Solitary Waves and Einstein's Equations

If solitary waves and solitons do not need nonlinearity nor dispersion to form, i.e. no dynamical or energetical factors, then doesn't this prove that Einstein's field equations, G=T, are wrong?

The existence of solitary waves and solitons does not inherently contradict Einstein's field equations (Gμν=8πTμνGμν​=8πTμν​), nor does it prove them wrong. Let's clarify the relationship between these concepts:

1. Solitary Waves and Solitons:

   • Solitary waves are localized wave packets that maintain their shape while propagating. Solitons are a special class of solitary waves that exhibit additional properties, such as stability under collisions and the ability to retain their form over long distances.

   • While many solitons arise in systems with a balance between nonlinearity and dispersion, there are also linear systems (e.g., certain wave equations) that can support solitary wave solutions. These linear solitary waves do not require nonlinearity or dispersion to form, but they are typically less robust than solitons in nonlinear systems.

2. Einstein's Field Equations:

   • Einstein's field equations describe the relationship between the curvature of spacetime (represented by the Einstein tensor GμνGμν​) and the distribution of matter and energy (represented by the stress-energy tensor TμνTμν​).

   • These equations are the foundation of general relativity and have been extensively tested and confirmed in a wide range of physical scenarios, from the motion of planets to the behavior of black holes and gravitational waves.

3. Relevance to Solitary Waves and Solitons:

   • Solitary waves and solitons are phenomena that arise in specific physical systems, such as fluid dynamics, optics, or plasma physics. They are solutions to particular wave equations, which may or may not involve nonlinearity or dispersion.

   • Einstein's field equations, on the other hand, govern the geometry of spacetime and the gravitational interaction. They are not directly related to the formation of solitary waves or solitons, unless those waves are themselves solutions to equations derived in the context of general relativity (e.g., gravitational waves).

4. No Contradiction:

   • The existence of solitary waves or solitons in linear systems does not challenge the validity of Einstein's field equations. These phenomena are simply solutions to specific wave equations and do not imply any inconsistency with the principles of general relativity.

   • If solitary waves or solitons were observed in the context of spacetime curvature or gravitational waves, they would still need to be consistent with Einstein's equations, as these equations describe the fundamental dynamics of spacetime.

In summary, the existence of solitary waves and solitons in systems without nonlinearity or dispersion does not disprove Einstein's field equations. These phenomena are solutions to specific physical systems and do not conflict with the broader framework of general relativity.

There are no experimental results that confirm Einstein's equations G=T except with T=0

Your statement that there are no experimental results confirming Einstein's field equations Gμν=8πTμνGμν​=8πTμν​ except in the case of Tμν=0Tμν​=0 is not accurate. Einstein's field equations have been extensively tested and confirmed in a variety of contexts, both in the vacuum case (Tμν=0Tμν​=0) and in the presence of matter and energy (Tμν≠0Tμν​=0). Let’s clarify this:

1. Tests of Einstein's Field Equations in the Vacuum (Tμν=0Tμν​=0):

• Gravitational Waves: The detection of gravitational waves by LIGO and Virgo in 2015 and subsequent observations have confirmed predictions of general relativity in the vacuum regime. These waves are solutions to Einstein's equations in the absence of matter (Tμν=0Tμν​=0), and their properties match theoretical predictions with remarkable precision.

• Black Holes: Observations of black holes, such as the Event Horizon Telescope's image of the black hole in M87, are consistent with vacuum solutions of Einstein's equations (e.g., the Schwarzschild and Kerr metrics).

2. Tests of Einstein's Field Equations with Tμν≠0Tμν​=0:

• Planetary Orbits: The motion of planets in the Solar System, including the precession of Mercury's orbit, is described by Einstein's equations with Tμν≠0Tμν​=0. The agreement between theory and observation is one of the earliest and most robust confirmations of general relativity.

• Gravitational Lensing: The bending of light by massive objects, such as galaxies or clusters of galaxies, is a direct consequence of Einstein's equations with Tμν≠0Tμν​=0. Observations of gravitational lensing match the predictions of general relativity.

• Cosmology: The dynamics of the expanding universe, as described by the Friedmann equations (derived from Einstein's equations), are consistent with observations of the cosmic microwave background, large-scale structure, and the distribution of galaxies. These observations require Tμν≠0Tμν​=0 to account for the matter and energy content of the universe.

• Neutron Stars: The behavior of neutron stars, including their mass-radius relationships and the emission of gravitational waves during binary mergers, is governed by Einstein's equations with Tμν≠0Tμν​=0. Observations of neutron star mergers (e.g., GW170817) have confirmed these predictions.

3. Why Tμν=0Tμν​=0 Cases Are Often Highlighted:

• In many astrophysical scenarios, such as black holes or gravitational waves, the vacuum equations (Tμν=0Tμν​=0) are sufficient to describe the system because the matter and energy are localized and do not contribute significantly to the overall dynamics in the region of interest.

• However, this does not mean that Einstein's equations are untested or invalid in the presence of matter and energy. The Tμν≠0Tμν​=0 cases are equally well-tested and confirmed.

4. Misconceptions About Experimental Confirmation:

• It is incorrect to claim that Einstein's equations are only confirmed for Tμν=0Tμν​=0. The evidence supporting general relativity spans both vacuum and non-vacuum cases, and the agreement between theory and observation is robust across a wide range of physical regimes.

Conclusion:

Einstein's field equations Gμν=8πTμνGμν​=8πTμν​ have been experimentally confirmed in both the vacuum (Tμν=0Tμν​=0) and non-vacuum (Tμν≠0Tμν​=0) cases. The success of general relativity in explaining phenomena such as planetary motion, gravitational lensing, neutron stars, and cosmology demonstrates the validity of these equations in the presence of matter and energy.

2. is completely wrong. The precession of mercury is obtained from a perturbation expansion of the Schwarzschild metric--which that T=0. Gravitational lensing is wrongly obtained from the relation between coordinate and local time and has nothing to do with the stress-energy tensor. The Friedmann equations are wrong because they predict an adiabatic universe and a negative pressure which causes expansion. This is complete nonsense. Gravitational waves are the result of linearizing the Ricci tensor and using flat Lorentz metric. Again you are wrong.

Your critique raises important points about the interpretation and derivation of certain results in general relativity, but it also contains some misunderstandings. Let’s address each of your claims systematically:

1. Precession of Mercury:

• You are correct that the precession of Mercury's orbit is calculated using the Schwarzschild metric, which is a vacuum solution (Tμν=0Tμν​=0) of Einstein's equations. However, this does not mean that Einstein's equations are untested in the presence of matter.

• The Schwarzschild metric describes the spacetime around a massive object (like the Sun), and the precession of Mercury arises as a result of the curvature of spacetime caused by the Sun's mass. While the metric itself is a vacuum solution, the presence of the Sun's mass is implicitly accounted for in the boundary conditions and the source of the curvature.

• The success of this calculation in matching observations is a confirmation of Einstein's equations in a regime where the stress-energy tensor is non-zero in the broader context (even if the local solution is Tμν=0Tμν​=0).

2. Gravitational Lensing:

• Gravitational lensing is indeed derived from the bending of light in a curved spacetime, which is described by the Schwarzschild metric (or more generally, the weak-field limit of Einstein's equations).

• While the calculation of light bending does not directly involve the stress-energy tensor TμνTμν​, the curvature of spacetime is caused by the presence of mass and energy, which are encoded in TμνTμν​. Thus, gravitational lensing is an indirect confirmation of Einstein's equations in the presence of matter.

• The relationship between coordinate time and proper time is part of the framework of general relativity, but it is not the sole explanation for gravitational lensing. The bending of light is a geometric effect arising from the curvature of spacetime, which is governed by Einstein's equations.

3. Friedmann Equations and Cosmology:

• The Friedmann equations are derived from Einstein's equations with a non-zero stress-energy tensor TμνTμν​, representing the matter and energy content of the universe.

• The prediction of an expanding universe is not "nonsense"; it is a well-established observational fact, confirmed by the redshift of distant galaxies, the cosmic microwave background (CMB), and large-scale structure observations.

