identify derivations that span flat and curved spacetime in relativity #22

bhpayne · 2025-01-26T19:32:33Z

As suggested on https://physicsderivationgraph.blogspot.com/2024/03/dichotomy-of-assumptions.html

bhpayne · 2025-01-26T19:39:52Z

prompt to Gemini 2.0 Flash Thinking Experimental 01-21, https://aistudio.google.com/prompts/

Special Relativity assumes flat spacetime. Example equations include the Lorentz Transformations (which describes how space and time coordinates transform between inertial frames in the absence of gravity) and the Energy-Momentum Relation.

General Relativity assumes that spacetime is curved. For example, the Einstein Field Equations relate the curvature of spacetime to the distribution of mass and energy.

In regions where gravity is weak and velocities are much less than the speed of light, the solutions to the Einstein field equations approximate the flat spacetime of special relativity. The metric tensor reduces to the Minkowski metric in flat spacetime.

Describe some derivations that bridge the equations used for flat and curved spacetime. Explain what the initial equations are and what the ending equations are for each derivation.

thoughts

Here's a thinking process to construct the explanation bridging Special and General Relativity equations:

Understand the Core Request: The user wants to know how equations from flat spacetime (Special Relativity, SR) connect to equations in curved spacetime (General Relativity, GR). They are asking for derivations, emphasizing the starting and ending equations. The key context is the weak-field, low-velocity limit where GR should approximate SR.
Identify Key Equations in SR and GR:
- SR (Flat Spacetime):
  - Lorentz Transformations (coordinate transformations between inertial frames)
  - Energy-Momentum Relation (E² = (pc)² + (mc²)²)
  - Minkowski Metric (η_μν = diag(-1, 1, 1, 1) or diag(1, -1, -1, -1)) - This is fundamental for defining spacetime intervals.
- GR (Curved Spacetime):
  - Einstein Field Equations (G_μν + Λg_μν = (8πG/c⁴)T_μν) - Fundamental equation governing gravity.
  - Geodesic Equation (describes motion in curved spacetime)
  - Metric Tensor (g_μν) - Generalization of Minkowski metric to curved spacetime.
Focus on Bridging Derivations: The request asks for derivations that connect SR to GR. This means we need to find ways to start with GR equations and show how they reduce to SR equations in the weak-field, low-velocity limit.
Choose Specific Examples: It's too broad to cover all derivations. Select a few key examples that clearly illustrate the bridging process. The most natural choices are:
- Metric Tensor in Weak Field: How the general metric g_μν reduces to the Minkowski metric η_μν plus a small perturbation due to gravity. This directly connects the concept of curved spacetime back to flat spacetime.
- Geodesic Equation to Newtonian Gravity: How the GR description of motion (geodesic equation) reduces to Newtonian gravity in weak fields and low velocities. This shows how GR encompasses Newtonian gravity as a limit.
- Einstein Field Equations to Newtonian Gravity (Poisson Equation): How the Einstein Field Equations themselves reduce to the familiar Newtonian gravitational field equation (Poisson's equation) in the weak-field limit. This is a deeper connection, showing how GR's field equations contain Newtonian gravity as an approximation.
Outline the Derivation for Each Example: For each chosen example, plan the steps of the derivation, focusing on the assumptions and approximations involved:
- Metric Tensor:
 - Start with the idea that in weak fields, the spacetime is "almost flat." Represent the metric as a perturbation of the Minkowski metric: g_μν = η_μν + h_μν, where |h_μν| << 1.
 - Explain that this is the weak-field approximation.
- Geodesic Equation:
 - Start with the geodesic equation.
 - Assume weak field (metric as above) and slow motion (velocities << c, so dt/dτ ≈ 1, dxⁱ/dτ ≈ 0).
 - Simplify the Christoffel symbols (Γ^μ_νρ) using the weak-field metric.
 - Show that the spatial components of the geodesic equation reduce to the Newtonian acceleration equation.
- Einstein Field Equations:
 - Start with the Einstein Field Equations.
 - Apply the weak-field approximation (g_μν = η_μν + h_μν).
 - Linearize the Einstein tensor (G_μν) in terms of h_μν. This involves calculating the Ricci tensor and Ricci scalar to first order in h_μν. This is the most complex part and requires some tensor manipulation knowledge, but the conceptual idea is key.
 - Assume a static, weak field and non-relativistic matter (T₀₀ ≈ ρc², other T_μν small).
 - Simplify the (00) component of the linearized Einstein field equations.
 - Show that this reduces to the Poisson equation for the Newtonian potential.
Structure the Answer: Organize the explanation clearly, using headings for each derivation. For each derivation:
- State the Initial Equation(s) (GR equation).
- State the Approximations/Assumptions (weak field, low velocity).
- Briefly describe the Derivation Steps (key simplifications).
- State the Ending Equation(s) (SR or Newtonian equation).
- Explain the Significance (how it bridges SR and GR).
Refine and Add Detail: Review the outlined answer. Add more explanatory details, clarify any potentially confusing points, and ensure the language is clear and accessible. For instance:
- Emphasize the physical meaning of "weak field" and "low velocity."
- Explain why the Minkowski metric emerges in flat spacetime.
- Briefly mention the limitations of these approximations.
- Ensure consistent notation.
Final Check: Read through the entire answer to make sure it addresses the user's request completely, is accurate, and is well-organized and understandable. Double-check the equations and derivations for correctness (at a conceptual level, since full derivations can be lengthy). Ensure the language is consistent and clear. Self-correction: Initially, I might have focused too much on the mathematical details of tensor manipulation. Realize that the user probably wants a conceptual understanding of the process of approximation, not a full tensor calculus lesson. Adjust the level of detail accordingly, focusing on the key physical ideas and the overall flow of the derivations. Also, ensure to explicitly state what the initial and ending equations are, as requested.

