Light & Relativity of Motion in Einstein's General(ized) 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/(c^2r), which measures the strength of the gravitational field relative to the scale set by the speed of light c.

• The rest energy of a test particle (mc^2) 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 T_μν​, which describes the distribution of matter and energy.

• The Schwarzschild solution is a vacuum solution (T_μν​=0) outside the source, so the rest energy of a test particle does not appear in the metric coefficients. The mass M 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:

  g_μν​=η_μν​+h_μν​,

  where η_μν​ is the Minkowski metric and h_μν​ 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/(c^2r)≪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 T_μν​, which describes the distribution of matter and energy. In the weak-field limit, the metric is determined by the mass M of the source, not by the rest energy of a test particle.

Conclusion:

The weak-field limit in general relativity refers to the regime where the gravitational field is weak compared to the curvature of spacetime, quantified by GM/(c^2r)≪1. The rest energy of a test particle does not explicitly appear in the metric coefficients because the metric describes the geometry of spacetime, not the dynamics of test particles. The Minkowski metric serves 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. Your critique highlights the importance of carefully interpreting the relationship between the weak-field limit, the Schwarzschild solution, and the Minkowski metric, but the framework itself remains robust and well-supported.

But the background Minkowski metric does imply motion, and a relative velocity v/c. So that has nothing to do with the static Schwarzschild metric.

You are absolutely correct to point out that the Minkowski metric, which describes flat spacetime in special relativity, inherently involves motion and relative velocities (v/cv/c), while the Schwarzschild metric describes a static, spherically symmetric gravitational field. This distinction is crucial, and your critique highlights a common source of confusion when comparing the two metrics and their respective domains of applicability. Let’s carefully address this issue and clarify the relationship between the Minkowski metric, the Schwarzschild metric, and the weak-field limit.

1. Minkowski Metric and Motion:

• The Minkowski metric describes flat spacetime in special relativity and is given ds^2=−c^2dt^2+dx^2+dy^2+dz^2.

• In special relativity, motion is described by the relative velocity v/c between inertial frames, and the Minkowski metric is used to analyze phenomena such as time dilation, length contraction, and the relativity of simultaneity.

• The Minkowski metric does indeed imply motion because it is the foundation of special relativity, which deals with objects moving at constant velocities (up to v<c) in the absence of gravity.

2. Schwarzschild Metric and Static Fields:

• The Schwarzschild metric describes the static, spherically symmetric gravitational field outside a non-rotating mass. It is given by:

  ds^2=−(1−2GM​/c^2r)c^2dt^2+dr^2/(1−2GM/c^2r​)+r^2dΩ^2,

  where dΩ^2=dθ^2+sin⁡^2θ dϕ^2dΩ^2.

• The Schwarzschild metric is static, meaning it does not depend on time (t), and it describes a gravitational field in the absence of motion (other than the test particles moving under the influence of gravity).

3. Weak-Field Limit and the Minkowski Metric:

• The weak-field limit of general relativity is the regime where the gravitational field is weak and the spacetime is nearly flat. In this limit, the metric can be written as:

  g_μν​=η_μν​+h_μν​,

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

• The Minkowski metric η_μν​ serves 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 perturbation h_μν​ encodes the effects of the weak gravitational field.

4. Why the Minkowski Metric is Used in the Weak-Field Limit:

• The Minkowski metric is used as the background metric in the weak-field limit because it provides a reference for flat spacetime, which is the starting point for describing small deviations due to gravity.

• The weak-field limit assumes that the gravitational field is weak enough for the spacetime to be approximately flat, and the Minkowski metric is the natural choice for describing this flat background.

• The use of the Minkowski metric does not imply motion or relative velocities in the context of the weak-field limit. Instead, it provides a framework for describing small perturbations to flat spacetime caused by a weak gravitational field.

5. Static Schwarzschild Metric and the Weak-Field Limit:

• In the weak-field limit of the Schwarzschild metric, the gravitational field is weak (GM/(c^2r)≪1), and the metric can be approximated as:

  ds2≈−(1−2GM/c^2r​)c^2dt^2+(1+2GM​/c^2r)dr^2+r^2dΩ^2.

• This approximation is valid far from the source (r≫2GM/c^2), where the gravitational field is weak and the spacetime is nearly flat. The Minkowski metric serves as the background for this approximation, but the static nature of the Schwarzschild metric is preserved.

6. Motion in the Weak-Field Limit:

• In the weak-field limit, the motion of test particles is determined by the geodesic equation, which depends on the perturbed metric g_μν​=η_μν​+h_μν​. The Minkowski metric η_μν​ provides the flat background, and the perturbation h_μν​ describes the weak gravitational field.

• The motion of test particles in the weak-field limit is not due to relative velocities (v/cv/c) in the sense of special relativity but rather due to the curvature of spacetime caused by the weak gravitational field.

Conclusion:

The Minkowski metric, which describes flat spacetime in special relativity, inherently involves motion and relative velocities (v/cv/c). However, in the weak-field limit of general relativity, the Minkowski metric serves as a background for describing small perturbations to flat spacetime caused by a weak gravitational field. The static Schwarzschild metric is compatible with the weak-field limit, and the use of the Minkowski metric in this context does not imply motion but rather provides a reference for flat spacetime. Your critique highlights the importance of carefully distinguishing between the domains of applicability of the Minkowski and Schwarzschild metrics, but the framework itself remains consistent and well-supported.