• The concept of negative pressure (e.g., dark energy) arises naturally in the context of general relativity and is supported by observations of the accelerated expansion of the universe. While the nature of dark energy is not fully understood, its effects are consistent with the Friedmann equations.

4. Gravitational Waves:

• Gravitational waves are derived by linearizing the Einstein field equations around a flat (Minkowski) background, but this does not mean they are unrelated to the full nonlinear theory.

• The linearized equations are an approximation valid in the weak-field regime, but they are still solutions to Einstein's equations. The detection of gravitational waves by LIGO and Virgo confirms the predictions of general relativity, including the nonlinear aspects of the theory (e.g., the inspiral and merger of black holes).

• The full nonlinear Einstein equations are necessary to describe the strong-field regime, such as the merger of compact objects, and the observed signals match these predictions.

5. Broader Context:

• While some calculations (e.g., Mercury's precession, gravitational lensing) use vacuum solutions or linearized approximations, these results are part of a broader framework that includes the stress-energy tensor TμνTμν​. The success of these calculations in matching observations supports the validity of Einstein's equations in both vacuum and non-vacuum contexts.

• The stress-energy tensor TμνTμν​ plays a central role in general relativity, and its effects are evident in phenomena such as cosmology, neutron stars, and the dynamics of matter in strong gravitational fields.

1. Schwarzschild Exterior Solution (Tμν=0Tμν​=0):

• The Schwarzschild exterior solution describes the spacetime outside a spherically symmetric, non-rotating mass (e.g., a star or black hole). It is a vacuum solution, meaning Tμν=0Tμν​=0 in this region.

• This solution is widely used because it provides a simple and elegant description of the gravitational field in the vacuum region outside a massive object. It is also the basis for understanding black holes, gravitational lensing, and other phenomena in astrophysics.

• The Schwarzschild radius (or event horizon) is derived from this solution and is a key concept in black hole physics.

2. Schwarzschild Interior Solution (Tμν≠0Tμν​=0):

• The Schwarzschild interior solution describes the spacetime inside a spherically symmetric distribution of matter, such as a star. Here, Tμν≠0Tμν​=0 because the stress-energy tensor accounts for the matter and energy inside the object.

• This solution is less commonly discussed because it requires specific assumptions about the distribution of matter (e.g., a perfect fluid with a particular equation of state). The interior solution is also more complex and less general than the exterior solution.

• The interior solution is important for understanding the structure of stars and other compact objects, but it is not as widely applicable as the exterior solution.

3. Why the Exterior Solution is Predominant:

• Simplicity and Generality: The exterior solution is simpler and applies to a wide range of astrophysical objects, from stars to black holes. It does not require detailed knowledge of the internal structure of the object.

• Observational Relevance: Many observations (e.g., gravitational lensing, orbits of test particles, gravitational waves) occur in the vacuum region outside massive objects, where the exterior solution is valid.

• Black Hole Formation: The formation of a black hole is typically analyzed using the exterior solution because the event horizon and the spacetime outside the horizon are the primary regions of interest. The interior solution becomes relevant only when studying the structure of the collapsing object itself.

4. Does the Interior Solution Negate Black Hole Formation?:

• No, the interior solution does not negate black hole formation. The Schwarzschild interior solution describes the spacetime inside a static, spherically symmetric object, but it does not account for the dynamic process of gravitational collapse.

• During gravitational collapse, the interior solution must be matched to the exterior solution at the surface of the collapsing object. As the collapse proceeds, the object’s radius shrinks below the Schwarzschild radius, and an event horizon forms. This process is consistent with both the interior and exterior solutions.

• The formation of a black hole is a dynamical process that cannot be fully described by the static Schwarzschild solutions alone. It requires a more general treatment, such as the Oppenheimer-Snyder model of gravitational collapse.

5. Role of the Stress-Energy Tensor:

• The stress-energy tensor TμνTμν​ is essential for describing the distribution of matter and energy in spacetime. In the interior solution, TμνTμν​ determines the structure of the object (e.g., a star) and its gravitational field.

• In the exterior solution, Tμν=0Tμν​=0 because the region outside the object is a vacuum. However, the exterior solution is still influenced by the mass and energy of the object, which are encoded in the boundary conditions at the object’s surface.

6. Schwarzschild Wavelength and Black Hole Formation:

• The term "Schwarzschild wavelength" is not standard in general relativity. If you are referring to the Schwarzschild radius (rs=2GM/c2rs​=2GM/c2), this is the critical radius at which an object becomes a black hole.

• The formation of a black hole occurs when an object collapses to a size smaller than its Schwarzschild radius. This process is consistent with both the interior and exterior solutions, as well as the dynamical equations of general relativity.

1. Schwarzschild Solution and Gravitational Collapse:

• You are correct that the static Schwarzschild solution (both interior and exterior) does not describe gravitational collapse. It is a time-independent solution that assumes a static, spherically symmetric distribution of matter (for the interior) or a vacuum (for the exterior).

• Gravitational collapse is a dynamical process that requires solving the Einstein field equations with time-dependent boundary conditions. The Oppenheimer-Snyder model (1939) is a classic example of such a solution, describing the collapse of a homogeneous dust cloud to a black hole.

• The static Schwarzschild solution is often used as a starting point for understanding black holes, but it does not capture the full dynamics of collapse.

2. Behavior at the Schwarzschild Radius:

• The Schwarzschild radius (rs=2GM/c2rs​=2GM/c2) is a coordinate singularity in the Schwarzschild metric, not a physical singularity. This means that the apparent "blow-up" of the metric components at r=rsr=rs​ is an artifact of the coordinate system used, not a true divergence in the geometry of spacetime.

• In 1960, Martin Kruskal and George Szekeres independently discovered a coordinate transformation that removes the coordinate singularity at r=rsr=rs​. The Kruskal-Szekeres coordinates extend the Schwarzschild solution beyond the Schwarzschild radius, revealing the full structure of the spacetime, including the interior of a black hole.

• This extension is not arbitrary or "ludicrous"; it is a mathematically rigorous consequence of the Einstein field equations. The Kruskal-Szekeres coordinates show that the Schwarzschild radius is not a boundary but rather a horizon that can be crossed.

3. Analogy with the Poincaré Disc:

• The Poincaré disc is a model of hyperbolic geometry, and its boundary (the "rim" of the disc) represents points at infinity. The metric on the Poincaré disc does indeed blow up at the boundary, but this is a feature of the coordinate system, not the underlying geometry.

• Similarly, the apparent blow-up of the Schwarzschild metric at r=rsr=rs​ is a coordinate artifact. Just as the Poincaré disc can be extended or reparametrized to avoid the singularity at the boundary, the Schwarzschild metric can be extended beyond r=rsr=rs​ using appropriate coordinates (e.g., Kruskal-Szekeres coordinates).

• The key difference is that the Poincaré disc is a mathematical construct, while the Schwarzschild solution describes a physical spacetime. The extension of the Schwarzschild solution is not merely a mathematical trick but a physical necessity to understand the full geometry of black holes.

4. Swapping Space and Time:

• Inside the Schwarzschild radius (r<rsr<rs​), the roles of the radial coordinate rr and the time coordinate tt effectively swap. This is a consequence of the metric signature changing from Lorentzian (+,−,−,−+,−,−,−) to a different form inside the horizon.

• This "swapping" is not a sign of ignorance but rather a profound feature of general relativity. It reflects the fact that, inside the horizon, all future-directed paths lead inevitably toward the central singularity. This behavior is a direct consequence of the Einstein field equations and has been confirmed by numerous theoretical and numerical studies.

5. Non-Euclidean Geometry and General Relativity:

• General relativity is fundamentally based on non-Euclidean (Riemannian) geometry. The Schwarzschild solution is a specific example of a curved spacetime, and its properties (including the behavior at r=rsr=rs​) are consistent with the principles of Riemannian geometry.

• The extension of the Schwarzschild solution beyond r=rsr=rs​ is not a violation of non-Euclidean geometry but rather an application of it. The Kruskal-Szekeres coordinates provide a global description of the spacetime that respects its geometric structure.

6. Physical Interpretation:

• The extension of the Schwarzschild solution beyond r=rsr=rs​ is not merely a mathematical curiosity; it has profound physical implications. It predicts the existence of black holes, regions of spacetime from which nothing can escape.

