About Gali Weinstein

I am specializing in the foundations of physics: special and general theories of relativity. Galina Weinstein.

Review of “Genius”: National Geographic

I watched the first episode of National Geographic’s drama. The episode jumps between a few different periods in Einstein’s life: Einstein the rebel and celebrity (“professor Einstein”) during the rise of Nazism in Germany and the young stubborn and rebel Einstein in Switzerland.

The first episode opens in 1922 Berlin, with the assassination of the German foreign minister Walter Rathenau. Then Einstein is getting intimate with his secretary Betty Neumann. Einstein is shown with his pants down. Subsequently his lover Betty is pressed against the blackboard and he holds the chalk on a blackboard full of chalk-drawn formulas.


Indeed, Einstein was not a saint. He was a liberal humanist and pacifist and he exploited his fame to advance anti-war cause and save Jews from the Nazi regime and oppose Nazism and later McCarthyism in America. However, he was constantly romantically entangled with other women. He divorced from Mileva whom he had mistreated when their marriage was on the rocks. He soon married his cousin, Elsa. He was not deeply in love with her and it seems she was eager to get married with Albert Einstein, the celebrity and genius, and he was simply drawn into marrying her. When Einstein hired Betty Neumann as his secretary, of course they immediately began an affair.


Elsa seemed to have turned a blind eye when he cheated because she enjoyed the attention and fame, being Albert Einstein’s wife, and he enjoyed the freedom to be with other women. He needed Elsa to take care of him and understand his needs (including his romantic needs with other women). First he married Mileva. He thought she would understand him, be his lover and sounding board, because she was a physics student. However, he finally realized that he needed a caregiver.

Geoffrey Rush as the older version of Albert Einstein is not exactly Einstein but he plays Einstein’s role very well. Einstein, however, was a quirky, weirdly shabby dressed genius:


He did not bother to shave, to comb his hair, to dress properly because he believed all this was a waste of time. He had a great sense of cynical humor and he was a rebel even as a grownup. The older Einstein of “Genius” does not exactly trap the special looks and personality of Einstein. Rush is more masculine than the real Einstein. Look at the photos of Albert Einstein:


History of the red star line.

In the BBC film Einstein and Eddington Einstein, the young genius, is played with lusty relish by Andy Serkis (New Scientist). I would combine the two actors, Rush and Serkis, and the result would be quite a good representation of Einstein.

Back to the 1922 blackboard.


What are these formulas?


Quirky formulas. These expressions neither look like equations of general relativity nor like a static universe line element – Einstein’s cosmological model from 1917 until 1930. Do they represent a version of Einstein’s unified field theory? In 1922 Einstein was only starting to develop his unified field theory. It seems the producers have copied random formulas from a certain document.

The National Geographic science consultant is the physicist Prof. Clifford Johnson.

Johnson tells what was it like advising to Genius. Here.


This is the Einstein tensor (Einstein’s field equations):




On the right-hand side one finds the value of:




Einstein though did not write his field equations in this form, at least not before 1919. And he added the condition that the above field equations are valid in unimodular coordinates:


This condition is not written on the blackboard.

There is no cosmological constant in these field equations. Until 1931 Einstein added the cosmological constant to his field equations.

And here on the bottom left-hand side of the photo, the right-hand side of the other blackboard, one sees the Ricci scalar:



The Ricci scalar is the second term on the left-hand side of the Einstein tensor.

When did the lecture take place?

The field equations on the blackboard:

Einstein tensor

do not include the cosmological constant. In 1917 Einstein modified his field equations to include the cosmological constant and he gave up this constant in 1931-1932. Hence, the lecture could either take place in Berlin 1916 or in California after 1931.

First, here Einstein had drawn on the blackboard the vanishing Ricci tensor:


Setting the Ricci tensor equal to zero is writing the vacuum field equations (field equations for the gravitational field in empty space).  Space is empty: There is no matter present and there are only gravitational fields. This perfectly makes sense if Einstein is lecturing in 1916. However, Einstein does not look like the young Albert Einstein. He thus must be lecturing in 1932 in California and not in Berlin.

Therefore, Einstein is lecturing in 1932 because in 1932 Einstein and de-Sitter suggested the Einstein de-Sitter model (a variant of Friedmann’s expanding universe) by assuming a universe with a cosmological constant equal to zero and “without introducing a curvature at all… we suppose the curvature to be zero” (i.e. a vanishing spatial curvature). In 1932 Einstein came to Pasadena and there with Willem de Sitter they worked on their joint paper. In Pasadena he thus asked whether the Ricci tensor could be set equal to zero:



In 1932 Einstein was 53 years old. He did not come back to Germany. Thus he could not have given a lecture in Berlin.

On the other side of the blackboard, at the bottom of the blackboard, on the right-hand side, one sees the Christoffel symbols (“the components of the gravitational field”). These should not vanish:



In the middle of the blackboard one sees the Minkowski spatial flat metric of special relativity:



The components of the metric tensor reduce to this Minkowski flat metric.

On top of the blackboard on the right-hand side, Einstein’s line element:


Beneath the Einstein tensor of general relativity one sees the time dilation formula from special relativity. It is not directly related to the Einstein tensor and especially it is written in a special relativistic form, i.e. in a coordinate-dependent form, not in a form of a metric theory of general relativity. It is thus completely unrelated to the other formulas on the blackboard:



Einstein wrote in the 1930s such formulas on blackboards. Generally, he never mixed on one blackboard special relativity (in coordinate-dependent form) with general relativity (in metric form).

The younger Einstein, Johnny Flynn is rather compelling. He captures Einstein’s charm quite well. The scene of the beam rider thought experiment:




was inspired by Carl Sagan’s memorable series Cosmos, the episode on the twin paradox:


Second episode: There are many historical inaccuracies in the second episode. Heinrich Friedrich Weber, Einstein’s physics professor is presented as someone who understands Einstein: The young Weber was also an impudent rebel. Poor Weber, he had to deal with Einstein, the rebel who flirted and mistreated Mileva Maric. Einstein thought that Weber’s courses where a masterpiece. Weber only wanted to help Einstein and suffered the consequences of being Einstein’s professor. That is the reason why he eventually turned on Einstein.

This is inaccurate. Weber seemed to have a particular dislike for Einstein. At the Zürich Polytechnic, Einstein could not easily bring himself to study what did not interest him and he skipped classes, especially those of mathematicians. He did not persuade Mileva to do the same thing (in the second episode one sees Einstein between the sheets persuading Mileva to skip classes). It seems to me pure invention.