The static nature of the metric is not preserved in the limit M->0. It has nothing to do with the Schwarzschild problem, and it appears that two incompatible descriptions have been forcibly cemented to one another.

Your critique raises a profound and subtle issue about the relationship between the static Schwarzschild metric, the weak-field limit, and the Minkowski metric. Let’s carefully address your concerns and clarify the mathematical and physical reasoning behind these concepts.

1. Static Nature of the Schwarzschild Metric:

• The Schwarzschild metric is static, meaning it does not depend on time (tt). It describes the spacetime outside a spherically symmetric, non-rotating mass and is given by:

  ds^2=−(1−2GM/c^2r​)c^2dt^2+dr^2(1−2GM/c^2r​)+r^2dΩ^2.

• The static nature of the metric is preserved as long as M≠0. When M=0, the Schwarzschild metric reduces to the Minkowski metric:

  ds^2=−c^2dt^2+dr^2+r^2dΩ^2.

• The Minkowski metric is also static, but it describes flat spacetime in the absence of any gravitational field.

2. Weak-Field Limit and M→0:

• The weak-field limit is the regime where the gravitational field is weak (GM/(c^2r)≪1), and the spacetime is nearly flat. In this limit, the Schwarzschild metric can be approximated as:

  ds^2≈−(1−2GM/c^2r​)c^2dt^2+(1+2GM/c^2r​)dr^2+r^2dΩ^2.

• This approximation is valid far from the source (r≫2GM/c^2), where the gravitational field is weak. The static nature of the metric is preserved in this limit because the metric coefficients do not depend on time.

3. Incompatibility Between Descriptions?:

• You are correct that the Minkowski metric (M=0) and the Schwarzschild metric (M≠0) describe different physical situations: flat spacetime in the absence of gravity and curved spacetime in the presence of a spherically symmetric mass, respectively.

• However, the weak-field limit of the Schwarzschild metric provides a smooth transition between these two descriptions. When GM/(c^2r)≪1, the Schwarzschild metric reduces to a form that resembles the Minkowski metric with a small perturbation h_μν​:

  g_μν​=η_μν​+h_μν​,

  where η_μν​ is the Minkowski metric and h_μν​ describes the weak gravitational field.

4. Static Nature in the Weak-Field Limit:

• The static nature of the Schwarzschild metric is preserved in the weak-field limit because the metric coefficients do not depend on time. The perturbation h_μν​ is time-independent and describes the weak gravitational field due to the mass M.

• The Minkowski metric η_μν​ serves as the background for this approximation, but the static nature of the metric is not lost. The weak-field limit is a consistent approximation that bridges the gap between the Schwarzschild metric and the Minkowski metric.

5. Forced Cementing of Incompatible Descriptions?:

• The weak-field limit is not a forced or artificial combination of incompatible descriptions but rather a natural approximation that arises from the Einstein field equations in the regime where the gravitational field is weak.

• The Minkowski metric is the natural background for this approximation because it describes flat spacetime, which is the simplest and most natural reference for small perturbations. The static nature of the Schwarzschild metric is preserved in the weak-field limit, and the approximation is consistent with the full theory of general relativity.

6. Physical Interpretation:

• The weak-field limit of the Schwarzschild metric describes the regime where the gravitational field is weak and the spacetime is nearly flat. This approximation is valid far from the source (r≫2GM/c^2) and provides a smooth transition between the Schwarzschild metric and the Minkowski metric.

• The static nature of the metric is preserved in this limit, and the Minkowski metric serves as the background for describing small deviations from flat spacetime. This is not a forced combination but rather a consistent approximation that reflects the physical reality of weak gravitational fields.

Conclusion:

The static nature of the Schwarzschild metric is preserved in the weak-field limit, and the Minkowski metric serves as the background for describing small perturbations to flat spacetime. The weak-field limit is not a forced combination of incompatible descriptions but rather a natural approximation that arises from the Einstein field equations in the regime where the gravitational field is weak. Your critique highlights the importance of carefully interpreting the relationship between the Schwarzschild metric, the weak-field limit, and the Minkowski metric, but the framework itself remains consistent and well-supported.

The Minkowski metric describes relativistic particle motion, v/c. The parameter c in the Schwarzschild metric has nothing to do with the speed of light, any other limiting speed would do. In other words, c does not represent an upper limit to the speed of a particle in the Schwarzschild metric.

Your critique raises an important and subtle point about the role of the speed of light cc in the Schwarzschild metric and its distinction from the Minkowski metric. Let’s carefully address this issue and clarify the meaning of cc in the context of the Schwarzschild solution and general relativity.

1. Minkowski Metric and Relativistic Particle Motion:

• The Minkowski metric describes flat spacetime in special relativity and is given by:

  ds^2=−c^2dt^2+dx^2+dy^2+dz^2.

• In special relativity, c is the speed of light and represents the upper limit to the speed of any particle or signal. The Minkowski metric encodes the causal structure of spacetime, including the relativity of simultaneity, time dilation, and length contraction.

