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

# Tag Archives: Einstein and general relativity

# 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 Indeed, in a 1920 unpublished draft of a paper for *Nature*, “Fundamental Ideas and Methods of the Theory of Relativity, Presented in Their Development”, Einstein wrote the above long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 g_{00} – of the metric tensor g_{mn}. 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:

- Einstein formulated an equivalence principle which is
*valid only locally*. Special relativity is valid locally and space-time is locally the Minkowski space-time. - 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-^{10}s 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 10^{10}.

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.

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.

# 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.

“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.

**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.**

# 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.**

# Some of the topics discussed in my first book, Einstein’s Pathway to the Special Theory of Relativity

People ask questions about Einstein’s special theory of relativity: How did Einstein come up with the theory of special relativity? What did he invent? What is the theory of special relativity? How did Einstein discover special relativity? Was Einstein the first to arrive at special relativity? Was Einstein the first to invent E = mc^{2}?

Did Poincaré publish special relativity before Einstein? Was Einstein’s special theory of relativity revolutionary for scientists of his day? How did the scientific community receive Einstein’s theory of special relativity when he published it? What were the initial reaction in the scientific community after Einstein had published his paper on special relativity?

In my book, *Einstein’s Pathway to the Special Theory of Relativity*, I try to answer these and many other questions.The topics discussed in my book are the following:

I start with Einstein’s childhood and school days.

I then discuss Einstein’s student days at the Zurich Polytechnic. Einstein the rebellious cannot take authority, the patent office, *Annus Mirabilis*, University of Bern and University of Zurich, Minkowski’s space-time formalism of special relativity.

Young Einstein, Aarau Class 1896

Additional topics treeated in my book are the following: Fizeau’s water tube experiment, Fresnel’s formula (Fresnel’s dragging coefficient), stellar aberration, and the Michelson and Michelson-Morley Experiments.

Albert Einstein at the Patent office

Mileva Marić and Einstein

Eduard Tete, Mileva Marić and Hans Albert

Einstein’s road to the special theory of relativity: Einstein first believes in the ether, he imagines the chasing a light beam thought experiment and the magnet and conductor thought experiment. Did Einstein respond to the Michelson and Morley experiment? Emission theory, Fizeau’s water tube experiment and ether drift experiments and Einstein’s path to special relativity; “The Step”.

Henri Poincaré’s possible influence on Einstein’s road to the special theory of relativity.

Einstein’s methodology and creativity, special principle of relativity and principle of constancy of the velocity of light, no signal moves beyond the speed of light, rigid body and special relativity, the meaning of distant simultaneity, clock synchronization, Lorentz contraction, challenges to Einstein’s connection of synchronisation and Lorentz contraction, Lorentz transformation with no light postulate, superluminal velocities, Laue’s derivation of Fresnel’s formula, the clock paradox and twin paradox, light quanta, mass-energy equivalence, variation of mass with velocity, Kaufmann’s experiments, the principles of relativity as heuristic principles, and Miller ether drift experiments.

The book also briefly discusses general relativity: Einstein’s 1920 “Geometry and Experience” talk (Einstein’s notion of practical geometry), equivalence principle, equivalence of gravitational and inertial mass, Galileo’s free fall, generalized principle of relativity, gravitational time dilation, the *Zurich Notebook*, theory of static gravitational fields, the metric tensor, the *Einstein-Besso manuscript*, Einstein-Grossmann *Entwurf* theory and *Entwurf *field equations, the hole argument, the inertio-gravitational field, Einstein’s general relativity: November 1915 field equations, general covariance and generally covariant field equations, the advance of Mercury’s perihelion, Schwarzschild’s solution and singularity, Mach’s principle, Einstein’s 1920 suggestion: Mach’s ether, Einstein’s static universe, the cosmological constant, de Sitter’s universe, and other topics in general relativity and cosmology which lead directly to my second book, *General Relativity Conflict and Rivalries*.

# Some of the topics discussed in my new book General Relativity Conflict and Rivalries

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

This book focuses on Albert Einstein and his interactions with, and responses to, various scientists, both famous and lesser-known. It takes as its starting point that the discussions between Einstein and other scientists all represented a contribution to the edifice of general relativity and relativistic cosmology. These scientists with whom Einstein implicitly or explicitly interacted form a complicated web of collaboration, which this study explores, focusing on their implicit and explicit responses to Einstein’s work.