Although Einstein had skipped classes and Weber’s lectures were old-fashioned (the latter did not provide the latest studies in physics, e.g. Maxwell’s theory), most of his time Einstein spent on his own studying Maxwell’s theory and learning at first hand the works of great pioneers in science and philosophy: Boltzmann, Helmholtz, Kirchhoff, Hertz, Mach. It is not true that most of his time he spent with Mileva between the sheets.

Eventually, Einstein finished first in his class in the intermediate exams, because he studied very hard on his own. He borrowed Marcel Grossmann’s notebooks and learned very hard from these notebooks. Second after him was his note taker Grossmann. Although Grossmann worked hard, he was not as genius as Einstein.

After obtaining the diploma, when he sought university positions all over Europe, Einstein was rebuffed because it seems that Weber was against him. Weber had a particular dislike for Einstein: Einstein thought that Weber’s lectures were a little old-fashioned, that he was a mediocre and not creative because he had essentially ceased doing scientific research before Einstein even entered the Polytechnic.  During Einstein’s years at the Polytechnic, Weber published only one scholarly paper. Einstein told Mileva that Weber lectures are a masterpiece. He later realized, however, that Weber’s lectures were a masterpiece in history of physics rather than in physics.

Indeed, Weber told Einstein: “You’re a clever fellow! But you have one fault. You won’t let anyone tell you a thing”.  However, he did not appreciate Einstein enough, i.e. he did not understand the rebel Einstein. By his distrust of authority Einstein had alienated Weber, but Weber could not understand Einstein.

As to the Chubby professor Jean Pernet. Einstein had no prospects with him. He was completely not fond of Einstein and he told Einstein he had no idea how difficult was the path of physics and that he should try some other field instead.

אודיסאת איינשטיין ליחסות הכללית Einstein’s Odyssey to General Relativity

מאמר שלי על דרכו של איינשטיין לתורת היחסות הכללית: אודיסאת איינשטיין ליחסות הכללית

סיינטיפיק אמריקן ישראל

“Einstein’s Odyssey to General Relativity”, Scientific American Israel

את המונח “אודיסאה” ליחסות הכללית טבע פרופ’ ג’ון סטצ’ל מאוניברסיטת בוסטון והוא מייצג את המסע המפרך של איינשטיין בדרכו ליחסות הכללית. ראו המאמר של סטצ’ל למטה

Odyssey to general relativity is John Stachel’s memorable phraseology. See:

Stachel, John (1979). “Einstein’s Odyssey: His Journey from Special to General Relativity”. In Einstein from B to Z, 2002.

I am sorry but this piece is in Hebrew. You can read my book General Relativity Conflict and Rivalries, my papers on Einstein and general relativity and a short summary below.


מפייסבוק: מארחים את ד”ר גלי וינשטיין לדבר על איינשטיין


My drawing of Einstein:      האיור שלי של איינשטיין

איינשטיין צעיר

And the original (I tried as hard as I could to draw a young Einstein…):       המקור


The article discusses the following topics:

1907. The Happiest thought of my life.


1907-1911. The equivalence principle and elevator experiments.


1911. Deflection of light and explaining deflection of light using an elevator thought experiment.


1911-1912 (1916). The disk thought experiment, gravitational time dilation and gravitational redshift.


1912. The disk thought experiment and non-Euclidean geometry.


1912. Einstein to Marcel Grossmann: “Grossmann, you must help me or else I’ll go crazy!”. Grossmann searched the literature, and brought the works of Bernhard Riemann, Gregorio Curbastro-Ricci, Tullio Levi-Civita and Elwin Bruno Christoffel to Einstein’s attention. With Grossmann’s help Einstein searched for gravitational field equations for the metric tensor in the Zurich Notebook.


1913-1914. The Entwurf theory. In 1913, Einstein and Michele Besso both tried to solve the new Entwurf field equations to find the perihelion advance of Mercury.

2October 1915. Einstein realizes there are problems with his 1914 Entwurf theory. November 1915. Einstein’s competition with David Hilbert.


November 1915. Four ground-breaking papers: Einstein presents the field equations of general relativity, finds the advance of the perihelion of Mercury and predicts that a ray of light passing near the Sun would undergo a deflection of amount 1.7 arc seconds.

General Relativity without the Equivalence principle?

I have skimmed through this book Handbook of Spacetime:



The following represents my impressions formulated after reading the sections about the equivalence principle.

I read this paper:




However, Einstein did not write this wonderful passage in the letter to Robert Lawson. Here is the letter to Lawson (Einstein to Lawson, 22 January 1920):


Einstein writes to Lawson in the above letter: “The article for Nature is almost finished, but it has unfortunately become so long that I very much doubt whether it could appear in Nature“. Indeed, in a 1920 unpublished draft of a paper for Nature, “Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development”, Einstein wrote the above long paragraph describing him in 1907 sitting in the Patent Office. He was brooding on special relativity, and suddenly there came to him the happiest thought of his life:




Let us analyze this passage. The man in free fall (elevator experiments): Special relativity is incorporated into general relativity as a model of space-time experienced by an observer in free fall, over short times and distances (locally):

Between 1905 and 1907, Einstein tried to extend the special theory of relativity so that it would explain gravitational phenomena. He reasoned that the most natural and simplest path to be taken was to correct the Newtonian gravitational field equation. Einstein also tried to adapt the Newtonian law of motion of the mass point in a gravitational field to the special theory of relativity. However, he found a contradiction with Galileo’s law of free fall, which states that all bodies are accelerated in the gravitational field in the same way (as long as air resistance is neglected). Einstein was sitting on a chair in my patent office in Bern and then suddenly a thought struck him: If a man falls freely, he would not feel his weight. This was the happiest thought of his life. He imagined an observer freely falling from the roof of a house; for the observer there is during the fall – at least in his immediate vicinity – no gravitational field. If the observer lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, regardless of their particular chemical and physical nature. The observer is therefore justified in interpreting his state as being (locally) at rest. Einstein’s 1907 breakthrough was to consider Galileo’s law of free fall as a powerful argument in favor of expanding the special principle of relativity to systems moving non-uniformly relative to each other. Einstein realized that he might be able to generalize and extend special relativity when guided by Galileo’s law of free fall. The Galilean law of free fall (or inertial mass is equal to gravitational mass) became known as the weak principle of equivalence.

Lewis Ryder explains: “Some writers distinguish two versions of the equivalence principle: the weak equivalence principle, which refers only to free fall in a gravitational field and is stated… as The worldline of a freely falling test body is independent of its composition or structure; and the strong equivalence principle, according to which no experiment in any area of physics should be able, locally, to distinguish a gravitational field from an accelerating frame”.