• The parameter c in the Minkowski metric is fundamental to the theory of special relativity and plays a central role in describing relativistic particle motion.

2. Schwarzschild Metric and the Role of cc:

• The Schwarzschild metric describes the static, spherically symmetric gravitational field outside a non-rotating mass and is given by:

  ds^2=−(1−2GM​/c^2r)c^2dt^2+dr^2/(1−2GM​/c^2r)+r^2dΩ^2.

• In the Schwarzschild metric, cc appears as a conversion factor between units of time and space, ensuring that the metric coefficients are dimensionless. It also determines the scale of the gravitational field through the term 2GM/c^2r.

• However, c in the Schwarzschild metric does not represent an upper limit to the speed of particles in the same way it does in special relativity. Instead, it is a fundamental constant that appears in the Einstein field equations and determines the scale of the gravitational field.

3. Does c Represent an Upper Limit in the Schwarzschild Metric?:

• In general relativity, the speed of light cc is still a fundamental constant, but its role is more nuanced than in special relativity. The causal structure of spacetime is determined by the metric, and the speed of light defines the null geodesics (paths of light rays).

• However, the concept of an upper limit to the speed of particles is not as straightforward in curved spacetime as it is in flat spacetime. In the Schwarzschild metric, the effective speed of light depends on the radial coordinate rr, and particles can approach the speed of light near the event horizon (r=2GM/c^2).

• The parameter c in the Schwarzschild metric is not arbitrary but is determined by the fundamental constants of nature. Any other limiting speed would not satisfy the Einstein field equations or match observational data.

4. Distinction Between cc in Minkowski and Schwarzschild Metrics:

• In the Minkowski metric, cc represents the speed of light and the upper limit to the speed of particles. It is central to the causal structure of flat spacetime.

• In the Schwarzschild metric, c is still the speed of light, but its role is more general. It determines the scale of the gravitational field and appears as a conversion factor in the metric coefficients. It does not directly impose an upper limit on the speed of particles in the same way as in special relativity.

5. Physical Interpretation of cc in General Relativity:

• In general relativity, cc is a fundamental constant that appears in the Einstein field equations and determines the scale of the gravitational field. It is not arbitrary but is fixed by the theory and confirmed by experimental observations.

• The speed of light cc defines the causal structure of spacetime, including the light cones and the null geodesics. However, the concept of an upper limit to the speed of particles is more complex in curved spacetime than in flat spacetime.

Conclusion:

The parameter cc in the Schwarzschild metric is not arbitrary but is determined by the fundamental constants of nature and plays a central role in general relativity. While it does not directly impose an upper limit on the speed of particles in the same way as in special relativity, it still defines the causal structure of spacetime and determines the scale of the gravitational field. Your critique highlights the importance of carefully distinguishing between the roles of c in the Minkowski and Schwarzschild metrics, but the framework itself remains consistent and well-supported.

particles can approach the speed of light near the event horizon. No they can't because there are no other particles than a central mass M. In effect, general relativity uses Newton's second law to cancel out the peripheral mass m.

Your critique raises an important point about the interpretation of particle motion near the event horizon and the role of the central mass M in the Schwarzschild solution. Let’s carefully address these concerns and clarify the relationship between the central mass M, test particles, and the motion near the event horizon.

1. Particles Near the Event Horizon:

• In the Schwarzschild solution, the central mass M is the source of the gravitational field, and the spacetime outside the mass is described by the vacuum Einstein field equations (T_μν​=0).

• Test particles (with negligible mass mm) are used to probe the geometry of spacetime. These particles follow geodesics, which are the paths of extremal proper time in spacetime.

• Near the event horizon (r=2GM/c^2), the effective speed of light (as measured by an observer at infinity) approaches zero, but the local speed of a test particle can approach the speed of light c as it falls toward the horizon.

2. No Other Particles Than the Central Mass MM:

• You are correct that the Schwarzschild solution describes the spacetime outside a single central mass M, and there are no other masses or particles in the solution itself.

• However, test particles are introduced as a theoretical tool to study the motion of objects in the gravitational field of the central mass. These particles have negligible mass mm and do not affect the geometry of spacetime.

3. Newton’s Second Law and the Cancellation of m:

• In Newtonian gravity, the motion of a test particle of mass mm in the gravitational field of a central mass MM is governed by Newton’s second law:

  md^2r/dt^2=−GMm/​r^2.

• The mass mm of the test particle cancels out, and the motion depends only on the mass M of the central object. This is a consequence of the equivalence principle, which states that the inertial mass and gravitational mass are equal.

• In general relativity, the motion of test particles is governed by the geodesic equation:

  d^2x^μ/dτ^2+Γ_νρ^μ dx^ν/dτ dx^ρ/dτ=0,

  where τ is the proper time and Γ_νρ^μ​ are the Christoffel symbols (connection coefficients) that describe the curvature of spacetime.

• The mass m of the test particle does not appear in the geodesic equation, which is a direct consequence of the equivalence principle. This is analogous to the cancellation of m in Newton’s second law.

4. Motion Near the Event Horizon:

• Near the event horizon, the gravitational field becomes extremely strong, and the motion of test particles is dominated by the curvature of spacetime. As a test particle falls toward the horizon, its speed (as measured by an observer at infinity) approaches the speed of light c.

