Update [08-04-2026]
It has passed 10 years by now since GW were detected from BHs merging.
The best way (or a very good way) to get acquainted with General Relativity is to get a hold of this books:

• The Meaning of Relativity - By A. Einstein (Fifth Edition)
  It has multiple chapters including: On geometry in general, Special Relativity, General Relativity, a first appendix with astronomical calculations, 2nd appendix with the Non-symmetric theory of relativity, Additional comments on each chapter
• RELATIVITY - The Special and the General Theory - A. EINSTEIN
  While the first edition was published in 1916, the author added several other chapters in the later editions including General Relativity and other areas it got experimental confirmation plus additional comments of unified field theory and astronomy
• Out of my later years - A. EINSTEIN
  This one has collected essays that were published during his lifetime on various philosophical things and science - (it has also a chapter describing again in simpler words the Einstein-Rosen bridge).

Here's also a list of papers Einstein wrote AFTER puting together the initial General Relativity theory (very useful, a must have for anyone going deep into GR and unified field theories - the base ground for any [sub]M-Theory & co. ):

• ANNALS OF MATHEMATICS Vol. 40, No. 4, October, 1939 - ON A STATIONARY SYSTEM WITH SPHERICAL SYMMETRY CONSISTING OF MANY GRAVITATING MASSES BY ALBERT EINSTEIN (Received May 10, 1939)
• ANNALS OF MATHEMATICS Vol. 39, No. 3, July, 1938 - ON A GENERALIZATION OF KALUZA'S THEORY OF ELECTRICITY BY A. EINSTEIN AND P. BERGMANN (Received April 8, 1938)
• ANNALS OF MATHEMATICS Vol. 44, No. 2, April, 1943 - ON THE NON-EXISTENCE OF REGULAR STATIONARY SOLUTIONS OF RELATIVISTIC FIELD EQUATIONS BY A. EINSTEIN AND WV. PAULI (Received January 4, 1943)
• The Particle Problem in the General Theory of Relativity A.Einstein and N.Rosen, Institude for Advanced Study, Princeton (Received May 8, 1935)
  (known for the initial description of the Einstein-Rosen bridge)
• ANNALS OF MATHEMATICS Vol. 39, No. 1, January, 1938 - THE GRAVITATIONAL EQUATIONS AND THE PROBLEM OF MOTION BY A. EINSTEIN, L. INFELD, AND B. HOFFMANN (Received June 16, 1937)
• ANNALS 0P MATHEMATICS Vol. 41, No. 2, April, 1940 - THE GRAVITATIONAL EQUATIONS AND THE PROBLEM OF MOTION. II BY A. EINSTEIN AND L. INFELD (Received May 29, 1939)
• ON THE MOTION OF PARTICLES IN THE GENERAL RELATIVITY THEORY - A. EINSTEIN and L. INFIELD (Received February 12, 1949)
• ANNALS OF MATHEMATICS Vol. 62, No. 1, July. 1955 Printed in U.S.A. - A NEW FORM OF THE GENERAL RELATIVISTIC FIELD EQUATIONS By A. EINSTEIN AND B. KAUFMAN (Received January 27, 1955)
• ANNALS OF MATHEMATICS Vol. 46, No. 4, October, 1945 - A GENERALIZATION OF THE RELATIVISTIC THEORY OF GRAVITATION BY ALBERT EINSTEIN (Received June 19, 1945)
• ANNALS OF MATHEMATICS Vol. 47, No. 4, October, 1946 - A GENERALIZATION OF THE RELATIVISTIC THEORY OF GRAVITATION, II By A. EINSTEIN AND E. G. STRAUS (Received January 24, 1946)

• A Summary of "On a Generalization of Kaluza's Theory of Electricity" by Einstein & Bergmann (1938):

  This paper by Einstein and Bergmann (1938) reimagines Kaluza’s 1921 five-dimensional theory, which unified gravity and electromagnetism by introducing an extra spatial dimension. However, the authors critically address limitations in Kaluza’s original framework, particularly through note (3), where they acknowledge their earlier failed attempts to find singularity-free solutions to the gravitational equations using Kaluza’s theory. These efforts were informed by their prior work on the Einstein-Rosen bridge (1935), a concept introduced in "The Particle Problem in the General Theory of Relativity" (Einstein & Rosen, 1935). In that paper, they proposed that elementary particles could be modeled as "bridges" connecting two sheets of space. However, they specifically concluded that such solutions were unattainable within Kaluza’s original cylindrical framework, where the fifth dimension was treated as a purely formal construct (i.e., compactified or periodic in nature). This failure—rooted in the geometric assumptions of Kaluza’s theory—motivated Einstein and Bergmann to develop their generalized version of Kaluza’s model.