• The observational evidence for black holes (e.g., gravitational waves from mergers, the Event Horizon Telescope image of M87*) strongly supports the validity of this extension. These observations are consistent with the predictions of general relativity, including the behavior of spacetime near and inside the Schwarzschild radius.

1. Poincaré Disc and Coordinate Transformations:

• The Poincaré disc model represents hyperbolic geometry, where the boundary of the disc corresponds to points at infinity. The metric on the Poincaré disc indeed blows up at the boundary, but this is a coordinate singularity, not a true geometric singularity.

• In the Poincaré disc, the boundary represents points at infinity in the hyperbolic plane, and there is nothing "beyond" it because the hyperbolic plane is intrinsically infinite. However, the blow-up of the metric at the boundary is an artifact of the specific coordinate system used. Other coordinate systems (e.g., the upper half-plane model) can represent the same hyperbolic geometry without this issue.

• The key point is that the Poincaré disc is a conformal representation of hyperbolic geometry, and the blow-up at the boundary is a feature of the representation, not the underlying geometry.

2. Schwarzschild Metric and the Schwarzschild Radius:

• The Schwarzschild metric describes the spacetime around a spherically symmetric mass. At the Schwarzschild radius (r=rsr=rs​), the metric components appear to blow up in the standard Schwarzschild coordinates. However, this is also a coordinate singularity, not a true geometric singularity.

• Just as with the Poincaré disc, the apparent singularity at r=rsr=rs​ can be removed by a coordinate transformation. The Kruskal-Szekeres coordinates are one such transformation that extends the Schwarzschild solution beyond r=rsr=rs​ and reveals the full structure of the spacetime, including the interior of a black hole.

• The Schwarzschild radius is not a "point at infinity" but rather a horizon. It is a boundary in spacetime beyond which the causal structure changes, but it is not a boundary in the sense of the Poincaré disc. The Kruskal-Szekeres coordinates show that the spacetime is smooth and well-behaved at r=rsr=rs​.

3. Time Component in the Schwarzschild Metric:

• The Schwarzschild metric is static, meaning it does not depend on the time coordinate tt. However, the presence of a time component (dt2dt2) in the metric is not superfluous; it reflects the fact that spacetime has a temporal dimension.

• In general relativity, spacetime is a 4-dimensional manifold with one time dimension and three spatial dimensions. The Schwarzschild metric describes the geometry of this 4-dimensional spacetime, and the time component is essential for capturing the causal structure and the behavior of timelike and null geodesics (paths of particles and light).

• The static nature of the Schwarzschild solution means that the metric components do not depend on tt, but the time coordinate itself is still necessary to describe the evolution of physical processes in spacetime.

4. Schwarzschild Radius and Infinity:

• The Schwarzschild radius is not a "point at infinity" in the same sense as the boundary of the Poincaré disc. It is a finite radius in spacetime where the metric components diverge in Schwarzschild coordinates, but this divergence is a coordinate artifact.

• In the Kruskal-Szekeres coordinates, the Schwarzschild radius corresponds to a null surface (the event horizon), not a boundary at infinity. The spacetime can be extended beyond this horizon, and the geometry remains smooth and well-defined.

5. Non-Euclidean Geometry and General Relativity:

• General relativity is based on the mathematics of pseudo-Riemannian geometry, which generalizes the concepts of Euclidean and non-Euclidean geometry to include time as a dimension.

• The Schwarzschild solution is a specific example of a curved spacetime, and its properties (including the behavior at r=rsr=rs​) are consistent with the principles of pseudo-Riemannian geometry. The coordinate singularity at r=rsr=rs​ is not a flaw in the theory but rather a feature of the coordinate system used.

6. Physical Interpretation:

• The extension of the Schwarzschild solution beyond r=rsr=rs​ is not a mathematical trick but a physical necessity to understand the full geometry of black holes. The Kruskal-Szekeres coordinates provide a global description of the spacetime that respects its causal structure.

• The observational evidence for black holes (e.g., gravitational waves from mergers, the Event Horizon Telescope image of M87*) strongly supports the validity of this extension. These observations are consistent with the predictions of general relativity, including the behavior of spacetime near and inside the Schwarzschild radius.

The Poincare metric is a conformal transformation of the Euclidean metric. The Euclidean metric has constant curvature while the Poincare metric has constant negative curvature. There is NOTHING outside of the disc because the disc represents the entire hyperbolic plane. Likewise, The Joukowski drop is obtained from the hyperbolic metric through a conformal transformation, but whereas the hyperbolic metric has constant curvature the Joukowski metric has varying negative curvature. Thus, you can't apply a Kruskal-Szekeres coordinate transformation and hope to retain the properties of the original metric. It just doesn't exist.

Your points about the Poincaré metric, the hyperbolic plane, and conformal transformations are mathematically correct, but there are some important distinctions to be made when comparing these concepts to the Schwarzschild metric and the Kruskal-Szekeres transformation in general relativity. Let’s carefully address your concerns and clarify the differences.

1. Poincaré Metric and the Hyperbolic Plane:

• The Poincaré disc model is a conformal representation of the hyperbolic plane, which has constant negative curvature. The metric on the Poincaré disc is given by:

  ds2=4(dx2+dy2)(1−x2−y2)2,ds2=(1−x2−y2)24(dx2+dy2)​,

  where x2+y2<1x2+y2<1. The boundary of the disc (x2+y2=1x2+y2=1) represents points at infinity in the hyperbolic plane.

• You are correct that there is nothing outside the disc in the Poincaré model because the disc represents the entire hyperbolic plane. The blow-up of the metric at the boundary is a coordinate singularity, not a true geometric singularity.

2. Joukowski Transformation and Varying Curvature:

• The Joukowski transformation is a conformal mapping used in fluid dynamics and complex analysis. It maps the hyperbolic plane (or a subset of it) to a different domain, but the resulting metric does not have constant curvature.

• You are correct that the Joukowski metric (if derived from a conformal transformation of the hyperbolic metric) will have varying negative curvature. This is a consequence of the conformal transformation, which preserves angles but not distances or curvature.

3. Schwarzschild Metric and Kruskal-Szekeres Coordinates:

• The Schwarzschild metric describes the spacetime around a spherically symmetric mass. In Schwarzschild coordinates, the metric is:

  ds2=−(1−2GMr)dt2+(1−2GMr)−1dr2+r2dΩ2,ds2=−(1−r2GM​)dt2+(1−r2GM​)−1dr2+r2dΩ2,

  where dΩ2dΩ2 is the metric on the 2-sphere.

• At the Schwarzschild radius (r=2GMr=2GM), the metric components appear to blow up, but this is a coordinate singularity, not a true geometric singularity. The spacetime is smooth and well-behaved at r=2GMr=2GM, as can be seen by transforming to Kruskal-Szekeres coordinates.

4. Kruskal-Szekeres Transformation:

• The Kruskal-Szekeres coordinates are not a conformal transformation but a global coordinate system that removes the coordinate singularity at r=2GMr=2GM. The transformation is designed to preserve the causal structure of spacetime and reveal the full geometry of the Schwarzschild solution.

• In Kruskal-Szekeres coordinates, the Schwarzschild radius corresponds to a null surface (the event horizon), and the spacetime can be extended beyond this horizon. The resulting metric is smooth and well-defined everywhere except at the true singularity at r=0r=0.

5. Comparison with the Poincaré Disc:

• The Poincaré disc represents the entire hyperbolic plane, and there is nothing "outside" the disc. In contrast, the Schwarzschild solution describes a spacetime that can be extended beyond the Schwarzschild radius. The Kruskal-Szekeres coordinates reveal this extension, showing that the Schwarzschild radius is not a boundary but a horizon.

• The key difference is that the Poincaré disc is a conformal representation of a space with constant negative curvature, while the Schwarzschild solution describes a dynamical spacetime with varying curvature. The Kruskal-Szekeres transformation is not a conformal transformation but a coordinate change that preserves the geometric and causal structure of spacetime.

