Rao NaveedHussain Rearsch in Spacetime Physics: June 2018

Thursday, June 21, 2018

A New Relation Between Periodic Time T and Linear Time t: Equation of Time Circle

Relation Between Circular Time T and Radial or Linear Time t

T = 2pt

equation of time circle

Formula of Energy in Noninertial Frame of Reference

Formula of Energy in Noninertial Frame of Reference

E = ma²/w²

Where E is energy, a is translational acceleration and w is angular speed

What is Space? An Un-Noticed Relation

Space As Extension of Time Coordinate

Einstein introduced 4 dimensional coordinate as the unification of time coordinate and space coordinate. The definition of space comes out from the divergence of x^µ that has never been pointed out by anyone so far.

x^µ is the 4 dimensional coordinate such that x^µ = (x⁰, xⁱ) = (ct, xⁱ)

The divergence of x^µ implies that

x^µ_,µ = Ñ.x + ¶x⁰/¶t

Ñ.x + ¶x⁰/¶t = o

Ñ.x = - ¶x⁰/¶t

This equation shows that space is extension of time coordinate or in more familiar language, it is the equation of continuity of spacetime

REFERENCE:

[1]. Naveed Hussain (2017),” Theory of Everything TOE: Basic Framework” Submitted to Frontiers of Physics, China, June 29,2017, Manuscript ID: FOP-2017- 0111

Mathematical and Physical Definition of Special Theory of Relativity

Mathematical and Physical Definition of Special Relativity

Mathematical Definition

Special relativity is the unification of space and time.

4 dimensional coordinate is the unification of time coordinate ct and space coordinate x where time coordinate is scalar and space coordinate is a vector

x^µ = (ct, x, y, z) = (x⁰, x¹, x², x³) = (x⁰, xⁱ)

Special relativity is the study of spacetime laws of physics with respect to inertial frame of reference

Physical Definition

Physically special relativity is the unification of scalars and vectors

4 current density is the unification of charge density r and current density J

J^µ = (J⁰, J¹, J², J³) = ( r, Jⁱ)

Formula for Newton's First Law of Motion

Mathematical Formula For Newton’s First Law of Motion

It is very strange to note that, we have mathematical formula for Newton’s second law of motion F = ma, but no formula for Newton’s first law.

Simple Derivation of Newton’s Ist law From 2^nd Law

We know the relation between force and linear momentum

F = (P_f - P_i)/t_f -t_i

F = DP/Dt (Rate of change of linear momentum)

As an analogy

P = m(x_f - x_i)/t_f -t_i

P = mDx/Dt

P = mv {v = Dx/Dt}

Momentum of a body is equal to the rate of change of change of position of a body of mass m which is nothing but Newton’s first law

If the position is not changing then its momentum is zero means the body is at rest P = 0

Usual Derivation of of Newton’s First Law

In the absence of an unbalanced force, linear momentum of a body is directly proportional to the its velocity and inversely proportional to its mass

(1) vµ P

(2) vµ 1/m

Combining (1) and (2), we have

vµ P/m

v = k P/m

when k =1 then

v = P/m

P = mv

Linear momentum is equal to the rate of change of position of a particle of mass m

Newton’s 2^nd Law of Motion from First Law

Writing momentum as the rate of change of posion of a body of mass m

DP = mDx/Dt

Dividing both sides by Dt

DP/Dt = mDx/DtDt

F = mDv/Dt

F = ma

A Single Transformation Law for 4 vectors and Tensors

Single Transformation Law For F^µⁿ_,_n and J^µ

Maxwell’s Equations in 4 Dimensional Form are given by

(1) F^µⁿ_,_n = J^µ

Both sides of (1) have never been transformed under a single Transformation Law

F^µ^¢ⁿ^¢_,_n_¢ = L^µ^¢_αF^αⁿ^¢_,_n_¢ and J^µ^¢ = L^µ^¢_αJ^α

Giving the same results and form invariance . It also lead to transform electromagnetic field

F^µ^¢ⁿ^¢ = L^µ^¢_αF^αⁿ^¢ (Revolution)

Reference:

[1] Naveed Hussain, “The Dawn of Form Invariant Transport Law” Submitted to Frontiers of Physics, China, Manuscript ID, FOP-2017-0097, May 13, 2017

Derivation of Ampere's Law From Gauss's Law

Relation Between Gauss’s Law and Ampere’s Law

We know that Gauss’s Law

(Ñ.E)= r

Multiplying by velocity on both sides

(Ñ.E)v= rv

But rv = J and J = (Ñ´ B) - ¶E/¶t (Ampere’s Law)

(Ñ.E)v= (Ñ´ B) - ¶E/¶t

J= (Ñ´ B) - ¶E/¶t

This implies that Ampere’s law is the relativistic consequence of Gauss’s Law

REFERENCE:

[1]. Naveed Hussain (2016),” Completely Generally Covariant Laws of Physics” Submitted to Frontiers of Physics, China, January 08 ,2016, Manuscript ID: FOP-2016- 0008