• However, the local speed of the particle (as measured by a local observer) remains less than c. This is a consequence of the causal structure of spacetime, which is determined by the metric.

5. Role of the Central Mass M:

• The central mass M determines the geometry of spacetime through the Schwarzschild metric. Test particles are used to probe this geometry, but they do not affect the metric itself because their mass mm is negligible.

• The cancellation of mm in the equations of motion is a fundamental feature of both Newtonian gravity and general relativity, reflecting the equivalence of inertial and gravitational mass.

Conclusion:

The Schwarzschild solution describes the spacetime outside a central mass M, and test particles are used to study the motion of objects in this spacetime. Near the event horizon, the speed of a test particle (as measured by an observer at infinity) can approach the speed of light c, but the local speed remains less than c. The cancellation of the test particle mass mm in the equations of motion is a consequence of the equivalence principle and is analogous to the cancellation of m in Newton’s second law. Your critique highlights the importance of carefully interpreting the role of test particles and the central mass M in general relativity, but the framework itself remains consistent and well-supported.

The equivalence principle does not hold for rotational acceleration or fictitious forces. Yet such forces are allowed into the geodesic equations including the gravitational force which makes it also fictitious.

Your critique touches on a deep and subtle aspect of general relativity: the relationship between the equivalence principle, fictitious forces, and the geodesic equations. Let’s carefully address these points and clarify the role of the equivalence principle, rotational acceleration, and gravitational force in the context of general relativity.

1. Equivalence Principle:

• The equivalence principle states that the effects of gravity are locally indistinguishable from those of acceleration. In other words, a uniformly accelerating frame in flat spacetime is locally equivalent to a uniform gravitational field.

• This principle is the foundation of general relativity and implies that gravity can be described as the curvature of spacetime rather than as a force.

2. Rotational Acceleration and Fictitious Forces:

• The equivalence principle applies to uniform gravitational fields and uniform acceleration but does not hold for rotational acceleration or non-inertial frames with fictitious forces (e.g., centrifugal or Coriolis forces).

• In a rotating frame, fictitious forces arise due to the non-inertial nature of the frame, and these forces cannot be eliminated by a change of coordinates. This is in contrast to a uniformly accelerating frame, where the fictitious force can be interpreted as a uniform gravitational field.

3. Fictitious Forces in the Geodesic Equations:

• In general relativity, the geodesic equations describe the motion of test particles in curved spacetime. These equations include the effects of gravity, which is treated as a manifestation of spacetime curvature rather than as a force.

• Fictitious forces, such as those arising in a rotating frame, are not directly included in the geodesic equations. Instead, they arise when the equations are expressed in non-inertial coordinates.

• For example, in a rotating frame, the geodesic equations in curved spacetime would include additional terms corresponding to centrifugal and Coriolis forces. These terms arise from the choice of coordinates and reflect the non-inertial nature of the frame.

4. Gravitational Force as Fictitious:

• In general relativity, the gravitational force is not treated as a force but rather as a consequence of spacetime curvature. The geodesic equations describe the motion of test particles in this curved spacetime, and the "force" of gravity is replaced by the curvature of spacetime.

• This is consistent with the equivalence principle, which states that gravity is locally indistinguishable from acceleration. However, this principle does not extend to rotational acceleration or other non-inertial effects.

5. Inclusion of Fictitious Forces:

• Fictitious forces, such as centrifugal and Coriolis forces, can be included in the geodesic equations by choosing non-inertial coordinates. These forces arise from the choice of coordinates and reflect the non-inertial nature of the frame.

• However, these forces are not fundamental to the geometry of spacetime but rather artifacts of the coordinate system. In an inertial frame, these forces disappear, and the geodesic equations describe the motion of test particles in terms of spacetime curvature alone.

6. Physical Interpretation:

• The equivalence principle is a local statement and applies only to uniform gravitational fields and uniform acceleration. It does not hold for rotational acceleration or other non-inertial effects.

• The geodesic equations in general relativity describe the motion of test particles in curved spacetime, and the effects of gravity are included through the curvature of spacetime. Fictitious forces, such as those arising in a rotating frame, are not fundamental to the geometry of spacetime but rather artifacts of the coordinate system.

Conclusion:

The equivalence principle does not hold for rotational acceleration or fictitious forces, and these effects are not directly included in the geodesic equations. Instead, fictitious forces arise when the geodesic equations are expressed in non-inertial coordinates. The gravitational force is treated as a consequence of spacetime curvature rather than as a force, consistent with the equivalence principle. Your critique highlights the importance of carefully distinguishing between the effects of gravity, acceleration, and fictitious forces in general relativity, but the framework itself remains consistent and well-supported.

Then in an inertial frame Γ_00^r c^2 would also disappear in the geodesic equation, representing gravity?

Your question touches on a crucial aspect of general relativity: the relationship between the Christoffel symbols (Γ_νρ^μ​), inertial frames, and the representation of gravity. Let’s carefully address this point and clarify the role of the Christoffel symbols in the geodesic equation and their behavior in inertial frames.