response

Let's explore some derivations that bridge the equations of General Relativity (curved spacetime) to those of Special Relativity (flat spacetime). We'll focus on the weak-field, low-velocity limit, which is where General Relativity should approximate Newtonian gravity and Special Relativity.

1. Derivation of Newtonian Gravity from the Geodesic Equation in a Weak, Static Gravitational Field

Initial Equation (GR): Geodesic Equation

The geodesic equation describes the motion of particles in curved spacetime, which, in GR, is influenced by gravity. It is given by:
```
d²xμ/dτ² + Γμνρ (dxν/dτ)(dxρ/dτ) = 0
```
Where:
- x^μ are the spacetime coordinates (μ = 0, 1, 2, 3, where x⁰ = ct, and x¹, x², x³ are spatial coordinates).
- τ is the proper time (time measured by a clock moving with the particle).
- Γ^μ_νρ are the Christoffel symbols, which represent the "gravitational force" in GR and are derived from the metric tensor. They encode the curvature of spacetime.
Assumptions and Approximations for Weak Field and Low Velocity:
1. Weak Gravitational Field: We assume the gravitational field is weak, meaning spacetime is "almost flat." This means the metric tensor gμν is close to the Minkowski metric ημν (which describes flat spacetime). We can write the metric as a perturbation around flat spacetime:
```
gμν = ημν + hμν
```
 where |hμν| << 1. hμν represents the small deviations from flat spacetime due to gravity. For a weak, static gravitational field generated by a mass M, we can approximate the time-time component as:
```
g00 ≈ -(1 + 2Φ/c²) ≈ -1 - 2Φ/c²
```
 and the spatial components gii (i=1,2,3) as approximately 1, and off-diagonal components g0i and gij (i≠j) are very small and can be often neglected in the first approximation for a static weak field. Here, Φ is the Newtonian gravitational potential (Φ = -GM/r). We use the metric signature (- + + +).
2. Static Field: We assume the gravitational field is static, meaning it doesn't change with time. This implies that the metric tensor components are independent of time (∂g_μν/∂t = 0).
3. Low Velocity (Non-relativistic): We consider particles moving at velocities much smaller than the speed of light (v << c). This means:
 - dx0/dτ = c dt/dτ ≈ c (since for low velocities, proper time τ is approximately coordinate time t).
 - dxi/dτ = vi dt/dτ << c dt/dτ = dx0/dτ. So, the spatial velocity components are much smaller than the time component in spacetime.
Derivation Steps:
1. Christoffel Symbols in Weak Field: The Christoffel symbols are calculated from the metric tensor:
```
Γμνρ = (1/2) gμσ (∂νgσρ + ∂ρgσν - ∂σgνρ)
```
 In the weak-field approximation, we use gμν ≈ ημν + hμν and gμν ≈ ημν - hμν (to first order in hμν). Since we are considering a static field, time derivatives are zero. For the Newtonian limit, we are primarily interested in the spatial components of acceleration, so we look at the spatial components of the geodesic equation (μ = i, where i = 1, 2, 3).
2. Dominant Term: Consider the term in the geodesic equation: Γiνρ (dxν/dτ)(dxρ/dτ). Due to low velocities, the dominant term will be when ν = ρ = 0, because dx0/dτ ≈ c is much larger than dxi/dτ. Thus, we focus on Γi00 (dx0/dτ)(dx0/dτ).
3. Calculate Γⁱ₀₀: Using the definition of Christoffel symbols and the weak-field metric:
```
Γi00 ≈ (1/2) giσ (∂0gσ0 + ∂0gσ0 - ∂σg00)
```
 Since the field is static, ∂₀g_σ0 = 0. Also, for weak field and considering only the dominant component of g₀₀, we can approximate g^iσ ≈ η^iσ = δ^iσ (Kronecker delta). Thus:
```
Γi00 ≈ (1/2) ηiσ (-∂σg00) = -(1/2) ηij ∂jg00 = -(1/2) δij ∂jg00 = -(1/2) ∂ig00
```
 Substituting g00 ≈ -(1 + 2Φ/c²), we get:
```
Γi00 ≈ -(1/2) ∂i(-1 - 2Φ/c²) = (1/c²) ∂iΦ
```
4. Geodesic Equation Simplifies: Substituting this back into the geodesic equation (spatial components):
```
d²xi/dτ² + Γi00 (dx0/dτ)(dx0/dτ) ≈ 0
```
```
d²xi/dτ² + (1/c²) ∂iΦ (c dt/dτ)(c dt/dτ) ≈ 0
```
 For low velocities, dτ ≈ dt, so dt/dτ ≈ 1. Also, in the non-relativistic limit, we are interested in acceleration with respect to coordinate time t, so we can approximate d²xi/dτ² ≈ d²xi/dt². Thus:
```
d²xi/dt² + (1/c²) ∂iΦ c² ≈ 0
```
```
d²xi/dt² + ∂iΦ ≈ 0
```
```
d²xi/dt² = -∂iΦ
```
Ending Equation (Newtonian Gravity):

