ד"ר צבי סקאר

1985, B.A. in Mathematics and Philosophy, University of Tennessee at Chattanooga, USA, graduated magna cum laude with Honors in Philosophy.                               

1989, M.A. in Mathematics, University of California, Berkeley, USA.  Thesis: On Sierpinski’s Sets, supervisor: Jaime Ihoda (Chaim Judah)

2000, Ph.D. in Mathematics, Bar Ilan University, Ramat Gan, ISRAEL.  Thesis: Compactifications of G-spaces, supervisor: Professor Hillel Furstenberg. 

 August 1988 – December 1989: Graduate Teaching Assistant, Department of Mathematics and Department of Astronomy, University of California, Berkeley, USA.

Courses: Discrete Mathematics, Introduction to Astronomy

 February 1993 – June 1993: Exercise grader (בודק תרגילים), Department of Mathematics, Bar Ilan University, Ramat Gan, ISRAEL

Course:  Mathematical Logic

October 1993 – September 1999: Graduate Teaching Assistant, Department of Mathematics, Bar Ilan University, Ramat Gan, ISRAEL

Courses: Calculus I and II, Topology, Logic for Computer Scientists

 

October 1997 – August 1999: Teaching assistant (מתרגל), Department of Applied Mathematics, Jerusalem College of Technology (Machon Lev), Jerusalem, ISRAEL

Courses: Calculus I and II, Complex Variables, Linear Algebra I, Linear Algebra for Business Majors, Differential Equations

  

 August 1999 – present: Lecturer (מרצה), Department of Applied Mathematics, Jerusalem College of Technology (Machon Lev), Jerusalem, ISRAEL

Courses: Precalculus, Calculus I and II, Linear Algebra I and II, Discrete Mathematics, Differential Equations, Probability, Statistics, Partial Differential Equations, Mathematical methods in physics; teaching assistant (מתרגל) for Seminar in Modern Physics (August 2004)

a. Summary of Past Research and Development Activities

 

1. Equivariant Topology: A G-space is G-Tychonoff if it can be equivariantly embedded into a compact Hausdorff G-space. A group G is called a V-group if every Tychonoff G-space is G-Tychonoff. In 1988, M. Megrelishvili constructed the first example of a non-V-group. This was the only known example until 1998, when M. Megrelishvili and I developed a systematic method of constructing non-V-groups.  We showed that the class of non-V-groups is large and proved the following theorem:

 

Theorem (Megrelishvili, Scarr)  If G is an -bounded topological group which is not locally precompact, then G is not a V-group. 

 

This theorem establishes the existence of monothetic (even cyclic) non-V-groups, answering a question of Megrelishvili.  We also obtained a characterization of locally compact groups in terms of G-normality. We also answered a question of E. Glasner by proving the following theorem:

 

Theorem  (Megrelishvili, Scarr)  Let G be a Polish group. Then G is a V-group if and only if G is locally compact.

 

1. Becker and A. Kechris proved that for any Polish group G, there is a universal Borel G-space which is a compact Polish G-space. The situation for topological G-spaces is strikingly different, as we proved in the following theorem, which had been conjectured by Kechris:

 

Theorem  (Megrelishvili, Scarr)  Let G be a Polish group. Then every Polish G-space can be topologically embedded into a compact Polish G-space if and only if G is locally compact.

 

The proof of this theorem uses the strong Choquet game from descriptive set theory.

 

We also studied the actions of non-Archimedean groups on zero-dimensional spaces and proved the following theorem:

 

Theorem  (Megrelishvili, Scarr)  Let G be a non-Archimedean and second countable group, and let X  be a compact, metrizable, zero-dimensional G­-space.  Let  be the natural (evaluation) action of the full group of homeomorphisms of the Cantor cube . Then

   (1)  there exists a topological group embedding , and

   (2)  there exists an embedding , equivariant with respect to , such that                  

          is an equivariant retract of  with respect to  and .