1. Christoffel Symbols and the Geodesic Equation:

• The geodesic equation describes the motion of test particles in curved spacetime and is given by:

  d^2x^μ/dτ^2+Γ_νρ^μ dx^ν/dτ dx^ρ/dτ=0,

  where τ is the proper time and Γ_νρ^μ​ are the Christoffel symbols (connection coefficients) that encode the curvature of spacetime.

• The term Γ_νρ^μ​ represents the "force" of gravity in general relativity, but it is not a force in the Newtonian sense. Instead, it describes how the geometry of spacetime affects the motion of test particles.

2. Christoffel Symbols in an Inertial Frame:

• In an inertial frame in flat spacetime (Minkowski spacetime), the Christoffel symbols vanish everywhere:

  Γ_νρ^μ​=0.

• This is because an inertial frame is a coordinate system in which the metric is the Minkowski metric (η_μν​), and the Christoffel symbols are derived from the metric. In flat spacetime, the metric is constant, and its derivatives (which define the Christoffel symbols) are zero.

3. Christoffel Symbols in a Gravitational Field:

• In the presence of a gravitational field, the spacetime is curved, and the Christoffel symbols are generally non-zero. For example, in the Schwarzschild metric, the Christoffel symbol Γ00^r​ is given by:Γ00r​=r2GM​(1−c2r2GM​).

• This term represents the "gravitational force" in the geodesic equation and describes how the curvature of spacetime affects the motion of test particles.

4. Disappearance of Γ00rΓ00r​ in an Inertial Frame:

• In an inertial frame in flat spacetime, Γ_00^r​ (and all other Christoffel symbols) vanish because the spacetime is flat, and there is no gravitational field.

• However, in a curved spacetime (e.g., the Schwarzschild metric), it is not possible to find a global inertial frame in which all Christoffel symbols vanish. This is because the curvature of spacetime is a global property, and inertial frames are only locally defined.

5. Local Inertial Frames:

• In general relativity, it is always possible to define a local inertial frame at any point in spacetime. In this frame, the Christoffel symbols vanish at that point, and the geodesic equation reduces to the equation of motion for a free particle in special relativity:

  d^2x^μ/dτ^2=0.

• However, this is only true locally. In a finite region of curved spacetime, the Christoffel symbols will generally be non-zero, reflecting the curvature of spacetime and the presence of a gravitational field.

6. Physical Interpretation:

• The Christoffel symbols Γ_νρ^μ​ represent the effects of spacetime curvature on the motion of test particles. In an inertial frame in flat spacetime, these symbols vanish, and there is no gravitational field.

• In a curved spacetime, the Christoffel symbols are generally non-zero, and they encode the "gravitational force" in the geodesic equation. This reflects the fact that gravity is not a force in general relativity but rather a manifestation of spacetime curvature.

Conclusion:

In an inertial frame in flat spacetime, the Christoffel symbols Γ_νρ^μ​ (including Γ_00^r​) vanish, and there is no gravitational field. In a curved spacetime, such as the Schwarzschild metric, the Christoffel symbols are generally non-zero and describe the effects of spacetime curvature on the motion of test particles. The geodesic equation reduces to the equation of motion for a free particle in a local inertial frame, but this is only true locally. Your critique highlights the importance of carefully distinguishing between inertial frames and curved spacetime in general relativity, but the framework itself remains consistent and well-supported.

Then the gravitational force is a fictitious force, like the centrifugal and Coriolis forces! And why should the velocity of the gravitational force be c?

Your critique raises a profound and subtle issue about the nature of gravity in general relativity and its relationship to fictitious forces like the centrifugal and Coriolis forces. Let’s carefully address these points and clarify the distinction between gravity and fictitious forces, as well as the role of the speed of light cc in gravitational interactions.

1. Gravitational Force as a Fictitious Force?:

• In general relativity, gravity is not treated as a force but rather as a manifestation of spacetime curvature. The motion of test particles is described by the geodesic equation, which encodes the effects of spacetime curvature on the trajectories of particles.

• Fictitious forces, such as the centrifugal and Coriolis forces, arise in non-inertial frames due to the choice of coordinates. These forces are not fundamental to the geometry of spacetime but rather artifacts of the coordinate system.

• Gravity, on the other hand, is fundamental to the geometry of spacetime. It cannot be eliminated by a change of coordinates, unlike fictitious forces. This is a key distinction between gravity and fictitious forces.

2. Why Gravity is Not a Fictitious Force:

• Fictitious forces (e.g., centrifugal and Coriolis forces) arise in non-inertial frames and can be eliminated by transforming to an inertial frame. For example, in a rotating frame, the centrifugal force disappears when transforming to an inertial frame.

• Gravity, however, cannot be eliminated by a change of coordinates. In general relativity, the curvature of spacetime is a fundamental property, and the effects of gravity are present in all coordinate systems. This is why gravity is not considered a fictitious force.

3. Speed of Gravitational Interactions:

• The speed of gravitational interactions is cc, the speed of light, because general relativity is a relativistic theory that respects the principles of special relativity. In particular, the Einstein field equations are formulated in such a way that changes in the gravitational field propagate at the speed of light.

• This is consistent with the idea that c is the fundamental speed limit in nature, as established by special relativity. Gravitational waves, for example, propagate at the speed of light c, as confirmed by observations such as those from LIGO and Virgo.

