S TEVEN B RYANT N ATURAL P HILOSOPHY A LLIANCE 14 TH A NNUAL C ONFERENCE U NIVERSITY OF C ONNECTICUT...
Transcript of S TEVEN B RYANT N ATURAL P HILOSOPHY A LLIANCE 14 TH A NNUAL C ONFERENCE U NIVERSITY OF C ONNECTICUT...
STEVEN BRYANT
NATURAL PHILOSOPHY ALLIANCE 14TH ANNUAL CONFERENCEUNIVERSITY OF CONNECTICUT
STORRS, CONNECTICUTMAY 20-25, 2007
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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Thank you