  Key Revisions and Additions:

  • Note (3) Context: The authors explicitly reference their earlier work on the Einstein-Rosen bridge (1935) in note (3), explaining that their attempts to construct a consistent field theory based on Kaluza’s cylindrical version (with the fifth dimension as a compact, periodic direction) were "fruitless" because they could not find solutions free from singularities. This failure was specific to Kaluza’s original framework, not their own generalized approach.
  • Timeline and Motivation: The Einstein-Rosen bridge (1935) and subsequent multi-body analysis by Einstein, Infeld, and Hoffmann (1937) highlighted the challenges of describing particles in general relativity. These results likely influenced Einstein and Bergmann’s decision to revisit Kaluza’s theory, aiming to resolve its shortcomings while retaining its geometric unification of gravity and electromagnetism. However, they did not abandon the idea of bridges entirely; instead, they sought a new geometric foundation where the fifth dimension had physical meaning (closed, not cylindrical), which might allow for consistent solutions.
  • Significance of Note (3): By linking their work to the Einstein-Rosen bridge, the authors clarify that their generalization of Kaluza’s theory was a direct response to unresolved issues in earlier approaches. The failure of Kaluza’s cylindrical model (which relied on artificial assumptions) pushed them to explore alternative geometries for the fifth dimension.

  Summary to our summary in a summary

  Einstein and Bergmann’s 1938 paper reimagines Kaluza’s five-dimensional theory by assigning physical reality to the fifth dimension, avoiding the artificial assumptions of Kaluza’s original cylindrical framework. Their work was driven by earlier failures (noted in (3)) to construct consistent solutions using Kaluza’s theory or the Einstein-Rosen bridge idea, which were unattainable under Kaluza’s compactified fifth dimension. By closing the fifth dimension geometrically and simplifying field equations, they aimed to create a more robust foundation for unifying gravity and electromagnetism—a step that, while incomplete, influenced later explorations of higher-dimensional physics.

  
  
  
  

• A Summary of "The Particle Problem in the General Theory of Relativity" (Einstein & Rosen, 1935)

  ^{Or on how to travel to other worlds and crush indigens with primitive sky-scrapers reaching out the sky in rainbows flags of crappy ideology , teaching them the value of worthless gold :)}

  This paper explores a new approach to modeling elementary particles within general relativity, avoiding singularities while using only gravitational ($g_{\mu\nu}$) and electromagnetic ($F_{\mu\nu}$) field variables. The authors propose that particles can be represented as "bridges" connecting two congruent sheets of space, a geometric solution to the problem of singularity-free particle representation.

  Key Concepts and Contributions

  • Problem with Singularities: Traditional solutions (e.g., Schwarzschild) contain singularities where the metric determinant $g$ vanishes. These singularities are problematic because they introduce arbitrariness into the theory. Einstein & Rosen argue that singularities should be excluded, as they undermine the consistency of field equations.
  • Modified Gravitational Equations: To eliminate singularities, the authors propose modifying the field equations by replacing $R_{\mu\nu}$ with $g^2 R_{\mu\nu}$. This avoids denominators and allows solutions valid for all finite points. Example: The Schwarzschild solution is reformulated using a new variable $u = \sqrt{r - 2m}$, leading to a regular solution describing two congruent space sheets connected by a "bridge" (a hypersurface where $g = 0$).
  • Bridges as Particles: A particle is modeled as a finite spatial bridge connecting the two sheets. For example:
    • A neutral particle (e.g., neutron or neutrino) corresponds to a bridge in a purely gravitational field.
    • An electrically charged particle (e.g., electron) requires both gravitational and electromagnetic fields, with the bridge representing the particle’s mass and charge.
    The theory naturally explains why negative mass particles cannot exist: modifying equations for negative $m$ fails to yield regular solutions.
  • Combining Gravity and Electromagnetism: By incorporating Maxwell’s equations into the field equations, the authors derive a unified framework where electrically charged particles are represented as bridges in a spacetime with two sheets. The solution for an electron (massless in this model) avoids singularities and uses only $g_{\mu\nu}$ and $F_{\mu\nu}$, without introducing new variables.
  • Implications for Matter and Motion: The theory unifies the field equations and the motion of particles, eliminating the need for separate geodesic postulates. Multiple particles correspond to solutions with multiple bridges, where each bridge represents a particle’s "charge" or "mass". The framework is logically self-contained but does not yet account for quantum phenomena, though the authors suggest it may be compatible with quantum restrictions on particle motion.