There are several formulations of the weak and the strong principles of equivalence in the literature. By far the most frequently used formulation of the strong principle of equivalence is Einstein’s 1912 local principle of equivalence: In a local free falling system special relativity is valid. (See my book General Relativity Conflict and Rivalries. Einstein’s Polemics with Physicists, 2015, for further details).

Nick Woodhouse explains:


in the chapter:


Hence Joshi says:


in the chapter:


Lewis Ryder


writes in the above paper:


(i.e. Einstein 1911 paper: “On the Influence of Gravitation on the Propagation of Light”). He formulates the equivalence principle in the following way: “In a freely falling (non-rotating) laboratory occupying a small region of spacetime, the local reference frames are inertial and the laws of physics are consistent with special relativity”. He then writes:


The equivalence principle enables us to find just one component g00 – of the metric tensor gmn. All components can be found (at least in principle) from the Einstein field equations. Ryder thus concludes that the equivalence principle is dispensable. I don’t quite agree with Ryder.

In my 2012 paper, “From the Berlin ‘Entwurf’ Field equations to the Einstein Tensor III: March 1916”, ArXiv: 1201.5358v1 [physics.hist-ph], 25 January, 2012 and also in my 2014 paper,  “Einstein, Schwarzschild, the Perihelion Motion of Mercury and the Rotating Disk Story”, ArXiv: 1411.7370v [physics.hist-ph], 26 Nov, 2014, I demonstrate the following:  On November 18, 1915, Einstein found approximate solutions to his November 11, 1915 field equations and explained the motion of the perihelion of Mercury. Einstein’s field equations cannot be solved in the general case, but can be solved in particular situations. Indeed, the first to offer an exact solution was Karl Schwarzschild. Schwarzschild found one line element, which satisfied the conditions imposed by Einstein on the gravitational field of the sun, as well as Einstein’s field equations from the November 11, 1915 paper. Schwarzschild sent Einstein a manuscript, in which he derived his exact solution of Einstein’s field equations. In January, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy, and a month later the paper was published. In March 1916 Einstein submitted to the Annalen der Physik a review article, “The Foundation of the General Theory of Relativity”, on the general theory of relativity. The paper was published two months later, in May 1916. The 1916 review article was written after Schwarzschild had found the complete exact solution to Einstein’s November 18, 1915 field equations. Even so, Einstein preferred not to base himself on Schwarzschild’s exact solution, and he returned to his first order approximate solution from November 18, 1915. In the final part of the 1916 review paper Einstein demonstrated that a gravitational field changes spatial dimensions and the clock period:


This equation is further explained in my 2012 paper (page. 56):


Neither did Einstein use the Schwarzschild solution nor was he guided by the  equivalence principle. He was rather using an approximate solution and the metric, the line element to arrive at the same factor he had obtained by assuming the heuristic equivalence principle. He thus demonstrated that the equivalence principle was a fundamental principle of his theory, because in 1912 he formulated an equivalence principle valid only locally  (see my book: General Relativity Conflict and Rivalries. Einstein’s Polemics with Physicists, 2015, p. 184). I further explain it below.

Ryder then explains: The equivalence principle is local (a complete cancelation of a gravitational field by an accelerating frame holds locally). However, over longer distances two objects in free fall at different places in a realistic gravitational field move toward each other and this does not happen in an accelerating elevator. The cancelation of the gravitational field by an accelerating field is thus not complete. According to general relativity this effect (tidal effect) is a consequence of the curvature of space-time:


Although the equivalence principle might have been a heuristic guide to Einstein in his route to the fully developed theory of general relativity, Ryder holds that it is now irrelevant.

I don’t agree with Ryder’s conclusion which resembles that of John Lighton Synge (and Hermann Bondi). Indeed the equivalence principle is not valid globally (i.e. for tidal effects). Although the strong equivalence principle can at best be valid locally, it is still crucial for the general theory of relativity:

  1. Einstein formulated an equivalence principle which is valid only locally. Special relativity is valid locally and space-time is locally the Minkowski space-time.
  2. The principle of equivalence is fundamental for a metric theory and for our understanding of curved space-time: Freely falling test bodies move along geodesic lines under the influence of gravity alone, they are subject to an inertio-gravitational field . The metric determines the single inertio-gravitational field (affine connection), and there is breakup into inertia and gravitation relative to the acceleration. According to the equivalence principle, the components of the affine connection vanish in local frames. John Stachel quotes a passage from Einstein’s letter to Max von Laue:


Stachel, John, “How Einstein Discovered General Relativity: A Historical Tale with Some Contemporary Morals”, Einstein B to Z, 2002.

Indeed Ryder quotes J. L. Synge :


Einstein’s equivalence principle was criticized by Synge:


Synge, J. L. (1960). Relativity: The General Theory (Amsterdam, The Netherlands: North Holland Publishing Co).

And Hermann Bondi reacted to Einstein’s principle of equivalence:


Bondi also said (‘NO SUCCESS LIKE FAILURE …’: EINSTEIN’S QUEST FOR GENERAL RELATIVITY, 1907–1920, Michel Janssen):



Other authors contributing to the Handbook of Spacetime write the following:

Graham S. Hall in his paper:


writes the following:


“The choice of a geodesic path (Einstein’s principle of equivalence) reflects the results of the experiments of Eötvös and others, which suggest that the path of a particle in a pure gravitational field is determined by its initial position and initial velocity”. This is not Einstein’s equivalence principle. This is the Galilean principle of equivalence or the weak equivalence principle.

And according to Vesselin Petkov:


the geodesic line is indeed a manifestation of Galileo’s free fall law:


Ryder presents tests for the equivalence principle. The operation of the global positioning system, the GPS, is a remarkable verification of the time dilation. The GPS system consists of an array of 24 satellites, which describe an orbit round the earth of radius 27,ooo km, and are 7000 km apart, and every 12 hours travel at about 4km/s.  Each satellite carries an atomic clock, and the purpose is to locate any point on the earth’s surface. This is done by sensing radio signals between the satellites and the receiver on the earth, with the times of transmission and reception recorded. The distances are then calculated. Only three satellites are needed to pinpoint the position of the receiver on the earth. Relativistic effects must be taken into account arising both from special relativity (time dilation: moving clocks on the satellites run slower than clocks at rest on the surface of the earth) and from general relativity (gravitational time dilation/gravitational frequency shift: when viewed from the surface of the Earth, clocks on the satellites appear to run faster than identical clocks on the surface of the earth). The combined effect (the special relativistic correction and the general relativistic correction) is that the clocks on the satellites run faster than identical clocks on the surface of the earth by 38.4 microseconds per day. The clocks thus need to be adjusted by about 4 x 10-10s per day. If this factor is not taken into account, the GPS system ceases to function after several hours. This provides a stunning verification of relativity, both special and general.

