Steven Bryant Natural Philosophy Alliance 14 th Annual Conference University of Connecticut

1. A Brute-Force Challenge to Einstein’s 1905 Derivation Reveals a Mathematical Inconsistency in the SRT Time Transformation Equation Steven Bryant Natural Philosophy Alliance 14th Annual Conference University of Connecticut Storrs, Connecticut May 20-25, 2007 Steve.Bryant@RelativityChallenge.Com www.RelativityChallenge.com

2. Objectives • Convince you that there is a mathematical problem in Einstein’s 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.

3. 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.

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

5. Input Values Transformation Equations Output Values Einstein’s transformation equations takes a set of input values and produces a set of output values.

6. Einstein performs several steps to create the equations that are then “normalized” to produce his final transformation equations. 1 2 3 4 Implicit Explicit Explicit Each equation is multiplied by: Einstein performs several steps to create the transformation equations. These steps are given in Section 3 of his 1905 paper.

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

8. Einstein performs four algebraic steps to produce his transformation. 1 2 3 4 Since Substitute with: Since Substitute with: Begin with: Since Substitute with: This slide must be true if you believe that Einstein’s SRT equations are right

9. Einstein’s 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

10. Validating Einstein’s derivations against the rules of algebra Input Values Statement Results 1 Note: x’=x-vt 2 We don’t get the same result Note: x’=x-vt 3 4

11. 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.

12. 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.

13. Dependent and Independent Variables 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. 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 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.

14. Fixing the problem The correction is to replace t with t’in setting up the partial differential equation , which produces the linear function… This new perspective explains each of these independently Understanding “why” this is the correction to the problem requires a new perspective

15. 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 Since Substitute with: Since Substitute with: Begin with: Since Substitute with: This revised derivation produces the SAME equation as Einstein

16. We can retest the new derivation to determine if a problem exists. Input Values Statement Results 1 Note: x’=x-vt 2 We get the same result Note: x’=x-vt 3 4

17. 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 When it simplifies as: When it simplifies as: Any new model must explain this corrected time equation.

18. Einstein’s Conflict between Independent and Dependent Variables 1 2 3 4 Since Substitute with: Since Substitute with: Begin 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’

19. Why is this a hard problem to identify? Equation Known New Findings 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.

20. 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

21. For Further Exploration…The Model of Complete and Incomplete Coordinate Systems (CICS) • 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. This mathematical finding requires a new model

22. For Further Exploration…The Model of Complete and Incomplete Coordinate Systems (CICS) System Equations Comparison Table !

23. For More Information www.RelativityChallenge.Com Website Podcasts Papers Yahoo! Groups FAQ Steven Bryant Steve.Bryant@RelativityChallenge.Com

24. Thank you