 

 

 

 

2. Covariant Uniform Acceleration: What is the correct defining equation for “uniform acceleration” in flat spacetime? The physical definition is universally agreed upon: uniform acceleration means “motion whose acceleration is constant in the comoving inertial frame.” But how can we best capture this definition mathematically? This question bothered Max Planck, who noticed that the obvious candidate  for a four-dimensional relativistic dynamics equation has no solution when the four-force F is a constant. Planck wrote to Albert Einstein about this problem. Einstein promptly submitted a correction to one of his papers. In the correction, Einstein states that the “concept ‘uniformly accelerated’ needs further clarification.”

 

     A better candidate is the covariant equation 

                                                       ,

where  is the four-acceleration. However, current techniques have produced only one-dimensional hyperbolic motion as solutions to this equation.  There are clearly some additional solutions, since this equation is covariant, while hyperbolic motion is not.

 

     Yaakov Friedman and I proposed the relativistic dynamics equation

                                                     

where u is the four-velocity and  is a rank 2 antisymmetric tensor. We then defined a motion to be uniformly accelerated if it satisfies equation (1) for constant A. We then computed explicit solutions of equation (1). The solutions are divided into four Lorentz-invariant classes: null, linear, rotational, and general.  For null acceleration, the worldline is cubic in the time. Linear acceleration covariantly extends one-dimensional hyperbolic motion, while rotational acceleration covariantly extends pure rotational motion.

 

     At this stage of the research, the following was still an open question:

 

Question 1  Does equation (1) model all uniformly accelerated motions?

 

We would eventually answer this question positively, but first we had to extend equation (1) to a uniformly accelerated frame.  The solutions   to

                                           

form an orthonormal basis to a one-parameter family of inertial frames which are instantaneously comoving to a uniformly accelerated observer. In this so-called generalized Fermi–Walker frame, the solutions to equation (1) have constant acceleration.

 

     Under B. Mashhoon’s Weak Hypothesis of Locality, we obtained explicit local spacetime transformations from a uniformly accelerated frame to an inertial frame. These transformations extend the Lorentz transformations. We also proved the following surprising result:

 

Theorem  (Friedman, Scarr)  Let  and  be uniformly accelerated frames with the same acceleration tensor .  Then the spacetime transformations between  and  are Lorentz.

 

Application  The spacetime transformations between an observer at rest on the Earth and an airplane flying at constant velocity are the Lorentz transformation, since both systems are uniformly accelerated with respect to the gravitational field of the Earth.

 

         

     Our results on spacetime transformations led to the following question:

 

Question 2  Do the spacetime transformations between uniformly accelerated systems form a group?

 

Although we have some preliminary indications that the answer is no, this question is still open.

 

     Our definition (equation (2)) of the uniformly accelerated frame is a system of uncoupled differential equations.  This means that they are relatively easy to solve.  B. Mashhoon, on the other hand, uses a coupled system defined by the Frenet frame.  The advantage of Mashhoon’s approach is that it is frame independent, while our definition is with respect to a particular inertial frame.  We showed, however, that the two approaches yield the same solutions.  This equivalence led to the answer to Question 1:

 

Theorem  (Friedman, Scarr)  A motion is uniformly accelerated if and only if it satisfies equation (1).

 

Thus, equation (1) provides a complete and covariant description of uniformly accelerated motion. 

 

     Continuing to develop the theory, we adapted the off-shell technique of L. Horwitz and C. Piron to obtain both velocity and acceleration transformations from a uniformly accelerated frame to an inertial frame.  We also obtained the time dilation between clocks in a uniformly accelerated frame.  The power series expansion of our time dilation formula contains all of the usual terms, but also an additional term that had only been obtained previously in Schwarzschild spacetime. We applied these results to the case of an accelerated charge and obtained the Lorentz-Abraham-Dirac equation

                                              .

 

     The next step was to extend our theory to curved spacetimes.  Given an arbitrary curved spacetime, we constructed a system of non-linear first-order differential equations which extends the geodesic equation and whose solutions are precisely the uniformly accelerated motions in the given spacetime.