Neil Ashby dedicates his paper to the GPS:


and gives a critical reason why the equivalence principle is indeed relevant. Consider again the GPS (global positioning system) or generally, Global navigation satellite systems (GNNS). For the GPS or GNNS, the only gravitational potential of significance is that of the earth itself. The earth and the satellites fall freely in the gravitational field of the sun (and external bodies in the solar system). Hence, according to the equivalence principle one can define a reference system which is locally very nearly inertial (with origin at the earth’s center of mass). In this locally inertial coordinate system (ECI) clocks can be synchronized using constancy of the speed of light (remember that special relativity is incorporated into general relativity as a model of space-time experienced locally by an observer in free fall):


One writes an approximate solution to Einstein’s field equation and obtains that clocks at rest on earth


run slow compared to clocks at rest at infinity by about seven parts in 1010.

Unless relativistic effects on clocks [clock synchronization; time dilation, the apparent slowing of moving clocks (STR); frequency shifts due to gravitation, gravitational redshift(GTR)] are taken into account, GPS will not work. GPS is thus a huge and remarkable laboratory for applications of the concepts of special and general relativity. In addition, Shapiro signal propagation delay (an additional general relativistic effect) and spatial curvature effects are significant and must be considered at the level of accuracy of 100 ps of delay. Ashby mentions another effect on earth that is exactly cancelled:


Wesson in this paper:


presents the standard explanation one would find in most recent textbooks on general relativity:


The Christoffel symbols are also used to define the Riemann tensor, which encodes all the relevant information about the gravitational field. However, the Riemann tensor has 20 independent components, and to obtain field equations to solve for the 10 elements of the metric tensor requires an object with the same number of components. This is provided by the contracted Ricci tensor. This is again contracted (taking its product with the metric tensor) to obtain the Ricci curvature scalar.  This gives a kind of measure of the average intensity of the gravitational field at a point in space-time. The combination of the Ricci tensor and the Ricci scalar is the Einstein tensor and it comprises the left hand-side of Einstein’s field equations.

At every space-time point there exist locally inertial reference frames, corresponding to locally flat coordinates carried by freely falling observers, in which the physics of general relativity is locally indistinguishable from that of special relativity. In physics textbooks this is indeed called the strong equivalence principle and it makes general relativity an extension of special relativity to a curved space-time.

Wesson then writes that general relativity is a theory of accelerations rather than forces and refers to the weak equivalence principle:


As said above, Einstein noted that if an observer in free fall lets go of any bodies, they remain relative to him, in a state of rest or uniform motion, regardless of their particular chemical and physical nature. This is the weak principle of equivalence: The worldline of a freely falling test body is independent of its composition or structure. The test body moves along a geodesic line. The geodesic equation is independent of the mass of the particle. No experiment whatsoever is able, locally, to distinguish a gravitational field from an accelerating system – the strong principle of equivalence (see Ryder above). A freely falling body is moving along a geodesic line. However, globally space-time is curved and this causes the body’s path to deviate from a geodesic line and to move along a non-geodesic line. Hence we speak of geodesics, manifolds, curvature of space-time, rather than forces.

José G. Pereira explains the difference between curvature and torsion (and force) (see paper here):


General relativity is based on the equivalence principle and geometry (curvature) replaces the concept of force. Trajectories are determined not by force equations but by geodesics:


How do we know that the equivalence principle is so fundamental?  Gravitational and inertial effects are mixed and cannot be separated in classical general relativity and the energy-momentum density of the gravitational field is a pseudo-tensor (and not a tensor):


General relativity is grounded on the equivalence principle. It includes the energy-momentum of both inertia and gravitation:


In 1928 Einstein proposed a geometrized unified field theory of gravitation and electromagnetism and invented teleparallelism. Einstein’s teleparallelism was a generalization of Elie Cartan’s 1922 idea. Picture20

According to Pereira et al: “In the general relativistic description of gravitation, geometry replaces the concept of force. This is possible because of the universal character of free fall, and would break down in its absence. On the other hand, the teleparallel version of general relativity is a gauge theory for the translation group and, as such, describes the gravitational interaction by a force similar to the Lorentz force of electromagnetism, a non-universal interaction. Relying on this analogy it is shown that, although the geometric description of general relativity necessarily requires the existence of the equivalence principle, the teleparallel gauge approach remains a consistent theory for gravitation in its absence”.

See his paper with R. Aldrovandi and K. H. Vu: “Gravitation Without the Equivalence Principle”, General Relativity and Gravitation 36, 2004, 101-110.

Petkov explains in his paper: (see further above)


the following:


The bottom line is that classical general relativity is fundamentally based on the equivalence principle. One cannot reject Einstein’s route to the theory of general relativity.








My “friend” Albert Einstein on Fakebook

One day you wake up and find that you are Albert Einstein’s Friend on Fakebook. When did I first become friends with Prof. Einstein on Fakebook? In 1903 when he got married. Wow I must be 120 years old now (until 130 years old!), a super-centenarian. I have never heard of Fakebook until now. However, if you ever wondered what Fakebook was, wonder no longer. On Fakebook you create a fake profile for a fictional or historical character for educational purposes. Fakebook is a tool that can be found on a website called “ClassTools”.

יום אחד גיליתי שאני חברה של אלברט איינשטיין ברשת חברתית ששמה פייקבוק. מתי נעשיתי חברה שלך פרופ’ איינשטיין בפייקבוק? כאשר הוא התחתן ב-1903. אני צריכה להיות עכשיו בת 120 (עד 130). למען האמת אף פעם לא שמעתי על פייקבוק עד עכשיו. אם תהיתם מהי פייקבוק, אז פייקבוק היא רשת שבה יוצרים פרופיל פיקטיבי (פייק) של דמות היסטורית או דמיונית למטרות חינוכיות. להלן הקיר של איינשטיין בפייקבוק

Below is Einstein’s wall on Fakebook:




Thomas Kuhn and Hilary Putnam. “Cut the pie any way you like, ‘meanings’ just ain’t in the head!”

This post is in memory of Prof. Hilary Putnam who died 4 days ago at the age of 89.

Thomas Kuhn’s first edition book, Structure of Scientific Revolutions is a very famous book, and it has caused quite a stir among philosophers of science since its first publication in 1962. Kuhn introduced the idea of a paradigm; a paradigm governs, in the first instance, not a subject matter but rather a group of practitioners. What do members of an isolated community of specialists share that accounts for the relative fullness of their professional communication and the relative unanimity of their professional judgment? To that question, the 1962 Structure licenses the answer, a paradigm or set of paradigms. Scientists would say they share a theory or set of theories.

However, there seemed to be philosophical problems with several themes and concepts in Kuhn’s 1962 work, especially with the “incommensurability thesis”. The thesis of incommensurability of scientific theories is such a controversial thesis in the philosophy of science and it has been criticized and challenged by many philosophers of science since 1962 (the year Kuhn had presented it). The initially heated debate about the concept of a paradigm has considerably cooled down because already in the postscript to the second edition of the Structure Kuhn suggested corrections to his original concept.

In the postscript Kuhn suggests a new term disciplinary matrix: disciplinary because it refers to the common possession of the practitioners of a particular discipline; “matrix” because it is composed of ordered elements of various sorts, each requiring further specification. Kuhn’s 1962 paradigms are constituents of the disciplinary matrix, and as such, they form a whole and function together. “Two groups, the members of which have systematically different sensations on receipt of the same stimuli, do in some sense live in different worlds”. There is no neutral algorithm for theory-choice, no systematic decision procedure which, properly applied, must lead each individual in the group to the same decision. Two men who perceive the same situation differently but nevertheless employ the same vocabulary in its discussion must be using words differently. They speak, that is, from incommensurable viewpoints.

In 1991, Kuhn explained that good reasons for belief could be supplied only by neutral observers, observers which are independent of influences of both beliefs and theories. These provide the stable Archimedean platform required to determine the truth or the probability of the particular belief, law, or theory to be evaluated (judgments). We cannot stand outside the theory and see the world as it is, i.e. compares our theories about the world with reality, the independent world. Only a fixed, rigid Archimedean platform can supply a base from which to measure the distance between current belief and true belief. However, we cannot connect our world of experience to the independent world, because the Archimedean platform outside of history, outside of time and space, is gone beyond recall. Hence, we have no neutral place to stand to evaluate whether or not our statements and propositions correspond to facts that obtain in the independent world.

Critics argued: Kuhn thinks that in order to be in a position to compare theories from older and more recent periods of normal science one needs a perspective external to each and indeed any era of science, but we never are able to escape from our current perspective.

Hilary Putnam argues that human beings are not in a position to judge whether or not our statements correspond to the independent world. To single out a correspondence between two domains, one needs some independent access to both domains. Kuhn talks as if each theory does refer-namely, to its own “world” of entities.

In 1988, in Dubbing and Redubbing Kuhn explained that a historian reading an out-of-date scientific text characteristically encounters passages that make no sense. Kuhn had this experience with Aristotle. It has been standard to ignore such passages or to dismiss them as the products of error, ignorance, or superstition, and that response is occasionally appropriate. Kuhn claims that the troublesome passages result in a different diagnosis. The apparent textual anomalies are artifacts, products of misunderstanding.

The historian has been understanding words and phrases in the text as he or she would if they had occurred in contemporary discourse. Through much of the text that way of reading proceeds without difficulty; most terms in the historian’s vocabulary are still used as they were by the author of the text. But some sets of interrelated terms are not, and it is the failure to isolate those terms and to discover how they were used that has permitted the passages in question to seem anomalous. Apparent anomaly is thus ordinary evidence of the need for local adjustment of the lexicon, and it often provides clues to the nature of the adjustment as well.

In his 1962 Structure Kuhn considered a pair of successive theories in the same historical line, Incommensurability meant that there was no common language into which both could be fully translated. Some statements constitutive of the older theory could not be stated in any language adequate to express its successor and vice versa. Incommensurability thus equals untranslatability, but what incommensurability bars is not quite the activity of professional translators. Kuhn adopts translation of that sort of Quine, Quine’s arguments for indeterminacy of translation: language is not universal. One must give up the assumption that anything expressible in one language, or by using one lexicon, can be expressed also in any other. Meaning must be abandoned. Anything that can be said in one language can, with sufficient imagination and effort, be understood by a speaker of another. What is a prerequisite to such understanding, however, is not a translation but language learning. Quine’s radical translator is, in fact, a language learner. Kuhn is suggesting that the problems of translating a scientific text, whether into a foreign tongue or into a later version of the language in which it was written, are far more like those of translating literature than has generally been supposed. In both cases the translator repeatedly encounters sentences that can be rendered in several alternative ways, none of which captures them completely. The content of alternative theories is unable to be compared due to the meaning variance of the terms employed by the theories. This leads to the reference problem: Different theories seem to employ the same terms to refer to different things. If reference remains stable through variation of conceptual content, no problem of theory comparison arises, since the reference is preserved even though terms may be associated with divergent conceptual content in the context of alternative theories.

Such a refined version of incommensurability was a target for attack. If it is impossible to compare theories either with respect to content or by means of common standards, then it is unclear how a decision between such theories may be made on an objective, rational basis. Such discontinuity in the transition between successive theories (discontinuity in reference), so that no term of a later theory refers to any entity referred to by an earlier theory, conflicts with scientific realism. The realist holds that successive theories in the same domain typically provide alternative descriptions of the same entities and that progress in science consists in an increase in truths known about a common set of entities. But if later theories refer to none of the same entities as earlier theories, then the realist account of scientific progress as an increase of truth about a common set of entities is untenable. The realist holds that the entities to which the terms of a theory refer exist independently of the theory, and that the world investigated by natural science is an objective reality which exists independently of human thought. But such assumption may not be shared by those anti-realist philosophers for whom the objects of reference and the world investigated by science depend in some way on human thought. Some anti-realist philosophers hold that the world and the objects it contains are constituted, either in whole or in part, by our theories, concepts or language. Such philosophers may deny that the terms of conceptually variant theories refer to the same objects since such theories constitute their own domains of reference. Hence, the question of whether later theories refer to the same entities as earlier theories raises metaphysical questions of a kind that tend to divide realism from anti-realism in the philosophy of science.