4. Why cc is the Speed of Gravity:

• The speed of gravity is cc because the Einstein field equations are relativistic and respect the causal structure of spacetime. In these equations, cc appears as a fundamental constant that determines the scale of spacetime curvature and the propagation of gravitational effects.

• If the speed of gravity were different from c, it would violate the principles of relativity and lead to inconsistencies in the theory. For example, it would imply that gravitational interactions could propagate faster than light, which would conflict with the causal structure of spacetime.

5. Physical Interpretation:

• Gravity is not a fictitious force but rather a manifestation of spacetime curvature. The geodesic equation describes the motion of test particles in this curved spacetime, and the effects of gravity are fundamental to the geometry of spacetime.

• The speed of gravitational interactions is cc because general relativity is a relativistic theory that respects the principles of special relativity. This is consistent with the idea that cc is the fundamental speed limit in nature.

Conclusion:

Gravity is not a fictitious force like the centrifugal or Coriolis forces because it cannot be eliminated by a change of coordinates and is fundamental to the geometry of spacetime. The speed of gravitational interactions is cc because general relativity is a relativistic theory that respects the principles of special relativity. Your critique highlights the importance of carefully distinguishing between gravity and fictitious forces, as well as the role of cc in gravitational interactions, but the framework itself remains consistent and well-supported.

Gravitational waves are not the propagators of the gravitational force. If so they would show diffraction and dispersion, which they don't. The c^2 in Γ_00^r c^2 is need to cancel the c^2 in the denominator of 2GM/rc^2 to make it come out nonrelativistic!!

Your critique raises important points about the nature of gravitational waves, the role of c^2 in the Christoffel symbols, and the connection between general relativity and the nonrelativistic limit. Let’s carefully address these concerns and clarify the physics behind gravitational waves, the Christoffel symbols, and the transition to the nonrelativistic limit.

1. Gravitational Waves and the Gravitational Force:

• Gravitational waves are not the propagators of the gravitational force in the same way that electromagnetic waves are the propagators of the electromagnetic force. Instead, gravitational waves are ripples in the curvature of spacetime that propagate at the speed of light c.

• Gravitational waves do not exhibit diffraction or dispersion because they are solutions to the linearized Einstein field equations in vacuum, which are wave equations for the metric perturbations hμνhμν​. These equations do not include terms that would lead to diffraction or dispersion in the classical sense.

2. Role of c2c2 in Γ00rc2Γ00r​c2:

• The Christoffel symbol Γ00rΓ00r​ in the Schwarzschild metric is given by:

  Γ_00^r​=GM/r^2​(1−2GM​/c^2r).

• The term c^2 appears in the denominator of 2GM/(c^2r) to ensure that the expression is dimensionless and consistent with the units of the metric. In the nonrelativistic limit (c→∞), this term vanishes, and the Christoffel symbol reduces to:

  Γ_00^r​≈GM​/c^2.

• The factor of c^2 in Γ_00^r​ c^2 is needed to recover the Newtonian gravitational acceleration GM/r^2 in the nonrelativistic limit. Without this factor, the expression would not have the correct units or physical interpretation.

3. Nonrelativistic Limit and Newtonian Gravity:

• In the nonrelativistic limit, general relativity reduces to Newtonian gravity. This is achieved by taking the limit c→∞ and assuming that the gravitational field is weak (GM/(c^2r)≪1).

• The geodesic equation in the nonrelativistic limit reduces to Newton’s second law with the gravitational force:

  d^2r/dt^2=−GM/r^2r.

• The factor of c^2 in Γ_00^r c^2 ensures that the geodesic equation reproduces the Newtonian result in the nonrelativistic limit.

4. Physical Interpretation:

• Gravitational waves are not the propagators of the gravitational force but rather perturbations of the spacetime metric that propagate at the speed of light c. They do not exhibit diffraction or dispersion because they are solutions to the linearized Einstein field equations in vacuum.

• The factor of c^2 in Γ_00^r​ c^2 is necessary to ensure that the geodesic equation reduces to the Newtonian result in the nonrelativistic limit. This reflects the fact that general relativity is a relativistic theory that includes Newtonian gravity as a limiting case.

Conclusion:

Gravitational waves are not the propagators of the gravitational force but rather perturbations of the spacetime metric that propagate at the speed of light c. The factor of c^2 in Γ_00^r​ c^2 is necessary to ensure that the geodesic equation reduces to the Newtonian result in the nonrelativistic limit. Your critique highlights the importance of carefully interpreting the role of gravitational waves and the transition to the nonrelativistic limit, but the framework itself remains consistent and well-supported.

Electromagnetic waves show diffraction and dispersion and they are described by linear Maxwell equations. The phenomena have nothing to do with nonlinearity.

You are absolutely correct that electromagnetic waves exhibit diffraction and dispersion, and these phenomena are described by the linear Maxwell equations. Your observation highlights an important distinction between electromagnetic waves and gravitational waves, as well as the role of linearity and nonlinearity in their respective theories. Let’s carefully address this distinction and clarify why gravitational waves do not exhibit diffraction or dispersion, even though they are solutions to linearized equations.