This improved a result of D. de la Fuente and A. Romero, whose corresponding equation models only hyperbolic motion.  We consider the particular case of radial motion in Schwarzschild spacetime and show that in this situation, there are no bounded orbits. 

 

 

3. Relativistic Rotations: 

 

The Problem   Consider a disk rotating with constant angular velocity with respect to an inertial frame.  Compute the spacetime transformations from the disk to the inertial frame.

 

This problem has been debated for over 100 years and continues to the present day. The recent book of Rizzi and Ruggerio makes it clear that there is still no universally accepted theory.  Moreover, many of the current approaches make arbitrary assumptions about the form of the transformation of the

radius of the disk, the radial coordinate. There is still no theory which derives the transformations from first principles.

 

     In our paper, Yaakov Friedman and I derive explicit spacetime transformations using only the basic tenets of Special Relativity, the inherent symmetries of the problem, and our previous results on uniform acceleration.  We do not make any arbitrary assumptions about the form of the transformations.

 

 

 

We avoid the horizon problem

 

     Our transformations imply that the speed, with respect to an inertial frame, of a rest point on the disk at any radius does not exceed the speed of light. This implies, contrary to standard reasoning, that

Special Relativity does not impose a limitation on the size of a rotating object. That is, we avoid

the horizon problem.  According to the horizon problem, global rigid rotations are forbidden in

relativistic theories, because at large distances, the angular velocity will exceed the speed of

light. In our approach, the horizon problem does not arise. We obtain a global reference frame, with no limit of the distance from the axis of rotation, and all speeds are bounded by the speed of light.

 

No time gap 

 

     We show how to synchronize the clocks on the disk, not only to each other, but also to

the clocks in the lab frame. Clock synchronization on a rotating disk was heretofore thought

impossible because a time gap invariably arises. We give a clear description of our synchronization procedure and explain how to avoid the time gap.  We were also lead to the following conjecture:

 

Conjecture  (Friedman, Scarr)  A system is uniformly accelerated if and only if all of the clocks in the system can be synchronized to each other, and the clocks will remain synchronized as long as the acceleration remains uniform.

 

This conjecture is still open.

 

4. Relativistic Newtonian Dynamics (RND)

 

     Bernhard Riemann was a strong advocate of a geometric approach to physics. His lifelong dream was to develop the geometry to unify the laws of electricity, magnetism, light and gravitation.  In fact, Riemann believed that the forces at play in a system determine the geometry of the system. In other words, force equals geometry.

 

     General Relativity (GR) is a direct application of “force equals geometry.”  Consider a particle freely falling towards the Earth.  In Newtonian Dynamics, the particle accelerates in response to the force of gravity, but in GR, the same particle has zero acceleration as it moves along a geodesic, a straight line in a curved spacetime.  Gravity is no longer a force.  It changes the geometry of spacetime from flat to curved and creates a new stage for particles to move on.   

 

     On the other hand, an electric field does not create a common stage on which all particles move.  Indeed, a neutral particle does not feel any electric force at all. The way spacetime curves due to an electric field depends on both the field and intrinsic properties of the object.  This leads to the following question:

 

Question  Can Riemann's principle of “force equals geometry” be applied to other forces?

 

In Relativistic Newtonian Dynamics (RND), the answer is “yes.” We first introduce the relativity of spacetime. This means that spacetime is an object-dependent notion. An object lives in its own spacetime, its own geometric world, which is defined by the forces which affect it. For example, in the vicinity of an electric field, a charged particle and a neutral particle exist in different worlds, in different spacetimes. In fact, for the neutral particle, the electric field does not exist. Likewise, in the vicinity of a magnet, a piece of iron and a piece of plastic live in two different worlds.

 

     The next new idea stems from the observation that an inanimate object has no internal mechanism with which to change its velocity. Hence, it has constant velocity, or zero acceleration, in its own world (spacetime). This lead us to formulate a new principle, the Generalized Principle of Inertia, which unifies Newton's first and second laws and states that:  An inanimate object moves freely, that is, with zero acceleration, in its own spacetime, whose geometry is determined by all of the forces affecting it.

