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 ELATIVITY C HALLENGE.C OM ELATIVITY C HALLENGE. COM A Brute-Force Challenge to Einsteins 1905 Derivation Reveals a Mathematical Inconsistency in the SRT Time Transformation Equation

Objectives Convince you that there is a mathematical problem in Einsteins 1905 SRT derivations Offer a mathematical correction that fixes the problem Explain why this problem has been so elusive Briefly introduce the Model of Complete and Incomplete Coordinate Systems.

Challenging SRT (What makes this presentation different?) Does not rely on paradoxes Does not rely on new terms or definitions Does not require the introduction of new variables Does not require that you change your understanding or point of view.

Preparing To See The Problem A belief that Einsteins 1905 derivation is sound A belief in the basic rules of algebra Agreement that Einstein is not above the rules of algebra.

Einsteins transformation equations takes a set of input values and produces a set of output values. Input ValuesOutput 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. Einstein performs several steps to create the transformation equations. These steps are given in Section 3 of his 1905 paper. Each equation is multiplied by: Each equation is multiplied by: 1234 Implicit Explicit

Two key pages in Section 3 of Einsteins Paper (Translated Version)

Einstein performs four algebraic steps to produce his transformation Begin with: Since Substitute with: Since Substitute with: Since Substitute with: This slide must be true if you believe that Einsteins SRT equations are right

Einsteins equations must follow algebraic rules 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 This slide must be true if you believe in the rules of algebra

Validating Einsteins derivations against the rules of algebra Note: x=x-vt StatementResults Input Values We dont 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 variable We can state: x is dependent upon x and t.

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… Understanding why this is the correction to the problem requires a new perspective This new perspective explains each of these independently

To correct Einsteins derivation, t is replaced with t (in his partial differential equation) followed by performing the four algebraic steps, resulting in the transformation Begin with: Since Substitute with: Since Substitute with: Since Substitute with: This revised derivation produces the SAME equation as Einstein

We can retest the new derivation to determine if a problem exists Note: x=x-vt StatementResults Input Values We get the same result

While the problem occurs during the derivation, it shows up in Einsteins equation. Einsteins original time transformation simplification is provided in column one and the corrected simplification is provided in column two. 12 When it simplifies as: When it simplifies as: Any new model must explain this corrected time equation.

Since Substitute with: Einsteins Conflict between Independent and Dependent Variables 1234 Begin with: Since Substitute with: Since Substitute with: 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. EquationKnownNew Findings

What does this mean? Einsteins 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 Einsteins 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) Einsteins 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) DomainNewtonSRTCICS Fixed-Point Equations Wave-Front Equations N/A One-Half Wave Oscillation Equations N/A Ether-Based Model Agrees with models interpretation of SRT experimental data (*) Developing System Equations Comparison Table !

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An Unusual Paragraph For Todays 3:00 Talk This is an unusual paragraph. I'm curious how quickly you can find out what is so unusual about it. You probably won't, at first, find anything particularly odd or unusual or in any way dissimilar to any ordinary composition. It looks so plain you would think nothing was wrong with it. In fact, nothing is wrong with it! But it is unusual. Why? Study it, and think about it, but you still may not find anything odd. But if you work at it a bit, you might find out! Try to do so without any coaching! No doubt, if you work at it for long, it may dawn on you. Who knows? Go to work and try your skill. Good luck. If you find it, dont say it. Thank you.

It aint what you dont know that gets you in trouble. Its what you know for sure that just aint so. M ARK T WAIN

Challenging SRT Identification of a specific fault (Ideally one that other can agree with) A change to correct the fault (Does not throw the baby out with the bath water) The new equations conform to the experimental evidence (This does not mean must agree with previous conclusions) A revised set of postulates and/or a revised theoretical explanation of the new equations. (Should offer fewer constraints and paradoxes than SRT).

Historical Pitfalls Challenging the paradoxes (e.g., twin paradox) Challenging the terms and definitions (e.g., simultaneity) Changing the math (e.g., introducing new variables) Introduction of a new perspective - first (e.g., redefining equations or variables). Whats needed is a challenge that avoids the historical pitfalls.

Why Must The Problem Be Corrected? Because the Xi derivation is not mathematically sound There must be algebraic traceability Because the Tau equation is not correct. The Tau equation must be set up properly in order for it to be simplified properly.

Some Interesting Questions Does this mean you go back to the Newtonian equations for everything? Why does Einstein arrive at the same equations in his other derivations? (e.g., Hotel puzzle) Are you saying that everyone who arrives at Einsteins equations are making the same mistake?