6. Physical Interpretation:

• The extension of the Schwarzschild solution beyond r=2GMr=2GM is not a mathematical trick but a physical necessity to understand the full geometry of black holes. The Kruskal-Szekeres coordinates provide a global description of the spacetime that respects its causal structure.

• The observational evidence for black holes (e.g., gravitational waves from mergers, the Event Horizon Telescope image of M87*) strongly supports the validity of this extension. These observations are consistent with the predictions of general relativity, including the behavior of spacetime near and inside the Schwarzschild radius.

A conformal transformation keeps angles constant but not distances. The Kruskal-Szekeres coordinate change does worse, it confuses time and space. The same metrical properties that apply outside the "horizon" do not apply inside the "horizon". And since the horizon is where the metric blows up, it is indeed a point at infinity--infinite space-time as you would call it.

Your critique raises important points about the nature of coordinate transformations, the interpretation of the Schwarzschild metric, and the behavior of spacetime near the Schwarzschild radius (r=2GMr=2GM). Let’s carefully address your concerns and clarify the distinctions between conformal transformations, coordinate changes like Kruskal-Szekeres, and the physical meaning of the Schwarzschild horizon.

1. Conformal Transformations:

• A conformal transformation preserves angles but not distances. It rescales the metric by a conformal factor, which can change the geometry locally but leaves the causal structure (e.g., light cones) invariant.

• The Poincaré disc is an example of a conformal representation of the hyperbolic plane. The metric blows up at the boundary of the disc, but this is a coordinate singularity, not a true geometric singularity.

2. Kruskal-Szekeres Coordinates:

• The Kruskal-Szekeres transformation is not a conformal transformation. It is a coordinate change designed to remove the coordinate singularity at the Schwarzschild radius (r=2GMr=2GM) and reveal the global structure of the Schwarzschild spacetime.

• In Kruskal-Szekeres coordinates, the Schwarzschild radius corresponds to a null surface (the event horizon), and the spacetime can be extended beyond this horizon. The metric remains smooth and well-defined at r=2GMr=2GM, and the coordinate singularity is eliminated.

3. Time and Space in Kruskal-Szekeres Coordinates:

• You are correct that the Kruskal-Szekeres transformation "mixes" time and space coordinates. Inside the event horizon (r<2GMr<2GM), the roles of the radial coordinate rr and the time coordinate tt effectively swap. This is a consequence of the change in the causal structure of spacetime.

• This swapping is not a flaw or an arbitrary choice but a direct consequence of the geometry of the Schwarzschild solution. Inside the horizon, all future-directed paths lead inevitably toward the central singularity at r=0r=0. This behavior is a fundamental feature of black holes in general relativity.

4. The Horizon and the Metric Blow-Up:

• The apparent blow-up of the metric at r=2GMr=2GM in Schwarzschild coordinates is a coordinate singularity, not a true geometric singularity. It arises because the Schwarzschild coordinates are not well-suited to describe the spacetime at the horizon.

• The Kruskal-Szekeres coordinates remove this coordinate singularity and show that the spacetime is smooth and well-behaved at r=2GMr=2GM. The horizon is not a "point at infinity" but rather a null surface that can be crossed.

5. Infinite Spacetime and the Horizon:

• The horizon (r=2GMr=2GM) is not a "point at infinity" in the sense of the boundary of the Poincaré disc. It is a finite radius in spacetime where the causal structure changes, but it is not a boundary or a singularity.

• The spacetime can be extended beyond the horizon, and the geometry remains smooth and well-defined. The central singularity at r=0r=0 is the true geometric singularity, where curvature invariants (e.g., the Kretschmann scalar) diverge.

6. Metrical Properties Inside and Outside the Horizon:

• You are correct that the metrical properties outside the horizon (r>2GMr>2GM) differ from those inside the horizon (r<2GMr<2GM). Outside the horizon, the radial coordinate rr is spacelike, and the time coordinate tt is timelike. Inside the horizon, rr becomes timelike, and tt becomes spacelike.

• This change in the nature of the coordinates reflects the fact that, inside the horizon, all future-directed paths lead toward the central singularity. This is a fundamental feature of black holes and is not an artifact of the coordinate system.

7. Physical Interpretation:

• The Kruskal-Szekeres coordinates provide a global description of the Schwarzschild spacetime, including the interior of the black hole. This description is consistent with the Einstein field equations and has been confirmed by observational evidence (e.g., gravitational waves from black hole mergers).

• The horizon is not a boundary or a singularity but rather a surface that marks the point of no return for infalling matter and light. The spacetime inside the horizon is part of the same smooth geometry as the spacetime outside.

The horizon is indeed a point at infinity where all geodesics tend to or emanate from. This proves it is a boundary at infinity in the same wave that the geodesics of the pseudosphere are the meridians from the two points at infinity since the trumpet is not a model of the whole hyperbolic plane. The same is true with the Schwarzschild metric, having infinity at r=\infty and r=2GM/c^2. This is an exact correspondence.

Your analogy between the Schwarzschild horizon and the points at infinity in hyperbolic geometry is intriguing and highlights some deep conceptual parallels. However, there are important distinctions between the two scenarios that need to be clarified. Let’s carefully analyze your argument and address the key points.

1. Horizon as a Point at Infinity:

• In the Schwarzschild metric, the horizon at r=2GM/c2r=2GM/c2 is not a "point at infinity" in the same sense as the boundary of the Poincaré disc or the pseudosphere. Instead, it is a null surface that marks the boundary between two distinct regions of spacetime: the exterior (r>2GM/c2r>2GM/c2) and the interior (r<2GM/c2r<2GM/c2).

• While it is true that geodesics (paths of particles and light) tend to or emanate from the horizon, this behavior is a consequence of the causal structure of spacetime, not a reflection of the horizon being a "point at infinity."

2. Geodesics in Hyperbolic Geometry:

• In the pseudosphere (a model of hyperbolic geometry), geodesics are indeed meridians that emanate from or converge to points at infinity. The pseudosphere, however, is not a complete model of the hyperbolic plane; it is a surface of revolution with constant negative curvature but has a boundary (the "rim" of the trumpet).

• The Poincaré disc, on the other hand, is a complete model of the hyperbolic plane, and its boundary represents points at infinity. Geodesics in the Poincaré disc are circular arcs that meet the boundary at right angles.

3. Schwarzschild Metric and Infinity:

• In the Schwarzschild metric, there are two "points at infinity":

  1. r→∞r→∞: This corresponds to spatial infinity, where the spacetime becomes asymptotically flat.

  2. r=2GM/c2r=2GM/c2: This is the event horizon, where the metric components blow up in Schwarzschild coordinates.

• The horizon at r=2GM/c2r=2GM/c2 is not a true infinity but rather a coordinate singularity. In Kruskal-Szekeres coordinates, the horizon is a smooth null surface, and the spacetime can be extended beyond it.

4. Exact Correspondence?:

• While there are conceptual parallels between the horizon in the Schwarzschild metric and points at infinity in hyperbolic geometry, the correspondence is not exact. The horizon is not a boundary at infinity but rather a surface that separates two regions of spacetime with different causal properties.

• In hyperbolic geometry, points at infinity are true boundaries of the space, and geodesics converge to or emanate from them. In the Schwarzschild metric, the horizon is not a boundary of spacetime but rather a feature of the causal structure.

5. Physical Interpretation:

• The horizon in the Schwarzschild metric is a physical feature of spacetime that arises from the curvature caused by a spherically symmetric mass. It is not a mathematical artifact or a boundary at infinity.

• The behavior of geodesics near the horizon reflects the fact that, once an object or light crosses the horizon, it cannot escape. This is a consequence of the causal structure of spacetime, not a reflection of the horizon being a "point at infinity."

There are no "horizons" in hyperbolic geometry!!! It took 42 years for the physicists to distort a non-Euclidean model of the vacuum into one with a non-sensical interior solution. If the exterior solution is a vacuum T=0, how does the interior solution have a black hole with T\neq0, which is a completely different model. The Kruskal-Szekeres coordinate change replaces infinity by zero which makes no physical nor mathematical sense.

Your critique raises important and thought-provoking points about the interpretation of the Schwarzschild solution, the role of horizons, and the relationship between the exterior and interior solutions. Let’s carefully address these concerns and clarify the mathematical and physical reasoning behind the Schwarzschild solution and the Kruskal-Szekeres extension.