 

     In GR, a freely-falling object in a gravitational field moves along a geodesic determined by the metric of the spacetime. With respect to this metric, the object's acceleration is zero. The Generalized Principle of Inertia extends this idea so that every object moves along a geodesic in its spacetime.  This geodesic is with respect to an appropriate metric, which we call the metric of the object's spacetime. The metric in GR is determined solely by the gravitational sources, while in our model, the metric of the object's spacetime depends on all of the forces affecting the object. In the case of static, conservative forces, the metric will depend only on the potentials of these forces.

 

     In the particular case of an object moving in a static, spherically symmetric, conservative force field, the explicit form of the metric of the object's spacetime may be derived from a set of Euler-Lagrange type equations and conservation laws, in addition to spherical symmetry and a carefully defined classical limit. For the gravitational force, this metric turns out to be the Schwarzschild metric. We thus obtain a simple derivation of the Schwarzschild metric, one that does not require the field equations of GR.  Furthermore, the resulting dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and exactly reproduces the following tests of GR:

 

• the gravitational time dilation,
• the anomalous precession of Mercury,
• the periastron advance of binary stars,
• gravitational lensing,
• the Shapiro time delay.

 

Thus, RND is a viable alternative to GR and, moreover, has a much simpler mathematical derivation. In addition, the techniques of RND are applicable to all conservative force fields, not just gravitation.

 

1. Summary of Current Research and Development Activities

 

GR versus RND   

 

We apply RND to a static, spherically symmetric gravitational field, without assuming that the metric is diagonal. Spherical symmetry implies that all off-diagonal components are zero except , the coefficient of drdt. We construct an infinite series of metrics, analytic everywhere except , for which .   The dynamics of these metrics all lead to the same trajectories as in the Schwarzschild model and therefore pass all classical test of GR. However, GR and RND predict different velocities on the trajectories, both for massive objects and massless particles. The total time for a radial round trip of light in RND is the same as in the Schwarzschild model, but RND allows for light rays to have different speeds propagating toward and away from the massive object. One of these metrics keeps the speed of light toward the object to be c. We are designing experiments, some of which may be possible on a chip, to test whether .

 

RND extends to multiple non-static forces, each of which obeys an inverse square law and whose field propagates at the speed of light.

 

 

    

 

 

1. Future Directions for Research and Development

 

1. Extending RND 

 

          RND needs to be extended to non-conservative forces, in particular, the magnetic force, and ultimately, the self-force.  We also want to extend RND to the microscopic region.  One possible approach is to use a complex scalar prepotential which is in fact a wave function.  We already have some preliminary results.

 

2. Relativity based on Symmetry

             

    We explore the role of symmetry in deriving the theory of Special Relativity (SR).  The Lorentz transformations are isometries preserving the Minkowski metric. One then defines velocity addition and shows that the domain  of all relativistically admissible velocities in SR is a bounded symmetric domain with respect to the group P of projective maps. The invariance of the metric between inertial frames enables one to use the Newtonian limit and the Lie algebra of P to derive a 3D relativistic dynamics equation.  This equation canonically embeds into a fully covariant 4D version. 

 

   If one describes relative motion using symmetric velocities, the corresponding bounded domain is symmetric with respect to the conformal maps. This description leads to a spin-half representation of the Lorentz group and is useful in obtaining analytic solutions of the relativistic dynamics equation.

 

   I will develop these ideas fully in a paper for the special issue of the journal Symmetry entitled “Relativity based on Symmetry.”  I plan to present the paper at the 2^nd International Conference on Symmetry in Benasque, Spain, in September 2019. 