  Significance and Legacy

  • Novelty: This work was one of the first attempts to model particles as geometric features (bridges) of spacetime, avoiding singularities.
  • Influence: The concept of "bridges" (later called Einstein-Rosen bridges or wormholes) became foundational for later theories in general relativity and quantum gravity.
  • Limitations: While the theory provides a consistent classical framework, it does not address quantum effects or the full many-body problem.

  Some Contextual Note on Kaluza’s Theory

  The authors’ work on bridges was motivated by earlier failures to construct singularity-free solutions using Kaluza’s cylindrical 5D model (1921). Their 1938 paper, "On a Generalization of Kaluza’s Theory", aimed to resolve these limitations by assigning physical reality to the fifth dimension, rather than treating it as an abstract formalism.

  Like a conclusion

  Einstein & Rosen’s 1935 paper proposed a groundbreaking geometric model for particles in general relativity, using bridges between two spacetime sheets to avoid singularities. While incomplete (it does not incorporate quantum mechanics), it laid the groundwork for later explorations of unified field theories and higher-dimensional physics. The concept of "bridges" remains a cornerstone in modern theoretical physics, influencing ideas about wormholes, quantum gravity, and the nature of elementary particles.

  
  
  
  

• (like a) Summary of "On the Non-Existence of Regular Stationary Solutions of Relativistic Field Equations" (Einstein & Pauli, 1943)

  This paper by A. Einstein and W. Pauli investigates whether the relativistic field equations (both in four-dimensional general relativity and five-dimensional generalizations) allow for non-singular stationary solutions representing fields with non-zero total mass or charge. The authors conclude that such solutions do not exist, reinforcing the difficulty of constructing consistent models of matter without singularities.

  Main Findings

  • No Non-Singular Stationary Solutions: The paper proves that the field equations of relativistic gravitational theory (including five-dimensional generalizations) do not admit any non-singular stationary solutions with non-zero total mass or charge. This applies to both 4D and 5D frameworks.
  • Asymptotic Analysis: The authors analyze the asymptotic behavior of solutions, requiring that the metric tensor $g_{\mu\nu}$ approaches a Euclidean (Minkowski) form at large distances. This implies the field must represent a particle with non-vanishing mass or charge.
  • Contradiction with Kaluza’s Theory: While Kaluza’s five-dimensional theory offers a geometric unification of gravity and electromagnetism, its foundation is flawed because the fifth dimension is treated differently from the others under coordinate transformations. This leads to an apparent rather than true unification.
  • Implications for Unified Field Theories: The result suggests that attempts to represent material particles as solutions of field equations based on Riemann’s tensor (as in general relativity) inevitably lead to singularities. This challenges the feasibility of unified field theories without introducing new mathematical structures or assumptions.

  Key Context and Methodology

  • Building on Earlier Work: The paper extends Einstein’s earlier proof (1941) that no solutions to the vacuum gravitational equations $R_{\mu\nu} = 0$ exist that are stationary, singularity-free, and asymptotically Euclidean. This was generalized to higher dimensions.
  • Analysis of Field Equations: The authors consider the five-dimensional metric tensor with asymptotic form: gik ≈ diag(A, A, A, B, D, C) where at least one component (A, B, C, D) has the form $-1 + \text{const}/r$ with a non-vanishing constant. This corresponds to a field with non-zero mass or charge.
  • Integral Theorems: By analyzing integral theorems derived from the field equations, the authors show that for solutions independent of time and asymptotically Euclidean, all components of the energy-momentum tensor must vanish. This implies that deviations from the Minkowski metric decay faster than $1/r$, ruling out non-zero mass or charge.