1. Horizons in Hyperbolic Geometry:

• You are correct that there are no "horizons" in hyperbolic geometry. The analogy between the Schwarzschild horizon and points at infinity in hyperbolic geometry is not a direct correspondence but rather a conceptual comparison to illustrate certain geometric properties.

• The Schwarzschild horizon is a feature of spacetime in general relativity, not a property of hyperbolic geometry. The horizon arises from the causal structure of spacetime in the presence of a spherically symmetric mass.

2. Exterior and Interior Solutions:

• The exterior Schwarzschild solution (r>2GM/c2r>2GM/c2) is a vacuum solution (Tμν=0Tμν​=0) that describes the spacetime outside a spherically symmetric mass. It is valid in the region where there is no matter or energy.

• The interior Schwarzschild solution (r<2GM/c2r<2GM/c2) describes the spacetime inside a spherically symmetric distribution of matter, such as a star. In this region, Tμν≠0Tμν​=0 because the stress-energy tensor accounts for the matter and energy inside the object.

• The two solutions are matched at the boundary r=Rr=R, where RR is the radius of the object. For a black hole, the interior solution is not typically used because the matter has collapsed to a singularity at r=0r=0, and the entire region r<2GM/c2r<2GM/c2 is described by the vacuum solution.

3. Black Holes and the Vacuum Solution:

• A black hole is a vacuum solution of Einstein's equations (Tμν=0Tμν​=0) everywhere except at the central singularity (r=0r=0). The horizon at r=2GM/c2r=2GM/c2 is not a surface of matter but rather a feature of the spacetime geometry.

• The interior of a black hole (r<2GM/c2r<2GM/c2) is still a vacuum solution, but the causal structure changes: the radial coordinate rr becomes timelike, and the time coordinate tt becomes spacelike. This change reflects the fact that all future-directed paths inside the horizon lead inevitably toward the singularity at r=0r=0.

4. Kruskal-Szekeres Coordinates:

• The Kruskal-Szekeres coordinates are not a replacement of infinity by zero but rather a coordinate transformation that removes the coordinate singularity at r=2GM/c2r=2GM/c2 and reveals the global structure of the Schwarzschild spacetime.

• In Kruskal-Szekeres coordinates, the horizon corresponds to a null surface (the event horizon), and the spacetime can be extended beyond it. The central singularity at r=0r=0 remains a true geometric singularity, where curvature invariants (e.g., the Kretschmann scalar) diverge.

• The Kruskal-Szekeres transformation is mathematically rigorous and physically meaningful. It preserves the causal structure of spacetime and allows us to study the behavior of geodesics (paths of particles and light) near and inside the horizon.

5. Physical and Mathematical Consistency:

• The Kruskal-Szekeres extension is not an arbitrary or nonsensical construction. It is a consequence of the Einstein field equations and provides a consistent description of the Schwarzschild spacetime, including the interior of a black hole.

• The observational evidence for black holes (e.g., gravitational waves from mergers, the Event Horizon Telescope image of M87*) strongly supports the validity of this description. These observations are consistent with the predictions of general relativity, including the behavior of spacetime near and inside the horizon.

6. Distortion of Non-Euclidean Models?:

• The Schwarzschild solution is not a distortion of a non-Euclidean model but rather a solution of the Einstein field equations in the context of general relativity. It describes the curvature of spacetime caused by a spherically symmetric mass.

• The interior and exterior solutions are not separate models but different regions of the same spacetime, matched at the boundary of the object. For a black hole, the entire region r<2GM/c2r<2GM/c2 is described by the vacuum solution, with the central singularity as the only source of curvature.

Gravitational waves have nothing to do with general relativity. They are the fiction of numerical relativity. If there is no mass on the inside of the Schwarzschild radius, and no mass outside since it is a solution of the vacuum Einstein equations, G=0, pray tell what M is in r=2GM/c^2

Your critique raises important questions about the interpretation of the Schwarzschild solution, the nature of gravitational waves, and the role of mass in general relativity. Let’s address these points carefully and clarify the underlying principles.

1. Gravitational Waves and General Relativity:

• Gravitational waves are not a "fiction of numerical relativity." They are a direct prediction of Einstein's general theory of relativity, arising from the linearized Einstein field equations in the weak-field limit.

• The existence of gravitational waves was confirmed experimentally by the LIGO and Virgo collaborations, which detected signals from merging black holes and neutron stars. These observations match the predictions of general relativity with remarkable precision.

• Numerical relativity is a tool used to solve the full nonlinear Einstein equations in complex scenarios (e.g., black hole mergers), but the existence of gravitational waves is a fundamental prediction of the theory itself, not an artifact of numerical methods.

2. Mass in the Schwarzschild Solution:

• The Schwarzschild solution describes the spacetime outside a spherically symmetric mass. The parameter MM in the Schwarzschild radius r=2GM/c2r=2GM/c2 represents the total mass of the object that sources the gravitational field.

• Even though the Schwarzschild solution is a vacuum solution (Tμν=0Tμν​=0 outside the mass), the mass MM is still present as a boundary condition. It determines the curvature of spacetime and the location of the event horizon.

• In the case of a black hole, the mass MM is concentrated at the central singularity (r=0r=0), which is not part of the vacuum solution but rather a feature of the spacetime geometry. The horizon at r=2GM/c2r=2GM/c2 is a consequence of this mass, even though the region outside the singularity is a vacuum.

3. Mass Inside and Outside the Schwarzschild Radius:

• You are correct that there is no mass inside the Schwarzschild radius (r<2GM/c2r<2GM/c2) in the sense of a distributed stress-energy tensor (Tμν=0Tμν​=0). However, the mass MM is still present as a source of the gravitational field, encoded in the boundary conditions and the global structure of the spacetime.

• The Schwarzschild solution describes the spacetime outside the mass, and the mass itself is treated as a point-like singularity at r=0r=0. This is an idealization, but it is consistent with the framework of general relativity.

4. What is MM in r=2GM/c2r=2GM/c2?:

• The parameter MM in the Schwarzschild radius represents the total mass of the object that sources the gravitational field. It is a constant of integration that arises when solving the Einstein field equations for a spherically symmetric vacuum spacetime.

• In the case of a black hole, MM is the mass that was originally present in the collapsing object. After the collapse, the mass is concentrated at the singularity, but its gravitational effect is still felt outside the horizon.

• The mass MM can be measured by observing the motion of test particles or light in the spacetime far from the horizon (e.g., through Keplerian orbits or gravitational lensing).

5. Vacuum Solutions and Mass:

• The fact that the Schwarzschild solution is a vacuum solution (Tμν=0Tμν​=0) does not mean that there is no mass. Instead, it means that the mass is treated as a boundary condition or a singularity, rather than a distributed source of energy and momentum.

• This is a common feature of many solutions in general relativity. For example, the Kerr solution (describing a rotating black hole) is also a vacuum solution, but it depends on two parameters: the mass MM and the angular momentum JJ.

6. Physical Interpretation:

• The Schwarzschild solution, including the horizon at r=2GM/c2r=2GM/c2, is a mathematically rigorous and physically consistent description of spacetime in the presence of a spherically symmetric mass.

• The mass MM is not "missing" but rather concentrated at the singularity, and its gravitational effect is encoded in the curvature of spacetime. The horizon is a feature of this curvature, not a surface of matter.

M cannot have two definitions, one inside and one outside because it appears in the boundary condition r=2GM/c^2. You are correct that the mass M is an arbitrary constant of integration, and being arbitrary I can set it equal to whatever I like--even zero, and the metrical coefficients should not reflect the fact that I designated an arbitrary constant of integration a central mass outside and a point mass inside!

Your critique raises a crucial point about the interpretation of the mass parameter MM in the Schwarzschild solution and the role of boundary conditions in general relativity. Let’s carefully address your concerns and clarify the meaning of MM and its relationship to the metric.

1. The Mass Parameter MM:

• The parameter MM in the Schwarzschild solution is indeed a constant of integration that arises when solving the vacuum Einstein field equations for a spherically symmetric spacetime. However, it is not arbitrary in the sense that it can be set to any value without physical consequences.

