S TEVEN B RYANT N ATURAL P HILOSOPHY A LLIANCE 14 TH A NNUAL C ONFERENCE U NIVERSITY OF C ONNECTICUT S TORRS, C ONNECTICUT M AY 20-25, 2007 S TEVE.B RYANT.

STEVEN BRYANT

NATURAL PHILOSOPHY ALLIANCE 14TH ANNUAL CONFERENCEUNIVERSITY OF CONNECTICUT

STORRS, CONNECTICUTMAY 20-25, 2007

[email protected]

A Brute-Force Challenge to Einstein’s 1905 Derivation Reveals a Mathematical

Inconsistency in the SRT Time Transformation Equation

Objectives

Convince you that there is a mathematical problem in Einstein’s 1905 SRT derivations

Offer a mathematical correction that fixes the problemExplain why this problem has been so elusiveBriefly introduce the Model of Complete and Incomplete

Coordinate Systems.

Challenging SRT(What makes this presentation different?)

Does not rely on paradoxesDoes not rely on new terms or definitionsDoes not require the introduction of new variablesDoes not require that you change your understanding or

point of view.

Preparing To See The Problem

A belief that Einstein’s 1905 derivation is soundA belief in the basic rules of algebraAgreement that Einstein is not above the rules of algebra.

Einstein’s transformation equations takes a set of input values and produces a set of output values.

t

z

y

x

2

2

2

2

2

1

1

cvcvx

t

z

ycv

vtx

Input Values Output ValuesTransformation Equations

Einstein performs several steps to create the equations that are then “normalized” to produce his final transformation equations.

Einstein performs several steps to create the transformation equations.

These steps are given in Section 3 of his 1905 paper.

Each equation

ismultiplied

by:

2

2

2

2

2

1

1

cvcvx

t

z

ycv

vtx

2

2

1c

v

1 2 3 4

2

2

2

2

2

2

2

2

2

1

1

1

1

cvcvx

t

cv

zcv

ycv

vtx

ImplicitExplicit Explicit

Two key pages in Section 3 of Einstein’s Paper(Translated Version)

Einstein performs four algebraic steps to produce his transformation.

1 2 3 4

2

2

22

2

22

1

)(

cv

vtx

vc

xc

vc

xvtc

c

22

2

22)(

vc

xc

vc

xvtc

c

)(22 vc

xvtc

c

c

Begin with: Since

Substitute with:

Since

Substitute with:

Since

Substitute with:

c )(22 vc

xvt

)(22 vc

xvt

vc

xt

vtxx

vtx

x

vc

x

t

This slide must be true if you believe that Einstein’s SRT equations are right

Einstein’s equations must follow algebraic rules

2

2

22

2

22

1

)(

cv

vtx

vc

xc

vc

xvtc

c

Each statement on the right-hand side must produce the same result , if we are going to be able to mathematically conclude that the left-hand side equals the right-hand side of the equation

2

2

1c

v

vtx

This slide must be true if you believe in the rules of algebra

Validating Einstein’s derivations against the rules of algebra

2

2

1cv

vtx

1

2

3

4

10

0

0

50

t

z

y

x

1

5

v

Note: x’=x-vt

Note: x’=x-vt

Statement Results

0

Input Values

22

2

vc

xc

)(22 vc

xvtc

c

580,924,997,210 orc

0

580,924,997,210 orc

We don’t get the same result

Dependent and Independent Variables

An Independent Variable is something that we provide or give. These are our “input values”

A Dependent Variable is something found as the result of a computation or equation.

Dependent and Independent Variables

x and t are the independent variables

x’ is the dependent variableWe can state: x’ is dependent

upon x and t.

vtxx

Dependent and Independent Variables

If we place x’=x-vt, it is clear that a point at rest in the system k must have a system of values x’, y, z, independent of time.

Einstein Says:

If we place x’=x-vt, it is clear that a point at rest in the system k must have a system of values x’, y, z, independent of time, where time is represented by t’.

Mathematically correct interpretation

If we place x’=x-vt, it is clear that a point at rest in the system k must have a system of values x’, y, z, independent of time, where time is represented by t.