This analysis uncovers latent undercurrents, indiscernible to other approaches to tracking the intellectual pathway of Einstein to his general theory of relativity. The interconnections and interactions presented here reveal the central figures who influenced Einstein during this intellectual period. Despite current approaches to history presupposing that the efforts of scientists such as Max Abraham and Gunnar Nordström, which differed from Einstein’s own views, be relegated to the background, this book shows that they all had an impact on the development of Einstein’s theories, stressing the limits of approaches focusing solely on Einstein. As such, *General Relativity Conflict and Rivalries* proves that the general theory of relativity was not developed as a single, coherent construction by an isolated, brooding individual, but, rather, that it came to fruition through Einstein’s conflicts and interactions with other scientists, and was consolidated by his creative processes during these exchanges.

Grossmann and Einstein

——————————————–

From Zurich to Berlin

- Einstein and
**Heinrich Zangger**(Einstein and**Michele Besso**) - From Zurich to Prague.
- Back to Zurich (Einstein and
**Marcel Grossmann**). - From Zurich to Berlin (Einstein and
**Max Planck**,**Erwin Freundlich**and others in Berlin) .

General Relativity between 1912 and 1916

- The Equivalence Principle.
- Einstein’s 1912 Polemic with
**Max Abraham**: Static Gravitational Field - Einstein’s 1912 Polemic with
**Gunnar Nordström**: Static Gravitational Field - Einstein’s 1912-1913 Collaboration with
**Marcel Grossmann**: Zurich Notebook to*Entwurf*Theory. - Einstein’s 1913-1914 Polemic with
**Nordström**: Scalar Theory versus Tensor Theory (Einstein and**Adriaan Fokker**). - Einstein’s Polemic with
**Gustav Mie**: Matter and Gravitation. - 1914 Collaboration with Grossmann and Final
*Entwurf*Theory. - Einstein’s Polemic with
**Tullio Levi-Civita**on the*Entwurf*Theory. - Einstein’s 1915 Competition with
**David Hilbert**and General Relativity - Einstein Answers
**Paul Ehrenfest**‘s Queries: 1916 General Relativity. - The Third Prediction of General Relativity: Gravitational RedShift
**Erich Kretschmann**‘s Critiques of Einstein’s Point Coincidence Argument (Einstein and**Élie****Cartan**).- Einstein and
**Mach**‘s Ideas. - Einstein’s Reaction to
**Karl Schwarzschild**‘s Solution (Einstein and**Nathan Rosen**and**Leopold Infeld**and others in Princeton). - The Fourth Classic Test of General Relativity: Light Delay.

General Relativity after 1916

- Einstein’s 1916 Polemic with
**Willem de Sitter**,**Levi-Civita**and**Nordström**on Gravitational Waves (Einstein and**Nathan Rosen**and**Leopold Infeld**and**Howard Percy Robertson**). - Einstein’s Polemic with
**de Sitter**: Matter World and Empty World. - Bending of Light and Gravitational Lens: Einstein and
**Arthur Stanley Eddington** - Einstein’s Interaction with
**Hermann Weyl**and the Cosmological Constant - Einstein’s 1920 Matter World, Mach’s Ether and the Dark Matter
- Einstein’s 1920 Polemic with
**Eddington**on de Sitter’s World. - Einstein’s Reaction to the
**Aleksandr Friedmann**Solution. - Einstein’s Reaction to the
**Georges Lemaître**Solution. **Edwin Hubble**‘s Experimental Results.- The Lemaître-Eddington Model
- Einstein and the Matter World: the Steady State Solution.
- Einstein’s Collaboration with
**de Sitter**. - Einstein’s Reaction to
**Lemaître**‘s Big Bang Model - Einstein’s Interaction with
**George Gamow**: Cosmological Constant is the Biggest Blunder - Einstein,
**Gödel**and Backward Time Travel

Einstein 1916

People ask many questions about Einstein’s general theory of relativity. For instance: How did Einstein come up with the theory of general relativity? What did he invent? What is the theory of general relativity? How did Einstein discover general relativity? How did he derive his theory? Why was Einstein the first to arrive at generally covariant field equations even though many lesser-known scientists worked on the gravitational problem?

Did David Hilbert publish the field equations of general relativity before Einstein? Was Einstein’s theory of relativity revolutionary for scientists of his day? How did the scientific community receive Einstein’s theory of general relativity when he published it? What were the initial reaction in the scientific community after Einstein had published his paper on relativity?