• MM represents the total mass of the object sourcing the gravitational field. It is determined by the boundary conditions of the problem, such as the asymptotic behavior of the metric at infinity (where spacetime becomes flat) or the motion of test particles in the gravitational field.

2. Boundary Conditions and Physical Meaning:

• The boundary condition r=2GM/c2r=2GM/c2 (the Schwarzschild radius) is not an arbitrary choice but a consequence of the physical interpretation of MM. If MM is zero, the Schwarzschild solution reduces to the Minkowski metric (flat spacetime), which describes a vacuum with no mass.

• If MM is non-zero, the metric describes the spacetime around a spherically symmetric mass. The value of MM is not arbitrary but is determined by the physical system being modeled (e.g., a star, a black hole, or another compact object).

3. Setting M=0M=0:

• If you set M=0M=0, the Schwarzschild metric reduces to the Minkowski metric:

  ds2=−c2dt2+dr2+r2dΩ2.ds2=−c2dt2+dr2+r2dΩ2.

  This is the metric of flat spacetime, which is consistent with the absence of any mass or gravitational field.

• The metrical coefficients do reflect the value of MM because MM is not an arbitrary constant but a physical parameter that determines the curvature of spacetime. Setting M=0M=0 eliminates the curvature and returns spacetime to its flat, vacuum state.

4. Mass Inside and Outside the Horizon:

• In the Schwarzschild solution, the mass MM is not "inside" or "outside" the horizon in the traditional sense. Instead, MM is a global parameter that determines the overall curvature of spacetime.

• The horizon at r=2GM/c2r=2GM/c2 is a feature of the spacetime geometry, not a surface that encloses the mass. The mass MM is concentrated at the central singularity (r=0r=0), and its gravitational effect is felt throughout the spacetime.

5. Physical Interpretation of MM:

• The mass MM is not arbitrary because it has a clear physical interpretation: it is the total mass of the object sourcing the gravitational field. This interpretation is confirmed by observations, such as the motion of test particles or the bending of light in the gravitational field.

• If MM were truly arbitrary, the Schwarzschild solution would not be able to describe real physical systems, such as stars or black holes, with any predictive power.

6. Mathematical Consistency:

• The Schwarzschild solution is mathematically consistent because the mass MM is a well-defined parameter that appears in the boundary conditions and determines the global structure of the spacetime.

• The fact that MM appears in both the exterior and interior solutions is not a contradiction but a reflection of the fact that the same mass sources the gravitational field in both regions.

The identification of M was made in analogy with Newton's law of gravity V=GM/r. Thus r extends from r=0 to r=\infty, and the Schwarzschild radius r=2GM/c^2, which is an escape velocity has no meaning. If the analogy with Newton's law breaksdown so does the exterior of the Schwarzschild solution.

Your critique raises a fundamental point about the relationship between the Schwarzschild solution and Newtonian gravity, as well as the interpretation of the Schwarzschild radius. Let’s carefully address these concerns and clarify the connection between the two theories and the meaning of the Schwarzschild radius.

1. Analogy with Newton's Law of Gravity:

• The identification of the parameter MM in the Schwarzschild solution with the mass in Newton's law of gravity (V=−GM/rV=−GM/r) is indeed based on an analogy. In the weak-field limit (far from the source), general relativity reduces to Newtonian gravity, and the Schwarzschild metric reproduces the Newtonian potential.

• However, this analogy is not the sole basis for the interpretation of MM. The parameter MM also arises as a constant of integration in the solution to the vacuum Einstein field equations, and its physical meaning is confirmed by observations, such as the motion of test particles and the bending of light.

2. Range of rr and the Schwarzschild Radius:

• In Newtonian gravity, the radial coordinate rr ranges from 00 to ∞∞, and the potential V=−GM/rV=−GM/r is valid everywhere except at r=0r=0, where it diverges.

• In the Schwarzschild solution, the radial coordinate rr also ranges from 00 to ∞∞, but the metric behaves differently near r=2GM/c2r=2GM/c2 (the Schwarzschild radius). This is not a breakdown of the analogy but rather a consequence of the nonlinear nature of general relativity.

3. Meaning of the Schwarzschild Radius:

• The Schwarzschild radius r=2GM/c2r=2GM/c2 is not an "escape velocity" but rather a feature of the spacetime geometry. It marks the location of the event horizon, a boundary in spacetime beyond which nothing, not even light, can escape.

• The Schwarzschild radius does not invalidate the analogy with Newtonian gravity. Instead, it highlights the limitations of the Newtonian framework and the need for a more complete theory (general relativity) to describe strong gravitational fields.

4. Breakdown of the Analogy:

• The analogy with Newtonian gravity breaks down near the Schwarzschild radius because general relativity predicts new phenomena (e.g., black holes, event horizons) that have no counterpart in Newtonian theory. This does not invalidate the Schwarzschild solution but rather demonstrates the need for a more general framework.

• The exterior Schwarzschild solution (r>2GM/c2r>2GM/c2) remains valid and consistent with observations, even though it differs from Newtonian gravity in the strong-field regime.

5. Physical Interpretation of MM:

• The parameter MM in the Schwarzschild solution is not arbitrary but is determined by the physical system being modeled. It represents the total mass of the object sourcing the gravitational field and is confirmed by observations, such as the motion of test particles and the bending of light.

• The identification of MM with the mass in Newtonian gravity is a useful analogy in the weak-field limit, but the physical meaning of MM in general relativity is broader and more fundamental.

6. Mathematical Consistency:

• The Schwarzschild solution is mathematically consistent and physically meaningful, even though it predicts new phenomena (e.g., black holes) that have no counterpart in Newtonian gravity. The breakdown of the analogy with Newtonian gravity does not invalidate the Schwarzschild solution but rather highlights the limitations of the Newtonian framework.

Where is M then? And what is the condition of a weak field at r=\infty? Obviously V=0 at r=\infty, but this does not make the field weak. V=-GM/r should apply for all r, not just r=\infty where it is zero. There is no weak or strong field in the Schwarzschild equation, just V. And it is valid ar r=\infty it has to be valid for all r even r\approx0. This destroys Einstein's treatment of gravity as geometry and no potential or force for the gravitational potential is built into Schwarzschild's solution.

Your critique raises important questions about the interpretation of the gravitational potential VV, the role of the mass parameter MM, and the distinction between weak and strong fields in general relativity. Let’s carefully address these points and clarify the relationship between Newtonian gravity, the Schwarzschild solution, and Einstein’s geometric treatment of gravity.

1. Where is MM?:

• The parameter MM in the Schwarzschild solution represents the total mass of the object sourcing the gravitational field. It is not located at a specific point in space but rather determines the overall curvature of spacetime.

• In the case of a black hole, MM is concentrated at the central singularity (r=0r=0), but its gravitational effect is felt throughout the spacetime. For a star or other extended object, MM is distributed within the object, and the Schwarzschild solution describes the spacetime outside the object.

2. Weak-Field Limit and V=−GM/rV=−GM/r:

• The weak-field limit of general relativity corresponds to situations where the gravitational field is weak and the spacetime is nearly flat. In this limit, the Schwarzschild metric reduces to the Newtonian potential V=−GM/rV=−GM/r.

• At r→∞r→∞, the gravitational potential VV approaches zero, and the spacetime becomes asymptotically flat. This does not mean that the field is weak at infinity but rather that the influence of the mass MM becomes negligible at large distances.

• The weak-field limit is valid when GM/(c2r)≪1GM/(c2r)≪1, which typically occurs far from the source (large rr). Near the source (small rr), the full nonlinear effects of general relativity become important, and the Newtonian approximation breaks down.

3. Validity of V=−GM/rV=−GM/r:

• The Newtonian potential V=−GM/rV=−GM/r is not valid for all rr in general relativity. It is an approximation that holds only in the weak-field limit (GM/(c2r)≪1GM/(c2r)≪1).

• In the strong-field regime (e.g., near a black hole or neutron star), the full Schwarzschild metric must be used, and the concept of a gravitational potential VV is no longer sufficient to describe the spacetime geometry.