Mathematically incorrect interpretation

Fixing the problem

The correction is to replace t with t’ in setting up the partial differential equation , which produces the linear function…

)(22 vc

xvt

Understanding “why” this is the correction to the problem requires a new perspective

This new perspective explains each of these independently

To correct Einstein’s derivation, t is replaced with t’ (in his partial differential equation) followed by performing the four algebraic steps, resulting in the transformation.

1 2 3 4

2

2

22

2

22

1

)(

cv

vtx

vc

xc

vc

xvtc

c

22

2

22)(

vc

xc

vc

xvtc

c

)(22 vc

xvtc

c

c

Begin with: Since

Substitute with:

Since

Substitute with:

Since

Substitute with:

c )(22 vc

xvt

)(22 vc

xvt

vc

xt

vtxx

vtx

x

vc

x

t

This revised derivation produces the SAME equation as Einstein

We can retest the new derivation to determine if a problem exists.

2

2

1cv

vtx

1

2

3

4

10

0

0

50

t

z

y

x

1

5

v

Note: x’=x-vt

Note: x’=x-vt

Statement Results

0

Input Values

22

2

vc

xc

)(22 vc

xvtc

c

0

0

0

We get the same result

While the problem occurs during the derivation, it shows up in Einstein’s equation. Einstein’s original time transformation simplification is provided in column one and the corrected simplification is provided in column two.

1 2

2

2

22

1

)(

cv

vtx

c

vc

xvt

2

2

2

22

1

)(

cvcvx

t

vc

xvt

When

it simplifies as:

When

it simplifies as:

)(22 vc

xvt

)(22 vc

xvt

Any new model must explain this corrected time equation.

Since

Substitute with:

vtxx

vtx

x

Einstein’s Conflict between Independent and Dependent Variables

1 2 3 4

2

2

22

2

22

1

)(

cv

vtx

vc

xc

vc

xvtc

c

22

2

22)(

vc

xc

vc

xvtc

c

)(22 vc

xvtc

c

c

Begin with: Since

Substitute with:

Since

Substitute with:

c )(22 vc

xvt

)(22 vc

xvt

vc

xt

vc

x

t

Einstein treats t as both a dependent and an independent variable, but produces the correct result only because he substitutes for t before he substitutes for x’

Why is this a hard problem to identify?

Einstein produces the correct equation

The problem is in the derivation, not in the final equation

Einstein properly simplifies his linear function

The problem occurs in the set-up of the partial differential equation, which happens prior to simplification.

2

2

1cv

vtx

2

2

2

1cvcvx

t

Equation Known New Findings

What does this mean?

Einstein’s theoretical model fails Requires a one-to-one mapping between input values and output values

(e.g., between points in the coordinate systems)

Any “alternative” theory that adopts Einstein’s time transformation must be reexamined Tau produces a different value

Requires the adoption of a new model (e.g., the model of Complete and Incomplete Coordinate Systems). Consistent with the experimental evidence New & different theoretical predictions New & different methods of applying the equations Mathematically sound (e.g., explains the revised equations). Consistent with Lorentz

Based on the concept of different types of coordinate systems, where the behaviors of phenomena is governed by the properties of the coordinate system

Extends (rather than throws away) Einstein’s original postulates Transformation Equations

Re-established the Newtonian transformation equations Extends the wave-front equations Explains and establishes the role of the “revised” transformation equations (covered in

this presentation)

Removes the SRT paradoxes and limitations.

For Further Exploration…The Model of Complete and Incomplete Coordinate Systems (CICS)

This mathematical finding requires a new model

For Further Exploration…The Model of Complete and Incomplete Coordinate Systems (CICS)

Domain Newton SRT CICS

Fixed-Point Equations

Wave-Front Equations

N/A

One-Half Wave Oscillation Equations

N/A N/A

Ether-Based Model Agrees with model’s interpretation of SRT experimental data (*)

Developing

System Equations Comparison Table

!

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Steven [email protected]

Thank you