Why did Einstein object so fiercely to Schwarzschild’s’ singularity (black holes)? Why did Einstein introduce the cosmological constant? Was it his biggest blunder? Why did Einstein suggest Mach’s principle? Is Mach’s principle wrong? And so forth.

In my book General Relativity Conflict and Rivalries: Einstein’s Polemics with Physicists I try to answer these and many other questions.

Between 1905 and 1907, Einstein first tried to extend the special theory of relativity and explain gravitational phenomena. This was the most natural and simplest path to be taken. These investigations did not fit in with Galileo’s law of free fall. This law, which may also be formulated as the law of the equality of inertial and gravitational mass, was illuminating Einstein, and he suspected that in it must lie the key to a deeper understanding of inertia and gravitation. He found “the happiest thought of my 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 “at rest”. Newton realized that Galileo’s law of free fall is connected with the equality of the inertial and gravitational mass; however, this connection was accidental. Einstein said that Galileo’s law of free fall can be viewed as Newton’s equality between inertial and gravitational mass, but for him the connection was not accidental. Einstein’s 1907 breakthrough was to consider Galileo’s law of free fall as a powerful argument in favor of expanding the principle of relativity to systems moving nonuniformly relative to each other. Einstein realized that he might be able to generalize the principle of relativity when guided by Galileo’s law of free fall; for if one body fell differently from all others in the gravitational field, then with the help of this body an observer in free fall (with all other bodies) could find out that he was falling in a gravitational field.

In June 1911, Einstein published his paper, “On the Influence of Gravitation on the Propagation of Light”. An important conclusion of this paper is that the velocity of light in a gravitational field is a function of the place. In December 1911, Max Abraham published a paper on gravitation at the basis of which was Einstein’s 1911 conclusion about a relationship between the variable velocity of light and the gravitational potential. In February 1912, Einstein published his work on static gravitational fields theory, which was based on his 1911 June theory. In March 1912, Einstein corrected his static gravitational fields paper, but Abraham claimed that Einstein borrowed his equations; however, it was actually Abraham who needed Einstein’s ideas and not the other way round. Einstein thought that Abraham converted to his theory of static fields while Abraham presumed exactly the opposite. Einstein then moved to Zurich and switched to new mathematical tools, the metric tensor as representing the gravitational potential. He examined various candidates for generally covariant field equations, and already considered the field equations of his general theory of relativity about three years before he published them in November 1915. However, he discarded these equations only to return to them more than three years later. Einstein’s 1912 theory of static fields finally led him to reject the generally covariant field equations and to develop limited generally covariant field equations.

Max Abraham

The Finnish physicist Gunnar Nordström developed a competing theory of gravitation to Einstein’s 1912-1913 gravitation theory. The equivalence principle was valid in his theory and it also satisfied red shift of the spectral lines from the sun. However, it was unable to supply the advance of the Perihelion of Mercury, such as Einstein’s theory; it led to a Perihelion like the one predicted by Newton’s law of gravity, and, it could not explain the deflection of light near the sun, because in Nordström’s theory the velocity of light was constant. Einstein’s 1913-1914 *Entwurf* theory of gravitation, the field equations of which were not generally covariant, remained without empirical support. Thus a decision in favor of one or the other theory – Einstein’s or Nordström’s – was impossible on empirical grounds. Einstein began to study Nordström’s theory from the theoretical point of view and he developed his own Einstein-Nordström theory on the basis of his conception of the natural interval. Eventually, in a joint 1914 paper with Lorentz’s student Adrian Fokker, Einstein showed that a generally covariant formalism is presented from which Nordström’s theory follows if a single assumption is made that it is possible to choose preferred systems of reference in such a way that the velocity of light is constant; and this was done after Einstein had failed to develop a generally covariant formulation for his own *Entwurf *theory.

Gunnar Nordström

After arriving back to Zurich in summer 1912, Einstein was looking for his old student friend Marcel Grossmann, who had meanwhile become a professor of mathematics in the Swiss Federal Polytechnic institute. He was immediately caught in the fire. So he arrived and he was indeed happy to collaborate on the problem of gravitation. Einstein’s collaboration with Marcel Grossmann led to two joint papers, the first entitled, “Entwurf einer verallgemeinerten Relativitätstheorie und einer Theorie der Gravitation” (“Outline of a Generalized Theory of Relativity and of a Theory of Gravitation”) is called by scholars the *Entwurf* paper.

