus 3D minus 3D is equal to a 2D configuration that is a quadratic form with only two variables.

But is it just maths or is there some correspondence to (excuse my being philosophical) reality? Aren't you talking about the extremely tiny, and aren't the velocities that approach relativistic taking place within particles themselves instead of at the particle level?

[AntonioLao]

The only physical requirement to achieve lightspeed is zero mass. Incidentally, in quantum mechanics the directional property of both position and linear momentum is made to disappear by taking their inner product which is non-commutative that is xp is not equal to px and usually known as the uncertainty principle. In place of their lost direction the idea of quantized intrinsic spin was created and with it predicted the existence of antimatter. If the directional property of time can be made to disappear then time will not flow and we can live forever.

[Bogie]

The only physical requirement to achieve lightspeed is zero mass. Incidentally, in quantum mechanics the directional property of both position and linear momentum is made to disappear by taking their inner product which is non-commutative that is xp is not equal to px and usually known as the uncertainty principle. In place of their lost direction the idea of quantized intrinsic spin was created and with it predicted the existence of antimatter. If the directional property of time can be made to disappear then time will not flow and we can live forever.

Fine but wouldn't we be talking about living as a stiff block of unresponsive protoplasm?

[AntonioLao]

We can live as pure energy and never gains or loses anymore energy. By the way, pure energy has zero viscosity and mass and moves constantly at maximum speed of light.

[Bogie]

We can live as pure energy and never gains or loses anymore energy. By the way, pure energy has zero viscosity and mass and moves constantly at maximum speed of light.

Are you saying that the mass disappears, is converted to pure energy, and the pure energy has no density? Isn't there some conservation problem with that?

[AntonioLao]

The radiation pressure is only a third of the total energy density of pure energy. The conservation is always implied by ��=����².