Critics have attacked Kuhn’s notion of semantic incommensurability. If we do take theories to be potential descriptions of the world, involving reference to worldly entities, kind, and properties, then the problems raised by incommensurability largely evaporate. Critics noted that causal theories of reference permit continuity of reference even through fairly radical theoretical change. Of course, the referentialist response shows only that reference can be retained. Consequently, it is only a partial defense of realism against semantic incommensurability. A further component of the defense of realism against incommensurability must be an epistemic one. For referentialism shows that a term can retain reference and hence that the relevant theories may be such that the later constitutes a better approximation to the truth than the earlier. Nonetheless, it may not be possible for philosophers or others to know that there has been such progress.


Hilary Putnam

Hilary Putnam’s criticism against Kuhn’s semantic incommensurability is embodied in the following notable fable from 1975. Putnam imagines our world has a Doppelgänger twin Earth, a planet just like our own. Let us suppose that somewhere in the galaxy there is a planet which we can call Twin Earth. Twin Earth is very much like Earth. People on Twin Earth speak English. People on Twin Earth who speak English, i.e. the ones who call themselves “Americans”, “Canadians”, “Englishmen” etc. There are a few differences regarding the standard English language on the earth. These differences themselves depend on some of the peculiarities of Twin Earth. One of the differences on Twin Earth compared to the conditions on Earth is that the liquid called “water” is not H2O but a different liquid with a very long and complicated chemical formula abbreviated xyz, indistinguishable from Water H2O at normal temperature and pressure. When Twin Earthians are thirsty they drink xyz. Water xyz tastes like water and it quenches thirst in the same way as Water H2O. The oceans and lakes, rivers and seas of Twin Earth contain water xyz and rain is consisted of the same liquid. The first contact of the Earthians with Twin Earth will produce a report: The word “water” has the same meaning on Earth and on Twin Earth. After the discovery that the liquid called “water” on Twin Earth is not H2O but xyz, the Earthians would correct their description: “On Twin Earth the word ‘water’ means xyz”. Hence the word “water” has two different meanings: on Twin Earth, what we call “water” (consisting of H2O molecules) simply isn’t water.

Let us return to the time when chemistry was not developed on Earth and on Twin Earth. Somewhere about the year 1750. The average Earthian (Oscar) did not know that water consisted of hydrogen and oxygen. The Twin Earthian (Twin-Oscar) did not know that “water” consisted of xyz. Hence, they would conclude that water and “water were the same and in 1750 no one on either Earth or Twin Earth could have distinguished water from “water”. Therefore, if Oscar and Twin-Oscar are not chemically or metallurgically educated, then there may be no difference at all in their psychological state when they use the word water and “water”.

Let us suppose we have a Doppelgänger on Twin Earth who is molecule for molecule identical to me. My Doppelgänger thinks the same verbalized thoughts as I do. In addition, he has the same sense data, the same dispositions, etc. Therefore, I and my Doppelgänger are in the same psychological state: It is absurd to claim that my Doppelgänger’s psychological state is one bit different from mine. My Doppelgänger cannot tell an elm from a beech tree. He means ‘beech’ when he says ‘elm’, and I on Earth mean elm when I say elm. Putnam concludes: “Cut the pie any way you like, “meanings” [that determine the extension of a term] just ain’t in the head! Hence, the psychological state of my Doppelgänger on Twin Earth and of myself on Earth cannot determine the extension (the group of things the term refers to) of any term.

According to Putnam, if we take into account only the individual psychological or cognitive state of Oscar and Twin-Oscar then water refers to “water” for Twin-Oscar. We do not have to rearrange in different ways the term water for Twin-Oscar. From a cognitive point of view, water is being understood by Oscar and Twin-Oscar as the same thing. Kuhn disagreed with Putnam.

Kuhn replied to Putnam: the report the visitors from Earth send home should not be about language but about chemistry. It must take some form like: “back to the drawing board! Something is badly wrong with the chemical theory, and that theory is incompatible with the existence of a substance with properties very nearly the same as water but described by an elaborate chemical formula. Such substance would be too heavy to evaporate at normal terrestrial temperatures. Its discovery and presence would demonstrate the fundamental errors in the chemical theory that gives meanings to compound names like H2O”.

In articles written between the late 1980s and early 1990s, Kuhn offered a new account of incommensurability, which localized meaning change to a restricted class of kind terms. These kind terms, together with their interconnections, form taxonomy that classifies the entities studied in a particular scientific field (taxonomic incommensurability). During a taxonomic change, some kind terms from the old taxonomy are preserved. But at the same time, some new kind terms are added, some old ones are deleted, and many others are rearranged in different ways. Meaning change happens only in a very restricted class of terms and there always exist unchanged concepts that may be used as the basis for rational comparison between rival paradigms. Through the localization of incommensurability, Kuhn hoped to deflect the charge of relativism. In 1993, Ian Hacking related this taxonomy to the world-change thesis: after a revolution, the world of individuals remains as it was, but scientists now work in a world of new kinds.

In 1983 and 1991, Kuhn redrew the picture of scientific revolutions. Changes in taxonomy capture the revolutionary features of paradigm shifts and the most important changes during scientific revolutions can now be conceptualized as taxonomic shifts. The role of taxonomy is not simply linguistic, but also cognitive and psychological. From a cognitive point of view, a taxonomy is a specific structure in the conceptual field defined by a frame. Kuhn sought to distinguish between translating a language (water into “water) and understanding it. Kuhn reasoned that while one might fail to translate from a foreign language into one’s own, it need not follow that one must fail to understand the other language. The combination of these two points yields a refined version of semantic incommensurability, on which translation failure is restricted to specialized vocabularies with a language, which is capable of being understood by rival theories.

Putnam thought that in particular, the individual psychological state of a person does not fix the extension or reference of a term; it is only the sociolinguistic state of the collective linguistic group to which the individual belongs that fixes the extension. Every linguistic community possesses at least some terms that are known only to a subset of the speakers who acquire the terms, and whose use by the other speakers depends upon a structured cooperation between them and the speakers in the relevant subsets.

Sorry no reference list is added here on the blog (to prevent student copying)


Men Explain Einstein to Me

If you are a woman, a man has probably at some point explained to you something about the area of your expertise. Rebecca Solnit tells about her own experience with a man who tried to explain (mansplain) a book she wrote to her.


Mansplainers usually assume you know less than you do because of your gender. Advices.



On March 14 the official Albert Einstein Facebook wall hosted Dr. Debbie Berebiches. This was a Facebook Live Event celebrating Einstein’s birthday. Dr. Berebiches answered questions about Einstein (I haven’t listened to her, I am sorry). Dr. Berebiches is the first Mexican woman to graduate with a Ph.D in physics from Stanford University. Yesterday an Israeli hi-tech veteran explained to her what was “the main contribution of general relativity to GPS”, a topic she clearly knows more about than him. She gave him a like and this was absolutely the right thing to do.