Marcel Grossmann

The *Entwurf* theory was already very close to Einstein’s general theory of relativity that he published in November 1915. The gravitational field is represented by a metric tensor, the mathematical apparatus of the theory is based on the work of Riemann, Christoffel, Ricci and Levi-Civita on differential covariants, and the action of gravity on other physical processes is represented by generally covariant equations (that is, in a form which remained unchanged under all coordinate transformations). However, there was a difference between the two theories, the *Entwurf* and general relativity. The *Entwurf* theory contained different field equations that represented the gravitational field, and these were not generally covariant.

Elwin Bruno Christoffel

Gregorio Curbastro Ricci

Bernhard Riemann

Indeed at first though – when Einstein first collaborated with Grossmann – he considered (in what scholars call the “Zurich Notebook) field equations that were very close to the ones he would eventually choose in November 1915:

(See visual explanation of the Zurich Notebook on John Norton’s website).

In 1913, Einstein thought for a while – or persuaded himself – that generally covariant field equations were not permissible; one must restrict the covariance of the equations. He introduced an ingenious argument – the Hole Argument – to demonstrate that generally covariant field equations were not permissible. The Hole Argument seemed to cause Einstein great satisfaction, or else he persuaded himself that he was satisfied. Having found the Hole argument, Einstein spent two years after 1913 looking for a non-generally covariant formulation of gravitational field equations.

Einstein’s collaboration with his close friends included Michele Besso as well. During a visit by Besso to Einstein in Zurich in June 1913 they both tried to solve the *Entwurf* field equations to find the perihelion advance of Mercury in the field of a static sun in what is known by the name, the *Einstein-Besso manuscript*. Besso was inducted by Einstein into the necessary calculations. The *Entwurf* theory predicted a perihelion advance of about 18” per century instead of 43” per century.

Michele Besso

Towards the end of 1915 Einstein abandoned the *Entwurf* theory, and with his new theory got the correct precession so quickly because he was able to apply the methods he had already worked out two years earlier with Besso. Einstein though did not acknowledge his earlier work with Besso.

Tulio Levi-Civita

Tullio Levi-Civita from Padua, one of the founders of tensor calculus, objected to a major problematic element in the *Entwurf *theory, which reflected its global problem: its field equations were restricted to an adapted coordinate system. Einstein proved that his gravitational tensor was a covariant tensor for adapted coordinate systems. In an exchange of letters and postcards that began in March 1915 and ended in May 1915, Levi-Civita presented his objections to Einstein’s above proof. Einstein tried to find ways to save his proof, and found it hard to give it up. Finally, Levi-Civita convinced Einstein about a fault in his arguments. Einstein realized that his *Entwurf *field equations of gravitation were entirely untenable. He discovered that Mercury’s perihelion’s motion was too small. In addition, he found that the equations were not covariant for transformations which corresponded to a uniform rotation of the reference system. Thus he came to the conviction that introducing the adapted coordinate system was a wrong path and that a more far-reaching covariance, preferably a general covariance, must be demanded.

Sometime in October 1915, Einstein dropped the Einstein-Grossman *Entwurf* theory. He adopted the postulate that his field equations were covariant with respect to arbitrary transformations of a determinant equal to 1 (unimodular transformations), and on November 4, 1915, he presented to the Prussian Academy of Sciences these new field equations. Einstein gradually expanded the range of the covariance of the field equations until November 25, 1915. On that day, Einstein presented to the Prussian Academy his final version to the gravitational field equations.

Einstein’s biographer Albrecht Fölsing explained: Einstein presented his field equations on November 25, 1915, but six days earlier, on November 20, Hilbert had derived the identical field equations for which Einstein had been searching such a long time. On November 18 Hilbert had sent Einstein a letter with a certain draft, and Fölsing asked about this possible draft: “Could Einstein, casting his eye over this paper, have discovered the term which was still lacking in his own equations, and thus ‘nostrified’ Hilbert?” Historical evidence support a scenario according to which Einstein discovered his final field equations by “casting his eye over” his own previous works. In November 4, 1915 Einstein wrote the components of the gravitational field and showed that a material point in a gravitational field moves on a geodesic line in space-time, the equation of which is written in terms of the Christoffel symbols. Einstein found it advantageous to use for the components of the gravitational field the Christoffel symbols. Einstein had already basically possessed the field equations in 1912, but had not recognized the formal importance of the Christoffel symbols as the components of the gravitational field. Einstein probably found the final form of the generally covariant field equations by manipulating his own (November 4, 1915) equations. Other historians’ findings seem to support the scenario according to which Einstein did not “nostrify” Hilbert.