  Connection to Einstein’s Note on Non-Symmetric Theory of Relativity

  In the last appendix of "The Meaning of Relativity" (5th edition), Einstein discusses the implications of his non-symmetric theory of relativity, noting that:

  "These can be interpreted and applied as the equations of conservation of momentum and energy. It should be noted that such conservation equations are never uniquely determined by the system of field equations. It is interesting that according to the equations the density of the energy current as well as the energy density vanish for a field that is independent of x₄. From this one can conclude that according to this theory a stationary field free from singularities can never represent a mass different from zero."

  This observation directly reinforces the conclusion of the 1943 paper by Einstein and Pauli. It explains why stationary, non-singular solutions with non-zero mass or charge are impossible: such fields would require energy density and current to vanish, violating the conditions for a non-zero mass. This aligns with quantum mechanics, which predicts that real energy is inherently time-dependent, challenging classical notions of static, singularity-free matter.

  Significance and Legacy

  • Challenge to Unified Field Theories: The result underscores the difficulty of constructing consistent unified field theories without singularities. It highlights a fundamental limitation in using Riemannian geometry alone to model matter and fields.
  • Influence on Later Work: This paper contributed to Einstein’s ongoing efforts to refine unified field theories, influencing later explorations of higher-dimensional models (e.g., Kaluza-Klein theory) and the need for alternative geometries or quantum frameworks.
  • Connection to Earlier Papers: The conclusion aligns with earlier work by Einstein and Rosen on "bridges" and the particle problem, reinforcing the idea that singularities are unavoidable in classical general relativity unless new mathematical tools (e.g., quantum gravity) are introduced.

  Conclusion in conclusion to the conclusion of the conclusion, that is a conclusion - I spoke !

  Einstein and Pauli’s 1943 paper rigorously demonstrates that regular stationary solutions with non-zero mass or charge do not exist in relativistic field equations, whether in four or five dimensions. This result highlights the inherent challenges in unifying gravity and electromagnetism within classical geometric frameworks and reinforces the need for new theoretical approaches—such as quantum field theory or string theory—to address the nature of matter and singularities in spacetime.

  
  
  
  

• Summary of "On the Motion of Particles in General Relativity Theory" (Einstein & Infeld, 1949)

  This paper by A. Einstein and L. Infeld addresses a fundamental problem in general relativity: how to derive the motion of particles directly from the field equations, without appealing to external assumptions or Newtonian mechanics. The authors aim to reconcile the equations of motion with the gravitational field equations in a unified framework.

  Key Concepts and Contributions

  • Introduction and Motivation:

    "The gravitational field manifests itself in the motion of bodies. Therefore the problem of determining the motion of such bodies from the field equations alone is of fundamental importance."

    Einstein and Infeld emphasize that general relativity must account for particle motion through its own field equations, not as an independent postulate. They note that earlier approaches had limitations, such as assuming mass independence or relying on inertial systems.
  • Comparison with Newtonian Mechanics:

    "In Newtonian mechanics, particles are represented as singularities of a scalar field In contrast, general relativity replaces Laplace’s equation $ \nabla^2\phi = 0 $ with the gravitational field equations $ R_{\mu\nu} = 0 $, which inherently satisfy the principle of general relativity. However, classical mechanics treats space-time as an independent entity, a conceptual weakness Newton himself recognized.

  • Non-Linearity and Interaction:

    "Only non-linear field equations can provide us with equations of motion because only non-linearity can express the interaction between singularities."

    The authors argue that linear field equations (like those in Newtonian gravity) cannot account for particle interactions, as they allow arbitrary superposition of fields without enforcing mutual influence. General relativity’s non-linear nature is essential for defining how particles "feel" each other’s gravitational effects.
  • Field Equations and Motion:

    "The reason why the gravitational field equations do provide us with equations of motion lies not in their non-linear character alone, but also in the fact that these equations are not independent from each other."