    

a. Peer-Reviewed Papers in Refereed Journals

 

Before latest appointment

 

1. Michael Megrelishvili and Tzvi Scarr, Constructing Tychonoff G-spaces which are not

         G-Tychonoff, Topology and its Applications 86 (1998), 69 – 81, 12 pages. Impact factor 0.549,

         citations 18, SCImago Q2

 

2. Tzvi Scarr, Topological and Borel compactifications of Polish G-spaces, Topology and its                   

         Applications 105 (2000), 113 – 119, 7 pages. Impact factor 0.549, citations 2,

         SCImago Q2

 

3. Michael Megrelishvili and Tzvi Scarr, The Equivariant Universality and Couniversality

         of the Cantor Cube, Fund. Math. 167 (2000), 269 – 275, 7 pages. Impact factor 0.609, citations

         10, SCImago Q3

 

4. Yaakov Friedman and Tzvi Scarr, Making the Relativistic Dynamics Equation

         Covariant: Explicit Solutions for Constant Force, Phys. Scr. 86 (2012) 7 pages. 065008.

         Impact factor 1.032, citations 15, SCImago Q3

 

5. Yaakov Friedman and Tzvi Scarr, Spacetime Transformations from a Uniformly

         Accelerated System, Phys. Scr. 87 (2013) 8 pages. 055004. Impact factor 1.296, citations 10,

         SCImago Q3

 

   Since latest appointment (October 2014)

 

6. Yaakov Friedman and Tzvi Scarr, Uniform Acceleration in General Relativity, Gen. Rel. Grav.  

        47:121 (2015) 19 pages. Impact factor 1.668, citations 14, SCImago Q2

 

7. Tzvi Scarr and Yaakov Friedman, Solutions for Uniform Acceleration in General Relativity, 

         Gen. Rel. Grav.  48:65 (2016) 9 pages. Impact factor 1.618, citations 4, SCImago Q2

 

8. Yaakov Friedman, Tzvi Scarr and Joseph Steiner, A geometric relativistic dynamics under any

        conservative force, Int. J. Geom. Meth. Mod. Phys. 16 (2019) 1950015. 17 pages. Impact

        factor 1.068, citations 0, SCImago Q2  

 

9. Yaakov Friedman and Tzvi Scarr, Relativity from the geometrization of Newtonian dynamics, 

        Europhys. Lett.  125 (2019) 49001 7 pages. Impact factor 1.957, citations, SCImago Q2 

 

 

1. Peer-Reviewed Papers in Refereed Conference Proceedings 

 

Before latest appointment

 

       1.Tzvi Scarr, Loops in Relativistic Dynamics, Proceedings of the Mile High Conference on    

         Quasigroups, Loops and Nonassociative Systems (July 2005) in Quasigroups

         and Related Systems 14 (2006), 91 – 109, 19 pages. Impact factor n/a, citations 0, SCImago

         n/a

 

2. Yaakov Friedman and Tzvi Scarr, Covariant Uniform Acceleration, Jour. Phys. 

         Conference Series 437 (2013) 012009, “The 8^th Biennial Conference on Classical and 

         Quantum Relativistic Dynamics of Particles and Fields (IARD 2012)” 27 pages. Impact

         factor n/a, citations 7, SCImago Q3

 

Since latest appointment (October 2014)   

 

3. Tzvi Scarr and Yaakov Friedman, A complete characterization of Relativistic Uniform   

           Acceleration, Jour. Phys. Conference Series of “The 10^th Biennial Conference

           on Classical and Quantum Relativistic Dynamics of Particles and Fields (IARD 2016)”

           12 pages. Impact factor n/a, citations 0, SCImago Q3

 

4. Yaakov Friedman and Tzvi Scarr, Geometrization of Newtonian Dynamics,

           Jour. Phys. Conference Series of “The 11^th Biennial Conference on Classical and

           Quantum Relativistic Dynamics of Particles and Fields (IARD 2018)” 19 pages. Impact

           factor n/a, citations 0, SCImago Q3

 

5. Yaakov Friedman, Tzvi Scarr, and Shmuel Stav, The Relativity of Spacetime and Geometric

          Relativistic Dynamics, submitted to Proceedings of the Fifteenth Marcel Grossman Meeting

          on General Relativity, edited by Elia Battistelli, Robert T. Jantzen, and Remo Ruffini, Open

          access e-book proceedings, World Scientific, Singapore, 2019

 

    

 

1. Books

 

      Before latest appointment 

 

1. Yaakov Friedman with the assistance of Tzvi Scarr, Physical Applications of

         Homogeneous Balls, Progress in Mathematical Physics, volume 40 (2005),

         Birkhauser, Boston, 278 pages.