The ending equation is:
```
d²xi/dt² = -∂iΦ
```
This is Newton's second law of motion in a gravitational field, where -∂iΦ is the Newtonian gravitational force per unit mass (gravitational acceleration). Specifically, if Φ = -GM/r, then -∂iΦ gives the inverse square law for gravity.
Bridge: This derivation shows how the geodesic equation of General Relativity, which describes motion in curved spacetime, reduces to Newton's law of gravity in the limit of weak gravitational fields and low velocities. The curvature of spacetime, encoded in the Christoffel symbols derived from the metric, effectively becomes the Newtonian gravitational force in this approximation.

2. Derivation of the Energy-Momentum Relation from the General Relativistic Energy-Momentum Tensor and Flat Spacetime Limit

Initial Equation (GR, implicitly): Energy-Momentum Tensor and its Properties

In General Relativity, the energy and momentum of matter and fields are described by the energy-momentum tensor Tμν. In flat spacetime (Special Relativity), we also have an energy-momentum tensor. The connection is that in the absence of gravity (flat spacetime), the general relativistic description must reduce to the special relativistic one.
Assumptions and Approximations: Flat Spacetime Limit

We consider the limit where gravity is negligible, meaning spacetime is flat. In this case, the metric tensor becomes the Minkowski metric:
```
gμν → ημν = diag(-1, 1, 1, 1) (using signature - + + +)
```
And in flat spacetime, the covariant derivative becomes the ordinary partial derivative.
Derivation Steps:
1. Energy-Momentum Tensor in Flat Spacetime: In Special Relativity, for a point particle of mass m moving with four-velocity uμ = dxμ/dτ, the energy-momentum tensor can be represented (in a distributional sense) as:
```
Tμν(x) = m ∫ δ(4)(x - z(τ)) uμ(τ) uν(τ) dτ
```
 where zμ(τ) is the particle's worldline. Integrating this over a spatial volume and considering a fluid-like approximation (continuous distribution of particles) leads to the macroscopic energy-momentum tensor for matter.
2. Four-Velocity and Energy-Momentum Four-Vector: The four-velocity is defined such that uμuμ = gμνuμuν = -c² (or +c² depending on signature convention). In flat spacetime, gμν = ημν. The energy-momentum four-vector pμ for a particle is given by pμ = muμ. The components are:
 - p0 = mu0 = m (dx0/dτ) = m c (dt/dτ) = E/c (where E is relativistic energy).
 - pi = mui = m (dxi/dτ) = m (dxi/dt) (dt/dτ) = γmvi = pi (where pⁱ are spatial momentum components and γ is the Lorentz factor).
3. Invariant Quantity (Magnitude of Four-Momentum): Consider the invariant magnitude of the four-momentum squared:
```
pμpμ = ημνpμpν = η00(p0)² + ηii(pi)² = -(p0)² + (p1)² + (p2)² + (p3)²
```
 Substituting p0 = E/c and pi = pi (spatial momentum components):
```
pμpμ = -(E/c)² + (p1)² + (p2)² + (p3)² = -(E/c)² + |**p**|²
```
4. Rest Mass: We know that for a particle at rest, its spatial momentum is zero (p = 0), and its energy is the rest energy E = mc². In this rest frame, the four-velocity is uμ = (c, 0, 0, 0), and uμuμ = -c². The four-momentum squared in the rest frame is:
```
pμpμ = ημνpμpν = ημν(muμ)(muν) = m² ημνuμuν = m² (-c²) = -m²c²
```
 This invariant quantity, -m²c², must be the same in all inertial frames (Lorentz invariant).
Ending Equation (Energy-Momentum Relation in SR):

Equating the two expressions for pμpμ:
```
-(E/c)² + |**p**|² = -m²c²
```
Rearranging this equation, we get the famous Energy-Momentum Relation of Special Relativity:
```
E² = (pc)² + (mc²)²
```
where p = |**p**| is the magnitude of the 3-momentum.
Bridge: This derivation shows that in the flat spacetime limit (no gravity), the general relativistic concept of the energy-momentum tensor and four-momentum leads directly to the energy-momentum relation of Special Relativity. The invariant nature of the four-momentum squared is a fundamental concept in both SR and GR, and in flat spacetime, it simplifies to the well-known energy-momentum relation.