He then explained things to me, about Einstein’s religion. I am terribly sorry. I may not be the number one expert in Einstein’s philosophy of religion and personal views… well not that he is a great expert in this topic. Subsequently he asked me: “Have you read anything Einstein wrote in German? How about the dice quote? How about general relativity? …” and gave me advices:


In fact I read many papers and manuscripts in German and when I wake up in the middle of a dream at night, I think I was speaking German to Albert Einstein in my dream.


I could have stumped him but I preferred not to do this and not to discuss these matters further with him. I don’t believe mansplainers will ever change. Hence, it is better not to stump them and simply ignore them and avoid arguing with them on the area of your expertise. A mansplainer’s advice is unsolicited so you are under no obligation to listen. Advice.

Einstein expressed his religious views in 1922:



and he also expressed his views in 1930:


This article is from the Einstein Encyclopedia:


The prediction of gravitational waves emerged as early as 1913

On February 11, 2016, The Max Planck Institute for the History of Science in Berlin published the following announcement: “One Hundred Years of Gravitational Waves: the long road from prediction to observation”:

“Collaborative work on the historiography 20th century physics by the Einstein Papers Project at Caltech, the Hebrew University of Jerusalem, and the Max Planck Institute for the History of Science carried out over many years has recently shown that the prediction of gravitational waves emerged as early as February 1916 from an exchange of letters between Albert Einstein and the astronomer Karl Schwarzschild . In these letters Einstein expressed skepticism about their existence. It is remarkable that their significant physical and mathematical work was carried out in the midst of a devastating war, while Schwarzschild served on the Eastern Front”.

Collaborative work by experts on the physics of Einstein from the Einstein papers Project, from the Max Planck Institute for the History of Science in Berlin: Prof. Jürgen Renn, Roberto Lalli and Alex Blum; and from the Hebrew University of Jerusalem the only representative is Prof. Hanoch Gutfreund, the academic director of the Albert Einstein Archives. Their main finding is therefore:

The prediction of gravitational waves emerged as early as February 1916 from an exchange of letters between Albert Einstein and the astronomer Karl Schwarzschild. However, from a historical point of view this is not quite accurate because Einstein reached the main idea of gravitational waves three years earlier, as I demonstrate below. Any way the group published two summaries of the study.

A summary was published in German:

“Als Einstein dann seine abschließende Arbeit zur allgemeinen Relativitätstheorie am 25. November 1915 der Preussischen Akademie in Berlin vorlegte, war die Frage, ob solche Wellen tatsächlich aus seiner Theorie folgen, noch offen. Einstein erwähnte das Thema zum ersten Mal in einem Brief, den er am 19. Februar 1916 an Karl Schwarzschild schickte. Nach einigen obskuren technischen Bemerkungen, stellte er lakonisch fest: „Es gibt also keine Gravitationswellen, welche Lichtwellen analog wären”.”

“Gravitationswellen – verloren und wiedergefunden” von Diana K. Buchwald, Hanoch Gutfreund und Jürgen Renn.

and also in English:

“When Einstein presented his theory of general relativity on Nov. 25, 1915 in Berlin, the question of whether such waves would constitute a consequence of his theory remained untouched. Einstein mentioned gravitational waves for the first time in a letter of 19 February 1916 to Karl Schwarzschild, a pioneer of astrophysics. After some obscure technical remarks, he laconically stated: “There are hence no gravitational waves that would be analogous to light waves”.”

“Gravitational Waves: Ripples in the Fabric of Spacetime Lost and Found” by Hanoch Gutfreund, Diana K. Buchwald and Jürgen Renn.

And here as well.

Hence, according to the three above authors Einstein mentioned gravitational waves for the first time in a letter of 19 February 1916 to Karl Schwarzschild. However, this is wrong . Einstein reached the main idea of gravitational waves three years earlier, which is not when the above group of scholars had thought the gravitational waves were mentioned for the first time. As early as  1913, Einstein started to think about gravitational waves when he worked on his Entwurf gravitation theory.

In the discussion after Einstein’s 1913 Vienna talk on the Entwurf theory, Max Born asked Einstein about the speed of propagation of gravitation, whether the speed would be that of the velocity of light. Here is Einstein’s reply:



In 1916, Einstein followed these steps and studied gravitational waves.

See my papers on gravitational waves (one and two) and my book for further information.

A Historical Note on Gravitational Waves

Dr. Roni Gross (press conference) holds Einstein’s general relativity paper from May 1916, “The Foundation of the General Theory of Relativity” (“Die Grundlage der allgemeinen Relativitätstheorie.” Annalen der Physik 49, 769-822). However, in this paper Einstein did not discover gravitational waves. Prof. Hanoch Gutfreund, the academic director of the Albert Einstein Archives, asked Dr. Rony Gross to present this document to the journalists.




Equations (52) and (53) from the original page on the right above:


are Einstein’s field equations for systems in unimodular coordinates. There are no gravitational waves here!

In his 1916 general relativity paper, “The Foundation of the General Theory of Relativity”, Einstein imposed a restrictive condition on his field equations. This condition is called unimodular coordinates.

Einstein presented the gravitational waves later in 1916, in a paper published under the title, “Approximate Integration of the Field Equations of Gravitation” (“Näherungsweise Integration der Feldgleichungen der Gravitation.” Königlich Preußische Akademie der Wissenschaften (Berlin). Sitzungsberichte, 688–696).

After the 1916 general relativity paper, Einstein succeeded in relinquishing the restrictive unimodular coordinates condition and in his new gravitational waves paper his equations were not restricted to systems in unimodular coordinates.


How did Einstein predict the existence of gravitational waves?

Einstein’s Discovery of Gravitational Waves 1916-1918

Einstein and Gravitational Waves 1936-1938

For further details on Einstein predicting gravitational waves read Chapter 3, section 1 in my new book: General Relativity Conflict and Rivalries, Einstein Polemics with physicists.