    The ten gravitational field equations contain only six effective functions due to coordinate freedom. This redundancy is resolved by the Bianchi identities, which ensure consistency and allow the derivation of particle motion laws from the field equations themselves.
  • New Approximation Method:

    "We regard the changes of the field in time as small compared with those in space."

    The authors propose a novel approximation technique where temporal changes are treated as perturbations. This approach leads to a manageable set of equations that can be solved iteratively, bridging classical and relativistic descriptions of motion.
  • Surface Integrals and Motion Derivation:

    "These surface integrals will depend only on the motion of the singularities and not on the shape of the surface."

    By integrating field quantities over surfaces surrounding particle singularities, Einstein and Infeld show how to deduce ordinary differential equations for particle motion directly from the partial differential field equations. This method avoids assumptions about mass independence or inertial frames.

  Connection to Previous Work

  The paper acknowledges earlier efforts by Einstein and Rosen on "bridges" and the particle problem, noting that those approaches also struggled with singularities. However, this 1949 work advances the field by:

  • Providing a self-contained derivation of motion from field equations, avoiding Newtonian postulates.
  • Highlighting the role of Bianchi identities in ensuring consistency between field equations and motion laws.
  • Introducing a new approximation method that simplifies solving complex relativistic interactions.

  Citations and Quotes

  "The classical principle of inertia becomes in relativity theory the principle of the geodesic line valid for a particle with infinitely small mass."

  — This reflects how general relativity generalizes Newtonian mechanics by replacing inertial motion with geodesic paths in curved spacetime.

  "Our aim is to investigate to what extent the field equations alone contain the equations of motion of particles; also to develop a method that will allow us to find these equations of motion up to an arbitrary approximation."

  — The paper’s central goal: unifying motion and field theory without external assumptions.

  "The ideas leading to the equations of motion are not easy and are mutually interwoven."

  — A candid acknowledgment of the complexity in deriving motion laws from first principles.

  Significance and Legacy

  • Fundamental to General Relativity: This paper clarifies how particle motion emerges naturally from gravitational field equations, reinforcing the theory’s self-consistency.
  • Bridge Between Classical and Quantum: The emphasis on non-linear interactions and surface integrals foreshadows later work in quantum field theory and string theory, where similar methods are used to derive particle behavior.
  • Methodological Innovation: The approximation method introduced here became a cornerstone for solving relativistic problems in astrophysics and cosmology.

  (yey, yet another conclusion to the) Conclusion

  Einstein and Infeld’s 1949 paper represents a pivotal step in general relativity, demonstrating how particle motion can be derived directly from the gravitational field equations. By emphasizing non-linearity, Bianchi identities, and surface integrals, they provide a rigorous framework that avoids Newtonian assumptions and inertial systems. As Einstein notes:

  "The laws of motion are supposed to be valid in an inertial system. Therefore space-time appears as an independent physical entity. The conceptual weakness of such a space-time background in the classical theory was already recognized by Newton."

  This work not only resolves longstanding issues but also sets the stage for modern theoretical physics, where spacetime and matter are inseparably linked.
  
  
  
  

• Summary of "On a Stationary System with Spherical Symmetry Consisting of Many Gravitating Masses" (Einstein, 1939)

  This paper by Albert Einstein investigates the nature of Schwarzschild singularities—points in spacetime where the gravitational field becomes infinite—and explores whether such singularities can exist in physical systems composed of real matter. By analyzing a system of many small gravitating masses moving under mutual gravitational influence, Einstein concludes that these singularities do not appear in physically realistic scenarios.

  Key Concepts and Contributions

  • Schwarzschild's Solution and Its Singularity:

    "It is noted that $g_{44} = \frac{1}{1 - 2M/r}$ vanishes for $r = 2M$. This means that a clock kept at this place would go at the rate zero."