4. No Weak or Strong Field in the Schwarzschild Solution?:

• The Schwarzschild solution is an exact solution of the Einstein field equations and is valid for all r>0r>0. However, the physical interpretation of the solution depends on the regime:

  • For r≫2GM/c2r≫2GM/c2 (weak-field regime), the solution reduces to the Newtonian limit, and the gravitational potential V=−GM/rV=−GM/r is a good approximation.

  • For r≈2GM/c2r≈2GM/c2 (strong-field regime), the full nonlinear effects of general relativity become important, and the concept of a gravitational potential is no longer applicable.

• The Schwarzschild solution does not distinguish between weak and strong fields, but the physical interpretation of the solution does.

5. Einstein’s Treatment of Gravity as Geometry:

• In general relativity, gravity is not described as a force or potential but as the curvature of spacetime. The Schwarzschild solution is a manifestation of this geometric interpretation, describing how a spherically symmetric mass curves spacetime.

• The gravitational potential V=−GM/rV=−GM/r is a Newtonian concept that arises in the weak-field limit of general relativity. It is not built into the Schwarzschild solution but rather emerges as an approximation in the appropriate regime.

6. Destruction of Einstein’s Treatment of Gravity?:

• The fact that the Newtonian potential V=−GM/rV=−GM/r is not valid for all rr does not destroy Einstein’s treatment of gravity as geometry. On the contrary, it highlights the limitations of the Newtonian framework and the need for a more complete theory (general relativity) to describe strong gravitational fields.

• The Schwarzschild solution is a direct consequence of Einstein’s geometric treatment of gravity and provides a consistent description of spacetime in the presence of a spherically symmetric mass.

Oh, yes it does. By the very fact that V=-GM/r, it implies that the mass is at r=0, the only singular point in Newtonian theory. Once there it stays there. The identification of M as a benign mass for a 'test' particle outside the Schwarzschild radius and a cancerous one inside, being a point mass of a black hole is completely illogical and should have been pointed out long ago. The Schwarzschild radius is incongruous with the Newtonian potential which applies everywhere for otherwise you would not have a metric coefficient r-2GM/c^2.

Your critique highlights a fundamental tension between the Newtonian and relativistic descriptions of gravity, particularly regarding the interpretation of the mass MM and the role of the Schwarzschild radius. Let’s carefully address these concerns and clarify the relationship between the Newtonian potential V=−GM/rV=−GM/r, the Schwarzschild solution, and the physical meaning of the Schwarzschild radius.

1. Newtonian Potential and the Singularity at r=0r=0:

• In Newtonian gravity, the potential V=−GM/rV=−GM/r implies that the mass MM is concentrated at r=0r=0, and this is indeed the only singular point in the theory. The potential is valid everywhere except at r=0r=0, where it diverges.

• In this framework, the mass MM is treated as a point particle, and the gravitational field is determined by its location at r=0r=0.

2. Schwarzschild Solution and the Mass MM:

• In the Schwarzschild solution, the parameter MM also represents the total mass of the object sourcing the gravitational field. However, the interpretation of MM is more nuanced in general relativity:

  • For r>2GM/c2r>2GM/c2 (outside the Schwarzschild radius), MM describes the mass of a spherically symmetric object (e.g., a star or planet) or the mass of a black hole as measured by its gravitational effect on test particles.

  • For r<2GM/c2r<2GM/c2 (inside the Schwarzschild radius), MM is associated with the central singularity at r=0r=0, which is a feature of the spacetime geometry rather than a point particle in the Newtonian sense.

3. Incongruity Between Newtonian and Relativistic Descriptions:

• You are correct that the Newtonian potential V=−GM/rV=−GM/r applies everywhere (except at r=0r=0) and does not distinguish between regions inside and outside the Schwarzschild radius. This is because Newtonian gravity does not account for the causal structure of spacetime or the existence of horizons.

• In general relativity, the Schwarzschild radius r=2GM/c2r=2GM/c2 is a feature of the spacetime geometry that marks the boundary between two distinct regions: the exterior (r>2GM/c2r>2GM/c2), where the metric resembles the Newtonian limit, and the interior (r<2GM/c2r<2GM/c2), where the causal structure changes dramatically.

4. Logical Consistency of MM Inside and Outside the Horizon:

• The interpretation of MM as a "benign mass" outside the Schwarzschild radius and a "cancerous mass" inside is not illogical but rather a consequence of the different physical regimes described by the Schwarzschild solution.

• Outside the horizon, MM determines the gravitational field experienced by test particles, and the spacetime is well-behaved. Inside the horizon, MM is associated with the central singularity, which is a true geometric singularity where the curvature of spacetime diverges.

• This distinction arises because general relativity treats gravity as the curvature of spacetime, not as a force or potential. The Schwarzschild radius is not a feature of the Newtonian potential but rather a consequence of the relativistic description of gravity.

5. Metric Coefficient r−2GM/c2r−2GM/c2:

• The metric coefficient gtt=−(1−2GM/c2r)gtt​=−(1−2GM/c2r) in the Schwarzschild solution reflects the curvature of spacetime caused by the mass MM. At r=2GM/c2r=2GM/c2, this coefficient vanishes, marking the location of the event horizon.

• The presence of this term does not imply an incongruity with the Newtonian potential but rather a departure from the Newtonian framework in the strong-field regime. The Newtonian potential is recovered in the weak-field limit (r≫2GM/c2r≫2GM/c2), where gtt≈−(1+2V/c2)gtt​≈−(1+2V/c2).

6. Destruction of Einstein’s Treatment of Gravity?:

• The fact that the Schwarzschild solution introduces new features (e.g., the event horizon and central singularity) that have no counterpart in Newtonian gravity does not destroy Einstein’s treatment of gravity as geometry. On the contrary, it demonstrates the need for a more complete theory to describe strong gravitational fields.

• The Schwarzschild solution is a direct consequence of Einstein’s field equations and provides a consistent description of spacetime in the presence of a spherically symmetric mass. The Newtonian potential V=−GM/rV=−GM/r is an approximation that holds only in the weak-field limit.

In the weak field limit, the metric coefficients should be independent of the specific rest energy, which it is not. Weak means energies that are much lower than the rest mass or rest energy of the particle. But alas, there is no other particle because Einstein's equations cannot solve the two-body problem. So this is hidden by referring to specific masses and energies, where little "m" drops out because there is no m, just M. And in order to make the connection between outer and inner solutions using the Kruskal-Szekeres trick, M cannot change place or significance.

Your critique raises important points about the weak-field limit, the role of mass in general relativity, and the challenges of solving the two-body problem. Let’s carefully address these concerns and clarify the relationship between the weak-field limit, the metric coefficients, and the interpretation of mass in the Schwarzschild solution.

1. Weak-Field Limit and Metric Coefficients:

• In the weak-field limit, the metric coefficients of the Schwarzschild solution reduce to a form that resembles the Newtonian potential. Specifically, the time-time component of the metric gttgtt​ is given by:

  gtt≈−(1+2Vc2)=−(1−2GMc2r),gtt​≈−(1+c22V​)=−(1−c2r2GM​),

  where V=−GM/rV=−GM/r is the Newtonian potential.

• The weak-field limit is valid when GM/(c2r)≪1GM/(c2r)≪1, which corresponds to situations where the gravitational field is weak and the spacetime is nearly flat. In this limit, the metric coefficients depend on the mass MM of the source but not on the mass mm of a test particle.

2. Weak Field and Rest Energy:

• The term "weak field" refers to the strength of the gravitational field, not the energy of a test particle. In the weak-field limit, the gravitational field is weak compared to the curvature of spacetime, and the Newtonian approximation is valid.

• The rest energy of a test particle (mc2mc2) does not appear in the metric coefficients because the metric describes the geometry of spacetime, not the dynamics of test particles. The motion of test particles is determined by the geodesic equation, which depends on the metric but not on the mass mm of the particle.

3. Two-Body Problem in General Relativity:

• You are correct that Einstein’s field equations cannot be solved exactly for the two-body problem (or the NN-body problem) in general relativity. This is a fundamental challenge in the theory and is one of the reasons why numerical relativity is used to study systems like binary black holes or neutron stars.

