About Gali Weinstein

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

Total Eclipse of the Sun and Deflection of light Rays

According to Einstein’s prediction, that is to say the deflection (bending) of light rays in the gravitational field of the Sun: those stars closest to the limb of the Sun during the eclipse are found to be displaced slightly by amounts that are inversely proportional to the distance of the stellar image from the Sun. The light from a star close to the limb of the Sun is bent inward, toward the Sun, as it passes through the Sun’s gravitational field. The image of the star appears to observers on the Earth to be shifted outward and away from the Sun.

The Universe and Dr. Einstein by Barnett (with forward by Einstein)


In 1915 Einstein calculated the angle between the actual path of the starlight, the true position of the star, and the apparent path of the ray of light, the star seen during the eclipse. He obtained a result: 1.7” (seconds of arc).

However, in 1911 and 1913 he derived a different result, actually he had obtained half of this result: 0.84” (seconds of arc).


Einstein’s letter to George Ellery Hale which illustrates starlight being deflected by the gravity of the Sun. Oct. 14, 1913. The Huntington Library, Art Collections, and Botanical Gardens. Here

During a total eclipse of the Sun, it is possible to take pictures of the field of stars surrounding the darkened location of the Sun, because during its occultation, the light emanating from the Sun does not interfere with visibility of fainter objects.

In the eclipse expedition of 1919 Sir Arthur Stanley Eddington and Charles Rundle Davidson went to find whether they could verify Einstein’s prediction of the deflection of starlight in the gravitational field of the Sun. Eddington and his assistant went to the island of Principe off the coast of Africa while Davidson and his assistant went to Sobral in North Brazil. In presenting their observations to the Royal Society of London in November 1919, the conclusion was that they verified Einstein’s prediction of deflection at the Sun’s limb to very good accuracy.


Sir Arthur Stanley Eddington. Source (internet, unknown). If anyone knows the source please leave a comment.

The pictures taken during the solar eclipse are compared with pictures of the same region of the heavens taken at night. An astronomer compares his photographs taken during a total eclipse of the Sun with check plates, that is to say with comparison plates of the same stars (the eclipse field) when the Sun has moved away.

In 1919 Eddington examined the check field of stars that was photographed at Oxford Observatory. It was nearly the same as that of the total eclipse field of stars, which was photographed at the small island belonging to Portugal, Principe, at the same altitude as in Oxford in order to ensure that any systematic error, due to imperfections of the telescopes or other causes, might affect both sets of plates equally. There were differences in scale though between the compared photographs. Eddington determined these differences of scale between Oxford and Principe. The primary purpose of the comparison was to check the possibility of systematic errors arising from the different conditions of observation at Oxford and Principe.

After comparing the Oxford and Principe check plates, Eddington concluded that the Oxford photographs show none of the displacements which are exhibited by the photographs of the eclipse field taken under precisely similar instrument conditions. Eddington inferred that the displacements in the latter case could only be attributed to presence of the eclipsed Sun in the field and not to systematic errors.

Eddington’s four values of deflection in Principe were: 1.94, 1.44, 1.55 and 1.67 seconds of arc. He calculated the mean of these to be: 1.65” (seconds of arc). He added corrections due to experimental errors and due to the fact that the four determinations involved only two eclipse plates. The final Principe result was: 1.61±0.30 seconds of arc. Eddington calculated the final Sobral result: 1.98±0.12 seconds of arc and concluded: “They evidently agree with Einstein’s predicted value 1.75 seconds of arc.



Photos taken at the Science Museum, London. Eddington’s original negative photo.

Final confirmation of Einstein’s prediction of the deflection of light near the Sun came from William Wallace Campbell and his assistant Robert J. Trumpler at the eclipse of September 22, 1922 in Australia. Campbell and Trumpler also compared the eclipsed plates with the photographs of the same stars taken at Tahiti four months before the eclipse. The observations with the first camera led to a stellar deflection of 1.82±0.15 seconds of arc for the light deflection at the Sun’s limb. The combined observations from the two instruments used by Campbell and Trumpler gave the value of 1.75±0.9 seconds of arc for the deflection at the Sun’s limb, which is in excellent agreement with the value predicted by Einstein’s theory.


For more photos see here.

For more information on the history of eclipse expeditions and Einstein’s general theory of relativity see my books:

General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists


Einstein’s Pathway to the Special Theory of Relativity (2nd Edition)






My new paper on Einstein and general relativity

didMy new paper on Einstein and the general theory of relativity:


My new book: Einstein’s Pathway to the Special Theory of Relativity (2nd Edition)


My new book Einstein’s Pathway to the Special Theory of Relativity (2nd Edition) is coming out in August 2017.

My new book is a comprehensive monograph on Albert Einstein’s Odyssey to Special and General Relativity.

It is the second edition of my first book, Einstein’s Pathway to the Special Theory of Relativity:


The book brings together the most recent studies regarding the discovery of Special Relativity between 1895 and 1905 and pertaining to the genesis of General Relativity between 1905 and 1918.

The book encompasses an in-depth historiographical analysis of Einstein’s theory of relativity and Einstein’s own derivations and philosophical perspectives of Einstein’s work.

The first chapter provides a narrative of Einstein’s early life until 1914 without resorting to hagiography.

The second chapter discusses Fin de siècle physics.

The third chapter deals with Einstein’s path to the Special Theory of Relativity and Henri Poincaré’s Dynamics of the Electron.

The fourth chapter focuses on the genesis of the General Theory of Relativity from 1905 until approximately 1922.

The fifth chapter centralizes on Einstein’s methodology and creativity, and on Poincaré’s philosophy.

The final chapter analyzes the sources.

The book is 660 pages long, a comprehensive study of Einstein’s discovery of special and general relativity and of Einstein’s cosmology.

I drew the cover of the book.

Einstein loved sailing and he owned a sailboat, which he called Tümmler (porpoise).


The cover of my new book Einstein’s Pathway to the Special Theory of Relativity (2nd Edition) shows Einstein, the young patent clerk wearing the patent office suit, the young man and the sea.




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: