Dear Dr. A, --Sent to Dr A on January 9, 2014--

The Lorentz factor or gamma, 1/sqrt(1-(v²/C²)), is common to time dilation and length contraction. It has innumerable applications. The equation (Ct)²=(Ct’)²+(vt)², based on a right triangle and the Pythagorean theorem, is used to formulate gamma. Fig. 3 details the calculation.

FOR THESE PURPOSES:

Inertial frame K’ is in motion relative to frame K at a speed v>0<C.

There are two identical light clocks. One is in frame K. One is in frame K’. Both are perpendicular to the direction of motion.

The vertical side of the right triangle is perpendicular to the direction of motion.

The horizontal side of the right triangle is parallel to the direction of motion.

DESCRIPTIONS:

The Right Triangle (Fig. 1):

The horizontal side, vt, is the distance that frame K’, and so the light clock in K’, advanced in the direction of motion relative to frame K at speed v in the time t.

The hypotenuse, Ct, is a one-way path of the light ray in the light clock in K’ as seen by an observer in frame K. Seen from frame K the light ray advances vertically in the light clock as the light clock advances horizontally relative to frame K. The length of the hypotenuse is the distance as seen by the observer in K that the light ray at speed C propagates in frame K in the time t.

The vertical side, Ct’, is the length of the light clock which is perpendicular to the direction of motion in frame K’. Within K’ a ray of light at speed C propagates the one-way length y’ of the light clock in the time t’, y’=Ct’.

The Light Clock in Frame K (Fig. 2):

A light clock identical to the one in frame K’ is in frame K. It is of length y and is perpendicular to the direction of motion. Within frame K a ray of light at speed C propagates the one-way length y of the light clock in the time t, y=Ct.

The Equation (Fig. 3):

The formulation of gamma uses the equation:

(Ct)²=(Ct’)²+(vt)²

Solved for t:

t=t’(1/sqrt(1-(v²/C²))) or

t=t’*gamma

ISSUES:

The Time Dilation Issue (Fig. 2):

The length of the light clock in frame K’ is Ct’, y’=Ct’.

The length of an identical light clock in frame K is y, y=Ct.

Both are perpendicular to the direction of motion.

Per SR y’=y

Then, Ct’=y’=y=Ct

Then, Ct’=Ct

Then, t=t’

As t’=t, the time t’ in frame K’ is not dilated relative to the time t in Frame K.

Per SR, t=t’*(gamma). However, mathematically, t=t’.

Time dilation as presented in SR is contradicted mathematically.

The Right Triangle Issue (Fig. 2):

As Ct’=Ct substitute Ct’, the length of the light clock in frame K’, with Ct the length of the identical light clock in frame K as the vertical side of the right triangle used to formulate gamma. Then the length of the vertical side of the right triangle and the length of the hypotenuse are both Ct. The hypotenuse is not the longest side of this right triangle. Per the laws of physics, the hypotenuse is the longest side of a right triangle. As such, the right triangle used to formulate gamma is invalid.

The Equation Issue:

Again, as Ct’=Ct, substitute Ct for Ct’ in the equation from which gamma is formulated.

The original equation:

(Ct)²=(Ct’)²+(vt)²

Substitute Ct for Ct’:

(Ct)²=(Ct)²+(vt)²

The equation used to formulate gamma is unbalanced and therefore, invalid.

From your expert perspective, are these issues substantive?

Thank you VERY much,

Butch

Thorntone Murray

January 9, 2014