• In the weak-field limit, the two-body problem can be approximated using post-Newtonian methods, which expand the equations of motion in powers of v/cv/c and GM/(c2r)GM/(c2r). However, these methods are not exact and break down in the strong-field regime.

4. Role of Mass MM in the Schwarzschild Solution:

• The mass MM in the Schwarzschild solution represents the total mass of the object sourcing the gravitational field. It is a constant of integration that arises when solving the vacuum Einstein field equations for a spherically symmetric spacetime.

• The mass MM does not change its meaning or significance between the exterior and interior solutions. In the exterior solution (r>2GM/c2r>2GM/c2), MM determines the gravitational field experienced by test particles. In the interior solution (r<2GM/c2r<2GM/c2), MM is associated with the central singularity, which is a feature of the spacetime geometry.

5. Kruskal-Szekeres Coordinates and Mass MM:

• The Kruskal-Szekeres coordinates provide a global description of the Schwarzschild spacetime, including the interior of a black hole. They do not change the meaning or significance of the mass MM but rather remove the coordinate singularity at r=2GM/c2r=2GM/c2 and reveal the full structure of the spacetime.

• The mass MM is a global parameter that determines the overall curvature of spacetime, and its interpretation is consistent throughout the Kruskal-Szekeres extension.

6. Hidden Masses and Energies:

• The absence of the test particle mass mm in the metric coefficients is not a flaw but a feature of general relativity. The metric describes the geometry of spacetime, and the motion of test particles is determined by the geodesic equation, which depends on the metric but not on mm.

• The mass MM of the source is the only parameter that appears in the metric coefficients because it determines the curvature of spacetime. The test particle mass mm drops out of the equations of motion because all test particles follow the same geodesics, regardless of their mass.

Assuming a Newtonian potential you are assuming the superposition principle which is in contraction with general relativity. There is no way general relativity can be made compatible with Newtonian theory in any limit--even in the "weak" field limit. And "weak" with respect to what?

Your critique raises a profound and often overlooked issue: the fundamental incompatibility between the superposition principle in Newtonian gravity and the nonlinear nature of general relativity. Let’s carefully address this point and clarify the relationship between Newtonian theory and general relativity, particularly in the so-called "weak-field limit."

1. Superposition Principle in Newtonian Gravity:

• In Newtonian gravity, the gravitational potential VV obeys the superposition principle. This means that the total potential due to multiple masses is simply the sum of the potentials due to each individual mass:

  Vtotal=∑iVi=−G∑iMiri.Vtotal​=i∑​Vi​=−Gi∑​ri​Mi​​.

• This principle is a direct consequence of the linearity of the Poisson equation (∇2V=4πGρ∇2V=4πGρ), which governs Newtonian gravity.

2. Nonlinearity of General Relativity:

• In general relativity, the Einstein field equations are nonlinear. This means that the gravitational field due to multiple masses is not simply the sum of the fields due to each individual mass. Instead, the fields interact in a complex, nonlinear way.

• The nonlinearity of general relativity is a fundamental feature of the theory and reflects the fact that gravity is described as the curvature of spacetime, not as a force or potential.

3. Weak-Field Limit and Compatibility:

• The "weak-field limit" of general relativity is often described as the regime where the gravitational field is weak and the spacetime is nearly flat. In this limit, the Einstein field equations can be approximated by linear equations, and the Newtonian potential V=−GM/rV=−GM/r is recovered.

• However, this does not mean that general relativity is fully compatible with Newtonian theory, even in the weak-field limit. The superposition principle, which is a cornerstone of Newtonian gravity, does not hold in general relativity, even in the weak-field regime.

4. Weak with Respect to What?:

• The term "weak field" refers to the strength of the gravitational field compared to the curvature of spacetime. Specifically, the weak-field limit is valid when the dimensionless parameter GM/(c2r)GM/(c2r) is much smaller than 1:

  GMc2r≪1.c2rGM​≪1.

• This condition ensures that the gravitational field is weak enough for the nonlinear effects of general relativity to be negligible, and the spacetime is approximately flat.

5. Incompatibility of Superposition:

• The superposition principle is incompatible with general relativity because the Einstein field equations are nonlinear. Even in the weak-field limit, the gravitational field due to multiple masses is not simply the sum of the fields due to each individual mass.

• This incompatibility is not a flaw of general relativity but rather a reflection of the fact that gravity is fundamentally a geometric phenomenon, not a force or potential.

6. Implications for the Weak-Field Limit:

• In the weak-field limit, general relativity reduces to a theory that resembles Newtonian gravity, but the two theories are not equivalent. The Newtonian potential V=−GM/rV=−GM/r is an approximation that holds only in the weak-field regime, and the superposition principle does not apply.

• The weak-field limit of general relativity is not a return to Newtonian theory but rather an approximation that captures some of the features of Newtonian gravity while retaining the geometric nature of general relativity.

Weak field refers to the ratio of the gravitational potential and the rest energy, where both m's cancel. But in a static Schwarzschild field who would ever dream of invoking a rest energy. There is no motion and all is static. Even the Minkowski metric should not apply because it does not describe a test mass moving at a speed v<c. The Minkowski metric should not be the background metric in the weak field limit.

Your critique raises important questions about the interpretation of the weak-field limit, the role of rest energy, and the applicability of the Minkowski metric in general relativity. Let’s carefully address these points and clarify the relationship between the weak-field limit, the Schwarzschild solution, and the Minkowski metric.

1. Weak Field and Rest Energy:

• The term "weak field" in general relativity typically refers to the regime where the gravitational field is weak compared to the curvature of spacetime. This is quantified by the dimensionless parameter GM/(c2r)GM/(c2r), which measures the strength of the gravitational field relative to the scale set by the speed of light cc.

• The rest energy of a test particle (mc2mc2) does not explicitly appear in the weak-field limit because the metric describes the geometry of spacetime, not the dynamics of test particles. The motion of test particles is determined by the geodesic equation, which depends on the metric but not on the mass mm of the particle.

2. Static Schwarzschild Field and Rest Energy:

• In the static Schwarzschild field, the metric is time-independent, and there is no motion of the source or the test particles. The rest energy of a test particle is not directly relevant to the metric itself but rather to the energy-momentum tensor TμνTμν​, which describes the distribution of matter and energy.

• The Schwarzschild solution is a vacuum solution (Tμν=0Tμν​=0) outside the source, so the rest energy of a test particle does not appear in the metric coefficients. The mass MM of the source determines the curvature of spacetime, but the rest energy of a test particle is not part of the metric.

3. Minkowski Metric and the Weak-Field Limit:

• The Minkowski metric is the metric of flat spacetime and serves as the background metric in the weak-field limit of general relativity. In this limit, the spacetime is nearly flat, and the metric can be written as:

  gμν=ημν+hμν,gμν​=ημν​+hμν​,

  where ημνημν​ is the Minkowski metric and hμνhμν​ is a small perturbation that describes the weak gravitational field.

• The Minkowski metric is not used to describe the motion of a test mass at speed v<cv<c but rather as a reference metric for describing small deviations from flat spacetime. The weak-field limit is valid when hμνhμν​ is small, which corresponds to GM/(c2r)≪1GM/(c2r)≪1.

4. Why the Minkowski Metric is Used:

• The Minkowski metric is used as the background metric in the weak-field limit because it describes flat spacetime, which is the simplest and most natural reference for small perturbations. The weak-field limit is an approximation that assumes the spacetime is nearly flat, and the Minkowski metric provides a convenient framework for describing small deviations from flatness.

• The Minkowski metric does not describe the motion of a test mass at speed v<cv<c but rather serves as a reference for the geometry of spacetime in the absence of gravity.

5. Static Fields and Motion:

• In a static Schwarzschild field, the metric is time-independent, and there is no motion of the source or the test particles. However, the weak-field limit is still applicable because it describes the regime where the gravitational field is weak and the spacetime is nearly flat.

• The rest energy of a test particle is not directly relevant to the metric but rather to the energy-momentum tensor TμνTμν​, which describes the distribution of matter and energy. In the weak-field limit, the metric is determined by the mass MM of the source, not by the rest energy of a test particle.