David Hilbert

In March 1916 Einstein submitted to the *Annalen der Physik* a review article on the general theory of relativity, “The Foundation of the General Theory of Relativity”. The paper was published two months later, in May 1916. In this paper Einstein presented a comprehensive general theory of relativity. In addition, in this paper Einstein presented the disk thought experiment. Einstein’s first mention of the rotating disk in print was in his paper dealing with the static gravitational fields of 1912; and after the 1912 paper, the rotating-disk thought experiment occurred in Einstein’s writings only in a 1916 review article on general relativity: He now understood that in the general theory of relativity the method of laying coordinates in the space-time continuum (in a definite manner) breaks down, and one cannot adapt coordinate systems to the four-dimensional space.

Further, Einstein avoided the Hole Argument quite naturally by the Point Coincidence Argument. The point being made in the 1916 Point-Coincidence Argument is, briefly, that unlike general relativity, in special relativity coordinates of space and time have direct physical meaning. Since all our physical experience can be ultimately reduced to such point coincidences, there is no immediate reason for preferring certain systems of coordinates to others, i.e., we arrive at the requirement of general covariance. In 1918 Einstein saw the need to define the principles on which general relativity was based. In his paper, “Principles of the General Theory of Relativity”, he wrote that his theory rests on three principles, which are not independent of each other. He formulated the principle of relativity in terms of the Point Coincidence Argument and added Mach’s principle.

On November 18, 1915 Einstein reported to the Prussian Academy that the perihelion motion of Mercury is explained by his new General Theory of Relativity: Einstein found approximate solutions to his November 11, 1915 field equations. Einstein’s field equations cannot be solved in the general case, but can be solved in particular situations. The first to offer such 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 18, 1915 paper. On December 22, 1915 Schwarzschild told Einstein that he reworked the calculation in his November 18 1915 paper of the Mercury perihelion. Subsequently Schwarzschild sent Einstein a manuscript, in which he derived his exact solution of Einstein’s field equations. On January 13, 1916, Einstein delivered Schwarzschild’s paper before the Prussian Academy, and a month later the paper was published. Einstein though objected to the Schwarzschild singularity in Schwarzschild’s solution.

Karl Schwarzschild

In 1917 Einstein introduced into his 1915 field equations a cosmological term having the cosmological constant as a coefficient, in order that the theory should yield a static universe. Einstein desired to eliminate absolute space from physics according to “Mach’s ideas”.

Ernst Mach

Willem De Sitter objected to the “world-matter” in Einstein’s world, and proposed a vacuum solution of Einstein’s field equations with the cosmological constant and with no “world-matter”. In 1920 the world-matter of Einstein’s world was equivalent to “Mach’s Ether”, a carrier of the effects of inertia.

Einstein, Paul Ehrenfest and De Sitter; Eddington and Hendrik Lorentz. Location: office of W. de Sitter in Leiden (The Netherlands). Date: 26 Sept. 1923

De Sitter’s 1917 solution predicted a spectral shift effect. In 1923 Arthur Stanley Eddington and Hermann Weyl adopted De Sitter’s model and studied this effect. Einstein objected to this “cosmological problem”. In 1922-1927, Alexander Friedmann and Georges Lemaitre published dynamical universe models.

Friedmann’s model with cosmological constant equal to zero was the simplest general relativity universe. Einstein was willing to accept the mathematics, but not the physics of a dynamical universe.

In 1929 Edwin Hubble announced the discovery that the actual universe is apparently expanding. In 1931 Einstein accepted Friedmann’s model with a cosmological constant equal to zero, which he previously abhorred; he claimed that one did not need the cosmological term anymore. It was very typical to Einstein that he used to do a theoretical work and he cared about experiments and observations.

Einstein in America.

In Princeton in 1949, Kurt Gödel found an exact solution to Einstein’s field equations. Gödel’s solution was a static and not an expanding universe. Gödel’s universe allows the existence of closed timelike curves (CTCs), paths through spacetime that, if followed, allow a time traveler to interact with his/her former self.

The book also discusses other topics: for instance, gravitational lensing, gravitational waves, Einstein-Rosen bridge, unified field theory and so forth.