איך מגלים גלי כבידה

אי שם זוג בינארי של חורים שחורים הסתובבו במסלול אליפטי והחלו לנוע זה לעבר זה. התדירות הזוויתית הסיבובית שלהם החלה לגדול ולגדול והמרחק ביניהם הלך וקטן. הם הסתובבו זה לעבר זה בעוד הם מעוותים את המרחב-זמן. בתהליך זה הם איבדו אנרגיה ופלטו גלי כבידה שנעו להם על פני מארג היקום במהירות האור. שני החורים השחורים לבסוף הגיעו למצב שבו הם התנגשו זה בזה והפכו לחור שחור אחד ענק. החור השחור הענק הזה איבד עוד ועוד אנרגיה ופלט גלי כבידה בקצב הולך וגובר

למרות שגלי כבידה נוצרים על ידי גופים מאוד מסיביים ביקום, כמו חורים שחורים שמתנגשים זה בזה ומתמזגים, גלי הכבידה עצמם הם תנודות מאוד זעירות בחלל. גלי הכבידה הם בעוצמה כה קטנה שלמעשה הם יגרמו להתמתחות ולכיווץ של המרחב-זמן בסדר גודל של 10 מיליונית מרוחבו של האטום. במצב הזה לגלות אותם זה כמו לחפש מחט בערמת שחת. קיוו מאוד לחפש גלי כבידה בעזרת

LIGO (Laser Interferometer Gravitational-wave Observatory)

אולם החיפושים העלו חרס. למען האמת היו עדויות עקיפות לקיומם של גלי כבידה מהתנהגות פולסרים, כוכבי נויטרון שמסתחררים במהירות רבה ומשחררים קרני אור. בתצפיות הפולסר מצאו איבוד אנרגיה כתוצאה מקרינת גלי כבידה, מה שזיכה את החוקרים שגילו זאת בפרס נובל ב-1993. אולם גילוי זה היה עקיף מכיוון שהחוקרים לא הצליחו למדוד ממש את גלי הכבידה ולבחון את גלי הכבידה. מכיוון שליגו לא גילה גלי כבידה בנו מכשיר משוכלל יותר שקרוי

Advanced LIGO

מכשיר זה הוא פי שלוש רגיש יותר מקודמו והוא מתוכנן לגלות גלי כבידה בטווח של מיליונית המיליונית…. מרוחבו של האטום

גלי הכבידה נקלטו במכשיר מאוד דומה למכשיר ששימש את מיקלסון ומורלי ב-1887 כדי לגלות האם כדור הארץ נע דרך האתר. מיקלסון ומורלי קיבלו תשובה שלילית אבל לא כך היה עם גלי הכבידה

אולם אותו מכשיר שנקרא האינטרפרומטר של מיקלסון הוביל לפני כמה ימים לתשובה חיובית בנוגע לגלי הכבידה. מאז 1887 עברו שנים רבות ובינתיים הומצא הלייזר ומשתמשים במקום בקרני אור בקרני לייזר באינטרפרומטר של מיקלסון והמכשיר עבר שכלולים רבים. לאחרונה אפילו המכשיר עבר שדרוגים נוספים ובספטמבר האחרון לאחר ששופץ ושוכלל הוא היה מוכן לגילוי גלי הכבידה

מהו האינטרפרומטר שבעזרתו גילו את גלי הכבידה? אור ממקור לייזר שולח שתי אלומות אור למפצל (מראה חצי מעבירה). שתי האלומות מתפצלות לשני כיוונים מאונכים זה לזה. אלומת לייזר אחת נעה במסלול אנכי של האינטרפרומטר והאלומה השנייה נעה לאורך המסלול השני. בקצה כל מסלול ממוקמת מראה מחזירה. אורכו של כל מסלול הוא 4 קילומטר. אבל אנחנו מדברים במונחים אסטרונומיים ומודדים גלי כבידה מגופים שמימיים. לכן 4 קילומטר הוא מרחק קטן מידי. הפתרון לבעיה הזו הוא לגרום לשתי אלומות הלייזר לנוע הלוך ושוב על כל מסלול מספר רב של פעמים בין מראות מחזירות. דבר זה גורם להיווצרות גלים עומדים בתדירות התהודה שלהם. זוהי שיטת הגברה של עוצמת הגלים שבה השתמשו החוקרים בניסוי



הסבר כאן


החוקרים מתארים במאמר שלהם שיטות שנועדו להתגבר על ההפרעות והרעשים שקשורים בפוטונים וככה להעלות את הרגישות של המכשיר על מנת לגלות את גלי הכבידה. אחת השיטות היא “שחזור אנרגיה”. החוקרים מקמו מראה משחזרת במסלול קרן הלייזר, שנועדה להחזיר את קרן הלייזר וליצור גלים עומדים בתדירות התהודה שלהם וככה להגביר את עוצמת הגלים

אחרי שכל קרן לייזר נעה מספר רב של פעמים על כל מסלול, שתי הקרניים מכל מסלול משתלבות יחד מחדש במפצל ומשם מגיעות לגלאי

אם לא עוברים באינטרפרומטר גלי כבידה, לאלומת האור לוקח זמן זהה לנוע הלוך ושוב לאורך שני המסלולים ואין הפרש פאזה בין שתי האלומות והן משתלבות יחד לכדי תבנית התאבכות הורסת בגלאי. מדובר בפוטו-גלאי ולכן כתוצאה לא מגלים שום אור בגלאי. במידה וגל כבידה חולף במהלך המסלול של אחת מאלומות האור, זמני המסע עבור קרני הלייזר בשני המסלולים משתנים ואז מתגלה סיגנל אור בגלאי

גלי כבידה הם רוחביים בדיוק כמו גלים אלקטרומגנטיים ולכן הם מתפשטים במאונך למישור שבו הם מתנודדים. גל כבידה שנע בניצב למישור הדף מעוות טבעת של חלקיקים לכדי אליפסה של חלקיקים. האליפסה הזו מוארכת בכיוון אחד בחצי מחזור תנודה של הגל ומוארכת בכיוון המאונך בחצי המחזור השני. ניתן למדוד את העיוות הזה שנגרם מהתנודה של הגל באינטרפרומטר. מסלול קרן האור האופקי יראה ארוך יותר ומסלול קרן האור האנכי יראה קצר יותר, כאשר גלי הכבידה מתרחבים בכיוון האופקי ומתכווצים בכיוון האנכי


אלומת אור שעוברת דרך גל הכבידה עוברת שינוי באורך הגל. פירושו למשל שאלומת האור הנעה על המסלול האופקי תעבור מסלול ארוך יותר מאשר האלומה האנכית. ככה נקבל הפרש פאזה בין שתי אלומות האור. כאשר בסוף שתי הקרניים האופקית והאנכית ישתלבו ויועברו לגלאי הוא יגלה השפעה של גלי הכבידה



I present and read several sections of my books on Einstein


I present my two books and read several sections of my books out loud:


Until February 29, 2016, you can all receive a generous discount when purchasing my book, General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists by Cambridge Scholars.