    Schwarzschild’s solution describes a static, spherically symmetric gravitational field. However, it contains a singularity at $r = 2M$ (the Schwarzschild radius), where the metric component $g_{44}$ diverges to infinity, implying time dilation becomes infinite and light cannot escape.
  • Physical Realism of Singularities:

    "The question arises whether it is possible to build up a field containing such singularities with the help of actual gravitating masses, or whether such regions with vanishing $ g_{44} $ do not exist in cases which have physical reality."

    Einstein questions whether these mathematical singularities correspond to real physical phenomena. He argues that no physical system can produce a true singularity because matter cannot be compressed indefinitely without violating relativity’s speed-of-light limit.
  • Modeling with Many Gravitating Particles:

    "This is a system resembling a spherical star cluster. Hereby we may proceed as if the field, in which the particles are moving, were produced by a continuous mass distribution of spherical symmetry."

    Einstein proposes modeling the system as a large number of small particles (like stars in a cluster) moving under mutual gravitational attraction. This approach avoids assuming an incompressible fluid (which is incompatible with relativity) and instead uses a realistic, dynamic system.
  • Non-Existence of Singularities in Physical Systems:

    "The result of the following consideration will be that it is impossible to make $ g_{44} $ zero anywhere, and that the total gravitating mass which may be produced by distributing particles within a given radius, always remains below a certain bound."

    By analyzing the motion of particles in circular orbits under mutual gravity, Einstein shows that $ g_{44} $ cannot become zero. The gravitational field remains finite, and the system’s mass is bounded, preventing the formation of true singularities.
  • Energy Content and Cluster Stability:

    "The energy content, i.e., the gravitating mass of the cluster, can be increased arbitrarily without destroying the cluster."

    Einstein finds that while the cluster’s mass can grow indefinitely by adding energy, this requires maintaining a specific balance between particle velocities and gravitational forces. However, compressing the cluster beyond a certain point would require particles to reach light speed, which is unphysical.

  Significance and Legacy

  • Resolution of Schwarzschild Singularities: The paper clarifies why Schwarzschild singularities do not occur in physical systems. Einstein concludes that matter cannot be concentrated arbitrarily, as this would require particles to move at or beyond the speed of light, violating relativity.
  • Connection to Realistic Systems: By modeling matter as a collection of free-moving particles rather than idealized fluids, Einstein provides a more physically plausible framework for understanding gravitational collapse and black holes.
  • Influence on Later Work: This paper arose from discussions with physicists like H. P. Robertson, V. Bargmann, and P. Bergmann on the mathematical and physical implications of Schwarzschild singularities. It laid groundwork for later studies on black hole formation and the cosmic censorship hypothesis.

  Citations and Quotes

  "The essential result of this investigation is a clear understanding as to why the 'Schwarzschild singularities' do not exist in physical reality."

  — Einstein’s conclusion that singularities cannot arise from real matter due to relativistic constraints.

  "Although the theory given here treats only clusters whose particles move along circular paths it does not seem to be subject to reasonable doubt that more general cases will have analogous results."

  — Einstein’s confidence in extending the findings to broader scenarios, reinforcing the universality of his conclusion.

  Super dupper Conclusion

  Einstein’s 1939 paper demonstrates that Schwarzschild singularities are not features of physical reality but artifacts of idealized models. By analyzing a system of many gravitating particles, he shows that matter cannot be compressed beyond limits imposed by relativity, preventing the formation of true singularities. This work remains foundational in understanding the nature of black holes and the behavior of gravitational fields under extreme conditions.

  
  
  
  

• Summary of Einstein’s Work on Relativity, Unified Field Theories, and Final Reflections (1952)

  This summary synthesizes key insights from Einstein’s later papers on general relativity, unified field theories, and his final reflections in the fifth edition of "Relativity: The Special and the General Theory" (1952). It highlights his ongoing struggle to unify gravity with electromagnetism, address singularities, and reconcile classical field theories with quantum mechanics.

  Key Themes Across Einstein’s Papers

  • Unified Field Theories:

    "It is common to all these attempts [to generalize relativity] to conceive physical reality as a field, and moreover, one which is a generalisation of the gravitational field..."

    Einstein sought to unify gravity and electromagnetism through geometric frameworks (e.g., Kaluza’s 5D theory, non-symmetric tensor models). However, he acknowledged limitations:

    "The question of the particular field law is secondary in the preceding general considerations. At the present time, the main question is whether a field theory of the kind here contemplated can lead to the goal at all."