  Before latest appointment

 

         2011-2012, German–Israel Foundation for Scientific Research and Development (GIF), 

           research assistant

Before latest appointment

 

September 2003 Completed the Dale Carnegie course on Interpersonal Relationships, Jerusalem

 

July 2004    Attended the “Non-commutative Geometry and Representation Theory in  

                    Mathematical Physics” Satellite Conference to the Fourth European

                    Congress of Mathematics, Karlstad University, Karlstad, Sweden

                    Title of Talk: Conformal Triumphs over Projective

 

July 2005   Attended the “Mile High Conference on Quasigroups, Loops and Nonassociative                    Systems” University of Denver, Denver, Colorado, USA

                   Title of Talk: Right versus Left: Relativistic Dynamics in the Einstein Loop

 

July 2006 Attended a COMSOL Multiphysics (FEMLAB) Minicourse, Bnei Brak

 

 

2010-2011 I wrote a 90-page חוברת for our Statistics courses (both the engineering course and the business course).  These notes include theory, examples, and solved problems.

 

2011-2012  TMPJ Theoretical Mathematics and Physics Jerusalem Seminar.

                    I gave seven our-hour lectures in this weekly seminar held at Machon Lev.

 

February 2012   I served on the Organizing Committee of the 2nd GIF Workshop “Exploring the Full

                   Range of Classical Electrodynamics: from Applied Physics to General Relativity”

                   Jerusalem College of Technology (JCT), Jerusalem, Israel

                   I also gave a talk: Covariant Uniform Acceleration

 

May 2012  Attended “The 8^th Biennial Conference on Classical and Quantum Relativistic

                   Dynamics of Particles and Fields (IARD 2012)” Galileo Galilei Institute

                   for Theoretical Physics, Florence, Italy

                   Title of Talk: Extending the Lorentz Transformations to Accelerated Systems         

 

December 2012  Attended “The 58^th Annual Israel Physical Society Meeting (IPS 2012)”

                   Title of Poster: Making the Relativistic Dynamics Equation Covariant

 

June 2013  Attended the 3rd GIF Workshop “Exploration of Electrodynamics” ZARM,

                   Universitat Bremen, Bremen, Germany

                   Title of Talk: Velocity and Acceleration Transformations and Time 

                   Dilation in a Uniformly Accelerated Frame           

 

July 2014   I served on the Organizing Committee of the 4th GIF Workshop “Exploring the Full   

                   Range of Classical Electrodynamics: from Applied Physics to General Relativity”

                   Jerusalem College of Technology (JCT), Jerusalem, Israel.

                   Title of Talk:  Relativistic Rotations         

 

 Since latest appointment

 

June 2016  Attended “The 10^th Biennial Conference on Classical and Quantum Relativistic

                   Dynamics of Particles and Fields (IARD 2016)” Ljubljana, Slovenia

                   Title of Talk: The Frenet Frame and Applications

 

February 2017 Attended workshop on “Rotating Systems in Relativity” Lev Academic Center,             

                   Jerusalem, Israel.

                   Title of Talk: Avoiding the Horizon Problem

 

June 2018  Attended “The 11^th Biennial Conference on Classical and Quantum Relativistic

                   Dynamics of Particles and Fields (IARD 2018)” Merida, Yucatan, Mexico

                   Title of Talk: Geometric Dynamics

 

September 2019  I have registered for the 2^nd International Conference on Symmetry in Benasque,

                   Spain and have submitted an abstract for my talk entitled “Special Relativity from

                   Symmetry.”