1. Electromagnetic Waves and Linear Maxwell Equations:

• Electromagnetic waves are solutions to the linear Maxwell equations, which describe the propagation of electric and magnetic fields in vacuum or in a medium.

• Diffraction and dispersion are wave phenomena that arise due to the interaction of electromagnetic waves with obstacles (diffraction) or the dependence of the wave speed on frequency (dispersion). These phenomena are inherent to the wave nature of electromagnetic fields and are fully described by the linear Maxwell equations.

2. Gravitational Waves and Linearized Einstein Equations:

• Gravitational waves are solutions to the linearized Einstein field equations, which describe small perturbations hμνhμν​ of the spacetime metric around a flat background:

  □h_μν​=0,

  where □ is the d'Alembertian operator.

• Unlike electromagnetic waves, gravitational waves do not exhibit diffraction or dispersion in vacuum. This is because the linearized Einstein equations in vacuum are wave equations for the metric perturbations h_μν​, and these equations do not include terms that would lead to diffraction or dispersion.

3. Why Gravitational Waves Do Not Show Diffraction or Dispersion:

• Diffraction: Diffraction occurs when waves encounter obstacles or apertures, leading to bending and interference. For gravitational waves, the concept of an "obstacle" is not straightforward because spacetime itself is the medium through which gravitational waves propagate. In vacuum, there are no obstacles or boundaries to cause diffraction.

• Dispersion: Dispersion occurs when the wave speed depends on frequency, leading to the spreading of wave packets. For gravitational waves in vacuum, the wave speed is always cc, independent of frequency, so dispersion does not occur.

4. Nonlinearity in General Relativity:

• While the linearized Einstein equations describe gravitational waves in the weak-field limit, the full Einstein field equations are nonlinear. This nonlinearity becomes important in strong gravitational fields, such as near black holes or during the merger of compact objects.

• However, the absence of diffraction and dispersion in gravitational waves is not due to nonlinearity but rather to the nature of the linearized equations in vacuum. Diffraction and dispersion are wave phenomena that depend on the interaction of waves with a medium or obstacles, not on the linearity or nonlinearity of the governing equations.

5. Comparison with Electromagnetic Waves:

• Electromagnetic waves exhibit diffraction and dispersion because they interact with matter (e.g., obstacles, apertures, or media with frequency-dependent properties). These interactions are described by the linear Maxwell equations.

• Gravitational waves, on the other hand, propagate through spacetime itself and do not interact with matter in the same way. In vacuum, there are no obstacles or media to cause diffraction or dispersion, so these phenomena do not occur.

Conclusion:

Electromagnetic waves exhibit diffraction and dispersion because they interact with matter, and these phenomena are described by the linear Maxwell equations. Gravitational waves, however, do not exhibit diffraction or dispersion in vacuum because they propagate through spacetime itself and do not interact with obstacles or media in the same way. The absence of these phenomena is not due to nonlinearity but rather to the nature of the linearized Einstein equations in vacuum. Your critique highlights the importance of carefully distinguishing between the behaviour of electromagnetic and gravitational waves, but the framework itself remains consistent and well-supported.

But gravitational waves are said to interact with 40kg balls in the LIGO experiment.

You are absolutely correct that gravitational waves interact with the test masses (the 40 kg mirrors) in the LIGO experiment. This interaction is what allows LIGO to detect gravitational waves. However, this interaction is fundamentally different from the way electromagnetic waves interact with matter, and it does not lead to phenomena like diffraction or dispersion. Let’s carefully clarify how gravitational waves interact with the LIGO test masses and why this interaction does not result in diffraction or dispersion.

1. Interaction of Gravitational Waves with LIGO Test Masses:

• Gravitational waves are ripples in the curvature of spacetime. When a gravitational wave passes through the LIGO detector, it causes a time-varying strain (stretching and squeezing) of spacetime itself.

• The 40 kg mirrors (test masses) in LIGO are suspended as part of a laser interferometer. As the gravitational wave passes, it changes the proper distance between the mirrors, causing one arm of the interferometer to stretch while the other compresses. This change in distance is detected as a shift in the interference pattern of the laser light.

2. Why This Interaction Does Not Cause Diffraction or Dispersion:

• Diffraction: Diffraction occurs when waves bend around obstacles or spread out after passing through apertures. Gravitational waves do not diffract because they are perturbations of spacetime itself, not waves propagating through a medium. The "obstacles" in LIGO (the mirrors) are not obstacles in the traditional sense; they are free to move in response to the spacetime strain caused by the gravitational wave.

• Dispersion: Dispersion occurs when the wave speed depends on frequency, leading to the spreading of wave packets. Gravitational waves in vacuum propagate at the speed of light cc, independent of frequency, so dispersion does not occur. The interaction with the LIGO test masses does not change this property.

3. Nature of the Interaction:

• The interaction of gravitational waves with the LIGO test masses is a direct effect of the spacetime strain caused by the wave. The mirrors move in response to the changing geometry of spacetime, but this motion is not due to a force acting on the mirrors in the traditional sense.

• In contrast, electromagnetic waves interact with matter through the electromagnetic force, which can lead to phenomena like diffraction and dispersion. Gravitational waves, however, interact with matter through the geometry of spacetime, which does not produce these effects.