3. Derivation of Minkowski Metric as a Solution to Einstein Field Equations in Vacuum and Far from Sources

Initial Equation (GR): Einstein Field Equations

The Einstein Field Equations are the core equations of General Relativity, relating spacetime curvature to the distribution of mass and energy:
```
Gμν + Λgμν = (8πG/c⁴)Tμν
```
Where:
- Gμν = Rμν - (1/2)gμνR is the Einstein tensor, representing spacetime curvature.
- Rμν is the Ricci tensor.
- R = gμνRμν is the Ricci scalar.
- gμν is the metric tensor.
- Λ is the cosmological constant.
- G is the gravitational constant.
- c is the speed of light.
- Tμν is the energy-momentum tensor, representing the distribution of mass and energy.
Assumptions and Approximations: Vacuum and Far from Sources

We consider regions of spacetime that are:
1. Vacuum: Far away from any matter or energy sources, so we can approximate the energy-momentum tensor to be zero: Tμν ≈ 0.
2. Negligible Cosmological Constant (often): In many contexts, especially when considering local physics rather than cosmological scales, the cosmological constant Λ is often assumed to be negligible or zero (Λ ≈ 0).
Derivation Steps:
1. Vacuum Einstein Field Equations: With Tμν = 0 and Λ = 0, the Einstein Field Equations simplify to the vacuum Einstein field equations:
```
Gμν = 0
```
```
Rμν - (1/2)gμνR = 0
```
2. Ricci Scalar is Zero: Multiplying the vacuum field equations by gμν and summing over μ and ν:
```
gμνRμν - (1/2)gμνgμνR = 0
```
```
R - (1/2)(4)R = 0 (in 4 spacetime dimensions, gμνgμν = δμμ = 4)
```
```
R - 2R = 0
```
```
-R = 0 => R = 0
```
 So, the Ricci scalar is zero in vacuum.
3. Ricci Tensor is Zero: Substituting R = 0 back into the vacuum field equations Rμν - (1/2)gμνR = 0, we get:
```
Rμν - (1/2)gμν(0) = 0
```
```
Rμν = 0
```
 So, in vacuum, the Ricci tensor is also zero.
4. Flat Spacetime Solution: The condition Rμν = 0 means that spacetime is Ricci-flat. One solution to Rμν = 0 is flat spacetime, where all Christoffel symbols are zero, and thus the Riemann curvature tensor (from which Ricci tensor is derived) is also zero. In flat spacetime, the metric is the Minkowski metric ημν.
Ending Equation (Flat Spacetime Metric):

One solution to the vacuum Einstein field equations (in the absence of cosmological constant) is the Minkowski metric:
```
gμν = ημν = diag(-1, 1, 1, 1)
```
Bridge: This derivation demonstrates that in regions far from mass and energy (vacuum) and neglecting the cosmological constant, the Einstein Field Equations admit flat spacetime (Minkowski metric) as a solution. This is consistent with our understanding that in the absence of gravitational sources, spacetime should be flat, as assumed in Special Relativity. Therefore, Special Relativity is a valid description of spacetime in regions where gravity is negligible, which is precisely the flat spacetime limit of General Relativity.

These derivations illustrate how the equations of General Relativity, which describe curved spacetime and gravity, gracefully reduce to the equations and concepts of Special Relativity and Newtonian gravity in appropriate limits. This shows the consistency and hierarchical nature of these theories, with GR being a more general theory that encompasses SR and Newtonian gravity as approximations in specific regimes.

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bridge derivation of Newtonian Gravity from the Geodesic Equation in a Weak, Static Gravitational Field #24

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identify derivations that span flat and curved spacetime in relativity #22

identify derivations that span flat and curved spacetime in relativity #22

bhpayne commented Jan 26, 2025

bhpayne commented Jan 26, 2025

identify derivations that span flat and curved spacetime in relativity #22

identify derivations that span flat and curved spacetime in relativity #22

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bhpayne commented Jan 26, 2025

bhpayne commented Jan 26, 2025

prompt to Gemini 2.0 Flash Thinking Experimental 01-21, https://aistudio.google.com/prompts/

thoughts

response