  • Singularities and Physical Reality:

    "The essential result of this investigation [on Schwarzschild singularities] is a clear understanding as to why the 'Schwarzschild singularities' do not exist in physical reality."

    Einstein argued that singularities arise only in idealized models, not in systems with real matter:

    "Matter cannot be concentrated arbitrarily. And this is due to the fact that otherwise the constituting particles would reach the velocity of light."

  • Motion of Particles and Field Equations:

    "Our aim is to investigate to what extent the field equations alone contain the equations of motion of particles..."

    Einstein and Infeld derived particle motion directly from field equations, avoiding Newtonian assumptions:

    "The laws of motion are supposed to be valid in an inertial system. Therefore space-time appears as an independent physical entity."

  Einstein’s Final Reflections (1952)

  In the note to the fifteenth edition of his book, Einstein revisited foundational questions about space, time, and the nature of physical reality. He emphasized that:

  "I have added, as a fifth appendix, a presentation of my views on the problem of space in general and the gradual modifications of our ideas on space resulting from the influence of the relativistic viewpoint."

  • Rejection of "Empty Space":

    "Physical objects are not in space, but these objects are spatially extended. In this way the concept ‘empty space’ loses its meaning."

    Einstein challenged the classical notion of space as an independent entity, aligning with his later work on field theories.
  • Field Theory and Quantum Mechanics:

    "The present-day generation of physicists is inclined to answer [the question of a complete field theory] in the negative. In conformity with the present form of the quantum theory, it believes that the state of a system cannot be specified directly..."

    Einstein acknowledged quantum mechanics’ influence but remained skeptical:

    "I think that such a far-reaching theoretical renunciation is not for the present justified by our actual knowledge, and that one should not desist from pursuing to the end the path of the relativistic field theory."

  Conclusion: Einstein’s Legacy and Final Wish

  Einstein’s final reflections in the 1952 edition underscore his unwavering commitment to relativistic field theories, even as quantum mechanics reshaped physics. He rejected the notion of "empty space" and insisted that physical reality must be described by fields, not abstract entities. His plea—"one should not desist from pursuing to the end the path of the relativistic field theory"—remains a testament to his belief in the unity of nature and the power of geometric principles to explain all forces.

  "I think that such a far-reaching theoretical renunciation is not for the present justified by our actual knowledge, and that one should not desist from pursuing to the end the path of the relativistic field theory."

  — Albert Einstein, Note to the Fifteenth Edition (1952)
  
  
  

Update [11-02-2016]
So it's official as of today 11-feb-2016 ! Gravitational waves have been detected from 2 merging blackholes.
LIGO detected in 2 separate locations (north-west of USA) and (south-east of USA) this cosmic event on September 2015
But today the scientific community gave them the recognition they deserved
It has been 100years since Einstein first published some paperwork regarding GW but only know they have been acknowleged and detected properly
by the scientific community.
I was very confident in Einstein's findings... I still am in more other things. Hope one day they'll come true !
Thanks Einstein - again!
https://youtu.be/B4XzLDM3Py8

ds2 = gμνdxμdxν

Γαμν = 1/2 gατ(∂gμτ/∂xν + ∂gντ/∂xμ - ∂gμν/∂xτ)

δAα = - Γαμν Aμ dxν

dx2α/ds2 = - Γαμν dxμ/ds dxν/ds

ΔAα = ∮δAα

ΔAα = - ∮Γαμν Aμ dxν

ΔAμ = 1/2 Rμσαβ Aσ fαβ

Rμσαβ = -∂Γμσα/∂xβ + ∂Γμσβ/∂xα + ΓμραΓρσβ - ΓμρβΓρσα

Rμν = -∂Γαμν/∂xα + ΓαμβΓβνα + ∂Γαμα/∂xν - ΓαμνΓβαβ

R = gμν Rμν

fαβ = ∮(xαdxβ - xβdxα)


Rμν - 1/2 gμν R = -kTμν

Rμν = -k(Tμν - 1/2 gμνT) = -kT*μν