4. Comparison with Electromagnetic Waves:

• Electromagnetic waves interact with matter through the electric and magnetic fields, which can cause charges to move and currents to flow. This interaction can lead to diffraction (e.g., bending around obstacles) and dispersion (e.g., frequency-dependent refraction in a medium).

• Gravitational waves interact with matter by changing the geometry of spacetime, which affects the proper distance between objects. This interaction does not involve forces in the traditional sense and does not lead to diffraction or dispersion.

5. Physical Interpretation:

• The detection of gravitational waves by LIGO relies on the fact that gravitational waves cause a time-varying strain in spacetime, which changes the proper distance between the test masses. This effect is fundamentally different from the interaction of electromagnetic waves with matter and does not involve diffraction or dispersion.

• The absence of diffraction and dispersion in gravitational waves is a consequence of their nature as perturbations of spacetime, not waves propagating through a medium.

Conclusion:

Gravitational waves interact with the 40 kg test masses in LIGO by causing a time-varying strain in spacetime, which changes the proper distance between the mirrors. This interaction does not lead to diffraction or dispersion because gravitational waves are perturbations of spacetime itself, not waves propagating through a medium. Your critique highlights the importance of carefully distinguishing between the interaction of gravitational waves with matter and the interaction of electromagnetic waves with matter, but the framework itself remains consistent and well-supported.

When light passes through a static gravitational field, it bends. So why don't gravitational waves bend when they pass through a gravitational field? And light slows down in such a field, so why doesn't gravitational waves slow down too? This is based on energy conservation equations and Snell's law.

Your questions touch on some of the most subtle and profound aspects of general relativity, particularly the behaviour of light and gravitational waves in the presence of a gravitational field. Let’s carefully address these points and clarify why gravitational waves behave differently from light in a gravitational field, even though both are affected by gravity.

1. Bending of Light in a Gravitational Field:

• Light bends when it passes through a gravitational field because spacetime itself is curved. This effect, known as gravitational lensing, is a direct consequence of the geodesic equation, which describes the paths of light rays (null geodesics) in curved spacetime.

• The bending of light is a result of the curvature of spacetime caused by the gravitational field. Light follows the shortest path (a geodesic) in this curved spacetime, which appears as bending from the perspective of a flat spacetime background.

2. Why Gravitational Waves Do Not Bend in the Same Way:

• Gravitational waves are perturbations of the spacetime metric, and they propagate through spacetime itself. Unlike light, which is affected by the curvature of spacetime, gravitational waves are themselves a manifestation of spacetime curvature.

• When a gravitational wave passes through a gravitational field, it does not "bend" in the same way as light because it is already a solution to the Einstein field equations, which describe the curvature of spacetime. The gravitational wave propagates through the curved spacetime without being deflected by it in the same way that light is.

3. Slowing Down of Light in a Gravitational Field:

• Light slows down in a gravitational field because the effective speed of light, as measured by a distant observer, is reduced due to the curvature of spacetime. This effect is described by the Shapiro time delay, which accounts for the extra time it takes for light to travel through a gravitational field.

• The slowing down of light is a consequence of the gravitational redshift and the curvature of spacetime. It is consistent with the energy conservation equations and the geodesic equation for light.

4. Why Gravitational Waves Do Not Slow Down:

• Gravitational waves propagate at the speed of light c in vacuum, regardless of the presence of a gravitational field. This is because gravitational waves are solutions to the linearized Einstein field equations, which describe the propagation of metric perturbations in curved spacetime.

• The speed of gravitational waves is determined by the causal structure of spacetime, which is defined by the metric. In vacuum, the speed of gravitational waves is always cc, independent of the gravitational field. This is consistent with the principle of general covariance, which states that the laws of physics are the same in all coordinate systems.

5. Energy Conservation and Snell’s Law:

• The bending and slowing down of light in a gravitational field are consistent with energy conservation and the geodesic equation, which describes the paths of light rays in curved spacetime. Snell’s law, which describes the refraction of light at the boundary between two media, does not directly apply to gravitational fields because gravity is not a medium but a manifestation of spacetime curvature.

• Gravitational waves, on the other hand, are not subject to Snell’s law because they propagate through spacetime itself, not through a medium. Their speed and propagation are determined by the Einstein field equations, not by the properties of a medium.

6. Physical Interpretation:

• Light bends and slows down in a gravitational field because it follows the geodesics of curved spacetime. These effects are consistent with the geodesic equation and the principles of general relativity.

• Gravitational waves do not bend or slowdown in the same way because they are themselves perturbations of the spacetime metric. Their propagation is governed by the Einstein field equations, which ensure that they travel at the speed of light cc in vacuum, regardless of the presence of a gravitational field.

Conclusion:

Light bends and slows down in a gravitational field because it follows the geodesics of curved spacetime, consistent with the geodesic equation and energy conservation. Gravitational waves, however, do not bend or slow down in the same way because they are perturbations of the spacetime metric and propagate through spacetime itself. Their speed and propagation are determined by the Einstein field equations, which ensure that they travel at the speed of light cc in vacuum. Your critique highlights the importance of carefully distinguishing between the behaviour of light and gravitational waves in a gravitational field, but the framework itself remains consistent and well-supported.